1 | % Complete documentation on the extended LaTeX markup used for Python |
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2 | % documentation is available in ''Documenting Python'', which is part |
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3 | % of the standard documentation for Python. It may be found online |
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4 | % at: |
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5 | % |
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6 | % http://www.python.org/doc/current/doc/doc.html |
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7 | |
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8 | %labels |
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9 | %Sections and subsections \label{sec: } |
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10 | %Chapters \label{ch: } |
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11 | %Equations \label{eq: } |
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12 | %Figures \label{fig: } |
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13 | |
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14 | % Is latex failing with; |
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15 | % 'modanuga_user_manual.ind' not found? |
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16 | % try this command-line |
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17 | % makeindex modanuga_user_manual.idx |
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18 | % To produce the modanuga_user_manual.ind file. |
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19 | |
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20 | %%%%%%%%%%%%%% TODO %%%%%%%%%%%%%%%% |
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21 | % |
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22 | % ensure_geospatial |
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23 | % ensure_absolute |
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24 | % set_geo_reference |
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25 | |
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26 | \documentclass{manual} |
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27 | |
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28 | \usepackage{graphicx} |
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29 | \usepackage[english]{babel} |
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30 | \usepackage{datetime} |
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31 | \usepackage[hang,small,bf]{caption} |
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32 | |
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33 | \input{definitions} |
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34 | |
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35 | %%%%% |
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36 | % Set penalties for widows, etc, very high |
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37 | %%%%% |
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38 | |
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39 | \widowpenalty=10000 |
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40 | \clubpenalty=10000 |
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41 | \raggedbottom |
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42 | |
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43 | \title{\anuga User Manual} |
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44 | \author{Geoscience Australia and the Australian National University} |
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45 | |
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46 | % Please at least include a long-lived email address; |
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47 | % the rest is at your discretion. |
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48 | \authoraddress{Geoscience Australia \\ |
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49 | Email: \email{ole.nielsen@ga.gov.au} |
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50 | } |
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51 | |
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52 | %Draft date |
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53 | |
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54 | % update before release! |
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55 | % Use an explicit date so that reformatting |
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56 | % doesn't cause a new date to be used. Setting |
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57 | % the date to \today can be used during draft |
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58 | % stages to make it easier to handle versions. |
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59 | |
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60 | \longdate % Make date format long using datetime.sty |
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61 | %\settimeformat{xxivtime} % 24 hour Format |
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62 | \settimeformat{oclock} % Verbose |
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63 | \date{\today, \ \currenttime} |
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64 | %\hyphenation{set\_datadir} |
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65 | |
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66 | \ifhtml |
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67 | \date{\today} % latex2html does not know about datetime |
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68 | \fi |
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69 | |
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70 | \input{version} % Get version info - this file may be modified by |
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71 | % update_anuga_user_manual.py - if not a dummy |
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72 | % will be used. |
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73 | |
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74 | %\release{1.0} % release version; this is used to define the |
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75 | % % \version macro |
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76 | |
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77 | \makeindex % tell \index to actually write the .idx file |
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78 | \makemodindex % If this contains a lot of module sections. |
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79 | |
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80 | \setcounter{tocdepth}{3} |
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81 | \setcounter{secnumdepth}{3} |
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82 | |
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83 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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84 | |
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85 | \begin{document} |
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86 | \maketitle |
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87 | |
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88 | % This makes the contents more accessible from the front page of the HTML. |
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89 | \ifhtml |
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90 | \chapter*{Front Matter\label{front}} |
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91 | \fi |
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92 | |
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93 | %Subversion keywords: |
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94 | % |
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95 | %$LastChangedDate: 2009-05-31 21:57:32 +0000 (Sun, 31 May 2009) $ |
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96 | %$LastChangedRevision: 7138 $ |
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97 | %$LastChangedBy: rwilson $ |
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98 | |
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99 | \input{copyright} |
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100 | |
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101 | \begin{abstract} |
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102 | \label{def:anuga} |
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103 | |
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104 | \noindent \anuga\index{\anuga} is a hydrodynamic modelling tool that |
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105 | allows users to model realistic flow problems in complex 2D geometries. |
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106 | Examples include dam breaks or the effects of natural hazards such |
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107 | as riverine flooding, storm surges and tsunami. |
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108 | |
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109 | The user must specify a study area represented by a mesh of triangular |
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110 | cells, the topography and bathymetry, frictional resistance, initial |
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111 | values for water level (called \emph{stage}\index{stage} within \anuga), |
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112 | boundary conditions and forces such as rainfall, stream flows, windstress or pressure gradients if applicable. |
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113 | |
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114 | \anuga tracks the evolution of water depth and horizontal momentum |
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115 | within each cell over time by solving the shallow water wave equation |
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116 | governing equation using a finite-volume method. |
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117 | |
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118 | \anuga also incorporates a mesh generator that |
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119 | allows the user to set up the geometry of the problem interactively as |
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120 | well as tools for interpolation and surface fitting, and a number of |
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121 | auxiliary tools for visualising and interrogating the model output. |
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122 | |
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123 | Most \anuga components are written in the object-oriented programming |
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124 | language Python and most users will interact with \anuga by writing |
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125 | small Python programs based on the \anuga library |
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126 | functions. Computationally intensive components are written for |
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127 | efficiency in C routines working directly with Python numpy structures. |
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128 | |
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129 | \end{abstract} |
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130 | |
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131 | \tableofcontents |
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132 | |
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133 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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134 | |
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135 | \chapter{Introduction} |
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136 | |
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137 | |
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138 | \section{Purpose} |
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139 | |
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140 | The purpose of this user manual is to introduce the new user to the |
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141 | inundation software system, describe what it can do and give step-by-step |
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142 | instructions for setting up and running hydrodynamic simulations. |
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143 | The stable release of \anuga and this manual are available on sourceforge ati |
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144 | \url{http://sourceforge.net/projects/anuga}. A snapshot of work in progress is |
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145 | available through the \anuga software repository at |
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146 | \url{https://datamining.anu.edu.au/svn/ga/anuga_core} |
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147 | where the more adventurous reader might like to go. |
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148 | |
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149 | This manual describes \anuga version 1.0. To check for later versions of this manual |
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150 | go to \url{https://datamining.anu.edu.au/anuga}. |
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151 | |
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152 | \section{Scope} |
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153 | |
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154 | This manual covers only what is needed to operate the software after |
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155 | installation and configuration. It does not includes instructions |
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156 | for installing the software or detailed API documentation, both of |
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157 | which will be covered in separate publications and by documentation |
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158 | in the source code. |
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159 | |
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160 | The latest installation instructions may be found at: |
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161 | \url{http://datamining.anu.edu.au/\~{}ole/anuga/user_manual/anuga_installation_guide.pdf}. |
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162 | |
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163 | \section{Audience} |
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164 | |
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165 | Readers are assumed to be familiar with the Python Programming language and |
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166 | its object oriented approach. |
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167 | Python tutorials include |
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168 | \url{http://docs.python.org/tut}, |
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169 | \url{http://www.sthurlow.com/python}, and |
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170 | %\url{http://datamining.anu.edu.au/\%7e ole/work/teaching/ctac2006/exercise1.pdf}. |
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171 | \url{http://datamining.anu.edu.au/\~{}ole/work/teaching/ctac2006/exercise1.pdf}. |
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172 | |
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173 | Readers also need to have a general understanding of scientific modelling, |
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174 | as well as enough programming experience to adapt the code to different |
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175 | requirements. |
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176 | |
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177 | \pagebreak |
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178 | |
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179 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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180 | |
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181 | \chapter{Background} |
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182 | |
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183 | Modelling the effects on the built environment of natural hazards such |
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184 | as riverine flooding, storm surges and tsunami is critical for |
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185 | understanding their economic and social impact on our urban |
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186 | communities. Geoscience Australia and the Australian National |
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187 | University are developing a hydrodynamic inundation modelling tool |
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188 | called \anuga to help simulate the impact of these hazards. |
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189 | |
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190 | The core of \anuga is the fluid dynamics module, called \code{shallow\_water}, |
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191 | which is based on a finite-volume method for solving the Shallow Water |
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192 | Wave Equation. The study area is represented by a mesh of triangular |
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193 | cells. By solving the governing equation within each cell, water |
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194 | depth and horizontal momentum are tracked over time. |
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195 | |
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196 | A major capability of \anuga is that it can model the process of |
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197 | wetting and drying as water enters and leaves an area. This means |
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198 | that it is suitable for simulating water flow onto a beach or dry land |
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199 | and around structures such as buildings. \anuga is also capable |
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200 | of modelling hydraulic jumps due to the ability of the finite-volume |
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201 | method to accommodate discontinuities in the solution\footnote{ |
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202 | While \anuga works with discontinuities in the conserved quantities stage, |
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203 | xmomentum and ymomentum, it does not allow discontinuities in the bed elevation.}. |
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204 | |
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205 | To set up a particular scenario the user specifies the geometry |
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206 | (bathymetry and topography), the initial water level (stage), |
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207 | boundary conditions such as tide, and any forcing terms that may |
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208 | drive the system such as rainfall, abstraction of water, wind stress or atmospheric pressure |
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209 | gradients. Gravity and frictional resistance from the different |
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210 | terrains in the model are represented by predefined forcing terms. |
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211 | See section \ref{sec:forcing terms} for details on forcing terms available in \anuga. |
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212 | |
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213 | The built-in mesh generator, called \code{graphical\_mesh\_generator}, |
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214 | allows the user to set up the geometry |
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215 | of the problem interactively and to identify boundary segments and |
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216 | regions using symbolic tags. These tags may then be used to set the |
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217 | actual boundary conditions and attributes for different regions |
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218 | (e.g.\ the Manning friction coefficient) for each simulation. |
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219 | |
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220 | Most \anuga components are written in the object-oriented programming |
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221 | language Python. Software written in Python can be produced quickly |
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222 | and can be readily adapted to changing requirements throughout its |
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223 | lifetime. Computationally intensive components are written for |
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224 | efficiency in C routines working directly with Python numeric |
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225 | structures. The animation tool developed for \anuga is based on |
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226 | OpenSceneGraph, an Open Source Software (OSS) component allowing high |
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227 | level interaction with sophisticated graphics primitives. |
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228 | See \cite{nielsen2005} for more background on \anuga. |
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229 | |
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230 | \chapter{Restrictions and limitations on \anuga} |
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231 | \label{ch:limitations} |
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232 | |
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233 | Although a powerful and flexible tool for hydrodynamic modelling, \anuga has a |
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234 | number of limitations that any potential user needs to be aware of. They are: |
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235 | |
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236 | \begin{itemize} |
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237 | \item The mathematical model is the 2D shallow water wave equation. |
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238 | As such it cannot resolve vertical convection and consequently not breaking |
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239 | waves or 3D turbulence (e.g.\ vorticity). |
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240 | %\item The surface is assumed to be open, e.g.\ \anuga cannot model |
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241 | %flow under ceilings or in pipes |
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242 | \item All spatial coordinates are assumed to be UTM (meters). As such, |
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243 | \anuga is unsuitable for modelling flows in areas larger than one UTM zone |
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244 | (6 degrees wide). |
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245 | \item Fluid is assumed to be inviscid -- i.e.\ no kinematic viscosity included. |
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246 | \item The finite volume is a very robust and flexible numerical technique, |
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247 | but it is not the fastest method around. If the geometry is sufficiently |
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248 | simple and if there is no need for wetting or drying, a finite-difference |
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249 | method may be able to solve the problem faster than \anuga. |
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250 | %\item Mesh resolutions near coastlines with steep gradients need to be... |
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251 | \item Frictional resistance is implemented using Manning's formula, but |
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252 | \anuga has not yet been fully validated in regard to bottom roughness. |
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253 | %\item \anuga contains no tsunami-genic functionality relating to earthquakes. |
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254 | \end{itemize} |
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255 | |
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256 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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257 | |
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258 | \chapter{Getting Started} |
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259 | \label{ch:getstarted} |
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260 | |
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261 | This section is designed to assist the reader to get started with |
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262 | \anuga by working through some examples. Two examples are discussed; |
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263 | the first is a simple example to illustrate many of the concepts, and |
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264 | the second is a more realistic example. |
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265 | |
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266 | |
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267 | \section{A Simple Example} |
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268 | \label{sec:simpleexample} |
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269 | |
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270 | \subsection{Overview} |
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271 | |
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272 | What follows is a discussion of the structure and operation of a |
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273 | script called \file{runup.py}. |
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274 | |
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275 | This example carries out the solution of the shallow-water wave |
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276 | equation in the simple case of a configuration comprising a flat |
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277 | bed, sloping at a fixed angle in one direction and having a |
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278 | constant depth across each line in the perpendicular direction. |
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279 | |
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280 | The example demonstrates the basic ideas involved in setting up a |
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281 | complex scenario. In general the user specifies the geometry |
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282 | (bathymetry and topography), the initial water level, boundary |
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283 | conditions such as tide, and any forcing terms that may drive the |
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284 | system such as rainfall, abstraction of water, wind stress or atmospheric pressure gradients. |
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285 | Frictional resistance from the different terrains in the model is |
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286 | represented by predefined forcing terms. In this example, the |
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287 | boundary is reflective on three sides and a time dependent wave on |
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288 | one side. |
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289 | |
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290 | The present example represents a simple scenario and does not |
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291 | include any forcing terms, nor is the data taken from a file as it |
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292 | would typically be. |
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293 | |
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294 | The conserved quantities involved in the |
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295 | problem are stage (absolute height of water surface), |
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296 | $x$-momentum and $y$-momentum. Other quantities |
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297 | involved in the computation are the friction and elevation. |
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298 | |
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299 | Water depth can be obtained through the equation: |
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300 | |
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301 | \begin{verbatim} |
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302 | depth = stage - elevation |
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303 | \end{verbatim} |
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304 | |
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305 | \subsection{Outline of the Program} |
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306 | |
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307 | In outline, \file{runup.py} performs the following steps: |
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308 | \begin{enumerate} |
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309 | \item Sets up a triangular mesh. |
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310 | \item Sets certain parameters governing the mode of |
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311 | operation of the model, specifying, for instance, |
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312 | where to store the model output. |
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313 | \item Inputs various quantities describing physical measurements, such |
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314 | as the elevation, to be specified at each mesh point (vertex). |
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315 | \item Sets up the boundary conditions. |
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316 | \item Carries out the evolution of the model through a series of time |
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317 | steps and outputs the results, providing a results file that can |
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318 | be viewed. |
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319 | \end{enumerate} |
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320 | |
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321 | \subsection{The Code} |
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322 | |
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323 | For reference we include below the complete code listing for |
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324 | \file{runup.py}. Subsequent paragraphs provide a |
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325 | 'commentary' that describes each step of the program and explains it |
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326 | significance. |
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327 | |
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328 | \label{ref:runup_py_code} |
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329 | \verbatiminput{demos/runup.py} |
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330 | |
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331 | \subsection{Establishing the Mesh}\index{mesh, establishing} |
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332 | |
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333 | The first task is to set up the triangular mesh to be used for the |
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334 | scenario. This is carried out through the statement: |
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335 | |
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336 | \begin{verbatim} |
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337 | points, vertices, boundary = rectangular_cross(10, 10) |
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338 | \end{verbatim} |
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339 | |
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340 | The function \function{rectangular_cross} is imported from a module |
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341 | \module{mesh\_factory} defined elsewhere. (\anuga also contains |
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342 | several other schemes that can be used for setting up meshes, but we |
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343 | shall not discuss these.) The above assignment sets up a $10 \times |
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344 | 10$ rectangular mesh, triangulated in a regular way. The assignment: |
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345 | |
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346 | \begin{verbatim} |
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347 | points, vertices, boundary = rectangular_cross(m, n) |
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348 | \end{verbatim} |
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349 | |
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350 | returns: |
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351 | |
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352 | \begin{itemize} |
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353 | \item a list \code{points} giving the coordinates of each mesh point, |
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354 | \item a list \code{vertices} specifying the three vertices of each triangle, and |
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355 | \item a dictionary \code{boundary} that stores the edges on |
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356 | the boundary and associates each with one of the symbolic tags \code{'left'}, \code{'right'}, |
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357 | \code{'top'} or \code{'bottom'}. |
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358 | \end{itemize} |
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359 | |
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360 | (For more details on symbolic tags, see page |
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361 | \pageref{ref:tagdescription}.) |
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362 | |
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363 | An example of a general unstructured mesh and the associated data |
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364 | structures \code{points}, \code{vertices} and \code{boundary} is |
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365 | given in Section \ref{sec:meshexample}. |
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366 | |
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367 | \subsection{Initialising the Domain} |
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368 | |
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369 | These variables are then used to set up a data structure |
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370 | \code{domain}, through the assignment: |
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371 | |
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372 | \begin{verbatim} |
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373 | domain = Domain(points, vertices, boundary) |
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374 | \end{verbatim} |
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375 | |
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376 | This creates an instance of the \class{Domain} class, which |
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377 | represents the domain of the simulation. Specific options are set at |
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378 | this point, including the basename for the output file and the |
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379 | directory to be used for data: |
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380 | |
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381 | \begin{verbatim} |
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382 | domain.set_name('runup') |
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383 | domain.set_datadir('.') |
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384 | \end{verbatim} |
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385 | |
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386 | In addition, the following statement could be used to state that |
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387 | quantities \code{stage}, \code{xmomentum} and \code{ymomentum} are |
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388 | to be stored: |
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389 | |
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390 | \begin{verbatim} |
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391 | domain.set_quantities_to_be_stored(['stage', 'xmomentum', 'ymomentum']) |
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392 | \end{verbatim} |
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393 | |
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394 | However, this is not necessary, as the above quantities are always stored by default. |
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395 | |
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396 | \subsection{Initial Conditions} |
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397 | |
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398 | The next task is to specify a number of quantities that we wish to |
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399 | set for each mesh point. The class \class{Domain} has a method |
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400 | \method{set\_quantity}, used to specify these quantities. It is a |
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401 | flexible method that allows the user to set quantities in a variety |
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402 | of ways -- using constants, functions, numeric arrays, expressions |
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403 | involving other quantities, or arbitrary data points with associated |
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404 | values, all of which can be passed as arguments. All quantities can |
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405 | be initialised using \method{set\_quantity}. For a conserved |
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406 | quantity (such as \code{stage, xmomentum, ymomentum}) this is called |
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407 | an \emph{initial condition}. However, other quantities that aren't |
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408 | updated by the equation are also assigned values using the same |
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409 | interface. The code in the present example demonstrates a number of |
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410 | forms in which we can invoke \method{set\_quantity}. |
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411 | |
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412 | \subsubsection{Elevation} |
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413 | |
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414 | The elevation, or height of the bed, is set using a function |
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415 | defined through the statements below, which is specific to this |
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416 | example and specifies a particularly simple initial configuration |
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417 | for demonstration purposes: |
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418 | |
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419 | \begin{verbatim} |
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420 | def topography(x, y): |
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421 | return -x/2 |
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422 | \end{verbatim} |
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423 | |
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424 | This simply associates an elevation with each point \code{(x, y)} of |
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425 | the plane. It specifies that the bed slopes linearly in the |
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426 | \code{x} direction, with slope $-\frac{1}{2}$, and is constant in |
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427 | the \code{y} direction. |
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428 | |
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429 | Once the function \function{topography} is specified, the quantity |
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430 | \code{elevation} is assigned through the simple statement: |
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431 | |
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432 | \begin{verbatim} |
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433 | domain.set_quantity('elevation', topography) |
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434 | \end{verbatim} |
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435 | |
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436 | NOTE: If using function to set \code{elevation} it must be vector |
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437 | compatible. For example, using square root will not work. |
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438 | |
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439 | \subsubsection{Friction} |
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440 | |
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441 | The assignment of the friction quantity (a forcing term) |
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442 | demonstrates another way we can use \method{set\_quantity} to set |
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443 | quantities -- namely, assign them to a constant numerical value: |
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444 | |
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445 | \begin{verbatim} |
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446 | domain.set_quantity('friction', 0.1) |
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447 | \end{verbatim} |
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448 | |
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449 | This specifies that the Manning friction coefficient is set to 0.1 |
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450 | at every mesh point. |
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451 | |
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452 | \subsubsection{Stage} |
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453 | |
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454 | The stage (the height of the water surface) is related to the |
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455 | elevation and the depth at any time by the equation: |
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456 | |
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457 | \begin{verbatim} |
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458 | stage = elevation + depth |
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459 | \end{verbatim} |
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460 | |
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461 | For this example, we simply assign a constant value to \code{stage}, |
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462 | using the statement: |
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463 | |
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464 | \begin{verbatim} |
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465 | domain.set_quantity('stage', -0.4) |
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466 | \end{verbatim} |
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467 | |
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468 | which specifies that the surface level is set to a height of $-0.4$, |
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469 | i.e.\ 0.4 units (metres) below the zero level. |
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470 | |
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471 | Although it is not necessary for this example, it may be useful to |
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472 | digress here and mention a variant to this requirement, which allows |
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473 | us to illustrate another way to use \method{set\_quantity} -- namely, |
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474 | incorporating an expression involving other quantities. Suppose, |
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475 | instead of setting a constant value for the stage, we wished to |
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476 | specify a constant value for the \emph{depth}. For such a case we |
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477 | need to specify that \code{stage} is everywhere obtained by adding |
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478 | that value to the value already specified for \code{elevation}. We |
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479 | would do this by means of the statements: |
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480 | |
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481 | \begin{verbatim} |
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482 | h = 0.05 # Constant depth |
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483 | domain.set_quantity('stage', expression='elevation + %f' % h) |
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484 | \end{verbatim} |
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485 | |
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486 | That is, the value of \code{stage} is set to $\code{h} = 0.05$ plus |
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487 | the value of \code{elevation} already defined. |
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488 | |
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489 | The reader will probably appreciate that this capability to |
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490 | incorporate expressions into statements using \method{set\_quantity} |
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491 | greatly expands its power. See Section \ref{sec:initial conditions} for more |
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492 | details. |
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493 | |
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494 | \subsection{Boundary Conditions}\index{boundary conditions} |
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495 | |
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496 | The boundary conditions are specified as follows: |
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497 | |
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498 | \begin{verbatim} |
---|
499 | Br = Reflective_boundary(domain) |
---|
500 | Bt = Transmissive_boundary(domain) |
---|
501 | Bd = Dirichlet_boundary([0.2, 0.0, 0.0]) |
---|
502 | Bw = Time_boundary(domain=domain, |
---|
503 | f=lambda t: [(0.1*sin(t*2*pi)-0.3)*exp(-t), 0.0, 0.0]) |
---|
504 | \end{verbatim} |
---|
505 | |
---|
506 | The effect of these statements is to set up a selection of different |
---|
507 | alternative boundary conditions and store them in variables that can be |
---|
508 | assigned as needed. Each boundary condition specifies the |
---|
509 | behaviour at a boundary in terms of the behaviour in neighbouring |
---|
510 | elements. The boundary conditions introduced here may be briefly described as |
---|
511 | follows: |
---|
512 | \begin{itemize} |
---|
513 | \item \textbf{Reflective boundary}\label{def:reflective boundary} |
---|
514 | Returns same \code{stage} as in its neighbour volume but momentum |
---|
515 | vector reversed 180 degrees (reflected). |
---|
516 | Specific to the shallow water equation as it works with the |
---|
517 | momentum quantities assumed to be the second and third conserved |
---|
518 | quantities. A reflective boundary condition models a solid wall. |
---|
519 | \item \textbf{Transmissive boundary}\label{def:transmissive boundary} |
---|
520 | Returns same conserved quantities as |
---|
521 | those present in its neighbour volume. This is one way of modelling |
---|
522 | outflow from a domain, but it should be used with caution if flow is |
---|
523 | not steady state as replication of momentum at the boundary |
---|
524 | may cause numerical instabilities propagating into the domain and |
---|
525 | eventually causing \anuga to crash. If this occurs, |
---|
526 | consider using e.g.\ a Dirichlet boundary condition with a stage value |
---|
527 | less than the elevation at the boundary. |
---|
528 | \item \textbf{Dirichlet boundary}\label{def:dirichlet boundary} Specifies |
---|
529 | constant values for stage, $x$-momentum and $y$-momentum at the boundary. |
---|
530 | \item \textbf{Time boundary}\label{def:time boundary} Like a Dirichlet |
---|
531 | boundary but with behaviour varying with time. |
---|
532 | \end{itemize} |
---|
533 | |
---|
534 | \label{ref:tagdescription}Before describing how these boundary |
---|
535 | conditions are assigned, we recall that a mesh is specified using |
---|
536 | three variables \code{points}, \code{vertices} and \code{boundary}. |
---|
537 | In the code we are discussing, these three variables are returned by |
---|
538 | the function \code{rectangular}. The example given in |
---|
539 | Section \ref{sec:realdataexample} illustrates another way of |
---|
540 | assigning the values, by means of the function |
---|
541 | \code{create_mesh_from_regions}. |
---|
542 | |
---|
543 | These variables store the data determining the mesh as follows. (You |
---|
544 | may find that the example given in Section \ref{sec:meshexample} |
---|
545 | helps to clarify the following discussion, even though that example |
---|
546 | is a \emph{non-rectangular} mesh.) |
---|
547 | \begin{itemize} |
---|
548 | \item The variable \code{points} stores a list of 2-tuples giving the |
---|
549 | coordinates of the mesh points. |
---|
550 | \item The variable \code{vertices} stores a list of 3-tuples of |
---|
551 | numbers, representing vertices of triangles in the mesh. In this |
---|
552 | list, the triangle whose vertices are \code{points[i]}, |
---|
553 | \code{points[j]}, \code{points[k]} is represented by the 3-tuple |
---|
554 | \code{(i, j, k)}. |
---|
555 | \item The variable \code{boundary} is a Python dictionary that |
---|
556 | not only stores the edges that make up the boundary but also assigns |
---|
557 | symbolic tags to these edges to distinguish different parts of the |
---|
558 | boundary. An edge with endpoints \code{points[i]} and |
---|
559 | \code{points[j]} is represented by the 2-tuple \code{(i, j)}. The |
---|
560 | keys for the dictionary are the 2-tuples \code{(i, j)} corresponding |
---|
561 | to boundary edges in the mesh, and the values are the tags are used |
---|
562 | to label them. In the present example, the value \code{boundary[(i, j)]} |
---|
563 | assigned to \code{(i, j)]} is one of the four tags |
---|
564 | \code{'left'}, \code{'right'}, \code{'top'} or \code{'bottom'}, |
---|
565 | depending on whether the boundary edge represented by \code{(i, j)} |
---|
566 | occurs at the left, right, top or bottom of the rectangle bounding |
---|
567 | the mesh. The function \code{rectangular} automatically assigns |
---|
568 | these tags to the boundary edges when it generates the mesh. |
---|
569 | \end{itemize} |
---|
570 | |
---|
571 | The tags provide the means to assign different boundary conditions |
---|
572 | to an edge depending on which part of the boundary it belongs to. |
---|
573 | (In Section \ref{sec:realdataexample} we describe an example that |
---|
574 | uses different boundary tags -- in general, the possible tags are entirely selectable by the user when generating the mesh and not |
---|
575 | limited to 'left', 'right', 'top' and 'bottom' as in this example.) |
---|
576 | All segments in bounding polygon must be tagged. If a tag is not supplied, the default tag name 'exterior' will be assigned by \anuga. |
---|
577 | |
---|
578 | Using the boundary objects described above, we assign a boundary |
---|
579 | condition to each part of the boundary by means of a statement like: |
---|
580 | |
---|
581 | \begin{verbatim} |
---|
582 | domain.set_boundary({'left': Br, 'right': Bw, 'top': Br, 'bottom': Br}) |
---|
583 | \end{verbatim} |
---|
584 | |
---|
585 | It is critical that all tags are associated with a boundary condition in this statement. |
---|
586 | If not the program will halt with a statement like: |
---|
587 | |
---|
588 | \begin{verbatim} |
---|
589 | Traceback (most recent call last): |
---|
590 | File "mesh_test.py", line 114, in ? |
---|
591 | domain.set_boundary({'west': Bi, 'east': Bo, 'north': Br, 'south': Br}) |
---|
592 | File "X:\inundation\sandpits\onielsen\anuga_core\source\anuga\abstract_2d_finite_volumes\domain.py", line 505, in set_boundary |
---|
593 | raise msg |
---|
594 | ERROR (domain.py): Tag "exterior" has not been bound to a boundary object. |
---|
595 | All boundary tags defined in domain must appear in the supplied dictionary. |
---|
596 | The tags are: ['ocean', 'east', 'north', 'exterior', 'south'] |
---|
597 | \end{verbatim} |
---|
598 | |
---|
599 | The command \code{set_boundary} stipulates that, in the current example, the right |
---|
600 | boundary varies with time, as defined by the lambda function, while the other |
---|
601 | boundaries are all reflective. |
---|
602 | |
---|
603 | The reader may wish to experiment by varying the choice of boundary |
---|
604 | types for one or more of the boundaries. (In the case of \code{Bd} |
---|
605 | and \code{Bw}, the three arguments in each case represent the |
---|
606 | \code{stage}, $x$-momentum and $y$-momentum, respectively.) |
---|
607 | |
---|
608 | \begin{verbatim} |
---|
609 | Bw = Time_boundary(domain=domain, f=lambda t: [(0.1*sin(t*2*pi)-0.3), 0.0, 0.0]) |
---|
610 | \end{verbatim} |
---|
611 | |
---|
612 | \subsection{Evolution}\index{evolution} |
---|
613 | |
---|
614 | The final statement: |
---|
615 | |
---|
616 | \begin{verbatim} |
---|
617 | for t in domain.evolve(yieldstep=0.1, duration=10.0): |
---|
618 | print domain.timestepping_statistics() |
---|
619 | \end{verbatim} |
---|
620 | |
---|
621 | causes the configuration of the domain to 'evolve', over a series of |
---|
622 | steps indicated by the values of \code{yieldstep} and |
---|
623 | \code{duration}, which can be altered as required. The value of |
---|
624 | \code{yieldstep} controls the time interval between successive model |
---|
625 | outputs. Behind the scenes more time steps are generally taken. |
---|
626 | |
---|
627 | \subsection{Output} |
---|
628 | |
---|
629 | The output is a NetCDF file with the extension \code{.sww}. It |
---|
630 | contains stage and momentum information and can be used with the |
---|
631 | \anuga viewer \code{animate} to generate a visual |
---|
632 | display (see Section \ref{sec:animate}). See Section \ref{sec:file formats} |
---|
633 | (page \pageref{sec:file formats}) for more on NetCDF and other file |
---|
634 | formats. |
---|
635 | |
---|
636 | The following is a listing of the screen output seen by the user |
---|
637 | when this example is run: |
---|
638 | |
---|
639 | \verbatiminput{examples/runupoutput.txt} |
---|
640 | |
---|
641 | |
---|
642 | \section{How to Run the Code} |
---|
643 | |
---|
644 | The code can be run in various ways: |
---|
645 | \begin{itemize} |
---|
646 | \item{from a Windows or Unix command line} as in\ \code{python runup.py} |
---|
647 | \item{within the Python IDLE environment} |
---|
648 | \item{within emacs} |
---|
649 | \item{within Windows, by double-clicking the \code{runup.py} |
---|
650 | file.} |
---|
651 | \end{itemize} |
---|
652 | |
---|
653 | |
---|
654 | \section{Exploring the Model Output} |
---|
655 | |
---|
656 | The following figures are screenshots from the \anuga visualisation |
---|
657 | tool \code{animate}. Figure \ref{fig:runupstart} shows the domain |
---|
658 | with water surface as specified by the initial condition, $t=0$. |
---|
659 | Figure \ref{fig:runup2} shows later snapshots for $t=2.3$ and |
---|
660 | $t=4$ where the system has been evolved and the wave is encroaching |
---|
661 | on the previously dry bed. |
---|
662 | |
---|
663 | \code{animate} is described in more detail in Section \ref{sec:animate}. |
---|
664 | |
---|
665 | \begin{figure}[htp] |
---|
666 | \centerline{\includegraphics[width=75mm, height=75mm] |
---|
667 | {graphics/bedslopestart.jpg}} |
---|
668 | \caption{Runup example viewed with the \anuga viewer} |
---|
669 | \label{fig:runupstart} |
---|
670 | \end{figure} |
---|
671 | |
---|
672 | \begin{figure}[htp] |
---|
673 | \centerline{ |
---|
674 | \includegraphics[width=75mm, height=75mm]{graphics/bedslopeduring.jpg} |
---|
675 | \includegraphics[width=75mm, height=75mm]{graphics/bedslopeend.jpg} |
---|
676 | } |
---|
677 | \caption{Runup example viewed with ANGUA viewer} |
---|
678 | \label{fig:runup2} |
---|
679 | \end{figure} |
---|
680 | |
---|
681 | \clearpage |
---|
682 | |
---|
683 | |
---|
684 | \section{A slightly more complex example} |
---|
685 | \label{sec:channelexample} |
---|
686 | |
---|
687 | \subsection{Overview} |
---|
688 | |
---|
689 | The next example is about waterflow in a channel with varying boundary conditions and |
---|
690 | more complex topographies. These examples build on the |
---|
691 | concepts introduced through the \file{runup.py} in Section \ref{sec:simpleexample}. |
---|
692 | The example will be built up through three progressively more complex scripts. |
---|
693 | |
---|
694 | \subsection{Overview} |
---|
695 | |
---|
696 | As in the case of \file{runup.py}, the actions carried |
---|
697 | out by the program can be organised according to this outline: |
---|
698 | \begin{enumerate} |
---|
699 | \item Set up a triangular mesh. |
---|
700 | \item Set certain parameters governing the mode of |
---|
701 | operation of the model -- specifying, for instance, where to store the |
---|
702 | model output. |
---|
703 | \item Set up initial conditions for various quantities such as the elevation, to be specified at each mesh point (vertex). |
---|
704 | \item Set up the boundary conditions. |
---|
705 | \item Carry out the evolution of the model through a series of time |
---|
706 | steps and output the results, providing a results file that can be |
---|
707 | viewed. |
---|
708 | \end{enumerate} |
---|
709 | |
---|
710 | \subsection{The Code} |
---|
711 | |
---|
712 | Here is the code for the first version of the channel flow \file{channel1.py}: |
---|
713 | |
---|
714 | \verbatiminput{demos/channel1.py} |
---|
715 | |
---|
716 | In discussing the details of this example, we follow the outline |
---|
717 | given above, discussing each major step of the code in turn. |
---|
718 | |
---|
719 | \subsection{Establishing the Mesh}\index{mesh, establishing} |
---|
720 | |
---|
721 | In this example we use a similar simple structured triangular mesh as in \file{runup.py} |
---|
722 | for simplicity, but this time we will use a symmetric one and also |
---|
723 | change the physical extent of the domain. The assignment: |
---|
724 | |
---|
725 | \begin{verbatim} |
---|
726 | points, vertices, boundary = rectangular_cross(m, n, len1=length, len2=width) |
---|
727 | \end{verbatim} |
---|
728 | |
---|
729 | returns an \code{mxn} mesh similar to the one used in the previous example, except that now the |
---|
730 | extent in the x and y directions are given by the value of \code{length} and \code{width} |
---|
731 | respectively. |
---|
732 | |
---|
733 | Defining \code{m} and \code{n} in terms of the extent as in this example provides a convenient way of |
---|
734 | controlling the resolution: By defining \code{dx} and \code{dy} to be the desired size of each |
---|
735 | hypothenuse in the mesh we can write the mesh generation as follows: |
---|
736 | |
---|
737 | \begin{verbatim} |
---|
738 | length = 10.0 |
---|
739 | width = 5.0 |
---|
740 | dx = dy = 1 # Resolution: Length of subdivisions on both axes |
---|
741 | |
---|
742 | points, vertices, boundary = rectangular_cross(int(length/dx), int(width/dy), |
---|
743 | len1=length, len2=width) |
---|
744 | \end{verbatim} |
---|
745 | |
---|
746 | which yields a mesh of length=10m, width=5m with 1m spacings. To increase the resolution, |
---|
747 | as we will later in this example, one merely decreases the values of \code{dx} and \code{dy}. |
---|
748 | |
---|
749 | The rest of this script is similar to the previous example on page \pageref{ref:runup_py_code}. |
---|
750 | % except for an application of the 'expression' form of \code{set\_quantity} where we use |
---|
751 | % the value of \code{elevation} to define the (dry) initial condition for \code{stage}: |
---|
752 | %\begin{verbatim} |
---|
753 | % domain.set_quantity('stage', expression='elevation') |
---|
754 | %\end{verbatim} |
---|
755 | |
---|
756 | |
---|
757 | \section{Model Output} |
---|
758 | |
---|
759 | The following figure is a screenshot from the \anuga visualisation |
---|
760 | tool \code{animate} of output from this example. |
---|
761 | |
---|
762 | \begin{figure}[htp] |
---|
763 | \centerline{\includegraphics[height=75mm] |
---|
764 | {graphics/channel1.png}}% |
---|
765 | \caption{Simple channel example viewed with the \anuga viewer.} |
---|
766 | \label{fig:channel1} |
---|
767 | \end{figure} |
---|
768 | |
---|
769 | \subsection{Changing boundary conditions on the fly} |
---|
770 | \label{sec:change boundary} |
---|
771 | |
---|
772 | Here is the code for the second version of the channel flow \file{channel2.py}: |
---|
773 | |
---|
774 | \verbatiminput{demos/channel2.py} |
---|
775 | |
---|
776 | This example differs from the first version in that a constant outflow boundary condition has |
---|
777 | been defined: |
---|
778 | |
---|
779 | \begin{verbatim} |
---|
780 | Bo = Dirichlet_boundary([-5, 0, 0]) # Outflow |
---|
781 | \end{verbatim} |
---|
782 | |
---|
783 | and that it is applied to the right hand side boundary when the water level there exceeds 0m. |
---|
784 | |
---|
785 | \begin{verbatim} |
---|
786 | for t in domain.evolve(yieldstep=0.2, finaltime=40.0): |
---|
787 | domain.write_time() |
---|
788 | |
---|
789 | if domain.get_quantity('stage').get_values(interpolation_points=[[10, 2.5]]) > 0: |
---|
790 | print 'Stage > 0: Changing to outflow boundary' |
---|
791 | domain.set_boundary({'right': Bo}) |
---|
792 | \end{verbatim} |
---|
793 | |
---|
794 | \label{sec:change boundary code} |
---|
795 | The \code{if} statement in the timestepping loop (\code{evolve}) gets the quantity |
---|
796 | \code{stage} and obtains the interpolated value at the point (10m, |
---|
797 | 2.5m) which is on the right boundary. If the stage exceeds 0m a |
---|
798 | message is printed and the old boundary condition at tag 'right' is |
---|
799 | replaced by the outflow boundary using the method: |
---|
800 | |
---|
801 | \begin{verbatim} |
---|
802 | domain.set_boundary({'right': Bo}) |
---|
803 | \end{verbatim} |
---|
804 | |
---|
805 | This type of dynamically varying boundary could for example be |
---|
806 | used to model the breakdown of a sluice door when water exceeds a certain level. |
---|
807 | |
---|
808 | \subsection{Output} |
---|
809 | |
---|
810 | The text output from this example looks like this: |
---|
811 | |
---|
812 | \begin{verbatim} |
---|
813 | ... |
---|
814 | Time = 15.4000, delta t in [0.03789902, 0.03789916], steps=6 (6) |
---|
815 | Time = 15.6000, delta t in [0.03789896, 0.03789908], steps=6 (6) |
---|
816 | Time = 15.8000, delta t in [0.03789891, 0.03789903], steps=6 (6) |
---|
817 | Stage > 0: Changing to outflow boundary |
---|
818 | Time = 16.0000, delta t in [0.02709050, 0.03789898], steps=6 (6) |
---|
819 | Time = 16.2000, delta t in [0.03789892, 0.03789904], steps=6 (6) |
---|
820 | ... |
---|
821 | \end{verbatim} |
---|
822 | |
---|
823 | \subsection{Flow through more complex topograhies} |
---|
824 | |
---|
825 | Here is the code for the third version of the channel flow \file{channel3.py}: |
---|
826 | |
---|
827 | \verbatiminput{demos/channel3.py} |
---|
828 | |
---|
829 | This example differs from the first two versions in that the topography |
---|
830 | contains obstacles. |
---|
831 | |
---|
832 | This is accomplished here by defining the function \code{topography} as follows: |
---|
833 | |
---|
834 | \begin{verbatim} |
---|
835 | def topography(x,y): |
---|
836 | """Complex topography defined by a function of vectors x and y.""" |
---|
837 | |
---|
838 | z = -x/10 |
---|
839 | |
---|
840 | N = len(x) |
---|
841 | for i in range(N): |
---|
842 | # Step |
---|
843 | if 10 < x[i] < 12: |
---|
844 | z[i] += 0.4 - 0.05*y[i] |
---|
845 | |
---|
846 | # Constriction |
---|
847 | if 27 < x[i] < 29 and y[i] > 3: |
---|
848 | z[i] += 2 |
---|
849 | |
---|
850 | # Pole |
---|
851 | if (x[i] - 34)**2 + (y[i] - 2)**2 < 0.4**2: |
---|
852 | z[i] += 2 |
---|
853 | |
---|
854 | return z |
---|
855 | \end{verbatim} |
---|
856 | |
---|
857 | In addition, changing the resolution to \code{dx = dy = 0.1} creates a finer mesh resolving the new features better. |
---|
858 | |
---|
859 | A screenshot of this model at time 15s is: |
---|
860 | \begin{figure}[htp] |
---|
861 | \centerline{\includegraphics[height=75mm] |
---|
862 | {graphics/channel3.png}} |
---|
863 | \caption{More complex flow in a channel} |
---|
864 | \label{fig:channel3} |
---|
865 | \end{figure} |
---|
866 | |
---|
867 | |
---|
868 | \section{An Example with Real Data} |
---|
869 | |
---|
870 | \label{sec:realdataexample} The following discussion builds on the |
---|
871 | concepts introduced through the \file{runup.py} example and |
---|
872 | introduces a second example, \file{runcairns.py}. This refers to |
---|
873 | a {\bf hypothetical} scenario using real-life data, |
---|
874 | in which the domain of interest surrounds the |
---|
875 | Cairns region. Two scenarios are given; firstly, a |
---|
876 | hypothetical tsunami wave is generated by a submarine mass failure |
---|
877 | situated on the edge of the continental shelf, and secondly, a fixed wave |
---|
878 | of given amplitude and period is introduced through the boundary. |
---|
879 | |
---|
880 | {\bf |
---|
881 | Each scenario has been designed to generate a tsunami which will |
---|
882 | inundate the Cairns region. To achieve this, suitably large |
---|
883 | parameters were chosen and were not based on any known tsunami sources |
---|
884 | or realistic amplitudes. |
---|
885 | } |
---|
886 | |
---|
887 | \subsection{Overview} |
---|
888 | As in the case of \file{runup.py}, the actions carried |
---|
889 | out by the program can be organised according to this outline: |
---|
890 | \begin{enumerate} |
---|
891 | \item Set up a triangular mesh. |
---|
892 | |
---|
893 | \item Set certain parameters governing the mode of |
---|
894 | operation of the model -- specifying, for instance, where to store the |
---|
895 | model output. |
---|
896 | |
---|
897 | \item Input various quantities describing physical measurements, such |
---|
898 | as the elevation, to be specified at each mesh point (vertex). |
---|
899 | |
---|
900 | \item Set up the boundary conditions. |
---|
901 | |
---|
902 | \item Carry out the evolution of the model through a series of time |
---|
903 | steps and output the results, providing a results file that can be |
---|
904 | visualised. |
---|
905 | \end{enumerate} |
---|
906 | |
---|
907 | \subsection{The Code} |
---|
908 | |
---|
909 | Here is the code for \file{runcairns.py}: |
---|
910 | |
---|
911 | \verbatiminput{demos/cairns/runcairns.py} |
---|
912 | |
---|
913 | In discussing the details of this example, we follow the outline |
---|
914 | given above, discussing each major step of the code in turn. |
---|
915 | |
---|
916 | \subsection{Establishing the Mesh}\index{mesh, establishing} |
---|
917 | |
---|
918 | One obvious way that the present example differs from |
---|
919 | \file{runup.py} is in the use of a more complex method to |
---|
920 | create the mesh. Instead of imposing a mesh structure on a |
---|
921 | rectangular grid, the technique used for this example involves |
---|
922 | building mesh structures inside polygons specified by the user, |
---|
923 | using a mesh-generator. |
---|
924 | |
---|
925 | The mesh-generator creates the mesh within a single |
---|
926 | polygon whose vertices are at geographical locations specified by |
---|
927 | the user. The user specifies the \emph{resolution} -- that is, the |
---|
928 | maximal area of a triangle used for triangulation -- and a triangular |
---|
929 | mesh is created inside the polygon using a mesh generation engine. |
---|
930 | On any given platform, the same mesh will be returned each time the |
---|
931 | script is run. |
---|
932 | |
---|
933 | Boundary tags are not restricted to \code{'left'}, \code{'bottom'}, |
---|
934 | \code{'right'} and \code{'top'}, as in the case of |
---|
935 | \file{runup.py}. Instead the user specifies a list of |
---|
936 | tags appropriate to the configuration being modelled. |
---|
937 | |
---|
938 | In addition, the mesh-generator provides a way to adapt to geographic or |
---|
939 | other features in the landscape, whose presence may require an |
---|
940 | increase in resolution. This is done by allowing the user to specify |
---|
941 | a number of \emph{interior polygons}, each with a specified |
---|
942 | resolution. It is also |
---|
943 | possible to specify one or more 'holes' -- that is, areas bounded by |
---|
944 | polygons in which no triangulation is required. |
---|
945 | |
---|
946 | In its general form, the mesh-generator takes for its input a bounding |
---|
947 | polygon and (optionally) a list of interior polygons. The user |
---|
948 | specifies resolutions, both for the bounding polygon and for each of |
---|
949 | the interior polygons. Given this data, the mesh-generator first creates a |
---|
950 | triangular mesh with varying resolution. |
---|
951 | |
---|
952 | The function used to implement this process is |
---|
953 | \function{create\_domain\_from\_regions} which creates a Domain object as |
---|
954 | well as a mesh file. Its arguments include the |
---|
955 | bounding polygon and its resolution, a list of boundary tags, and a |
---|
956 | list of pairs \code{[polygon, resolution]} specifying the interior |
---|
957 | polygons and their resolutions. |
---|
958 | |
---|
959 | The resulting mesh is output to a \emph{mesh file}\index{mesh |
---|
960 | file}\label{def:mesh file}. This term is used to describe a file of |
---|
961 | a specific format used to store the data specifying a mesh. (There |
---|
962 | are in fact two possible formats for such a file: it can either be a |
---|
963 | binary file, with extension \code{.msh}, or an ASCII file, with |
---|
964 | extension \code{.tsh}. In the present case, the binary file format |
---|
965 | \code{.msh} is used. See Section \ref{sec:file formats} (page |
---|
966 | \pageref{sec:file formats}) for more on file formats. |
---|
967 | |
---|
968 | In practice, the details of the polygons used are read from a |
---|
969 | separate file \file{project.py}. Here is a complete listing of |
---|
970 | \file{project.py}: |
---|
971 | |
---|
972 | \verbatiminput{demos/cairns/project.py} |
---|
973 | |
---|
974 | Figure \ref{fig:cairns3d} illustrates the landscape of the region |
---|
975 | for the Cairns example. Understanding the landscape is important in |
---|
976 | determining the location and resolution of interior polygons. The |
---|
977 | supporting data is found in the ASCII grid, \code{cairns.asc}, which |
---|
978 | has been sourced from the publically available Australian Bathymetry |
---|
979 | and Topography Grid 2005, \cite{grid250}. The required resolution |
---|
980 | for inundation modelling will depend on the underlying topography and |
---|
981 | bathymetry; as the terrain becomes more complex, the desired resolution |
---|
982 | would decrease to the order of tens of metres. |
---|
983 | |
---|
984 | \clearpage |
---|
985 | |
---|
986 | \begin{figure}[htp] |
---|
987 | \centerline{\includegraphics[scale=0.5]{graphics/cairns3.jpg}} |
---|
988 | \caption{Landscape of the Cairns scenario.} |
---|
989 | \label{fig:cairns3d} |
---|
990 | \end{figure} |
---|
991 | |
---|
992 | The following statements are used to read in the specific polygons |
---|
993 | from \code{project.cairns} and assign a defined resolution to |
---|
994 | each polygon. |
---|
995 | |
---|
996 | \begin{verbatim} |
---|
997 | islands_res = 100000 |
---|
998 | cairns_res = 100000 |
---|
999 | shallow_res = 500000 |
---|
1000 | interior_regions = [[project.poly_cairns, cairns_res], |
---|
1001 | [project.poly_island0, islands_res], |
---|
1002 | [project.poly_island1, islands_res], |
---|
1003 | [project.poly_island2, islands_res], |
---|
1004 | [project.poly_island3, islands_res], |
---|
1005 | [project.poly_shallow, shallow_res]] |
---|
1006 | \end{verbatim} |
---|
1007 | |
---|
1008 | Figure \ref{fig:cairnspolys} |
---|
1009 | illustrates the polygons used for the Cairns scenario. |
---|
1010 | |
---|
1011 | \clearpage |
---|
1012 | |
---|
1013 | \begin{figure}[htp] |
---|
1014 | \centerline{\includegraphics[scale=0.5] |
---|
1015 | {graphics/cairnsmodel.jpg}} |
---|
1016 | \caption{Interior and bounding polygons for the Cairns example.} |
---|
1017 | \label{fig:cairnspolys} |
---|
1018 | \end{figure} |
---|
1019 | |
---|
1020 | The statement: |
---|
1021 | |
---|
1022 | \begin{verbatim} |
---|
1023 | remainder_res = 10000000 |
---|
1024 | domain = create_domain_from_regions(project.bounding_polygon, |
---|
1025 | boundary_tags={'top': [0], |
---|
1026 | 'ocean_east': [1], |
---|
1027 | 'bottom': [2], |
---|
1028 | 'onshore': [3]}, |
---|
1029 | maximum_triangle_area=project.default_res, |
---|
1030 | mesh_filename=project.meshname, |
---|
1031 | interior_regions=project.interior_regions, |
---|
1032 | use_cache=True, |
---|
1033 | verbose=True) |
---|
1034 | \end{verbatim} |
---|
1035 | |
---|
1036 | is then used to create the mesh, taking the bounding polygon to be |
---|
1037 | the polygon \code{bounding\_polygon} specified in \file{project.py}. |
---|
1038 | The argument \code{boundary\_tags} assigns a dictionary, whose keys |
---|
1039 | are the names of the boundary tags used for the bounding |
---|
1040 | polygon -- \code{'top'}, \code{'ocean\_east'}, \code{'bottom'}, and |
---|
1041 | \code{'onshore'} -- and whose values identify the indices of the |
---|
1042 | segments associated with each of these tags. |
---|
1043 | The polygon may be arranged either clock-wise or counter clock-wise and the |
---|
1044 | indices refer to edges in the order they appear: Edge 0 connects vertex 0 and vertex 1, edge 1 connects vertex 1 and 2; and so forth. |
---|
1045 | (Here, the values associated with each boundary tag are one-element lists, but they can have as many indices as there are edges) |
---|
1046 | If polygons intersect, or edges coincide (or are even very close) the resolution may be undefined in some regions. |
---|
1047 | Use the underlying mesh interface for such cases |
---|
1048 | (see Chapter \ref{sec:mesh interface}). |
---|
1049 | If a segment is omitted in the tags definition an Exception is raised. |
---|
1050 | |
---|
1051 | Note that every point on each polygon defining the mesh will be used as vertices in triangles. |
---|
1052 | Consequently, polygons with points very close together will cause triangles with very small |
---|
1053 | areas to be generated irrespective of the requested resolution. |
---|
1054 | Make sure points on polygons are spaced to be no closer than the smallest resolution requested. |
---|
1055 | |
---|
1056 | \subsection{Initialising the Domain} |
---|
1057 | |
---|
1058 | Since we used \code{create_domain_from_regions} to create the mesh file, we do not need to |
---|
1059 | create the domain explicitly, as the above function does both mesh and domain creation. |
---|
1060 | |
---|
1061 | The following statements specify a basename and data directory, and |
---|
1062 | sets a minimum storable height, which helps with visualisation and post-processing |
---|
1063 | if one wants to remove water less than 1cm deep (for instance). |
---|
1064 | |
---|
1065 | \begin{verbatim} |
---|
1066 | domain.set_name('cairns_' + project.scenario) # Name of SWW file |
---|
1067 | domain.set_datadir('.') # Store SWW output here |
---|
1068 | domain.set_minimum_storable_height(0.01) # Store only depth > 1cm |
---|
1069 | \end{verbatim} |
---|
1070 | |
---|
1071 | \subsection{Initial Conditions} |
---|
1072 | |
---|
1073 | Quantities for \file{runcairns.py} are set |
---|
1074 | using similar methods to those in \file{runup.py}. However, |
---|
1075 | in this case, many of the values are read from the auxiliary file |
---|
1076 | \file{project.py} or, in the case of \code{elevation}, from an |
---|
1077 | auxiliary points file. |
---|
1078 | |
---|
1079 | \subsubsection{Stage} |
---|
1080 | |
---|
1081 | The stage is initially set to 0.0 (i.e.\ Mean Sea Level) by the following statements: |
---|
1082 | |
---|
1083 | \begin{verbatim} |
---|
1084 | tide = 0.0 |
---|
1085 | domain.set_quantity('stage', tide) |
---|
1086 | \end{verbatim} |
---|
1087 | |
---|
1088 | It could also take the value of the highest astronomical tide. |
---|
1089 | |
---|
1090 | %For the scenario we are modelling in this case, we use a callable |
---|
1091 | %object \code{tsunami_source}, assigned by means of a function |
---|
1092 | %\function{slide\_tsunami}. This is similar to how we set elevation in |
---|
1093 | %\file{runup.py} using a function -- however, in this case the |
---|
1094 | %function is both more complex and more interesting. |
---|
1095 | |
---|
1096 | %The function returns the water displacement for all \code{x} and |
---|
1097 | %\code{y} in the domain. The water displacement is a double Gaussian |
---|
1098 | %function that depends on the characteristics of the slide (length, |
---|
1099 | %width, thickness, slope, etc), its location (origin) and the depth at that |
---|
1100 | %location. For this example, we choose to apply the slide function |
---|
1101 | %at a specified time into the simulation. {\bf Note, the parameters used |
---|
1102 | %in this example have been deliberately chosen to generate a suitably |
---|
1103 | %large amplitude tsunami which would inundate the Cairns region.} |
---|
1104 | |
---|
1105 | \subsubsection{Friction} |
---|
1106 | |
---|
1107 | We assign the friction exactly as we did for \file{runup.py}: |
---|
1108 | |
---|
1109 | \begin{verbatim} |
---|
1110 | domain.set_quantity('friction', 0.0) |
---|
1111 | \end{verbatim} |
---|
1112 | |
---|
1113 | \subsubsection{Elevation} |
---|
1114 | |
---|
1115 | The elevation is specified by reading data from a file with a name derived from |
---|
1116 | \code{project.demname} with the \code{.pts} extension: |
---|
1117 | |
---|
1118 | \begin{verbatim} |
---|
1119 | domain.set_quantity('elevation', |
---|
1120 | filename=project.demname + '.pts', |
---|
1121 | use_cache=True, |
---|
1122 | verbose=True, |
---|
1123 | alpha=0.1) |
---|
1124 | \end{verbatim} |
---|
1125 | |
---|
1126 | The \code{alpha} parameter controls how smooth the elevation surface |
---|
1127 | should be. See section \ref{class:alpha_shape}, page \pageref{class:alpha_shape}. |
---|
1128 | |
---|
1129 | Setting \code{cache=True} allows \anuga to save the result in order |
---|
1130 | to make subsequent runs faster. |
---|
1131 | |
---|
1132 | Using \code{verbose=True} tells the function to write diagnostics to |
---|
1133 | the screen. |
---|
1134 | |
---|
1135 | \subsection{Boundary Conditions}\index{boundary conditions} |
---|
1136 | |
---|
1137 | Setting boundaries follows a similar pattern to the one used for |
---|
1138 | \file{runup.py}, except that in this case we need to associate a |
---|
1139 | boundary type with each of the |
---|
1140 | boundary tag names introduced when we established the mesh. In place of the four |
---|
1141 | boundary types introduced for \file{runup.py}, we use the reflective |
---|
1142 | boundary for each of the tagged segments defined by \code{create_domain_from_regions}: |
---|
1143 | |
---|
1144 | \begin{verbatim} |
---|
1145 | Bd = Dirichlet_boundary([tide,0,0]) # Mean water level |
---|
1146 | Bs = Transmissive_stage_zero_momentum_boundary(domain) # Neutral boundary |
---|
1147 | |
---|
1148 | if project.scenario == 'fixed_wave': |
---|
1149 | # Huge 50m wave starting after 60 seconds and lasting 1 hour. |
---|
1150 | Bw = Time_boundary(domain=domain, |
---|
1151 | function=lambda t: [(60<t<3660)*50, 0, 0]) |
---|
1152 | domain.set_boundary({'ocean_east': Bw, |
---|
1153 | 'bottom': Bs, |
---|
1154 | 'onshore': Bd, |
---|
1155 | 'top': Bs}) |
---|
1156 | |
---|
1157 | if project.scenario == 'slide': |
---|
1158 | # Boundary conditions for slide scenario |
---|
1159 | domain.set_boundary({'ocean_east': Bd, |
---|
1160 | 'bottom': Bd, |
---|
1161 | 'onshore': Bd, |
---|
1162 | 'top': Bd}) |
---|
1163 | \end{verbatim} |
---|
1164 | |
---|
1165 | Note that we use different boundary conditions depending on the \code{scenario} |
---|
1166 | defined in \file{project.py}. |
---|
1167 | |
---|
1168 | \subsection{Evolution} |
---|
1169 | |
---|
1170 | With the basics established, the running of the 'evolve' step is |
---|
1171 | very similar to the corresponding step in \file{runup.py}, except we have different \code{evolve} |
---|
1172 | loops for the two scenarios. |
---|
1173 | |
---|
1174 | For the slide scenario, the simulation is run for an intial 60 seconds, at which time |
---|
1175 | the slide occurs. We use the function \function{tsunami_source} to adjust \code{stage} |
---|
1176 | values. We then run the simulation until 5000 seconds with the output stored |
---|
1177 | every ten seconds. |
---|
1178 | |
---|
1179 | \begin{verbatim} |
---|
1180 | if project.scenario == 'slide': |
---|
1181 | # Initial run without any event |
---|
1182 | for t in domain.evolve(yieldstep=10, finaltime=60): |
---|
1183 | print domain.timestepping_statistics() |
---|
1184 | print domain.boundary_statistics(tags='ocean_east') |
---|
1185 | |
---|
1186 | # Add slide to water surface |
---|
1187 | if allclose(t, 60): |
---|
1188 | domain.add_quantity('stage', tsunami_source) |
---|
1189 | |
---|
1190 | # Continue propagating wave |
---|
1191 | for t in domain.evolve(yieldstep=10, finaltime=5000, |
---|
1192 | skip_initial_step=True): |
---|
1193 | print domain.timestepping_statistics() |
---|
1194 | print domain.boundary_statistics(tags='ocean_east') |
---|
1195 | |
---|
1196 | if project.scenario == 'fixed_wave': |
---|
1197 | # Save every two mins leading up to wave approaching land |
---|
1198 | for t in domain.evolve(yieldstep=120, finaltime=5000): |
---|
1199 | print domain.timestepping_statistics() |
---|
1200 | print domain.boundary_statistics(tags='ocean_east') |
---|
1201 | |
---|
1202 | # Save every 30 secs as wave starts inundating ashore |
---|
1203 | for t in domain.evolve(yieldstep=10, finaltime=10000, |
---|
1204 | skip_initial_step=True): |
---|
1205 | print domain.timestepping_statistics() |
---|
1206 | print domain.boundary_statistics(tags='ocean_east') |
---|
1207 | \end{verbatim} |
---|
1208 | |
---|
1209 | For the fixed wave scenario, the simulation is run to 10000 seconds, |
---|
1210 | with the first half of the simulation stored at two minute intervals, |
---|
1211 | and the second half of the simulation stored at ten second intervals. |
---|
1212 | This functionality is especially convenient as it allows the detailed |
---|
1213 | parts of the simulation to be viewed at higher time resolution. |
---|
1214 | |
---|
1215 | \section{Exploring the Model Output} |
---|
1216 | |
---|
1217 | Now that the scenario has been run, the user can view the output in a number of ways. |
---|
1218 | As described earlier, the user may run \code{animate} to view a three-dimensional representation |
---|
1219 | of the simulation. |
---|
1220 | |
---|
1221 | The user may also be interested in a maximum inundation map. This simply shows the |
---|
1222 | maximum water depth over the domain and is achieved with the function \code{sww2dem} |
---|
1223 | described in Section \ref{sec:basicfileconversions}). |
---|
1224 | \file{ExportResults.py} demonstrates how this function can be used: |
---|
1225 | |
---|
1226 | \verbatiminput{demos/cairns/ExportResults.py} |
---|
1227 | |
---|
1228 | The script generates a maximum water depth ASCII grid at a defined |
---|
1229 | resolution (here 100 m$^2$) which can then be viewed in a GIS environment, for |
---|
1230 | example. The parameters used in the function are defined in \file{project.py}. |
---|
1231 | Figures \ref{fig:maxdepthcairnsslide} and \ref{fig:maxdepthcairnsfixedwave} show |
---|
1232 | the maximum water depth within the defined region for the slide and fixed wave scenario |
---|
1233 | respectively. {\bf Note, these inundation maps have been based on purely hypothetical |
---|
1234 | scenarios and were designed explicitly for demonstration purposes only.} |
---|
1235 | The user could develop a maximum absolute momentum or other expressions which can be |
---|
1236 | derived from the quantities. |
---|
1237 | It must be noted here that depth is more meaningful when the elevation is positive |
---|
1238 | (\code{depth} = \code{stage} $-$ \code{elevation}) as it describes the water height |
---|
1239 | above the available elevation. When the elevation is negative, depth is meauring the |
---|
1240 | water height from the sea floor. With this in mind, maximum inundation maps are |
---|
1241 | typically "clipped" to the coastline. However, the data input here did not contain a |
---|
1242 | coastline. |
---|
1243 | |
---|
1244 | \clearpage |
---|
1245 | |
---|
1246 | \begin{figure}[htp] |
---|
1247 | \centerline{\includegraphics[scale=0.5]{graphics/slidedepth.jpg}} |
---|
1248 | \caption{Maximum inundation map for the Cairns slide scenario. \bf Note, this |
---|
1249 | inundation map has been based on a purely hypothetical scenario which was |
---|
1250 | designed explictiy for demonstration purposes only.} |
---|
1251 | \label{fig:maxdepthcairnsslide} |
---|
1252 | \end{figure} |
---|
1253 | |
---|
1254 | \clearpage |
---|
1255 | |
---|
1256 | \begin{figure}[htp] |
---|
1257 | \centerline{\includegraphics[scale=0.5]{graphics/fixedwavedepth.jpg}} |
---|
1258 | \caption{Maximum inundation map for the Cairns fixed wave scenario. |
---|
1259 | \bf Note, this inundation map has been based on a purely hypothetical scenario which was |
---|
1260 | designed explictiy for demonstration purposes only.} |
---|
1261 | \label{fig:maxdepthcairnsfixedwave} |
---|
1262 | \end{figure} |
---|
1263 | |
---|
1264 | \clearpage |
---|
1265 | |
---|
1266 | The user may also be interested in interrogating the solution at a particular spatial |
---|
1267 | location to understand the behaviour of the system through time. To do this, the user |
---|
1268 | must first define the locations of interest. A number of locations have been |
---|
1269 | identified for the Cairns scenario, as shown in Figure \ref{fig:cairnsgauges}. |
---|
1270 | |
---|
1271 | \begin{figure}[htp] |
---|
1272 | \centerline{\includegraphics[scale=0.5]{graphics/cairnsgauges.jpg}} |
---|
1273 | \caption{Point locations to show time series information for the Cairns scenario.} |
---|
1274 | \label{fig:cairnsgauges} |
---|
1275 | \end{figure} |
---|
1276 | |
---|
1277 | These locations |
---|
1278 | must be stored in either a .csv or .txt file. The corresponding .csv file for |
---|
1279 | the gauges shown in Figure \ref{fig:cairnsgauges} is \file{gauges.csv}: |
---|
1280 | |
---|
1281 | \verbatiminput{demos/cairns/gauges.csv} |
---|
1282 | |
---|
1283 | Header information has been included to identify the location in terms of eastings and |
---|
1284 | northings, and each gauge is given a name. The elevation column can be zero here. |
---|
1285 | This information is then passed to the function \code{sww2csv_gauges} (shown in |
---|
1286 | \file{GetTimeseries.py} which generates the csv files for each point location. The CSV files |
---|
1287 | can then be used in \code{csv2timeseries_graphs} to create the timeseries plot for each desired |
---|
1288 | quantity. \code{csv2timeseries_graphs} relies on \code{pylab} to be installed which is not part |
---|
1289 | of the standard \code{anuga} release, however it can be downloaded and installed from \code{http://matplotlib.sourceforge.net/} |
---|
1290 | |
---|
1291 | \verbatiminput{demos/cairns/GetTimeseries.py} |
---|
1292 | |
---|
1293 | Here, the time series for the quantities stage, depth and speed will be generated for |
---|
1294 | each gauge defined in the gauge file. As described earlier, depth is more meaningful |
---|
1295 | for onshore gauges, and stage is more appropriate for offshore gauges. |
---|
1296 | |
---|
1297 | As an example output, |
---|
1298 | Figure \ref{fig:reef} shows the time series for the quantity stage for the |
---|
1299 | Elford Reef location for each scenario (the elevation at this location is negative, |
---|
1300 | therefore stage is the more appropriate quantity to plot). Note the large negative stage value when the slide was |
---|
1301 | introduced. This is due to the double gaussian form of the initial surface |
---|
1302 | displacement of the slide. By contrast, the time series for depth is shown for the onshore location of the Cairns |
---|
1303 | Airport in Figure \ref{fig:airportboth}. |
---|
1304 | |
---|
1305 | \begin{figure}[htp] |
---|
1306 | \centerline{\includegraphics[scale=0.5]{graphics/gaugeElfordReefstage.png}} |
---|
1307 | \caption{Time series information of the quantity stage for the Elford Reef location for the |
---|
1308 | fixed wave and slide scenario.} |
---|
1309 | \label{fig:reef} |
---|
1310 | \end{figure} |
---|
1311 | |
---|
1312 | \begin{figure}[htp] |
---|
1313 | \centerline{\includegraphics[scale=0.5]{graphics/gaugeCairnsAirportdepth.png}} |
---|
1314 | \caption{Time series information of the quantity depth for the Cairns Airport |
---|
1315 | location for the slide and fixed wave scenario.} |
---|
1316 | \label{fig:airportboth} |
---|
1317 | \end{figure} |
---|
1318 | |
---|
1319 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
---|
1320 | |
---|
1321 | \chapter{\anuga Public Interface} |
---|
1322 | \label{ch:interface} |
---|
1323 | |
---|
1324 | This chapter gives an overview of the features of \anuga available |
---|
1325 | to the user at the public interface. These are grouped under the |
---|
1326 | following headings, which correspond to the outline of the examples |
---|
1327 | described in Chapter \ref{ch:getstarted}: |
---|
1328 | \begin{itemize} |
---|
1329 | \item Establishing the Mesh: Section \ref{sec:establishing the mesh} |
---|
1330 | \item Initialising the Domain: Section \ref{sec:initialising the domain} |
---|
1331 | % \item Specifying the Quantities: Section \ref{sec:quantities} |
---|
1332 | \item Initial Conditions: Section \ref{sec:initial conditions} |
---|
1333 | \item Boundary Conditions: Section \ref{sec:boundary conditions} |
---|
1334 | \item Forcing Terms: Section \ref{sec:forcing terms} |
---|
1335 | \item Evolution: Section \ref{sec:evolution} |
---|
1336 | \end{itemize} |
---|
1337 | |
---|
1338 | The listings are intended merely to give the reader an idea of what |
---|
1339 | each feature is, where to find it and how it can be used -- they do |
---|
1340 | not give full specifications; for these the reader |
---|
1341 | may consult the code. The code for every function or class contains |
---|
1342 | a documentation string, or 'docstring', that specifies the precise |
---|
1343 | syntax for its use. This appears immediately after the line |
---|
1344 | introducing the code, between two sets of triple quotes. |
---|
1345 | |
---|
1346 | Each listing also describes the location of the module in which |
---|
1347 | the code for the feature being described can be found. All modules |
---|
1348 | are in the folder \file{inundation} or one of its subfolders, and the |
---|
1349 | location of each module is described relative to \file{inundation}. Rather |
---|
1350 | than using pathnames, whose syntax depends on the operating system, |
---|
1351 | we use the format adopted for importing the function or class for |
---|
1352 | use in Python code. For example, suppose we wish to specify that the |
---|
1353 | function \function{create\_mesh\_from\_regions} is in a module called |
---|
1354 | \module{mesh\_interface} in a subfolder of \module{inundation} called |
---|
1355 | \code{pmesh}. In Linux or Unix syntax, the pathname of the file |
---|
1356 | containing the function, relative to \file{inundation}, would be: |
---|
1357 | |
---|
1358 | \begin{verbatim} |
---|
1359 | pmesh/mesh_interface.py |
---|
1360 | \end{verbatim} |
---|
1361 | |
---|
1362 | \label{sec:mesh interface} |
---|
1363 | while in Windows syntax it would be: |
---|
1364 | |
---|
1365 | \begin{verbatim} |
---|
1366 | pmesh\mesh_interface.py |
---|
1367 | \end{verbatim} |
---|
1368 | |
---|
1369 | Rather than using either of these forms, in this chapter we specify |
---|
1370 | the location simply as \code{pmesh.mesh_interface}, in keeping with |
---|
1371 | the usage in the Python statement for importing the function, |
---|
1372 | namely: |
---|
1373 | |
---|
1374 | \begin{verbatim} |
---|
1375 | from pmesh.mesh_interface import create_mesh_from_regions |
---|
1376 | \end{verbatim} |
---|
1377 | |
---|
1378 | Each listing details the full set of parameters for the class or |
---|
1379 | function; however, the description is generally limited to the most |
---|
1380 | important parameters and the reader is again referred to the code |
---|
1381 | for more details. |
---|
1382 | |
---|
1383 | The following parameters are common to many functions and classes |
---|
1384 | and are omitted from the descriptions given below: |
---|
1385 | |
---|
1386 | %\begin{tabular}{ll} |
---|
1387 | \begin{tabular}{p{2.0cm} p{14.0cm}} |
---|
1388 | \emph{use\_cache} & Specifies whether caching is to be used for improved performance. |
---|
1389 | See Section \ref{sec:caching} for details on the underlying caching functionality\\ |
---|
1390 | \emph{verbose} & If \code{True}, provides detailed terminal output to the user\\ |
---|
1391 | \end{tabular} |
---|
1392 | |
---|
1393 | |
---|
1394 | \section{Mesh Generation}\index{Mesh!generation} |
---|
1395 | \label{sec:establishing the mesh} |
---|
1396 | Before discussing the part of the interface relating to mesh |
---|
1397 | generation, we begin with a description of a simple example of a |
---|
1398 | mesh and use it to describe how mesh data is stored. |
---|
1399 | |
---|
1400 | \label{sec:meshexample} Figure \ref{fig:simplemesh} represents a |
---|
1401 | very simple mesh comprising just 11 points and 10 triangles. |
---|
1402 | |
---|
1403 | \begin{figure}[htp] |
---|
1404 | \begin{center} |
---|
1405 | \includegraphics[width=90mm, height=90mm]{triangularmesh.jpg} |
---|
1406 | \end{center} |
---|
1407 | \caption{A simple mesh} |
---|
1408 | \label{fig:simplemesh} |
---|
1409 | \end{figure} |
---|
1410 | |
---|
1411 | \clearpage |
---|
1412 | |
---|
1413 | The variables \code{points}, \code{triangles} and \code{boundary} |
---|
1414 | represent the data displayed in Figure \ref{fig:simplemesh} as |
---|
1415 | follows. The list \code{points} stores the coordinates of the |
---|
1416 | points, and may be displayed schematically as in Table \ref{tab:points}. |
---|
1417 | |
---|
1418 | \begin{table}[htp] |
---|
1419 | \begin{center} |
---|
1420 | \begin{tabular}[t]{|c|cc|} \hline |
---|
1421 | index & \code{x} & \code{y}\\ \hline |
---|
1422 | 0 & 1 & 1\\ |
---|
1423 | 1 & 4 & 2\\ |
---|
1424 | 2 & 8 & 1\\ |
---|
1425 | 3 & 1 & 3\\ |
---|
1426 | 4 & 5 & 5\\ |
---|
1427 | 5 & 8 & 6\\ |
---|
1428 | 6 & 11 & 5\\ |
---|
1429 | 7 & 3 & 6\\ |
---|
1430 | 8 & 1 & 8\\ |
---|
1431 | 9 & 4 & 9\\ |
---|
1432 | 10 & 10 & 7\\ \hline |
---|
1433 | \end{tabular} |
---|
1434 | \end{center} |
---|
1435 | \caption{Point coordinates for mesh in Figure \protect \ref{fig:simplemesh}} |
---|
1436 | \label{tab:points} |
---|
1437 | \end{table} |
---|
1438 | |
---|
1439 | The list \code{triangles} specifies the triangles that make up the |
---|
1440 | mesh. It does this by specifying, for each triangle, the indices |
---|
1441 | (the numbers shown in the first column above) that correspond to the |
---|
1442 | three points at the triangles vertices, taken in an anti-clockwise order |
---|
1443 | around the triangle. Thus, in the example shown in Figure |
---|
1444 | \ref{fig:simplemesh}, the variable \code{triangles} contains the |
---|
1445 | entries shown in Table \ref{tab:triangles}. The starting point is |
---|
1446 | arbitrary so triangle $(0,1,3)$ is considered the same as $(1,3,0)$ |
---|
1447 | and $(3,0,1)$. |
---|
1448 | |
---|
1449 | \begin{table}[htp] |
---|
1450 | \begin{center} |
---|
1451 | \begin{tabular}{|c|ccc|} |
---|
1452 | \hline |
---|
1453 | index & \multicolumn{3}{c|}{\code{points}}\\ |
---|
1454 | \hline |
---|
1455 | 0 & 0 & 1 & 3\\ |
---|
1456 | 1 & 1 & 2 & 4\\ |
---|
1457 | 2 & 2 & 5 & 4\\ |
---|
1458 | 3 & 2 & 6 & 5\\ |
---|
1459 | 4 & 4 & 5 & 9\\ |
---|
1460 | 5 & 4 & 9 & 7\\ |
---|
1461 | 6 & 3 & 4 & 7\\ |
---|
1462 | 7 & 7 & 9 & 8\\ |
---|
1463 | 8 & 1 & 4 & 3\\ |
---|
1464 | 9 & 5 & 10 & 9\\ |
---|
1465 | \hline |
---|
1466 | \end{tabular} |
---|
1467 | \end{center} |
---|
1468 | |
---|
1469 | \caption{Triangles for mesh in Figure \protect \ref{fig:simplemesh}} |
---|
1470 | \label{tab:triangles} |
---|
1471 | \end{table} |
---|
1472 | |
---|
1473 | Finally, the variable \code{boundary} identifies the boundary |
---|
1474 | triangles and associates a tag with each. |
---|
1475 | |
---|
1476 | % \refmodindex[pmesh.meshinterface]{pmesh.mesh\_interface} |
---|
1477 | \label{sec:meshgeneration} |
---|
1478 | |
---|
1479 | \begin{funcdesc}{create_mesh_from_regions}{bounding_polygon, |
---|
1480 | boundary_tags, |
---|
1481 | maximum_triangle_area=None, |
---|
1482 | filename=None, |
---|
1483 | interior_regions=None, |
---|
1484 | interior_holes=None, |
---|
1485 | poly_geo_reference=None, |
---|
1486 | mesh_geo_reference=None, |
---|
1487 | minimum_triangle_angle=28.0, |
---|
1488 | fail_if_polygons_outside=True, |
---|
1489 | use_cache=False, |
---|
1490 | verbose=True} |
---|
1491 | Module: \module{pmesh.mesh\_interface} |
---|
1492 | |
---|
1493 | This function allows a user to initiate the automatic creation of a |
---|
1494 | mesh inside a specified polygon (input \code{bounding_polygon}). |
---|
1495 | Among the parameters that can be set are the \emph{resolution} |
---|
1496 | (maximal area for any triangle in the mesh) and the minimal angle |
---|
1497 | allowable in any triangle. The user can specify a number of internal |
---|
1498 | polygons within each of which the resolution of the mesh can be |
---|
1499 | specified. \code{interior_regions} is a paired list containing the |
---|
1500 | interior polygon and its resolution. Additionally, the user specifies |
---|
1501 | a list of boundary tags, one for each edge of the bounding polygon. |
---|
1502 | |
---|
1503 | \textbf{WARNING}. Note that the dictionary structure used for the |
---|
1504 | parameter \code{boundary\_tags} is different from that used for the |
---|
1505 | variable \code{boundary} that occurs in the specification of a mesh. |
---|
1506 | In the case of \code{boundary}, the tags are the \emph{values} of |
---|
1507 | the dictionary, whereas in the case of \code{boundary_tags}, the |
---|
1508 | tags are the \emph{keys} and the \emph{value} corresponding to a |
---|
1509 | particular tag is a list of numbers identifying boundary edges |
---|
1510 | labelled with that tag. Because of this, it is theoretically |
---|
1511 | possible to assign the same edge to more than one tag. However, an |
---|
1512 | attempt to do this will cause an error. |
---|
1513 | |
---|
1514 | \textbf{WARNING}. Do not have polygon lines cross or be on-top of each |
---|
1515 | other. This can result in regions of unspecified resolutions. Do |
---|
1516 | not have polygon close to each other. This can result in the area |
---|
1517 | between the polygons having small triangles. For more control |
---|
1518 | over the mesh outline use the methods described below. |
---|
1519 | |
---|
1520 | \end{funcdesc} |
---|
1521 | |
---|
1522 | \begin{funcdesc}{create_domain_from_regions}{bounding_polygon, |
---|
1523 | boundary_tags, |
---|
1524 | maximum_triangle_area=None, |
---|
1525 | mesh_filename=None, |
---|
1526 | interior_regions=None, |
---|
1527 | interior_holes=None, |
---|
1528 | poly_geo_reference=None, |
---|
1529 | mesh_geo_reference=None, |
---|
1530 | minimum_triangle_angle=28.0, |
---|
1531 | fail_if_polygons_outside=True, |
---|
1532 | use_cache=False, |
---|
1533 | verbose=True} |
---|
1534 | |
---|
1535 | Module: \module{interface.py} |
---|
1536 | |
---|
1537 | This higher-level function allows a user to create a domain (and associated mesh) |
---|
1538 | inside a specified polygon. |
---|
1539 | |
---|
1540 | \code{bounding_polygon} is a list of points in Eastings and Northings, |
---|
1541 | relative to the zone stated in \code{poly_geo_reference} if specified. |
---|
1542 | Otherwise points are just x, y coordinates with no particular |
---|
1543 | association to any location. |
---|
1544 | |
---|
1545 | \code{boundary_tags} is a dictionary of symbolic tags. For every tag there |
---|
1546 | is a list of indices referring to segments associated with that tag. |
---|
1547 | If a segment is omitted it will be assigned the default tag ''. |
---|
1548 | |
---|
1549 | \code{maximum_triangle_area} is the maximal area per triangle |
---|
1550 | for the bounding polygon, excluding the interior regions. |
---|
1551 | |
---|
1552 | \code{mesh_filename} is the name of the file to contain the generated |
---|
1553 | mesh data. |
---|
1554 | |
---|
1555 | \code{interior_regions} is a list of tuples consisting of (polygon, |
---|
1556 | resolution) for each region to be separately refined. Do not have |
---|
1557 | polygon lines cross or be on-top of each other. Also do not have |
---|
1558 | polygons close to each other. |
---|
1559 | |
---|
1560 | \code{poly_geo_reference} is the geo_reference of the bounding polygon and |
---|
1561 | the interior polygons. |
---|
1562 | If none, assume absolute. Please pass one though, since absolute |
---|
1563 | references have a zone. |
---|
1564 | |
---|
1565 | \code{mesh_geo_reference} is the geo_reference of the mesh to be created. |
---|
1566 | If none is given one will be automatically generated. It will use |
---|
1567 | the lower left hand corner of bounding_polygon (absolute) |
---|
1568 | as the x and y values for the geo_ref. |
---|
1569 | |
---|
1570 | \code{minimum_triangle_angle} is the minimum angle allowed for each generated triangle. |
---|
1571 | This controls the \emph{slimness} allowed for a triangle. |
---|
1572 | |
---|
1573 | \code{fail_if_polygons_outside} -- if True (the default) an Exception in thrown |
---|
1574 | if interior polygons fall outside the bounding polygon. If False, these |
---|
1575 | will be ignored and execution continues. |
---|
1576 | |
---|
1577 | \textbf{WARNING}. Note that the dictionary structure used for the |
---|
1578 | parameter \code{boundary_tags} is different from that used for the |
---|
1579 | variable \code{boundary} that occurs in the specification of a mesh. |
---|
1580 | In the case of \code{boundary}, the tags are the \emph{values} of |
---|
1581 | the dictionary, whereas in the case of \code{boundary_tags}, the |
---|
1582 | tags are the \emph{keys} and the \emph{value} corresponding to a |
---|
1583 | particular tag is a list of numbers identifying boundary edges |
---|
1584 | labelled with that tag. Because of this, it is theoretically |
---|
1585 | possible to assign the same edge to more than one tag. However, an |
---|
1586 | attempt to do this will cause an error. |
---|
1587 | |
---|
1588 | \textbf{WARNING}. Do not have polygon lines cross or be on-top of each |
---|
1589 | other. This can result in regions of unspecified resolutions. Do |
---|
1590 | not have polygon close to each other. This can result in the area |
---|
1591 | between the polygons having small triangles. For more control |
---|
1592 | over the mesh outline use the methods described below. |
---|
1593 | |
---|
1594 | \end{funcdesc} |
---|
1595 | |
---|
1596 | \subsection{Advanced mesh generation} |
---|
1597 | |
---|
1598 | For more control over the creation of the mesh outline, use the |
---|
1599 | methods of the class \class{Mesh}. |
---|
1600 | |
---|
1601 | \begin{classdesc}{Mesh}{userSegments=None, |
---|
1602 | userVertices=None, |
---|
1603 | holes=None, |
---|
1604 | regions=None, |
---|
1605 | geo_reference=None} |
---|
1606 | Module: \module{pmesh.mesh} |
---|
1607 | |
---|
1608 | A class used to build a mesh outline and generate a two-dimensional |
---|
1609 | triangular mesh. The mesh outline is used to describe features on the |
---|
1610 | mesh, such as the mesh boundary. Many of this class's methods are used |
---|
1611 | to build a mesh outline, such as \code{add_vertices()} and |
---|
1612 | \code{add_region_from_polygon()}. |
---|
1613 | |
---|
1614 | \code{userSegments} and \code{userVertices} define the outline enclosing the mesh. |
---|
1615 | |
---|
1616 | \code{holes} describes any regions inside the mesh that are not to be included in the mesh. |
---|
1617 | |
---|
1618 | \code{geo_reference} defines the geo_reference to which all point information is relative. |
---|
1619 | If \code{geo_reference} is \code{None} then the default geo_reference is used. |
---|
1620 | \end{classdesc} |
---|
1621 | |
---|
1622 | \subsubsection{Key Methods of Class Mesh} |
---|
1623 | |
---|
1624 | \begin{methoddesc}{\emph{<mesh>}.add_hole}{x, y, geo_reference=None} |
---|
1625 | Module: \module{pmesh.mesh} |
---|
1626 | |
---|
1627 | This method adds a hole to the mesh outline. |
---|
1628 | |
---|
1629 | \code{x} and \code{y} define a point on the already defined hole boundary. |
---|
1630 | |
---|
1631 | If \code{geo_reference} is not supplied the points are assumed to be absolute. |
---|
1632 | \end{methoddesc} |
---|
1633 | |
---|
1634 | \begin{methoddesc}{\emph{<mesh>}.add_hole_from_polygon}{polygon, |
---|
1635 | segment_tags=None, |
---|
1636 | geo_reference=None} |
---|
1637 | Module: \module{pmesh.mesh} |
---|
1638 | |
---|
1639 | This method is used to add a 'hole' within a region -- that is, to |
---|
1640 | define a interior region where the triangular mesh will not be |
---|
1641 | generated -- to a \class{Mesh} instance. The region boundary is described by |
---|
1642 | the polygon passed in. Additionally, the user specifies a list of |
---|
1643 | boundary tags, one for each edge of the bounding polygon. |
---|
1644 | |
---|
1645 | \code{polygon} is the polygon that defines the hole to be added to the \code{<mesh>}. |
---|
1646 | |
---|
1647 | \code{segment_tags} -- ?? |
---|
1648 | |
---|
1649 | If \code{geo_reference} is \code{None} then the default \code{geo_reference} is used. |
---|
1650 | \end{methoddesc} |
---|
1651 | |
---|
1652 | \begin{methoddesc}{\emph{<mesh>}.add_points_and_segments}{points, |
---|
1653 | segments=None, |
---|
1654 | segment_tags=None} |
---|
1655 | Module: \module{pmesh.mesh} |
---|
1656 | |
---|
1657 | This adds points and segments connecting the points to a mesh. |
---|
1658 | |
---|
1659 | \code{points} is a list of points. |
---|
1660 | |
---|
1661 | \code{segments} is a list of segments. Each segment is defined by the start and end |
---|
1662 | of the line by its point index, e.g.\ use \code{segments = [[0,1],[1,2]]} to make a |
---|
1663 | polyline between points 0, 1 and 2. |
---|
1664 | |
---|
1665 | \code{segment_tags} may be used to optionally define a tag for each segment. |
---|
1666 | \end{methoddesc} |
---|
1667 | |
---|
1668 | \begin{methoddesc}{\emph{<mesh>}.add_region}{x,y, geo_reference=None, tag=None} |
---|
1669 | Module: \module{pmesh.mesh} |
---|
1670 | |
---|
1671 | This method adds a region to a mesh outline. |
---|
1672 | |
---|
1673 | \code{x} and \code{y} define a point on the already-defined region that is to |
---|
1674 | be added to the mesh. |
---|
1675 | |
---|
1676 | If \code{geo_reference} is not supplied the points data is assumed to be absolute. |
---|
1677 | |
---|
1678 | \code{tag} -- ?? |
---|
1679 | |
---|
1680 | A region instance is returned. This can be used to set the resolution of the added region. |
---|
1681 | \end{methoddesc} |
---|
1682 | |
---|
1683 | \begin{methoddesc}{\emph{<mesh>}.add_region_from_polygon}{polygon, |
---|
1684 | segment_tags=None, |
---|
1685 | max_triangle_area=None, |
---|
1686 | geo_reference=None, |
---|
1687 | region_tag=None} |
---|
1688 | Module: \module{pmesh.mesh} |
---|
1689 | |
---|
1690 | This method adds a region to a |
---|
1691 | \class{Mesh} instance. Regions are commonly used to describe an area |
---|
1692 | with an increased density of triangles by setting \code{max_triangle_area}. |
---|
1693 | |
---|
1694 | \code{polygon} describes the region boundary to add to the \code{<mesh>}. |
---|
1695 | |
---|
1696 | \code{segment_tags} specifies a list of segment tags, one for each edge of the |
---|
1697 | bounding polygon. |
---|
1698 | |
---|
1699 | If \code{geo_reference} is not supplied the points data is assumed to be absolute. |
---|
1700 | |
---|
1701 | \code{region_tag} sets the region tag. |
---|
1702 | \end{methoddesc} |
---|
1703 | |
---|
1704 | \begin{methoddesc}{\emph{<mesh>}.add_vertices}{point_data} |
---|
1705 | Module: \module{pmesh.mesh} |
---|
1706 | |
---|
1707 | Add user vertices to a mesh. |
---|
1708 | |
---|
1709 | \code{point_data} is the list of point data, and can be a list of (x,y) values, |
---|
1710 | a numeric array or a geospatial_data instance. |
---|
1711 | \end{methoddesc} |
---|
1712 | |
---|
1713 | \begin{methoddesc}{\emph{<mesh>}.auto_segment}{alpha=None, |
---|
1714 | raw_boundary=True, |
---|
1715 | remove_holes=False, |
---|
1716 | smooth_indents=False, |
---|
1717 | expand_pinch=False} |
---|
1718 | Module: \module{pmesh.mesh} |
---|
1719 | |
---|
1720 | Add segments between some of the user vertices to give the vertices an |
---|
1721 | outline. The outline is an alpha shape. This method is |
---|
1722 | useful since a set of user vertices need to be outlined by segments |
---|
1723 | before generate_mesh is called. |
---|
1724 | |
---|
1725 | \code{alpha} determines the $smoothness$ of the alpha shape. |
---|
1726 | |
---|
1727 | \code{raw_boundary}, if \code{True} instructs the function to return the raw |
---|
1728 | boundary, i.e.\ the regular edges of the alpha shape. |
---|
1729 | |
---|
1730 | \code{remove_holes}, if \code{True} enables a filter to remove small holes |
---|
1731 | (small is defined by boundary_points_fraction). |
---|
1732 | |
---|
1733 | \code{smooth_indents}, if \code{True} removes sharp triangular indents |
---|
1734 | in the boundary. |
---|
1735 | |
---|
1736 | \code{expand_pinch}, if \code{True} tests for pinch-off and corrects |
---|
1737 | (i.e.\ a boundary vertex with more than two edges). |
---|
1738 | \end{methoddesc} |
---|
1739 | |
---|
1740 | \begin{methoddesc}{\emph{<mesh>}.export_mesh_file}{ofile} |
---|
1741 | Module: \module{pmesh.mesh} |
---|
1742 | |
---|
1743 | This method is used to save a mesh to a file. |
---|
1744 | |
---|
1745 | \code{ofile} is the name of the mesh file to be written, including the extension. |
---|
1746 | Use the extension \code{.msh} for the file to be in NetCDF format and |
---|
1747 | \code{.tsh} for the file to be ASCII format. |
---|
1748 | \end{methoddesc} |
---|
1749 | |
---|
1750 | \begin{methoddesc}{\emph{<mesh>}.generate_mesh}{maximum_triangle_area="", |
---|
1751 | minimum_triangle_angle=28.0, |
---|
1752 | verbose=True} |
---|
1753 | Module: \module{pmesh.mesh} |
---|
1754 | |
---|
1755 | This method is used to generate the triangular mesh. |
---|
1756 | |
---|
1757 | \code{maximum_triangle_area} sets the maximum area of any triangle in the mesh. |
---|
1758 | |
---|
1759 | \code{minimum_triangle_angle} sets the minimum area of any triangle in the mesh. |
---|
1760 | |
---|
1761 | These two parameters can be used to control the triangle density. |
---|
1762 | \end{methoddesc} |
---|
1763 | |
---|
1764 | \begin{methoddesc}{\emph{<mesh>}.import_ungenerate_file}{ofile, |
---|
1765 | tag=None, |
---|
1766 | region_tag=None} |
---|
1767 | Module: \module{pmesh.mesh} |
---|
1768 | |
---|
1769 | This method is used to import a polygon file in the ungenerate format, |
---|
1770 | which is used by arcGIS. The polygons from the file are converted to |
---|
1771 | vertices and segments. |
---|
1772 | |
---|
1773 | \code{ofile} is the name of the polygon file. |
---|
1774 | |
---|
1775 | \code{tag} is the tag given to all the polygon's segments. |
---|
1776 | If \code{tag} is not supplied then the segment will not effect the water |
---|
1777 | flow, it will only effect the mesh generation. |
---|
1778 | |
---|
1779 | \code{region_tag} is the tag given to all the polygon's segments. If |
---|
1780 | it is a string the tag will be assigned to all regions. If it |
---|
1781 | is a list the first value in the list will be applied to the first |
---|
1782 | polygon etc. |
---|
1783 | |
---|
1784 | This function can be used to import building footprints. |
---|
1785 | \end{methoddesc} |
---|
1786 | |
---|
1787 | |
---|
1788 | \section{Initialising the Domain}\index{Initialising the Domain} |
---|
1789 | \label{sec:initialising the domain} |
---|
1790 | |
---|
1791 | \begin{classdesc}{Domain}{source=None, |
---|
1792 | triangles=None, |
---|
1793 | boundary=None, |
---|
1794 | conserved_quantities=None, |
---|
1795 | other_quantities=None, |
---|
1796 | tagged_elements=None, |
---|
1797 | geo_reference=None, |
---|
1798 | use_inscribed_circle=False, |
---|
1799 | mesh_filename=None, |
---|
1800 | use_cache=False, |
---|
1801 | verbose=False, |
---|
1802 | full_send_dict=None, |
---|
1803 | ghost_recv_dict=None, |
---|
1804 | processor=0, |
---|
1805 | numproc=1, |
---|
1806 | number_of_full_nodes=None, |
---|
1807 | number_of_full_triangles=None} |
---|
1808 | Module: \refmodule{abstract_2d_finite_volumes.domain} |
---|
1809 | |
---|
1810 | This class is used to create an instance of a structure used to |
---|
1811 | store and manipulate data associated with a mesh. The mesh is |
---|
1812 | specified either by assigning the name of a mesh file to |
---|
1813 | \code{source} or by specifying the points, triangle and boundary of the |
---|
1814 | mesh. |
---|
1815 | \end{classdesc} |
---|
1816 | |
---|
1817 | \subsection{Key Methods of Domain} |
---|
1818 | |
---|
1819 | \begin{methoddesc}{\emph{<domain>}.set_name}{name} |
---|
1820 | Module: \refmodule{abstract_2d_finite_volumes.domain}, |
---|
1821 | page \pageref{mod:domain} |
---|
1822 | |
---|
1823 | \code{name} is used to name the domain. The \code{name} is also used to identify the output SWW file. |
---|
1824 | If no name is assigned to a domain, the assumed name is \code{'domain'}. |
---|
1825 | \end{methoddesc} |
---|
1826 | |
---|
1827 | \begin{methoddesc}{\emph{<domain>}.get_name}{} |
---|
1828 | Module: \module{abstract_2d_finite_volumes.domain} |
---|
1829 | |
---|
1830 | Returns the name assigned to the domain by \code{set_name()}. If no name has been |
---|
1831 | assigned, returns \code{'domain'}. |
---|
1832 | \end{methoddesc} |
---|
1833 | |
---|
1834 | \begin{methoddesc}{\emph{<domain>}.set_datadir}{path} |
---|
1835 | Module: \module{abstract_2d_finite_volumes.domain} |
---|
1836 | |
---|
1837 | \code{path} specifies the path to the directory used to store SWW files. |
---|
1838 | |
---|
1839 | Before this method is used to set the SWW directory path, the assumed directory |
---|
1840 | path is \code{default_datadir} specified in \code{config.py}. |
---|
1841 | |
---|
1842 | Since different operating systems use different formats for specifying pathnames |
---|
1843 | it is necessary to specify path separators using the Python code \code{os.sep} rather than |
---|
1844 | the operating-specific ones such as '$\slash$' or '$\backslash$'. |
---|
1845 | For this to work you will need to include the statement \code{import os} |
---|
1846 | in your code, before the first use of \code{set_datadir()}. |
---|
1847 | |
---|
1848 | For example, to set the data directory to a subdirectory |
---|
1849 | \code{data} of the directory \code{project}, you could use |
---|
1850 | the statements: |
---|
1851 | |
---|
1852 | \begin{verbatim} |
---|
1853 | import os |
---|
1854 | domain.set_datadir{'project' + os.sep + 'data'} |
---|
1855 | \end{verbatim} |
---|
1856 | \end{methoddesc} |
---|
1857 | |
---|
1858 | \begin{methoddesc}{\emph{<domain>}.get_datadir}{} |
---|
1859 | Module: \module{abstract_2d_finite_volumes.domain} |
---|
1860 | |
---|
1861 | Returns the path to the directory where SWW files will be stored. |
---|
1862 | |
---|
1863 | If the path has not previously been set with \code{set_datadir()} this method |
---|
1864 | will return the value \code{default_datadir} specified in \code{config.py}. |
---|
1865 | \end{methoddesc} |
---|
1866 | |
---|
1867 | \begin{methoddesc}{\emph{<domain>}.set_minimum_allowed_height}{minimum_allowed_height} |
---|
1868 | Module: \module{shallow_water.shallow_water_domain} |
---|
1869 | |
---|
1870 | Set the minimum depth (in metres) that will be recognised in |
---|
1871 | the numerical scheme (including limiters and flux computations) |
---|
1872 | |
---|
1873 | \code{minimum_allowed_height} is the new minimum allowed height value. |
---|
1874 | |
---|
1875 | Default value is $10^{-3}$ metre, but by setting this to a greater value, |
---|
1876 | e.g.\ for large scale simulations, the computation time can be |
---|
1877 | significantly reduced. |
---|
1878 | \end{methoddesc} |
---|
1879 | |
---|
1880 | \begin{methoddesc}{\emph{<domain>}.set_minimum_storable_height}{minimum_storable_height} |
---|
1881 | Module: \module{shallow_water.shallow_water_domain} |
---|
1882 | |
---|
1883 | Sets the minimum depth that will be recognised when writing |
---|
1884 | to an SWW file. This is useful for removing thin water layers |
---|
1885 | that seems to be caused by friction creep. |
---|
1886 | |
---|
1887 | \code{minimum_storable_height} is the new minimum storable height value. |
---|
1888 | \end{methoddesc} |
---|
1889 | |
---|
1890 | \begin{methoddesc}{\emph{<domain>}.set_maximum_allowed_speed}{maximum_allowed_speed} |
---|
1891 | Module: \module{shallow_water.shallow_water_domain} |
---|
1892 | |
---|
1893 | Set the maximum particle speed that is allowed in water |
---|
1894 | shallower than \code{minimum_allowed_height}. This is useful for |
---|
1895 | controlling speeds in very thin layers of water and at the same time |
---|
1896 | allow some movement avoiding pooling of water. |
---|
1897 | |
---|
1898 | \code{maximum_allowed_speed} sets the maximum allowed speed value. |
---|
1899 | \end{methoddesc} |
---|
1900 | |
---|
1901 | \begin{methoddesc}{\emph{<domain>}.set_time}{time=0.0} |
---|
1902 | Module: \module{abstract_2d_finite_volumes.domain} |
---|
1903 | |
---|
1904 | \code{time} sets the initial time, in seconds, for the simulation. The |
---|
1905 | default is 0.0. |
---|
1906 | \end{methoddesc} |
---|
1907 | |
---|
1908 | \begin{methoddesc}{\emph{<domain>}.set_default_order}{n} |
---|
1909 | Module: \module{abstract_2d_finite_volumes.domain} |
---|
1910 | |
---|
1911 | Sets the default (spatial) order to the value specified by |
---|
1912 | \code{n}, which must be either 1 or 2. (Assigning any other value |
---|
1913 | to \code{n} will cause an error.) |
---|
1914 | \end{methoddesc} |
---|
1915 | |
---|
1916 | \begin{methoddesc}{\emph{<domain>}.set_store_vertices_uniquely}{flag, reduction=None} |
---|
1917 | Module: \module{shallow_water.shallow_water_domain} |
---|
1918 | |
---|
1919 | Decide whether vertex values should be stored uniquely as |
---|
1920 | computed in the model or whether they should be reduced to one |
---|
1921 | value per vertex using averaging. |
---|
1922 | |
---|
1923 | \code{flag} may be \code{True} (meaning do not smooth vertex values) or \code{False}. |
---|
1924 | |
---|
1925 | \code{reduction} defines the smoothing operation if \code{flag} is \code{True}. If not |
---|
1926 | supplied, \code{reduction} is assumed to be \code{mean}. |
---|
1927 | |
---|
1928 | Triangles stored in the SWW file can be discontinuous reflecting |
---|
1929 | the internal representation of the finite-volume scheme |
---|
1930 | (this is a feature allowing for arbitrary steepness of the water surface gradient |
---|
1931 | as well as the momentum gradients). |
---|
1932 | However, for visual purposes and also for use with \code{Field_boundary} |
---|
1933 | (and \code{File_boundary}), it is often desirable to store triangles |
---|
1934 | with values at each vertex point as the average of the potentially |
---|
1935 | discontinuous numbers found at vertices of different triangles sharing the |
---|
1936 | same vertex location. |
---|
1937 | |
---|
1938 | Storing one way or the other is controlled in \anuga through the method |
---|
1939 | \code{<domain>.store_vertices_uniquely()}. Options are: |
---|
1940 | \begin{itemize} |
---|
1941 | \item \code{<domain>.store_vertices_uniquely(True)}: Allow discontinuities in the SWW file |
---|
1942 | \item \code{<domain>.store_vertices_uniquely(False)}: (Default). |
---|
1943 | Average values |
---|
1944 | to ensure continuity in SWW file. The latter also makes for smaller |
---|
1945 | SWW files. |
---|
1946 | \end{itemize} |
---|
1947 | |
---|
1948 | Note that when model data in the SWW file are averaged (i.e.\ not stored uniquely), |
---|
1949 | then there will most likely be a small discrepancy between values extracted from the SWW |
---|
1950 | file and the same data stored in the model domain. This must be borne in mind when comparing |
---|
1951 | data from the SWW files with that of the model internally. |
---|
1952 | \end{methoddesc} |
---|
1953 | |
---|
1954 | % Structural methods |
---|
1955 | \begin{methoddesc}{\emph{<domain>}.get_nodes}{absolute=False} |
---|
1956 | Module: \module{abstract_2d_finite_volumes.domain} |
---|
1957 | |
---|
1958 | Return x,y coordinates of all nodes in the domain mesh. The nodes are ordered |
---|
1959 | in an \code{Nx2} array where N is the number of nodes. This is the same format |
---|
1960 | they were provided in the constructor i.e.\ without any duplication. |
---|
1961 | |
---|
1962 | \code{absolute} is a boolean which determines whether coordinates |
---|
1963 | are to be made absolute by taking georeference into account. |
---|
1964 | Default is \code{False} as many parts of \anuga expect relative coordinates. |
---|
1965 | \end{methoddesc} |
---|
1966 | |
---|
1967 | \begin{methoddesc}{\emph{<domain>}.get_vertex_coordinates}{absolute=False} |
---|
1968 | Module: \module{abstract_2d_finite_volumes.domain} |
---|
1969 | |
---|
1970 | \label{pg:get vertex coordinates} |
---|
1971 | Return vertex coordinates for all triangles as a \code{3*Mx2} array |
---|
1972 | where the jth vertex of the ith triangle is located in row 3*i+j and |
---|
1973 | M is the number of triangles in the mesh. |
---|
1974 | |
---|
1975 | \code{absolute} is a boolean which determines whether coordinates |
---|
1976 | are to be made absolute by taking georeference into account. |
---|
1977 | Default is \code{False} as many parts of \anuga expect relative coordinates. |
---|
1978 | \end{methoddesc} |
---|
1979 | |
---|
1980 | \begin{methoddesc}{\emph{<domain>}.get_centroid_coordinates}{absolute=False} |
---|
1981 | Module: \module{abstract_2d_finite_volumes.domain} |
---|
1982 | |
---|
1983 | Return centroid coordinates for all triangles as an \code{Mx2} array. |
---|
1984 | |
---|
1985 | \code{absolute} is a boolean which determines whether coordinates |
---|
1986 | are to be made absolute by taking georeference into account. |
---|
1987 | Default is \code{False} as many parts of \anuga expect relative coordinates. |
---|
1988 | \end{methoddesc} |
---|
1989 | |
---|
1990 | \begin{methoddesc}{\emph{<domain>}.get_triangles}{indices=None} |
---|
1991 | Module: \module{abstract_2d_finite_volumes.domain} |
---|
1992 | |
---|
1993 | Return an \code{Mx3} integer array where M is the number of triangles. |
---|
1994 | Each row corresponds to one triangle and the three entries are |
---|
1995 | indices into the mesh nodes which can be obtained using the method |
---|
1996 | \code{get_nodes()}. |
---|
1997 | |
---|
1998 | \code{indices}, if specified, is the set of triangle \code{id}s of interest. |
---|
1999 | \end{methoddesc} |
---|
2000 | |
---|
2001 | \begin{methoddesc}{\emph{<domain>}.get_disconnected_triangles}{} |
---|
2002 | Module: \module{abstract_2d_finite_volumes.domain} |
---|
2003 | |
---|
2004 | Get the domain mesh based on nodes obtained from \code{get_vertex_coordinates()}. |
---|
2005 | |
---|
2006 | Returns an \code{Mx3} array of integers where each row corresponds to |
---|
2007 | a triangle. A triangle is a triplet of indices into |
---|
2008 | point coordinates obtained from \code{get_vertex_coordinates()} and each |
---|
2009 | index appears only once. |
---|
2010 | |
---|
2011 | This provides a mesh where no triangles share nodes |
---|
2012 | (hence the name disconnected triangles) and different |
---|
2013 | nodes may have the same coordinates. |
---|
2014 | |
---|
2015 | This version of the mesh is useful for storing meshes with |
---|
2016 | discontinuities at each node and is e.g.\ used for storing |
---|
2017 | data in SWW files. |
---|
2018 | |
---|
2019 | The triangles created will have the format: |
---|
2020 | |
---|
2021 | \begin{verbatim} |
---|
2022 | [[0,1,2], |
---|
2023 | [3,4,5], |
---|
2024 | [6,7,8], |
---|
2025 | ... |
---|
2026 | [3*M-3 3*M-2 3*M-1]] |
---|
2027 | \end{verbatim} |
---|
2028 | \end{methoddesc} |
---|
2029 | |
---|
2030 | |
---|
2031 | \section{Initial Conditions}\index{Initial Conditions} |
---|
2032 | \label{sec:initial conditions} |
---|
2033 | In standard usage of partial differential equations, initial conditions |
---|
2034 | refers to the values associated to the system variables (the conserved |
---|
2035 | quantities here) for \code{time = 0}. In setting up a scenario script |
---|
2036 | as described in Sections \ref{sec:simpleexample} and \ref{sec:realdataexample}, |
---|
2037 | \code{set_quantity} is used to define the initial conditions of variables |
---|
2038 | other than the conserved quantities, such as friction. Here, we use the terminology |
---|
2039 | of initial conditions to refer to initial values for variables which need |
---|
2040 | prescription to solve the shallow water wave equation. Further, it must be noted |
---|
2041 | that \code{set_quantity()} does not necessarily have to be used in the initial |
---|
2042 | condition setting; it can be used at any time throughout the simulation. |
---|
2043 | |
---|
2044 | \begin{methoddesc}{\emph{<domain>}.set_quantity}{numeric=None, |
---|
2045 | quantity=None, |
---|
2046 | function=None, |
---|
2047 | geospatial_data=None, |
---|
2048 | filename=None, |
---|
2049 | attribute_name=None, |
---|
2050 | alpha=None, |
---|
2051 | location='vertices', |
---|
2052 | polygon=None, |
---|
2053 | indices=None, |
---|
2054 | smooth=False, |
---|
2055 | verbose=False, |
---|
2056 | use_cache=False} |
---|
2057 | Module: \module{abstract_2d_finite_volumes.domain} \\ |
---|
2058 | (This method passes off to \module{abstract_2d_finite_volumes.quantity.set_values()}) |
---|
2059 | |
---|
2060 | This function is used to assign values to individual quantities for a |
---|
2061 | domain. It is very flexible and can be used with many data types: a |
---|
2062 | statement of the form \code{\emph{<domain>}.set_quantity(name, x)} can be used |
---|
2063 | to define a quantity having the name \code{name}, where the other |
---|
2064 | argument \code{x} can be any of the following: |
---|
2065 | |
---|
2066 | \begin{itemize} |
---|
2067 | \item a number, in which case all vertices in the mesh gets that for |
---|
2068 | the quantity in question |
---|
2069 | \item a list of numbers or a numeric array ordered the same way as the mesh vertices |
---|
2070 | \item a function (e.g.\ see the samples introduced in Chapter 2) |
---|
2071 | \item an expression composed of other quantities and numbers, arrays, lists (for |
---|
2072 | example, a linear combination of quantities, such as |
---|
2073 | \code{\emph{<domain>}.set_quantity('stage','elevation'+x))} |
---|
2074 | \item the name of a file from which the data can be read. In this case, the optional |
---|
2075 | argument \code{attribute_name} will select which attribute to use from the file. If left out, |
---|
2076 | \code{set_quantity()} will pick one. This is useful in cases where there is only one attribute |
---|
2077 | \item a geospatial dataset (See Section \ref{sec:geospatial}). |
---|
2078 | Optional argument \code{attribute_name} applies here as with files |
---|
2079 | \end{itemize} |
---|
2080 | |
---|
2081 | Exactly one of the arguments \code{numeric}, \code{quantity}, \code{function}, |
---|
2082 | \code{geospatial_data} and \code{filename} must be present. |
---|
2083 | |
---|
2084 | \code{set_quantity()} will look at the type of the \code{numeric} and |
---|
2085 | determine what action to take. |
---|
2086 | |
---|
2087 | Values can also be set using the appropriate keyword arguments. |
---|
2088 | If \code{x} is a function, for example, \code{domain.set_quantity(name, x)}, \code{domain.set_quantity(name, numeric=x)}, |
---|
2089 | and \code{domain.set_quantity(name, function=x)} are all equivalent. |
---|
2090 | |
---|
2091 | Other optional arguments are: |
---|
2092 | \begin{itemize} |
---|
2093 | \item \code{indices} which is a list of ids of triangles to which \code{set_quantity()} |
---|
2094 | should apply its assignment of values. |
---|
2095 | \item \code{location} determines which part of the triangles to assign to. |
---|
2096 | Options are 'vertices' (the default), 'edges', 'unique vertices', and 'centroids'. |
---|
2097 | If 'vertices' is used, edge and centroid values are automatically computed as the |
---|
2098 | appropriate averages. This option ensures continuity of the surface. |
---|
2099 | If, on the other hand, 'centroids' is used, vertex and edge values will be set to the |
---|
2100 | same value effectively creating a piecewise constant surface with possible |
---|
2101 | discontinuities at the edges. |
---|
2102 | \end{itemize} |
---|
2103 | |
---|
2104 | \anuga provides a number of predefined initial conditions to be used |
---|
2105 | with \code{set_quantity()}. See for example callable object \code{slump_tsunami} below. |
---|
2106 | \end{methoddesc} |
---|
2107 | |
---|
2108 | \begin{methoddesc}{\emph{<domain>}.add_quantity}{numeric=None, |
---|
2109 | quantity=None, |
---|
2110 | function=None, |
---|
2111 | geospatial_data=None, |
---|
2112 | filename=None, |
---|
2113 | attribute_name=None, |
---|
2114 | alpha=None, |
---|
2115 | location='vertices', |
---|
2116 | polygon=None, |
---|
2117 | indices=None, |
---|
2118 | smooth=False, |
---|
2119 | verbose=False, |
---|
2120 | use_cache=False} |
---|
2121 | Module: \module{abstract_2d_finite_volumes.domain} \\ |
---|
2122 | (passes off to \module{abstract_2d_finite_volumes.domain.set_quantity()}) |
---|
2123 | |
---|
2124 | \label{add quantity} |
---|
2125 | This function is used to \emph{add} values to individual quantities for a |
---|
2126 | domain. It has the same syntax as \code{\emph{<domain>}.set_quantity(name, x)}. |
---|
2127 | |
---|
2128 | A typical use of this function is to add structures to an existing elevation model: |
---|
2129 | |
---|
2130 | \begin{verbatim} |
---|
2131 | # Create digital elevation model from points file |
---|
2132 | domain.set_quantity('elevation', filename='elevation_file.pts, verbose=True) |
---|
2133 | |
---|
2134 | # Add buildings from file |
---|
2135 | building_polygons, building_heights = csv2building_polygons(building_file) |
---|
2136 | |
---|
2137 | B = [] |
---|
2138 | for key in building_polygons: |
---|
2139 | poly = building_polygons[key] |
---|
2140 | elev = building_heights[key] |
---|
2141 | B.append((poly, elev)) |
---|
2142 | |
---|
2143 | domain.add_quantity('elevation', Polygon_function(B, default=0.0)) |
---|
2144 | \end{verbatim} |
---|
2145 | \end{methoddesc} |
---|
2146 | |
---|
2147 | \begin{methoddesc}{\emph{<domain>}.set_region}{tag, quantity, X, location='vertices'} |
---|
2148 | Module: \module{abstract_2d_finite_volumes.domain} \\ |
---|
2149 | (see also \module{abstract_2d_finite_volumes.quantity.set_values}) |
---|
2150 | |
---|
2151 | This function is used to assign values to individual quantities given |
---|
2152 | a regional tag. It is similar to \code{set_quantity()}. |
---|
2153 | |
---|
2154 | For example, if in the mesh-generator a regional tag of 'ditch' was |
---|
2155 | used, \code{set_region()} can be used to set elevation of this region to |
---|
2156 | -10m. \code{X} is the constant or function to be applied to the \code{quantity}, |
---|
2157 | over the tagged region. \code{location} describes how the values will be |
---|
2158 | applied. Options are 'vertices' (the default), 'edges', 'unique |
---|
2159 | vertices', and 'centroids'. |
---|
2160 | |
---|
2161 | This method can also be called with a list of region objects. This is |
---|
2162 | useful for adding quantities in regions, and having one quantity |
---|
2163 | value based on another quantity. See \module{abstract_2d_finite_volumes.region} for |
---|
2164 | more details. |
---|
2165 | \end{methoddesc} |
---|
2166 | |
---|
2167 | \begin{funcdesc}{slump_tsunami}{length, depth, slope, width=None, thickness=None, |
---|
2168 | radius=None, dphi=0.48, x0=0.0, y0=0.0, alpha=0.0, |
---|
2169 | gravity=9.8, gamma=1.85, |
---|
2170 | massco=1, dragco=1, frictionco=0, |
---|
2171 | dx=None, kappa=3.0, kappad=1.0, zsmall=0.01, scale=None, |
---|
2172 | domain=None, |
---|
2173 | verbose=False} |
---|
2174 | Module: \module{shallow\_water.smf} |
---|
2175 | |
---|
2176 | This function returns a callable object representing an initial water |
---|
2177 | displacement generated by a submarine sediment failure. These failures can take the form of |
---|
2178 | a submarine slump or slide. In the case of a slide, use \code{slide_tsunami} instead. |
---|
2179 | |
---|
2180 | \code{length} is the length of the slide or slump. |
---|
2181 | |
---|
2182 | \code{depth} is the water depth to the centre of the sediment mass. |
---|
2183 | |
---|
2184 | \code{slope} is the bathymetric slope. |
---|
2185 | |
---|
2186 | Other slump or slide parameters can be included if they are known. |
---|
2187 | \end{funcdesc} |
---|
2188 | |
---|
2189 | \begin{funcdesc}{\emph{<callable_object>} = file_function}{filename, |
---|
2190 | domain=None, |
---|
2191 | quantities=None, |
---|
2192 | interpolation_points=None, |
---|
2193 | time_thinning=1, |
---|
2194 | time_limit=None, |
---|
2195 | verbose=False, |
---|
2196 | use_cache=False, |
---|
2197 | boundary_polygon=None} |
---|
2198 | Module: \module{abstract_2d_finite_volumes.util} |
---|
2199 | |
---|
2200 | Reads the time history of spatial data for specified interpolation points from |
---|
2201 | a NetCDF file and returns a callable object. Values returned from the \code{\emph{<callable_object>}} |
---|
2202 | are interpolated values based on the input file using the underlying \code{interpolation_function}. |
---|
2203 | |
---|
2204 | \code{filename} is the name of the input file. Could be an SWW or STS file. |
---|
2205 | |
---|
2206 | \code{quantities} is either the name of a single quantity to be |
---|
2207 | interpolated or a list of such quantity names. In the second case, the resulting |
---|
2208 | function will return a tuple of values -- one for each quantity. |
---|
2209 | |
---|
2210 | \code{interpolation_points} is a list of absolute coordinates or a |
---|
2211 | geospatial object for points at which values are sought. |
---|
2212 | |
---|
2213 | \code{boundary_polygon} is a list of coordinates specifying the vertices of the boundary. |
---|
2214 | This must be the same polygon as used when calling \code{create_mesh_from_regions()}. |
---|
2215 | This argument can only be used when reading boundary data from an STS format file. |
---|
2216 | |
---|
2217 | The model time stored within the file function can be accessed using |
---|
2218 | the method \code{\emph{<callable_object>}.get_time()} |
---|
2219 | |
---|
2220 | The underlying algorithm used is as follows:\\ |
---|
2221 | Given a time series (i.e.\ a series of values associated with |
---|
2222 | different times), whose values are either just numbers, a set of |
---|
2223 | numbers defined at the vertices of a triangular mesh (such as those |
---|
2224 | stored in SWW files) or a set of |
---|
2225 | numbers defined at a number of points on the boundary (such as those |
---|
2226 | stored in STS files), \code{Interpolation_function()} is used to |
---|
2227 | create a callable object that interpolates a value for an arbitrary |
---|
2228 | time \code{t} within the model limits and possibly a point \code{(x, y)} |
---|
2229 | within a mesh region. |
---|
2230 | |
---|
2231 | The actual time series at which data is available is specified by |
---|
2232 | means of an array \code{time} of monotonically increasing times. The |
---|
2233 | quantities containing the values to be interpolated are specified in |
---|
2234 | an array -- or dictionary of arrays (used in conjunction with the |
---|
2235 | optional argument \code{quantity_names}) -- called |
---|
2236 | \code{quantities}. The optional arguments \code{vertex_coordinates} |
---|
2237 | and \code{triangles} represent the spatial mesh associated with the |
---|
2238 | quantity arrays. If omitted the function must be created using an STS file |
---|
2239 | or a TMS file. |
---|
2240 | |
---|
2241 | Since, in practice, values need to be computed at specified points, |
---|
2242 | the syntax allows the user to specify, once and for all, a list |
---|
2243 | \code{interpolation_points} of points at which values are required. |
---|
2244 | In this case, the function may be called using the form \code{\emph{<callable_object>}(t, id)}, |
---|
2245 | where \code{id} is an index for the list \code{interpolation_points}. |
---|
2246 | \end{funcdesc} |
---|
2247 | |
---|
2248 | \begin{classdesc}{\emph{<callable_object>} = Interpolation_function}{time, |
---|
2249 | quantities, |
---|
2250 | quantity_names=None, |
---|
2251 | vertex_coordinates=None, |
---|
2252 | triangles=None, |
---|
2253 | interpolation_points=None, |
---|
2254 | time_thinning=1, |
---|
2255 | verbose=False, |
---|
2256 | gauge_neighbour_id=None} |
---|
2257 | Module: \module{fit_interpolate.interpolate} |
---|
2258 | |
---|
2259 | Given a time series (i.e.\ a series of values associated with |
---|
2260 | different times) whose values are either just numbers or a set of |
---|
2261 | numbers defined at the vertices of a triangular mesh (such as those |
---|
2262 | stored in SWW files), \code{Interpolation_function} is used to |
---|
2263 | create a callable object that interpolates a value for an arbitrary |
---|
2264 | time \code{t} within the model limits and possibly a point \code{(x, y)} |
---|
2265 | within a mesh region. |
---|
2266 | |
---|
2267 | The actual time series at which data is available is specified by |
---|
2268 | means of an array \code{time} of monotonically increasing times. The |
---|
2269 | quantities containing the values to be interpolated are specified in |
---|
2270 | an array -- or dictionary of arrays (used in conjunction with the |
---|
2271 | optional argument \code{quantity\_names}) -- called |
---|
2272 | \code{quantities}. The optional arguments \code{vertex_coordinates} |
---|
2273 | and \code{triangles} represent the spatial mesh associated with the |
---|
2274 | quantity arrays. If omitted the function created by |
---|
2275 | \code{Interpolation_function} will be a function of \code{t} only. |
---|
2276 | |
---|
2277 | Since, in practice, values need to be computed at specified points, |
---|
2278 | the syntax allows the user to specify, once and for all, a list |
---|
2279 | \code{interpolation_points} of points at which values are required. |
---|
2280 | In this case, the function may be called using the form \code{f(t, id)}, |
---|
2281 | where \code{id} is an index for the list \code{interpolation_points}. |
---|
2282 | \end{classdesc} |
---|
2283 | |
---|
2284 | |
---|
2285 | \section{Boundary Conditions}\index{boundary conditions} |
---|
2286 | \label{sec:boundary conditions} |
---|
2287 | |
---|
2288 | \anuga provides a large number of predefined boundary conditions, |
---|
2289 | represented by objects such as \code{Reflective_boundary(domain)} and |
---|
2290 | \code{Dirichlet_boundary([0.2, 0.0, 0.0])}, described in the examples |
---|
2291 | in Chapter 2. Alternatively, you may prefer to ''roll your own'', |
---|
2292 | following the method explained in Section \ref{sec:roll your own}. |
---|
2293 | |
---|
2294 | These boundary objects may be used with the function \code{set_boundary} described below |
---|
2295 | to assign boundary conditions according to the tags used to label boundary segments. |
---|
2296 | |
---|
2297 | \begin{methoddesc}{\emph{<domain>}.set_boundary}{boundary_map} |
---|
2298 | Module: \module{abstract_2d_finite_volumes.domain} |
---|
2299 | |
---|
2300 | This function allows you to assign a boundary object (corresponding to a |
---|
2301 | pre-defined or user-specified boundary condition) to every boundary segment that |
---|
2302 | has been assigned a particular tag. |
---|
2303 | |
---|
2304 | \code{boundary_map} is a dictionary of boundary objects keyed by symbolic tags. |
---|
2305 | \end{methoddesc} |
---|
2306 | |
---|
2307 | \begin{methoddesc} {\emph{<domain>}.get_boundary_tags}{} |
---|
2308 | Module: \module{abstract\_2d\_finite\_volumes.domain} |
---|
2309 | |
---|
2310 | Returns a list of the available boundary tags. |
---|
2311 | \end{methoddesc} |
---|
2312 | |
---|
2313 | \subsection{Predefined boundary conditions} |
---|
2314 | |
---|
2315 | \begin{classdesc}{Reflective_boundary}{domain=None} |
---|
2316 | Module: \module{shallow_water} |
---|
2317 | |
---|
2318 | Reflective boundary returns same conserved quantities as those present in |
---|
2319 | the neighbouring volume but reflected. |
---|
2320 | |
---|
2321 | This class is specific to the shallow water equation as it works with the |
---|
2322 | momentum quantities assumed to be the second and third conserved quantities. |
---|
2323 | \end{classdesc} |
---|
2324 | |
---|
2325 | \begin{classdesc}{Transmissive_boundary}{domain=None} |
---|
2326 | \label{pg: transmissive boundary} |
---|
2327 | Module: \module{abstract_2d_finite_volumes.generic_boundary_conditions} |
---|
2328 | |
---|
2329 | A transmissive boundary returns the same conserved quantities as |
---|
2330 | those present in the neighbouring volume. |
---|
2331 | |
---|
2332 | The underlying domain must be specified when the boundary is instantiated. |
---|
2333 | \end{classdesc} |
---|
2334 | |
---|
2335 | \begin{classdesc}{Dirichlet_boundary}{conserved_quantities=None} |
---|
2336 | Module: \module{abstract_2d_finite_volumes.generic_boundary_conditions} |
---|
2337 | |
---|
2338 | A Dirichlet boundary returns constant values for each of conserved |
---|
2339 | quantities. In the example of \code{Dirichlet_boundary([0.2, 0.0, 0.0])}, |
---|
2340 | the \code{stage} value at the boundary is 0.2 and the \code{xmomentum} and |
---|
2341 | \code{ymomentum} at the boundary are set to 0.0. The list must contain |
---|
2342 | a value for each conserved quantity. |
---|
2343 | \end{classdesc} |
---|
2344 | |
---|
2345 | \begin{classdesc}{Time_boundary}{domain=None, |
---|
2346 | function=None, |
---|
2347 | default_boundary=None, |
---|
2348 | verbose=False} |
---|
2349 | Module: \module{abstract_2d_finite_volumes.generic_boundary_conditions} |
---|
2350 | |
---|
2351 | A time-dependent boundary returns values for the conserved |
---|
2352 | quantities as a function of time \code{function(t)}. The user must specify |
---|
2353 | the domain to get access to the model time. |
---|
2354 | |
---|
2355 | Optional argument \code{default_boundary} can be used to specify another boundary object |
---|
2356 | to be used in case model time exceeds the time available in the file used by \code{File_boundary}. |
---|
2357 | The \code{default_boundary} could be a simple Dirichlet condition or |
---|
2358 | even another \code{Time_boundary} typically using data pertaining to another time interval. |
---|
2359 | \end{classdesc} |
---|
2360 | |
---|
2361 | \begin{classdesc}{File_boundary}{filename, |
---|
2362 | domain, |
---|
2363 | time_thinning=1, |
---|
2364 | time_limit=None, |
---|
2365 | boundary_polygon=None, |
---|
2366 | default_boundary=None, |
---|
2367 | use_cache=False, |
---|
2368 | verbose=False} |
---|
2369 | Module: \module{abstract_2d_finite_volumes.generic_boundary_conditions} |
---|
2370 | |
---|
2371 | This method may be used if the user wishes to apply a SWW file, STS file or |
---|
2372 | a time series file (TMS) to a boundary segment or segments. |
---|
2373 | The boundary values are obtained from a file and interpolated to the |
---|
2374 | appropriate segments for each conserved quantity. |
---|
2375 | |
---|
2376 | Optional argument \code{default_boundary} can be used to specify another boundary object |
---|
2377 | to be used in case model time exceeds the time available in the file used by \code{File_boundary}. |
---|
2378 | The \code{default_boundary} could be a simple Dirichlet condition or |
---|
2379 | even another \code{File_boundary} typically using data pertaining to another time interval. |
---|
2380 | \end{classdesc} |
---|
2381 | |
---|
2382 | \begin{classdesc}{Field_boundary}{filename, |
---|
2383 | domain, |
---|
2384 | mean_stage=0.0, |
---|
2385 | time_thinning=1, |
---|
2386 | time_limit=None, |
---|
2387 | boundary_polygon=None, |
---|
2388 | default_boundary=None, |
---|
2389 | use_cache=False, |
---|
2390 | verbose=False} |
---|
2391 | Module: \module{shallow_water.shallow_water_domain} |
---|
2392 | |
---|
2393 | This method works in the same way as \code{File_boundary} except that it |
---|
2394 | allows for the value of stage to be offset by a constant specified in the |
---|
2395 | keyword argument \code{mean_stage}. |
---|
2396 | |
---|
2397 | This functionality allows for models to be run for a range of tides using |
---|
2398 | the same boundary information (from STS, SWW or TMS files). The tidal value |
---|
2399 | for each run would then be specified in the keyword argument \code{mean_stage}. |
---|
2400 | If \code{mean_stage} = 0.0, \code{Field_boundary} and \code{File_boundary} |
---|
2401 | behave identically. |
---|
2402 | |
---|
2403 | Note that if the optional argument \code{default_boundary} is specified |
---|
2404 | its stage value will be adjusted by \code{mean_stage} just like the values |
---|
2405 | obtained from the file. |
---|
2406 | |
---|
2407 | See \code{File_boundary} for further details. |
---|
2408 | \end{classdesc} |
---|
2409 | |
---|
2410 | \begin{classdesc}{Transmissive_momentum_set_stage_boundary}{domain=None, |
---|
2411 | function=None} |
---|
2412 | Module: \module{shallow_water.shallow_water_domain} |
---|
2413 | \label{pg: transmissive momentum set stage boundary} |
---|
2414 | |
---|
2415 | This boundary returns the same momentum conserved quantities as |
---|
2416 | those present in its neighbour volume but sets stage as in a \code{Time_boundary}. |
---|
2417 | The underlying domain must be specified when boundary is instantiated. |
---|
2418 | |
---|
2419 | This type of boundary is useful when stage is known at the boundary as a |
---|
2420 | function of time, but momenta (or speeds) aren't. |
---|
2421 | |
---|
2422 | This class is specific to the shallow water equation as it works with the |
---|
2423 | momentum quantities assumed to be the second and third conserved quantities. |
---|
2424 | |
---|
2425 | In some circumstances, this boundary condition may cause numerical instabilities for similar |
---|
2426 | reasons as what has been observed with the simple fully transmissive boundary condition |
---|
2427 | (see Page \pageref{pg: transmissive boundary}). |
---|
2428 | This could for example be the case if a planar wave is reflected out through this boundary. |
---|
2429 | \end{classdesc} |
---|
2430 | |
---|
2431 | \begin{classdesc}{Transmissive_stage_zero_momentum_boundary}{domain=None} |
---|
2432 | Module: \module{shallow_water} |
---|
2433 | \label{pg: transmissive stage zero momentum boundary} |
---|
2434 | |
---|
2435 | This boundary returns same stage conserved quantities as |
---|
2436 | those present in its neighbour volume but sets momentum to zero. |
---|
2437 | The underlying domain must be specified when boundary is instantiated |
---|
2438 | |
---|
2439 | This type of boundary is useful when stage is known at the boundary as a |
---|
2440 | function of time, but momentum should be set to zero. This is for example |
---|
2441 | the case where a boundary is needed in the ocean on the two sides perpendicular |
---|
2442 | to the coast to maintain the wave height of the incoming wave. |
---|
2443 | |
---|
2444 | This class is specific to the shallow water equation as it works with the |
---|
2445 | momentum quantities assumed to be the second and third conserved quantities. |
---|
2446 | |
---|
2447 | This boundary condition should not cause the numerical instabilities seen with the transmissive momentum |
---|
2448 | boundary conditions (see Page \pageref{pg: transmissive boundary} and |
---|
2449 | Page \pageref{pg: transmissive momentum set stage boundary}). |
---|
2450 | \end{classdesc} |
---|
2451 | |
---|
2452 | \begin{classdesc}{Dirichlet_discharge_boundary}{domain=None, stage0=None, wh0=None} |
---|
2453 | Module: \module{shallow_water.shallow_water_domain} |
---|
2454 | |
---|
2455 | \code{stage0} sets the stage. |
---|
2456 | |
---|
2457 | \code{wh0} sets momentum in the inward normal direction. |
---|
2458 | \end{classdesc} |
---|
2459 | |
---|
2460 | \subsection{User-defined boundary conditions} |
---|
2461 | \label{sec:roll your own} |
---|
2462 | |
---|
2463 | All boundary classes must inherit from the generic boundary class |
---|
2464 | \code{Boundary} and have a method called \code{evaluate()} which must |
---|
2465 | take as inputs \code{self}, \code{vol_id} and \code{edge_id} where \code{self} refers to the |
---|
2466 | object itself and \code{vol_id} and \code{edge_id} are integers referring to |
---|
2467 | particular edges. The method must return a list of three floating point |
---|
2468 | numbers representing values for \code{stage}, |
---|
2469 | \code{xmomentum} and \code{ymomentum}, respectively. |
---|
2470 | |
---|
2471 | The constructor of a particular boundary class may be used to specify |
---|
2472 | particular values or flags to be used by the \code{evaluate()} method. |
---|
2473 | Please refer to the source code for the existing boundary conditions |
---|
2474 | for examples of how to implement boundary conditions. |
---|
2475 | |
---|
2476 | |
---|
2477 | \section{Forcing Terms}\index{Forcing terms} |
---|
2478 | \label{sec:forcing terms} |
---|
2479 | |
---|
2480 | \anuga provides a number of predefined forcing functions to be used with simulations. |
---|
2481 | Gravity and friction are always calculated using the elevation and friction quantities, |
---|
2482 | but the user may additionally add forcing terms to the list |
---|
2483 | \code{domain.forcing_terms} and have them affect the model. |
---|
2484 | |
---|
2485 | Currently, predefined forcing terms are: \\ |
---|
2486 | \begin{classdesc}{General_forcing}{domain, |
---|
2487 | quantity_name, |
---|
2488 | rate=0.0, |
---|
2489 | center=None, |
---|
2490 | radius=None, |
---|
2491 | polygon=None, |
---|
2492 | default_rate=None, |
---|
2493 | verbose=False} |
---|
2494 | Module: \module{shallow_water.shallow_water_domain} |
---|
2495 | |
---|
2496 | This is a general class to modify any quantity according to a given rate of change. |
---|
2497 | Other specific forcing terms are based on this class but it can be used by itself as well (e.g.\ to modify momentum). |
---|
2498 | |
---|
2499 | \code{domain} is the domain being evolved. |
---|
2500 | |
---|
2501 | \code{quantity_name} is the name of the quantity that will be affected by this forcing term. |
---|
2502 | |
---|
2503 | \code{rate} is the rate at which the quantity should change. This can be either a constant or a |
---|
2504 | function of time. Positive values indicate increases, negative values indicate decreases. |
---|
2505 | The parameter \code{rate} can be \code{None} at initialisation but must be specified |
---|
2506 | before a forcing term is applied (i.e.\ simulation has started). |
---|
2507 | The default value is 0.0 -- i.e.\ no forcing. |
---|
2508 | |
---|
2509 | \code{center} and \code{ radius} optionally restrict forcing to a circle with given center and radius. |
---|
2510 | |
---|
2511 | \code{polygon} optionally restricts forcing to an area enclosed by the given polygon. |
---|
2512 | |
---|
2513 | Note: specifying \code{center}, \code{radius} and \code{polygon} will cause an exception to be thrown. |
---|
2514 | Moreover, if the specified polygon or circle does not lie fully within the mesh boundary an Exception will be thrown. |
---|
2515 | |
---|
2516 | Example: |
---|
2517 | |
---|
2518 | \begin{verbatim} |
---|
2519 | P = [[x0, y0], [x1, y0], [x1, y1], [x0, y1]] # Square polygon |
---|
2520 | |
---|
2521 | xmom = General_forcing(domain, 'xmomentum', polygon=P) |
---|
2522 | ymom = General_forcing(domain, 'ymomentum', polygon=P) |
---|
2523 | |
---|
2524 | xmom.rate = f |
---|
2525 | ymom.rate = g |
---|
2526 | |
---|
2527 | domain.forcing_terms.append(xmom) |
---|
2528 | domain.forcing_terms.append(ymom) |
---|
2529 | \end{verbatim} |
---|
2530 | |
---|
2531 | Here, \code{f} and \code{g} are assumed to be defined as functions of time providing |
---|
2532 | a time dependent rate of change for xmomentum and ymomentum respectively. |
---|
2533 | \code{P} is assumed to be the polygon, specified as a list of points. |
---|
2534 | \end{classdesc} |
---|
2535 | |
---|
2536 | \begin{classdesc}{Inflow}{domain, |
---|
2537 | rate=0.0, |
---|
2538 | center=None, radius=None, |
---|
2539 | polygon=None, |
---|
2540 | default_rate=None, |
---|
2541 | verbose=False} |
---|
2542 | Module: \module{shallow_water.shallow_water_domain} |
---|
2543 | |
---|
2544 | This is a general class for inflow and abstraction of water according to a given rate of change. |
---|
2545 | This class will always modify the \code{stage} quantity. |
---|
2546 | |
---|
2547 | Inflow is based on the \code{General_forcing} class so the functionality is similar. |
---|
2548 | |
---|
2549 | \code{domain} is the domain being evolved. |
---|
2550 | |
---|
2551 | \code{rate} is the flow rate ($m^3/s$) at which the quantity should change. This can be either a constant or a |
---|
2552 | function of time. Positive values indicate inflow, negative values indicate outflow. |
---|
2553 | Note: The specified flow will be divided by the area of the inflow region and then applied to update the |
---|
2554 | stage quantity. |
---|
2555 | |
---|
2556 | \code{center} and \code{ radius} optionally restrict forcing to a circle with given center and radius. |
---|
2557 | |
---|
2558 | \code{polygon} optionally restricts forcing to an area enclosed by the given polygon. |
---|
2559 | |
---|
2560 | Example: |
---|
2561 | |
---|
2562 | \begin{verbatim} |
---|
2563 | hydrograph = Inflow(center=(320, 300), radius=10, |
---|
2564 | rate=file_function('QPMF_Rot_Sub13.tms')) |
---|
2565 | |
---|
2566 | domain.forcing_terms.append(hydrograph) |
---|
2567 | \end{verbatim} |
---|
2568 | |
---|
2569 | Here, \code{'QPMF_Rot_Sub13.tms'} is assumed to be a NetCDF file in the TMS format defining a timeseries for a hydrograph. |
---|
2570 | \end{classdesc} |
---|
2571 | |
---|
2572 | \begin{classdesc}{Rainfall}{domain, |
---|
2573 | rate=0.0, |
---|
2574 | center=None, |
---|
2575 | radius=None, |
---|
2576 | polygon=None, |
---|
2577 | default_rate=None, |
---|
2578 | verbose=False} |
---|
2579 | Module: \module{shallow_water.shallow_water_domain} |
---|
2580 | |
---|
2581 | This is a general class for implementing rainfall over the domain, possibly restricted to a given circle or polygon. |
---|
2582 | This class will always modify the \code{stage} quantity. |
---|
2583 | |
---|
2584 | Rainfall is based on the \code{General_forcing} class so the functionality is similar. |
---|
2585 | |
---|
2586 | \code{domain} is the domain being evolved. |
---|
2587 | |
---|
2588 | \code{rate} is the total rain rate over the specified domain. |
---|
2589 | Note: Raingauge Data needs to reflect the time step. |
---|
2590 | For example, if rain gauge is \code{mm} read every \code{dt} seconds, then the input |
---|
2591 | here is as \code{mm/dt} so 10 mm in 5 minutes becomes |
---|
2592 | 10/(5x60) = 0.0333mm/s. This parameter can be either a constant or a |
---|
2593 | function of time. Positive values indicate rain being added (or be used for general infiltration), |
---|
2594 | negative values indicate outflow at the specified rate (presumably this could model evaporation or abstraction). |
---|
2595 | |
---|
2596 | \code{center} and \code{ radius} optionally restrict forcing to a circle with given center and radius. |
---|
2597 | |
---|
2598 | \code{polygon} optionally restricts forcing to an area enclosed by the given polygon. |
---|
2599 | |
---|
2600 | Example: |
---|
2601 | |
---|
2602 | \begin{verbatim} |
---|
2603 | catchmentrainfall = Rainfall(rate=file_function('Q100_2hr_Rain.tms')) |
---|
2604 | domain.forcing_terms.append(catchmentrainfall) |
---|
2605 | \end{verbatim} |
---|
2606 | |
---|
2607 | Here, \code{'Q100_2hr_Rain.tms'} is assumed to be a NetCDF file in the TMS format defining a timeseries for the rainfall. |
---|
2608 | \end{classdesc} |
---|
2609 | |
---|
2610 | \begin{classdesc}{Culvert_flow}{domain, |
---|
2611 | culvert_description_filename=None, |
---|
2612 | culvert_routine=None, |
---|
2613 | end_point0=None, |
---|
2614 | end_point1=None, |
---|
2615 | enquiry_point0=None, |
---|
2616 | enquiry_point1=None, |
---|
2617 | type='box', |
---|
2618 | width=None, |
---|
2619 | height=None, |
---|
2620 | length=None, |
---|
2621 | number_of_barrels=1, |
---|
2622 | trigger_depth=0.01, |
---|
2623 | manning=None, |
---|
2624 | sum_loss=None, |
---|
2625 | use_velocity_head=False, |
---|
2626 | use_momentum_jet=False, |
---|
2627 | label=None, |
---|
2628 | description=None, |
---|
2629 | update_interval=None, |
---|
2630 | log_file=False, |
---|
2631 | discharge_hydrograph=False, |
---|
2632 | verbose=False} |
---|
2633 | Module: \module{culvert_flows.culvert_class} |
---|
2634 | |
---|
2635 | This is a general class for implementing flow through a culvert. |
---|
2636 | This class modifies the quantities \code{stage}, \code{xmomentum} and \code{ymomentum} in areas at both ends of the culvert. |
---|
2637 | |
---|
2638 | The \code{Culvert_flow} forcing term uses \code{Inflow} and \code{General_forcing} to update the quantities. |
---|
2639 | The flow direction is determined on-the-fly so openings are referenced simple as opening0 and opening1 |
---|
2640 | with either being able to take the role as Inflow or Outflow. |
---|
2641 | |
---|
2642 | The Culvert_flow class takes as input: |
---|
2643 | \begin{itemize} |
---|
2644 | \item \code{domain}: a reference to the domain being evolved |
---|
2645 | \item \code{culvert_description_filename}: |
---|
2646 | \item \code{culvert_routine}: |
---|
2647 | \item \code{end_point0}: Coordinates of one opening |
---|
2648 | \item \code{end_point1}: Coordinates of other opening |
---|
2649 | \item \code{enquiry_point0}: |
---|
2650 | \item \code{enquiry_point1}: |
---|
2651 | \item \code{type}: (default is 'box') |
---|
2652 | \item \code{width}: |
---|
2653 | \item \code{height}: |
---|
2654 | \item \code{length}: |
---|
2655 | \item \code{number_of_barrels}: Number of identical pipes in the culvert (default is 1) |
---|
2656 | \item \code{trigger_depth}: (default is 0.01) |
---|
2657 | \item \code{manning}: Mannings Roughness for Culvert |
---|
2658 | \item \code{sum_loss}: |
---|
2659 | \item \code{use_velocity_head}: |
---|
2660 | \item \code{use_momentum_jet}: |
---|
2661 | \item \code{label}: Short text naming the culvert |
---|
2662 | \item \code{description}: Text describing the culvert |
---|
2663 | \item \code{update_interval}: |
---|
2664 | \item \code{log_file}: |
---|
2665 | \item \code{discharge_hydrograph}: |
---|
2666 | \end{itemize} |
---|
2667 | |
---|
2668 | The user can specify different culvert routines. Hower \anuga currently provides only one, namely the |
---|
2669 | \code{boyd_generalised_culvert_model} as used in the example below: |
---|
2670 | |
---|
2671 | \begin{verbatim} |
---|
2672 | from anuga.culvert_flows.culvert_class import Culvert_flow |
---|
2673 | from anuga.culvert_flows.culvert_routines import boyd_generalised_culvert_model |
---|
2674 | |
---|
2675 | culvert1 = Culvert_flow(domain, |
---|
2676 | label='Culvert No. 1', |
---|
2677 | description='This culvert is a test unit 1.2m Wide by 0.75m High', |
---|
2678 | end_point0=[9.0, 2.5], |
---|
2679 | end_point1=[13.0, 2.5], |
---|
2680 | width=1.20, |
---|
2681 | height=0.75, |
---|
2682 | culvert_routine=boyd_generalised_culvert_model, |
---|
2683 | number_of_barrels=1, |
---|
2684 | verbose=True) |
---|
2685 | |
---|
2686 | culvert2 = Culvert_flow(domain, |
---|
2687 | label='Culvert No. 2', |
---|
2688 | description='This culvert is a circular test with d=1.2m', |
---|
2689 | end_point0=[9.0, 1.5], |
---|
2690 | end_point1=[30.0, 3.5], |
---|
2691 | diameter=1.20, |
---|
2692 | invert_level0=7, |
---|
2693 | culvert_routine=boyd_generalised_culvert_model, |
---|
2694 | number_of_barrels=1, |
---|
2695 | verbose=True) |
---|
2696 | |
---|
2697 | domain.forcing_terms.append(culvert1) |
---|
2698 | domain.forcing_terms.append(culvert2) |
---|
2699 | \end{verbatim} |
---|
2700 | \end{classdesc} |
---|
2701 | |
---|
2702 | |
---|
2703 | \section{Evolution}\index{evolution} |
---|
2704 | \label{sec:evolution} |
---|
2705 | |
---|
2706 | \begin{methoddesc}{\emph{<domain>}.evolve}{yieldstep=None, |
---|
2707 | finaltime=None, |
---|
2708 | duration=None, |
---|
2709 | skip_initial_step=False} |
---|
2710 | Module: \module{abstract_2d_finite_volumes.domain} |
---|
2711 | |
---|
2712 | This method is invoked once all the |
---|
2713 | preliminaries have been completed, and causes the model to progress |
---|
2714 | through successive steps in its evolution, storing results and |
---|
2715 | outputting statistics whenever a user-specified period |
---|
2716 | \code{yieldstep} is completed. Generally during this period the |
---|
2717 | model will evolve through several steps internally |
---|
2718 | as the method forces the water speed to be calculated |
---|
2719 | on successive new cells. |
---|
2720 | |
---|
2721 | \code{yieldstep} is the interval in seconds between yields where results are |
---|
2722 | stored, statistics written and the domain is inspected or possibly modified. |
---|
2723 | If omitted an internal predefined \code{yieldstep} is used. Internally, smaller |
---|
2724 | timesteps may be taken. |
---|
2725 | |
---|
2726 | \code{duration} is the duration of the simulation in seconds. |
---|
2727 | |
---|
2728 | \code{finaltime} is the time in seconds where simulation should end. This is currently |
---|
2729 | relative time, so it's the same as \code{duration}. If both \code{duration} and |
---|
2730 | \code{finaltime} are given an exception is thrown. |
---|
2731 | |
---|
2732 | \code{skip_initial_step} is a boolean flag that decides whether the first |
---|
2733 | yield step is skipped or not. This is useful for example to avoid |
---|
2734 | duplicate steps when multiple evolve processes are dove tailed. |
---|
2735 | |
---|
2736 | The code specified by the user in the block following the evolve statement is |
---|
2737 | only executed once every \code{yieldstep} even though |
---|
2738 | \anuga typically will take many more internal steps behind the scenes. |
---|
2739 | |
---|
2740 | You can include \method{evolve} in a statement of the type: |
---|
2741 | |
---|
2742 | \begin{verbatim} |
---|
2743 | for t in domain.evolve(yieldstep, finaltime): |
---|
2744 | <Do something with domain and t> |
---|
2745 | \end{verbatim} |
---|
2746 | \end{methoddesc} |
---|
2747 | |
---|
2748 | \subsection{Diagnostics} |
---|
2749 | \label{sec:diagnostics} |
---|
2750 | |
---|
2751 | \begin{methoddesc}{\emph{<domain>}.statistics}{} |
---|
2752 | Module: \module{abstract\_2d\_finite\_volumes.domain} |
---|
2753 | |
---|
2754 | Outputs statistics about the mesh within the \code{Domain}. |
---|
2755 | \end{methoddesc} |
---|
2756 | |
---|
2757 | \begin{methoddesc}{\emph{<domain>}.timestepping_statistics}{track_speeds=False, triangle_id=None} |
---|
2758 | Module: \module{abstract_2d_finite_volumes.domain} |
---|
2759 | |
---|
2760 | Returns a string of the following type for each timestep:\\ |
---|
2761 | \code{Time = 0.9000, delta t in [0.00598964, 0.01177388], steps=12 (12)} |
---|
2762 | |
---|
2763 | Here the numbers in \code{steps=12 (12)} indicate the number of steps taken and |
---|
2764 | the number of first-order steps, respectively. |
---|
2765 | |
---|
2766 | The optional keyword argument \code{track_speeds} will |
---|
2767 | generate a histogram of speeds generated by each triangle if set to \code{True}. The |
---|
2768 | speeds relate to the size of the timesteps used by \anuga and |
---|
2769 | this diagnostics may help pinpoint problem areas where excessive speeds |
---|
2770 | are generated. |
---|
2771 | |
---|
2772 | The optional keyword argument \code{triangle_id} can be used to specify a particular |
---|
2773 | triangle rather than the one with the largest speed. |
---|
2774 | \end{methoddesc} |
---|
2775 | |
---|
2776 | \begin{methoddesc}{\emph{<domain>}.boundary_statistics}{quantities=None, |
---|
2777 | tags=None} |
---|
2778 | Module: \module{abstract_2d_finite_volumes.domain} |
---|
2779 | |
---|
2780 | Generates output about boundary forcing at each timestep. |
---|
2781 | |
---|
2782 | \code{quantities} names the quantities to be reported -- may be \code{None}, |
---|
2783 | a string or a list of strings. |
---|
2784 | |
---|
2785 | \code{tags} names the tags to be reported -- may be either None, a string or a list of strings. |
---|
2786 | |
---|
2787 | When \code{quantities = 'stage'} and \code{tags = ['top', 'bottom']} |
---|
2788 | will return a string like: |
---|
2789 | |
---|
2790 | \begin{verbatim} |
---|
2791 | Boundary values at time 0.5000: |
---|
2792 | top: |
---|
2793 | stage in [ -0.25821218, -0.02499998] |
---|
2794 | bottom: |
---|
2795 | stage in [ -0.27098821, -0.02499974] |
---|
2796 | \end{verbatim} |
---|
2797 | \end{methoddesc} |
---|
2798 | |
---|
2799 | \begin{methoddesc}{Q = \emph{<domain>}.get_quantity}{name, |
---|
2800 | location='vertices', |
---|
2801 | indices=None} |
---|
2802 | Module: \module{abstract_2d_finite_volumes.domain} |
---|
2803 | |
---|
2804 | This function returns a Quantity object Q. |
---|
2805 | Access to its values should be done through \code{Q.get_values()} documented on Page \pageref{pg:get values}. |
---|
2806 | |
---|
2807 | \code{name} is the name of the quantity to retrieve. |
---|
2808 | |
---|
2809 | \code{location} is |
---|
2810 | |
---|
2811 | \code{indices} is |
---|
2812 | \end{methoddesc} |
---|
2813 | |
---|
2814 | \begin{methoddesc}{\emph{<domain>}.set_quantities_to_be_monitored}{quantity, |
---|
2815 | polygon=None, |
---|
2816 | time_interval=None} |
---|
2817 | Module: \module{abstract\_2d\_finite\_volumes.domain} |
---|
2818 | |
---|
2819 | Selects quantities and derived quantities for which extrema attained at internal timesteps |
---|
2820 | will be collected. |
---|
2821 | |
---|
2822 | \code{quantity} specifies the quantity or quantities to be monitored and must be either: |
---|
2823 | \begin{itemize} |
---|
2824 | \item the name of a quantity or derived quantity such as 'stage-elevation', |
---|
2825 | \item a list of quantity names, or |
---|
2826 | \item \code{None}. |
---|
2827 | \end{itemize} |
---|
2828 | |
---|
2829 | \code{polygon} can be used to monitor only triangles inside the polygon. Otherwise |
---|
2830 | all triangles will be included. |
---|
2831 | |
---|
2832 | \code{time_interval} will restrict monitoring to time steps in the interval. Otherwise |
---|
2833 | all timesteps will be included. |
---|
2834 | |
---|
2835 | Information can be tracked in the evolve loop by printing \code{quantity_statistics} and |
---|
2836 | collected data will be stored in the SWW file. |
---|
2837 | \end{methoddesc} |
---|
2838 | |
---|
2839 | \begin{methoddesc}{\emph{<domain>}.quantity_statistics}{precision='\%.4f'} |
---|
2840 | Module: \module{abstract_2d_finite_volumes.domain} |
---|
2841 | |
---|
2842 | Reports on extrema attained by selected quantities. |
---|
2843 | |
---|
2844 | Returns a string of the following type for each timestep: |
---|
2845 | |
---|
2846 | \begin{verbatim} |
---|
2847 | Monitored quantities at time 1.0000: |
---|
2848 | stage-elevation: |
---|
2849 | values since time = 0.00 in [0.00000000, 0.30000000] |
---|
2850 | minimum attained at time = 0.00000000, location = (0.16666667, 0.33333333) |
---|
2851 | maximum attained at time = 0.00000000, location = (0.83333333, 0.16666667) |
---|
2852 | ymomentum: |
---|
2853 | values since time = 0.00 in [0.00000000, 0.06241221] |
---|
2854 | minimum attained at time = 0.00000000, location = (0.33333333, 0.16666667) |
---|
2855 | maximum attained at time = 0.22472667, location = (0.83333333, 0.66666667) |
---|
2856 | xmomentum: |
---|
2857 | values since time = 0.00 in [-0.06062178, 0.47886313] |
---|
2858 | minimum attained at time = 0.00000000, location = (0.16666667, 0.33333333) |
---|
2859 | maximum attained at time = 0.35103646, location = (0.83333333, 0.16666667) |
---|
2860 | \end{verbatim} |
---|
2861 | |
---|
2862 | The quantities (and derived quantities) listed here must be selected at model |
---|
2863 | initialisation time using the method \code{domain.set_quantities_to_be_monitored()}. |
---|
2864 | |
---|
2865 | The optional keyword argument \code{precision='\%.4f'} will |
---|
2866 | determine the precision used for floating point values in the output. |
---|
2867 | This diagnostics helps track extrema attained by the selected quantities |
---|
2868 | at every internal timestep. |
---|
2869 | |
---|
2870 | These values are also stored in the SWW file for post-processing. |
---|
2871 | \end{methoddesc} |
---|
2872 | |
---|
2873 | \begin{methoddesc}{\emph{<quantity>}.get_values}{interpolation_points=None, |
---|
2874 | location='vertices', |
---|
2875 | indices=None, |
---|
2876 | use_cache=False, |
---|
2877 | verbose=False} |
---|
2878 | \label{pg:get values} |
---|
2879 | Module: \module{abstract_2d_finite_volumes.quantity} |
---|
2880 | |
---|
2881 | Extract values for quantity as a numeric array. |
---|
2882 | |
---|
2883 | \code{interpolation_points} is a list of (x, y) coordinates where the value is |
---|
2884 | sought (using interpolation). If \code{interpolation_points} is given, values |
---|
2885 | for \code{location} and \code{indices} are ignored. |
---|
2886 | Assume either an absolute UTM coordinates or geospatial data object. |
---|
2887 | |
---|
2888 | \code{location} specifies where values are to be stored. |
---|
2889 | Permissible options are 'vertices', 'edges', 'centroids' or 'unique vertices'. |
---|
2890 | |
---|
2891 | The returned values will have the leading dimension equal to length of the \code{indices} list or |
---|
2892 | N (all values) if \code{indices} is \code{None}. |
---|
2893 | |
---|
2894 | If \code{location} is 'centroids' the dimension of returned |
---|
2895 | values will be a list or a numeric array of length N, N being |
---|
2896 | the number of elements. |
---|
2897 | |
---|
2898 | If \code{location} is 'vertices' or 'edges' the dimension of |
---|
2899 | returned values will be of dimension \code{Nx3}. |
---|
2900 | |
---|
2901 | If \code{location} is 'unique vertices' the average value at |
---|
2902 | each vertex will be returned and the dimension of returned values |
---|
2903 | will be a 1d array of length "number of vertices" |
---|
2904 | |
---|
2905 | \code{indices} is the set of element ids that the operation applies to. |
---|
2906 | |
---|
2907 | The values will be stored in elements following their internal ordering. |
---|
2908 | \end{methoddesc} |
---|
2909 | |
---|
2910 | \begin{methoddesc}{\emph{<quantity>}.set_values}{numeric=None, |
---|
2911 | quantity=None, |
---|
2912 | function=None, |
---|
2913 | geospatial_data=None, |
---|
2914 | filename=None, |
---|
2915 | attribute_name=None, |
---|
2916 | alpha=None, |
---|
2917 | location='vertices', |
---|
2918 | polygon=None, |
---|
2919 | indices=None, |
---|
2920 | smooth=False, |
---|
2921 | verbose=False, |
---|
2922 | use_cache=False} |
---|
2923 | Module: \module{abstract_2d_finite_volumes.quantity} |
---|
2924 | |
---|
2925 | Assign values to a quantity object. |
---|
2926 | |
---|
2927 | This method works the same way as \code{set_quantity()} except that it doesn't take |
---|
2928 | a quantity name as the first argument since it is applied directly to the quantity. |
---|
2929 | Use \code{set_values} is used to assign |
---|
2930 | values to a new quantity that has been created but which is |
---|
2931 | not part of the domain's predefined quantities. |
---|
2932 | |
---|
2933 | \code{location} is ?? |
---|
2934 | |
---|
2935 | \code{indices} is ?? |
---|
2936 | |
---|
2937 | The method \code{set_values()} is always called by \code{set_quantity()} |
---|
2938 | behind the scenes. |
---|
2939 | \end{methoddesc} |
---|
2940 | |
---|
2941 | \begin{methoddesc}{\emph{<quantity>}.get_integral}{} |
---|
2942 | Module: \module{abstract_2d_finite_volumes.quantity} |
---|
2943 | |
---|
2944 | Return the computed integral over the entire domain for the quantity. |
---|
2945 | \end{methoddesc} |
---|
2946 | |
---|
2947 | \begin{methoddesc}{\emph{<quantity>}.get_maximum_value}{indices=None} |
---|
2948 | Module: \module{abstract_2d_finite_volumes.quantity} |
---|
2949 | |
---|
2950 | Return the maximum value of a quantity (on centroids). |
---|
2951 | |
---|
2952 | \code{indices} is the optional set of element \code{id}s that |
---|
2953 | the operation applies to. |
---|
2954 | |
---|
2955 | We do not seek the maximum at vertices as each vertex can |
---|
2956 | have multiple values -- one for each triangle sharing it. |
---|
2957 | \end{methoddesc} |
---|
2958 | |
---|
2959 | \begin{methoddesc}{\emph{<quantity>}.get_maximum_location}{indices=None} |
---|
2960 | Module: \module{abstract_2d_finite_volumes.quantity} |
---|
2961 | |
---|
2962 | Return the location of the maximum value of a quantity (on centroids). |
---|
2963 | |
---|
2964 | \code{indices} is the optional set of element \code{id}s that |
---|
2965 | the operation applies to. |
---|
2966 | |
---|
2967 | We do not seek the maximum at vertices as each vertex can |
---|
2968 | have multiple values -- one for each triangle sharing it. |
---|
2969 | |
---|
2970 | If there are multiple cells with the same maximum value, the |
---|
2971 | first cell encountered in the triangle array is returned. |
---|
2972 | \end{methoddesc} |
---|
2973 | |
---|
2974 | \begin{methoddesc}{\emph{<domain>}.get_wet_elements}{indices=None} |
---|
2975 | Module: \module{shallow_water.shallow_water_domain} |
---|
2976 | |
---|
2977 | Returns the indices for elements where h $>$ minimum_allowed_height |
---|
2978 | |
---|
2979 | \code{indices} is the optional set of element \code{id}s that |
---|
2980 | the operation applies to. |
---|
2981 | \end{methoddesc} |
---|
2982 | |
---|
2983 | \begin{methoddesc}{\emph{<domain>}.get_maximum_inundation_elevation}{indices=None} |
---|
2984 | Module: \module{shallow_water.shallow_water_domain} |
---|
2985 | |
---|
2986 | Return highest elevation where h $>$ 0. |
---|
2987 | |
---|
2988 | \code{indices} is the optional set of element \code{id}s that |
---|
2989 | the operation applies to. |
---|
2990 | |
---|
2991 | Example to find maximum runup elevation: |
---|
2992 | \begin{verbatim} |
---|
2993 | z = domain.get_maximum_inundation_elevation() |
---|
2994 | \end{verbatim} |
---|
2995 | \end{methoddesc} |
---|
2996 | |
---|
2997 | \begin{methoddesc}{\emph{<domain>}.get_maximum_inundation_location}{indices=None} |
---|
2998 | Module: \module{shallow_water.shallow_water_domain} |
---|
2999 | |
---|
3000 | Return location (x,y) of highest elevation where h $>$ 0. |
---|
3001 | |
---|
3002 | \code{indices} is the optional set of element \code{id}s that |
---|
3003 | the operation applies to. |
---|
3004 | |
---|
3005 | Example to find maximum runup location: |
---|
3006 | \begin{verbatim} |
---|
3007 | x, y = domain.get_maximum_inundation_location() |
---|
3008 | \end{verbatim} |
---|
3009 | \end{methoddesc} |
---|
3010 | |
---|
3011 | |
---|
3012 | \section{Queries of SWW model output files} |
---|
3013 | After a model has been run, it is often useful to extract various information from the SWW |
---|
3014 | output file (see Section \ref{sec:sww format}). This is typically more convenient than using the |
---|
3015 | diagnostics described in Section \ref{sec:diagnostics} which rely on the model running -- something |
---|
3016 | that can be very time consuming. The SWW files are easy and quick to read and offer information |
---|
3017 | about the model results such as runup heights, time histories of selected quantities, |
---|
3018 | flow through cross sections and much more. |
---|
3019 | |
---|
3020 | \begin{funcdesc}{elevation = get_maximum_inundation_elevation}{filename, |
---|
3021 | polygon=None, |
---|
3022 | time_interval=None, |
---|
3023 | verbose=False} |
---|
3024 | Module: \module{shallow_water.data_manager} |
---|
3025 | |
---|
3026 | Return the highest elevation where depth is positive ($h > 0$). |
---|
3027 | |
---|
3028 | \code{filename} is the path to a NetCDF SWW file containing \anuga model output. |
---|
3029 | |
---|
3030 | \code{polygon} restricts the query to the points that lie within the polygon. |
---|
3031 | |
---|
3032 | \code {time_interval} restricts the query to within the time interval. |
---|
3033 | |
---|
3034 | Example to find maximum runup elevation: |
---|
3035 | |
---|
3036 | \begin{verbatim} |
---|
3037 | max_runup = get_maximum_inundation_elevation(filename) |
---|
3038 | \end{verbatim} |
---|
3039 | |
---|
3040 | If no inundation is found (within the \code{polygon} and \code{time_interval}, if specified) |
---|
3041 | the return value is \code{None}. This indicates "No Runup" or "Everything is dry". |
---|
3042 | \end{funcdesc} |
---|
3043 | |
---|
3044 | \begin{funcdesc}{(x, y) = get_maximum_inundation_location}{filename, |
---|
3045 | polygon=None, |
---|
3046 | time_interval=None, |
---|
3047 | verbose=False} |
---|
3048 | Module: \module{shallow_water.data_manager} |
---|
3049 | |
---|
3050 | Return location (x,y) of the highest elevation where depth is positive ($h > 0$). |
---|
3051 | |
---|
3052 | \code{filename} is the path to a NetCDF SWW file containing \anuga model output. |
---|
3053 | |
---|
3054 | \code{polygon} restricts the query to the points that lie within the polygon. |
---|
3055 | |
---|
3056 | \code {time_interval} restricts the query to within the time interval. |
---|
3057 | |
---|
3058 | Example to find maximum runup location: |
---|
3059 | |
---|
3060 | \begin{verbatim} |
---|
3061 | max_runup_location = get_maximum_inundation_location(filename) |
---|
3062 | \end{verbatim} |
---|
3063 | |
---|
3064 | If no inundation is found (within the \code{polygon} and \code{time_interval}, if specified) |
---|
3065 | the return value is \code{None}. This indicates "No Runup" or "Everything is dry". |
---|
3066 | is \code{None}. This indicates "No Runup" or "Everything is dry". |
---|
3067 | \end{funcdesc} |
---|
3068 | |
---|
3069 | \begin{funcdesc}{sww2timeseries}{swwfiles, |
---|
3070 | gauge_filename, |
---|
3071 | production_dirs, |
---|
3072 | report=None, |
---|
3073 | reportname=None, |
---|
3074 | plot_quantity=None, |
---|
3075 | generate_fig=False, |
---|
3076 | surface=None, |
---|
3077 | time_min=None, |
---|
3078 | time_max=None, |
---|
3079 | time_thinning=1, |
---|
3080 | time_unit=None, |
---|
3081 | title_on=None, |
---|
3082 | use_cache=False, |
---|
3083 | verbose=False} |
---|
3084 | Module: \module{abstract_2d_finite_volumes.util} |
---|
3085 | |
---|
3086 | Read a set of SWW files and plot the time series for the prescribed quantities |
---|
3087 | at defined gauge locations and prescribed time range. |
---|
3088 | |
---|
3089 | \code{swwfiles} is a dictionary of SWW files with keys of \code{label_id}. |
---|
3090 | |
---|
3091 | \code{gauge_filename} is the path to a file containing gauge data. |
---|
3092 | |
---|
3093 | THIS NEEDS MORE WORK. WORK ON FUNCTION __DOC__ STRING, IF NOTHING ELSE! |
---|
3094 | \end{funcdesc} |
---|
3095 | |
---|
3096 | \begin{funcdesc}{(time, Q) = get_flow_through_cross_section}{filename, polyline, verbose=False} |
---|
3097 | Module: \module{shallow_water.data_manager} |
---|
3098 | |
---|
3099 | Obtain flow ($m^3/s$) perpendicular to specified cross section. |
---|
3100 | |
---|
3101 | \code{filename} is the path to the SWW file. |
---|
3102 | |
---|
3103 | \code{polyline} is the representation of the desired cross section -- it may contain |
---|
3104 | multiple sections allowing for complex shapes. Assumes absolute UTM coordinates. |
---|
3105 | |
---|
3106 | Returns a tuple \code{time, Q} where: |
---|
3107 | |
---|
3108 | \code{time} is all the stored times in the SWW file. |
---|
3109 | |
---|
3110 | \code{Q} is a hydrograph of total flow across the given segments for all stored times. |
---|
3111 | |
---|
3112 | The normal flow is computed for each triangle intersected by the \code{polyline} and |
---|
3113 | added up. If multiple segments at different angles are specified the normal flows |
---|
3114 | may partially cancel each other. |
---|
3115 | |
---|
3116 | Example to find flow through cross section: |
---|
3117 | |
---|
3118 | \begin{verbatim} |
---|
3119 | cross_section = [[x, 0], [x, width]] |
---|
3120 | time, Q = get_flow_through_cross_section(filename, cross_section) |
---|
3121 | \end{verbatim} |
---|
3122 | \end{funcdesc} |
---|
3123 | |
---|
3124 | |
---|
3125 | \section{Other} |
---|
3126 | \begin{methoddesc}{quantity = \emph{<domain>}.create_quantity_from_expression}{string} |
---|
3127 | Module: \module{abstract_2d_finite_volumes.domain} |
---|
3128 | |
---|
3129 | Create a new quantity from other quantities in the domain using an arbitrary expression. |
---|
3130 | |
---|
3131 | \code{string} is a string containing an arbitrary quantity expression. |
---|
3132 | |
---|
3133 | Returns the new \code{Quantity} object. |
---|
3134 | |
---|
3135 | Handy for creating derived quantities on-the-fly: |
---|
3136 | |
---|
3137 | \begin{verbatim} |
---|
3138 | Depth = domain.create_quantity_from_expression('stage-elevation') |
---|
3139 | |
---|
3140 | exp = '(xmomentum*xmomentum + ymomentum*ymomentum)**0.5' |
---|
3141 | Absolute_momentum = domain.create_quantity_from_expression(exp) |
---|
3142 | \end{verbatim} |
---|
3143 | |
---|
3144 | %See also \file{Analytical_solution_circular_hydraulic_jump.py} for an example. |
---|
3145 | \end{methoddesc} |
---|
3146 | |
---|
3147 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
---|
3148 | |
---|
3149 | \chapter{\anuga System Architecture} |
---|
3150 | |
---|
3151 | \section{File Formats} |
---|
3152 | \label{sec:file formats} |
---|
3153 | |
---|
3154 | \anuga makes use of a number of different file formats. The |
---|
3155 | following table lists all these formats, which are described in more |
---|
3156 | detail in the paragraphs below. |
---|
3157 | |
---|
3158 | \begin{center} |
---|
3159 | \begin{tabular}{|ll|} \hline |
---|
3160 | \textbf{Extension} & \textbf{Description} \\ |
---|
3161 | \hline\hline |
---|
3162 | \code{.sww} & NetCDF format for storing model output with mesh information \code{f(t,x,y)}\\ |
---|
3163 | \code{.sts} & NetCDF format for storing model ouput \code{f(t,x,y)} without mesh information\\ |
---|
3164 | \code{.tms} & NetCDF format for storing time series \code{f(t)}\\ |
---|
3165 | \code{.csv/.txt} & ASCII format for storing arbitrary points and associated attributes\\ |
---|
3166 | \code{.pts} & NetCDF format for storing arbitrary points and associated attributes\\ |
---|
3167 | \code{.asc} & ASCII format of regular DEMs as output from ArcView\\ |
---|
3168 | \code{.prj} & Associated ArcView file giving more metadata for \code{.asc} format\\ |
---|
3169 | \code{.ers} & ERMapper header format of regular DEMs for ArcView\\ |
---|
3170 | \code{.dem} & NetCDF representation of regular DEM data\\ |
---|
3171 | \code{.tsh} & ASCII format for storing meshes and associated boundary and region info\\ |
---|
3172 | \code{.msh} & NetCDF format for storing meshes and associated boundary and region info\\ |
---|
3173 | \code{.nc} & Native ferret NetCDF format\\ |
---|
3174 | \code{.geo} & Houdinis ASCII geometry format (?) \\ \par \hline |
---|
3175 | \end{tabular} |
---|
3176 | \end{center} |
---|
3177 | |
---|
3178 | The above table shows the file extensions used to identify the |
---|
3179 | formats of files. However, typically, in referring to a format we |
---|
3180 | capitalise the extension and omit the initial full stop -- thus, we |
---|
3181 | refer to 'SWW files' or 'PRJ files', not 'sww files' or '.prj files'. |
---|
3182 | |
---|
3183 | \bigskip |
---|
3184 | |
---|
3185 | A typical dataflow can be described as follows: |
---|
3186 | |
---|
3187 | SOMETHING MISSING HERE!? |
---|
3188 | |
---|
3189 | \subsection{Manually Created Files} |
---|
3190 | |
---|
3191 | \begin{tabular}{ll} |
---|
3192 | ASC, PRJ & Digital elevation models (gridded)\\ |
---|
3193 | NC & Model outputs for use as boundary conditions (e.g.\ from MOST) |
---|
3194 | \end{tabular} |
---|
3195 | |
---|
3196 | \subsection{Automatically Created Files} |
---|
3197 | |
---|
3198 | \begin{tabular}{ll} |
---|
3199 | ASC, PRJ $\rightarrow$ DEM $\rightarrow$ PTS & Convert DEMs to native \code{.pts} file\\ |
---|
3200 | NC $\rightarrow$ SWW & Convert MOST boundary files to boundary \code{.sww}\\ |
---|
3201 | PTS + TSH $\rightarrow$ TSH with elevation & Least squares fit\\ |
---|
3202 | TSH $\rightarrow$ SWW & Convert TSH to \code{.sww}-viewable using \code{animate}\\ |
---|
3203 | TSH + Boundary SWW $\rightarrow$ SWW & Simulation using \code{\anuga}\\ |
---|
3204 | Polygonal mesh outline $\rightarrow$ & TSH or MSH |
---|
3205 | \end{tabular} |
---|
3206 | |
---|
3207 | \bigskip |
---|
3208 | |
---|
3209 | \subsection{SWW, STS and TMS Formats} |
---|
3210 | \label{sec:sww format} |
---|
3211 | The SWW, STS and TMS formats are all NetCDF formats and are of key importance for \anuga. |
---|
3212 | |
---|
3213 | An SWW file is used for storing \anuga output and therefore pertains |
---|
3214 | to a set of points and a set of times at which a model is evaluated. |
---|
3215 | It contains, in addition to dimension information, the following |
---|
3216 | variables: |
---|
3217 | |
---|
3218 | \begin{itemize} |
---|
3219 | \item \code{x} and \code{y}: coordinates of the points, represented as numeric arrays |
---|
3220 | \item \code{elevation}: a numeric array storing bed-elevations |
---|
3221 | \item \code{volumes}: a list specifying the points at the vertices of each of the triangles |
---|
3222 | % Refer here to the example to be provided in describing the simple example |
---|
3223 | \item \code{time}: a numeric array containing times for model evaluation |
---|
3224 | \end{itemize} |
---|
3225 | |
---|
3226 | The contents of an SWW file may be viewed using the anuga viewer \code{animate}, |
---|
3227 | which creates an on-screen visialisation. See section \ref{sec:animate} |
---|
3228 | (page \pageref{sec:animate}) in Appendix \ref{ch:supportingtools} for more on \code{animate}. |
---|
3229 | |
---|
3230 | Alternatively, there are tools, such as \code{ncdump}, that allow |
---|
3231 | you to convert a NetCDF file into a readable format such as the |
---|
3232 | Class Definition Language (CDL). The following is an excerpt from a |
---|
3233 | CDL representation of the output file \file{runup.sww} generated |
---|
3234 | from running the simple example \file{runup.py} of Chapter \ref{ch:getstarted}: |
---|
3235 | |
---|
3236 | %FIXME (Ole): Should put in example with nonzero xllcorner, yllcorner |
---|
3237 | \verbatiminput{examples/bedslopeexcerpt.cdl} |
---|
3238 | |
---|
3239 | The SWW format is used not only for output but also serves as input |
---|
3240 | for functions such as \function{file\_boundary} and |
---|
3241 | \function{file\_function}, described in Chapter \ref{ch:interface}. |
---|
3242 | |
---|
3243 | An STS file is used for storing a set of points and associated times. |
---|
3244 | It contains, in addition to dimension information, the following |
---|
3245 | variables: |
---|
3246 | \begin{itemize} |
---|
3247 | \item \code{x} and \code{y}: coordinates of the points, represented as numeric arrays |
---|
3248 | \item \code{permutation}: Original indices of the points as specified by the optional \code{ordering_file} |
---|
3249 | (see the function \code{urs2sts()} in Section \ref{sec:basicfileconversions}) |
---|
3250 | \item \code{elevation}: a numeric array storing bed-elevations |
---|
3251 | % Refer here to the example to be provided in describing the simple example |
---|
3252 | \item \code{time}: a numeric array containing times for model evaluation |
---|
3253 | \end{itemize} |
---|
3254 | |
---|
3255 | The only difference between the STS format and the SWW format is the former does |
---|
3256 | not contain a list specifying the points at the vertices of each of the triangles |
---|
3257 | (\code{volumes}). Consequently information (arrays) stored within an STS file such |
---|
3258 | as \code{elevation} can be accessed in exactly the same way as it would be extracted |
---|
3259 | from an SWW file. |
---|
3260 | |
---|
3261 | A TMS file is used to store time series data that is independent of position. |
---|
3262 | |
---|
3263 | \subsection{Mesh File Formats} |
---|
3264 | |
---|
3265 | A mesh file is a file that has a specific format suited to |
---|
3266 | triangular meshes and their outlines. A mesh file can have one of |
---|
3267 | two formats: it can be either a TSH file, which is an ASCII file, or |
---|
3268 | an MSH file, which is a NetCDF file. A mesh file can be generated |
---|
3269 | from the function \function{create_mesh_from_regions()} (see |
---|
3270 | Section \ref{sec:meshgeneration}) and be used to initialise a domain. |
---|
3271 | |
---|
3272 | A mesh file can define the outline of the mesh -- the vertices and |
---|
3273 | line segments that enclose the region in which the mesh is |
---|
3274 | created -- and the triangular mesh itself, which is specified by |
---|
3275 | listing the triangles and their vertices, and the segments, which |
---|
3276 | are those sides of the triangles that are associated with boundary |
---|
3277 | conditions. |
---|
3278 | |
---|
3279 | In addition, a mesh file may contain 'holes' and/or 'regions'. A |
---|
3280 | hole represents an area where no mesh is to be created, while a |
---|
3281 | region is a labelled area used for defining properties of a mesh, |
---|
3282 | such as friction values. A hole or region is specified by a point |
---|
3283 | and bounded by a number of segments that enclose that point. |
---|
3284 | |
---|
3285 | A mesh file can also contain a georeference, which describes an |
---|
3286 | offset to be applied to $x$ and $y$ values -- e.g.\ to the vertices. |
---|
3287 | |
---|
3288 | \subsection{Formats for Storing Arbitrary Points and Attributes} |
---|
3289 | |
---|
3290 | A CSV/TXT file is used to store data representing |
---|
3291 | arbitrary numerical attributes associated with a set of points. |
---|
3292 | |
---|
3293 | The format for an CSV/TXT file is:\\ |
---|
3294 | \\ |
---|
3295 | first line: \code{[column names]}\\ |
---|
3296 | other lines: \code{[x value], [y value], [attributes]}\\ |
---|
3297 | |
---|
3298 | for example: |
---|
3299 | |
---|
3300 | \begin{verbatim} |
---|
3301 | x, y, elevation, friction |
---|
3302 | 0.6, 0.7, 4.9, 0.3 |
---|
3303 | 1.9, 2.8, 5.0, 0.3 |
---|
3304 | 2.7, 2.4, 5.2, 0.3 |
---|
3305 | \end{verbatim} |
---|
3306 | |
---|
3307 | The delimiter is a comma. The first two columns are assumed to |
---|
3308 | be $x$ and $y$ coordinates. |
---|
3309 | |
---|
3310 | A PTS file is a NetCDF representation of the data held in an points CSV |
---|
3311 | file. If the data is associated with a set of $N$ points, then the |
---|
3312 | data is stored using an $N \times 2$ numeric array of float |
---|
3313 | variables for the points and an $N \times 1$ numeric array for each |
---|
3314 | attribute. |
---|
3315 | |
---|
3316 | \subsection{ArcView Formats} |
---|
3317 | |
---|
3318 | Files of the three formats ASC, PRJ and ERS are all associated with |
---|
3319 | data from ArcView. |
---|
3320 | |
---|
3321 | An ASC file is an ASCII representation of DEM output from ArcView. |
---|
3322 | It contains a header with the following format: |
---|
3323 | |
---|
3324 | \begin{tabular}{l l} |
---|
3325 | \code{ncols} & \code{753}\\ |
---|
3326 | \code{nrows} & \code{766}\\ |
---|
3327 | \code{xllcorner} & \code{314036.58727982}\\ |
---|
3328 | \code{yllcorner} & \code{6224951.2960092}\\ |
---|
3329 | \code{cellsize} & \code{100}\\ |
---|
3330 | \code{NODATA_value} & \code{-9999} |
---|
3331 | \end{tabular} |
---|
3332 | |
---|
3333 | The remainder of the file contains the elevation data for each grid point |
---|
3334 | in the grid defined by the above information. |
---|
3335 | |
---|
3336 | A PRJ file is an ArcView file used in conjunction with an ASC file |
---|
3337 | to represent metadata for a DEM. |
---|
3338 | |
---|
3339 | \subsection{DEM Format} |
---|
3340 | |
---|
3341 | A DEM file in \anuga is a NetCDF representation of regular DEM data. |
---|
3342 | |
---|
3343 | \subsection{Other Formats} |
---|
3344 | |
---|
3345 | \subsection{Basic File Conversions} |
---|
3346 | \label{sec:basicfileconversions} |
---|
3347 | |
---|
3348 | \begin{funcdesc}{sww2dem}{(basename_in, |
---|
3349 | basename_out=None, |
---|
3350 | quantity=None, |
---|
3351 | timestep=None, |
---|
3352 | reduction=None, |
---|
3353 | cellsize=10, |
---|
3354 | number_of_decimal_places=None, |
---|
3355 | NODATA_value=-9999, |
---|
3356 | easting_min=None, |
---|
3357 | easting_max=None, |
---|
3358 | northing_min=None, |
---|
3359 | northing_max=None, |
---|
3360 | verbose=False, |
---|
3361 | origin=None, |
---|
3362 | datum='WGS84', |
---|
3363 | format='ers', |
---|
3364 | block_size=None} |
---|
3365 | Module: \module{shallow_water.data_manager} |
---|
3366 | |
---|
3367 | Takes data from an SWW file \code{basename_in} and converts it to DEM format (ASC or |
---|
3368 | ERS) of a desired grid size \code{cellsize} in metres. The user can select how |
---|
3369 | many decimal places the output data is represented with by using \code{number_of_decimal_places}, |
---|
3370 | with the default being 3. |
---|
3371 | |
---|
3372 | The $easting$ and $northing$ values are used if the user wishes to determine the output |
---|
3373 | within a specified rectangular area. The \code{reduction} input refers to a function |
---|
3374 | to reduce the quantities over all time step of the SWW file, e.g.\ maximum. |
---|
3375 | \end{funcdesc} |
---|
3376 | |
---|
3377 | \begin{funcdesc}{dem2pts}{basename_in, basename_out=None, |
---|
3378 | easting_min=None, easting_max=None, |
---|
3379 | northing_min=None, northing_max=None, |
---|
3380 | use_cache=False, verbose=False} |
---|
3381 | Module: \module{shallow\_water.data\_manager} |
---|
3382 | |
---|
3383 | Takes DEM data (a NetCDF file representation of data from a regular Digital |
---|
3384 | Elevation Model) and converts it to PTS format. |
---|
3385 | \end{funcdesc} |
---|
3386 | |
---|
3387 | \begin{funcdesc}{urs2sts}{basename_in, basename_out=None, |
---|
3388 | weights=None, verbose=False, |
---|
3389 | origin=None,mean_stage=0.0, |
---|
3390 | zscale=1.0, ordering_filename=None} |
---|
3391 | Module: \module{shallow\_water.data\_manager} |
---|
3392 | |
---|
3393 | Takes URS data (timeseries data in mux2 format) and converts it to STS format. |
---|
3394 | The optional filename \code{ordering\_filename} specifies the permutation of indices |
---|
3395 | of points to select along with their longitudes and latitudes. This permutation will also be |
---|
3396 | stored in the STS file. If absent, all points are taken and the permutation will be trivial, |
---|
3397 | i.e.\ $0, 1, \ldots, N-1$, where $N$ is the total number of points stored. |
---|
3398 | \end{funcdesc} |
---|
3399 | |
---|
3400 | \begin{funcdesc}{csv2building\_polygons}{file\_name, floor\_height=3} |
---|
3401 | Module: \module{shallow\_water.data\_manager} |
---|
3402 | |
---|
3403 | Convert CSV files of the form: |
---|
3404 | |
---|
3405 | \begin{verbatim} |
---|
3406 | easting,northing,id,floors |
---|
3407 | 422664.22,870785.46,2,0 |
---|
3408 | 422672.48,870780.14,2,0 |
---|
3409 | 422668.17,870772.62,2,0 |
---|
3410 | 422660.35,870777.17,2,0 |
---|
3411 | 422664.22,870785.46,2,0 |
---|
3412 | 422661.30,871215.06,3,1 |
---|
3413 | 422667.50,871215.70,3,1 |
---|
3414 | 422668.30,871204.86,3,1 |
---|
3415 | 422662.21,871204.33,3,1 |
---|
3416 | 422661.30,871215.06,3,1 |
---|
3417 | \end{verbatim} |
---|
3418 | |
---|
3419 | to a dictionary of polygons with \code{id} as key. |
---|
3420 | The associated number of \code{floors} are converted to m above MSL and |
---|
3421 | returned as a separate dictionary also keyed by \code{id}. |
---|
3422 | |
---|
3423 | Optional parameter \code{floor_height} is the height of each building story. |
---|
3424 | |
---|
3425 | These can e.g.\ be converted to a \code{Polygon_function} for use with \code{add_quantity} |
---|
3426 | as shown on page \pageref{add quantity}. |
---|
3427 | \end{funcdesc} |
---|
3428 | |
---|
3429 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
---|
3430 | |
---|
3431 | \chapter{\anuga mathematical background} |
---|
3432 | \label{cd:mathematical background} |
---|
3433 | |
---|
3434 | |
---|
3435 | \section{Introduction} |
---|
3436 | |
---|
3437 | This chapter outlines the mathematics underpinning \anuga. |
---|
3438 | |
---|
3439 | |
---|
3440 | \section{Model} |
---|
3441 | \label{sec:model} |
---|
3442 | |
---|
3443 | The shallow water wave equations are a system of differential |
---|
3444 | conservation equations which describe the flow of a thin layer of |
---|
3445 | fluid over terrain. The form of the equations are: |
---|
3446 | \[ |
---|
3447 | \frac{\partial \UU}{\partial t}+\frac{\partial \EE}{\partial |
---|
3448 | x}+\frac{\partial \GG}{\partial y}=\SSS |
---|
3449 | \] |
---|
3450 | where $\UU=\left[ {{\begin{array}{*{20}c} |
---|
3451 | h & {uh} & {vh} \\ |
---|
3452 | \end{array} }} \right]^T$ is the vector of conserved quantities; water depth |
---|
3453 | $h$, $x$-momentum $uh$ and $y$-momentum $vh$. Other quantities |
---|
3454 | entering the system are bed elevation $z$ and stage (absolute water |
---|
3455 | level) $w$, where the relation $w = z + h$ holds true at all times. |
---|
3456 | The fluxes in the $x$ and $y$ directions, $\EE$ and $\GG$ are given |
---|
3457 | by |
---|
3458 | \[ |
---|
3459 | \EE=\left[ {{\begin{array}{*{20}c} |
---|
3460 | {uh} \hfill \\ |
---|
3461 | {u^2h+gh^2/2} \hfill \\ |
---|
3462 | {uvh} \hfill \\ |
---|
3463 | \end{array} }} \right]\mbox{ and }\GG=\left[ {{\begin{array}{*{20}c} |
---|
3464 | {vh} \hfill \\ |
---|
3465 | {vuh} \hfill \\ |
---|
3466 | {v^2h+gh^2/2} \hfill \\ |
---|
3467 | \end{array} }} \right] |
---|
3468 | \] |
---|
3469 | and the source term (which includes gravity and friction) is given |
---|
3470 | by |
---|
3471 | \[ |
---|
3472 | \SSS=\left[ {{\begin{array}{*{20}c} |
---|
3473 | 0 \hfill \\ |
---|
3474 | -{gh(z_{x} + S_{fx} )} \hfill \\ |
---|
3475 | -{gh(z_{y} + S_{fy} )} \hfill \\ |
---|
3476 | \end{array} }} \right] |
---|
3477 | \] |
---|
3478 | where $S_f$ is the bed friction. The friction term is modelled using |
---|
3479 | Manning's resistance law |
---|
3480 | \[ |
---|
3481 | S_{fx} =\frac{u\eta ^2\sqrt {u^2+v^2} }{h^{4/3}}\mbox{ and }S_{fy} |
---|
3482 | =\frac{v\eta ^2\sqrt {u^2+v^2} }{h^{4/3}} |
---|
3483 | \] |
---|
3484 | in which $\eta$ is the Manning resistance coefficient. |
---|
3485 | The model does not currently include consideration of kinematic viscosity or dispersion. |
---|
3486 | |
---|
3487 | As demonstrated in our papers, \cite{ZR1999,nielsen2005} these |
---|
3488 | equations and their implementation in \anuga provide a reliable |
---|
3489 | model of general flows associated with inundation such as dam breaks |
---|
3490 | and tsunamis. |
---|
3491 | |
---|
3492 | |
---|
3493 | \section{Finite Volume Method} |
---|
3494 | \label{sec:fvm} |
---|
3495 | |
---|
3496 | We use a finite-volume method for solving the shallow water wave |
---|
3497 | equations \cite{ZR1999}. The study area is represented by a mesh of |
---|
3498 | triangular cells as in Figure~\ref{fig:mesh} in which the conserved |
---|
3499 | quantities of water depth $h$, and horizontal momentum $(uh, vh)$, |
---|
3500 | in each volume are to be determined. The size of the triangles may |
---|
3501 | be varied within the mesh to allow greater resolution in regions of |
---|
3502 | particular interest. |
---|
3503 | |
---|
3504 | \begin{figure}[htp] \begin{center} |
---|
3505 | \includegraphics[width=8.0cm,keepaspectratio=true]{graphics/step-five} |
---|
3506 | \caption{Triangular mesh used in our finite volume method. Conserved |
---|
3507 | quantities $h$, $uh$ and $vh$ are associated with the centroid of |
---|
3508 | each triangular cell.} |
---|
3509 | \label{fig:mesh} |
---|
3510 | \end{center} \end{figure} |
---|
3511 | |
---|
3512 | The equations constituting the finite-volume method are obtained by |
---|
3513 | integrating the differential conservation equations over each |
---|
3514 | triangular cell of the mesh. Introducing some notation we use $i$ to |
---|
3515 | refer to the $i$th triangular cell $T_i$, and ${\cal N}(i)$ to the |
---|
3516 | set of indices referring to the cells neighbouring the $i$th cell. |
---|
3517 | Then $A_i$ is the area of the $i$th triangular cell and $l_{ij}$ is |
---|
3518 | the length of the edge between the $i$th and $j$th cells. |
---|
3519 | |
---|
3520 | By applying the divergence theorem we obtain for each volume an |
---|
3521 | equation which describes the rate of change of the average of the |
---|
3522 | conserved quantities within each cell, in terms of the fluxes across |
---|
3523 | the edges of the cells and the effect of the source terms. In |
---|
3524 | particular, rate equations associated with each cell have the form |
---|
3525 | $$ |
---|
3526 | \frac{d\UU_i }{dt}+ \frac1{A_i}\sum\limits_{j\in{\cal N}(i)} \HH_{ij} l_{ij} = \SSS_i |
---|
3527 | $$ |
---|
3528 | where |
---|
3529 | \begin{itemize} |
---|
3530 | \item $\UU_i$ the vector of conserved quantities averaged over the $i$th cell, |
---|
3531 | \item $\SSS_i$ is the source term associated with the $i$th cell, and |
---|
3532 | \item $\HH_{ij}$ is the outward normal flux of material across the \textit{ij}th edge. |
---|
3533 | \end{itemize} |
---|
3534 | |
---|
3535 | %\item $l_{ij}$ is the length of the edge between the $i$th and $j$th |
---|
3536 | %cells |
---|
3537 | %\item $m_{ij}$ is the midpoint of |
---|
3538 | %the \textit{ij}th edge, |
---|
3539 | %\item |
---|
3540 | %$\mathbf{n}_{ij} = (n_{ij,1} , n_{ij,2})$is the outward pointing |
---|
3541 | %normal along the \textit{ij}th edge, and The |
---|
3542 | |
---|
3543 | The flux $\HH_{ij}$ is evaluated using a numerical flux function |
---|
3544 | $\HH(\cdot, \cdot ; \ \cdot)$ which is consistent with the shallow |
---|
3545 | water flux in the sense that for all conservation vectors $\UU$ and normal vectors $\nn$ |
---|
3546 | $$ |
---|
3547 | H(\UU,\UU;\ \nn) = \EE(\UU) n_1 + \GG(\UU) n_2 . |
---|
3548 | $$ |
---|
3549 | |
---|
3550 | Then |
---|
3551 | $$ |
---|
3552 | \HH_{ij} = \HH(\UU_i(m_{ij}), |
---|
3553 | \UU_j(m_{ij}); \mathbf{n}_{ij}) |
---|
3554 | $$ |
---|
3555 | where $m_{ij}$ is the midpoint of the \textit{ij}th edge and |
---|
3556 | $\mathbf{n}_{ij}$ is the outward pointing normal, with respect to the $i$th cell, on the |
---|
3557 | \textit{ij}th edge. The function $\UU_i(x)$ for $x \in |
---|
3558 | T_i$ is obtained from the vector $\UU_k$ of conserved average values for the $i$th and |
---|
3559 | neighbouring cells. |
---|
3560 | |
---|
3561 | We use a second order reconstruction to produce a piece-wise linear |
---|
3562 | function construction of the conserved quantities for all $x \in |
---|
3563 | T_i$ for each cell (see Figure~\ref{fig:mesh:reconstruct}). This |
---|
3564 | function is allowed to be discontinuous across the edges of the |
---|
3565 | cells, but the slope of this function is limited to avoid |
---|
3566 | artificially introduced oscillations. |
---|
3567 | |
---|
3568 | Godunov's method (see \cite{Toro1992}) involves calculating the |
---|
3569 | numerical flux function $\HH(\cdot, \cdot ; \ \cdot)$ by exactly |
---|
3570 | solving the corresponding one dimensional Riemann problem normal to |
---|
3571 | the edge. We use the central-upwind scheme of \cite{KurNP2001} to |
---|
3572 | calculate an approximation of the flux across each edge. |
---|
3573 | |
---|
3574 | \begin{figure}[htp] \begin{center} |
---|
3575 | \includegraphics[width=8.0cm,keepaspectratio=true]{graphics/step-reconstruct} |
---|
3576 | \caption{From the values of the conserved quantities at the centroid |
---|
3577 | of the cell and its neighbouring cells, a discontinuous piecewise |
---|
3578 | linear reconstruction of the conserved quantities is obtained.} |
---|
3579 | \label{fig:mesh:reconstruct} |
---|
3580 | \end{center} \end{figure} |
---|
3581 | |
---|
3582 | In the computations presented in this paper we use an explicit Euler |
---|
3583 | time stepping method with variable timestepping adapted to the |
---|
3584 | observed CFL condition: |
---|
3585 | |
---|
3586 | \begin{equation} |
---|
3587 | \Delta t = \min_{k,i=[0,1,2]} \min \left( \frac{r_k}{v_{k,i}}, \frac{r_{n_{k,i}}}{v_{k,i}} \right ) |
---|
3588 | \label{eq:CFL condition} |
---|
3589 | \end{equation} |
---|
3590 | where $r_k$ is the radius of the $k$'th triangle and $v_{k,i}$ is the maximal velocity across |
---|
3591 | edge joining triangle $k$ and it's $i$'th neighbour, triangle $n_{k,i}$, as calculated by the |
---|
3592 | numerical flux function |
---|
3593 | using the central upwind scheme of \cite{KurNP2001}. The symbol $r_{n_{k,i}}$ denotes the radius |
---|
3594 | of the $i$'th neighbour of triangle $k$. The radii are calculated as radii of the inscribed circles |
---|
3595 | of each triangle. |
---|
3596 | |
---|
3597 | |
---|
3598 | \section{Flux limiting} |
---|
3599 | |
---|
3600 | The shallow water equations are solved numerically using a |
---|
3601 | finite volume method on an unstructured triangular grid. |
---|
3602 | The upwind central scheme due to Kurganov and Petrova is used as an |
---|
3603 | approximate Riemann solver for the computation of inviscid flux functions. |
---|
3604 | This makes it possible to handle discontinuous solutions. |
---|
3605 | |
---|
3606 | To alleviate the problems associated with numerical instabilities due to |
---|
3607 | small water depths near a wet/dry boundary we employ a new flux limiter that |
---|
3608 | ensures that unphysical fluxes are never encounted. |
---|
3609 | |
---|
3610 | Let $u$ and $v$ be the velocity components in the $x$ and $y$ direction, |
---|
3611 | $w$ the absolute water level (stage) and |
---|
3612 | $z$ the bed elevation. The latter are assumed to be relative to the |
---|
3613 | same height datum. |
---|
3614 | The conserved quantities tracked by \anuga are momentum in the |
---|
3615 | $x$-direction ($\mu = uh$), momentum in the $y$-direction ($\nu = vh$) |
---|
3616 | and depth ($h = w-z$). |
---|
3617 | |
---|
3618 | The flux calculation requires access to the velocity vector $(u, v)$ |
---|
3619 | where each component is obtained as $u = \mu/h$ and $v = \nu/h$ respectively. |
---|
3620 | In the presence of very small water depths, these calculations become |
---|
3621 | numerically unreliable and will typically cause unphysical speeds. |
---|
3622 | |
---|
3623 | We have employed a flux limiter which replaces the calculations above with |
---|
3624 | the limited approximations. |
---|
3625 | \begin{equation} |
---|
3626 | \hat{u} = \frac{\mu}{h + h_0/h}, \bigskip \hat{v} = \frac{\nu}{h + h_0/h}, |
---|
3627 | \end{equation} |
---|
3628 | where $h_0$ is a regularisation parameter that controls the minimal |
---|
3629 | magnitude of the denominator. Taking the limits we have for $\hat{u}$ |
---|
3630 | \[ |
---|
3631 | \lim_{h \rightarrow 0} \hat{u} = |
---|
3632 | \lim_{h \rightarrow 0} \frac{\mu}{h + h_0/h} = 0 |
---|
3633 | \] |
---|
3634 | and |
---|
3635 | \[ |
---|
3636 | \lim_{h \rightarrow \infty} \hat{u} = |
---|
3637 | \lim_{h \rightarrow \infty} \frac{\mu}{h + h_0/h} = \frac{\mu}{h} = u |
---|
3638 | \] |
---|
3639 | with similar results for $\hat{v}$. |
---|
3640 | |
---|
3641 | The maximal value of $\hat{u}$ is attained when $h+h_0/h$ is minimal or (by differentiating the denominator) |
---|
3642 | \[ |
---|
3643 | 1 - h_0/h^2 = 0 |
---|
3644 | \] |
---|
3645 | or |
---|
3646 | \[ |
---|
3647 | h_0 = h^2 |
---|
3648 | \] |
---|
3649 | |
---|
3650 | \anuga has a global parameter $H_0$ that controls the minimal depth which |
---|
3651 | is considered in the various equations. This parameter is typically set to |
---|
3652 | $10^{-3}$. Setting |
---|
3653 | \[ |
---|
3654 | h_0 = H_0^2 |
---|
3655 | \] |
---|
3656 | provides a reasonable balance between accurracy and stability. In fact, |
---|
3657 | setting $h=H_0$ will scale the predicted speed by a factor of $0.5$: |
---|
3658 | \[ |
---|
3659 | \left[ \frac{\mu}{h + h_0/h} \right]_{h = H_0} = |
---|
3660 | \left[ \frac{\mu}{H_0 + H_0^2/H_0} \right] = |
---|
3661 | \frac{\mu}{2 H_0} = \frac{\mu}{2 h} = \frac{u}{2} |
---|
3662 | \] |
---|
3663 | In general, for multiples of the minimal depth $N H_0$ one obtains |
---|
3664 | \begin{equation} |
---|
3665 | \left[ \frac{\mu}{h + h_0/h} \right]_{h = N H_0} = |
---|
3666 | \frac{\mu}{N H_0 + H_0/N} = |
---|
3667 | \frac{\mu}{h (1 + 1/N^2)} |
---|
3668 | \label{eq:flux limit multiple} |
---|
3669 | \end{equation} |
---|
3670 | which converges quadratically to the true value with the multiple N. |
---|
3671 | |
---|
3672 | Although this equation can be used for any depth, we have restricted its use to depths less than $10 * H_0$ (or 1 cm) to computational resources. |
---|
3673 | According to Equation \ref{eq:flux limit multiple} this cutoff |
---|
3674 | affects the calculated velocity by less than 1 \%. |
---|
3675 | |
---|
3676 | %The developed numerical model has been applied to several test cases |
---|
3677 | %as well as to real flows. numeric tests prove the robustness and accuracy of the model. |
---|
3678 | |
---|
3679 | |
---|
3680 | \section{Slope limiting} |
---|
3681 | A multidimensional slope-limiting technique is employed to achieve second-order spatial |
---|
3682 | accuracy and to prevent spurious oscillations. This is using the MinMod limiter and is |
---|
3683 | documented elsewhere. |
---|
3684 | |
---|
3685 | However close to the bed, the limiter must ensure that no negative depths occur. |
---|
3686 | On the other hand, in deep water, the bed topography should be ignored for the |
---|
3687 | purpose of the limiter. |
---|
3688 | |
---|
3689 | Let $w, z, h$ be the stage, bed elevation and depth at the centroid and |
---|
3690 | let $w_i, z_i, h_i$ be the stage, bed elevation and depth at vertex $i$. |
---|
3691 | Define the minimal depth across all vertices as $\hmin$ as |
---|
3692 | \[ |
---|
3693 | \hmin = \min_i h_i |
---|
3694 | \] |
---|
3695 | |
---|
3696 | Let $\tilde{w_i}$ be the stage obtained from a gradient limiter |
---|
3697 | limiting on stage only. The corresponding depth is then defined as |
---|
3698 | \[ |
---|
3699 | \tilde{h_i} = \tilde{w_i} - z_i |
---|
3700 | \] |
---|
3701 | We would use this limiter in deep water which we will define (somewhat boldly) |
---|
3702 | as |
---|
3703 | \[ |
---|
3704 | \hmin \ge \epsilon |
---|
3705 | \] |
---|
3706 | |
---|
3707 | Similarly, let $\bar{w_i}$ be the stage obtained from a gradient |
---|
3708 | limiter limiting on depth respecting the bed slope. |
---|
3709 | The corresponding depth is defined as |
---|
3710 | \[ |
---|
3711 | \bar{h_i} = \bar{w_i} - z_i |
---|
3712 | \] |
---|
3713 | |
---|
3714 | We introduce the concept of a balanced stage $w_i$ which is obtained as |
---|
3715 | the linear combination |
---|
3716 | |
---|
3717 | \[ |
---|
3718 | w_i = \alpha \tilde{w_i} + (1-\alpha) \bar{w_i} |
---|
3719 | \] |
---|
3720 | or |
---|
3721 | \[ |
---|
3722 | w_i = z_i + \alpha \tilde{h_i} + (1-\alpha) \bar{h_i} |
---|
3723 | \] |
---|
3724 | where $\alpha \in [0, 1]$. |
---|
3725 | |
---|
3726 | Since $\tilde{w_i}$ is obtained in 'deep' water where the bedslope |
---|
3727 | is ignored we have immediately that |
---|
3728 | \[ |
---|
3729 | \alpha = 1 \mbox{ for } \hmin \ge \epsilon %or dz=0 |
---|
3730 | \] |
---|
3731 | %where the maximal bed elevation range $dz$ is defined as |
---|
3732 | %\[ |
---|
3733 | % dz = \max_i |z_i - z| |
---|
3734 | %\] |
---|
3735 | |
---|
3736 | If $\hmin < \epsilon$ we want to use the 'shallow' limiter just enough that |
---|
3737 | no negative depths occur. Formally, we will require that |
---|
3738 | \[ |
---|
3739 | \alpha \tilde{h_i} + (1-\alpha) \bar{h_i} > \epsilon, \forall i |
---|
3740 | \] |
---|
3741 | or |
---|
3742 | \begin{equation} |
---|
3743 | \alpha(\tilde{h_i} - \bar{h_i}) > \epsilon - \bar{h_i}, \forall i |
---|
3744 | \label{eq:limiter bound} |
---|
3745 | \end{equation} |
---|
3746 | |
---|
3747 | There are two cases: |
---|
3748 | \begin{enumerate} |
---|
3749 | \item $\bar{h_i} \le \tilde{h_i}$: The deep water (limited using stage) |
---|
3750 | vertex is at least as far away from the bed than the shallow water |
---|
3751 | (limited using depth). In this case we won't need any contribution from |
---|
3752 | $\bar{h_i}$ and can accept any $\alpha$. |
---|
3753 | |
---|
3754 | E.g.\ $\alpha=1$ reduces Equation \ref{eq:limiter bound} to |
---|
3755 | \[ |
---|
3756 | \tilde{h_i} > \epsilon |
---|
3757 | \] |
---|
3758 | whereas $\alpha=0$ yields |
---|
3759 | \[ |
---|
3760 | \bar{h_i} > \epsilon |
---|
3761 | \] |
---|
3762 | all well and good. |
---|
3763 | \item $\bar{h_i} > \tilde{h_i}$: In this case the the deep water vertex is |
---|
3764 | closer to the bed than the shallow water vertex or even below the bed. |
---|
3765 | In this case we need to find an $\alpha$ that will ensure a positive depth. |
---|
3766 | Rearranging Equation \ref{eq:limiter bound} and solving for $\alpha$ one |
---|
3767 | obtains the bound |
---|
3768 | \[ |
---|
3769 | \alpha < \frac{\epsilon - \bar{h_i}}{\tilde{h_i} - \bar{h_i}}, \forall i |
---|
3770 | \] |
---|
3771 | \end{enumerate} |
---|
3772 | |
---|
3773 | Ensuring Equation \ref{eq:limiter bound} holds true for all vertices one |
---|
3774 | arrives at the definition |
---|
3775 | \[ |
---|
3776 | \alpha = \min_{i} \frac{\bar{h_i} - \epsilon}{\bar{h_i} - \tilde{h_i}} |
---|
3777 | \] |
---|
3778 | which will guarantee that no vertex 'cuts' through the bed. Finally, should |
---|
3779 | $\bar{h_i} < \epsilon$ and therefore $\alpha < 0$, we suggest setting |
---|
3780 | $\alpha=0$ and similarly capping $\alpha$ at 1 just in case. |
---|
3781 | |
---|
3782 | %Furthermore, |
---|
3783 | %dropping the $\epsilon$ ensures that alpha is always positive and also |
---|
3784 | %provides a numerical safety {??) |
---|
3785 | |
---|
3786 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
---|
3787 | |
---|
3788 | \chapter{Basic \anuga Assumptions} |
---|
3789 | |
---|
3790 | |
---|
3791 | \section{Time} |
---|
3792 | |
---|
3793 | Physical model time cannot be earlier than 1 Jan 1970 00:00:00. |
---|
3794 | If one wished to recreate scenarios prior to that date it must be done |
---|
3795 | using some relative time (e.g.\ 0). |
---|
3796 | |
---|
3797 | The \anuga domain has an attribute \code{starttime} which is used in cases where the |
---|
3798 | simulation should be started later than the beginning of some input data such as those |
---|
3799 | obtained from boundaries or forcing functions (hydrographs, file_boundary etc). |
---|
3800 | |
---|
3801 | The \code{domain.startime} may be adjusted in \code{file_boundary} in the case the |
---|
3802 | input data does not itself start until a later time. |
---|
3803 | |
---|
3804 | |
---|
3805 | \section{Spatial data} |
---|
3806 | |
---|
3807 | \subsection{Projection} |
---|
3808 | All spatial data relates to the WGS84 datum (or GDA94) and assumes a projection |
---|
3809 | into UTM with false easting of 500000 and false northing of |
---|
3810 | 1000000 on the southern hemisphere (0 on the northern hemisphere). |
---|
3811 | All locations must consequently be specified in Cartesian coordinates |
---|
3812 | (eastings, northings) or (x,y) where the unit is metres. |
---|
3813 | Alternative projections can be used, but \anuga does have the concept of a UTM zone |
---|
3814 | that must be the same for all coordinates in the model. |
---|
3815 | |
---|
3816 | \subsection{Internal coordinates} |
---|
3817 | It is important to realise that for numerical precision \anuga uses coordinates that are relative |
---|
3818 | to the lower left node of the rectangle containing the mesh ($x_{\mbox{min}}$, $y_{\mbox{min}}$). |
---|
3819 | This origin is referred to internally as \code{xllcorner}, \code{yllcorner} following the ESRI ASCII grid notation. |
---|
3820 | The SWW file format also includes \code{xllcorner}, \code{yllcorner} and any coordinate in the file should be adjusted |
---|
3821 | by adding this origin. See Section \ref{sec:sww format}. |
---|
3822 | |
---|
3823 | Throughout the \anuga interface, functions have optional boolean arguments \code{absolute} which controls |
---|
3824 | whether spatial data received is using the internal representation (\code{absolute=False}) or the |
---|
3825 | user coordinate set (\code{absolute=True}). See e.g.\ \code{get_vertex_coordinates()} on \pageref{pg:get vertex coordinates}. |
---|
3826 | |
---|
3827 | DEMs, meshes and boundary conditions can have different origins. However, the internal representation in \anuga |
---|
3828 | will use the origin of the mesh. |
---|
3829 | |
---|
3830 | \subsection{Polygons} |
---|
3831 | When generating a mesh it is assumed that polygons do not cross. |
---|
3832 | Having polygons that cross can cause bad meshes to be produced or the mesh generation itself may fail. |
---|
3833 | |
---|
3834 | %OLD |
---|
3835 | %The dataflow is: (See data_manager.py and from scenarios) |
---|
3836 | % |
---|
3837 | % |
---|
3838 | %Simulation scenarios |
---|
3839 | %--------------------% |
---|
3840 | %% |
---|
3841 | % |
---|
3842 | %Sub directories contain scrips and derived files for each simulation. |
---|
3843 | %The directory ../source_data contains large source files such as |
---|
3844 | %DEMs provided externally as well as MOST tsunami simulations to be used |
---|
3845 | %as boundary conditions. |
---|
3846 | % |
---|
3847 | %Manual steps are: |
---|
3848 | % Creation of DEMs from argcview (.asc + .prj) |
---|
3849 | % Creation of mesh from pmesh (.tsh) |
---|
3850 | % Creation of tsunami simulations from MOST (.nc) |
---|
3851 | %% |
---|
3852 | % |
---|
3853 | %Typical scripted steps are% |
---|
3854 | % |
---|
3855 | % prepare_dem.py: Convert asc and prj files supplied by arcview to |
---|
3856 | % native dem and pts formats% |
---|
3857 | % |
---|
3858 | % prepare_pts.py: Convert netcdf output from MOST to an SWW file suitable |
---|
3859 | % as boundary condition% |
---|
3860 | % |
---|
3861 | % prepare_mesh.py: Merge DEM (pts) and mesh (tsh) using least squares |
---|
3862 | % smoothing. The outputs are tsh files with elevation data.% |
---|
3863 | % |
---|
3864 | % run_simulation.py: Use the above together with various parameters to |
---|
3865 | % run inundation simulation. |
---|
3866 | |
---|
3867 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
---|
3868 | |
---|
3869 | \appendix |
---|
3870 | |
---|
3871 | |
---|
3872 | \chapter{Supporting Tools} |
---|
3873 | \label{ch:supportingtools} |
---|
3874 | |
---|
3875 | This section describes a number of supporting tools, supplied with \anuga, that offer a |
---|
3876 | variety of types of functionality and can enhance the basic capabilities of \anuga. |
---|
3877 | |
---|
3878 | |
---|
3879 | \section{caching} |
---|
3880 | \label{sec:caching} |
---|
3881 | |
---|
3882 | The \code{cache} function is used to provide supervised caching of function |
---|
3883 | results. A Python function call of the form: |
---|
3884 | |
---|
3885 | \begin{verbatim} |
---|
3886 | result = func(arg1, ..., argn) |
---|
3887 | \end{verbatim} |
---|
3888 | |
---|
3889 | can be replaced by: |
---|
3890 | |
---|
3891 | \begin{verbatim} |
---|
3892 | from caching import cache |
---|
3893 | result = cache(func,(arg1, ..., argn)) |
---|
3894 | \end{verbatim} |
---|
3895 | |
---|
3896 | which returns the same output but reuses cached |
---|
3897 | results if the function has been computed previously in the same context. |
---|
3898 | \code{result} and the arguments can be simple types, tuples, list, dictionaries or |
---|
3899 | objects, but not unhashable types such as functions or open file objects. |
---|
3900 | The function \code{func} may be a member function of an object or a module. |
---|
3901 | |
---|
3902 | This type of caching is particularly useful for computationally intensive |
---|
3903 | functions with few frequently used combinations of input arguments. Note that |
---|
3904 | if the inputs or output are very large caching may not save time because |
---|
3905 | disc access may dominate the execution time. |
---|
3906 | |
---|
3907 | If the function definition changes after a result has been cached, this will be |
---|
3908 | detected by examining the functions bytecode and the function will be recomputed. |
---|
3909 | However, caching will not detect changes in modules used by \code{func}. |
---|
3910 | In this case the cache must be cleared manually. |
---|
3911 | |
---|
3912 | Options are set by means of the function \code{set_option(key, value)}, |
---|
3913 | where \code{key} is a key associated with a |
---|
3914 | Python dictionary \code{options}. This dictionary stores settings such as the name of |
---|
3915 | the directory used, the maximum number of cached files allowed, and so on. |
---|
3916 | |
---|
3917 | The \code{cache} function allows the user also to specify a list of dependent files. If any of these |
---|
3918 | have been changed, the function is recomputed and the results stored again. |
---|
3919 | |
---|
3920 | %Other features include support for compression and a capability to \ldots |
---|
3921 | |
---|
3922 | USAGE: \nopagebreak |
---|
3923 | |
---|
3924 | \begin{verbatim} |
---|
3925 | result = cache(func, args, kwargs, dependencies, cachedir, verbose, |
---|
3926 | compression, evaluate, test, return_filename) |
---|
3927 | \end{verbatim} |
---|
3928 | |
---|
3929 | |
---|
3930 | \pagebreak |
---|
3931 | \section{\anuga viewer -- animate} |
---|
3932 | \label{sec:animate} |
---|
3933 | |
---|
3934 | The output generated by \anuga may be viewed by |
---|
3935 | means of the visualisation tool \code{animate}, which takes an |
---|
3936 | SWW file generated by \anuga and creates a visual representation |
---|
3937 | of the data. Examples may be seen in Figures \ref{fig:runupstart} |
---|
3938 | and \ref{fig:runup2}. To view an SWW file with |
---|
3939 | \code{animate} in the Windows environment, you can simply drag the |
---|
3940 | icon representing the file over an icon on the desktop for the |
---|
3941 | \code{animate} executable file (or a shortcut to it), or set up a |
---|
3942 | file association to make files with the extension \code{.sww} open |
---|
3943 | with \code{animate}. Alternatively, you can operate \code{animate} |
---|
3944 | from the command line. |
---|
3945 | |
---|
3946 | Upon starting the viewer, you will see an interactive moving-picture |
---|
3947 | display. You can use the keyboard and mouse to slow down, speed up or |
---|
3948 | stop the display, change the viewing position or carry out a number |
---|
3949 | of other simple operations. Help is also displayed when you press |
---|
3950 | the \code{h} key. |
---|
3951 | |
---|
3952 | The main keys operating the interactive screen are: |
---|
3953 | \begin{center} |
---|
3954 | \begin{tabular}{|ll|} \hline |
---|
3955 | \code{h} & toggle on-screen help display \\ |
---|
3956 | \code{w} & toggle wireframe \\ |
---|
3957 | space bar & start/stop\\ |
---|
3958 | up/down arrows & increase/decrease speed\\ |
---|
3959 | left/right arrows & direction in time \emph{(when running)}\\ |
---|
3960 | & step through simulation \emph{(when stopped)}\\ |
---|
3961 | left mouse button & rotate\\ |
---|
3962 | middle mouse button & pan\\ |
---|
3963 | right mouse button & zoom\\ \hline |
---|
3964 | \end{tabular} |
---|
3965 | \end{center} |
---|
3966 | |
---|
3967 | % \vfill |
---|
3968 | |
---|
3969 | The following describes how to operate \code{animate} from the command line: |
---|
3970 | |
---|
3971 | Usage: \code{animate [options] swwfile \ldots}\\ \nopagebreak |
---|
3972 | where: \\ \nopagebreak |
---|
3973 | \begin{tabular}{ll} |
---|
3974 | \code{--display <type>} & \code{MONITOR | POWERWALL | REALITY\_CENTER |}\\ |
---|
3975 | & \code{HEAD\_MOUNTED\_DISPLAY}\\ |
---|
3976 | \code{--rgba} & Request a RGBA colour buffer visual\\ |
---|
3977 | \code{--stencil} & Request a stencil buffer visual\\ |
---|
3978 | \code{--stereo} & Use default stereo mode which is \code{ANAGLYPHIC} if not \\ |
---|
3979 | & overridden by environmental variable\\ |
---|
3980 | \code{--stereo <mode>} & \code{ANAGLYPHIC | QUAD\_BUFFER | HORIZONTAL\_SPLIT |}\\ |
---|
3981 | & \code{VERTICAL\_SPLIT | LEFT\_EYE | RIGHT\_EYE |}\\ |
---|
3982 | & \code{ON | OFF} \\ |
---|
3983 | \code{-alphamax <float 0-1>} & Maximum transparency clamp value\\ |
---|
3984 | \code{-alphamin <float 0-1>} & Transparency value at \code{hmin}\\ |
---|
3985 | \code{-cullangle <float angle 0-90>} & Cull triangles steeper than this value\\ |
---|
3986 | \code{-help} & Display this information\\ |
---|
3987 | \code{-hmax <float>} & Height above which transparency is set to |
---|
3988 | \code{alphamax}\\ |
---|
3989 | \code{-hmin <float>} & Height below which transparency is set to |
---|
3990 | zero\\ |
---|
3991 | \code{-lightpos <float>,<float>,<float>} & $x,y,z$ of bedslope directional light ($z$ is |
---|
3992 | up, default is overhead)\\ |
---|
3993 | \code{-loop} & Repeated (looped) playback of \code{.swm} files\\ |
---|
3994 | \code{-movie <dirname>} & Save numbered images to named directory and |
---|
3995 | quit\\ |
---|
3996 | \code{-nosky} & Omit background sky\\ |
---|
3997 | \code{-scale <float>} & Vertical scale factor\\ |
---|
3998 | \code{-texture <file>} & Image to use for bedslope topography\\ |
---|
3999 | \code{-tps <rate>} & Timesteps per second\\ |
---|
4000 | \code{-version} & Revision number and creation (not compile) |
---|
4001 | date\\ |
---|
4002 | \end{tabular} |
---|
4003 | |
---|
4004 | |
---|
4005 | \pagebreak |
---|
4006 | \section{utilities/polygons} |
---|
4007 | |
---|
4008 | \declaremodule{standard}{utilities.polygon} |
---|
4009 | \refmodindex{utilities.polygon} |
---|
4010 | |
---|
4011 | \begin{classdesc}{<callable> = Polygon_function}{regions, |
---|
4012 | default=0.0, |
---|
4013 | geo_reference=None} |
---|
4014 | Module: \code{utilities.polygon} |
---|
4015 | |
---|
4016 | Creates a callable object that returns one of a specified list of values when |
---|
4017 | evaluated at a point \code{(x, y)}, depending on which polygon, from a specified list of polygons, the |
---|
4018 | point belongs to. |
---|
4019 | |
---|
4020 | \code{regions} is a list of pairs |
---|
4021 | \code{(P, v)}, where each \code{P} is a polygon and each \code{v} |
---|
4022 | is either a constant value or a function of coordinates \code{x} |
---|
4023 | and \code{y}, specifying the return value for a point inside \code{P}. |
---|
4024 | |
---|
4025 | \code{default} may be used to specify a value (or a function) |
---|
4026 | for a point not lying inside any of the specified polygons. |
---|
4027 | |
---|
4028 | When a point lies in more than one polygon, the return value is taken to |
---|
4029 | be the value for whichever of these polygon appears later in the list. |
---|
4030 | |
---|
4031 | %FIXME (Howard): CAN x, y BE VECTORS? |
---|
4032 | |
---|
4033 | \code{geo_reference} refers to the status of points |
---|
4034 | that are passed into the function. Typically they will be relative to |
---|
4035 | some origin. |
---|
4036 | |
---|
4037 | Typical usage may take the form: |
---|
4038 | |
---|
4039 | \begin{verbatim} |
---|
4040 | set_quantity('elevation', |
---|
4041 | Polygon_function([(P1, v1), (P2, v2)], |
---|
4042 | default=v3, |
---|
4043 | geo_reference=domain.geo_reference)) |
---|
4044 | \end{verbatim} |
---|
4045 | \end{classdesc} |
---|
4046 | |
---|
4047 | \begin{funcdesc}{<polygon> = read_polygon}{filename, split=','} |
---|
4048 | Module: \code{utilities.polygon} |
---|
4049 | |
---|
4050 | Reads the specified file and returns a polygon. |
---|
4051 | Each line of the file must contain exactly two numbers, separated by a delimiter, which are interpreted |
---|
4052 | as coordinates of one vertex of the polygon. |
---|
4053 | |
---|
4054 | \code{filename} is the path to the file to read. |
---|
4055 | |
---|
4056 | \code{split} sets the delimiter character between the numbers on one line of |
---|
4057 | the file. If not specified, the delimiter is the ',' character. |
---|
4058 | \end{funcdesc} |
---|
4059 | |
---|
4060 | \label{ref:function_populate_polygon} |
---|
4061 | \begin{funcdesc}{populate_polygon}{polygon, number_of_points, seed=None, exclude=None} |
---|
4062 | Module: \code{utilities.polygon} |
---|
4063 | |
---|
4064 | Populates the interior of the specified polygon with the specified number of points, |
---|
4065 | selected by means of a uniform distribution function. |
---|
4066 | |
---|
4067 | \code{polygon} is the polygon to populate. |
---|
4068 | |
---|
4069 | \code{number_of_points} is the (optional) number of points. |
---|
4070 | |
---|
4071 | \code{seed}is the optional seed for random number generator. |
---|
4072 | |
---|
4073 | \code{exclude} is a list of polygons (inside main polygon) from where points should be excluded. |
---|
4074 | \end{funcdesc} |
---|
4075 | |
---|
4076 | \label{ref:function_point_in_polygon} |
---|
4077 | \begin{funcdesc}{<point> = point_in_polygon}{polygon, delta=1e-8} |
---|
4078 | Module: \code{utilities.polygon} |
---|
4079 | |
---|
4080 | Returns a point inside the specified polygon and close to the edge. The distance between |
---|
4081 | the returned point and the nearest point of the polygon is less than $\sqrt{2}$ times the |
---|
4082 | second argument \code{delta}, which is taken as $10^{-8}$ by default. |
---|
4083 | \end{funcdesc} |
---|
4084 | |
---|
4085 | \label{ref:function_inside_polygon} |
---|
4086 | \begin{funcdesc}{<array> = inside_polygon}{points, polygon, closed=True, verbose=False} |
---|
4087 | Module: \code{utilities.polygon} |
---|
4088 | |
---|
4089 | Get a set of points that lie inside a given polygon. |
---|
4090 | |
---|
4091 | \code{points} is the list of points to test. |
---|
4092 | |
---|
4093 | \code{polygon} is the polygon to test the points against. |
---|
4094 | |
---|
4095 | \code{closed} specifies if points on the polygon edge are considered to be inside |
---|
4096 | or outside the polygon -- \code{True} means they are inside. |
---|
4097 | |
---|
4098 | Returns a numeric array comprising the indices of the points in the list |
---|
4099 | that lie inside the polygon. If none of the points are inside, returns |
---|
4100 | \code{zeros((0,), 'l') (ie, an empty numeric array)}. |
---|
4101 | |
---|
4102 | Compare with \code{outside_polygon()}, page \pageref{ref:function_outside_polygon}. |
---|
4103 | \end{funcdesc} |
---|
4104 | |
---|
4105 | \label{ref:function_outside_polygon} |
---|
4106 | \begin{funcdesc}{<array> = outside_polygon}{points, polygon, closed=True, verbose=False} |
---|
4107 | Module: \code{utilities.polygon} |
---|
4108 | |
---|
4109 | Get a set of points that lie outside a given polygon. |
---|
4110 | |
---|
4111 | \code{points} is the list of points to test. |
---|
4112 | |
---|
4113 | \code{polygon} is the polygon to test the points against. |
---|
4114 | |
---|
4115 | \code{closed} specifies if points on the polygon edge are considered to be outside |
---|
4116 | or inside the polygon -- \code{True} means they are outside. |
---|
4117 | |
---|
4118 | Returns a numeric array comprising the indices of the points in the list |
---|
4119 | that lie outside the polygon. If none of the points are outside, returns |
---|
4120 | \code{zeros((0,), 'l')} (ie, an empty numeric array). |
---|
4121 | |
---|
4122 | Compare with \code{inside_polygon()}, page \pageref{ref:function_inside_polygon}. |
---|
4123 | \end{funcdesc} |
---|
4124 | |
---|
4125 | \label{ref:function_is_inside_polygon} |
---|
4126 | \begin{funcdesc}{<boolean> = is_inside_polygon}{point, polygon, closed=True, verbose=False} |
---|
4127 | Module: \code{utilities.polygon} |
---|
4128 | |
---|
4129 | Determines if a single point is inside a polygon. |
---|
4130 | |
---|
4131 | \code{point} is the point to test. |
---|
4132 | |
---|
4133 | \code{polygon} is the polygon to test \code{point} against. |
---|
4134 | |
---|
4135 | \code{closed} is a flag that forces the function to consider a point on the polygon |
---|
4136 | edge to be inside or outside -- if \code{True} a point on the edge is considered inside the |
---|
4137 | polygon. |
---|
4138 | |
---|
4139 | Returns \code{True} if \code{point} is inside \code{polygon}. |
---|
4140 | |
---|
4141 | Compare with \code{inside_polygon()}, page \pageref{ref:function_inside_polygon}. |
---|
4142 | \end{funcdesc} |
---|
4143 | |
---|
4144 | \label{ref:function_is_outside_polygon} |
---|
4145 | \begin{funcdesc}{<boolean> = is_outside_polygon}{point, polygon, closed=True, verbose=False, |
---|
4146 | % points_geo_ref=None, polygon_geo_ref=None |
---|
4147 | } |
---|
4148 | Module: \code{utilities.polygon} |
---|
4149 | |
---|
4150 | Determines if a single point is outside a polygon. |
---|
4151 | |
---|
4152 | \code{point} is the point to test. |
---|
4153 | |
---|
4154 | \code{polygon} is the polygon to test \code{point} against. |
---|
4155 | |
---|
4156 | \code{closed} is a flag that forces the function to consider a point on the polygon |
---|
4157 | edge to be outside or inside -- if \code{True} a point on the edge is considered inside the |
---|
4158 | polygon. |
---|
4159 | |
---|
4160 | %\code{points_geo_ref} is ?? |
---|
4161 | % |
---|
4162 | %\code{polygon_geo_ref} is ?? |
---|
4163 | |
---|
4164 | Compare with \code{outside_polygon()}, page \pageref{ref:function_outside_polygon}. |
---|
4165 | \end{funcdesc} |
---|
4166 | |
---|
4167 | \label{ref:function_point_on_line} |
---|
4168 | \begin{funcdesc}{<boolean> = point_on_line}{point, line, rtol=1.0e-5, atol=1.0e-8} |
---|
4169 | Module: \code{utilities.polygon} |
---|
4170 | |
---|
4171 | Determine if a point is on a line to some tolerance. The line is considered to |
---|
4172 | extend past its end-points. |
---|
4173 | |
---|
4174 | \code{point} is the point to test ([$x$, $y$]). |
---|
4175 | |
---|
4176 | \code{line} is the line to test \code{point} against ([[$x1$,$y1$], [$x2$,$y2$]]). |
---|
4177 | |
---|
4178 | \code{rtol} is the relative tolerance to use when testing for coincidence. |
---|
4179 | |
---|
4180 | \code{atol} is the absolute tolerance to use when testing for coincidence. |
---|
4181 | |
---|
4182 | Returns \code{True} if the point is on the line, else \code{False}. |
---|
4183 | \end{funcdesc} |
---|
4184 | |
---|
4185 | \label{ref:function_separate_points_by_polygon} |
---|
4186 | \begin{funcdesc}{(indices, count) = separate_points_by_polygon}{points, polygon, |
---|
4187 | closed=True, |
---|
4188 | check_input=True, |
---|
4189 | verbose=False} |
---|
4190 | \indexedcode{separate_points_by_polygon} |
---|
4191 | Module: \code{utilities.polygon} |
---|
4192 | |
---|
4193 | Separate a set of points into points that are inside and outside a polygon. |
---|
4194 | |
---|
4195 | \code{points} is a list of points to separate. |
---|
4196 | |
---|
4197 | \code{polygon} is the polygon used to separate the points. |
---|
4198 | |
---|
4199 | \code{closed} determines whether points on the polygon edge should be |
---|
4200 | regarded as inside or outside the polygon. \code{True} means they are inside. |
---|
4201 | |
---|
4202 | \code{check_input} specifies whether the input parameters are checked -- \code{True} |
---|
4203 | means check the input parameters. |
---|
4204 | |
---|
4205 | The function returns a tuple \code{(indices, count)} where \code{indices} is a list of |
---|
4206 | point $indices$ from the input \code{points} list, with the indices of points inside the |
---|
4207 | polygon at the left and indices of points outside the polygon listed at the right. The |
---|
4208 | \code{count} value is the count (from the left) of the indices of the points $inside$ the |
---|
4209 | polygon. |
---|
4210 | \end{funcdesc} |
---|
4211 | |
---|
4212 | \begin{funcdesc}{<area> = polygon_area}{polygon} |
---|
4213 | Module: \code{utilities.polygon} |
---|
4214 | |
---|
4215 | Returns area of an arbitrary polygon (reference http://mathworld.wolfram.com/PolygonArea.html). |
---|
4216 | \end{funcdesc} |
---|
4217 | |
---|
4218 | \begin{funcdesc}{[$x_{min}$, $x_{max}$, $y_{min}$, $y_{max}$] = plot_polygons} |
---|
4219 | {polygons_points, style=None, |
---|
4220 | figname=None, label=None, verbose=False} |
---|
4221 | Module: \code{utilities.polygon} |
---|
4222 | |
---|
4223 | Plot a list of polygons to a file. |
---|
4224 | |
---|
4225 | \code{polygons_points} is a list of polygons to plot. |
---|
4226 | |
---|
4227 | \code{style} is a list of style strings to be applied to the corresponding polygon |
---|
4228 | in \code{polygons_points}. A polygon can be closed for plotting purposes by assigning |
---|
4229 | the style string 'line' to it in the appropriate place in the \code{style} list. |
---|
4230 | The default style is 'line'. |
---|
4231 | |
---|
4232 | \code{figname} is the path to the file to save the plot in. If not specified, use |
---|
4233 | \file{test_image.png}. |
---|
4234 | |
---|
4235 | The function returns a list containing the minimum and maximum of the points in all the |
---|
4236 | input polygons, i.e.\ \code{[$x_{min}$, $x_{max}$, $y_{min}$, $y_{max}$]}. |
---|
4237 | \end{funcdesc} |
---|
4238 | |
---|
4239 | |
---|
4240 | \pagebreak |
---|
4241 | \section{coordinate_transforms} |
---|
4242 | |
---|
4243 | |
---|
4244 | \pagebreak |
---|
4245 | \section{geospatial_data} |
---|
4246 | \label{sec:geospatial} |
---|
4247 | |
---|
4248 | This describes a class that represents arbitrary point data in UTM |
---|
4249 | coordinates along with named attribute values. |
---|
4250 | |
---|
4251 | %FIXME (Ole): This gives a LaTeX error |
---|
4252 | %\declaremodule{standard}{geospatial_data} |
---|
4253 | %\refmodindex{geospatial_data} |
---|
4254 | |
---|
4255 | \begin{classdesc}{Geospatial_data} |
---|
4256 | {data_points=None, |
---|
4257 | attributes=None, |
---|
4258 | geo_reference=None, |
---|
4259 | default_attribute_name=None, |
---|
4260 | file_name=None, |
---|
4261 | latitudes=None, |
---|
4262 | longitudes=None, |
---|
4263 | points_are_lats_longs=False, |
---|
4264 | max_read_lines=None, |
---|
4265 | load_file_now=True, |
---|
4266 | verbose=False} |
---|
4267 | Module: \code{geospatial_data.geospatial_data} |
---|
4268 | |
---|
4269 | This class is used to store a set of data points and associated |
---|
4270 | attributes, allowing these to be manipulated by methods defined for |
---|
4271 | the class. |
---|
4272 | |
---|
4273 | The data points are specified either by reading them from a NetCDF |
---|
4274 | or CSV file, identified through the parameter \code{file_name}, or |
---|
4275 | by providing their \code{x}- and \code{y}-coordinates in metres, |
---|
4276 | either as a sequence of 2-tuples of floats or as an $M \times 2$ |
---|
4277 | numeric array of floats, where $M$ is the number of points. |
---|
4278 | |
---|
4279 | Coordinates are interpreted relative to the origin specified by the |
---|
4280 | object \code{geo_reference}, which contains data indicating the UTM |
---|
4281 | zone, easting and northing. If \code{geo_reference} is not |
---|
4282 | specified, a default is used. |
---|
4283 | |
---|
4284 | Attributes are specified through the parameter \code{attributes}, |
---|
4285 | set either to a list or array of length $M$ or to a dictionary whose |
---|
4286 | keys are the attribute names and whose values are lists or arrays of |
---|
4287 | length $M$. One of the attributes may be specified as the default |
---|
4288 | attribute, by assigning its name to \code{default_attribute_name}. |
---|
4289 | If no value is specified, the default attribute is taken to be the |
---|
4290 | first one. |
---|
4291 | |
---|
4292 | Note that the \code{Geospatial_data} object currently reads entire datasets |
---|
4293 | into memory i.e.\ no memomry blocking takes place. |
---|
4294 | For this we refer to the \code{set_quantity()} method which will read PTS and CSV |
---|
4295 | files into \anuga using memory blocking allowing large files to be used. |
---|
4296 | \end{classdesc} |
---|
4297 | |
---|
4298 | \begin{methoddesc}{\emph{<Geospatial_data>}.import_points_file} |
---|
4299 | {file_name, delimiter=None, verbose=False} |
---|
4300 | Module: \code{geospatial_data.geospatial_data} |
---|
4301 | |
---|
4302 | Import a TXT, CSV or PTS points data file into a code{Geospatial_data} object. |
---|
4303 | |
---|
4304 | \code{file_name} is the path to a TXT, CSV or PTS points data file. |
---|
4305 | |
---|
4306 | \code{delimiter} is currently unused. |
---|
4307 | \end{methoddesc} |
---|
4308 | |
---|
4309 | \begin{methoddesc}{\emph{<Geospatial_data>}.export_points_file}{file_name, absolute=True, |
---|
4310 | as_lat_long=False, isSouthHemisphere=True} |
---|
4311 | Module: \code{geospatial_data.geospatial_data} |
---|
4312 | |
---|
4313 | Export a CSV or PTS points data file from a \code{Geospatial_data} object. |
---|
4314 | |
---|
4315 | \code{file_name} is the path to the CSV or PTS points file to write. |
---|
4316 | |
---|
4317 | \code{absolute} determines if the exported data is absolute or relative to the |
---|
4318 | \code{Geospatial_data} object geo_reference. If \code{True} the exported |
---|
4319 | data is absolute. |
---|
4320 | |
---|
4321 | \code{as_lat_long} exports the points data as latitudes and longitudes if \code{True}. |
---|
4322 | |
---|
4323 | \code{isSouthHemisphere} has effect only if \code{as_lat_long} is \code{True} and causes |
---|
4324 | latitude/longitude values to be for the southern (\code{True}) or northern hemispheres |
---|
4325 | (\code{False}). |
---|
4326 | \end{methoddesc} |
---|
4327 | |
---|
4328 | \begin{methoddesc}{points = \emph{<Geospatial_data>}.get_data_points} |
---|
4329 | {absolute=True, geo_reference=None, |
---|
4330 | as_lat_long=False, isSouthHemisphere=True} |
---|
4331 | Module: \code{geospatial_data.geospatial_data} |
---|
4332 | |
---|
4333 | Get the coordinates for all the data points as an $N \times 2$ array. |
---|
4334 | |
---|
4335 | \code{absolute} determines if the exported data is absolute or relative to the |
---|
4336 | \code{Geospatial_data} object geo_reference. If \code{True} the exported |
---|
4337 | data is absolute. |
---|
4338 | |
---|
4339 | \code{geo_reference} is the geo_reference the points are relative to, if supplied. |
---|
4340 | |
---|
4341 | \code{as_lat_long} exports the points data as latitudes and longitudes if \code{True}. |
---|
4342 | |
---|
4343 | \code{isSouthHemisphere} has effect only if \code{as_lat_long} is \code{True} and causes |
---|
4344 | latitude/longitude values to be for the southern (\code{True}) or northern hemispheres |
---|
4345 | (\code{False}). |
---|
4346 | \end{methoddesc} |
---|
4347 | |
---|
4348 | \begin{methoddesc}{\emph{<Geospatial_data>}.set_attributes}{attributes} |
---|
4349 | Module: \code{geospatial_data.geospatial_data} |
---|
4350 | |
---|
4351 | Set the attributes for a \code{Geospatial_data} object. |
---|
4352 | |
---|
4353 | \code{attributes} is the new value for the object's attributes. May be a dictionary or \code{None}. |
---|
4354 | \end{methoddesc} |
---|
4355 | |
---|
4356 | \begin{methoddesc}{atributes = \emph{<Geospatial_data>}.get_attributes}{attribute_name=None} |
---|
4357 | Module: \code{geospatial_data.geospatial_data} |
---|
4358 | |
---|
4359 | Get a named attribute from a \code{Geospatial_data} object. |
---|
4360 | |
---|
4361 | \code{attribute_name} is the name of the desired attribute. If \code{None}, return |
---|
4362 | the default attribute. |
---|
4363 | \end{methoddesc} |
---|
4364 | |
---|
4365 | \begin{methoddesc}{\emph{<Geospatial_data>}.get_all_attributes}{} |
---|
4366 | Module: \code{geospatial_data.geospatial_data} |
---|
4367 | |
---|
4368 | Get all attributes of a \code{Geospatial_data} object. |
---|
4369 | |
---|
4370 | Returns \code{None} or the attributes dictionary (which may be empty). |
---|
4371 | \end{methoddesc} |
---|
4372 | |
---|
4373 | \begin{methoddesc}{\emph{<Geospatial_data>}.set_default_attribute_name}{default_attribute_name} |
---|
4374 | Module: \code{geospatial_data.geospatial_data} |
---|
4375 | |
---|
4376 | Set the default attribute name of a \code{Geospatial_data} object. |
---|
4377 | |
---|
4378 | \code{default_attribute_name} is the new default attribute name. |
---|
4379 | \end{methoddesc} |
---|
4380 | |
---|
4381 | \begin{methoddesc}{\emph{<Geospatial_data>}.set_geo_reference}{geo_reference} |
---|
4382 | Module: \code{geospatial_data.geospatial_data} |
---|
4383 | |
---|
4384 | Set the internal geo_reference of a \code{Geospatial_data} object. |
---|
4385 | |
---|
4386 | \code{geo_reference} is the new internal geo_reference for the object. |
---|
4387 | If \code{None} will use the default geo_reference. |
---|
4388 | |
---|
4389 | If the \code{Geospatial_data} object already has an internal geo_reference |
---|
4390 | then the points data will be changed to use the new geo_reference. |
---|
4391 | \end{methoddesc} |
---|
4392 | |
---|
4393 | \begin{methoddesc}{\emph{<Geospatial_data>}.__add__}{other} |
---|
4394 | Module: \code{geospatial_data.geospatial_data} |
---|
4395 | |
---|
4396 | The \code{__add__()} method is defined so it is possible to add two |
---|
4397 | \code{Geospatial_data} objects. |
---|
4398 | \end{methoddesc} |
---|
4399 | |
---|
4400 | \label{ref:function_clip} |
---|
4401 | \begin{methoddesc}{geospatial = \emph{<Geospatial_data>}.clip}{polygon, closed=True, verbose=False} |
---|
4402 | Module: \code{geospatial_data.geospatial_data} |
---|
4403 | |
---|
4404 | Clip a \code{Geospatial_data} object with a polygon. |
---|
4405 | |
---|
4406 | \code{polygon} is the polygon to clip the \code{Geospatial_data} object with. |
---|
4407 | This may be a list of points, an $N \times 2$ array or a \code{Geospatial_data} |
---|
4408 | object. |
---|
4409 | |
---|
4410 | \code{closed} determines whether points on the \code{polygon} edge are inside (\code{True}) |
---|
4411 | or outside (\code{False}) the polygon. |
---|
4412 | |
---|
4413 | Returns a new \code{Geospatial_data} object representing points inside the |
---|
4414 | |
---|
4415 | Compare with \code{clip_outside()}, page \pageref{ref:function_clip_outside}. |
---|
4416 | specified polygon. |
---|
4417 | \end{methoddesc} |
---|
4418 | |
---|
4419 | \label{ref:function_clip_outside} |
---|
4420 | \begin{methoddesc}{geospatial = \emph{<Geospatial_data>}.clip_outside} |
---|
4421 | {polygon, closed=True, verbose=False} |
---|
4422 | Module: \code{geospatial_data.geospatial_data} |
---|
4423 | |
---|
4424 | Clip a \code{Geospatial_data} object with a polygon. |
---|
4425 | |
---|
4426 | \code{polygon} is the polygon to clip the \code{Geospatial_data} object with. |
---|
4427 | This may be a list of points, an $N \times 2$ array or a \code{Geospatial_data} |
---|
4428 | object. |
---|
4429 | |
---|
4430 | \code{closed} determines whether points on the \code{polygon} edge are inside (\code{True}) |
---|
4431 | or outside (\code{False}) the polygon. |
---|
4432 | |
---|
4433 | Returns a new \code{Geospatial_data} object representing points outside the |
---|
4434 | specified polygon. |
---|
4435 | |
---|
4436 | Compare with \code{clip()}, page \pageref{ref:function_clip}. |
---|
4437 | \end{methoddesc} |
---|
4438 | |
---|
4439 | \begin{methoddesc}{(g1, g2) = \emph{<Geospatial_data>}.split} |
---|
4440 | {factor=0.5, seed_num=None, verbose=False} |
---|
4441 | Module: \code{geospatial_data.geospatial_data} |
---|
4442 | |
---|
4443 | Split a \code{Geospatial_data} object into two objects of predetermined ratios. |
---|
4444 | |
---|
4445 | \code{factor} is the ratio of the size of the first returned object to the |
---|
4446 | original object. If '0.5' is supplied, the two resulting objects will be |
---|
4447 | of equal size. |
---|
4448 | |
---|
4449 | \code{seed_num}, if supplied, will be the random number generator seed used for |
---|
4450 | the split. |
---|
4451 | |
---|
4452 | Points of the two new geospatial_data object are selected RANDOMLY. |
---|
4453 | |
---|
4454 | Returns two geospatial_data objects that are disjoint sets of the original. |
---|
4455 | \end{methoddesc} |
---|
4456 | |
---|
4457 | \subsection{Miscellaneous Functions} |
---|
4458 | |
---|
4459 | The functions here are not \code{Geospatial_data} object methods, but are used with them. |
---|
4460 | |
---|
4461 | \begin{methoddesc}{X = find_optimal_smoothing_parameter} |
---|
4462 | {data_file, |
---|
4463 | alpha_list=None, |
---|
4464 | mesh_file=None, |
---|
4465 | boundary_poly=None, |
---|
4466 | mesh_resolution=100000, |
---|
4467 | north_boundary=None, |
---|
4468 | south_boundary=None, |
---|
4469 | east_boundary=None, |
---|
4470 | west_boundary=None, |
---|
4471 | plot_name='all_alphas', |
---|
4472 | split_factor=0.1, |
---|
4473 | seed_num=None, |
---|
4474 | cache=False, |
---|
4475 | verbose=False} |
---|
4476 | Module: \code{geospatial_data.geospatial_data} |
---|
4477 | |
---|
4478 | Calculate the minimum covariance from a set of points in a file. It does this |
---|
4479 | by removing a small random sample of points from \code{data_file} and creating |
---|
4480 | models with different alpha values from \code{alpha_list} and cross validates |
---|
4481 | the predicted value to the previously removed point data. Returns the |
---|
4482 | alpha value which has the smallest covariance. |
---|
4483 | |
---|
4484 | \code{data_file} is the input data file and must not contain points outside |
---|
4485 | the boundaries defined and is either a PTS, TXT or CSV file. |
---|
4486 | |
---|
4487 | \code{alpha_list} is the list of alpha values to use. |
---|
4488 | |
---|
4489 | \code{mesh_file} is the path to a mesh file to create (if supplied). |
---|
4490 | If \code{None} a mesh file will be created (named \file{temp.msh}). |
---|
4491 | NOTE: if there is a \code{mesh_resolution} defined or any boundaries are defined, |
---|
4492 | any input \code{mesh_file} value is ignored. |
---|
4493 | |
---|
4494 | \code{mesh_resolution} is the maximum area size for a triangle. |
---|
4495 | |
---|
4496 | \code{north_boundary}\\ |
---|
4497 | \code{south_boundary}\\ |
---|
4498 | \code{east_boundary}\\ |
---|
4499 | \code{west_boundary} are the boundary values to use. |
---|
4500 | |
---|
4501 | \code{plot_name} is the path name of the plot file to write. |
---|
4502 | |
---|
4503 | \code{seed_num} is the random number generator seed to use. |
---|
4504 | |
---|
4505 | The function returns a tuple \code{(min_covar, alpha)} where \code{min_covar} is |
---|
4506 | the minumum normalised covariance and \code{alpha} is the alpha value that |
---|
4507 | created it. A plot file is also written. |
---|
4508 | |
---|
4509 | This is an example of function usage: \nopagebreak |
---|
4510 | |
---|
4511 | \begin{verbatim} |
---|
4512 | convariance_value, alpha = $\backslash$ |
---|
4513 | find_optimal_smoothing_parameter(data_file=fileName, |
---|
4514 | alpha_list=[0.0001, 0.01, 1], |
---|
4515 | mesh_file=None, |
---|
4516 | mesh_resolution=3, |
---|
4517 | north_boundary=5, |
---|
4518 | south_boundary=-5, |
---|
4519 | east_boundary=5, |
---|
4520 | west_boundary=-5, |
---|
4521 | plot_name='all_alphas', |
---|
4522 | seed_num=100000, |
---|
4523 | verbose=False) |
---|
4524 | \end{verbatim} |
---|
4525 | |
---|
4526 | NOTE: The function will not work if the \code{data_file} extent is greater than the |
---|
4527 | \code{boundary_poly} polygon or any of the boundaries, e.g.\ \code{north_boundary}, etc. |
---|
4528 | \end{methoddesc} |
---|
4529 | |
---|
4530 | |
---|
4531 | \pagebreak |
---|
4532 | \section{Graphical Mesh Generator GUI} |
---|
4533 | The program \code{graphical_mesh_generator.py} in the \code{pmesh} module |
---|
4534 | allows the user to set up the mesh of the problem interactively. |
---|
4535 | It can be used to build the outline of a mesh or to visualise a mesh |
---|
4536 | automatically generated. |
---|
4537 | |
---|
4538 | Graphical Mesh Generator will let the user select various modes. The |
---|
4539 | current allowable modes are $vertex$, $segment$, $hole$ or $region$. The mode |
---|
4540 | describes what sort of object is added or selected in response to |
---|
4541 | mouse clicks. When changing modes any prior selected objects become |
---|
4542 | deselected. |
---|
4543 | |
---|
4544 | In general the left mouse button will add an object and the right |
---|
4545 | mouse button will select an object. A selected object can de deleted |
---|
4546 | by pressing the the middle mouse button (scroll bar). |
---|
4547 | |
---|
4548 | |
---|
4549 | \pagebreak |
---|
4550 | \section{class Alpha_Shape} |
---|
4551 | \emph{Alpha shapes} are used to generate close-fitting boundaries |
---|
4552 | around sets of points. The alpha shape algorithm produces a shape |
---|
4553 | that approximates to the 'shape formed by the points' -- or the shape |
---|
4554 | that would be seen by viewing the points from a coarse enough |
---|
4555 | resolution. For the simplest types of point sets, the alpha shape |
---|
4556 | reduces to the more precise notion of the convex hull. However, for |
---|
4557 | many sets of points the convex hull does not provide a close fit and |
---|
4558 | the alpha shape usually fits more closely to the original point set, |
---|
4559 | offering a better approximation to the shape being sought. |
---|
4560 | |
---|
4561 | In \anuga, an alpha shape is used to generate a polygonal boundary |
---|
4562 | around a set of points before mesh generation. The algorithm uses a |
---|
4563 | parameter $\alpha$ that can be adjusted to make the resultant shape |
---|
4564 | resemble the shape suggested by intuition more closely. An alpha |
---|
4565 | shape can serve as an initial boundary approximation that the user |
---|
4566 | can adjust as needed. |
---|
4567 | |
---|
4568 | The following paragraphs describe the class used to model an alpha |
---|
4569 | shape and some of the important methods and attributes associated |
---|
4570 | with instances of this class. |
---|
4571 | |
---|
4572 | \label{class:alpha_shape} |
---|
4573 | \begin{classdesc}{Alpha_Shape}{points, alpha=None} |
---|
4574 | Module: \code{alpha_shape} |
---|
4575 | |
---|
4576 | Instantiate an instance of the \code{Alpha_Shape} class. |
---|
4577 | |
---|
4578 | \code{points} is an $N \times 2$ list of points (\code{[[x1, y1],[x2, y2]}\ldots\code{]}). |
---|
4579 | |
---|
4580 | \code{alpha} is the 'fitting' parameter. |
---|
4581 | \end{classdesc} |
---|
4582 | |
---|
4583 | \begin{funcdesc}{alpha_shape_via_files}{point_file, boundary_file, alpha= None} |
---|
4584 | Module: \code{alpha_shape} |
---|
4585 | |
---|
4586 | This function reads points from the specified point file |
---|
4587 | \code{point_file}, computes the associated alpha shape (either |
---|
4588 | using the specified value for \code{alpha} or, if no value is |
---|
4589 | specified, automatically setting it to an optimal value) and outputs |
---|
4590 | the boundary to a file named \code{boundary_file}. This output file |
---|
4591 | lists the coordinates \code{(x, y)} of each point in the boundary, |
---|
4592 | using one line per point. |
---|
4593 | \end{funcdesc} |
---|
4594 | |
---|
4595 | \label{ref:method_set_boundary_type} |
---|
4596 | \begin{methoddesc}{\emph{<Alpha_shape>}.set_boundary_type} |
---|
4597 | {raw_boundary=True, |
---|
4598 | remove_holes=False, |
---|
4599 | smooth_indents=False, |
---|
4600 | expand_pinch=False, |
---|
4601 | boundary_points_fraction=0.2} |
---|
4602 | Module: \code{alpha_shape} |
---|
4603 | |
---|
4604 | This function sets internal state that controls how the \code{Alpha_shape} |
---|
4605 | boundary is presented or exported. |
---|
4606 | |
---|
4607 | \code{raw_boundary} sets the type to $raw$ if \code{True}, |
---|
4608 | i.e.\ the regular edges of the alpha shape. |
---|
4609 | |
---|
4610 | \code{remove_holes}, if \code{True} removes small holes ('small' is defined by |
---|
4611 | \code{boundary_points_fraction}). |
---|
4612 | |
---|
4613 | \code{smooth_indents}, if \code{True} removes sharp triangular indents in |
---|
4614 | the boundary. |
---|
4615 | |
---|
4616 | \code{expand_pinch}, if \code{True} tests for pinch-off and |
---|
4617 | corrects -- preventing a boundary vertex from having more than two edges. |
---|
4618 | \end{methoddesc} |
---|
4619 | |
---|
4620 | \label{ref:method_get_boundary} |
---|
4621 | \begin{methoddesc}{boundary = \emph{<Alpha_shape>}.get_boundary}{} |
---|
4622 | Module: \code{alpha_shape} |
---|
4623 | |
---|
4624 | Returns a list of tuples representing the boundary of the alpha |
---|
4625 | shape. Each tuple represents a segment in the boundary by providing |
---|
4626 | the indices of its two endpoints. |
---|
4627 | |
---|
4628 | See \code{set_boundary_type()}, page \pageref{ref:method_set_boundary_type}. |
---|
4629 | \end{methoddesc} |
---|
4630 | |
---|
4631 | \label{ref:method_write_boundary} |
---|
4632 | \begin{methoddesc}{\emph{<Alpha_shape>}.write_boundary}{file_name} |
---|
4633 | Module: \code{alpha_shape} |
---|
4634 | |
---|
4635 | Writes the list of 2-tuples returned by \code{get_boundary()} to the |
---|
4636 | file \code{file_name}, using one line per tuple. |
---|
4637 | |
---|
4638 | See \code{set_boundary_type()}, page \pageref{ref:method_set_boundary_type}. \\ |
---|
4639 | See \code{get_boundary()}, page \pageref{ref:method_get_boundary}. |
---|
4640 | \end{methoddesc} |
---|
4641 | |
---|
4642 | |
---|
4643 | \pagebreak |
---|
4644 | \section{Numerical Tools} |
---|
4645 | |
---|
4646 | The following table describes some useful numerical functions that |
---|
4647 | may be found in the module \module{utilities.numerical\_tools}: |
---|
4648 | |
---|
4649 | \begin{tabular}{|p{7.4cm} p{8.6cm}|} |
---|
4650 | \hline |
---|
4651 | \code{angle(v1, v2=None)} & Angle between two-dimensional vectors |
---|
4652 | \code{v1} and \code{v2}, or between \code{v1} and the $x$-axis if |
---|
4653 | \code{v2} is \code{None}. Value is in range $0$ to $2\pi$. \\ |
---|
4654 | & \\ |
---|
4655 | \code{normal\_vector(v)} & Normal vector to \code{v}.\\ |
---|
4656 | & \\ |
---|
4657 | \code{mean(x)} & Mean value of a vector \code{x}.\\ |
---|
4658 | & \\ |
---|
4659 | \code{cov(x, y=None)} & Covariance of vectors \code{x} and \code{y}. |
---|
4660 | If \code{y} is \code{None}, returns \code{cov(x, x)}.\\ |
---|
4661 | & \\ |
---|
4662 | \code{err(x, y=0, n=2, relative=True)} & Relative error of $\parallel$\code{x}$-$\code{y}$\parallel$ |
---|
4663 | to $\parallel$\code{y}$\parallel$ (2-norm if \code{n} = 2 or |
---|
4664 | Max norm if \code{n} = \code{None}). If denominator evaluates |
---|
4665 | to zero or if \code{y} is omitted or if \code{relative=False}, |
---|
4666 | absolute error is returned.\\ |
---|
4667 | & \\ |
---|
4668 | \code{norm(x)} & 2-norm of \code{x}.\\ |
---|
4669 | & \\ |
---|
4670 | \code{corr(x, y=None)} & Correlation of \code{x} and \code{y}. If |
---|
4671 | \code{y} is \code{None} returns autocorrelation of \code{x}.\\ |
---|
4672 | & \\ |
---|
4673 | \code{ensure\_numeric(A, typecode=None)} & Returns a numeric array for any sequence \code{A}. If |
---|
4674 | \code{A} is already a numeric array it will be returned |
---|
4675 | unaltered. Otherwise, an attempt is made to convert |
---|
4676 | it to a numeric array. (Needed because \code{array(A)} can |
---|
4677 | cause memory overflow.)\\ |
---|
4678 | & \\ |
---|
4679 | \code{histogram(a, bins, relative=False)} & Standard histogram. If \code{relative} is \code{True}, |
---|
4680 | values will be normalised against the total and thus |
---|
4681 | represent frequencies rather than counts.\\ |
---|
4682 | & \\ |
---|
4683 | \code{create\_bins(data, number\_of\_bins=None)} & Safely create bins for use with histogram. |
---|
4684 | If \code{data} contains only one point |
---|
4685 | or is constant, one bin will be created. |
---|
4686 | If \code{number\_of\_bins} is omitted, |
---|
4687 | 10 bins will be created.\\ |
---|
4688 | \hline |
---|
4689 | \end{tabular} |
---|
4690 | |
---|
4691 | |
---|
4692 | \section{Finding the Optimal Alpha Value} |
---|
4693 | |
---|
4694 | The function ???? |
---|
4695 | more to come very soon |
---|
4696 | |
---|
4697 | %?% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
---|
4698 | %?% |
---|
4699 | %?% \chapter{Modules available in \anuga} |
---|
4700 | %?% |
---|
4701 | %?% |
---|
4702 | %?% %abstract_2d_finite_volumes |
---|
4703 | %?% |
---|
4704 | %?% \section{\module{abstract_2d_finite_volumes.domain}} |
---|
4705 | %?% Generic module for 2D triangular domains for finite-volume computations of conservation laws |
---|
4706 | %?% \declaremodule[domain]{}{domain} |
---|
4707 | %?% \label{mod:domain} |
---|
4708 | %?% |
---|
4709 | %?% |
---|
4710 | %?% \section{\module{abstract_2d_finite_volumes.ermapper_grids}} |
---|
4711 | %?% \declaremodule[ermappergrids]{}{ermapper_grids} |
---|
4712 | %?% \label{mod:ermapper_grids} |
---|
4713 | %?% |
---|
4714 | %?% |
---|
4715 | %?% \section{\module{abstract_2d_finite_volumes.general_mesh} } |
---|
4716 | %?% \declaremodule[generalmesh]{}{general_mesh} |
---|
4717 | %?% \label{mod:general_mesh} |
---|
4718 | %?% |
---|
4719 | %?% |
---|
4720 | %?% \section{\module{abstract_2d_finite_volumes.generic_boundary_conditions} } |
---|
4721 | %?% \declaremodule[genericboundaryconditions]{}{generic_boundary_conditions} |
---|
4722 | %?% \label{mod:generic_boundary_conditions} |
---|
4723 | %?% |
---|
4724 | %?% |
---|
4725 | %?% \section{\module{abstract_2d_finite_volumes.mesh_factory} } |
---|
4726 | %?% \declaremodule[meshfactory]{}{mesh_factory} |
---|
4727 | %?% \label{mod:mesh_factory} |
---|
4728 | %?% |
---|
4729 | %?% |
---|
4730 | %?% \section{\module{abstract_2d_finite_volumes.mesh_factory} } |
---|
4731 | %?% \declaremodule[meshfactory]{}{mesh_factory} |
---|
4732 | %?% \label{mod:mesh_factory} |
---|
4733 | %?% |
---|
4734 | %?% |
---|
4735 | %?% \section{\module{abstract_2d_finite_volumes.neighbour_mesh} } |
---|
4736 | %?% \declaremodule[neighbourmesh]{}{neighbour_mesh} |
---|
4737 | %?% \label{mod:neighbour_mesh} |
---|
4738 | %?% |
---|
4739 | %?% |
---|
4740 | %?% \section{\module{abstract_2d_finite_volumes.pmesh2domain} } |
---|
4741 | %?% \declaremodule[pmesh2domain]{}{pmesh2domain} |
---|
4742 | %?% \label{mod:pmesh2domain} |
---|
4743 | %?% |
---|
4744 | %?% |
---|
4745 | %?% \section{\module{abstract_2d_finite_volumes.quantity}} |
---|
4746 | %?% \declaremodule{}{quantity} |
---|
4747 | %?% \label{mod:quantity} |
---|
4748 | %?% |
---|
4749 | %?% \begin{verbatim} |
---|
4750 | %?% Class Quantity - Implements values at each triangular element |
---|
4751 | %?% |
---|
4752 | %?% To create: |
---|
4753 | %?% |
---|
4754 | %?% Quantity(domain, vertex_values) |
---|
4755 | %?% |
---|
4756 | %?% domain: Associated domain structure. Required. |
---|
4757 | %?% |
---|
4758 | %?% vertex_values: Nx3 array of values at each vertex for each element. |
---|
4759 | %?% Default None |
---|
4760 | %?% |
---|
4761 | %?% If vertex_values are None Create array of zeros compatible with domain. |
---|
4762 | %?% Otherwise check that it is compatible with dimenions of domain. |
---|
4763 | %?% Otherwise raise an exception |
---|
4764 | %?% \end{verbatim} |
---|
4765 | %?% |
---|
4766 | %?% |
---|
4767 | %?% \section{\module{abstract_2d_finite_volumes.region} } |
---|
4768 | %?% \declaremodule[region]{}{region} |
---|
4769 | %?% \label{mod:region} |
---|
4770 | %?% |
---|
4771 | %?% |
---|
4772 | %?% \section{\module{abstract_2d_finite_volumes.util} } |
---|
4773 | %?% \declaremodule[util]{}{util} |
---|
4774 | %?% \label{mod:util} |
---|
4775 | %?% |
---|
4776 | %?% |
---|
4777 | %?% advection |
---|
4778 | %?% alpha_shape |
---|
4779 | %?% caching |
---|
4780 | %?% coordinate_transforms |
---|
4781 | %?% culvert_flows |
---|
4782 | %?% damage_modelling |
---|
4783 | %?% euler |
---|
4784 | %?% fit_interpolate |
---|
4785 | %?% geospatial_data |
---|
4786 | %?% lib |
---|
4787 | %?% load_mesh |
---|
4788 | %?% mesh_engine |
---|
4789 | %?% pmesh |
---|
4790 | %?% SConstruct |
---|
4791 | %?% shallow_water |
---|
4792 | %?% utilities |
---|
4793 | %?% |
---|
4794 | %?% |
---|
4795 | %?% \section{\module{shallow\_water}} |
---|
4796 | %?% |
---|
4797 | %?% 2D triangular domains for finite-volume |
---|
4798 | %?% computations of the shallow water wave equation. |
---|
4799 | %?% This module contains a specialisation of class Domain from module domain.py consisting of methods specific to the Shallow Water |
---|
4800 | %?% Wave Equation |
---|
4801 | %?% |
---|
4802 | %?% \declaremodule[shallowwater]{}{shallow\_water} |
---|
4803 | %?% \label{mod:shallowwater} |
---|
4804 | |
---|
4805 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
---|
4806 | |
---|
4807 | \chapter{\anuga Full-scale Validations} |
---|
4808 | |
---|
4809 | |
---|
4810 | \section{Overview} |
---|
4811 | |
---|
4812 | There are some long-running validations that are not included in the small-scale |
---|
4813 | validations that run when you execute the \module{validate_all.py} |
---|
4814 | script in the \module{anuga_validation/automated_validation_test} directory. |
---|
4815 | These validations are not run automatically since they can take a large amount |
---|
4816 | of time and require an internet connection and user input. |
---|
4817 | |
---|
4818 | |
---|
4819 | \section{Patong Beach} |
---|
4820 | |
---|
4821 | The Patong Beach validation is run from the \module{automated_validation_tests/patong_beach_validation} |
---|
4822 | directory. Just execute the \module{validate_patong.py} script in that directory. |
---|
4823 | This will run a Patong Beach simulation and compare the generated SWW file with a |
---|
4824 | known good copy of that file. |
---|
4825 | |
---|
4826 | The script attempts to refresh the validation data files from master copies held |
---|
4827 | on the Sourceforge project site. If you don't have an internet connection you |
---|
4828 | may still run the validation, as long as you have the required files. |
---|
4829 | |
---|
4830 | You may download the validation data files by hand and then run the validation. |
---|
4831 | Just go to the \anuga Sourceforge project download page at |
---|
4832 | \module{http://sourceforge.net/project/showfiles.php?group_id=172848} and select |
---|
4833 | the \module{validation_data} package, \module{patong-1.0} release. You need the |
---|
4834 | \module{data.tgz} file and one or more of the \module{patong.sww.\{BASIC|TRIAL|FINAL\}.tgz} |
---|
4835 | files. |
---|
4836 | |
---|
4837 | The BASIC validation is the quickest and the FINAL validation is the slowest. |
---|
4838 | The \module{validate.py} script will use whatever files you have, BASIC first and |
---|
4839 | FINAL last. |
---|
4840 | |
---|
4841 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
---|
4842 | |
---|
4843 | \chapter{Frequently Asked Questions} |
---|
4844 | |
---|
4845 | The Frequently Asked Questions have been move to the online FAQ at: \\ |
---|
4846 | \url{https://datamining.anu.edu.au/anuga/wiki/FrequentlyAskedQuestions} |
---|
4847 | |
---|
4848 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
---|
4849 | |
---|
4850 | \chapter{Glossary} |
---|
4851 | |
---|
4852 | \begin{tabular}{|lp{10cm}|c|} \hline |
---|
4853 | \emph{Term} & \emph{Definition} & \emph{Page}\\ |
---|
4854 | \hline |
---|
4855 | \indexedbold{\anuga} & Name of software (joint development between ANU and GA) & \pageref{def:anuga}\\ |
---|
4856 | \indexedbold{bathymetry} & offshore elevation & \\ |
---|
4857 | \indexedbold{conserved quantity} & conserved (stage, x and y momentum) & \\ |
---|
4858 | % \indexedbold{domain} & The domain of a function is the set of all input values to the function.& \\ |
---|
4859 | \indexedbold{Digital Elevation Model (DEM)} & DEMs are digital files consisting of points of elevations, |
---|
4860 | sampled systematically at equally spaced intervals.& \\ |
---|
4861 | \indexedbold{Dirichlet boundary} & A boundary condition imposed on a differential equation that specifies |
---|
4862 | the values the solution is to take on the boundary of the domain. |
---|
4863 | & \pageref{def:dirichlet boundary}\\ |
---|
4864 | \indexedbold{edge} & A triangular cell within the computational mesh can be depicted |
---|
4865 | as a set of vertices joined by lines (the edges). & \\ |
---|
4866 | \indexedbold{elevation} & refers to bathymetry and topography & \\ |
---|
4867 | \indexedbold{evolution} & integration of the shallow water wave equations over time & \\ |
---|
4868 | \indexedbold{finite volume method} & The method evaluates the terms in the shallow water wave equation as |
---|
4869 | fluxes at the surfaces of each finite volume. Because the flux entering |
---|
4870 | a given volume is identical to that leaving the adjacent volume, these |
---|
4871 | methods are conservative. Another advantage of the finite volume method is |
---|
4872 | that it is easily formulated to allow for unstructured meshes. The method |
---|
4873 | is used in many computational fluid dynamics packages. & \\ |
---|
4874 | \indexedbold{forcing term} & &\\ |
---|
4875 | \indexedbold{flux} & the amount of flow through the volume per unit time & \\ |
---|
4876 | \indexedbold{grid} & Evenly spaced mesh & \\ |
---|
4877 | \indexedbold{latitude} & The angular distance on a mericlear north and south of the equator, expressed in degrees and minutes. & \\ |
---|
4878 | \indexedbold{longitude} & The angular distance east or west, between the meridian of a particular place on Earth |
---|
4879 | and that of the Prime Meridian (located in Greenwich, England) expressed in degrees or time.& \\ |
---|
4880 | \indexedbold{Manning friction coefficient} & &\\ |
---|
4881 | \indexedbold{mesh} & Triangulation of domain &\\ |
---|
4882 | \indexedbold{mesh file} & A TSH or MSH file & \pageref{def:mesh file}\\ |
---|
4883 | \indexedbold{NetCDF} & &\\ |
---|
4884 | \indexedbold{node} & A point at which edges meet & \\ |
---|
4885 | \indexedbold{northing} & A rectangular (x,y) coordinate measurement of distance north from a north-south |
---|
4886 | reference line, usually a meridian used as the axis of origin within a map zone |
---|
4887 | or projection. Northing is a UTM (Universal Transverse Mercator) coordinate. & \\ |
---|
4888 | \indexedbold{points file} & A PTS or CSV file & \\ |
---|
4889 | \hline |
---|
4890 | \end{tabular} |
---|
4891 | |
---|
4892 | \begin{tabular}{|lp{10cm}|c|} |
---|
4893 | \hline |
---|
4894 | \indexedbold{polygon} & A sequence of points in the plane. \anuga represents a polygon either as a list |
---|
4895 | consisting of Python tuples or lists of length 2 or as an $N \times 2$ numeric array, |
---|
4896 | where $N$ is the number of points. |
---|
4897 | |
---|
4898 | The unit square, for example, would be represented either as \code{[ [0,0], [1,0], [1,1], [0,1] ]} |
---|
4899 | or as \code{array( [0,0], [1,0], [1,1], [0,1] )}. |
---|
4900 | |
---|
4901 | NOTE: For details refer to the module \module{utilities/polygon.py}. & \\ |
---|
4902 | \indexedbold{resolution} & The maximal area of a triangular cell in a mesh & \\ |
---|
4903 | \indexedbold{reflective boundary} & Models a solid wall. Returns same conserved quantities as those present in the |
---|
4904 | neighbouring volume but reflected. Specific to the shallow water equation as |
---|
4905 | it works with the momentum quantities assumed to be the second and third |
---|
4906 | conserved quantities. & \pageref{def:reflective boundary}\\ |
---|
4907 | \indexedbold{stage} & &\\ |
---|
4908 | % \indexedbold{try this} |
---|
4909 | \indexedbold{animate} & visualisation tool used with \anuga & \pageref{sec:animate}\\ |
---|
4910 | \indexedbold{time boundary} & Returns values for the conserved quantities as a function of time. |
---|
4911 | The user must specify the domain to get access to the model time. |
---|
4912 | & \pageref{def:time boundary}\\ |
---|
4913 | \indexedbold{topography} & onshore elevation &\\ |
---|
4914 | \indexedbold{transmissive boundary} & & \pageref{def:transmissive boundary}\\ |
---|
4915 | \indexedbold{vertex} & A point at which edges meet. & \\ |
---|
4916 | \indexedbold{xmomentum} & conserved quantity (note, two-dimensional SWW equations say only |
---|
4917 | \code{x} and \code{y} and NOT \code{z}) &\\ |
---|
4918 | \indexedbold{ymomentum} & conserved quantity & \\ |
---|
4919 | \hline |
---|
4920 | \end{tabular} |
---|
4921 | |
---|
4922 | %The \code{\e appendix} markup need not be repeated for additional |
---|
4923 | %appendices. |
---|
4924 | |
---|
4925 | % |
---|
4926 | % The ugly "%begin{latexonly}" pseudo-environments are really just to |
---|
4927 | % keep LaTeX2HTML quiet during the \renewcommand{} macros; they're |
---|
4928 | % not really valuable. |
---|
4929 | % |
---|
4930 | % If you don't want the Module Index, you can remove all of this up |
---|
4931 | % until the second \input line. |
---|
4932 | % |
---|
4933 | |
---|
4934 | %?% %begin{latexonly} |
---|
4935 | %?% %\renewcommand{\indexname}{Module Index} |
---|
4936 | %?% %end{latexonly} |
---|
4937 | %?% \input{mod\jobname.ind} % Module Index |
---|
4938 | |
---|
4939 | %begin{latexonly} |
---|
4940 | %\renewcommand{\indexname}{Index} |
---|
4941 | %end{latexonly} |
---|
4942 | \input{\jobname.ind} % Index |
---|
4943 | |
---|
4944 | |
---|
4945 | \begin{thebibliography}{99} |
---|
4946 | %\begin{thebibliography} |
---|
4947 | \bibitem[nielsen2005]{nielsen2005} |
---|
4948 | {\it Hydrodynamic modelling of coastal inundation}. |
---|
4949 | Nielsen, O., S. Roberts, D. Gray, A. McPherson and A. Hitchman. |
---|
4950 | In Zerger, A. and Argent, R.M. (eds) MODSIM 2005 International Congress on |
---|
4951 | Modelling and Simulation. Modelling and Simulation Society of Australia and |
---|
4952 | New Zealand, December 2005, pp. 518-523. ISBN: 0-9758400-2-9.\\ |
---|
4953 | http://www.mssanz.org.au/modsim05/papers/nielsen.pdf |
---|
4954 | |
---|
4955 | \bibitem[grid250]{grid250} |
---|
4956 | Australian Bathymetry and Topography Grid, June 2005. |
---|
4957 | Webster, M.A. and Petkovic, P. |
---|
4958 | Geoscience Australia Record 2005/12. ISBN: 1-920871-46-2.\\ |
---|
4959 | http://www.ga.gov.au/meta/ANZCW0703008022.html |
---|
4960 | |
---|
4961 | \bibitem[ZR1999]{ZR1999} |
---|
4962 | \newblock {Catastrophic Collapse of Water Supply Reservoirs in Urban Areas}. |
---|
4963 | \newblock C.~Zoppou and S.~Roberts. |
---|
4964 | \newblock {\em ASCE J. Hydraulic Engineering}, 125(7):686--695, 1999. |
---|
4965 | |
---|
4966 | \bibitem[Toro1999]{Toro1992} |
---|
4967 | \newblock Riemann problems and the waf method for solving the two-dimensional |
---|
4968 | shallow water equations. |
---|
4969 | \newblock E.~F. Toro. |
---|
4970 | \newblock {\em Philosophical Transactions of the Royal Society, Series A}, |
---|
4971 | 338:43--68, 1992. |
---|
4972 | |
---|
4973 | \bibitem{KurNP2001} |
---|
4974 | \newblock Semidiscrete central-upwind schemes for hyperbolic conservation laws |
---|
4975 | and hamilton-jacobi equations. |
---|
4976 | \newblock A.~Kurganov, S.~Noelle, and G.~Petrova. |
---|
4977 | \newblock {\em SIAM Journal of Scientific Computing}, 23(3):707--740, 2001. |
---|
4978 | \end{thebibliography} |
---|
4979 | % \end{thebibliography}{99} |
---|
4980 | |
---|
4981 | \end{document} |
---|