1 | """Class Domain - |
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2 | 1D interval domains for finite-volume computations of |
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3 | the shallow water wave equation. |
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4 | |
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5 | This module contains a specialisation of class Domain from module domain.py |
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6 | consisting of methods specific to the Shallow Water Wave Equation |
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7 | |
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8 | |
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9 | U_t + E_x = S |
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10 | |
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11 | where |
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12 | |
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13 | U = [w, uh] |
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14 | E = [uh, u^2h + gh^2/2] |
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15 | S represents source terms forcing the system |
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16 | (e.g. gravity, friction, wind stress, ...) |
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17 | |
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18 | and _t, _x, _y denote the derivative with respect to t, x and y respectiely. |
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19 | |
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20 | The quantities are |
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21 | |
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22 | symbol variable name explanation |
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23 | x x horizontal distance from origin [m] |
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24 | z elevation elevation of bed on which flow is modelled [m] |
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25 | h height water height above z [m] |
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26 | w stage absolute water level, w = z+h [m] |
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27 | u speed in the x direction [m/s] |
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28 | uh xmomentum momentum in the x direction [m^2/s] |
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29 | |
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30 | eta mannings friction coefficient [to appear] |
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31 | nu wind stress coefficient [to appear] |
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32 | |
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33 | The conserved quantities are w, uh |
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34 | |
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35 | For details see e.g. |
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36 | Christopher Zoppou and Stephen Roberts, |
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37 | Catastrophic Collapse of Water Supply Reservoirs in Urban Areas, |
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38 | Journal of Hydraulic Engineering, vol. 127, No. 7 July 1999 |
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39 | |
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40 | |
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41 | John Jakeman, Ole Nielsen, Stephen Roberts, Duncan Gray, Christopher Zoppou |
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42 | Geoscience Australia, 2006 |
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43 | """ |
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44 | |
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45 | import numpy |
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46 | |
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47 | from flow_1d.generic_1d.generic_domain import Generic_domain |
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48 | from sww_boundary_conditions import * |
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49 | from sww_forcing_terms import * |
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50 | |
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51 | #Shallow water domain |
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52 | class Domain(Generic_domain): |
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53 | |
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54 | def __init__(self, coordinates, boundary = None, tagged_elements = None): |
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55 | |
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56 | conserved_quantities = ['stage', 'xmomentum'] |
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57 | evolved_quantities = ['stage', 'xmomentum', 'elevation', 'height', 'velocity'] |
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58 | other_quantities = ['friction'] |
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59 | Generic_domain.__init__(self, |
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60 | coordinates = coordinates, |
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61 | boundary = boundary, |
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62 | conserved_quantities = conserved_quantities, |
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63 | evolved_quantities = evolved_quantities, |
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64 | other_quantities = other_quantities, |
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65 | tagged_elements = tagged_elements) |
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66 | |
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67 | from flow_1d.config import minimum_allowed_height, g, h0 |
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68 | self.minimum_allowed_height = minimum_allowed_height |
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69 | self.g = g |
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70 | self.h0 = h0 |
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71 | |
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72 | #forcing terms |
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73 | self.forcing_terms.append(gravity) |
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74 | #self.forcing_terms.append(manning_friction) |
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75 | |
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76 | |
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77 | #Stored output |
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78 | self.store = True |
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79 | self.format = 'sww' |
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80 | self.smooth = True |
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81 | |
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82 | #Evolve parametrs |
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83 | self.cfl = 1.0 |
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84 | |
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85 | #Reduction operation for get_vertex_values |
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86 | from flow_1d.utilities.util import mean |
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87 | self.reduction = mean |
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88 | #self.reduction = min #Looks better near steep slopes |
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89 | |
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90 | self.quantities_to_be_stored = ['stage','xmomentum'] |
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91 | |
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92 | self.__doc__ = 'shallow_water_domain' |
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93 | |
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94 | |
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95 | def set_quantities_to_be_stored(self, q): |
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96 | """Specify which quantities will be stored in the sww file. |
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97 | |
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98 | q must be either: |
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99 | - the name of a quantity |
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100 | - a list of quantity names |
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101 | - None |
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102 | |
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103 | In the two first cases, the named quantities will be stored at each |
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104 | yieldstep |
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105 | (This is in addition to the quantities elevation and friction) |
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106 | If q is None, storage will be switched off altogether. |
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107 | """ |
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108 | |
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109 | |
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110 | if q is None: |
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111 | self.quantities_to_be_stored = [] |
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112 | self.store = False |
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113 | return |
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114 | |
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115 | if isinstance(q, basestring): |
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116 | q = [q] # Turn argument into a list |
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117 | |
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118 | #Check correcness |
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119 | for quantity_name in q: |
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120 | msg = 'Quantity %s is not a valid conserved quantity' %quantity_name |
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121 | assert quantity_name in self.conserved_quantities, msg |
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122 | |
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123 | self.quantities_to_be_stored = q |
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124 | |
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125 | |
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126 | |
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127 | def check_integrity(self): |
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128 | Generic_Domain.check_integrity(self) |
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129 | #Check that we are solving the shallow water wave equation |
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130 | |
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131 | msg = 'First conserved quantity must be "stage"' |
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132 | assert self.conserved_quantities[0] == 'stage', msg |
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133 | msg = 'Second conserved quantity must be "xmomentum"' |
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134 | assert self.conserved_quantities[1] == 'xmomentum', msg |
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135 | |
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136 | |
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137 | |
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138 | def compute_fluxes(self): |
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139 | #Call correct module function |
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140 | #(either from this module or C-extension) |
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141 | |
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142 | import sys |
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143 | |
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144 | |
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145 | timestep = float(sys.maxint) |
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146 | |
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147 | stage = self.quantities['stage'] |
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148 | xmom = self.quantities['xmomentum'] |
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149 | bed = self.quantities['elevation'] |
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150 | height = self.quantities['height'] |
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151 | velocity = self.quantities['velocity'] |
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152 | |
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153 | |
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154 | from sww_comp_flux_ext import compute_fluxes_ext_short |
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155 | |
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156 | #self.flux_timestep = compute_fluxes_ext(timestep,self,stage,xmom,bed,height,velocity) |
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157 | |
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158 | |
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159 | self.flux_timestep = compute_fluxes_ext_short(timestep,self,stage,xmom,bed) |
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160 | |
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161 | |
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162 | |
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163 | def distribute_to_vertices_and_edges(self): |
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164 | #Call correct module function |
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165 | #(either from this module or C-extension) |
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166 | distribute_to_vertices_and_edges(self) |
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167 | |
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168 | def evolve(self, yieldstep = None, finaltime = None, duration = None, |
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169 | skip_initial_step = False): |
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170 | """Specialisation of basic evolve method from parent class |
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171 | """ |
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172 | |
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173 | #Call basic machinery from parent class |
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174 | for t in Generic_domain.evolve(self, yieldstep, finaltime,duration, |
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175 | skip_initial_step): |
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176 | |
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177 | #Pass control on to outer loop for more specific actions |
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178 | yield(t) |
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179 | |
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180 | def initialise_storage(self): |
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181 | """Create and initialise self.writer object for storing data. |
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182 | Also, save x and bed elevation |
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183 | """ |
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184 | |
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185 | import data_manager |
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186 | |
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187 | #Initialise writer |
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188 | self.writer = data_manager.get_dataobject(self, mode = 'w') |
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189 | |
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190 | #Store vertices and connectivity |
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191 | self.writer.store_connectivity() |
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192 | |
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193 | |
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194 | def store_timestep(self, name): |
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195 | """Store named quantity and time. |
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196 | |
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197 | Precondition: |
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198 | self.write has been initialised |
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199 | """ |
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200 | self.writer.store_timestep(name) |
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201 | |
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202 | |
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203 | #=============== End of Shallow Water Domain =============================== |
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204 | |
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205 | |
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206 | # Module functions for gradient limiting (distribute_to_vertices_and_edges) |
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207 | |
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208 | def distribute_to_vertices_and_edges(domain): |
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209 | """Distribution from centroids to vertices specific to the |
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210 | shallow water wave |
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211 | equation. |
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212 | |
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213 | It will ensure that h (w-z) is always non-negative even in the |
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214 | presence of steep bed-slopes by taking a weighted average between shallow |
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215 | and deep cases. |
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216 | |
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217 | In addition, all conserved quantities get distributed as per either a |
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218 | constant (order==1) or a piecewise linear function (order==2). |
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219 | |
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220 | FIXME: more explanation about removal of artificial variability etc |
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221 | |
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222 | Precondition: |
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223 | All quantities defined at centroids and bed elevation defined at |
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224 | vertices. |
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225 | |
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226 | Postcondition |
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227 | Conserved quantities defined at vertices |
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228 | |
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229 | """ |
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230 | |
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231 | #from config import optimised_gradient_limiter |
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232 | |
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233 | #Remove very thin layers of water |
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234 | #protect_against_infinitesimal_and_negative_heights(domain) |
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235 | |
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236 | import sys |
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237 | from config import epsilon, h0 |
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238 | |
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239 | N = domain.number_of_elements |
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240 | |
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241 | #Shortcuts |
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242 | Stage = domain.quantities['stage'] |
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243 | Xmom = domain.quantities['xmomentum'] |
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244 | Bed = domain.quantities['elevation'] |
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245 | Height = domain.quantities['height'] |
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246 | Velocity = domain.quantities['velocity'] |
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247 | |
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248 | #Arrays |
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249 | w_C = Stage.centroid_values |
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250 | uh_C = Xmom.centroid_values |
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251 | z_C = Bed.centroid_values |
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252 | h_C = Height.centroid_values |
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253 | u_C = Velocity.centroid_values |
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254 | |
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255 | #print id(h_C) |
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256 | #FIXME replace with numpy.where |
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257 | for i in range(N): |
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258 | h_C[i] = w_C[i] - z_C[i] |
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259 | if h_C[i] <= 0.0: |
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260 | #print 'h_C[%d]= %15.5e\n' % (i,h_C[i]) |
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261 | h_C[i] = 1.0e-15 |
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262 | w_C[i] = z_C[i] |
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263 | uh_C[i] = 0.0 |
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264 | |
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265 | |
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266 | ## for i in range(len(h_C)): |
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267 | ## if h_C[i] < epsilon: |
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268 | ## u_C[i] = 0.0 #Could have been negative |
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269 | ## h_C[i] = 0.0 |
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270 | ## else: |
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271 | |
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272 | u_C[:,] = uh_C/(h_C + h0/h_C) |
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273 | |
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274 | for name in [ 'velocity', 'stage' ]: |
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275 | Q = domain.quantities[name] |
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276 | if domain.order == 1: |
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277 | Q.extrapolate_first_order() |
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278 | elif domain.order == 2: |
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279 | #print "add extrapolate second order to shallow water" |
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280 | #if name != 'height': |
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281 | Q.extrapolate_second_order() |
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282 | #Q.limit() |
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283 | else: |
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284 | raise 'Unknown order' |
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285 | |
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286 | stage_V = domain.quantities['stage'].vertex_values |
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287 | bed_V = domain.quantities['elevation'].vertex_values |
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288 | h_V = domain.quantities['height'].vertex_values |
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289 | u_V = domain.quantities['velocity'].vertex_values |
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290 | xmom_V = domain.quantities['xmomentum'].vertex_values |
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291 | |
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292 | h_V[:,:] = stage_V - bed_V |
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293 | for i in range(len(h_C)): |
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294 | for j in range(2): |
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295 | if h_V[i,j] < 0.0 : |
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296 | #print 'h_V[%d,%d] = %f \n' % (i,j,h_V[i,j]) |
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297 | dh = h_V[i,j] |
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298 | h_V[i,j] = 0.0 |
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299 | stage_V[i,j] = bed_V[i,j] |
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300 | h_V[i,(j+1)%2] = h_V[i,(j+1)%2] + dh |
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301 | stage_V[i,(j+1)%2] = stage_V[i,(j+1)%2] + dh |
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302 | |
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303 | xmom_V[:,:] = u_V * h_V |
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304 | |
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305 | return |
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306 | # |
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307 | |
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308 | |
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309 | |
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310 | |
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311 | |
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312 | |
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313 | |
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314 | # |
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315 | def protect_against_infinitesimal_and_negative_heights(domain): |
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316 | """Protect against infinitesimal heights and associated high velocities |
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317 | """ |
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318 | |
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319 | #Shortcuts |
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320 | wc = domain.quantities['stage'].centroid_values |
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321 | zc = domain.quantities['elevation'].centroid_values |
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322 | xmomc = domain.quantities['xmomentum'].centroid_values |
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323 | hc = wc - zc #Water depths at centroids |
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324 | |
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325 | zv = domain.quantities['elevation'].vertex_values |
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326 | wv = domain.quantities['stage'].vertex_values |
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327 | hv = wv-zv |
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328 | xmomv = domain.quantities['xmomentum'].vertex_values |
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329 | #remove the above two lines and corresponding code below |
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330 | |
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331 | #Update |
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332 | #FIXME replace with numpy.where |
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333 | for k in range(domain.number_of_elements): |
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334 | |
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335 | if hc[k] < domain.minimum_allowed_height: |
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336 | #Control stage |
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337 | if hc[k] < domain.epsilon: |
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338 | wc[k] = zc[k] # Contain 'lost mass' error |
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339 | wv[k,0] = zv[k,0] |
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340 | wv[k,1] = zv[k,1] |
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341 | |
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342 | xmomc[k] = 0.0 |
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343 | |
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344 | #N = domain.number_of_elements |
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345 | #if (k == 0) | (k==N-1): |
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346 | # wc[k] = zc[k] # Contain 'lost mass' error |
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347 | # wv[k,0] = zv[k,0] |
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348 | # wv[k,1] = zv[k,1] |
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349 | |
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350 | def h_limiter(domain): |
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351 | """Limit slopes for each volume to eliminate artificial variance |
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352 | introduced by e.g. second order extrapolator |
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353 | |
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354 | limit on h = w-z |
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355 | |
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356 | This limiter depends on two quantities (w,z) so it resides within |
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357 | this module rather than within quantity.py |
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358 | """ |
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359 | |
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360 | N = domain.number_of_elements |
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361 | beta_h = domain.beta_h |
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362 | |
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363 | #Shortcuts |
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364 | wc = domain.quantities['stage'].centroid_values |
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365 | zc = domain.quantities['elevation'].centroid_values |
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366 | hc = wc - zc |
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367 | |
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368 | wv = domain.quantities['stage'].vertex_values |
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369 | zv = domain.quantities['elevation'].vertex_values |
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370 | hv = wv-zv |
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371 | |
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372 | hvbar = zeros(hv.shape, numpy.float) #h-limited values |
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373 | |
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374 | #Find min and max of this and neighbour's centroid values |
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375 | hmax = zeros(hc.shape, numpy.float) |
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376 | hmin = zeros(hc.shape, numpy.float) |
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377 | |
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378 | for k in range(N): |
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379 | hmax[k] = hmin[k] = hc[k] |
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380 | #for i in range(3): |
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381 | for i in range(2): |
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382 | n = domain.neighbours[k,i] |
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383 | if n >= 0: |
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384 | hn = hc[n] #Neighbour's centroid value |
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385 | |
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386 | hmin[k] = min(hmin[k], hn) |
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387 | hmax[k] = max(hmax[k], hn) |
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388 | |
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389 | |
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390 | #Diffences between centroids and maxima/minima |
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391 | dhmax = hmax - hc |
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392 | dhmin = hmin - hc |
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393 | |
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394 | #Deltas between vertex and centroid values |
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395 | dh = zeros(hv.shape, numpy.float) |
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396 | #for i in range(3): |
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397 | for i in range(2): |
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398 | dh[:,i] = hv[:,i] - hc |
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399 | |
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400 | #Phi limiter |
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401 | for k in range(N): |
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402 | |
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403 | #Find the gradient limiter (phi) across vertices |
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404 | phi = 1.0 |
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405 | #for i in range(3): |
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406 | for i in range(2): |
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407 | r = 1.0 |
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408 | if (dh[k,i] > 0): r = dhmax[k]/dh[k,i] |
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409 | if (dh[k,i] < 0): r = dhmin[k]/dh[k,i] |
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410 | |
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411 | phi = min( min(r*beta_h, 1), phi ) |
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412 | |
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413 | #Then update using phi limiter |
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414 | #for i in range(3): |
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415 | for i in range(2): |
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416 | hvbar[k,i] = hc[k] + phi*dh[k,i] |
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417 | |
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418 | return hvbar |
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419 | |
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420 | def balance_deep_and_shallow(domain): |
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421 | """Compute linear combination between stage as computed by |
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422 | gradient-limiters limiting using w, and stage computed by |
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423 | gradient-limiters limiting using h (h-limiter). |
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424 | The former takes precedence when heights are large compared to the |
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425 | bed slope while the latter takes precedence when heights are |
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426 | relatively small. Anything in between is computed as a balanced |
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427 | linear combination in order to avoid numpyal disturbances which |
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428 | would otherwise appear as a result of hard switching between |
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429 | modes. |
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430 | |
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431 | The h-limiter is always applied irrespective of the order. |
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432 | """ |
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433 | |
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434 | #Shortcuts |
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435 | wc = domain.quantities['stage'].centroid_values |
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436 | zc = domain.quantities['elevation'].centroid_values |
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437 | hc = wc - zc |
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438 | |
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439 | wv = domain.quantities['stage'].vertex_values |
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440 | zv = domain.quantities['elevation'].vertex_values |
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441 | hv = wv-zv |
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442 | |
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443 | #Limit h |
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444 | hvbar = h_limiter(domain) |
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445 | |
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446 | for k in range(domain.number_of_elements): |
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447 | #Compute maximal variation in bed elevation |
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448 | # This quantitiy is |
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449 | # dz = max_i abs(z_i - z_c) |
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450 | # and it is independent of dimension |
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451 | # In the 1d case zc = (z0+z1)/2 |
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452 | # In the 2d case zc = (z0+z1+z2)/3 |
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453 | |
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454 | dz = max(abs(zv[k,0]-zc[k]), |
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455 | abs(zv[k,1]-zc[k]))#, |
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456 | # abs(zv[k,2]-zc[k])) |
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457 | |
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458 | |
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459 | hmin = min( hv[k,:] ) |
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460 | |
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461 | #Create alpha in [0,1], where alpha==0 means using the h-limited |
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462 | #stage and alpha==1 means using the w-limited stage as |
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463 | #computed by the gradient limiter (both 1st or 2nd order) |
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464 | |
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465 | #If hmin > dz/2 then alpha = 1 and the bed will have no effect |
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466 | #If hmin < 0 then alpha = 0 reverting to constant height above bed. |
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467 | |
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468 | if dz > 0.0: |
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469 | alpha = max( min( 2*hmin/dz, 1.0), 0.0 ) |
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470 | else: |
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471 | #Flat bed |
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472 | alpha = 1.0 |
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473 | |
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474 | alpha = 0.0 |
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475 | #Let |
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476 | # |
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477 | # wvi be the w-limited stage (wvi = zvi + hvi) |
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478 | # wvi- be the h-limited state (wvi- = zvi + hvi-) |
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479 | # |
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480 | # |
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481 | #where i=0,1,2 denotes the vertex ids |
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482 | # |
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483 | #Weighted balance between w-limited and h-limited stage is |
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484 | # |
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485 | # wvi := (1-alpha)*(zvi+hvi-) + alpha*(zvi+hvi) |
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486 | # |
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487 | #It follows that the updated wvi is |
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488 | # wvi := zvi + (1-alpha)*hvi- + alpha*hvi |
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489 | # |
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490 | # Momentum is balanced between constant and limited |
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491 | |
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492 | |
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493 | #for i in range(3): |
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494 | # wv[k,i] = zv[k,i] + hvbar[k,i] |
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495 | |
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496 | #return |
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497 | |
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498 | if alpha < 1: |
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499 | |
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500 | #for i in range(3): |
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501 | for i in range(2): |
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502 | wv[k,i] = zv[k,i] + (1.0-alpha)*hvbar[k,i] + alpha*hv[k,i] |
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503 | |
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504 | #Momentums at centroids |
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505 | xmomc = domain.quantities['xmomentum'].centroid_values |
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506 | # ymomc = domain.quantities['ymomentum'].centroid_values |
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507 | |
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508 | #Momentums at vertices |
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509 | xmomv = domain.quantities['xmomentum'].vertex_values |
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510 | # ymomv = domain.quantities['ymomentum'].vertex_values |
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511 | |
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512 | # Update momentum as a linear combination of |
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513 | # xmomc and ymomc (shallow) and momentum |
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514 | # from extrapolator xmomv and ymomv (deep). |
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515 | xmomv[k,:] = (1.0-alpha)*xmomc[k] + alpha*xmomv[k,:] |
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516 | # ymomv[k,:] = (1-alpha)*ymomc[k] + alpha*ymomv[k,:] |
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517 | |
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518 | |
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519 | |
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