[7839] | 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 flow_1d.sww_flow.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 flow_1d.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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