1 | """Class Domain - |
---|
2 | 1D interval domains for finite-volume computations of |
---|
3 | the shallow water wave equation. |
---|
4 | |
---|
5 | This module contains a specialisation of class Domain from module domain.py |
---|
6 | consisting of methods specific to the Shallow Water Wave Equation |
---|
7 | |
---|
8 | |
---|
9 | U_t + E_x = S |
---|
10 | |
---|
11 | where |
---|
12 | |
---|
13 | U = [w, uh] |
---|
14 | E = [uh, u^2h + gh^2/2] |
---|
15 | S represents source terms forcing the system |
---|
16 | (e.g. gravity, friction, wind stress, ...) |
---|
17 | |
---|
18 | and _t, _x, _y denote the derivative with respect to t, x and y respectiely. |
---|
19 | |
---|
20 | The quantities are |
---|
21 | |
---|
22 | symbol variable name explanation |
---|
23 | x x horizontal distance from origin [m] |
---|
24 | z elevation elevation of bed on which flow is modelled [m] |
---|
25 | h height water height above z [m] |
---|
26 | w stage absolute water level, w = z+h [m] |
---|
27 | u speed in the x direction [m/s] |
---|
28 | uh xmomentum momentum in the x direction [m^2/s] |
---|
29 | |
---|
30 | eta mannings friction coefficient [to appear] |
---|
31 | nu wind stress coefficient [to appear] |
---|
32 | |
---|
33 | The conserved quantities are w, uh |
---|
34 | |
---|
35 | For details see e.g. |
---|
36 | Christopher Zoppou and Stephen Roberts, |
---|
37 | Catastrophic Collapse of Water Supply Reservoirs in Urban Areas, |
---|
38 | Journal of Hydraulic Engineering, vol. 127, No. 7 July 1999 |
---|
39 | |
---|
40 | |
---|
41 | John Jakeman, Ole Nielsen, Stephen Roberts, Duncan Gray, Christopher Zoppou |
---|
42 | Geoscience Australia, 2006 |
---|
43 | """ |
---|
44 | |
---|
45 | import numpy |
---|
46 | |
---|
47 | from anuga_1d.base.generic_domain import Generic_domain |
---|
48 | from sww_boundary_conditions import * |
---|
49 | from sww_forcing_terms import * |
---|
50 | |
---|
51 | #Shallow water domain |
---|
52 | class Domain(Generic_domain): |
---|
53 | |
---|
54 | def __init__(self, coordinates, boundary = None, tagged_elements = None): |
---|
55 | |
---|
56 | conserved_quantities = ['stage', 'xmomentum'] |
---|
57 | evolved_quantities = ['stage', 'xmomentum', 'elevation', 'height', 'velocity'] |
---|
58 | other_quantities = ['friction'] |
---|
59 | Generic_domain.__init__(self, |
---|
60 | coordinates = coordinates, |
---|
61 | boundary = boundary, |
---|
62 | conserved_quantities = conserved_quantities, |
---|
63 | evolved_quantities = evolved_quantities, |
---|
64 | other_quantities = other_quantities, |
---|
65 | tagged_elements = tagged_elements) |
---|
66 | |
---|
67 | from anuga_1d.config import minimum_allowed_height, g, h0 |
---|
68 | self.minimum_allowed_height = minimum_allowed_height |
---|
69 | self.g = g |
---|
70 | self.h0 = h0 |
---|
71 | |
---|
72 | #forcing terms |
---|
73 | self.forcing_terms.append(gravity) |
---|
74 | #self.forcing_terms.append(manning_friction) |
---|
75 | |
---|
76 | |
---|
77 | #Stored output |
---|
78 | self.store = True |
---|
79 | self.format = 'sww' |
---|
80 | self.smooth = True |
---|
81 | |
---|
82 | #Evolve parametrs |
---|
83 | self.cfl = 1.0 |
---|
84 | |
---|
85 | #Reduction operation for get_vertex_values |
---|
86 | from anuga_1d.base.util import mean |
---|
87 | self.reduction = mean |
---|
88 | #self.reduction = min #Looks better near steep slopes |
---|
89 | |
---|
90 | self.quantities_to_be_stored = ['stage','xmomentum'] |
---|
91 | |
---|
92 | self.__doc__ = 'sww_domain' |
---|
93 | |
---|
94 | |
---|
95 | def set_quantities_to_be_stored(self, q): |
---|
96 | """Specify which quantities will be stored in the sww file. |
---|
97 | |
---|
98 | q must be either: |
---|
99 | - the name of a quantity |
---|
100 | - a list of quantity names |
---|
101 | - None |
---|
102 | |
---|
103 | In the two first cases, the named quantities will be stored at each |
---|
104 | yieldstep |
---|
105 | (This is in addition to the quantities elevation and friction) |
---|
106 | If q is None, storage will be switched off altogether. |
---|
107 | """ |
---|
108 | |
---|
109 | |
---|
110 | if q is None: |
---|
111 | self.quantities_to_be_stored = [] |
---|
112 | self.store = False |
---|
113 | return |
---|
114 | |
---|
115 | if isinstance(q, basestring): |
---|
116 | q = [q] # Turn argument into a list |
---|
117 | |
---|
118 | #Check correcness |
---|
119 | for quantity_name in q: |
---|
120 | msg = 'Quantity %s is not a valid conserved quantity' %quantity_name |
---|
121 | assert quantity_name in self.conserved_quantities, msg |
---|
122 | |
---|
123 | self.quantities_to_be_stored = q |
---|
124 | |
---|
125 | |
---|
126 | |
---|
127 | def check_integrity(self): |
---|
128 | Generic_Domain.check_integrity(self) |
---|
129 | #Check that we are solving the shallow water wave equation |
---|
130 | |
---|
131 | msg = 'First conserved quantity must be "stage"' |
---|
132 | assert self.conserved_quantities[0] == 'stage', msg |
---|
133 | msg = 'Second conserved quantity must be "xmomentum"' |
---|
134 | assert self.conserved_quantities[1] == 'xmomentum', msg |
---|
135 | |
---|
136 | |
---|
137 | |
---|
138 | def compute_fluxes(self): |
---|
139 | #Call correct module function |
---|
140 | #(either from this module or C-extension) |
---|
141 | |
---|
142 | import sys |
---|
143 | |
---|
144 | |
---|
145 | timestep = float(sys.maxint) |
---|
146 | |
---|
147 | stage = self.quantities['stage'] |
---|
148 | xmom = self.quantities['xmomentum'] |
---|
149 | bed = self.quantities['elevation'] |
---|
150 | height = self.quantities['height'] |
---|
151 | velocity = self.quantities['velocity'] |
---|
152 | |
---|
153 | |
---|
154 | from anuga_1d.sww.sww_comp_flux_ext import compute_fluxes_ext_short |
---|
155 | |
---|
156 | #self.flux_timestep = compute_fluxes_ext(timestep,self,stage,xmom,bed,height,velocity) |
---|
157 | |
---|
158 | |
---|
159 | self.flux_timestep = compute_fluxes_ext_short(timestep,self,stage,xmom,bed) |
---|
160 | |
---|
161 | |
---|
162 | |
---|
163 | def distribute_to_vertices_and_edges(self): |
---|
164 | #Call correct module function |
---|
165 | #(either from this module or C-extension) |
---|
166 | distribute_to_vertices_and_edges(self) |
---|
167 | |
---|
168 | def evolve(self, yieldstep = None, finaltime = None, duration = None, |
---|
169 | skip_initial_step = False): |
---|
170 | """Specialisation of basic evolve method from parent class |
---|
171 | """ |
---|
172 | |
---|
173 | #Call basic machinery from parent class |
---|
174 | for t in Generic_domain.evolve(self, yieldstep, finaltime,duration, |
---|
175 | skip_initial_step): |
---|
176 | |
---|
177 | #Pass control on to outer loop for more specific actions |
---|
178 | yield(t) |
---|
179 | |
---|
180 | |
---|
181 | |
---|
182 | |
---|
183 | |
---|
184 | #=============== End of Shallow Water Domain =============================== |
---|
185 | |
---|
186 | |
---|
187 | # Module functions for gradient limiting (distribute_to_vertices_and_edges) |
---|
188 | |
---|
189 | def distribute_to_vertices_and_edges(domain): |
---|
190 | """Distribution from centroids to vertices specific to the |
---|
191 | shallow water wave |
---|
192 | equation. |
---|
193 | |
---|
194 | It will ensure that h (w-z) is always non-negative even in the |
---|
195 | presence of steep bed-slopes by taking a weighted average between shallow |
---|
196 | and deep cases. |
---|
197 | |
---|
198 | In addition, all conserved quantities get distributed as per either a |
---|
199 | constant (order==1) or a piecewise linear function (order==2). |
---|
200 | |
---|
201 | FIXME: more explanation about removal of artificial variability etc |
---|
202 | |
---|
203 | Precondition: |
---|
204 | All quantities defined at centroids and bed elevation defined at |
---|
205 | vertices. |
---|
206 | |
---|
207 | Postcondition |
---|
208 | Conserved quantities defined at vertices |
---|
209 | |
---|
210 | """ |
---|
211 | |
---|
212 | #from config import optimised_gradient_limiter |
---|
213 | |
---|
214 | #Remove very thin layers of water |
---|
215 | #protect_against_infinitesimal_and_negative_heights(domain) |
---|
216 | |
---|
217 | import sys |
---|
218 | from anuga_1d.config import epsilon, h0 |
---|
219 | |
---|
220 | N = domain.number_of_elements |
---|
221 | |
---|
222 | #Shortcuts |
---|
223 | Stage = domain.quantities['stage'] |
---|
224 | Xmom = domain.quantities['xmomentum'] |
---|
225 | Bed = domain.quantities['elevation'] |
---|
226 | Height = domain.quantities['height'] |
---|
227 | Velocity = domain.quantities['velocity'] |
---|
228 | |
---|
229 | #Arrays |
---|
230 | w_C = Stage.centroid_values |
---|
231 | uh_C = Xmom.centroid_values |
---|
232 | z_C = Bed.centroid_values |
---|
233 | h_C = Height.centroid_values |
---|
234 | u_C = Velocity.centroid_values |
---|
235 | |
---|
236 | #Calculate height (and fix negatives)better be non-negative! |
---|
237 | h_C[:] = w_C - z_C |
---|
238 | u_C[:] = uh_C/(h_C + h0/h_C) |
---|
239 | |
---|
240 | for name in [ 'velocity', 'stage' ]: |
---|
241 | Q = domain.quantities[name] |
---|
242 | if domain.order == 1: |
---|
243 | Q.extrapolate_first_order() |
---|
244 | elif domain.order == 2: |
---|
245 | #print "add extrapolate second order to shallow water" |
---|
246 | #if name != 'height': |
---|
247 | Q.extrapolate_second_order() |
---|
248 | #Q.limit() |
---|
249 | else: |
---|
250 | raise 'Unknown order' |
---|
251 | |
---|
252 | |
---|
253 | stage_V = domain.quantities['stage'].vertex_values |
---|
254 | bed_V = domain.quantities['elevation'].vertex_values |
---|
255 | h_V = domain.quantities['height'].vertex_values |
---|
256 | u_V = domain.quantities['velocity'].vertex_values |
---|
257 | xmom_V = domain.quantities['xmomentum'].vertex_values |
---|
258 | |
---|
259 | h_V[:,:] = stage_V - bed_V |
---|
260 | |
---|
261 | # # protect from edge values going negative |
---|
262 | # h_V[:,1] = numpy.where(h_V[:,0] < 0.0 , h_V[:,1]-h_V[:,0], h_V[:,1]) |
---|
263 | # h_V[:,0] = numpy.where(h_V[:,0] < 0.0 , 0.0, h_V[:,0]) |
---|
264 | # |
---|
265 | # h_V[:,0] = numpy.where(h_V[:,1] < 0.0 , h_V[:,0]-h_V[:,1], h_V[:,0]) |
---|
266 | # h_V[:,1] = numpy.where(h_V[:,1] < 0.0 , 0.0, h_V[:,1]) |
---|
267 | # |
---|
268 | # |
---|
269 | # stage_V[:,:] = bed_V + h_V |
---|
270 | xmom_V[:,:] = u_V * h_V |
---|
271 | |
---|
272 | return |
---|
273 | # |
---|
274 | |
---|
275 | |
---|
276 | |
---|
277 | |
---|
278 | |
---|
279 | |
---|
280 | |
---|
281 | # |
---|
282 | def protect_against_infinitesimal_and_negative_heights(domain): |
---|
283 | """Protect against infinitesimal heights and associated high velocities |
---|
284 | """ |
---|
285 | |
---|
286 | #Shortcuts |
---|
287 | wc = domain.quantities['stage'].centroid_values |
---|
288 | zc = domain.quantities['elevation'].centroid_values |
---|
289 | xmomc = domain.quantities['xmomentum'].centroid_values |
---|
290 | hc = wc - zc #Water depths at centroids |
---|
291 | |
---|
292 | zv = domain.quantities['elevation'].vertex_values |
---|
293 | wv = domain.quantities['stage'].vertex_values |
---|
294 | hv = wv-zv |
---|
295 | xmomv = domain.quantities['xmomentum'].vertex_values |
---|
296 | #remove the above two lines and corresponding code below |
---|
297 | |
---|
298 | #Update |
---|
299 | #FIXME replace with numpy.where |
---|
300 | for k in range(domain.number_of_elements): |
---|
301 | |
---|
302 | if hc[k] < domain.minimum_allowed_height: |
---|
303 | #Control stage |
---|
304 | if hc[k] < domain.epsilon: |
---|
305 | wc[k] = zc[k] # Contain 'lost mass' error |
---|
306 | wv[k,0] = zv[k,0] |
---|
307 | wv[k,1] = zv[k,1] |
---|
308 | |
---|
309 | xmomc[k] = 0.0 |
---|
310 | |
---|
311 | #N = domain.number_of_elements |
---|
312 | #if (k == 0) | (k==N-1): |
---|
313 | # wc[k] = zc[k] # Contain 'lost mass' error |
---|
314 | # wv[k,0] = zv[k,0] |
---|
315 | # wv[k,1] = zv[k,1] |
---|
316 | |
---|
317 | def h_limiter(domain): |
---|
318 | """Limit slopes for each volume to eliminate artificial variance |
---|
319 | introduced by e.g. second order extrapolator |
---|
320 | |
---|
321 | limit on h = w-z |
---|
322 | |
---|
323 | This limiter depends on two quantities (w,z) so it resides within |
---|
324 | this module rather than within quantity.py |
---|
325 | """ |
---|
326 | |
---|
327 | N = domain.number_of_elements |
---|
328 | beta_h = domain.beta_h |
---|
329 | |
---|
330 | #Shortcuts |
---|
331 | wc = domain.quantities['stage'].centroid_values |
---|
332 | zc = domain.quantities['elevation'].centroid_values |
---|
333 | hc = wc - zc |
---|
334 | |
---|
335 | wv = domain.quantities['stage'].vertex_values |
---|
336 | zv = domain.quantities['elevation'].vertex_values |
---|
337 | hv = wv-zv |
---|
338 | |
---|
339 | hvbar = zeros(hv.shape, numpy.float) #h-limited values |
---|
340 | |
---|
341 | #Find min and max of this and neighbour's centroid values |
---|
342 | hmax = zeros(hc.shape, numpy.float) |
---|
343 | hmin = zeros(hc.shape, numpy.float) |
---|
344 | |
---|
345 | for k in range(N): |
---|
346 | hmax[k] = hmin[k] = hc[k] |
---|
347 | #for i in range(3): |
---|
348 | for i in range(2): |
---|
349 | n = domain.neighbours[k,i] |
---|
350 | if n >= 0: |
---|
351 | hn = hc[n] #Neighbour's centroid value |
---|
352 | |
---|
353 | hmin[k] = min(hmin[k], hn) |
---|
354 | hmax[k] = max(hmax[k], hn) |
---|
355 | |
---|
356 | |
---|
357 | #Diffences between centroids and maxima/minima |
---|
358 | dhmax = hmax - hc |
---|
359 | dhmin = hmin - hc |
---|
360 | |
---|
361 | #Deltas between vertex and centroid values |
---|
362 | dh = zeros(hv.shape, numpy.float) |
---|
363 | #for i in range(3): |
---|
364 | for i in range(2): |
---|
365 | dh[:,i] = hv[:,i] - hc |
---|
366 | |
---|
367 | #Phi limiter |
---|
368 | for k in range(N): |
---|
369 | |
---|
370 | #Find the gradient limiter (phi) across vertices |
---|
371 | phi = 1.0 |
---|
372 | #for i in range(3): |
---|
373 | for i in range(2): |
---|
374 | r = 1.0 |
---|
375 | if (dh[k,i] > 0): r = dhmax[k]/dh[k,i] |
---|
376 | if (dh[k,i] < 0): r = dhmin[k]/dh[k,i] |
---|
377 | |
---|
378 | phi = min( min(r*beta_h, 1), phi ) |
---|
379 | |
---|
380 | #Then update using phi limiter |
---|
381 | #for i in range(3): |
---|
382 | for i in range(2): |
---|
383 | hvbar[k,i] = hc[k] + phi*dh[k,i] |
---|
384 | |
---|
385 | return hvbar |
---|
386 | |
---|
387 | def balance_deep_and_shallow(domain): |
---|
388 | """Compute linear combination between stage as computed by |
---|
389 | gradient-limiters limiting using w, and stage computed by |
---|
390 | gradient-limiters limiting using h (h-limiter). |
---|
391 | The former takes precedence when heights are large compared to the |
---|
392 | bed slope while the latter takes precedence when heights are |
---|
393 | relatively small. Anything in between is computed as a balanced |
---|
394 | linear combination in order to avoid numpyal disturbances which |
---|
395 | would otherwise appear as a result of hard switching between |
---|
396 | modes. |
---|
397 | |
---|
398 | The h-limiter is always applied irrespective of the order. |
---|
399 | """ |
---|
400 | |
---|
401 | #Shortcuts |
---|
402 | wc = domain.quantities['stage'].centroid_values |
---|
403 | zc = domain.quantities['elevation'].centroid_values |
---|
404 | hc = wc - zc |
---|
405 | |
---|
406 | wv = domain.quantities['stage'].vertex_values |
---|
407 | zv = domain.quantities['elevation'].vertex_values |
---|
408 | hv = wv-zv |
---|
409 | |
---|
410 | #Limit h |
---|
411 | hvbar = h_limiter(domain) |
---|
412 | |
---|
413 | for k in range(domain.number_of_elements): |
---|
414 | #Compute maximal variation in bed elevation |
---|
415 | # This quantitiy is |
---|
416 | # dz = max_i abs(z_i - z_c) |
---|
417 | # and it is independent of dimension |
---|
418 | # In the 1d case zc = (z0+z1)/2 |
---|
419 | # In the 2d case zc = (z0+z1+z2)/3 |
---|
420 | |
---|
421 | dz = max(abs(zv[k,0]-zc[k]), |
---|
422 | abs(zv[k,1]-zc[k]))#, |
---|
423 | # abs(zv[k,2]-zc[k])) |
---|
424 | |
---|
425 | |
---|
426 | hmin = min( hv[k,:] ) |
---|
427 | |
---|
428 | #Create alpha in [0,1], where alpha==0 means using the h-limited |
---|
429 | #stage and alpha==1 means using the w-limited stage as |
---|
430 | #computed by the gradient limiter (both 1st or 2nd order) |
---|
431 | |
---|
432 | #If hmin > dz/2 then alpha = 1 and the bed will have no effect |
---|
433 | #If hmin < 0 then alpha = 0 reverting to constant height above bed. |
---|
434 | |
---|
435 | if dz > 0.0: |
---|
436 | alpha = max( min( 2*hmin/dz, 1.0), 0.0 ) |
---|
437 | else: |
---|
438 | #Flat bed |
---|
439 | alpha = 1.0 |
---|
440 | |
---|
441 | alpha = 0.0 |
---|
442 | #Let |
---|
443 | # |
---|
444 | # wvi be the w-limited stage (wvi = zvi + hvi) |
---|
445 | # wvi- be the h-limited state (wvi- = zvi + hvi-) |
---|
446 | # |
---|
447 | # |
---|
448 | #where i=0,1,2 denotes the vertex ids |
---|
449 | # |
---|
450 | #Weighted balance between w-limited and h-limited stage is |
---|
451 | # |
---|
452 | # wvi := (1-alpha)*(zvi+hvi-) + alpha*(zvi+hvi) |
---|
453 | # |
---|
454 | #It follows that the updated wvi is |
---|
455 | # wvi := zvi + (1-alpha)*hvi- + alpha*hvi |
---|
456 | # |
---|
457 | # Momentum is balanced between constant and limited |
---|
458 | |
---|
459 | |
---|
460 | #for i in range(3): |
---|
461 | # wv[k,i] = zv[k,i] + hvbar[k,i] |
---|
462 | |
---|
463 | #return |
---|
464 | |
---|
465 | if alpha < 1: |
---|
466 | |
---|
467 | #for i in range(3): |
---|
468 | for i in range(2): |
---|
469 | wv[k,i] = zv[k,i] + (1.0-alpha)*hvbar[k,i] + alpha*hv[k,i] |
---|
470 | |
---|
471 | #Momentums at centroids |
---|
472 | xmomc = domain.quantities['xmomentum'].centroid_values |
---|
473 | # ymomc = domain.quantities['ymomentum'].centroid_values |
---|
474 | |
---|
475 | #Momentums at vertices |
---|
476 | xmomv = domain.quantities['xmomentum'].vertex_values |
---|
477 | # ymomv = domain.quantities['ymomentum'].vertex_values |
---|
478 | |
---|
479 | # Update momentum as a linear combination of |
---|
480 | # xmomc and ymomc (shallow) and momentum |
---|
481 | # from extrapolator xmomv and ymomv (deep). |
---|
482 | xmomv[k,:] = (1.0-alpha)*xmomc[k] + alpha*xmomv[k,:] |
---|
483 | # ymomv[k,:] = (1-alpha)*ymomc[k] + alpha*ymomv[k,:] |
---|
484 | |
---|
485 | |
---|
486 | |
---|