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Contents of /trunk/escript/py_src/flows.py

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Revision 3074 - (show annotations)
Tue Jul 27 01:47:45 2010 UTC (9 years, 2 months ago) by gross
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tolerance for PCG relaxed.
1 # -*- coding: utf-8 -*-
2 ########################################################
3 #
4 # Copyright (c) 2003-2010 by University of Queensland
5 # Earth Systems Science Computational Center (ESSCC)
6 # http://www.uq.edu.au/esscc
7 #
8 # Primary Business: Queensland, Australia
9 # Licensed under the Open Software License version 3.0
10 # http://www.opensource.org/licenses/osl-3.0.php
11 #
12 ########################################################
13
14 __copyright__="""Copyright (c) 2003-2010 by University of Queensland
15 Earth Systems Science Computational Center (ESSCC)
16 http://www.uq.edu.au/esscc
17 Primary Business: Queensland, Australia"""
18 __license__="""Licensed under the Open Software License version 3.0
19 http://www.opensource.org/licenses/osl-3.0.php"""
20 __url__="https://launchpad.net/escript-finley"
21
22 """
23 Some models for flow
24
25 :var __author__: name of author
26 :var __copyright__: copyrights
27 :var __license__: licence agreement
28 :var __url__: url entry point on documentation
29 :var __version__: version
30 :var __date__: date of the version
31 """
32
33 __author__="Lutz Gross, l.gross@uq.edu.au"
34
35 from escript import *
36 import util
37 from linearPDEs import LinearPDE, LinearPDESystem, LinearSinglePDE, SolverOptions
38 from pdetools import HomogeneousSaddlePointProblem,Projector, ArithmeticTuple, PCG, NegativeNorm, GMRES
39
40 class DarcyFlow(object):
41 """
42 solves the problem
43
44 *u_i+k_{ij}*p_{,j} = g_i*
45 *u_{i,i} = f*
46
47 where *p* represents the pressure and *u* the Darcy flux. *k* represents the permeability,
48
49 :note: The problem is solved in a least squares formulation.
50 """
51
52 def __init__(self, domain, useReduced=False, adaptSubTolerance=True, solveForFlux=False):
53 """
54 initializes the Darcy flux problem
55 :param domain: domain of the problem
56 :type domain: `Domain`
57 :param useReduced: uses reduced oreder on flux and pressure
58 :type useReduced: ``bool``
59 :param adaptSubTolerance: switches on automatic subtolerance selection
60 :type adaptSubTolerance: ``bool``
61 :param solveForFlux: if True the solver solves for the flux (do not use!)
62 :type solveForFlux: ``bool``
63 """
64 self.domain=domain
65 self.solveForFlux=solveForFlux
66 self.useReduced=useReduced
67 self.__adaptSubTolerance=adaptSubTolerance
68 self.verbose=False
69
70 self.__pde_k=LinearPDESystem(domain)
71 self.__pde_k.setSymmetryOn()
72 if self.useReduced: self.__pde_k.setReducedOrderOn()
73
74 self.__pde_p=LinearSinglePDE(domain)
75 self.__pde_p.setSymmetryOn()
76 if self.useReduced: self.__pde_p.setReducedOrderOn()
77
78 self.__pde_l=LinearSinglePDE(domain) # this is here for getSolverOptionsWeighting
79 # self.__pde_l.setSymmetryOn()
80 # if self.useReduced: self.__pde_l.setReducedOrderOn()
81 self.setTolerance()
82 self.setAbsoluteTolerance()
83 self.__f=Scalar(0,self.__pde_p.getFunctionSpaceForCoefficient("X"))
84 self.__g=Vector(0,self.__pde_p.getFunctionSpaceForCoefficient("Y"))
85
86 def getSolverOptionsFlux(self):
87 """
88 Returns the solver options used to solve the flux problems
89
90 *K^{-1} u=F*
91
92 :return: `SolverOptions`
93 """
94 return self.__pde_k.getSolverOptions()
95
96 def setSolverOptionsFlux(self, options=None):
97 """
98 Sets the solver options used to solve the flux problems
99
100 *K^{-1}u=F*
101
102 If ``options`` is not present, the options are reset to default
103
104 :param options: `SolverOptions`
105 :note: if the adaption of subtolerance is choosen, the tolerance set by ``options`` will be overwritten before the solver is called.
106 """
107 return self.__pde_v.setSolverOptions(options)
108
109 def getSolverOptionsPressure(self):
110 """
111 Returns the solver options used to solve the pressure problems
112
113 *(Q^* K Q)p=-Q^*G*
114
115 :return: `SolverOptions`
116 """
117 return self.__pde_p.getSolverOptions()
118
119 def setSolverOptionsPressure(self, options=None):
120 """
121 Sets the solver options used to solve the pressure problems
122
123 *(Q^* K Q)p=-Q^*G*
124
125 If ``options`` is not present, the options are reset to default
126
127 :param options: `SolverOptions`
128 :note: if the adaption of subtolerance is choosen, the tolerance set by ``options`` will be overwritten before the solver is called.
129 """
130 return self.__pde_p.setSolverOptions(options)
131
132 def getSolverOptionsWeighting(self):
133 """
134 Returns the solver options used to solve the pressure problems
135
136 *(D K D^*)p=-D F*
137
138 :return: `SolverOptions`
139 """
140 return self.__pde_l.getSolverOptions()
141
142 def setSolverOptionsWeighting(self, options=None):
143 """
144 Sets the solver options used to solve the pressure problems
145
146 *(D K D^*)p=-D F*
147
148 If ``options`` is not present, the options are reset to default
149
150 :param options: `SolverOptions`
151 :note: if the adaption of subtolerance is choosen, the tolerance set by ``options`` will be overwritten before the solver is called.
152 """
153 return self.__pde_l.setSolverOptions(options)
154
155
156 def setValue(self,f=None, g=None, location_of_fixed_pressure=None, location_of_fixed_flux=None, permeability=None):
157 """
158 assigns values to model parameters
159
160 :param f: volumetic sources/sinks
161 :type f: scalar value on the domain (e.g. `Data`)
162 :param g: flux sources/sinks
163 :type g: vector values on the domain (e.g. `Data`)
164 :param location_of_fixed_pressure: mask for locations where pressure is fixed
165 :type location_of_fixed_pressure: scalar value on the domain (e.g. `Data`)
166 :param location_of_fixed_flux: mask for locations where flux is fixed.
167 :type location_of_fixed_flux: vector values on the domain (e.g. `Data`)
168 :param permeability: permeability tensor. If scalar ``s`` is given the tensor with
169 ``s`` on the main diagonal is used.
170 :type permeability: scalar or tensor values on the domain (e.g. `Data`)
171 :note: the values of parameters which are not set by calling ``setValue`` are not altered.
172 :note: at any point on the boundary of the domain the pressure (``location_of_fixed_pressure`` >0)
173 or the normal component of the flux (``location_of_fixed_flux[i]>0`` if direction of the normal
174 is along the *x_i* axis.
175 """
176 if f !=None:
177 f=util.interpolate(f, self.__pde_p.getFunctionSpaceForCoefficient("X"))
178 if f.isEmpty():
179 f=Scalar(0,self.__pde_p.getFunctionSpaceForCoefficient("X"))
180 else:
181 if f.getRank()>0: raise ValueError,"illegal rank of f."
182 self.__f=f
183 if g !=None:
184 g=util.interpolate(g, self.__pde_p.getFunctionSpaceForCoefficient("Y"))
185 if g.isEmpty():
186 g=Vector(0,self.__pde_p.getFunctionSpaceForCoefficient("Y"))
187 else:
188 if not g.getShape()==(self.domain.getDim(),):
189 raise ValueError,"illegal shape of g"
190 self.__g=g
191 if location_of_fixed_pressure!=None:
192 self.__pde_p.setValue(q=location_of_fixed_pressure)
193 #self.__pde_l.setValue(q=location_of_fixed_pressure)
194 if location_of_fixed_flux!=None:
195 self.__pde_k.setValue(q=location_of_fixed_flux)
196
197 if permeability!=None:
198 perm=util.interpolate(permeability,self.__pde_p.getFunctionSpaceForCoefficient("A"))
199 if perm.getRank()==0:
200 perm_inv=(1./perm)*util.kronecker(self.domain.getDim())
201 perm=perm*util.kronecker(self.domain.getDim())
202 elif perm.getRank()==2:
203 perm_inv=util.inverse(perm)
204 else:
205 raise ValueError,"illegal rank of permeability."
206
207 self.__permeability=perm
208 self.__permeability_inv=perm_inv
209 self.__l =(util.longestEdge(self.domain)**2*util.length(self.__permeability_inv))/10
210 #self.__l =(self.domain.getSize()**2*util.length(self.__permeability_inv))/10
211 if self.solveForFlux:
212 self.__pde_k.setValue(D=self.__permeability_inv)
213 else:
214 self.__pde_k.setValue(D=self.__permeability_inv, A=self.__l*util.outer(util.kronecker(self.domain),util.kronecker(self.domain)))
215 self.__pde_p.setValue(A=self.__permeability)
216 #self.__pde_l.setValue(D=1/self.__l)
217 #self.__pde_l.setValue(A=self.__permeability)
218
219 def __applWeight(self, v, f=None):
220 # solves L p = f-Dv with p = 0
221 if self.getSolverOptionsWeighting().isVerbose() or self.verbose: print "DarcyFlux: Applying weighting operator"
222 if f == None:
223 return -util.div(v)*self.__l
224 else:
225 return (f-util.div(v))*self.__l
226 # if f == None:
227 # self.__pde_l.setValue(Y=-util.div(v))
228 # else:
229 # return (f-util.div(v))/self.__l
230 # return self.__pde_l.getSolution()
231
232 def __getPressure(self, v, p0, g=None):
233 # solves (G*KG)p = G^(g-v) with p = p0 where location_of_fixed_pressure>0
234 if self.getSolverOptionsPressure().isVerbose() or self.verbose: print "DarcyFlux: Pressure update"
235 if g == None:
236 self.__pde_p.setValue(X=-v, r=p0)
237 else:
238 self.__pde_p.setValue(X=g-v, r=p0)
239 p=self.__pde_p.getSolution()
240 return p
241
242 def __Aprod_v(self,dv):
243 # calculates: (a,b,c) = (K^{-1}(dv + KG * dp), L^{-1}Ddv, dp) with (G*KG)dp = - G^*dv
244 dp=self.__getPressure(dv, p0=Data()) # dp = (G*KG)^{-1} (0-G^*dv)
245 a=util.tensor_mult(self.__permeability_inv,dv)+util.grad(dp) # a= K^{-1}u+G*dp
246 b= - self.__applWeight(dv) # b = - (D K D^*)^{-1} (0-Dv)
247 return ArithmeticTuple(a,b,-dp)
248
249 def __Msolve_PCG_v(self,r):
250 # K^{-1} u = r[0] + D^*r[1] = K^{-1}(dv + KG * dp) + D^*L^{-1}Ddv
251 if self.getSolverOptionsFlux().isVerbose() or self.verbose: print "DarcyFlux: Applying preconditioner"
252 self.__pde_k.setValue(X=r[1]*util.kronecker(self.domain), Y=r[0], r=Data())
253 # self.__pde_p.getOperator().saveMM("prec.mm")
254 return self.__pde_k.getSolution()
255
256 def __inner_PCG_v(self,v,r):
257 return util.integrate(util.inner(v,r[0])+util.div(v)*r[1])
258
259 def __Aprod_p(self,dp):
260 if self.getSolverOptionsFlux().isVerbose(): print "DarcyFlux: Applying operator"
261 Gdp=util.grad(dp)
262 self.__pde_k.setValue(Y=-Gdp,X=Data(), r=Data())
263 du=self.__pde_k.getSolution()
264 # self.__pde_v.getOperator().saveMM("proj.mm")
265 return ArithmeticTuple(util.tensor_mult(self.__permeability,Gdp),-du)
266
267 def __getFlux(self,p, v0, f=None, g=None):
268 # solves (K^{-1}+D^*L^{-1} D) v = D^*L^{-1}f + K^{-1}g - Gp
269 if f!=None:
270 self.__pde_k.setValue(X=self.__applWeight(v0*0,self.__f)*util.kronecker(self.domain))
271 self.__pde_k.setValue(r=v0)
272 g2=util.tensor_mult(self.__permeability_inv,g)
273 if p == None:
274 self.__pde_k.setValue(Y=g2)
275 else:
276 self.__pde_k.setValue(Y=g2-util.grad(p))
277 return self.__pde_k.getSolution()
278
279 #v=self.__getFlux(p, u0, f=self.__f, g=g2)
280 def __Msolve_PCG_p(self,r):
281 if self.getSolverOptionsPressure().isVerbose(): print "DarcyFlux: Applying preconditioner"
282 self.__pde_p.setValue(X=r[0]-r[1], Y=Data(), r=Data(), y=Data())
283 # self.__pde_p.getOperator().saveMM("prec.mm")
284 return self.__pde_p.getSolution()
285
286 def __inner_PCG_p(self,p,r):
287 return util.integrate(util.inner(util.grad(p), r[0]-r[1]))
288
289 def __L2(self,v):
290 return util.sqrt(util.integrate(util.length(util.interpolate(v,Function(self.domain)))**2))
291
292 def solve(self,u0,p0, max_iter=100, verbose=False, max_num_corrections=10):
293 """
294 solves the problem.
295
296 The iteration is terminated if the residual norm is less then self.getTolerance().
297
298 :param u0: initial guess for the flux. At locations in the domain marked by ``location_of_fixed_flux`` the value of ``u0`` is kept unchanged.
299 :type u0: vector value on the domain (e.g. `Data`).
300 :param p0: initial guess for the pressure. At locations in the domain marked by ``location_of_fixed_pressure`` the value of ``p0`` is kept unchanged.
301 :type p0: scalar value on the domain (e.g. `Data`).
302 :param verbose: if set some information on iteration progress are printed
303 :type verbose: ``bool``
304 :return: flux and pressure
305 :rtype: ``tuple`` of `Data`.
306
307 :note: The problem is solved as a least squares form
308 *(K^[-1]+D^* (DKD^*)^[-1] D)u+G p=D^* (DKD^*)^[-1] f + K^[-1]g*
309 *G^*u+G^* K Gp=G^*g*
310
311 where *D* is the *div* operator and *(Gp)_i=p_{,i}* for the permeability *K=k_{ij}*.
312 """
313 self.verbose=verbose
314 rtol=self.getTolerance()
315 atol=self.getAbsoluteTolerance()
316 self.setSubProblemTolerance()
317 num_corrections=0
318 converged=False
319 norm_r=None
320
321 # Eliminate the hydrostatic pressure:
322 if self.verbose: print "DarcyFlux: calculate hydrostatic pressure component."
323 self.__pde_p.setValue(X=self.__g, r=p0, y=-util.inner(self.domain.getNormal(),u0))
324 p0=self.__pde_p.getSolution()
325 g2=self.__g - util.tensor_mult(self.__permeability, util.grad(p0))
326 norm_g2=util.integrate(util.inner(g2,util.tensor_mult(self.__permeability_inv,g2)))**0.5
327
328 p=p0*0
329 if self.solveForFlux:
330 v=u0.copy()
331 else:
332 v=self.__getFlux(p, u0, f=self.__f, g=g2)
333
334 while not converged and norm_g2 > 0:
335 Gp=util.grad(p)
336 KGp=util.tensor_mult(self.__permeability,Gp)
337 if self.verbose:
338 def_p=g2-(v+KGp)
339 def_v=self.__f-util.div(v)
340 print "DarcyFlux: L2: g-v-K*grad(p) = %e (v = %e)."%(self.__L2(def_p),self.__L2(v))
341 print "DarcyFlux: L2: f-div(v) = %e (grad(v) = %e)."%(self.__L2(def_v),self.__L2(util.grad(v)))
342 print "DarcyFlux: K^{-1}-norm of v = %e."%util.integrate(util.inner(v,util.tensor_mult(self.__permeability_inv,v)))**0.5
343 print "DarcyFlux: K^{-1}-norm of g2 = %e."%norm_g2
344 print "DarcyFlux: K-norm of grad(dp) = %e."%util.integrate(util.inner(Gp,KGp))**0.5
345 ATOL=atol+rtol*norm_g2
346 if self.verbose: print "DarcyFlux: absolute tolerance ATOL = %e."%(ATOL,)
347 if norm_r == None or norm_r>ATOL:
348 if num_corrections>max_num_corrections:
349 raise ValueError,"maximum number of correction steps reached."
350
351 if self.solveForFlux:
352 # initial residual is r=K^{-1}*(g-v-K*Gp)+D^*L^{-1}(f-Du)
353 v,r, norm_r=PCG(ArithmeticTuple(util.tensor_mult(self.__permeability_inv,g2-v)-Gp,self.__applWeight(v,self.__f),p),
354 self.__Aprod_v,
355 v,
356 self.__Msolve_PCG_v,
357 self.__inner_PCG_v,
358 atol=ATOL, rtol=0.,iter_max=max_iter, verbose=self.verbose)
359 p=r[2]
360 else:
361 # initial residual is r=G^*(g2-KGp - v)
362 p,r, norm_r=PCG(ArithmeticTuple(g2-KGp,v),
363 self.__Aprod_p,
364 p,
365 self.__Msolve_PCG_p,
366 self.__inner_PCG_p,
367 atol=ATOL, rtol=0.,iter_max=max_iter, verbose=self.verbose)
368 v=r[1]
369 if self.verbose: print "DarcyFlux: residual norm = %e."%norm_r
370 num_corrections+=1
371 else:
372 if self.verbose: print "DarcyFlux: stopping criterium reached."
373 converged=True
374 return v,p+p0
375 def setTolerance(self,rtol=1e-4):
376 """
377 sets the relative tolerance ``rtol`` used to terminate the solution process. The iteration is terminated if
378
379 *|g-v-K gard(p)|_PCG <= atol + rtol * |K^{1/2}g2|_0*
380
381 where ``atol`` is an absolut tolerance (see `setAbsoluteTolerance`).
382
383 :param rtol: relative tolerance for the pressure
384 :type rtol: non-negative ``float``
385 """
386 if rtol<0:
387 raise ValueError,"Relative tolerance needs to be non-negative."
388 self.__rtol=rtol
389 def getTolerance(self):
390 """
391 returns the relative tolerance
392 :return: current relative tolerance
393 :rtype: ``float``
394 """
395 return self.__rtol
396
397 def setAbsoluteTolerance(self,atol=0.):
398 """
399 sets the absolute tolerance ``atol`` used to terminate the solution process. The iteration is terminated if
400
401 *|g-v-K gard(p)|_PCG <= atol + rtol * |K^{1/2}g2|_0*
402
403
404 where ``rtol`` is an absolut tolerance (see `setTolerance`), *|f|^2 = integrate(length(f)^2)* and *(Qp)_i=k_{ij}p_{,j}* for the permeability *k_{ij}*.
405
406 :param atol: absolute tolerance for the pressure
407 :type atol: non-negative ``float``
408 """
409 if atol<0:
410 raise ValueError,"Absolute tolerance needs to be non-negative."
411 self.__atol=atol
412 def getAbsoluteTolerance(self):
413 """
414 returns the absolute tolerance
415 :return: current absolute tolerance
416 :rtype: ``float``
417 """
418 return self.__atol
419 def getSubProblemTolerance(self):
420 """
421 Returns a suitable subtolerance
422 :type: ``float``
423 """
424 return max(util.EPSILON**(0.5),self.getTolerance()**2)
425
426 def setSubProblemTolerance(self):
427 """
428 Sets the relative tolerance to solve the subproblem(s) if subtolerance adaption is selected.
429 """
430 if self.__adaptSubTolerance:
431 sub_tol=self.getSubProblemTolerance()
432 self.getSolverOptionsFlux().setTolerance(sub_tol)
433 self.getSolverOptionsFlux().setAbsoluteTolerance(0.)
434 self.getSolverOptionsPressure().setTolerance(sub_tol)
435 self.getSolverOptionsPressure().setAbsoluteTolerance(0.)
436 self.getSolverOptionsWeighting().setTolerance(sub_tol)
437 self.getSolverOptionsWeighting().setAbsoluteTolerance(0.)
438 if self.verbose: print "DarcyFlux: relative subtolerance is set to %e."%sub_tol
439
440
441 class DarcyFlowOld(object):
442 """
443 solves the problem
444
445 *u_i+k_{ij}*p_{,j} = g_i*
446 *u_{i,i} = f*
447
448 where *p* represents the pressure and *u* the Darcy flux. *k* represents the permeability,
449
450 :note: The problem is solved in a least squares formulation.
451 """
452
453 def __init__(self, domain, weight=None, useReduced=False, adaptSubTolerance=True):
454 """
455 initializes the Darcy flux problem
456 :param domain: domain of the problem
457 :type domain: `Domain`
458 :param useReduced: uses reduced oreder on flux and pressure
459 :type useReduced: ``bool``
460 :param adaptSubTolerance: switches on automatic subtolerance selection
461 :type adaptSubTolerance: ``bool``
462 """
463 self.domain=domain
464 if weight == None:
465 s=self.domain.getSize()
466 self.__l=(3.*util.longestEdge(self.domain)*s/util.sup(s))**2
467 # self.__l=(3.*util.longestEdge(self.domain))**2
468 #self.__l=(0.1*util.longestEdge(self.domain)*s/util.sup(s))**2
469 else:
470 self.__l=weight
471 self.__pde_v=LinearPDESystem(domain)
472 if useReduced: self.__pde_v.setReducedOrderOn()
473 self.__pde_v.setSymmetryOn()
474 self.__pde_v.setValue(D=util.kronecker(domain), A=self.__l*util.outer(util.kronecker(domain),util.kronecker(domain)))
475 self.__pde_p=LinearSinglePDE(domain)
476 self.__pde_p.setSymmetryOn()
477 if useReduced: self.__pde_p.setReducedOrderOn()
478 self.__f=Scalar(0,self.__pde_v.getFunctionSpaceForCoefficient("X"))
479 self.__g=Vector(0,self.__pde_v.getFunctionSpaceForCoefficient("Y"))
480 self.setTolerance()
481 self.setAbsoluteTolerance()
482 self.__adaptSubTolerance=adaptSubTolerance
483 self.verbose=False
484 def getSolverOptionsFlux(self):
485 """
486 Returns the solver options used to solve the flux problems
487
488 *(I+D^*D)u=F*
489
490 :return: `SolverOptions`
491 """
492 return self.__pde_v.getSolverOptions()
493 def setSolverOptionsFlux(self, options=None):
494 """
495 Sets the solver options used to solve the flux problems
496
497 *(I+D^*D)u=F*
498
499 If ``options`` is not present, the options are reset to default
500 :param options: `SolverOptions`
501 :note: if the adaption of subtolerance is choosen, the tolerance set by ``options`` will be overwritten before the solver is called.
502 """
503 return self.__pde_v.setSolverOptions(options)
504 def getSolverOptionsPressure(self):
505 """
506 Returns the solver options used to solve the pressure problems
507
508 *(Q^*Q)p=Q^*G*
509
510 :return: `SolverOptions`
511 """
512 return self.__pde_p.getSolverOptions()
513 def setSolverOptionsPressure(self, options=None):
514 """
515 Sets the solver options used to solve the pressure problems
516
517 *(Q^*Q)p=Q^*G*
518
519 If ``options`` is not present, the options are reset to default
520 :param options: `SolverOptions`
521 :note: if the adaption of subtolerance is choosen, the tolerance set by ``options`` will be overwritten before the solver is called.
522 """
523 return self.__pde_p.setSolverOptions(options)
524
525 def setValue(self,f=None, g=None, location_of_fixed_pressure=None, location_of_fixed_flux=None, permeability=None):
526 """
527 assigns values to model parameters
528
529 :param f: volumetic sources/sinks
530 :type f: scalar value on the domain (e.g. `Data`)
531 :param g: flux sources/sinks
532 :type g: vector values on the domain (e.g. `Data`)
533 :param location_of_fixed_pressure: mask for locations where pressure is fixed
534 :type location_of_fixed_pressure: scalar value on the domain (e.g. `Data`)
535 :param location_of_fixed_flux: mask for locations where flux is fixed.
536 :type location_of_fixed_flux: vector values on the domain (e.g. `Data`)
537 :param permeability: permeability tensor. If scalar ``s`` is given the tensor with
538 ``s`` on the main diagonal is used. If vector ``v`` is given the tensor with
539 ``v`` on the main diagonal is used.
540 :type permeability: scalar, vector or tensor values on the domain (e.g. `Data`)
541
542 :note: the values of parameters which are not set by calling ``setValue`` are not altered.
543 :note: at any point on the boundary of the domain the pressure (``location_of_fixed_pressure`` >0)
544 or the normal component of the flux (``location_of_fixed_flux[i]>0`` if direction of the normal
545 is along the *x_i* axis.
546 """
547 if f !=None:
548 f=util.interpolate(f, self.__pde_v.getFunctionSpaceForCoefficient("X"))
549 if f.isEmpty():
550 f=Scalar(0,self.__pde_v.getFunctionSpaceForCoefficient("X"))
551 else:
552 if f.getRank()>0: raise ValueError,"illegal rank of f."
553 self.__f=f
554 if g !=None:
555 g=util.interpolate(g, self.__pde_p.getFunctionSpaceForCoefficient("Y"))
556 if g.isEmpty():
557 g=Vector(0,self.__pde_v.getFunctionSpaceForCoefficient("Y"))
558 else:
559 if not g.getShape()==(self.domain.getDim(),):
560 raise ValueError,"illegal shape of g"
561 self.__g=g
562
563 if location_of_fixed_pressure!=None: self.__pde_p.setValue(q=location_of_fixed_pressure)
564 if location_of_fixed_flux!=None: self.__pde_v.setValue(q=location_of_fixed_flux)
565
566 if permeability!=None:
567 perm=util.interpolate(permeability,self.__pde_p.getFunctionSpaceForCoefficient("A"))
568 if perm.getRank()==0:
569 perm=perm*util.kronecker(self.domain.getDim())
570 elif perm.getRank()==1:
571 perm, perm2=Tensor(0.,self.__pde_p.getFunctionSpaceForCoefficient("A")), perm
572 for i in range(self.domain.getDim()): perm[i,i]=perm2[i]
573 elif perm.getRank()==2:
574 pass
575 else:
576 raise ValueError,"illegal rank of permeability."
577 self.__permeability=perm
578 self.__pde_p.setValue(A=util.transposed_tensor_mult(self.__permeability,self.__permeability))
579
580 def setTolerance(self,rtol=1e-4):
581 """
582 sets the relative tolerance ``rtol`` used to terminate the solution process. The iteration is terminated if
583
584 *|g-v-Qp| <= atol + rtol * min( max( |g-v|, |Qp| ), max( |v|, |g-Qp| ) )*
585
586 where ``atol`` is an absolut tolerance (see `setAbsoluteTolerance`), *|f|^2 = integrate(length(f)^2)* and *(Qp)_i=k_{ij}p_{,j}* for the permeability *k_{ij}*.
587
588 :param rtol: relative tolerance for the pressure
589 :type rtol: non-negative ``float``
590 """
591 if rtol<0:
592 raise ValueError,"Relative tolerance needs to be non-negative."
593 self.__rtol=rtol
594 def getTolerance(self):
595 """
596 returns the relative tolerance
597
598 :return: current relative tolerance
599 :rtype: ``float``
600 """
601 return self.__rtol
602
603 def setAbsoluteTolerance(self,atol=0.):
604 """
605 sets the absolute tolerance ``atol`` used to terminate the solution process. The iteration is terminated if
606
607 *|g-v-Qp| <= atol + rtol * min( max( |g-v|, |Qp| ), max( |v|, |g-Qp| ) )*
608
609 where ``rtol`` is an absolut tolerance (see `setTolerance`), *|f|^2 = integrate(length(f)^2)* and *(Qp)_i=k_{ij}p_{,j}* for the permeability *k_{ij}*.
610
611 :param atol: absolute tolerance for the pressure
612 :type atol: non-negative ``float``
613 """
614 if atol<0:
615 raise ValueError,"Absolute tolerance needs to be non-negative."
616 self.__atol=atol
617 def getAbsoluteTolerance(self):
618 """
619 returns the absolute tolerance
620
621 :return: current absolute tolerance
622 :rtype: ``float``
623 """
624 return self.__atol
625 def getSubProblemTolerance(self):
626 """
627 Returns a suitable subtolerance
628 @type: ``float``
629 """
630 return max(util.EPSILON**(0.75),self.getTolerance()**2)
631 def setSubProblemTolerance(self):
632 """
633 Sets the relative tolerance to solve the subproblem(s) if subtolerance adaption is selected.
634 """
635 if self.__adaptSubTolerance:
636 sub_tol=self.getSubProblemTolerance()
637 self.getSolverOptionsFlux().setTolerance(sub_tol)
638 self.getSolverOptionsFlux().setAbsoluteTolerance(0.)
639 self.getSolverOptionsPressure().setTolerance(sub_tol)
640 self.getSolverOptionsPressure().setAbsoluteTolerance(0.)
641 if self.verbose: print "DarcyFlux: relative subtolerance is set to %e."%sub_tol
642
643 def solve(self,u0,p0, max_iter=100, verbose=False, max_num_corrections=10):
644 """
645 solves the problem.
646
647 The iteration is terminated if the residual norm is less then self.getTolerance().
648
649 :param u0: initial guess for the flux. At locations in the domain marked by ``location_of_fixed_flux`` the value of ``u0`` is kept unchanged.
650 :type u0: vector value on the domain (e.g. `Data`).
651 :param p0: initial guess for the pressure. At locations in the domain marked by ``location_of_fixed_pressure`` the value of ``p0`` is kept unchanged.
652 :type p0: scalar value on the domain (e.g. `Data`).
653 :param verbose: if set some information on iteration progress are printed
654 :type verbose: ``bool``
655 :return: flux and pressure
656 :rtype: ``tuple`` of `Data`.
657
658 :note: The problem is solved as a least squares form
659
660 *(I+D^*D)u+Qp=D^*f+g*
661 *Q^*u+Q^*Qp=Q^*g*
662
663 where *D* is the *div* operator and *(Qp)_i=k_{ij}p_{,j}* for the permeability *k_{ij}*.
664 We eliminate the flux form the problem by setting
665
666 *u=(I+D^*D)^{-1}(D^*f-g-Qp)* with u=u0 on location_of_fixed_flux
667
668 form the first equation. Inserted into the second equation we get
669
670 *Q^*(I-(I+D^*D)^{-1})Qp= Q^*(g-(I+D^*D)^{-1}(D^*f+g))* with p=p0 on location_of_fixed_pressure
671
672 which is solved using the PCG method (precondition is *Q^*Q*). In each iteration step
673 PDEs with operator *I+D^*D* and with *Q^*Q* needs to be solved using a sub iteration scheme.
674 """
675 self.verbose=verbose
676 rtol=self.getTolerance()
677 atol=self.getAbsoluteTolerance()
678 self.setSubProblemTolerance()
679 num_corrections=0
680 converged=False
681 p=p0
682 norm_r=None
683 while not converged:
684 v=self.getFlux(p, fixed_flux=u0)
685 Qp=self.__Q(p)
686 norm_v=self.__L2(v)
687 norm_Qp=self.__L2(Qp)
688 if norm_v == 0.:
689 if norm_Qp == 0.:
690 return v,p
691 else:
692 fac=norm_Qp
693 else:
694 if norm_Qp == 0.:
695 fac=norm_v
696 else:
697 fac=2./(1./norm_v+1./norm_Qp)
698 ATOL=(atol+rtol*fac)
699 if self.verbose:
700 print "DarcyFlux: L2 norm of v = %e."%norm_v
701 print "DarcyFlux: L2 norm of k.util.grad(p) = %e."%norm_Qp
702 print "DarcyFlux: L2 defect u = %e."%(util.integrate(util.length(self.__g-util.interpolate(v,Function(self.domain))-Qp)**2)**(0.5),)
703 print "DarcyFlux: L2 defect div(v) = %e."%(util.integrate((self.__f-util.div(v))**2)**(0.5),)
704 print "DarcyFlux: absolute tolerance ATOL = %e."%ATOL
705 if norm_r == None or norm_r>ATOL:
706 if num_corrections>max_num_corrections:
707 raise ValueError,"maximum number of correction steps reached."
708 p,r, norm_r=PCG(self.__g-util.interpolate(v,Function(self.domain))-Qp,self.__Aprod,p,self.__Msolve_PCG,self.__inner_PCG,atol=0.5*ATOL, rtol=0.,iter_max=max_iter, verbose=self.verbose)
709 num_corrections+=1
710 else:
711 converged=True
712 return v,p
713 def __L2(self,v):
714 return util.sqrt(util.integrate(util.length(util.interpolate(v,Function(self.domain)))**2))
715
716 def __Q(self,p):
717 return util.tensor_mult(self.__permeability,util.grad(p))
718
719 def __Aprod(self,dp):
720 if self.getSolverOptionsFlux().isVerbose(): print "DarcyFlux: Applying operator"
721 Qdp=self.__Q(dp)
722 self.__pde_v.setValue(Y=-Qdp,X=Data(), r=Data())
723 du=self.__pde_v.getSolution()
724 # self.__pde_v.getOperator().saveMM("proj.mm")
725 return Qdp+du
726 def __inner_GMRES(self,r,s):
727 return util.integrate(util.inner(r,s))
728
729 def __inner_PCG(self,p,r):
730 return util.integrate(util.inner(self.__Q(p), r))
731
732 def __Msolve_PCG(self,r):
733 if self.getSolverOptionsPressure().isVerbose(): print "DarcyFlux: Applying preconditioner"
734 self.__pde_p.setValue(X=util.transposed_tensor_mult(self.__permeability,r), Y=Data(), r=Data())
735 # self.__pde_p.getOperator().saveMM("prec.mm")
736 return self.__pde_p.getSolution()
737
738 def getFlux(self,p=None, fixed_flux=Data()):
739 """
740 returns the flux for a given pressure ``p`` where the flux is equal to ``fixed_flux``
741 on locations where ``location_of_fixed_flux`` is positive (see `setValue`).
742 Note that ``g`` and ``f`` are used, see `setValue`.
743
744 :param p: pressure.
745 :type p: scalar value on the domain (e.g. `Data`).
746 :param fixed_flux: flux on the locations of the domain marked be ``location_of_fixed_flux``.
747 :type fixed_flux: vector values on the domain (e.g. `Data`).
748 :return: flux
749 :rtype: `Data`
750 :note: the method uses the least squares solution *u=(I+D^*D)^{-1}(D^*f-g-Qp)* where *D* is the *div* operator and *(Qp)_i=k_{ij}p_{,j}*
751 for the permeability *k_{ij}*
752 """
753 self.setSubProblemTolerance()
754 g=self.__g
755 f=self.__f
756 self.__pde_v.setValue(X=self.__l*f*util.kronecker(self.domain), r=fixed_flux)
757 if p == None:
758 self.__pde_v.setValue(Y=g)
759 else:
760 self.__pde_v.setValue(Y=g-self.__Q(p))
761 return self.__pde_v.getSolution()
762
763 class StokesProblemCartesian(HomogeneousSaddlePointProblem):
764 """
765 solves
766
767 -(eta*(u_{i,j}+u_{j,i}))_j + p_i = f_i-stress_{ij,j}
768 u_{i,i}=0
769
770 u=0 where fixed_u_mask>0
771 eta*(u_{i,j}+u_{j,i})*n_j-p*n_i=surface_stress +stress_{ij}n_j
772
773 if surface_stress is not given 0 is assumed.
774
775 typical usage:
776
777 sp=StokesProblemCartesian(domain)
778 sp.setTolerance()
779 sp.initialize(...)
780 v,p=sp.solve(v0,p0)
781 """
782 def __init__(self,domain,**kwargs):
783 """
784 initialize the Stokes Problem
785
786 The approximation spaces used for velocity (=Solution(domain)) and pressure (=ReducedSolution(domain)) must be
787 LBB complient, for instance using quadratic and linear approximation on the same element or using linear approximation
788 with macro elements for the pressure.
789
790 :param domain: domain of the problem.
791 :type domain: `Domain`
792 """
793 HomogeneousSaddlePointProblem.__init__(self,**kwargs)
794 self.domain=domain
795 self.__pde_u=LinearPDE(domain,numEquations=self.domain.getDim(),numSolutions=self.domain.getDim())
796 self.__pde_u.setSymmetryOn()
797
798 self.__pde_prec=LinearPDE(domain)
799 self.__pde_prec.setReducedOrderOn()
800 self.__pde_prec.setSymmetryOn()
801
802 self.__pde_proj=LinearPDE(domain)
803 self.__pde_proj.setReducedOrderOn()
804 self.__pde_proj.setValue(D=1)
805 self.__pde_proj.setSymmetryOn()
806
807 def getSolverOptionsVelocity(self):
808 """
809 returns the solver options used solve the equation for velocity.
810
811 :rtype: `SolverOptions`
812 """
813 return self.__pde_u.getSolverOptions()
814 def setSolverOptionsVelocity(self, options=None):
815 """
816 set the solver options for solving the equation for velocity.
817
818 :param options: new solver options
819 :type options: `SolverOptions`
820 """
821 self.__pde_u.setSolverOptions(options)
822 def getSolverOptionsPressure(self):
823 """
824 returns the solver options used solve the equation for pressure.
825 :rtype: `SolverOptions`
826 """
827 return self.__pde_prec.getSolverOptions()
828 def setSolverOptionsPressure(self, options=None):
829 """
830 set the solver options for solving the equation for pressure.
831 :param options: new solver options
832 :type options: `SolverOptions`
833 """
834 self.__pde_prec.setSolverOptions(options)
835
836 def setSolverOptionsDiv(self, options=None):
837 """
838 set the solver options for solving the equation to project the divergence of
839 the velocity onto the function space of presure.
840
841 :param options: new solver options
842 :type options: `SolverOptions`
843 """
844 self.__pde_proj.setSolverOptions(options)
845 def getSolverOptionsDiv(self):
846 """
847 returns the solver options for solving the equation to project the divergence of
848 the velocity onto the function space of presure.
849
850 :rtype: `SolverOptions`
851 """
852 return self.__pde_proj.getSolverOptions()
853
854 def updateStokesEquation(self, v, p):
855 """
856 updates the Stokes equation to consider dependencies from ``v`` and ``p``
857 :note: This method can be overwritten by a subclass. Use `setStokesEquation` to set new values.
858 """
859 pass
860 def setStokesEquation(self, f=None,fixed_u_mask=None,eta=None,surface_stress=None,stress=None, restoration_factor=None):
861 """
862 assigns new values to the model parameters.
863
864 :param f: external force
865 :type f: `Vector` object in `FunctionSpace` `Function` or similar
866 :param fixed_u_mask: mask of locations with fixed velocity.
867 :type fixed_u_mask: `Vector` object on `FunctionSpace` `Solution` or similar
868 :param eta: viscosity
869 :type eta: `Scalar` object on `FunctionSpace` `Function` or similar
870 :param surface_stress: normal surface stress
871 :type surface_stress: `Vector` object on `FunctionSpace` `FunctionOnBoundary` or similar
872 :param stress: initial stress
873 :type stress: `Tensor` object on `FunctionSpace` `Function` or similar
874 """
875 if eta !=None:
876 k=util.kronecker(self.domain.getDim())
877 kk=util.outer(k,k)
878 self.eta=util.interpolate(eta, Function(self.domain))
879 self.__pde_prec.setValue(D=1/self.eta)
880 self.__pde_u.setValue(A=self.eta*(util.swap_axes(kk,0,3)+util.swap_axes(kk,1,3)))
881 if restoration_factor!=None:
882 n=self.domain.getNormal()
883 self.__pde_u.setValue(d=restoration_factor*util.outer(n,n))
884 if fixed_u_mask!=None:
885 self.__pde_u.setValue(q=fixed_u_mask)
886 if f!=None: self.__f=f
887 if surface_stress!=None: self.__surface_stress=surface_stress
888 if stress!=None: self.__stress=stress
889
890 def initialize(self,f=Data(),fixed_u_mask=Data(),eta=1,surface_stress=Data(),stress=Data(), restoration_factor=0):
891 """
892 assigns values to the model parameters
893
894 :param f: external force
895 :type f: `Vector` object in `FunctionSpace` `Function` or similar
896 :param fixed_u_mask: mask of locations with fixed velocity.
897 :type fixed_u_mask: `Vector` object on `FunctionSpace` `Solution` or similar
898 :param eta: viscosity
899 :type eta: `Scalar` object on `FunctionSpace` `Function` or similar
900 :param surface_stress: normal surface stress
901 :type surface_stress: `Vector` object on `FunctionSpace` `FunctionOnBoundary` or similar
902 :param stress: initial stress
903 :type stress: `Tensor` object on `FunctionSpace` `Function` or similar
904 """
905 self.setStokesEquation(f,fixed_u_mask, eta, surface_stress, stress, restoration_factor)
906
907 def Bv(self,v,tol):
908 """
909 returns inner product of element p and div(v)
910
911 :param v: a residual
912 :return: inner product of element p and div(v)
913 :rtype: ``float``
914 """
915 self.__pde_proj.setValue(Y=-util.div(v))
916 self.getSolverOptionsDiv().setTolerance(tol)
917 self.getSolverOptionsDiv().setAbsoluteTolerance(0.)
918 out=self.__pde_proj.getSolution()
919 return out
920
921 def inner_pBv(self,p,Bv):
922 """
923 returns inner product of element p and Bv=-div(v)
924
925 :param p: a pressure increment
926 :param Bv: a residual
927 :return: inner product of element p and Bv=-div(v)
928 :rtype: ``float``
929 """
930 return util.integrate(util.interpolate(p,Function(self.domain))*util.interpolate(Bv,Function(self.domain)))
931
932 def inner_p(self,p0,p1):
933 """
934 Returns inner product of p0 and p1
935
936 :param p0: a pressure
937 :param p1: a pressure
938 :return: inner product of p0 and p1
939 :rtype: ``float``
940 """
941 s0=util.interpolate(p0,Function(self.domain))
942 s1=util.interpolate(p1,Function(self.domain))
943 return util.integrate(s0*s1)
944
945 def norm_v(self,v):
946 """
947 returns the norm of v
948
949 :param v: a velovity
950 :return: norm of v
951 :rtype: non-negative ``float``
952 """
953 return util.sqrt(util.integrate(util.length(util.grad(v))**2))
954
955
956 def getDV(self, p, v, tol):
957 """
958 return the value for v for a given p (overwrite)
959
960 :param p: a pressure
961 :param v: a initial guess for the value v to return.
962 :return: dv given as *Adv=(f-Av-B^*p)*
963 """
964 self.updateStokesEquation(v,p)
965 self.__pde_u.setValue(Y=self.__f, y=self.__surface_stress)
966 self.getSolverOptionsVelocity().setTolerance(tol)
967 self.getSolverOptionsVelocity().setAbsoluteTolerance(0.)
968 if self.__stress.isEmpty():
969 self.__pde_u.setValue(X=p*util.kronecker(self.domain)-2*self.eta*util.symmetric(util.grad(v)))
970 else:
971 self.__pde_u.setValue(X=self.__stress+p*util.kronecker(self.domain)-2*self.eta*util.symmetric(util.grad(v)))
972 out=self.__pde_u.getSolution()
973 return out
974
975 def norm_Bv(self,Bv):
976 """
977 Returns Bv (overwrite).
978
979 :rtype: equal to the type of p
980 :note: boundary conditions on p should be zero!
981 """
982 return util.sqrt(util.integrate(util.interpolate(Bv,Function(self.domain))**2))
983
984 def solve_AinvBt(self,p, tol):
985 """
986 Solves *Av=B^*p* with accuracy `tol`
987
988 :param p: a pressure increment
989 :return: the solution of *Av=B^*p*
990 :note: boundary conditions on v should be zero!
991 """
992 self.__pde_u.setValue(Y=Data(), y=Data(), X=-p*util.kronecker(self.domain))
993 out=self.__pde_u.getSolution()
994 return out
995
996 def solve_prec(self,Bv, tol):
997 """
998 applies preconditioner for for *BA^{-1}B^** to *Bv*
999 with accuracy `self.getSubProblemTolerance()`
1000
1001 :param Bv: velocity increment
1002 :return: *p=P(Bv)* where *P^{-1}* is an approximation of *BA^{-1}B^ * )*
1003 :note: boundary conditions on p are zero.
1004 """
1005 self.__pde_prec.setValue(Y=Bv)
1006 self.getSolverOptionsPressure().setTolerance(tol)
1007 self.getSolverOptionsPressure().setAbsoluteTolerance(0.)
1008 out=self.__pde_prec.getSolution()
1009 return out

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