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Revision 2288 - (show annotations)
Tue Feb 24 06:11:48 2009 UTC (10 years, 6 months ago) by gross
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a new idea for the sacling factor
1 ########################################################
2 #
3 # Copyright (c) 2003-2008 by University of Queensland
4 # Earth Systems Science Computational Center (ESSCC)
5 # http://www.uq.edu.au/esscc
6 #
7 # Primary Business: Queensland, Australia
8 # Licensed under the Open Software License version 3.0
9 # http://www.opensource.org/licenses/osl-3.0.php
10 #
11 ########################################################
12
13 __copyright__="""Copyright (c) 2003-2008 by University of Queensland
14 Earth Systems Science Computational Center (ESSCC)
15 http://www.uq.edu.au/esscc
16 Primary Business: Queensland, Australia"""
17 __license__="""Licensed under the Open Software License version 3.0
18 http://www.opensource.org/licenses/osl-3.0.php"""
19 __url__="http://www.uq.edu.au/esscc/escript-finley"
20
21 """
22 Some models for flow
23
24 @var __author__: name of author
25 @var __copyright__: copyrights
26 @var __license__: licence agreement
27 @var __url__: url entry point on documentation
28 @var __version__: version
29 @var __date__: date of the version
30 """
31
32 __author__="Lutz Gross, l.gross@uq.edu.au"
33
34 from escript import *
35 import util
36 from linearPDEs import LinearPDE, LinearPDESystem, LinearSinglePDE
37 from pdetools import HomogeneousSaddlePointProblem,Projector, ArithmeticTuple, PCG, NegativeNorm, GMRES
38
39 class DarcyFlow(object):
40 """
41 solves the problem
42
43 M{u_i+k_{ij}*p_{,j} = g_i}
44 M{u_{i,i} = f}
45
46 where M{p} represents the pressure and M{u} the Darcy flux. M{k} represents the permeability,
47
48 @note: The problem is solved in a least squares formulation.
49 """
50
51 def __init__(self, domain,useReduced=False):
52 """
53 initializes the Darcy flux problem
54 @param domain: domain of the problem
55 @type domain: L{Domain}
56 """
57 self.domain=domain
58 self.__l=util.longestEdge(self.domain)**2
59 self.__pde_v=LinearPDESystem(domain)
60 if useReduced: self.__pde_v.setReducedOrderOn()
61 self.__pde_v.setSymmetryOn()
62 self.__pde_v.setValue(D=util.kronecker(domain), A=self.__l*util.outer(util.kronecker(domain),util.kronecker(domain)))
63 self.__pde_p=LinearSinglePDE(domain)
64 self.__pde_p.setSymmetryOn()
65 if useReduced: self.__pde_p.setReducedOrderOn()
66 self.__f=Scalar(0,self.__pde_v.getFunctionSpaceForCoefficient("X"))
67 self.__g=Vector(0,self.__pde_v.getFunctionSpaceForCoefficient("Y"))
68 self.setTolerance()
69 self.setAbsoluteTolerance()
70 self.setSubProblemTolerance()
71
72 def setValue(self,f=None, g=None, location_of_fixed_pressure=None, location_of_fixed_flux=None, permeability=None):
73 """
74 assigns values to model parameters
75
76 @param f: volumetic sources/sinks
77 @type f: scalar value on the domain (e.g. L{Data})
78 @param g: flux sources/sinks
79 @type g: vector values on the domain (e.g. L{Data})
80 @param location_of_fixed_pressure: mask for locations where pressure is fixed
81 @type location_of_fixed_pressure: scalar value on the domain (e.g. L{Data})
82 @param location_of_fixed_flux: mask for locations where flux is fixed.
83 @type location_of_fixed_flux: vector values on the domain (e.g. L{Data})
84 @param permeability: permeability tensor. If scalar C{s} is given the tensor with
85 C{s} on the main diagonal is used. If vector C{v} is given the tensor with
86 C{v} on the main diagonal is used.
87 @type permeability: scalar, vector or tensor values on the domain (e.g. L{Data})
88
89 @note: the values of parameters which are not set by calling C{setValue} are not altered.
90 @note: at any point on the boundary of the domain the pressure (C{location_of_fixed_pressure} >0)
91 or the normal component of the flux (C{location_of_fixed_flux[i]>0} if direction of the normal
92 is along the M{x_i} axis.
93 """
94 if f !=None:
95 f=util.interpolate(f, self.__pde_v.getFunctionSpaceForCoefficient("X"))
96 if f.isEmpty():
97 f=Scalar(0,self.__pde_v.getFunctionSpaceForCoefficient("X"))
98 else:
99 if f.getRank()>0: raise ValueError,"illegal rank of f."
100 self.__f=f
101 if g !=None:
102 g=util.interpolate(g, self.__pde_p.getFunctionSpaceForCoefficient("Y"))
103 if g.isEmpty():
104 g=Vector(0,self.__pde_v.getFunctionSpaceForCoefficient("Y"))
105 else:
106 if not g.getShape()==(self.domain.getDim(),):
107 raise ValueError,"illegal shape of g"
108 self.__g=g
109
110 if location_of_fixed_pressure!=None: self.__pde_p.setValue(q=location_of_fixed_pressure)
111 if location_of_fixed_flux!=None: self.__pde_v.setValue(q=location_of_fixed_flux)
112
113 if permeability!=None:
114 perm=util.interpolate(permeability,self.__pde_p.getFunctionSpaceForCoefficient("A"))
115 if perm.getRank()==0:
116 perm=perm*util.kronecker(self.domain.getDim())
117 elif perm.getRank()==1:
118 perm, perm2=Tensor(0.,self.__pde_p.getFunctionSpaceForCoefficient("A")), perm
119 for i in range(self.domain.getDim()): perm[i,i]=perm2[i]
120 elif perm.getRank()==2:
121 pass
122 else:
123 raise ValueError,"illegal rank of permeability."
124 self.__permeability=perm
125 self.__pde_p.setValue(A=util.transposed_tensor_mult(self.__permeability,self.__permeability))
126
127 def setTolerance(self,rtol=1e-4):
128 """
129 sets the relative tolerance C{rtol} used to terminate the solution process. The iteration is terminated if
130
131 M{|g-v-Qp| <= atol + rtol * min( max( |g-v|, |Qp| ), max( |v|, |g-Qp| ) ) }
132
133 where C{atol} is an absolut tolerance (see L{setAbsoluteTolerance}), M{|f|^2 = integrate(length(f)^2)} and M{(Qp)_i=k_{ij}p_{,j}} for the permeability M{k_{ij}}.
134
135 @param rtol: relative tolerance for the pressure
136 @type rtol: non-negative C{float}
137 """
138 if rtol<0:
139 raise ValueError,"Relative tolerance needs to be non-negative."
140 self.__rtol=rtol
141 def getTolerance(self):
142 """
143 returns the relative tolerance
144
145 @return: current relative tolerance
146 @rtype: C{float}
147 """
148 return self.__rtol
149
150 def setAbsoluteTolerance(self,atol=0.):
151 """
152 sets the absolute tolerance C{atol} used to terminate the solution process. The iteration is terminated if
153
154 M{|g-v-Qp| <= atol + rtol * min( max( |g-v|, |Qp| ), max( |v|, |g-Qp| ) ) }
155
156 where C{rtol} is an absolut tolerance (see L{setTolerance}), M{|f|^2 = integrate(length(f)^2)} and M{(Qp)_i=k_{ij}p_{,j}} for the permeability M{k_{ij}}.
157
158 @param atol: absolute tolerance for the pressure
159 @type atol: non-negative C{float}
160 """
161 if atol<0:
162 raise ValueError,"Absolute tolerance needs to be non-negative."
163 self.__atol=atol
164 def getAbsoluteTolerance(self):
165 """
166 returns the absolute tolerance
167
168 @return: current absolute tolerance
169 @rtype: C{float}
170 """
171 return self.__atol
172
173 def setSubProblemTolerance(self,rtol=None):
174 """
175 Sets the relative tolerance to solve the subproblem(s). If C{rtol} is not present
176 C{self.getTolerance()**2} is used.
177
178 @param rtol: relative tolerence
179 @type rtol: positive C{float}
180 """
181 if rtol == None:
182 if self.getTolerance()<=0.:
183 raise ValueError,"A positive relative tolerance must be set."
184 self.__sub_tol=max(util.EPSILON**(0.75),self.getTolerance()**2)
185 else:
186 if rtol<=0:
187 raise ValueError,"sub-problem tolerance must be positive."
188 self.__sub_tol=max(util.EPSILON**(0.75),rtol)
189
190 def getSubProblemTolerance(self):
191 """
192 Returns the subproblem reduction factor.
193
194 @return: subproblem reduction factor
195 @rtype: C{float}
196 """
197 return self.__sub_tol
198
199 def solve(self,u0,p0, max_iter=100, verbose=False, show_details=False, max_num_corrections=10):
200 """
201 solves the problem.
202
203 The iteration is terminated if the residual norm is less then self.getTolerance().
204
205 @param u0: initial guess for the flux. At locations in the domain marked by C{location_of_fixed_flux} the value of C{u0} is kept unchanged.
206 @type u0: vector value on the domain (e.g. L{Data}).
207 @param p0: initial guess for the pressure. At locations in the domain marked by C{location_of_fixed_pressure} the value of C{p0} is kept unchanged.
208 @type p0: scalar value on the domain (e.g. L{Data}).
209 @param verbose: if set some information on iteration progress are printed
210 @type verbose: C{bool}
211 @param show_details: if set information on the subiteration process are printed.
212 @type show_details: C{bool}
213 @return: flux and pressure
214 @rtype: C{tuple} of L{Data}.
215
216 @note: The problem is solved as a least squares form
217
218 M{(I+D^*D)u+Qp=D^*f+g}
219 M{Q^*u+Q^*Qp=Q^*g}
220
221 where M{D} is the M{div} operator and M{(Qp)_i=k_{ij}p_{,j}} for the permeability M{k_{ij}}.
222 We eliminate the flux form the problem by setting
223
224 M{u=(I+D^*D)^{-1}(D^*f-g-Qp)} with u=u0 on location_of_fixed_flux
225
226 form the first equation. Inserted into the second equation we get
227
228 M{Q^*(I-(I+D^*D)^{-1})Qp= Q^*(g-(I+D^*D)^{-1}(D^*f+g))} with p=p0 on location_of_fixed_pressure
229
230 which is solved using the PCG method (precondition is M{Q^*Q}). In each iteration step
231 PDEs with operator M{I+D^*D} and with M{Q^*Q} needs to be solved using a sub iteration scheme.
232 """
233 self.verbose=verbose or True
234 self.show_details= show_details and self.verbose
235 rtol=self.getTolerance()
236 atol=self.getAbsoluteTolerance()
237 if self.verbose: print "DarcyFlux: initial sub tolerance = %e"%self.getSubProblemTolerance()
238
239 num_corrections=0
240 converged=False
241 p=p0
242 norm_r=None
243 while not converged:
244 v=self.getFlux(p, fixed_flux=u0, show_details=self.show_details)
245 Qp=self.__Q(p)
246 norm_v=self.__L2(v)
247 norm_Qp=self.__L2(Qp)
248 if norm_v == 0.:
249 if norm_Qp == 0.:
250 return v,p
251 else:
252 fac=norm_Qp
253 else:
254 if norm_Qp == 0.:
255 fac=norm_v
256 else:
257 fac=2./(1./norm_v+1./norm_Qp)
258 ATOL=(atol+rtol*fac)
259 if self.verbose:
260 print "DarcyFlux: L2 norm of v = %e."%norm_v
261 print "DarcyFlux: L2 norm of k.grad(p) = %e."%norm_Qp
262 print "DarcyFlux: absolute tolerance ATOL = %e."%ATOL
263 if norm_r == None or norm_r>ATOL:
264 if num_corrections>max_num_corrections:
265 raise ValueError,"maximum number of correction steps reached."
266 p,r, norm_r=PCG(self.__g-util.interpolate(v,Function(self.domain))-Qp,self.__Aprod,p,self.__Msolve_PCG,self.__inner_PCG,atol=0.1*ATOL, rtol=0.,iter_max=max_iter, verbose=self.verbose)
267 num_corrections+=1
268 else:
269 converged=True
270 return v,p
271 #
272 #
273 # r_hat=g-util.interpolate(v,Function(self.domain))-Qp
274 # #===========================================================================
275 # norm_r_hat=self.__L2(r_hat)
276 # norm_v=self.__L2(v)
277 # norm_g=self.__L2(g)
278 # norm_gv=self.__L2(g-v)
279 # norm_Qp=self.__L2(Qp)
280 # norm_gQp=self.__L2(g-Qp)
281 # fac=min(max(norm_v,norm_gQp),max(norm_Qp,norm_gv))
282 # fac=min(norm_v,norm_Qp,norm_gv)
283 # norm_r_hat_PCG=util.sqrt(self.__inner_PCG(self.__Msolve_PCG(r_hat),r_hat))
284 # print "norm_r_hat = ",norm_r_hat,norm_r_hat_PCG, norm_r_hat_PCG/norm_r_hat
285 # if r!=None:
286 # print "diff = ",self.__L2(r-r_hat)/norm_r_hat
287 # sub_tol=min(rtol/self.__L2(r-r_hat)*norm_r_hat,1.)*self.getSubProblemTolerance()
288 # self.setSubProblemTolerance(sub_tol)
289 # print "subtol_new=",self.getSubProblemTolerance()
290 # print "norm_v = ",norm_v
291 # print "norm_gv = ",norm_gv
292 # print "norm_Qp = ",norm_Qp
293 # print "norm_gQp = ",norm_gQp
294 # print "norm_g = ",norm_g
295 # print "max(norm_v,norm_gQp)=",max(norm_v,norm_gQp)
296 # print "max(norm_Qp,norm_gv)=",max(norm_Qp,norm_gv)
297 # if fac == 0:
298 # if self.verbose: print "DarcyFlux: trivial case!"
299 # return v,p
300 # #===============================================================================
301 # # norm_v=util.sqrt(self.__inner_PCG(self.__Msolve_PCG(v),v))
302 # # norm_Qp=self.__L2(Qp)
303 # norm_r_hat=util.sqrt(self.__inner_PCG(self.__Msolve_PCG(r_hat),r_hat))
304 # # print "**** norm_v, norm_Qp :",norm_v,norm_Qp
305 #
306 # ATOL=(atol+rtol*2./(1./norm_v+1./norm_Qp))
307 # if self.verbose:
308 # print "DarcyFlux: residual = %e"%norm_r_hat
309 # print "DarcyFlux: absolute tolerance ATOL = %e."%ATOL
310 # if norm_r_hat <= ATOL:
311 # print "DarcyFlux: iteration finalized."
312 # converged=True
313 # else:
314 # # p=GMRES(r_hat,self.__Aprod, p, self.__inner_GMRES, atol=ATOL, rtol=0., iter_max=max_iter, iter_restart=20, verbose=self.verbose,P_R=self.__Msolve_PCG)
315 # # p,r=PCG(r_hat,self.__Aprod,p,self.__Msolve_PCG,self.__inner_PCG,atol=ATOL*min(0.1,norm_r_hat_PCG/norm_r_hat), rtol=0.,iter_max=max_iter, verbose=self.verbose)
316 # p,r, norm_r=PCG(r_hat,self.__Aprod,p,self.__Msolve_PCG,self.__inner_PCG,atol=0.1*ATOL, rtol=0.,iter_max=max_iter, verbose=self.verbose)
317 # print "norm_r =",norm_r
318 # return v,p
319 def __L2(self,v):
320 return util.sqrt(util.integrate(util.length(util.interpolate(v,Function(self.domain)))**2))
321
322 def __Q(self,p):
323 return util.tensor_mult(self.__permeability,util.grad(p))
324
325 def __Aprod(self,dp):
326 self.__pde_v.setTolerance(self.getSubProblemTolerance())
327 if self.show_details: print "DarcyFlux: Applying operator"
328 Qdp=self.__Q(dp)
329 self.__pde_v.setValue(Y=-Qdp,X=Data(), r=Data())
330 du=self.__pde_v.getSolution(verbose=self.show_details)
331 return Qdp+du
332 def __inner_GMRES(self,r,s):
333 return util.integrate(util.inner(r,s))
334
335 def __inner_PCG(self,p,r):
336 return util.integrate(util.inner(self.__Q(p), r))
337
338 def __Msolve_PCG(self,r):
339 self.__pde_p.setTolerance(self.getSubProblemTolerance())
340 if self.show_details: print "DarcyFlux: Applying preconditioner"
341 self.__pde_p.setValue(X=util.transposed_tensor_mult(self.__permeability,r), Y=Data(), r=Data())
342 return self.__pde_p.getSolution(verbose=self.show_details)
343
344 def getFlux(self,p=None, fixed_flux=Data(), show_details=False):
345 """
346 returns the flux for a given pressure C{p} where the flux is equal to C{fixed_flux}
347 on locations where C{location_of_fixed_flux} is positive (see L{setValue}).
348 Note that C{g} and C{f} are used, see L{setValue}.
349
350 @param p: pressure.
351 @type p: scalar value on the domain (e.g. L{Data}).
352 @param fixed_flux: flux on the locations of the domain marked be C{location_of_fixed_flux}.
353 @type fixed_flux: vector values on the domain (e.g. L{Data}).
354 @param tol: relative tolerance to be used.
355 @type tol: positive C{float}.
356 @return: flux
357 @rtype: L{Data}
358 @note: the method uses the least squares solution M{u=(I+D^*D)^{-1}(D^*f-g-Qp)} where M{D} is the M{div} operator and M{(Qp)_i=k_{ij}p_{,j}}
359 for the permeability M{k_{ij}}
360 """
361 self.__pde_v.setTolerance(self.getSubProblemTolerance())
362 g=self.__g
363 f=self.__f
364 self.__pde_v.setValue(X=self.__l*f*util.kronecker(self.domain), r=fixed_flux)
365 if p == None:
366 self.__pde_v.setValue(Y=g)
367 else:
368 self.__pde_v.setValue(Y=g-self.__Q(p))
369 return self.__pde_v.getSolution(verbose=show_details)
370
371 class StokesProblemCartesian(HomogeneousSaddlePointProblem):
372 """
373 solves
374
375 -(eta*(u_{i,j}+u_{j,i}))_j + p_i = f_i-stress_{ij,j}
376 u_{i,i}=0
377
378 u=0 where fixed_u_mask>0
379 eta*(u_{i,j}+u_{j,i})*n_j-p*n_i=surface_stress +stress_{ij}n_j
380
381 if surface_stress is not given 0 is assumed.
382
383 typical usage:
384
385 sp=StokesProblemCartesian(domain)
386 sp.setTolerance()
387 sp.initialize(...)
388 v,p=sp.solve(v0,p0)
389 """
390 def __init__(self,domain,**kwargs):
391 """
392 initialize the Stokes Problem
393
394 @param domain: domain of the problem. The approximation order needs to be two.
395 @type domain: L{Domain}
396 @warning: The apprximation order needs to be two otherwise you may see oscilations in the pressure.
397 """
398 HomogeneousSaddlePointProblem.__init__(self,**kwargs)
399 self.domain=domain
400 self.vol=util.integrate(1.,Function(self.domain))
401 self.__pde_u=LinearPDE(domain,numEquations=self.domain.getDim(),numSolutions=self.domain.getDim())
402 self.__pde_u.setSymmetryOn()
403 # self.__pde_u.setSolverMethod(self.__pde_u.DIRECT)
404 # self.__pde_u.setSolverMethod(preconditioner=LinearPDE.RILU)
405
406 self.__pde_prec=LinearPDE(domain)
407 self.__pde_prec.setReducedOrderOn()
408 # self.__pde_prec.setSolverMethod(self.__pde_prec.LUMPING)
409 self.__pde_prec.setSymmetryOn()
410
411 def initialize(self,f=Data(),fixed_u_mask=Data(),eta=1,surface_stress=Data(),stress=Data()):
412 """
413 assigns values to the model parameters
414
415 @param f: external force
416 @type f: L{Vector} object in L{FunctionSpace} L{Function} or similar
417 @param fixed_u_mask: mask of locations with fixed velocity.
418 @type fixed_u_mask: L{Vector} object on L{FunctionSpace} L{Solution} or similar
419 @param eta: viscosity
420 @type eta: L{Scalar} object on L{FunctionSpace} L{Function} or similar
421 @param surface_stress: normal surface stress
422 @type eta: L{Vector} object on L{FunctionSpace} L{FunctionOnBoundary} or similar
423 @param stress: initial stress
424 @type stress: L{Tensor} object on L{FunctionSpace} L{Function} or similar
425 @note: All values needs to be set.
426
427 """
428 self.eta=eta
429 A =self.__pde_u.createCoefficient("A")
430 self.__pde_u.setValue(A=Data())
431 for i in range(self.domain.getDim()):
432 for j in range(self.domain.getDim()):
433 A[i,j,j,i] += 1.
434 A[i,j,i,j] += 1.
435 self.__pde_prec.setValue(D=1/self.eta)
436 self.__pde_u.setValue(A=A*self.eta,q=fixed_u_mask)
437 self.__f=f
438 self.__surface_stress=surface_stress
439 self.__stress=stress
440
441 def inner_pBv(self,p,v):
442 """
443 returns inner product of element p and div(v)
444
445 @param p: a pressure increment
446 @param v: a residual
447 @return: inner product of element p and div(v)
448 @rtype: C{float}
449 """
450 return util.integrate(-p*util.div(v))
451
452 def inner_p(self,p0,p1):
453 """
454 Returns inner product of p0 and p1
455
456 @param p0: a pressure
457 @param p1: a pressure
458 @return: inner product of p0 and p1
459 @rtype: C{float}
460 """
461 s0=util.interpolate(p0/self.eta,Function(self.domain))
462 s1=util.interpolate(p1/self.eta,Function(self.domain))
463 return util.integrate(s0*s1)
464
465 def norm_v(self,v):
466 """
467 returns the norm of v
468
469 @param v: a velovity
470 @return: norm of v
471 @rtype: non-negative C{float}
472 """
473 return util.sqrt(util.integrate(util.length(util.grad(v))))
474
475 def getV(self, p, v0):
476 """
477 return the value for v for a given p (overwrite)
478
479 @param p: a pressure
480 @param v0: a initial guess for the value v to return.
481 @return: v given as M{v= A^{-1} (f-B^*p)}
482 """
483 self.__pde_u.setTolerance(self.getSubProblemTolerance())
484 self.__pde_u.setValue(Y=self.__f, y=self.__surface_stress, r=v0)
485 if self.__stress.isEmpty():
486 self.__pde_u.setValue(X=p*util.kronecker(self.domain))
487 else:
488 self.__pde_u.setValue(X=self.__stress+p*util.kronecker(self.domain))
489 out=self.__pde_u.getSolution(verbose=self.show_details)
490 return out
491
492
493 raise NotImplementedError,"no v calculation implemented."
494
495
496 def norm_Bv(self,v):
497 """
498 Returns Bv (overwrite).
499
500 @rtype: equal to the type of p
501 @note: boundary conditions on p should be zero!
502 """
503 return util.sqrt(util.integrate(util.div(v)**2))
504
505 def solve_AinvBt(self,p):
506 """
507 Solves M{Av=B^*p} with accuracy L{self.getSubProblemTolerance()}
508
509 @param p: a pressure increment
510 @return: the solution of M{Av=B^*p}
511 @note: boundary conditions on v should be zero!
512 """
513 self.__pde_u.setTolerance(self.getSubProblemTolerance())
514 self.__pde_u.setValue(Y=Data(), y=Data(), r=Data(),X=-p*util.kronecker(self.domain))
515 out=self.__pde_u.getSolution(verbose=self.show_details)
516 return out
517
518 def solve_precB(self,v):
519 """
520 applies preconditioner for for M{BA^{-1}B^*} to M{Bv}
521 with accuracy L{self.getSubProblemTolerance()} (overwrite).
522
523 @param v: velocity increment
524 @return: M{p=P(Bv)} where M{P^{-1}} is an approximation of M{BA^{-1}B^*}
525 @note: boundary conditions on p are zero.
526 """
527 self.__pde_prec.setValue(Y=-util.div(v))
528 self.__pde_prec.setTolerance(self.getSubProblemTolerance())
529 return self.__pde_prec.getSolution(verbose=self.show_details)

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