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Revision 2208 - (show annotations)
Mon Jan 12 06:37:07 2009 UTC (10 years, 8 months ago) by gross
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more work on the dary solver 


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
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.__pde_v=LinearPDESystem(domain)
59 if useReduced: self.__pde_v.setReducedOrderOn()
60 self.__pde_v.setSymmetryOn()
61 self.__pde_v.setValue(D=util.kronecker(domain), A=util.outer(util.kronecker(domain),util.kronecker(domain)))
62 self.__pde_p=LinearSinglePDE(domain)
63 self.__pde_p.setSymmetryOn()
64 if useReduced: self.__pde_p.setReducedOrderOn()
65 self.__f=Scalar(0,self.__pde_v.getFunctionSpaceForCoefficient("X"))
66 self.__g=Vector(0,self.__pde_v.getFunctionSpaceForCoefficient("Y"))
67 self.__ATOL= None
68
69 def setValue(self,f=None, g=None, location_of_fixed_pressure=None, location_of_fixed_flux=None, permeability=None):
70 """
71 assigns values to model parameters
72
73 @param f: volumetic sources/sinks
74 @type f: scalar value on the domain (e.g. L{Data})
75 @param g: flux sources/sinks
76 @type g: vector values on the domain (e.g. L{Data})
77 @param location_of_fixed_pressure: mask for locations where pressure is fixed
78 @type location_of_fixed_pressure: scalar value on the domain (e.g. L{Data})
79 @param location_of_fixed_flux: mask for locations where flux is fixed.
80 @type location_of_fixed_flux: vector values on the domain (e.g. L{Data})
81 @param permeability: permeability tensor. If scalar C{s} is given the tensor with
82 C{s} on the main diagonal is used. If vector C{v} is given the tensor with
83 C{v} on the main diagonal is used.
84 @type permeability: scalar, vector or tensor values on the domain (e.g. L{Data})
85
86 @note: the values of parameters which are not set by calling C{setValue} are not altered.
87 @note: at any point on the boundary of the domain the pressure (C{location_of_fixed_pressure} >0)
88 or the normal component of the flux (C{location_of_fixed_flux[i]>0} if direction of the normal
89 is along the M{x_i} axis.
90 """
91 if f !=None:
92 f=util.interpolate(f, self.__pde_v.getFunctionSpaceForCoefficient("X"))
93 if f.isEmpty():
94 f=Scalar(0,self.__pde_v.getFunctionSpaceForCoefficient("X"))
95 else:
96 if f.getRank()>0: raise ValueError,"illegal rank of f."
97 self.f=f
98 if g !=None:
99 g=util.interpolate(g, self.__pde_p.getFunctionSpaceForCoefficient("Y"))
100 if g.isEmpty():
101 g=Vector(0,self.__pde_v.getFunctionSpaceForCoefficient("Y"))
102 else:
103 if not g.getShape()==(self.domain.getDim(),):
104 raise ValueError,"illegal shape of g"
105 self.__g=g
106
107 if location_of_fixed_pressure!=None: self.__pde_p.setValue(q=location_of_fixed_pressure)
108 if location_of_fixed_flux!=None: self.__pde_v.setValue(q=location_of_fixed_flux)
109
110 if permeability!=None:
111 perm=util.interpolate(permeability,self.__pde_p.getFunctionSpaceForCoefficient("A"))
112 if perm.getRank()==0:
113 perm=perm*util.kronecker(self.domain.getDim())
114 elif perm.getRank()==1:
115 perm, perm2=Tensor(0.,self.__pde_p.getFunctionSpaceForCoefficient("A")), perm
116 for i in range(self.domain.getDim()): perm[i,i]=perm2[i]
117 elif perm.getRank()==2:
118 pass
119 else:
120 raise ValueError,"illegal rank of permeability."
121 self.__permeability=perm
122 self.__pde_p.setValue(A=util.transposed_tensor_mult(self.__permeability,self.__permeability))
123
124
125 def getFlux(self,p=None, fixed_flux=Data(),tol=1.e-8, show_details=False):
126 """
127 returns the flux for a given pressure C{p} where the flux is equal to C{fixed_flux}
128 on locations where C{location_of_fixed_flux} is positive (see L{setValue}).
129 Note that C{g} and C{f} are used, see L{setValue}.
130
131 @param p: pressure.
132 @type p: scalar value on the domain (e.g. L{Data}).
133 @param fixed_flux: flux on the locations of the domain marked be C{location_of_fixed_flux}.
134 @type fixed_flux: vector values on the domain (e.g. L{Data}).
135 @param tol: relative tolerance to be used.
136 @type tol: positive C{float}.
137 @return: flux
138 @rtype: L{Data}
139 @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}}
140 for the permeability M{k_{ij}}
141 """
142 self.__pde_v.setTolerance(tol)
143 g=self.__g
144 f=self.__f
145 self.__pde_v.setValue(X=f*util.kronecker(self.domain), r=fixed_flux)
146 if p == None:
147 self.__pde_v.setValue(Y=g)
148 else:
149 self.__pde_v.setValue(Y=g-util.tensor_mult(self.__permeability,util.grad(p)))
150 return self.__pde_v.getSolution(verbose=show_details)
151
152 def getPressure(self,v=None, fixed_pressure=Data(),tol=1.e-8, show_details=False):
153 """
154 returns the pressure for a given flux C{v} where the pressure is equal to C{fixed_pressure}
155 on locations where C{location_of_fixed_pressure} is positive (see L{setValue}).
156 Note that C{g} is used, see L{setValue}.
157
158 @param v: flux.
159 @type v: vector-valued on the domain (e.g. L{Data}).
160 @param fixed_pressure: pressure on the locations of the domain marked be C{location_of_fixed_pressure}.
161 @type fixed_pressure: vector values on the domain (e.g. L{Data}).
162 @param tol: relative tolerance to be used.
163 @type tol: positive C{float}.
164 @return: pressure
165 @rtype: L{Data}
166 @note: the method uses the least squares solution M{p=(Q^*Q)^{-1}Q^*(g-u)} where and M{(Qp)_i=k_{ij}p_{,j}}
167 for the permeability M{k_{ij}}
168 """
169 self.__pde_v.setTolerance(tol)
170 g=self.__g
171 self.__pde_p.setValue(r=fixed_pressure)
172 if v == None:
173 self.__pde_p.setValue(X=util.transposed_tensor_mult(self.__permeability,g-v))
174 else:
175 self.__pde_p.setValue(X=util.transposed_tensor_mult(self.__permeability,g))
176 return self.__pde_p.getSolution(verbose=show_details)
177
178 def setTolerance(self,atol=0,rtol=1e-8,p_ref=None,v_ref=None):
179 """
180 set the tolerance C{ATOL} used to terminate the solution process. It is used
181
182 M{ATOL = atol + rtol * max( |g-v_ref|, |Qp_ref| )}
183
184 where M{|f|^2 = integrate(length(f)^2)} and M{(Qp)_i=k_{ij}p_{,j}} for the permeability M{k_{ij}}. If C{v_ref} or C{p_ref} is not present zero is assumed.
185
186 The iteration is terminated if for the current approximation C{p}, flux C{v=(I+D^*D)^{-1}(D^*f-g-Qp)} and their residual
187
188 M{r=Q^*(g-Qp-v)}
189
190 the condition
191
192 M{<(Q^*Q)^{-1} r,r> <= ATOL}
193
194 holds. M{D} is the M{div} operator and M{(Qp)_i=k_{ij}p_{,j}} for the permeability M{k_{ij}}
195
196 @param atol: absolute tolerance for the pressure
197 @type atol: non-negative C{float}
198 @param rtol: relative tolerance for the pressure
199 @type rtol: non-negative C{float}
200 @param p_ref: reference pressure. If not present zero is used. You may use physical arguments to set a resonable value for C{p_ref}, use the
201 L{getPressure} method or use the value from a previous time step.
202 @type p_ref: scalar value on the domain (e.g. L{Data}).
203 @param v_ref: reference velocity. If not present zero is used. You may use physical arguments to set a resonable value for C{v_ref}, use the
204 L{getFlux} method or use the value from a previous time step.
205 @type v_ref: vector-valued on the domain (e.g. L{Data}).
206 @return: used absolute tolerance.
207 @rtype: positive C{float}
208 """
209 g=self.__g
210 if not v_ref == None:
211 f1=util.integrate(util.length(util.interpolate(g-v_ref,Function(self.domain)))**2)
212 else:
213 f1=util.integrate(util.length(util.interpolate(g))**2)
214 if not p_ref == None:
215 f2=util.integrate(util.length(util.tensor_mult(self.__permeability,util.grad(p_ref)))**2)
216 else:
217 f2=0
218 self.__ATOL= atol + rtol * util.sqrt(max(f1,f2))
219 if self.__ATOL<=0:
220 raise ValueError,"Positive tolerance (=%e) is expected."%self.__ATOL
221 return self.__ATOL
222
223 def getTolerance(self):
224 """
225 returns the current tolerance.
226
227 @return: used absolute tolerance.
228 @rtype: positive C{float}
229 """
230 if self.__ATOL==None:
231 raise ValueError,"no tolerance is defined."
232 return self.__ATOL
233
234 def solve(self,u0,p0, max_iter=100, verbose=False, show_details=False, sub_rtol=1.e-8):
235 """
236 solves the problem.
237
238 The iteration is terminated if the residual norm is less then self.getTolerance().
239
240 @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.
241 @type u0: vector value on the domain (e.g. L{Data}).
242 @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.
243 @type p0: scalar value on the domain (e.g. L{Data}).
244 @param sub_rtol: tolerance to be used in the sub iteration. It is recommended that M{sub_rtol<rtol*5.e-3}
245 @type sub_rtol: positive-negative C{float}
246 @param verbose: if set some information on iteration progress are printed
247 @type verbose: C{bool}
248 @param show_details: if set information on the subiteration process are printed.
249 @type show_details: C{bool}
250 @return: flux and pressure
251 @rtype: C{tuple} of L{Data}.
252
253 @note: The problem is solved as a least squares form
254
255 M{(I+D^*D)u+Qp=D^*f+g}
256 M{Q^*u+Q^*Qp=Q^*g}
257
258 where M{D} is the M{div} operator and M{(Qp)_i=k_{ij}p_{,j}} for the permeability M{k_{ij}}.
259 We eliminate the flux form the problem by setting
260
261 M{u=(I+D^*D)^{-1}(D^*f-g-Qp)} with u=u0 on location_of_fixed_flux
262
263 form the first equation. Inserted into the second equation we get
264
265 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
266
267 which is solved using the PCG method (precondition is M{Q^*Q}). In each iteration step
268 PDEs with operator M{I+D^*D} and with M{Q^*Q} needs to be solved using a sub iteration scheme.
269 """
270 self.verbose=verbose
271 self.show_details= show_details and self.verbose
272 self.__pde_v.setTolerance(sub_rtol)
273 self.__pde_p.setTolerance(sub_rtol)
274 ATOL=self.getTolerance()
275 if self.verbose: print "DarcyFlux: absolute tolerance = %e"%ATOL
276 #########################################################################################################################
277 #
278 # we solve:
279 #
280 # Q^*(I-(I+D^*D)^{-1})Q dp = Q^* (g-u0-Qp0 - (I+D^*D)^{-1} ( D^*(f-Du0)+g-u0-Qp0) )
281 #
282 # residual is
283 #
284 # r= Q^* (g-u0-Qp0 - (I+D^*D)^{-1} ( D^*(f-Du0)+g-u0-Qp0) - Q dp +(I+D^*D)^{-1})Q dp ) = Q^* (g - Qp - v)
285 #
286 # with v = (I+D^*D)^{-1} (D^*f+g-Qp) including BC
287 #
288 # we use (g - Qp, v) to represent the residual. not that
289 #
290 # dr(dp)=( -Q(dp), dv) with dv = - (I+D^*D)^{-1} Q(dp)
291 #
292 # while the initial residual is
293 #
294 # r0=( g - Qp0, v00) with v00=(I+D^*D)^{-1} (D^*f+g-Qp0) including BC
295 #
296 d0=self.__g-util.tensor_mult(self.__permeability,util.grad(p0))
297 self.__pde_v.setValue(Y=d0, X=self.__f*util.kronecker(self.domain), r=u0)
298 v00=self.__pde_v.getSolution(verbose=show_details)
299 if self.verbose: print "DarcyFlux: range of initial flux = ",util.inf(v00), util.sup(v00)
300 self.__pde_v.setValue(r=Data())
301 # start CG
302 r=ArithmeticTuple(d0, v00)
303 p,r=PCG(r,self.__Aprod_PCG,p0,self.__Msolve_PCG,self.__inner_PCG,atol=ATOL, rtol=0.,iter_max=max_iter, verbose=self.verbose)
304 return r[1],p
305
306 def __Aprod_PCG(self,dp):
307 if self.show_details: print "DarcyFlux: Applying operator"
308 # -dr(dp) = (Qdp,du) where du = (I+D^*D)^{-1} (Qdp)
309 mQdp=util.tensor_mult(self.__permeability,util.grad(dp))
310 self.__pde_v.setValue(Y=mQdp,X=Data(), r=Data())
311 du=self.__pde_v.getSolution(verbose=self.show_details)
312 return ArithmeticTuple(mQdp,du)
313
314 def __inner_PCG(self,p,r):
315 a=util.tensor_mult(self.__permeability,util.grad(p))
316 f0=util.integrate(util.inner(a,r[0]))
317 f1=util.integrate(util.inner(a,r[1]))
318 # print "__inner_PCG:",f0,f1,"->",f0-f1
319 return f0-f1
320
321 def __Msolve_PCG(self,r):
322 if self.show_details: print "DarcyFlux: Applying preconditioner"
323 self.__pde_p.setValue(X=util.transposed_tensor_mult(self.__permeability,r[0]-r[1]), r=Data())
324 return self.__pde_p.getSolution(verbose=self.show_details)
325
326 class StokesProblemCartesian(HomogeneousSaddlePointProblem):
327 """
328 solves
329
330 -(eta*(u_{i,j}+u_{j,i}))_j + p_i = f_i-stress_{ij,j}
331 u_{i,i}=0
332
333 u=0 where fixed_u_mask>0
334 eta*(u_{i,j}+u_{j,i})*n_j-p*n_i=surface_stress +stress_{ij}n_j
335
336 if surface_stress is not given 0 is assumed.
337
338 typical usage:
339
340 sp=StokesProblemCartesian(domain)
341 sp.setTolerance()
342 sp.initialize(...)
343 v,p=sp.solve(v0,p0)
344 """
345 def __init__(self,domain,**kwargs):
346 """
347 initialize the Stokes Problem
348
349 @param domain: domain of the problem. The approximation order needs to be two.
350 @type domain: L{Domain}
351 @warning: The apprximation order needs to be two otherwise you may see oscilations in the pressure.
352 """
353 HomogeneousSaddlePointProblem.__init__(self,**kwargs)
354 self.domain=domain
355 self.vol=util.integrate(1.,Function(self.domain))
356 self.__pde_u=LinearPDE(domain,numEquations=self.domain.getDim(),numSolutions=self.domain.getDim())
357 self.__pde_u.setSymmetryOn()
358 # self.__pde_u.setSolverMethod(self.__pde_u.DIRECT)
359 # self.__pde_u.setSolverMethod(preconditioner=LinearPDE.RILU)
360
361 self.__pde_prec=LinearPDE(domain)
362 self.__pde_prec.setReducedOrderOn()
363 # self.__pde_prec.setSolverMethod(self.__pde_prec.LUMPING)
364 self.__pde_prec.setSymmetryOn()
365
366 self.__pde_proj=LinearPDE(domain)
367 self.__pde_proj.setReducedOrderOn()
368 self.__pde_proj.setSymmetryOn()
369 self.__pde_proj.setValue(D=1.)
370
371 def initialize(self,f=Data(),fixed_u_mask=Data(),eta=1,surface_stress=Data(),stress=Data()):
372 """
373 assigns values to the model parameters
374
375 @param f: external force
376 @type f: L{Vector} object in L{FunctionSpace} L{Function} or similar
377 @param fixed_u_mask: mask of locations with fixed velocity.
378 @type fixed_u_mask: L{Vector} object on L{FunctionSpace} L{Solution} or similar
379 @param eta: viscosity
380 @type eta: L{Scalar} object on L{FunctionSpace} L{Function} or similar
381 @param surface_stress: normal surface stress
382 @type eta: L{Vector} object on L{FunctionSpace} L{FunctionOnBoundary} or similar
383 @param stress: initial stress
384 @type stress: L{Tensor} object on L{FunctionSpace} L{Function} or similar
385 @note: All values needs to be set.
386
387 """
388 self.eta=eta
389 A =self.__pde_u.createCoefficient("A")
390 self.__pde_u.setValue(A=Data())
391 for i in range(self.domain.getDim()):
392 for j in range(self.domain.getDim()):
393 A[i,j,j,i] += 1.
394 A[i,j,i,j] += 1.
395 self.__pde_prec.setValue(D=1/self.eta)
396 self.__pde_u.setValue(A=A*self.eta,q=fixed_u_mask,Y=f,y=surface_stress)
397 self.__stress=stress
398
399 def B(self,v):
400 """
401 returns div(v)
402 @rtype: equal to the type of p
403
404 @note: boundary conditions on p should be zero!
405 """
406 if self.show_details: print "apply divergence:"
407 self.__pde_proj.setValue(Y=-util.div(v))
408 self.__pde_proj.setTolerance(self.getSubProblemTolerance())
409 return self.__pde_proj.getSolution(verbose=self.show_details)
410
411 def inner_pBv(self,p,Bv):
412 """
413 returns inner product of element p and Bv (overwrite)
414
415 @type p: equal to the type of p
416 @type Bv: equal to the type of result of operator B
417 @rtype: C{float}
418
419 @rtype: equal to the type of p
420 """
421 s0=util.interpolate(p,Function(self.domain))
422 s1=util.interpolate(Bv,Function(self.domain))
423 return util.integrate(s0*s1)
424
425 def inner_p(self,p0,p1):
426 """
427 returns inner product of element p0 and p1 (overwrite)
428
429 @type p0: equal to the type of p
430 @type p1: equal to the type of p
431 @rtype: C{float}
432
433 @rtype: equal to the type of p
434 """
435 s0=util.interpolate(p0/self.eta,Function(self.domain))
436 s1=util.interpolate(p1/self.eta,Function(self.domain))
437 return util.integrate(s0*s1)
438
439 def inner_v(self,v0,v1):
440 """
441 returns inner product of two element v0 and v1 (overwrite)
442
443 @type v0: equal to the type of v
444 @type v1: equal to the type of v
445 @rtype: C{float}
446
447 @rtype: equal to the type of v
448 """
449 gv0=util.grad(v0)
450 gv1=util.grad(v1)
451 return util.integrate(util.inner(gv0,gv1))
452
453 def solve_A(self,u,p):
454 """
455 solves Av=f-Au-B^*p (v=0 on fixed_u_mask)
456 """
457 if self.show_details: print "solve for velocity:"
458 self.__pde_u.setTolerance(self.getSubProblemTolerance())
459 if self.__stress.isEmpty():
460 self.__pde_u.setValue(X=-2*self.eta*util.symmetric(util.grad(u))+p*util.kronecker(self.domain))
461 else:
462 self.__pde_u.setValue(X=self.__stress-2*self.eta*util.symmetric(util.grad(u))+p*util.kronecker(self.domain))
463 out=self.__pde_u.getSolution(verbose=self.show_details)
464 return out
465
466 def solve_prec(self,p):
467 if self.show_details: print "apply preconditioner:"
468 self.__pde_prec.setTolerance(self.getSubProblemTolerance())
469 self.__pde_prec.setValue(Y=p)
470 q=self.__pde_prec.getSolution(verbose=self.show_details)
471 return q

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