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Revision 2156 - (show annotations)
Mon Dec 15 05:09:02 2008 UTC (10 years, 9 months ago) by gross
File MIME type: text/x-python
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some modifications to the iterative solver to make them easier to use. 
There are also improved versions of the Darcy flux solver and the incompressible 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):
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 self.__pde_v.setValue(D=util.kronecker(domain), A=util.outer(util.kronecker(domain),util.kronecker(domain)))
60 self.__pde_v.setSymmetryOn()
61 self.__pde_p=LinearSinglePDE(domain)
62 self.__pde_p.setSymmetryOn()
63 self.__f=Scalar(0,self.__pde_v.getFunctionSpaceForCoefficient("X"))
64 self.__g=Vector(0,self.__pde_v.getFunctionSpaceForCoefficient("Y"))
65
66 def setValue(self,f=None, g=None, location_of_fixed_pressure=None, location_of_fixed_flux=None, permeability=None):
67 """
68 assigns values to model parameters
69
70 @param f: volumetic sources/sinks
71 @type f: scalar value on the domain (e.g. L{Data})
72 @param g: flux sources/sinks
73 @type g: vector values on the domain (e.g. L{Data})
74 @param location_of_fixed_pressure: mask for locations where pressure is fixed
75 @type location_of_fixed_pressure: scalar value on the domain (e.g. L{Data})
76 @param location_of_fixed_flux: mask for locations where flux is fixed.
77 @type location_of_fixed_flux: vector values on the domain (e.g. L{Data})
78 @param permeability: permeability tensor. If scalar C{s} is given the tensor with
79 C{s} on the main diagonal is used. If vector C{v} is given the tensor with
80 C{v} on the main diagonal is used.
81 @type permeability: scalar, vector or tensor values on the domain (e.g. L{Data})
82
83 @note: the values of parameters which are not set by calling C{setValue} are not altered.
84 @note: at any point on the boundary of the domain the pressure (C{location_of_fixed_pressure} >0)
85 or the normal component of the flux (C{location_of_fixed_flux[i]>0} if direction of the normal
86 is along the M{x_i} axis.
87 """
88 if f !=None:
89 f=util.interpolate(f, self.__pde_v.getFunctionSpaceForCoefficient("X"))
90 if f.isEmpty():
91 f=Scalar(0,self.__pde_v.getFunctionSpaceForCoefficient("X"))
92 else:
93 if f.getRank()>0: raise ValueError,"illegal rank of f."
94 self.f=f
95 if g !=None:
96 g=util.interpolate(g, self.__pde_p.getFunctionSpaceForCoefficient("Y"))
97 if g.isEmpty():
98 g=Vector(0,self.__pde_v.getFunctionSpaceForCoefficient("Y"))
99 else:
100 if not g.getShape()==(self.domain.getDim(),):
101 raise ValueError,"illegal shape of g"
102 self.__g=g
103
104 if location_of_fixed_pressure!=None: self.__pde_p.setValue(q=location_of_fixed_pressure)
105 if location_of_fixed_flux!=None: self.__pde_v.setValue(q=location_of_fixed_flux)
106
107 if permeability!=None:
108 perm=util.interpolate(permeability,self.__pde_p.getFunctionSpaceForCoefficient("A"))
109 if perm.getRank()==0:
110 perm=perm*util.kronecker(self.domain.getDim())
111 elif perm.getRank()==1:
112 perm, perm2=Tensor(0.,self.__pde_p.getFunctionSpaceForCoefficient("A")), perm
113 for i in range(self.domain.getDim()): perm[i,i]=perm2[i]
114 elif perm.getRank()==2:
115 pass
116 else:
117 raise ValueError,"illegal rank of permeability."
118 self.__permeability=perm
119 self.__pde_p.setValue(A=util.transposed_tensor_mult(self.__permeability,self.__permeability))
120
121
122 def getFlux(self,p, fixed_flux=Data(),tol=1.e-8, show_details=False):
123 """
124 returns the flux for a given pressure C{p} where the flux is equal to C{fixed_flux}
125 on locations where C{location_of_fixed_flux} is positive (see L{setValue}).
126 Note that C{g} and C{f} are used, L{setValue}.
127
128 @param p: pressure.
129 @type p: scalar value on the domain (e.g. L{Data}).
130 @param fixed_flux: flux on the locations of the domain marked be C{location_of_fixed_flux}.
131 @type fixed_flux: vector values on the domain (e.g. L{Data}).
132 @param tol: relative tolerance to be used.
133 @type tol: positive float.
134 @return: flux
135 @rtype: L{Data}
136 @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}}
137 for the permeability M{k_{ij}}
138 """
139 self.__pde_v.setTolerance(tol)
140 self.__pde_v.setValue(Y=self.__g, X=self.__f*util.kronecker(self.domain), r=fixed_flux)
141 return self.__pde_v.getSolution(verbose=show_details)
142
143 def solve(self,u0,p0,atol=0,rtol=1e-8, max_iter=100, verbose=False, show_details=False, sub_rtol=1.e-8):
144 """
145 solves the problem.
146
147 The iteration is terminated if the error in the pressure is less then C{rtol * |q| + atol} where
148 C{|q|} denotes the norm of the right hand side (see escript user's guide for details).
149
150 @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.
151 @type u0: vector value on the domain (e.g. L{Data}).
152 @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.
153 @type p0: scalar value on the domain (e.g. L{Data}).
154 @param atol: absolute tolerance for the pressure
155 @type atol: non-negative C{float}
156 @param rtol: relative tolerance for the pressure
157 @type rtol: non-negative C{float}
158 @param sub_rtol: tolerance to be used in the sub iteration. It is recommended that M{sub_rtol<rtol*5.e-3}
159 @type sub_rtol: positive-negative C{float}
160 @param verbose: if set some information on iteration progress are printed
161 @type verbose: C{bool}
162 @param show_details: if set information on the subiteration process are printed.
163 @type show_details: C{bool}
164 @return: flux and pressure
165 @rtype: C{tuple} of L{Data}.
166
167 @note: The problem is solved as a least squares form
168
169 M{(I+D^*D)u+Qp=D^*f+g}
170 M{Q^*u+Q^*Qp=Q^*g}
171
172 where M{D} is the M{div} operator and M{(Qp)_i=k_{ij}p_{,j}} for the permeability M{k_{ij}}.
173 We eliminate the flux form the problem by setting
174
175 M{u=(I+D^*D)^{-1}(D^*f-g-Qp)} with u=u0 on location_of_fixed_flux
176
177 form the first equation. Inserted into the second equation we get
178
179 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
180
181 which is solved using the PCG method (precondition is M{Q^*Q}). In each iteration step
182 PDEs with operator M{I+D^*D} and with M{Q^*Q} needs to be solved using a sub iteration scheme.
183 """
184 self.verbose=verbose
185 self.show_details= show_details and self.verbose
186 self.__pde_v.setTolerance(sub_rtol)
187 self.__pde_p.setTolerance(sub_rtol)
188 u2=u0*self.__pde_v.getCoefficient("q")
189 #
190 # first the reference velocity is calculated from
191 #
192 # (I+D^*D)u_ref=D^*f+g (including bundray conditions for u)
193 #
194 self.__pde_v.setValue(Y=self.__g, X=self.__f*util.kronecker(self.domain), r=u0)
195 u_ref=self.__pde_v.getSolution(verbose=show_details)
196 if self.verbose: print "DarcyFlux: maximum reference flux = ",util.Lsup(u_ref)
197 self.__pde_v.setValue(r=Data())
198 #
199 # and then we calculate a reference pressure
200 #
201 # Q^*Qp_ref=Q^*g-Q^*u_ref ((including bundray conditions for p)
202 #
203 self.__pde_p.setValue(X=util.transposed_tensor_mult(self.__permeability,(self.__g-u_ref)), r=p0)
204 p_ref=self.__pde_p.getSolution(verbose=self.show_details)
205 if self.verbose: print "DarcyFlux: maximum reference pressure = ",util.Lsup(p_ref)
206 self.__pde_p.setValue(r=Data())
207 #
208 # (I+D^*D)du + Qdp = - Qp_ref u=du+u_ref
209 # Q^*du + Q^*Qdp = Q^*g-Q^*u_ref-Q^*Qp_ref=0 p=dp+pref
210 #
211 # du= -(I+D^*D)^(-1} Q(p_ref+dp) u = u_ref+du
212 #
213 # => Q^*(I-(I+D^*D)^(-1})Q dp = Q^*(I+D^*D)^(-1} Qp_ref
214 # or Q^*(I-(I+D^*D)^(-1})Q p = Q^*Qp_ref
215 #
216 # r= Q^*( (I+D^*D)^(-1} Qp_ref - Q dp + (I+D^*D)^(-1})Q dp) = Q^*(-du-Q dp)
217 # with du=-(I+D^*D)^(-1} Q(p_ref+dp)
218 #
219 # we use the (du,Qdp) to represent the resudual
220 # Q^*Q is a preconditioner
221 #
222 # <(Q^*Q)^{-1}r,r> -> right hand side norm is <Qp_ref,Qp_ref>
223 #
224 Qp_ref=util.tensor_mult(self.__permeability,util.grad(p_ref))
225 norm_rhs=util.sqrt(util.integrate(util.inner(Qp_ref,Qp_ref)))
226 ATOL=max(norm_rhs*rtol +atol, 200. * util.EPSILON * norm_rhs)
227 if not ATOL>0:
228 raise ValueError,"Negative absolute tolerance (rtol = %e, norm right hand side =%, atol =%e)."%(rtol, norm_rhs, atol)
229 if self.verbose: print "DarcyFlux: norm of right hand side = %e (absolute tolerance = %e)"%(norm_rhs,ATOL)
230 #
231 # caclulate the initial residual
232 #
233 self.__pde_v.setValue(X=Data(), Y=-util.tensor_mult(self.__permeability,util.grad(p0)), r=Data())
234 du=self.__pde_v.getSolution(verbose=show_details)
235 r=ArithmeticTuple(util.tensor_mult(self.__permeability,util.grad(p0-p_ref)), du)
236 dp,r=PCG(r,self.__Aprod_PCG,p0,self.__Msolve_PCG,self.__inner_PCG,atol=ATOL, rtol=0.,iter_max=max_iter, verbose=self.verbose)
237 util.saveVTK("d.vtu",p=dp,p_ref=p_ref)
238 return u_ref+r[1],dp
239
240 def __Aprod_PCG(self,p):
241 if self.show_details: print "DarcyFlux: Applying operator"
242 Qp=util.tensor_mult(self.__permeability,util.grad(p))
243 self.__pde_v.setValue(Y=Qp,X=Data())
244 w=self.__pde_v.getSolution(verbose=self.show_details)
245 return ArithmeticTuple(-Qp,w)
246
247 def __inner_PCG(self,p,r):
248 a=util.tensor_mult(self.__permeability,util.grad(p))
249 out=-util.integrate(util.inner(a,r[0]+r[1]))
250 return out
251
252 def __Msolve_PCG(self,r):
253 if self.show_details: print "DarcyFlux: Applying preconditioner"
254 self.__pde_p.setValue(X=-util.transposed_tensor_mult(self.__permeability,r[0]+r[1]))
255 return self.__pde_p.getSolution(verbose=self.show_details)
256
257 class StokesProblemCartesian(HomogeneousSaddlePointProblem):
258 """
259 solves
260
261 -(eta*(u_{i,j}+u_{j,i}))_j + p_i = f_i-stress_{ij,j}
262 u_{i,i}=0
263
264 u=0 where fixed_u_mask>0
265 eta*(u_{i,j}+u_{j,i})*n_j-p*n_i=surface_stress +stress_{ij}n_j
266
267 if surface_stress is not given 0 is assumed.
268
269 typical usage:
270
271 sp=StokesProblemCartesian(domain)
272 sp.setTolerance()
273 sp.initialize(...)
274 v,p=sp.solve(v0,p0)
275 """
276 def __init__(self,domain,**kwargs):
277 """
278 initialize the Stokes Problem
279
280 @param domain: domain of the problem. The approximation order needs to be two.
281 @type domain: L{Domain}
282 @warning: The apprximation order needs to be two otherwise you may see oscilations in the pressure.
283 """
284 HomogeneousSaddlePointProblem.__init__(self,**kwargs)
285 self.domain=domain
286 self.vol=util.integrate(1.,Function(self.domain))
287 self.__pde_u=LinearPDE(domain,numEquations=self.domain.getDim(),numSolutions=self.domain.getDim())
288 self.__pde_u.setSymmetryOn()
289 # self.__pde_u.setSolverMethod(self.__pde_u.DIRECT)
290 # self.__pde_u.setSolverMethod(preconditioner=LinearPDE.RILU)
291
292 self.__pde_prec=LinearPDE(domain)
293 self.__pde_prec.setReducedOrderOn()
294 # self.__pde_prec.setSolverMethod(self.__pde_prec.LUMPING)
295 self.__pde_prec.setSymmetryOn()
296
297 self.__pde_proj=LinearPDE(domain)
298 self.__pde_proj.setReducedOrderOn()
299 self.__pde_proj.setSymmetryOn()
300 self.__pde_proj.setValue(D=1.)
301
302 def initialize(self,f=Data(),fixed_u_mask=Data(),eta=1,surface_stress=Data(),stress=Data()):
303 """
304 assigns values to the model parameters
305
306 @param f: external force
307 @type f: L{Vector} object in L{FunctionSpace} L{Function} or similar
308 @param fixed_u_mask: mask of locations with fixed velocity.
309 @type fixed_u_mask: L{Vector} object on L{FunctionSpace} L{Solution} or similar
310 @param eta: viscosity
311 @type eta: L{Scalar} object on L{FunctionSpace} L{Function} or similar
312 @param surface_stress: normal surface stress
313 @type eta: L{Vector} object on L{FunctionSpace} L{FunctionOnBoundary} or similar
314 @param stress: initial stress
315 @type stress: L{Tensor} object on L{FunctionSpace} L{Function} or similar
316 @note: All values needs to be set.
317
318 """
319 self.eta=eta
320 A =self.__pde_u.createCoefficient("A")
321 self.__pde_u.setValue(A=Data())
322 for i in range(self.domain.getDim()):
323 for j in range(self.domain.getDim()):
324 A[i,j,j,i] += 1.
325 A[i,j,i,j] += 1.
326 self.__pde_prec.setValue(D=1/self.eta)
327 self.__pde_u.setValue(A=A*self.eta,q=fixed_u_mask,Y=f,y=surface_stress)
328 self.__stress=stress
329
330 def B(self,v):
331 """
332 returns div(v)
333 @rtype: equal to the type of p
334
335 @note: boundary conditions on p should be zero!
336 """
337 if self.show_details: print "apply divergence:"
338 self.__pde_proj.setValue(Y=-util.div(v))
339 self.__pde_proj.setTolerance(self.getSubProblemTolerance())
340 return self.__pde_proj.getSolution(verbose=self.show_details)
341
342 def inner_pBv(self,p,Bv):
343 """
344 returns inner product of element p and Bv (overwrite)
345
346 @type p: equal to the type of p
347 @type Bv: equal to the type of result of operator B
348 @rtype: C{float}
349
350 @rtype: equal to the type of p
351 """
352 s0=util.interpolate(p,Function(self.domain))
353 s1=util.interpolate(Bv,Function(self.domain))
354 return util.integrate(s0*s1)
355
356 def inner_p(self,p0,p1):
357 """
358 returns inner product of element p0 and p1 (overwrite)
359
360 @type p0: equal to the type of p
361 @type p1: equal to the type of p
362 @rtype: C{float}
363
364 @rtype: equal to the type of p
365 """
366 s0=util.interpolate(p0/self.eta,Function(self.domain))
367 s1=util.interpolate(p1/self.eta,Function(self.domain))
368 return util.integrate(s0*s1)
369
370 def inner_v(self,v0,v1):
371 """
372 returns inner product of two element v0 and v1 (overwrite)
373
374 @type v0: equal to the type of v
375 @type v1: equal to the type of v
376 @rtype: C{float}
377
378 @rtype: equal to the type of v
379 """
380 gv0=util.grad(v0)
381 gv1=util.grad(v1)
382 return util.integrate(util.inner(gv0,gv1))
383
384 def solve_A(self,u,p):
385 """
386 solves Av=f-Au-B^*p (v=0 on fixed_u_mask)
387 """
388 if self.show_details: print "solve for velocity:"
389 self.__pde_u.setTolerance(self.getSubProblemTolerance())
390 if self.__stress.isEmpty():
391 self.__pde_u.setValue(X=-2*self.eta*util.symmetric(util.grad(u))+p*util.kronecker(self.domain))
392 else:
393 self.__pde_u.setValue(X=self.__stress-2*self.eta*util.symmetric(util.grad(u))+p*util.kronecker(self.domain))
394 out=self.__pde_u.getSolution(verbose=self.show_details)
395 return out
396
397 def solve_prec(self,p):
398 if self.show_details: print "apply preconditioner:"
399 self.__pde_prec.setTolerance(self.getSubProblemTolerance())
400 self.__pde_prec.setValue(Y=p)
401 q=self.__pde_prec.getSolution(verbose=self.show_details)
402 return q

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