# Diff of /trunk/escriptcore/py_src/flows.py

revision 1481 by artak, Wed Apr 9 00:45:47 2008 UTC revision 2620 by gross, Thu Aug 20 06:24:00 2009 UTC
# Line 1  Line 1
1  # \$Id:\$  ########################################################
2  #  #
3  #######################################################  # Copyright (c) 2003-2009 by University of Queensland
4    # Earth Systems Science Computational Center (ESSCC)
5    # http://www.uq.edu.au/esscc
6  #  #
7  #       Copyright 2008 by University of Queensland  # Primary Business: Queensland, Australia
#
#######################################################
10  #  #
11    ########################################################
12
14    Earth Systems Science Computational Center (ESSCC)
15    http://www.uq.edu.au/esscc
20
21  """  """
22  Some models for flow  Some models for flow
# Line 24  Some models for flow Line 30  Some models for flow
30  """  """
31
32  __author__="Lutz Gross, l.gross@uq.edu.au"  __author__="Lutz Gross, l.gross@uq.edu.au"
http://www.access.edu.au
__url__="http://www.iservo.edu.au/esys"
__version__="\$Revision:\$"
__date__="\$Date:\$"
33
34  from escript import *  from escript import *
35  import util  import util
36  from linearPDEs import LinearPDE  from linearPDEs import LinearPDE, LinearPDESystem, LinearSinglePDE, SolverOptions
37  from pdetools import HomogeneousSaddlePointProblem  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, weight=None, useReduced=False, adaptSubTolerance=True):
52            """
53            initializes the Darcy flux problem
54            @param domain: domain of the problem
55            @type domain: L{Domain}
56        @param useReduced: uses reduced oreder on flux and pressure
57        @type useReduced: C{bool}
58        @param adaptSubTolerance: switches on automatic subtolerance selection
60            """
61            self.domain=domain
62            if weight == None:
63               s=self.domain.getSize()
64               self.__l=(3.*util.longestEdge(self.domain)*s/util.sup(s))**2
65               # self.__l=(3.*util.longestEdge(self.domain))**2
66               # self.__l=(0.1*util.longestEdge(self.domain)*s/util.sup(s))**2
67            else:
68               self.__l=weight
69            self.__pde_v=LinearPDESystem(domain)
70            if useReduced: self.__pde_v.setReducedOrderOn()
71            self.__pde_v.setSymmetryOn()
72            self.__pde_v.setValue(D=util.kronecker(domain), A=self.__l*util.outer(util.kronecker(domain),util.kronecker(domain)))
73            self.__pde_p=LinearSinglePDE(domain)
74            self.__pde_p.setSymmetryOn()
75            if useReduced: self.__pde_p.setReducedOrderOn()
76            self.__f=Scalar(0,self.__pde_v.getFunctionSpaceForCoefficient("X"))
77            self.__g=Vector(0,self.__pde_v.getFunctionSpaceForCoefficient("Y"))
78            self.setTolerance()
79            self.setAbsoluteTolerance()
81        self.verbose=False
82        def getSolverOptionsFlux(self):
83        """
84        Returns the solver options used to solve the flux problems
85
86        M{(I+D^*D)u=F}
87
88        @return: L{SolverOptions}
89        """
90        return self.__pde_v.getSolverOptions()
91        def setSolverOptionsFlux(self, options=None):
92        """
93        Sets the solver options used to solve the flux problems
94
95        M{(I+D^*D)u=F}
96
97        If C{options} is not present, the options are reset to default
98        @param options: L{SolverOptions}
99        @note: if the adaption of subtolerance is choosen, the tolerance set by C{options} will be overwritten before the solver is called.
100        """
101        return self.__pde_v.setSolverOptions(options)
102        def getSolverOptionsPressure(self):
103        """
104        Returns the solver options used to solve the pressure problems
105
106        M{(Q^*Q)p=Q^*G}
107
108        @return: L{SolverOptions}
109        """
110        return self.__pde_p.getSolverOptions()
111        def setSolverOptionsPressure(self, options=None):
112        """
113        Sets the solver options used to solve the pressure problems
114
115        M{(Q^*Q)p=Q^*G}
116
117        If C{options} is not present, the options are reset to default
118        @param options: L{SolverOptions}
119        @note: if the adaption of subtolerance is choosen, the tolerance set by C{options} will be overwritten before the solver is called.
120        """
121        return self.__pde_p.setSolverOptions(options)
122
123        def setValue(self,f=None, g=None, location_of_fixed_pressure=None, location_of_fixed_flux=None, permeability=None):
124            """
125            assigns values to model parameters
126
127            @param f: volumetic sources/sinks
128            @type f: scalar value on the domain (e.g. L{Data})
129            @param g: flux sources/sinks
130            @type g: vector values on the domain (e.g. L{Data})
131            @param location_of_fixed_pressure: mask for locations where pressure is fixed
132            @type location_of_fixed_pressure: scalar value on the domain (e.g. L{Data})
133            @param location_of_fixed_flux:  mask for locations where flux is fixed.
134            @type location_of_fixed_flux: vector values on the domain (e.g. L{Data})
135            @param permeability: permeability tensor. If scalar C{s} is given the tensor with
136                                 C{s} on the main diagonal is used. If vector C{v} is given the tensor with
137                                 C{v} on the main diagonal is used.
138            @type permeability: scalar, vector or tensor values on the domain (e.g. L{Data})
139
140            @note: the values of parameters which are not set by calling C{setValue} are not altered.
141            @note: at any point on the boundary of the domain the pressure (C{location_of_fixed_pressure} >0)
142                   or the normal component of the flux (C{location_of_fixed_flux[i]>0} if direction of the normal
143                   is along the M{x_i} axis.
144            """
145            if f !=None:
146               f=util.interpolate(f, self.__pde_v.getFunctionSpaceForCoefficient("X"))
147               if f.isEmpty():
148                   f=Scalar(0,self.__pde_v.getFunctionSpaceForCoefficient("X"))
149               else:
150                   if f.getRank()>0: raise ValueError,"illegal rank of f."
151               self.__f=f
152            if g !=None:
153               g=util.interpolate(g, self.__pde_p.getFunctionSpaceForCoefficient("Y"))
154               if g.isEmpty():
155                 g=Vector(0,self.__pde_v.getFunctionSpaceForCoefficient("Y"))
156               else:
157                 if not g.getShape()==(self.domain.getDim(),):
158                   raise ValueError,"illegal shape of g"
159               self.__g=g
160
161            if location_of_fixed_pressure!=None: self.__pde_p.setValue(q=location_of_fixed_pressure)
162            if location_of_fixed_flux!=None: self.__pde_v.setValue(q=location_of_fixed_flux)
163
164            if permeability!=None:
165               perm=util.interpolate(permeability,self.__pde_p.getFunctionSpaceForCoefficient("A"))
166               if perm.getRank()==0:
167                   perm=perm*util.kronecker(self.domain.getDim())
168               elif perm.getRank()==1:
169                   perm, perm2=Tensor(0.,self.__pde_p.getFunctionSpaceForCoefficient("A")), perm
170                   for i in range(self.domain.getDim()): perm[i,i]=perm2[i]
171               elif perm.getRank()==2:
172                  pass
173               else:
174                  raise ValueError,"illegal rank of permeability."
175               self.__permeability=perm
176               self.__pde_p.setValue(A=util.transposed_tensor_mult(self.__permeability,self.__permeability))
177
178        def setTolerance(self,rtol=1e-4):
179            """
180            sets the relative tolerance C{rtol} used to terminate the solution process. The iteration is terminated if
181
182            M{|g-v-Qp| <= atol + rtol * min( max( |g-v|, |Qp| ), max( |v|, |g-Qp| ) ) }
183
184            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}}.
185
186            @param rtol: relative tolerance for the pressure
187            @type rtol: non-negative C{float}
188            """
189            if rtol<0:
190                raise ValueError,"Relative tolerance needs to be non-negative."
191            self.__rtol=rtol
192        def getTolerance(self):
193            """
194            returns the relative tolerance
195
196            @return: current relative tolerance
197            @rtype: C{float}
198            """
199            return self.__rtol
200
201        def setAbsoluteTolerance(self,atol=0.):
202            """
203            sets the absolute tolerance C{atol} used to terminate the solution process. The iteration is terminated if
204
205            M{|g-v-Qp| <= atol + rtol * min( max( |g-v|, |Qp| ), max( |v|, |g-Qp| ) ) }
206
207            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}}.
208
209            @param atol: absolute tolerance for the pressure
210            @type atol: non-negative C{float}
211            """
212            if atol<0:
213                raise ValueError,"Absolute tolerance needs to be non-negative."
214            self.__atol=atol
215        def getAbsoluteTolerance(self):
216           """
217           returns the absolute tolerance
218
219           @return: current absolute tolerance
220           @rtype: C{float}
221           """
222           return self.__atol
223        def getSubProblemTolerance(self):
224        """
225        Returns a suitable subtolerance
226        @type: C{float}
227        """
228        return max(util.EPSILON**(0.75),self.getTolerance()**2)
229        def setSubProblemTolerance(self):
230             """
231             Sets the relative tolerance to solve the subproblem(s) if subtolerance adaption is selected.
232             """
234             sub_tol=self.getSubProblemTolerance()
235                 self.getSolverOptionsFlux().setTolerance(sub_tol)
236             self.getSolverOptionsFlux().setAbsoluteTolerance(0.)
237             self.getSolverOptionsPressure().setTolerance(sub_tol)
238             self.getSolverOptionsPressure().setAbsoluteTolerance(0.)
239             if self.verbose: print "DarcyFlux: relative subtolerance is set to %e."%sub_tol
240
241        def solve(self,u0,p0, max_iter=100, verbose=False, max_num_corrections=10):
242             """
243             solves the problem.
244
245             The iteration is terminated if the residual norm is less then self.getTolerance().
246
247             @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.
248             @type u0: vector value on the domain (e.g. L{Data}).
249             @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.
250             @type p0: scalar value on the domain (e.g. L{Data}).
251             @param verbose: if set some information on iteration progress are printed
252             @type verbose: C{bool}
253             @return: flux and pressure
254             @rtype: C{tuple} of L{Data}.
255
256             @note: The problem is solved as a least squares form
257
258             M{(I+D^*D)u+Qp=D^*f+g}
259             M{Q^*u+Q^*Qp=Q^*g}
260
261             where M{D} is the M{div} operator and M{(Qp)_i=k_{ij}p_{,j}} for the permeability M{k_{ij}}.
262             We eliminate the flux form the problem by setting
263
264             M{u=(I+D^*D)^{-1}(D^*f-g-Qp)} with u=u0 on location_of_fixed_flux
265
266             form the first equation. Inserted into the second equation we get
267
268             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
269
270             which is solved using the PCG method (precondition is M{Q^*Q}). In each iteration step
271             PDEs with operator M{I+D^*D} and with M{Q^*Q} needs to be solved using a sub iteration scheme.
272             """
273             self.verbose=verbose
274             rtol=self.getTolerance()
275             atol=self.getAbsoluteTolerance()
276         self.setSubProblemTolerance()
277
278             num_corrections=0
279             converged=False
280             p=p0
281             norm_r=None
282             while not converged:
283                   v=self.getFlux(p, fixed_flux=u0)
284                   Qp=self.__Q(p)
285                   norm_v=self.__L2(v)
286                   norm_Qp=self.__L2(Qp)
287                   if norm_v == 0.:
288                      if norm_Qp == 0.:
289                         return v,p
290                      else:
291                        fac=norm_Qp
292                   else:
293                      if norm_Qp == 0.:
294                        fac=norm_v
295                      else:
296                        fac=2./(1./norm_v+1./norm_Qp)
297                   ATOL=(atol+rtol*fac)
298                   if self.verbose:
299                        print "DarcyFlux: L2 norm of v = %e."%norm_v
300                        print "DarcyFlux: L2 norm of k.grad(p) = %e."%norm_Qp
301                        print "DarcyFlux: L2 defect u = %e."%(util.integrate(util.length(self.__g-util.interpolate(v,Function(self.domain))-Qp)**2)**(0.5),)
302                        print "DarcyFlux: L2 defect div(v) = %e."%(util.integrate((self.__f-util.div(v))**2)**(0.5),)
303                        print "DarcyFlux: absolute tolerance ATOL = %e."%ATOL
304                   if norm_r == None or norm_r>ATOL:
305                       if num_corrections>max_num_corrections:
306                             raise ValueError,"maximum number of correction steps reached."
307                       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)
308                       num_corrections+=1
309                   else:
310                       converged=True
311             return v,p
312        def __L2(self,v):
313             return util.sqrt(util.integrate(util.length(util.interpolate(v,Function(self.domain)))**2))
314
315        def __Q(self,p):
317
318        def __Aprod(self,dp):
319              if self.getSolverOptionsFlux().isVerbose(): print "DarcyFlux: Applying operator"
320              Qdp=self.__Q(dp)
321              self.__pde_v.setValue(Y=-Qdp,X=Data(), r=Data())
322              du=self.__pde_v.getSolution()
323              # self.__pde_v.getOperator().saveMM("proj.mm")
324              return Qdp+du
325        def __inner_GMRES(self,r,s):
326             return util.integrate(util.inner(r,s))
327
328        def __inner_PCG(self,p,r):
329             return util.integrate(util.inner(self.__Q(p), r))
330
331        def __Msolve_PCG(self,r):
332          if self.getSolverOptionsPressure().isVerbose(): print "DarcyFlux: Applying preconditioner"
333              self.__pde_p.setValue(X=util.transposed_tensor_mult(self.__permeability,r), Y=Data(), r=Data())
334              # self.__pde_p.getOperator().saveMM("prec.mm")
335              return self.__pde_p.getSolution()
336
337        def getFlux(self,p=None, fixed_flux=Data()):
338            """
339            returns the flux for a given pressure C{p} where the flux is equal to C{fixed_flux}
340            on locations where C{location_of_fixed_flux} is positive (see L{setValue}).
341            Note that C{g} and C{f} are used, see L{setValue}.
342
343            @param p: pressure.
344            @type p: scalar value on the domain (e.g. L{Data}).
345            @param fixed_flux: flux on the locations of the domain marked be C{location_of_fixed_flux}.
346            @type fixed_flux: vector values on the domain (e.g. L{Data}).
347            @param tol: relative tolerance to be used.
348            @type tol: positive C{float}.
349            @return: flux
350            @rtype: L{Data}
351            @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}}
352                   for the permeability M{k_{ij}}
353            """
354        self.setSubProblemTolerance()
355            g=self.__g
356            f=self.__f
357            self.__pde_v.setValue(X=self.__l*f*util.kronecker(self.domain), r=fixed_flux)
358            if p == None:
359               self.__pde_v.setValue(Y=g)
360            else:
361               self.__pde_v.setValue(Y=g-self.__Q(p))
362            return self.__pde_v.getSolution()
363
365        """       """
366        solves       solves
367
368            -(eta*(u_{i,j}+u_{j,i}))_j - p_i = f_i            -(eta*(u_{i,j}+u_{j,i}))_j + p_i = f_i-stress_{ij,j}
369                  u_{i,i}=0                  u_{i,i}=0
370
372            eta*(u_{i,j}+u_{j,i})*n_j=surface_stress            eta*(u_{i,j}+u_{j,i})*n_j-p*n_i=surface_stress +stress_{ij}n_j
373
374        if surface_stress is not give 0 is assumed.       if surface_stress is not given 0 is assumed.
375
376        typical usage:       typical usage:
377
378              sp=StokesProblemCartesian(domain)              sp=StokesProblemCartesian(domain)
379              sp.setTolerance()              sp.setTolerance()
380              sp.initialize(...)              sp.initialize(...)
381              v,p=sp.solve(v0,p0)              v,p=sp.solve(v0,p0)
382        """       """
383        def __init__(self,domain,**kwargs):       def __init__(self,domain,adaptSubTolerance=True, **kwargs):
385             initialize the Stokes Problem
386
387             @param domain: domain of the problem. The approximation order needs to be two.
388             @type domain: L{Domain}
389         @param adaptSubTolerance: If True the tolerance for subproblem is set automatically.
391             @warning: The apprximation order needs to be two otherwise you may see oscilations in the pressure.
392             """
394           self.domain=domain           self.domain=domain
395           self.vol=util.integrate(1.,Function(self.domain))           self.vol=util.integrate(1.,Function(self.domain))
396           self.__pde_u=LinearPDE(domain,numEquations=self.domain.getDim(),numSolutions=self.domain.getDim())           self.__pde_u=LinearPDE(domain,numEquations=self.domain.getDim(),numSolutions=self.domain.getDim())
397           self.__pde_u.setSymmetryOn()           self.__pde_u.setSymmetryOn()
398           self.__pde_u.setSolverMethod(preconditioner=LinearPDE.ILU0)

399           self.__pde_prec=LinearPDE(domain)           self.__pde_prec=LinearPDE(domain)
400           self.__pde_prec.setReducedOrderOn()           self.__pde_prec.setReducedOrderOn()
401           self.__pde_prec.setSymmetryOn()           self.__pde_prec.setSymmetryOn()
402
403           self.__pde_proj=LinearPDE(domain)           self.__pde_proj=LinearPDE(domain)
404           self.__pde_proj.setReducedOrderOn()           self.__pde_proj.setReducedOrderOn()
405         self.__pde_proj.setValue(D=1)
406           self.__pde_proj.setSymmetryOn()           self.__pde_proj.setSymmetryOn()
self.__pde_proj.setValue(D=1.)
407
409             """
410         returns the solver options used  solve the equation for velocity.
411
412         @rtype: L{SolverOptions}
413         """
414         return self.__pde_u.getSolverOptions()
415         def setSolverOptionsVelocity(self, options=None):
416             """
417         set the solver options for solving the equation for velocity.
418
419         @param options: new solver  options
420         @type options: L{SolverOptions}
421         """
422             self.__pde_u.setSolverOptions(options)
423         def getSolverOptionsPressure(self):
424             """
425         returns the solver options used  solve the equation for pressure.
426         @rtype: L{SolverOptions}
427         """
428         return self.__pde_prec.getSolverOptions()
429         def setSolverOptionsPressure(self, options=None):
430             """
431         set the solver options for solving the equation for pressure.
432         @param options: new solver  options
433         @type options: L{SolverOptions}
434         """
435         self.__pde_prec.setSolverOptions(options)
436
437         def setSolverOptionsDiv(self, options=None):
438             """
439         set the solver options for solving the equation to project the divergence of
440         the velocity onto the function space of presure.
441
442         @param options: new solver options
443         @type options: L{SolverOptions}
444         """
445         self.__pde_prec.setSolverOptions(options)
446         def getSolverOptionsDiv(self):
447             """
448         returns the solver options for solving the equation to project the divergence of
449         the velocity onto the function space of presure.
450
451         @rtype: L{SolverOptions}
452         """
453         return self.__pde_prec.getSolverOptions()
454         def setSubProblemTolerance(self):
455             """
456         Updates the tolerance for subproblems
457             """
459                 sub_tol=self.getSubProblemTolerance()
460             self.getSolverOptionsDiv().setTolerance(sub_tol)
461             self.getSolverOptionsDiv().setAbsoluteTolerance(0.)
462             self.getSolverOptionsPressure().setTolerance(sub_tol)
463             self.getSolverOptionsPressure().setAbsoluteTolerance(0.)
464             self.getSolverOptionsVelocity().setTolerance(sub_tol)
465             self.getSolverOptionsVelocity().setAbsoluteTolerance(0.)
466
467
469            """
470            assigns values to the model parameters
471
472            @param f: external force
473            @type f: L{Vector} object in L{FunctionSpace} L{Function} or similar
475            @type fixed_u_mask: L{Vector} object on L{FunctionSpace} L{Solution} or similar
476            @param eta: viscosity
477            @type eta: L{Scalar} object on L{FunctionSpace} L{Function} or similar
478            @param surface_stress: normal surface stress
479            @type eta: L{Vector} object on L{FunctionSpace} L{FunctionOnBoundary} or similar
480            @param stress: initial stress
481        @type stress: L{Tensor} object on L{FunctionSpace} L{Function} or similar
482            @note: All values needs to be set.
483            """
484          self.eta=eta          self.eta=eta
485          A =self.__pde_u.createCoefficientOfGeneralPDE("A")          A =self.__pde_u.createCoefficient("A")
486      self.__pde_u.setValue(A=Data())      self.__pde_u.setValue(A=Data())
487          for i in range(self.domain.getDim()):          for i in range(self.domain.getDim()):
488          for j in range(self.domain.getDim()):          for j in range(self.domain.getDim()):
489              A[i,j,j,i] += 1.              A[i,j,j,i] += 1.
490              A[i,j,i,j] += 1.              A[i,j,i,j] += 1.
491      self.__pde_prec.setValue(D=1./self.eta)          n=self.domain.getNormal()
494            self.__f=f
495            self.__surface_stress=surface_stress
496            self.__stress=stress
497
498         def Bv(self,v):
499             """
500             returns inner product of element p and div(v)
501
502             @param p: a pressure increment
503             @param v: a residual
504             @return: inner product of element p and div(v)
505             @rtype: C{float}
506             """
507             self.__pde_proj.setValue(Y=-util.div(v))
508             return self.__pde_proj.getSolution()
509
510        def B(self,arg):       def inner_pBv(self,p,Bv):
511           d=util.div(arg)           """
512           self.__pde_proj.setValue(Y=d)           returns inner product of element p and Bv=-div(v)
513           self.__pde_proj.setTolerance(self.getSubProblemTolerance())
514           return self.__pde_proj.getSolution(verbose=self.show_details)           @param p: a pressure increment
515             @param v: a residual
516        def inner(self,p0,p1):           @return: inner product of element p and Bv=-div(v)
517           s0=util.interpolate(p0,Function(self.domain))           @rtype: C{float}
518           s1=util.interpolate(p1,Function(self.domain))           """
519             return util.integrate(util.interpolate(p,Function(self.domain))*util.interpolate(Bv,Function(self.domain)))
520
521         def inner_p(self,p0,p1):
522             """
523             Returns inner product of p0 and p1
524
525             @param p0: a pressure
526             @param p1: a pressure
527             @return: inner product of p0 and p1
528             @rtype: C{float}
529             """
530             s0=util.interpolate(p0/self.eta,Function(self.domain))
531             s1=util.interpolate(p1/self.eta,Function(self.domain))
532           return util.integrate(s0*s1)           return util.integrate(s0*s1)
533
534        def getStress(self,u):       def norm_v(self,v):
536           return 2.*self.eta*util.symmetric(mg)           returns the norm of v
537
538        def solve_A(self,u,p):           @param v: a velovity
539           """           @return: norm of v
540           solves Av=f-Au-B^*p (v=0 on fixed_u_mask)           @rtype: non-negative C{float}
541           """           """
543           self.__pde_u.setValue(X=-self.getStress(u)-p*util.kronecker(self.domain))
544           return  self.__pde_u.getSolution(verbose=self.show_details)       def getV(self, p, v0):
545             """
546        def solve_prec(self,p):           return the value for v for a given p (overwrite)
547           self.__pde_prec.setTolerance(self.getSubProblemTolerance())
548           self.__pde_prec.setValue(Y=p)           @param p: a pressure
549           q=self.__pde_prec.getSolution(verbose=self.show_details)           @param v0: a initial guess for the value v to return.
550           return q           @return: v given as M{v= A^{-1} (f-B^*p)}
551        def stoppingcriterium(self,Bv,v,p):           """
552            n_r=util.sqrt(self.inner(Bv,Bv))           self.__pde_u.setValue(Y=self.__f, y=self.__surface_stress, r=v0)
553            n_v=util.Lsup(v)           if self.__stress.isEmpty():
554            if self.verbose: print "PCG step %s: L2(div(v)) = %s, Lsup(v)=%s"%(self.iter,n_r,n_v)              self.__pde_u.setValue(X=p*util.kronecker(self.domain))
555            self.iter+=1           else:
556            if n_r <= self.vol**(1./2.-1./self.domain.getDim())*n_v*self.getTolerance():              self.__pde_u.setValue(X=self.__stress+p*util.kronecker(self.domain))
557                if self.verbose: print "PCG terminated after %s steps."%self.iter           out=self.__pde_u.getSolution()
558                return True           return  out
559            else:
560                return False       def norm_Bv(self,Bv):
561        def stoppingcriterium_GMRES(self,norm_r,norm_b):          """
562            if self.verbose: print "GMRES step %s: L2(r) = %s, L2(b)*TOL=%s"%(self.iter,norm_r,norm_b*self.getTolerance())          Returns Bv (overwrite).
563            self.iter+=1
564            if norm_r <= norm_b*self.getTolerance():          @rtype: equal to the type of p
565                if self.verbose: print "GMRES terminated after %s steps."%self.iter          @note: boundary conditions on p should be zero!
566                return True          """
567            else:          return util.sqrt(util.integrate(util.interpolate(Bv,Function(self.domain))**2))
568                return False
569         def solve_AinvBt(self,p):
570        def stoppingcriterium_MINRES(self,norm_r,norm_Ax):           """
571            if self.verbose: print "MINRES step %s: L2(r) = %s, L2(b)*TOL=%s"%(self.iter,norm_r,norm_Ax*self.getTolerance())           Solves M{Av=B^*p} with accuracy L{self.getSubProblemTolerance()}
572            self.iter+=1
573            if norm_r <= norm_Ax*self.getTolerance():           @param p: a pressure increment
574                if self.verbose: print "MINRES terminated after %s steps."%self.iter           @return: the solution of M{Av=B^*p}
575                return True           @note: boundary conditions on v should be zero!
576            else:           """
577                return False           self.__pde_u.setValue(Y=Data(), y=Data(), r=Data(),X=-p*util.kronecker(self.domain))
578             out=self.__pde_u.getSolution()
579             return  out
580
581         def solve_prec(self,Bv):
582             """
583             applies preconditioner for for M{BA^{-1}B^*} to M{Bv}
584             with accuracy L{self.getSubProblemTolerance()}
585
586             @param v: velocity increment
587             @return: M{p=P(Bv)} where M{P^{-1}} is an approximation of M{BA^{-1}B^*}
588             @note: boundary conditions on p are zero.
589             """
590             self.__pde_prec.setValue(Y=Bv)
591             return self.__pde_prec.getSolution()

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