/[escript]/trunk/escript/py_src/flows.py
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revision 1673 by gross, Thu Jul 24 22:28:50 2008 UTC revision 3515 by gross, Thu May 19 08:20:57 2011 UTC
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1  # $Id:$  # -*- 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  #       Copyright 2008 by University of Queensland  # Primary Business: Queensland, Australia
9  #  # Licensed under the Open Software License version 3.0
10  #                http://esscc.uq.edu.au  # http://www.opensource.org/licenses/osl-3.0.php
 #        Primary Business: Queensland, Australia  
 #  Licensed under the Open Software License version 3.0  
 #     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  Some models for flow
24    
25  @var __author__: name of author  :var __author__: name of author
26  @var __copyright__: copyrights  :var __copyright__: copyrights
27  @var __license__: licence agreement  :var __license__: licence agreement
28  @var __url__: url entry point on documentation  :var __url__: url entry point on documentation
29  @var __version__: version  :var __version__: version
30  @var __date__: date of the version  :var __date__: date of the version
31  """  """
32    
33  __author__="Lutz Gross, l.gross@uq.edu.au"  __author__="Lutz Gross, l.gross@uq.edu.au"
 __copyright__="""  Copyright (c) 2008 by ACcESS MNRF  
                     http://www.access.edu.au  
                 Primary Business: Queensland, Australia"""  
 __license__="""Licensed under the Open Software License version 3.0  
              http://www.opensource.org/licenses/osl-3.0.php"""  
 __url__="http://www.iservo.edu.au/esys"  
 __version__="$Revision:$"  
 __date__="$Date:$"  
34    
35  from escript import *  import escript
36  import util  import util
37  from linearPDEs import LinearPDE  from linearPDEs import LinearPDE, LinearPDESystem, LinearSinglePDE, SolverOptions
38  from pdetools import HomogeneousSaddlePointProblem,Projector  from pdetools import HomogeneousSaddlePointProblem,Projector, ArithmeticTuple, PCG, NegativeNorm, GMRES
39    
40  class StokesProblemCartesian_DC(HomogeneousSaddlePointProblem):  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       :cvar SIMPLE: simple solver
50       :cvar POST: solver using global postprocessing of flux
51       :cvar STAB: solver uses (non-symmetric) stabilization
52       :cvar SYMSTAB: solver uses symmetric stabilization
53       """
54       SIMPLE="SIMPLE"
55       POST="POST"
56       STAB="STAB"
57       SYMSTAB="SYMSTAB"
58       def __init__(self, domain, useReduced=False, solver="SYMSTAB", verbose=False, w=1.):
59        """        """
60        solves        initializes the Darcy flux problem
61          :param domain: domain of the problem
62            -(eta*(u_{i,j}+u_{j,i}))_j - p_i = f_i        :type domain: `Domain`
63                  u_{i,i}=0        :param useReduced: uses reduced oreder on flux and pressure
64          :type useReduced: ``bool``
65            u=0 where  fixed_u_mask>0        :param solver: solver method
66            eta*(u_{i,j}+u_{j,i})*n_j=surface_stress        :type solver: in [`DarcyFlow.SIMPLE`, `DarcyFlow.POST', `DarcyFlow.STAB`, `DarcyFlow.SYMSTAB` ]
67          :param verbose: if ``True`` some information on the iteration progress are printed.
68          :type verbose: ``bool``
69          :param w: weighting factor for `DarcyFlow.POST` solver
70          :type w: ``float``
71          
72          """
73          self.domain=domain
74          self.solver=solver
75          self.useReduced=useReduced
76          self.verbose=verbose
77          self.scale=1.
78          
79          
80          self.__pde_v=LinearPDESystem(domain)
81          self.__pde_v.setSymmetryOn()
82          if self.useReduced: self.__pde_v.setReducedOrderOn()
83          self.__pde_p=LinearSinglePDE(domain)
84          self.__pde_p.setSymmetryOn()
85          if self.useReduced: self.__pde_p.setReducedOrderOn()
86          
87          if self.solver  == self.SIMPLE:
88         if self.verbose: print "DarcyFlow: simple solver is used."
89             self.__pde_v.setValue(D=util.kronecker(self.domain.getDim()))
90          elif self.solver  == self.POST:
91         self.w=w
92         if util.inf(w)<0.:
93            raise ValueError,"Weighting factor must be non-negative."
94         if self.verbose: print "DarcyFlow: global postprocessing of flux is used."
95          elif self.solver  == self.STAB:
96          if self.verbose: print "DarcyFlow: (non-symmetric) stabilization is used."
97          elif  self.solver  == self.SYMSTAB:
98          if self.verbose: print "DarcyFlow: symmetric stabilization is used."
99          else:
100        raise ValueError,"unknown solver %s"%self.solver
101          self.__f=escript.Scalar(0,self.__pde_p.getFunctionSpaceForCoefficient("X"))
102          self.__g=escript.Vector(0,self.__pde_v.getFunctionSpaceForCoefficient("Y"))
103          self.location_of_fixed_pressure = escript.Scalar(0, self.__pde_p.getFunctionSpaceForCoefficient("q"))
104          self.location_of_fixed_flux = escript.Vector(0, self.__pde_v.getFunctionSpaceForCoefficient("q"))
105          self.setTolerance()
106        
107            
108       def setValue(self,f=None, g=None, location_of_fixed_pressure=None, location_of_fixed_flux=None, permeability=None):
109          """
110          assigns values to model parameters
111    
112        if surface_stress is not give 0 is assumed.        :param f: volumetic sources/sinks
113          :type f: scalar value on the domain (e.g. `escript.Data`)
114          :param g: flux sources/sinks
115          :type g: vector values on the domain (e.g. `escript.Data`)
116          :param location_of_fixed_pressure: mask for locations where pressure is fixed
117          :type location_of_fixed_pressure: scalar value on the domain (e.g. `escript.Data`)
118          :param location_of_fixed_flux:  mask for locations where flux is fixed.
119          :type location_of_fixed_flux: vector values on the domain (e.g. `escript.Data`)
120          :param permeability: permeability tensor. If scalar ``s`` is given the tensor with ``s`` on the main diagonal is used.
121          :type permeability: scalar or symmetric tensor values on the domain (e.g. `escript.Data`)
122    
123          :note: the values of parameters which are not set by calling ``setValue`` are not altered.
124          :note: at any point on the boundary of the domain the pressure
125                 (``location_of_fixed_pressure`` >0) or the normal component of the
126                 flux (``location_of_fixed_flux[i]>0``) if direction of the normal
127                 is along the *x_i* axis.
128    
129        typical usage:        """
130          if location_of_fixed_pressure!=None:
131               self.location_of_fixed_pressure=util.wherePositive(location_of_fixed_pressure)
132               self.__pde_p.setValue(q=self.location_of_fixed_pressure)
133          if location_of_fixed_flux!=None:
134              self.location_of_fixed_flux=util.wherePositive(location_of_fixed_flux)
135              self.__pde_v.setValue(q=self.location_of_fixed_flux)
136          
137                
138          # pressure  is rescaled by the factor 1/self.scale
139          if permeability!=None:
140        
141         perm=util.interpolate(permeability,self.__pde_v.getFunctionSpaceForCoefficient("A"))
142             V=util.vol(self.domain)
143             l=V**(1./self.domain.getDim())
144            
145         if perm.getRank()==0:
146            perm_inv=(1./perm)
147                self.scale=util.integrate(perm_inv)/V*l
148            perm_inv=perm_inv*((1./self.scale)*util.kronecker(self.domain.getDim()))
149            perm=perm*(self.scale*util.kronecker(self.domain.getDim()))
150            
151            
152         elif perm.getRank()==2:
153            perm_inv=util.inverse(perm)
154                self.scale=util.sqrt(util.integrate(util.length(perm_inv)**2)/V)*l
155            perm_inv*=(1./self.scale)
156            perm=perm*self.scale
157         else:
158            raise ValueError,"illegal rank of permeability."
159            
160         self.__permeability=perm
161         self.__permeability_inv=perm_inv
162         if self.verbose: print "DarcyFlow: scaling factor for pressure is %e."%self.scale
163        
164         if self.solver  == self.SIMPLE:
165            self.__pde_p.setValue(A=self.__permeability)
166         elif self.solver  == self.POST:
167            self.__pde_p.setValue(A=self.__permeability)
168            k=util.kronecker(self.domain.getDim())
169            self.lamb = self.w*util.length(perm_inv)*l
170            self.__pde_v.setValue(D=self.__permeability_inv, A=self.lamb*self.domain.getSize()*util.outer(k,k))
171         elif self.solver  == self.STAB:
172            self.__pde_p.setValue(A=0.5*self.__permeability)
173            self.__pde_v.setValue(D=0.5*self.__permeability_inv)
174         elif  self.solver  == self.SYMSTAB:
175            self.__pde_p.setValue(A=0.5*self.__permeability)
176            self.__pde_v.setValue(D=0.5*self.__permeability_inv)
177    
178          if g != None:
179        g=util.interpolate(g, self.__pde_v.getFunctionSpaceForCoefficient("Y"))
180        if g.isEmpty():
181              g=Vector(0,self.__pde_v.getFunctionSpaceForCoefficient("Y"))
182        else:
183            if not g.getShape()==(self.domain.getDim(),): raise ValueError,"illegal shape of g"
184        self.__g=g
185          if f !=None:
186         f=util.interpolate(f, self.__pde_p.getFunctionSpaceForCoefficient("Y"))
187         if f.isEmpty():      
188              f=Scalar(0,self.__pde_p.getFunctionSpaceForCoefficient("Y"))
189         else:
190             if f.getRank()>0: raise ValueError,"illegal rank of f."
191         self.__f=f
192       def getSolverOptionsFlux(self):
193          """
194          Returns the solver options used to solve the flux problems
195          :return: `SolverOptions`
196          """
197          return self.__pde_v.getSolverOptions()
198          
199       def setSolverOptionsFlux(self, options=None):
200          """
201          Sets the solver options used to solve the flux problems
202          If ``options`` is not present, the options are reset to default
203          :param options: `SolverOptions`
204          """
205          return self.__pde_v.setSolverOptions(options)
206        
207       def getSolverOptionsPressure(self):
208          """
209          Returns the solver options used to solve the pressure problems
210          :return: `SolverOptions`
211          """
212          return self.__pde_p.getSolverOptions()
213          
214       def setSolverOptionsPressure(self, options=None):
215          """
216          Sets the solver options used to solve the pressure problems
217          If ``options`` is not present, the options are reset to default
218          
219          :param options: `SolverOptions`
220          :note: if the adaption of subtolerance is choosen, the tolerance set by ``options`` will be overwritten before the solver is called.
221          """
222          return self.__pde_p.setSolverOptions(options)
223          
224       def setTolerance(self,rtol=1e-4):
225          """
226          sets the relative tolerance ``rtol`` for the pressure for the stabelized solvers.
227          
228          :param rtol: relative tolerance for the pressure
229          :type rtol: non-negative ``float``
230          """
231          if rtol<0:
232         raise ValueError,"Relative tolerance needs to be non-negative."
233          self.__rtol=rtol
234          
235       def getTolerance(self):
236          """
237          returns the relative tolerance
238          :return: current relative tolerance
239          :rtype: ``float``
240          """
241          return self.__rtol
242          
243       def solve(self,u0,p0, max_iter=100, iter_restart=20):
244          """
245          solves the problem.
246          
247          The iteration is terminated if the residual norm is less then self.getTolerance().
248    
249          :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.
250          :type u0: vector value on the domain (e.g. `escript.Data`).
251          :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.
252          :type p0: scalar value on the domain (e.g. `escript.Data`).
253          :param max_iter: maximum number of (outer) iteration steps for the stabilization solvers,
254          :type max_iter: ``int``
255          :param iter_restart: number of steps after which the iteration is restarted. The larger ``iter_restart`` the larger the required memory.
256                               A small value for ``iter_restart`` may require a large number of iteration steps or may even lead to a failure
257                               of the iteration. ``iter_restart`` is relevant for the stabilization solvers only.
258          :type iter_restart: ``int``
259          :return: flux and pressure
260          :rtype: ``tuple`` of `escript.Data`.
261    
             sp=StokesProblemCartesian(domain)  
             sp.setTolerance()  
             sp.initialize(...)  
             v,p=sp.solve(v0,p0)  
262        """        """
263        def __init__(self,domain,**kwargs):        # rescale initial guess:
264           HomogeneousSaddlePointProblem.__init__(self,**kwargs)        p0=p0/self.scale
265           self.domain=domain        if self.solver  == self.SIMPLE or self.solver  == self.POST :
266           self.vol=util.integrate(1.,Function(self.domain))          self.__pde_p.setValue(X=self.__g ,
267           self.__pde_u=LinearPDE(domain,numEquations=self.domain.getDim(),numSolutions=self.domain.getDim())                                Y=self.__f,
268           self.__pde_u.setSymmetryOn()                                y= - util.inner(self.domain.getNormal(),u0 * self.location_of_fixed_flux),
269           # self.__pde_u.setSolverMethod(preconditioner=LinearPDE.ILU0)                                r=p0)
270            p=self.__pde_p.getSolution()
271            u = self.getFlux(p, u0)
272          elif  self.solver  == self.STAB:
273        u,p = self.__solve_STAB(u0,p0, max_iter, iter_restart)
274          elif  self.solver  == self.SYMSTAB:
275        u,p = self.__solve_SYMSTAB(u0,p0, max_iter, iter_restart)
276        
277          if self.verbose:
278            KGp=util.tensor_mult(self.__permeability,util.grad(p))
279            def_p=self.__g-(u+KGp)
280            def_v=self.__f-util.div(u, self.__pde_v.getFunctionSpaceForCoefficient("X"))
281            print "DarcyFlux: |g-u-K*grad(p)|_2 = %e (|u|_2 = %e)."%(self.__L2(def_p),self.__L2(u))
282            print "DarcyFlux: |f-div(u)|_2 = %e (|grad(u)|_2 = %e)."%(self.__L2(def_v),self.__L2(util.grad(u)))
283          #rescale result
284          p=p*self.scale
285          return u,p
286          
287       def getFlux(self,p, u0=None):
288            """
289            returns the flux for a given pressure ``p`` where the flux is equal to ``u0``
290            on locations where ``location_of_fixed_flux`` is positive (see `setValue`).
291            Notice that ``g`` and ``f`` are used, see `setValue`.
292    
293            :param p: pressure.
294            :type p: scalar value on the domain (e.g. `escript.Data`).
295            :param u0: flux on the locations of the domain marked be ``location_of_fixed_flux``.
296            :type u0: vector values on the domain (e.g. `escript.Data`) or ``None``
297            :return: flux
298            :rtype: `escript.Data`
299            """
300            if self.solver  == self.SIMPLE or self.solver  == self.POST  :
301                KGp=util.tensor_mult(self.__permeability,util.grad(p))
302                self.__pde_v.setValue(Y=self.__g-KGp, X=escript.Data())
303                if u0 == None:
304               self.__pde_v.setValue(r=escript.Data())
305            else:
306               self.__pde_v.setValue(r=u0)
307                u= self.__pde_v.getSolution()
308        elif self.solver  == self.POST:
309                self.__pde_v.setValue(Y=util.tensor_mult(self.__permeability_inv,self.__g)-util.grad(p),
310                                      X=self.lamb * self.__f * util.kronecker(self.domain.getDim()))
311                if u0 == None:
312               self.__pde_v.setValue(r=escript.Data())
313            else:
314               self.__pde_v.setValue(r=u0)
315                u= self.__pde_v.getSolution()
316        elif self.solver  == self.STAB:
317             gp=util.grad(p)
318             self.__pde_v.setValue(Y=0.5*(util.tensor_mult(self.__permeability_inv,self.__g)+gp),
319                                   X= p * util.kronecker(self.domain.getDim()),
320                                   y= - p * self.domain.getNormal())                          
321             if u0 == None:
322               self.__pde_v.setValue(r=escript.Data())
323             else:
324               self.__pde_v.setValue(r=u0)
325             u= self.__pde_v.getSolution()
326        elif  self.solver  == self.SYMSTAB:
327             gp=util.grad(p)
328             self.__pde_v.setValue(Y=0.5*(util.tensor_mult(self.__permeability_inv,self.__g)-gp),
329                                   X= escript.Data() ,
330                                   y= escript.Data() )                          
331             if u0 == None:
332               self.__pde_v.setValue(r=escript.Data())
333             else:
334               self.__pde_v.setValue(r=u0)
335             u= self.__pde_v.getSolution()
336        return u
337          
338        
339       def __solve_STAB(self, u0, p0, max_iter, iter_restart):
340              # p0 is used as an initial guess
341          u=self.getFlux(p0, u0)  
342              self.__pde_p.setValue( Y=self.__f-util.div(u),
343                                     X=0.5*(self.__g - u - util.tensor_mult(self.__permeability,util.grad(p0)) ),
344                                     y= escript.Data(),
345                                     r=escript.Data())
346    
347          dp=self.__pde_p.getSolution()
348          p=GMRES(dp,
349                  self.__STAB_Aprod,
350              p0,
351              self.__inner,
352              atol=self.__norm(p0+dp)*self.getTolerance() ,
353              rtol=0.,
354              iter_max=max_iter,
355              iter_restart=iter_restart,
356              verbose=self.verbose,P_R=None)
357                            
358           # self.__pde_proj=LinearPDE(domain,numEquations=1,numSolutions=1)            u=self.getFlux(p, u0)
359           # self.__pde_proj.setReducedOrderOn()            return u,p
          # self.__pde_proj.setSymmetryOn()  
          # self.__pde_proj.setSolverMethod(LinearPDE.LUMPING)  
   
       def initialize(self,f=Data(),fixed_u_mask=Data(),eta=1,surface_stress=Data()):  
         self.eta=eta  
         A =self.__pde_u.createCoefficientOfGeneralPDE("A")  
     self.__pde_u.setValue(A=Data())  
         for i in range(self.domain.getDim()):  
         for j in range(self.domain.getDim()):  
             A[i,j,j,i] += 1.  
             A[i,j,i,j] += 1.  
         # self.__inv_eta=util.interpolate(self.eta,ReducedFunction(self.domain))  
         self.__pde_u.setValue(A=A*self.eta,q=fixed_u_mask,Y=f,y=surface_stress)  
   
         # self.__pde_proj.setValue(D=1/eta)  
         # self.__pde_proj.setValue(Y=1.)  
         # self.__inv_eta=util.interpolate(self.__pde_proj.getSolution(),ReducedFunction(self.domain))  
         self.__inv_eta=util.interpolate(self.eta,ReducedFunction(self.domain))  
   
       def B(self,arg):  
          a=util.div(arg, ReducedFunction(self.domain))  
          return a-util.integrate(a)/self.vol  
   
       def inner(self,p0,p1):  
          return util.integrate(p0*p1)  
          # return util.integrate(1/self.__inv_eta**2*p0*p1)  
   
       def getStress(self,u):  
          mg=util.grad(u)  
          return 2.*self.eta*util.symmetric(mg)  
       def getEtaEffective(self):  
          return self.eta  
   
       def solve_A(self,u,p):  
          """  
          solves Av=f-Au-B^*p (v=0 on fixed_u_mask)  
          """  
          self.__pde_u.setTolerance(self.getSubProblemTolerance())  
          self.__pde_u.setValue(X=-self.getStress(u),X_reduced=-p*util.kronecker(self.domain))  
          return  self.__pde_u.getSolution(verbose=self.show_details)  
   
   
       def solve_prec(self,p):  
         a=self.__inv_eta*p  
         return a-util.integrate(a)/self.vol  
   
       def stoppingcriterium(self,Bv,v,p):  
           n_r=util.sqrt(self.inner(Bv,Bv))  
           n_v=util.sqrt(util.integrate(util.length(util.grad(v))**2))  
           if self.verbose: print "PCG step %s: L2(div(v)) = %s, L2(grad(v))=%s"%(self.iter,n_r,n_v) , util.Lsup(v)  
           if self.iter == 0: self.__n_v=n_v;  
           self.__n_v, n_v_old =n_v, self.__n_v  
           self.iter+=1  
           if self.iter>1 and n_r <= n_v*self.getTolerance() and abs(n_v_old-self.__n_v) <= n_v * self.getTolerance():  
               if self.verbose: print "PCG terminated after %s steps."%self.iter  
               return True  
           else:  
               return False  
       def stoppingcriterium2(self,norm_r,norm_b,solver='GMRES',TOL=None):  
       if TOL==None:  
              TOL=self.getTolerance()  
           if self.verbose: print "%s step %s: L2(r) = %s, L2(b)*TOL=%s"%(solver,self.iter,norm_r,norm_b*TOL)  
           self.iter+=1  
             
           if norm_r <= norm_b*TOL:  
               if self.verbose: print "%s terminated after %s steps."%(solver,self.iter)  
               return True  
           else:  
               return False  
360    
361       def __solve_SYMSTAB(self, u0, p0, max_iter, iter_restart):
362              # p0 is used as an initial guess
363          u=self.getFlux(p0, u0)  
364              self.__pde_p.setValue( Y= self.__f,
365                                     X=  0.5*(self.__g + u - util.tensor_mult(self.__permeability,util.grad(p0)) ),
366                                     y=  -  util.inner(self.domain.getNormal(), u),
367                                     r=escript.Data())
368          dp=self.__pde_p.getSolution()
369          
370          p=GMRES(dp,
371                  self.__SYMSTAB_Aprod,
372              p0,
373              self.__inner,
374              atol=self.__norm(p0+dp)*self.getTolerance() ,
375              rtol=0.,
376              iter_max=max_iter,
377              iter_restart=iter_restart,
378              verbose=self.verbose,P_R=None)
379                
380              u=self.getFlux(p, u0)
381              return u,p
382    
383       def __L2(self,v):
384             return util.sqrt(util.integrate(util.length(util.interpolate(v,escript.Function(self.domain)))**2))      
385      
386       def __norm(self,r):
387             return util.sqrt(self.__inner(r,r))
388            
389       def __inner(self,r,s):
390             return util.integrate(util.inner(r,s), escript.Function(self.domain))
391            
392       def __STAB_Aprod(self,p):
393          gp=util.grad(p)
394          self.__pde_v.setValue(Y=-0.5*gp,
395                                X=-p*util.kronecker(self.__pde_v.getDomain()),
396                                y= p * self.domain.getNormal(),  
397                                r=escript.Data())
398          u = -self.__pde_v.getSolution()
399          self.__pde_p.setValue(Y=util.div(u),
400                                X=0.5*(u+util.tensor_mult(self.__permeability,gp)),
401                                y=escript.Data(),
402                                r=escript.Data())
403        
404          return  self.__pde_p.getSolution()
405      
406       def __SYMSTAB_Aprod(self,p):
407          gp=util.grad(p)
408          self.__pde_v.setValue(Y=0.5*gp ,
409                                X=escript.Data(),
410                                y=escript.Data(),  
411                                r=escript.Data())
412          u = -self.__pde_v.getSolution()
413          self.__pde_p.setValue(Y=escript.Data(),
414                                X=0.5*(-u+util.tensor_mult(self.__permeability,gp)),
415                                y=escript.Data(),
416                                r=escript.Data())
417        
418          return  self.__pde_p.getSolution()
419          
420    
421  class StokesProblemCartesian(HomogeneousSaddlePointProblem):  class StokesProblemCartesian(HomogeneousSaddlePointProblem):
422        """       """
423        solves       solves
424    
425            -(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}
426                  u_{i,i}=0                  u_{i,i}=0
427    
428            u=0 where  fixed_u_mask>0            u=0 where  fixed_u_mask>0
429            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
430    
431        if surface_stress is not give 0 is assumed.       if surface_stress is not given 0 is assumed.
432    
433        typical usage:       typical usage:
434    
435              sp=StokesProblemCartesian(domain)              sp=StokesProblemCartesian(domain)
436              sp.setTolerance()              sp.setTolerance()
437              sp.initialize(...)              sp.initialize(...)
438              v,p=sp.solve(v0,p0)              v,p=sp.solve(v0,p0)
439        """       """
440        def __init__(self,domain,**kwargs):       def __init__(self,domain,**kwargs):
441             """
442             initialize the Stokes Problem
443    
444             The approximation spaces used for velocity (=Solution(domain)) and pressure (=ReducedSolution(domain)) must be
445             LBB complient, for instance using quadratic and linear approximation on the same element or using linear approximation
446             with macro elements for the pressure.
447    
448             :param domain: domain of the problem.
449             :type domain: `Domain`
450             """
451           HomogeneousSaddlePointProblem.__init__(self,**kwargs)           HomogeneousSaddlePointProblem.__init__(self,**kwargs)
452           self.domain=domain           self.domain=domain
453           self.vol=util.integrate(1.,Function(self.domain))           self.__pde_v=LinearPDE(domain,numEquations=self.domain.getDim(),numSolutions=self.domain.getDim())
454           self.__pde_u=LinearPDE(domain,numEquations=self.domain.getDim(),numSolutions=self.domain.getDim())           self.__pde_v.setSymmetryOn()
455           self.__pde_u.setSymmetryOn()      
          # self.__pde_u.setSolverMethod(preconditioner=LinearPDE.ILU0)  
               
456           self.__pde_prec=LinearPDE(domain)           self.__pde_prec=LinearPDE(domain)
457           self.__pde_prec.setReducedOrderOn()           self.__pde_prec.setReducedOrderOn()
458           self.__pde_prec.setSymmetryOn()           self.__pde_prec.setSymmetryOn()
459    
460           self.__pde_proj=LinearPDE(domain)           self.__pde_proj=LinearPDE(domain)
461           self.__pde_proj.setReducedOrderOn()           self.__pde_proj.setReducedOrderOn()
462         self.__pde_proj.setValue(D=1)
463           self.__pde_proj.setSymmetryOn()           self.__pde_proj.setSymmetryOn()
          self.__pde_proj.setValue(D=1.)  
464    
465        def initialize(self,f=Data(),fixed_u_mask=Data(),eta=1,surface_stress=Data()):       def getSolverOptionsVelocity(self):
466          self.eta=eta           """
467          A =self.__pde_u.createCoefficientOfGeneralPDE("A")       returns the solver options used  solve the equation for velocity.
468      self.__pde_u.setValue(A=Data())      
469          for i in range(self.domain.getDim()):       :rtype: `SolverOptions`
470          for j in range(self.domain.getDim()):       """
471              A[i,j,j,i] += 1.       return self.__pde_v.getSolverOptions()
472              A[i,j,i,j] += 1.       def setSolverOptionsVelocity(self, options=None):
473      self.__pde_prec.setValue(D=1/self.eta)           """
474          self.__pde_u.setValue(A=A*self.eta,q=fixed_u_mask,Y=f,y=surface_stress)       set the solver options for solving the equation for velocity.
475        
476        def B(self,arg):       :param options: new solver  options
477           d=util.div(arg)       :type options: `SolverOptions`
478           self.__pde_proj.setValue(Y=d)       """
479           self.__pde_proj.setTolerance(self.getSubProblemTolerance())           self.__pde_v.setSolverOptions(options)
480           return self.__pde_proj.getSolution(verbose=self.show_details)       def getSolverOptionsPressure(self):
481             """
482        def inner(self,p0,p1):       returns the solver options used  solve the equation for pressure.
483           s0=util.interpolate(p0,Function(self.domain))       :rtype: `SolverOptions`
484           s1=util.interpolate(p1,Function(self.domain))       """
485         return self.__pde_prec.getSolverOptions()
486         def setSolverOptionsPressure(self, options=None):
487             """
488         set the solver options for solving the equation for pressure.
489         :param options: new solver  options
490         :type options: `SolverOptions`
491         """
492         self.__pde_prec.setSolverOptions(options)
493    
494         def setSolverOptionsDiv(self, options=None):
495             """
496         set the solver options for solving the equation to project the divergence of
497         the velocity onto the function space of presure.
498        
499         :param options: new solver options
500         :type options: `SolverOptions`
501         """
502         self.__pde_proj.setSolverOptions(options)
503         def getSolverOptionsDiv(self):
504             """
505         returns the solver options for solving the equation to project the divergence of
506         the velocity onto the function space of presure.
507        
508         :rtype: `SolverOptions`
509         """
510         return self.__pde_proj.getSolverOptions()
511    
512         def updateStokesEquation(self, v, p):
513             """
514             updates the Stokes equation to consider dependencies from ``v`` and ``p``
515             :note: This method can be overwritten by a subclass. Use `setStokesEquation` to set new values.
516             """
517             pass
518         def setStokesEquation(self, f=None,fixed_u_mask=None,eta=None,surface_stress=None,stress=None, restoration_factor=None):
519            """
520            assigns new values to the model parameters.
521    
522            :param f: external force
523            :type f: `Vector` object in `FunctionSpace` `Function` or similar
524            :param fixed_u_mask: mask of locations with fixed velocity.
525            :type fixed_u_mask: `Vector` object on `FunctionSpace` `Solution` or similar
526            :param eta: viscosity
527            :type eta: `Scalar` object on `FunctionSpace` `Function` or similar
528            :param surface_stress: normal surface stress
529            :type surface_stress: `Vector` object on `FunctionSpace` `FunctionOnBoundary` or similar
530            :param stress: initial stress
531        :type stress: `Tensor` object on `FunctionSpace` `Function` or similar
532            """
533            if eta !=None:
534                k=util.kronecker(self.domain.getDim())
535                kk=util.outer(k,k)
536                self.eta=util.interpolate(eta, escript.Function(self.domain))
537            self.__pde_prec.setValue(D=1/self.eta)
538                self.__pde_v.setValue(A=self.eta*(util.swap_axes(kk,0,3)+util.swap_axes(kk,1,3)))
539            if restoration_factor!=None:
540                n=self.domain.getNormal()
541                self.__pde_v.setValue(d=restoration_factor*util.outer(n,n))
542            if fixed_u_mask!=None:
543                self.__pde_v.setValue(q=fixed_u_mask)
544            if f!=None: self.__f=f
545            if surface_stress!=None: self.__surface_stress=surface_stress
546            if stress!=None: self.__stress=stress
547    
548         def initialize(self,f=escript.Data(),fixed_u_mask=escript.Data(),eta=1,surface_stress=escript.Data(),stress=escript.Data(), restoration_factor=0):
549            """
550            assigns values to the model parameters
551    
552            :param f: external force
553            :type f: `Vector` object in `FunctionSpace` `Function` or similar
554            :param fixed_u_mask: mask of locations with fixed velocity.
555            :type fixed_u_mask: `Vector` object on `FunctionSpace` `Solution` or similar
556            :param eta: viscosity
557            :type eta: `Scalar` object on `FunctionSpace` `Function` or similar
558            :param surface_stress: normal surface stress
559            :type surface_stress: `Vector` object on `FunctionSpace` `FunctionOnBoundary` or similar
560            :param stress: initial stress
561        :type stress: `Tensor` object on `FunctionSpace` `Function` or similar
562            """
563            self.setStokesEquation(f,fixed_u_mask, eta, surface_stress, stress, restoration_factor)
564    
565         def Bv(self,v,tol):
566             """
567             returns inner product of element p and div(v)
568    
569             :param v: a residual
570             :return: inner product of element p and div(v)
571             :rtype: ``float``
572             """
573             self.__pde_proj.setValue(Y=-util.div(v))
574         self.getSolverOptionsDiv().setTolerance(tol)
575         self.getSolverOptionsDiv().setAbsoluteTolerance(0.)
576             out=self.__pde_proj.getSolution()
577             return out
578    
579         def inner_pBv(self,p,Bv):
580             """
581             returns inner product of element p and Bv=-div(v)
582    
583             :param p: a pressure increment
584             :param Bv: a residual
585             :return: inner product of element p and Bv=-div(v)
586             :rtype: ``float``
587             """
588             return util.integrate(util.interpolate(p,escript.Function(self.domain))*util.interpolate(Bv, escript.Function(self.domain)))
589    
590         def inner_p(self,p0,p1):
591             """
592             Returns inner product of p0 and p1
593    
594             :param p0: a pressure
595             :param p1: a pressure
596             :return: inner product of p0 and p1
597             :rtype: ``float``
598             """
599             s0=util.interpolate(p0, escript.Function(self.domain))
600             s1=util.interpolate(p1, escript.Function(self.domain))
601           return util.integrate(s0*s1)           return util.integrate(s0*s1)
602    
603        def inner_a(self,a0,a1):       def norm_v(self,v):
604           p0=util.interpolate(a0[1],Function(self.domain))           """
605           p1=util.interpolate(a1[1],Function(self.domain))           returns the norm of v
606           alfa=(1/self.vol)*util.integrate(p0)  
607           beta=(1/self.vol)*util.integrate(p1)           :param v: a velovity
608       v0=util.grad(a0[0])           :return: norm of v
609       v1=util.grad(a1[0])           :rtype: non-negative ``float``
610           return util.integrate((p0-alfa)*(p1-beta)+((1/self.eta)**2)*util.inner(v0,v1))           """
611             return util.sqrt(util.integrate(util.length(util.grad(v))**2))
   
       def getStress(self,u):  
          mg=util.grad(u)  
          return 2.*self.eta*util.symmetric(mg)  
       def getEtaEffective(self):  
          return self.eta  
   
       def solve_A(self,u,p):  
          """  
          solves Av=f-Au-B^*p (v=0 on fixed_u_mask)  
          """  
          self.__pde_u.setTolerance(self.getSubProblemTolerance())  
          self.__pde_u.setValue(X=-self.getStress(u)-p*util.kronecker(self.domain))  
          return  self.__pde_u.getSolution(verbose=self.show_details)  
   
   
       def solve_prec(self,p):  
      #proj=Projector(domain=self.domain, reduce = True, fast=False)  
          self.__pde_prec.setTolerance(self.getSubProblemTolerance())  
          self.__pde_prec.setValue(Y=p)  
          q=self.__pde_prec.getSolution(verbose=self.show_details)  
          return q  
   
       def solve_prec1(self,p):  
      #proj=Projector(domain=self.domain, reduce = True, fast=False)  
          self.__pde_prec.setTolerance(self.getSubProblemTolerance())  
          self.__pde_prec.setValue(Y=p)  
          q=self.__pde_prec.getSolution(verbose=self.show_details)  
      q0=util.interpolate(q,Function(self.domain))  
          print util.inf(q*q0),util.sup(q*q0)  
          q-=(1/self.vol)*util.integrate(q0)  
          print util.inf(q*q0),util.sup(q*q0)  
          return q  
   
       def stoppingcriterium(self,Bv,v,p):  
           n_r=util.sqrt(self.inner(Bv,Bv))  
           n_v=util.sqrt(util.integrate(util.length(util.grad(v))**2))  
           if self.verbose: print "PCG step %s: L2(div(v)) = %s, L2(grad(v))=%s"%(self.iter,n_r,n_v)  
           if self.iter == 0: self.__n_v=n_v;  
           self.__n_v, n_v_old =n_v, self.__n_v  
           self.iter+=1  
           if self.iter>1 and n_r <= n_v*self.getTolerance() and abs(n_v_old-self.__n_v) <= n_v * self.getTolerance():  
               if self.verbose: print "PCG terminated after %s steps."%self.iter  
               return True  
           else:  
               return False  
       def stoppingcriterium2(self,norm_r,norm_b,solver='GMRES',TOL=None):  
       if TOL==None:  
              TOL=self.getTolerance()  
           if self.verbose: print "%s step %s: L2(r) = %s, L2(b)*TOL=%s"%(solver,self.iter,norm_r,norm_b*TOL)  
           self.iter+=1  
             
           if norm_r <= norm_b*TOL:  
               if self.verbose: print "%s terminated after %s steps."%(solver,self.iter)  
               return True  
           else:  
               return False  
612    
613    
614         def getDV(self, p, v, tol):
615             """
616             return the value for v for a given p (overwrite)
617    
618             :param p: a pressure
619             :param v: a initial guess for the value v to return.
620             :return: dv given as *Adv=(f-Av-B^*p)*
621             """
622             self.updateStokesEquation(v,p)
623             self.__pde_v.setValue(Y=self.__f, y=self.__surface_stress)
624         self.getSolverOptionsVelocity().setTolerance(tol)
625         self.getSolverOptionsVelocity().setAbsoluteTolerance(0.)
626             if self.__stress.isEmpty():
627                self.__pde_v.setValue(X=p*util.kronecker(self.domain)-2*self.eta*util.symmetric(util.grad(v)))
628             else:
629                self.__pde_v.setValue(X=self.__stress+p*util.kronecker(self.domain)-2*self.eta*util.symmetric(util.grad(v)))
630             out=self.__pde_v.getSolution()
631             return  out
632    
633         def norm_Bv(self,Bv):
634            """
635            Returns Bv (overwrite).
636    
637            :rtype: equal to the type of p
638            :note: boundary conditions on p should be zero!
639            """
640            return util.sqrt(util.integrate(util.interpolate(Bv, escript.Function(self.domain))**2))
641    
642         def solve_AinvBt(self,p, tol):
643             """
644             Solves *Av=B^*p* with accuracy `tol`
645    
646             :param p: a pressure increment
647             :return: the solution of *Av=B^*p*
648             :note: boundary conditions on v should be zero!
649             """
650             self.__pde_v.setValue(Y=escript.Data(), y=escript.Data(), X=-p*util.kronecker(self.domain))
651             out=self.__pde_v.getSolution()
652             return  out
653    
654         def solve_prec(self,Bv, tol):
655             """
656             applies preconditioner for for *BA^{-1}B^** to *Bv*
657             with accuracy `self.getSubProblemTolerance()`
658    
659             :param Bv: velocity increment
660             :return: *p=P(Bv)* where *P^{-1}* is an approximation of *BA^{-1}B^ * )*
661             :note: boundary conditions on p are zero.
662             """
663             self.__pde_prec.setValue(Y=Bv)
664         self.getSolverOptionsPressure().setTolerance(tol)
665         self.getSolverOptionsPressure().setAbsoluteTolerance(0.)
666             out=self.__pde_prec.getSolution()
667             return out

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