/[escript]/trunk/escriptcore/py_src/flows.py
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revision 2349 by gross, Mon Mar 30 08:14:23 2009 UTC revision 3852 by jfenwick, Thu Mar 1 05:34:52 2012 UTC
# Line 1  Line 1 
1    # -*- coding: utf-8 -*-
2  ########################################################  ########################################################
3  #  #
4  # Copyright (c) 2003-2008 by University of Queensland  # Copyright (c) 2003-2010 by University of Queensland
5  # Earth Systems Science Computational Center (ESSCC)  # Earth Systems Science Computational Center (ESSCC)
6  # http://www.uq.edu.au/esscc  # http://www.uq.edu.au/esscc
7  #  #
# Line 10  Line 11 
11  #  #
12  ########################################################  ########################################################
13    
14  __copyright__="""Copyright (c) 2003-2008 by University of Queensland  __copyright__="""Copyright (c) 2003-2010 by University of Queensland
15  Earth Systems Science Computational Center (ESSCC)  Earth Systems Science Computational Center (ESSCC)
16  http://www.uq.edu.au/esscc  http://www.uq.edu.au/esscc
17  Primary Business: Queensland, Australia"""  Primary Business: Queensland, Australia"""
# Line 21  __url__="https://launchpad.net/escript-f Line 22  __url__="https://launchpad.net/escript-f
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"
34    
35  from escript import *  from . import escript
36  import util  from . import util
37  from linearPDEs import LinearPDE, LinearPDESystem, LinearSinglePDE  from .linearPDEs import LinearPDE, LinearPDESystem, LinearSinglePDE, SolverOptions
38  from pdetools import HomogeneousSaddlePointProblem,Projector, ArithmeticTuple, PCG, NegativeNorm, GMRES  from .pdetools import HomogeneousSaddlePointProblem,Projector, ArithmeticTuple, PCG, NegativeNorm, GMRES
39    
40  class DarcyFlow(object):  class DarcyFlow(object):
41      """     """
42      solves the problem     solves the problem
43      
44      M{u_i+k_{ij}*p_{,j} = g_i}     *u_i+k_{ij}*p_{,j} = g_i*
45      M{u_{i,i} = f}     *u_{i,i} = f*
46      
47      where M{p} represents the pressure and M{u} the Darcy flux. M{k} represents the permeability,     where *p* represents the pressure and *u* the Darcy flux. *k* represents the permeability,
48      
49      @note: The problem is solved in a least squares formulation.     :cvar EVAL: direct pressure gradient evaluation for flux
50      """     :cvar POST: global postprocessing of flux by solving the PDE *K_{ij} u_j + (w * K * l u_{k,k})_{,i}= - p_{,j} + K_{ij} g_j*
51                   where *l* is the length scale, *K* is the inverse of the permeability tensor, and *w* is a positive weighting factor.
52      def __init__(self, domain,useReduced=False):     :cvar SMOOTH: global smoothing by solving the PDE *K_{ij} u_j= - p_{,j} + K_{ij} g_j*
53          """     """
54          initializes the Darcy flux problem     EVAL="EVAL"
55          @param domain: domain of the problem     SIMPLE="EVAL"
56          @type domain: L{Domain}     POST="POST"
57          """     SMOOTH="SMOOTH"
58          self.domain=domain     def __init__(self, domain, useReduced=False, solver="POST", verbose=False, w=1.):
59          self.__l=util.longestEdge(self.domain)**2        """
60          self.__pde_v=LinearPDESystem(domain)        initializes the Darcy flux problem
61          if useReduced: self.__pde_v.setReducedOrderOn()        :param domain: domain of the problem
62          self.__pde_v.setSymmetryOn()        :type domain: `Domain`
63          self.__pde_v.setValue(D=util.kronecker(domain), A=self.__l*util.outer(util.kronecker(domain),util.kronecker(domain)))        :param useReduced: uses reduced oreder on flux and pressure
64          self.__pde_p=LinearSinglePDE(domain)        :type useReduced: ``bool``
65          self.__pde_p.setSymmetryOn()        :param solver: solver method
66          if useReduced: self.__pde_p.setReducedOrderOn()        :type solver: in [`DarcyFlow.EVAL`, `DarcyFlow.POST',  `DarcyFlow.SMOOTH' ]
67          self.__f=Scalar(0,self.__pde_v.getFunctionSpaceForCoefficient("X"))        :param verbose: if ``True`` some information on the iteration progress are printed.
68          self.__g=Vector(0,self.__pde_v.getFunctionSpaceForCoefficient("Y"))        :type verbose: ``bool``
69          self.setTolerance()        :param w: weighting factor for `DarcyFlow.POST` solver
70          self.setAbsoluteTolerance()        :type w: ``float``
71          self.setSubProblemTolerance()        
72          """
73      def setValue(self,f=None, g=None, location_of_fixed_pressure=None, location_of_fixed_flux=None, permeability=None):        if not solver in [DarcyFlow.EVAL, DarcyFlow.POST,  DarcyFlow.SMOOTH ] :
74          """            raise ValueError("unknown solver %d."%solver)
75          assigns values to model parameters  
76          self.domain=domain
77          @param f: volumetic sources/sinks        self.solver=solver
78          @type f: scalar value on the domain (e.g. L{Data})        self.useReduced=useReduced
79          @param g: flux sources/sinks        self.verbose=verbose
80          @type g: vector values on the domain (e.g. L{Data})        self.l=None
81          @param location_of_fixed_pressure: mask for locations where pressure is fixed        self.w=None
82          @type location_of_fixed_pressure: scalar value on the domain (e.g. L{Data})      
83          @param location_of_fixed_flux:  mask for locations where flux is fixed.        self.__pde_p=LinearSinglePDE(domain)
84          @type location_of_fixed_flux: vector values on the domain (e.g. L{Data})        self.__pde_p.setSymmetryOn()
85          @param permeability: permeability tensor. If scalar C{s} is given the tensor with        if self.useReduced: self.__pde_p.setReducedOrderOn()
86                               C{s} on the main diagonal is used. If vector C{v} is given the tensor with  
87                               C{v} on the main diagonal is used.        if self.solver  == self.EVAL:
88          @type permeability: scalar, vector or tensor values on the domain (e.g. L{Data})           self.__pde_v=None
89             if self.verbose: print("DarcyFlow: simple solver is used.")
90          @note: the values of parameters which are not set by calling C{setValue} are not altered.  
91          @note: at any point on the boundary of the domain the pressure (C{location_of_fixed_pressure} >0)        elif self.solver  == self.POST:
92                 or the normal component of the flux (C{location_of_fixed_flux[i]>0} if direction of the normal           if util.inf(w)<0.:
93                 is along the M{x_i} axis.              raise ValueError("Weighting factor must be non-negative.")
94          """           if self.verbose: print("DarcyFlow: global postprocessing of flux is used.")
95          if f !=None:           self.__pde_v=LinearPDESystem(domain)
96             f=util.interpolate(f, self.__pde_v.getFunctionSpaceForCoefficient("X"))           self.__pde_v.setSymmetryOn()
97             if f.isEmpty():           if self.useReduced: self.__pde_v.setReducedOrderOn()
98                 f=Scalar(0,self.__pde_v.getFunctionSpaceForCoefficient("X"))           self.w=w
99             else:           self.l=util.vol(self.domain)**(1./self.domain.getDim()) # length scale
100                 if f.getRank()>0: raise ValueError,"illegal rank of f."  
101             self.__f=f        elif self.solver  == self.SMOOTH:
102          if g !=None:           self.__pde_v=LinearPDESystem(domain)
103             g=util.interpolate(g, self.__pde_p.getFunctionSpaceForCoefficient("Y"))           self.__pde_v.setSymmetryOn()
104             if g.isEmpty():           if self.useReduced: self.__pde_v.setReducedOrderOn()
105               g=Vector(0,self.__pde_v.getFunctionSpaceForCoefficient("Y"))           if self.verbose: print("DarcyFlow: flux smoothing is used.")
106             else:           self.w=0
107               if not g.getShape()==(self.domain.getDim(),):  
108                 raise ValueError,"illegal shape of g"        self.__f=escript.Scalar(0,self.__pde_p.getFunctionSpaceForCoefficient("X"))
109             self.__g=g        self.__g=escript.Vector(0,self.__pde_p.getFunctionSpaceForCoefficient("Y"))
110          self.location_of_fixed_pressure = escript.Scalar(0, self.__pde_p.getFunctionSpaceForCoefficient("q"))
111          if location_of_fixed_pressure!=None: self.__pde_p.setValue(q=location_of_fixed_pressure)        self.location_of_fixed_flux = escript.Vector(0, self.__pde_p.getFunctionSpaceForCoefficient("q"))
112          if location_of_fixed_flux!=None: self.__pde_v.setValue(q=location_of_fixed_flux)        self.perm_scale=1.
113        
114          if permeability!=None:          
115             perm=util.interpolate(permeability,self.__pde_p.getFunctionSpaceForCoefficient("A"))     def setValue(self,f=None, g=None, location_of_fixed_pressure=None, location_of_fixed_flux=None, permeability=None):
116             if perm.getRank()==0:        """
117                 perm=perm*util.kronecker(self.domain.getDim())        assigns values to model parameters
118             elif perm.getRank()==1:  
119                 perm, perm2=Tensor(0.,self.__pde_p.getFunctionSpaceForCoefficient("A")), perm        :param f: volumetic sources/sinks
120                 for i in range(self.domain.getDim()): perm[i,i]=perm2[i]        :type f: scalar value on the domain (e.g. `escript.Data`)
121             elif perm.getRank()==2:        :param g: flux sources/sinks
122                pass        :type g: vector values on the domain (e.g. `escript.Data`)
123             else:        :param location_of_fixed_pressure: mask for locations where pressure is fixed
124                raise ValueError,"illegal rank of permeability."        :type location_of_fixed_pressure: scalar value on the domain (e.g. `escript.Data`)
125             self.__permeability=perm        :param location_of_fixed_flux:  mask for locations where flux is fixed.
126             self.__pde_p.setValue(A=util.transposed_tensor_mult(self.__permeability,self.__permeability))        :type location_of_fixed_flux: vector values on the domain (e.g. `escript.Data`)
127          :param permeability: permeability tensor. If scalar ``s`` is given the tensor with ``s`` on the main diagonal is used.
128      def setTolerance(self,rtol=1e-4):        :type permeability: scalar or symmetric tensor values on the domain (e.g. `escript.Data`)
129          """  
130          sets the relative tolerance C{rtol} used to terminate the solution process. The iteration is terminated if        :note: the values of parameters which are not set by calling ``setValue`` are not altered.
131          :note: at any point on the boundary of the domain the pressure
132          M{|g-v-Qp| <= atol + rtol * min( max( |g-v|, |Qp| ), max( |v|, |g-Qp| ) ) }               (``location_of_fixed_pressure`` >0) or the normal component of the
133                 flux (``location_of_fixed_flux[i]>0``) if direction of the normal
134          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}}.               is along the *x_i* axis.
135    
136          @param rtol: relative tolerance for the pressure        """
137          @type rtol: non-negative C{float}        if location_of_fixed_pressure!=None:
138          """             self.location_of_fixed_pressure=util.wherePositive(location_of_fixed_pressure)
139          if rtol<0:             self.__pde_p.setValue(q=self.location_of_fixed_pressure)
140              raise ValueError,"Relative tolerance needs to be non-negative."        if location_of_fixed_flux!=None:
141          self.__rtol=rtol            self.location_of_fixed_flux=util.wherePositive(location_of_fixed_flux)
142      def getTolerance(self):            if not self.__pde_v == None: self.__pde_v.setValue(q=self.location_of_fixed_flux)
143          """              
144          returns the relative tolerance        if permeability!=None:
145        
146          @return: current relative tolerance           perm=util.interpolate(permeability,self.__pde_p.getFunctionSpaceForCoefficient("A"))
147          @rtype: C{float}           self.perm_scale=util.Lsup(util.length(perm))
148          """           if self.verbose: print(("DarcyFlow: permeability scaling factor = %e."%self.perm_scale))
149          return self.__rtol           perm=perm*(1./self.perm_scale)
150            
151      def setAbsoluteTolerance(self,atol=0.):           if perm.getRank()==0:
152          """  
153          sets the absolute tolerance C{atol} used to terminate the solution process. The iteration is terminated if              perm_inv=(1./perm)
154                perm_inv=perm_inv*util.kronecker(self.domain.getDim())
155          M{|g-v-Qp| <= atol + rtol * min( max( |g-v|, |Qp| ), max( |v|, |g-Qp| ) ) }              perm=perm*util.kronecker(self.domain.getDim())
156            
157          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}}.          
158             elif perm.getRank()==2:
159          @param atol: absolute tolerance for the pressure              perm_inv=util.inverse(perm)
         @type atol: non-negative C{float}  
         """  
         if atol<0:  
             raise ValueError,"Absolute tolerance needs to be non-negative."  
         self.__atol=atol  
     def getAbsoluteTolerance(self):  
        """  
        returns the absolute tolerance  
         
        @return: current absolute tolerance  
        @rtype: C{float}  
        """  
        return self.__atol  
   
     def setSubProblemTolerance(self,rtol=None):  
          """  
          Sets the relative tolerance to solve the subproblem(s). If C{rtol} is not present  
          C{self.getTolerance()**2} is used.  
   
          @param rtol: relative tolerence  
          @type rtol: positive C{float}  
          """  
          if rtol == None:  
               if self.getTolerance()<=0.:  
                   raise ValueError,"A positive relative tolerance must be set."  
               self.__sub_tol=max(util.EPSILON**(0.75),self.getTolerance()**2)  
160           else:           else:
161               if rtol<=0:              raise ValueError("illegal rank of permeability.")
162                   raise ValueError,"sub-problem tolerance must be positive."          
163               self.__sub_tol=max(util.EPSILON**(0.75),rtol)           self.__permeability=perm
164             self.__permeability_inv=perm_inv
165      def getSubProblemTolerance(self):      
166           """           #====================
167           Returns the subproblem reduction factor.           self.__pde_p.setValue(A=self.__permeability)
168             if self.solver  == self.EVAL:
169           @return: subproblem reduction factor                pass # no extra work required
170           @rtype: C{float}           elif self.solver  == self.POST:
171           """                k=util.kronecker(self.domain.getDim())
172           return self.__sub_tol                self.omega = self.w*util.length(perm_inv)*self.l*self.domain.getSize()
173                  self.__pde_v.setValue(D=self.__permeability_inv, A=self.omega*util.outer(k,k))
174      def solve(self,u0,p0, max_iter=100, verbose=False, show_details=False, max_num_corrections=10):           elif self.solver  == self.SMOOTH:
175           """              self.__pde_v.setValue(D=self.__permeability_inv)
176           solves the problem.  
177          if g != None:
178           The iteration is terminated if the residual norm is less then self.getTolerance().          g=util.interpolate(g, self.__pde_p.getFunctionSpaceForCoefficient("Y"))
179            if g.isEmpty():
180           @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.               g=Vector(0,self.__pde_p.getFunctionSpaceForCoefficient("Y"))
          @type u0: vector value on the domain (e.g. L{Data}).  
          @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.  
          @type p0: scalar value on the domain (e.g. L{Data}).  
          @param verbose: if set some information on iteration progress are printed  
          @type verbose: C{bool}  
          @param show_details:  if set information on the subiteration process are printed.  
          @type show_details: C{bool}  
          @return: flux and pressure  
          @rtype: C{tuple} of L{Data}.  
   
          @note: The problem is solved as a least squares form  
   
          M{(I+D^*D)u+Qp=D^*f+g}  
          M{Q^*u+Q^*Qp=Q^*g}  
   
          where M{D} is the M{div} operator and M{(Qp)_i=k_{ij}p_{,j}} for the permeability M{k_{ij}}.  
          We eliminate the flux form the problem by setting  
   
          M{u=(I+D^*D)^{-1}(D^*f-g-Qp)} with u=u0 on location_of_fixed_flux  
   
          form the first equation. Inserted into the second equation we get  
   
          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  
   
          which is solved using the PCG method (precondition is M{Q^*Q}). In each iteration step  
          PDEs with operator M{I+D^*D} and with M{Q^*Q} needs to be solved using a sub iteration scheme.  
          """  
          self.verbose=verbose or True  
          self.show_details= show_details and self.verbose  
          rtol=self.getTolerance()  
          atol=self.getAbsoluteTolerance()  
          if self.verbose: print "DarcyFlux: initial sub tolerance = %e"%self.getSubProblemTolerance()  
   
          num_corrections=0  
          converged=False  
          p=p0  
          norm_r=None  
          while not converged:  
                v=self.getFlux(p, fixed_flux=u0, show_details=self.show_details)  
                Qp=self.__Q(p)  
                norm_v=self.__L2(v)  
                norm_Qp=self.__L2(Qp)  
                if norm_v == 0.:  
                   if norm_Qp == 0.:  
                      return v,p  
                   else:  
                     fac=norm_Qp  
                else:  
                   if norm_Qp == 0.:  
                     fac=norm_v  
                   else:  
                     fac=2./(1./norm_v+1./norm_Qp)  
                ATOL=(atol+rtol*fac)  
                if self.verbose:  
                     print "DarcyFlux: L2 norm of v = %e."%norm_v  
                     print "DarcyFlux: L2 norm of k.grad(p) = %e."%norm_Qp  
                     print "DarcyFlux: absolute tolerance ATOL = %e."%ATOL  
                if norm_r == None or norm_r>ATOL:  
                    if num_corrections>max_num_corrections:  
                          raise ValueError,"maximum number of correction steps reached."  
                    p,r, norm_r=PCG(self.__g-util.interpolate(v,Function(self.domain))-Qp,self.__Aprod,p,self.__Msolve_PCG,self.__inner_PCG,atol=0.1*ATOL, rtol=0.,iter_max=max_iter, verbose=self.verbose)  
                    num_corrections+=1  
                else:  
                    converged=True  
          return v,p  
 #  
 #                
 #               r_hat=g-util.interpolate(v,Function(self.domain))-Qp  
 #               #===========================================================================  
 #               norm_r_hat=self.__L2(r_hat)  
 #               norm_v=self.__L2(v)  
 #               norm_g=self.__L2(g)  
 #               norm_gv=self.__L2(g-v)  
 #               norm_Qp=self.__L2(Qp)  
 #               norm_gQp=self.__L2(g-Qp)  
 #               fac=min(max(norm_v,norm_gQp),max(norm_Qp,norm_gv))  
 #               fac=min(norm_v,norm_Qp,norm_gv)  
 #               norm_r_hat_PCG=util.sqrt(self.__inner_PCG(self.__Msolve_PCG(r_hat),r_hat))  
 #               print "norm_r_hat = ",norm_r_hat,norm_r_hat_PCG, norm_r_hat_PCG/norm_r_hat  
 #               if r!=None:  
 #                   print "diff = ",self.__L2(r-r_hat)/norm_r_hat  
 #                   sub_tol=min(rtol/self.__L2(r-r_hat)*norm_r_hat,1.)*self.getSubProblemTolerance()  
 #                   self.setSubProblemTolerance(sub_tol)  
 #                   print "subtol_new=",self.getSubProblemTolerance()  
 #               print "norm_v = ",norm_v  
 #               print "norm_gv = ",norm_gv  
 #               print "norm_Qp = ",norm_Qp  
 #               print "norm_gQp = ",norm_gQp  
 #               print "norm_g = ",norm_g  
 #               print "max(norm_v,norm_gQp)=",max(norm_v,norm_gQp)  
 #               print "max(norm_Qp,norm_gv)=",max(norm_Qp,norm_gv)  
 #               if fac == 0:  
 #                   if self.verbose: print "DarcyFlux: trivial case!"  
 #                   return v,p  
 #               #===============================================================================  
 #               # norm_v=util.sqrt(self.__inner_PCG(self.__Msolve_PCG(v),v))  
 #               # norm_Qp=self.__L2(Qp)  
 #               norm_r_hat=util.sqrt(self.__inner_PCG(self.__Msolve_PCG(r_hat),r_hat))  
 #               # print "**** norm_v, norm_Qp :",norm_v,norm_Qp  
 #  
 #               ATOL=(atol+rtol*2./(1./norm_v+1./norm_Qp))  
 #               if self.verbose:  
 #                   print "DarcyFlux: residual = %e"%norm_r_hat  
 #                   print "DarcyFlux: absolute tolerance ATOL = %e."%ATOL  
 #               if norm_r_hat <= ATOL:  
 #                   print "DarcyFlux: iteration finalized."  
 #                   converged=True  
 #               else:  
 #                   # p=GMRES(r_hat,self.__Aprod, p, self.__inner_GMRES, atol=ATOL, rtol=0., iter_max=max_iter, iter_restart=20, verbose=self.verbose,P_R=self.__Msolve_PCG)  
 #                   # p,r=PCG(r_hat,self.__Aprod,p,self.__Msolve_PCG,self.__inner_PCG,atol=ATOL*min(0.1,norm_r_hat_PCG/norm_r_hat), rtol=0.,iter_max=max_iter, verbose=self.verbose)  
 #                   p,r, norm_r=PCG(r_hat,self.__Aprod,p,self.__Msolve_PCG,self.__inner_PCG,atol=0.1*ATOL, rtol=0.,iter_max=max_iter, verbose=self.verbose)  
 #               print "norm_r =",norm_r  
 #         return v,p  
     def __L2(self,v):  
          return util.sqrt(util.integrate(util.length(util.interpolate(v,Function(self.domain)))**2))  
   
     def __Q(self,p):  
           return util.tensor_mult(self.__permeability,util.grad(p))  
   
     def __Aprod(self,dp):  
           self.__pde_v.setTolerance(self.getSubProblemTolerance())  
           if self.show_details: print "DarcyFlux: Applying operator"  
           Qdp=self.__Q(dp)  
           self.__pde_v.setValue(Y=-Qdp,X=Data(), r=Data())  
           du=self.__pde_v.getSolution(verbose=self.show_details, iter_max = 100000)  
           return Qdp+du  
     def __inner_GMRES(self,r,s):  
          return util.integrate(util.inner(r,s))  
   
     def __inner_PCG(self,p,r):  
          return util.integrate(util.inner(self.__Q(p), r))  
   
     def __Msolve_PCG(self,r):  
           self.__pde_p.setTolerance(self.getSubProblemTolerance())  
           if self.show_details: print "DarcyFlux: Applying preconditioner"  
           self.__pde_p.setValue(X=util.transposed_tensor_mult(self.__permeability,r), Y=Data(), r=Data())  
           return self.__pde_p.getSolution(verbose=self.show_details, iter_max = 100000)  
   
     def getFlux(self,p=None, fixed_flux=Data(), show_details=False):  
         """  
         returns the flux for a given pressure C{p} where the flux is equal to C{fixed_flux}  
         on locations where C{location_of_fixed_flux} is positive (see L{setValue}).  
         Note that C{g} and C{f} are used, see L{setValue}.  
   
         @param p: pressure.  
         @type p: scalar value on the domain (e.g. L{Data}).  
         @param fixed_flux: flux on the locations of the domain marked be C{location_of_fixed_flux}.  
         @type fixed_flux: vector values on the domain (e.g. L{Data}).  
         @param tol: relative tolerance to be used.  
         @type tol: positive C{float}.  
         @return: flux  
         @rtype: L{Data}  
         @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}}  
                for the permeability M{k_{ij}}  
         """  
         self.__pde_v.setTolerance(self.getSubProblemTolerance())  
         g=self.__g  
         f=self.__f  
         self.__pde_v.setValue(X=self.__l*f*util.kronecker(self.domain), r=fixed_flux)  
         if p == None:  
            self.__pde_v.setValue(Y=g)  
181          else:          else:
182             self.__pde_v.setValue(Y=g-self.__Q(p))               if not g.getShape()==(self.domain.getDim(),): raise ValueError("illegal shape of g")
183          return self.__pde_v.getSolution(verbose=show_details, iter_max=100000)          self.__g=g
184          if f !=None:
185             f=util.interpolate(f, self.__pde_p.getFunctionSpaceForCoefficient("Y"))
186             if f.isEmpty():      
187                 f=Scalar(0,self.__pde_p.getFunctionSpaceForCoefficient("Y"))
188             else:
189                 if f.getRank()>0: raise ValueError("illegal rank of f.")
190             self.__f=f
191    
192       def getSolverOptionsFlux(self):
193          """
194          Returns the solver options used to solve the flux problems
195          :return: `SolverOptions`
196          """
197          if self.__pde_v == None:
198              return None
199          else:
200              return self.__pde_v.getSolverOptions()
201          
202       def setSolverOptionsFlux(self, options=None):
203          """
204          Sets the solver options used to solve the flux problems
205          If ``options`` is not present, the options are reset to default
206          :param options: `SolverOptions`
207          """
208          if not self.__pde_v == None:
209              self.__pde_v.setSolverOptions(options)
210        
211       def getSolverOptionsPressure(self):
212          """
213          Returns the solver options used to solve the pressure problems
214          :return: `SolverOptions`
215          """
216          return self.__pde_p.getSolverOptions()
217          
218       def setSolverOptionsPressure(self, options=None):
219          """
220          Sets the solver options used to solve the pressure problems
221          If ``options`` is not present, the options are reset to default
222          
223          :param options: `SolverOptions`
224          :note: if the adaption of subtolerance is choosen, the tolerance set by ``options`` will be overwritten before the solver is called.
225          """
226          return self.__pde_p.setSolverOptions(options)
227          
228       def solve(self, u0, p0):
229          """
230          solves the problem.
231          
232          :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.
233          :type u0: vector value on the domain (e.g. `escript.Data`).
234          :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.
235          :type p0: scalar value on the domain (e.g. `escript.Data`).
236          :return: flux and pressure
237          :rtype: ``tuple`` of `escript.Data`.
238    
239          """
240          self.__pde_p.setValue(X=self.__g * 1./self.perm_scale,
241                                Y=self.__f * 1./self.perm_scale,
242                                y= - util.inner(self.domain.getNormal(),u0 * self.location_of_fixed_flux * 1./self.perm_scale ),
243                                r=p0)
244          p=self.__pde_p.getSolution()
245          u = self.getFlux(p, u0)
246          return u,p
247          
248       def getFlux(self,p, u0=None):
249            """
250            returns the flux for a given pressure ``p`` where the flux is equal to ``u0``
251            on locations where ``location_of_fixed_flux`` is positive (see `setValue`).
252            Notice that ``g`` is used, see `setValue`.
253    
254            :param p: pressure.
255            :type p: scalar value on the domain (e.g. `escript.Data`).
256            :param u0: flux on the locations of the domain marked be ``location_of_fixed_flux``.
257            :type u0: vector values on the domain (e.g. `escript.Data`) or ``None``
258            :return: flux
259            :rtype: `escript.Data`
260            """
261            if self.solver  == self.EVAL:
262               u = self.__g-self.perm_scale * util.tensor_mult(self.__permeability,util.grad(p))
263            elif self.solver  == self.POST or self.solver  == self.SMOOTH:
264                self.__pde_v.setValue(Y=util.tensor_mult(self.__permeability_inv,self.__g * 1./self.perm_scale)-util.grad(p))
265                if u0 == None:
266                   self.__pde_v.setValue(r=escript.Data())
267                else:
268                   if not isinstance(u0, escript.Data) : u0 = escript.Vector(u0, escript.Solution(self.domain))
269                   self.__pde_v.setValue(r=1./self.perm_scale * u0)
270                   u= self.__pde_v.getSolution() * self.perm_scale
271            return u
272          
273  class StokesProblemCartesian(HomogeneousSaddlePointProblem):  class StokesProblemCartesian(HomogeneousSaddlePointProblem):
274       """       """
275       solves       solves
# Line 386  class StokesProblemCartesian(Homogeneous Line 288  class StokesProblemCartesian(Homogeneous
288              sp.setTolerance()              sp.setTolerance()
289              sp.initialize(...)              sp.initialize(...)
290              v,p=sp.solve(v0,p0)              v,p=sp.solve(v0,p0)
291                sp.setStokesEquation(...) # new values for some parameters
292                v1,p1=sp.solve(v,p)
293       """       """
294       def __init__(self,domain,**kwargs):       def __init__(self,domain,**kwargs):
295           """           """
296           initialize the Stokes Problem           initialize the Stokes Problem
297    
298           @param domain: domain of the problem. The approximation order needs to be two.           The approximation spaces used for velocity (=Solution(domain)) and pressure (=ReducedSolution(domain)) must be
299           @type domain: L{Domain}           LBB complient, for instance using quadratic and linear approximation on the same element or using linear approximation
300           @warning: The apprximation order needs to be two otherwise you may see oscilations in the pressure.           with macro elements for the pressure.
301    
302             :param domain: domain of the problem.
303             :type domain: `Domain`
304           """           """
305           HomogeneousSaddlePointProblem.__init__(self,**kwargs)           HomogeneousSaddlePointProblem.__init__(self,**kwargs)
306           self.domain=domain           self.domain=domain
307           self.vol=util.integrate(1.,Function(self.domain))           self.__pde_v=LinearPDE(domain,numEquations=self.domain.getDim(),numSolutions=self.domain.getDim())
308           self.__pde_u=LinearPDE(domain,numEquations=self.domain.getDim(),numSolutions=self.domain.getDim())           self.__pde_v.setSymmetryOn()
309           self.__pde_u.setSymmetryOn()      
          # self.__pde_u.setSolverMethod(self.__pde_u.DIRECT)  
          # self.__pde_u.setSolverMethod(preconditioner=LinearPDE.RILU)  
   
310           self.__pde_prec=LinearPDE(domain)           self.__pde_prec=LinearPDE(domain)
311           self.__pde_prec.setReducedOrderOn()           self.__pde_prec.setReducedOrderOn()
          # self.__pde_prec.setSolverMethod(self.__pde_prec.LUMPING)  
312           self.__pde_prec.setSymmetryOn()           self.__pde_prec.setSymmetryOn()
313    
314       def initialize(self,f=Data(),fixed_u_mask=Data(),eta=1,surface_stress=Data(),stress=Data()):           self.__pde_proj=LinearPDE(domain)
315             self.__pde_proj.setReducedOrderOn()
316             self.__pde_proj.setValue(D=1)
317             self.__pde_proj.setSymmetryOn()
318    
319         def getSolverOptionsVelocity(self):
320             """
321         returns the solver options used  solve the equation for velocity.
322        
323         :rtype: `SolverOptions`
324         """
325             return self.__pde_v.getSolverOptions()
326         def setSolverOptionsVelocity(self, options=None):
327             """
328         set the solver options for solving the equation for velocity.
329        
330         :param options: new solver  options
331         :type options: `SolverOptions`
332         """
333             self.__pde_v.setSolverOptions(options)
334         def getSolverOptionsPressure(self):
335             """
336         returns the solver options used  solve the equation for pressure.
337         :rtype: `SolverOptions`
338         """
339             return self.__pde_prec.getSolverOptions()
340         def setSolverOptionsPressure(self, options=None):
341             """
342         set the solver options for solving the equation for pressure.
343         :param options: new solver  options
344         :type options: `SolverOptions`
345         """
346             self.__pde_prec.setSolverOptions(options)
347    
348         def setSolverOptionsDiv(self, options=None):
349             """
350         set the solver options for solving the equation to project the divergence of
351         the velocity onto the function space of presure.
352        
353         :param options: new solver options
354         :type options: `SolverOptions`
355         """
356             self.__pde_proj.setSolverOptions(options)
357         def getSolverOptionsDiv(self):
358             """
359         returns the solver options for solving the equation to project the divergence of
360         the velocity onto the function space of presure.
361        
362         :rtype: `SolverOptions`
363         """
364             return self.__pde_proj.getSolverOptions()
365    
366         def updateStokesEquation(self, v, p):
367             """
368             updates the Stokes equation to consider dependencies from ``v`` and ``p``
369             :note: This method can be overwritten by a subclass. Use `setStokesEquation` to set new values to model parameters.
370             """
371             pass
372         def setStokesEquation(self, f=None,fixed_u_mask=None,eta=None,surface_stress=None,stress=None, restoration_factor=None):
373            """
374            assigns new values to the model parameters.
375    
376            :param f: external force
377            :type f: `Vector` object in `FunctionSpace` `Function` or similar
378            :param fixed_u_mask: mask of locations with fixed velocity.
379            :type fixed_u_mask: `Vector` object on `FunctionSpace` `Solution` or similar
380            :param eta: viscosity
381            :type eta: `Scalar` object on `FunctionSpace` `Function` or similar
382            :param surface_stress: normal surface stress
383            :type surface_stress: `Vector` object on `FunctionSpace` `FunctionOnBoundary` or similar
384            :param stress: initial stress
385        :type stress: `Tensor` object on `FunctionSpace` `Function` or similar
386            """
387            if eta !=None:
388                k=util.kronecker(self.domain.getDim())
389                kk=util.outer(k,k)
390                self.eta=util.interpolate(eta, escript.Function(self.domain))
391                self.__pde_prec.setValue(D=1/self.eta)
392                self.__pde_v.setValue(A=self.eta*(util.swap_axes(kk,0,3)+util.swap_axes(kk,1,3)))
393            if restoration_factor!=None:
394                n=self.domain.getNormal()
395                self.__pde_v.setValue(d=restoration_factor*util.outer(n,n))
396            if fixed_u_mask!=None:
397                self.__pde_v.setValue(q=fixed_u_mask)
398            if f!=None: self.__f=f
399            if surface_stress!=None: self.__surface_stress=surface_stress
400            if stress!=None: self.__stress=stress
401    
402         def initialize(self,f=escript.Data(),fixed_u_mask=escript.Data(),eta=1,surface_stress=escript.Data(),stress=escript.Data(), restoration_factor=0):
403          """          """
404          assigns values to the model parameters          assigns values to the model parameters
405    
406          @param f: external force          :param f: external force
407          @type f: L{Vector} object in L{FunctionSpace} L{Function} or similar          :type f: `Vector` object in `FunctionSpace` `Function` or similar
408          @param fixed_u_mask: mask of locations with fixed velocity.          :param fixed_u_mask: mask of locations with fixed velocity.
409          @type fixed_u_mask: L{Vector} object on L{FunctionSpace} L{Solution} or similar          :type fixed_u_mask: `Vector` object on `FunctionSpace` `Solution` or similar
410          @param eta: viscosity          :param eta: viscosity
411          @type eta: L{Scalar} object on L{FunctionSpace} L{Function} or similar          :type eta: `Scalar` object on `FunctionSpace` `Function` or similar
412          @param surface_stress: normal surface stress          :param surface_stress: normal surface stress
413          @type eta: L{Vector} object on L{FunctionSpace} L{FunctionOnBoundary} or similar          :type surface_stress: `Vector` object on `FunctionSpace` `FunctionOnBoundary` or similar
414          @param stress: initial stress          :param stress: initial stress
415      @type stress: L{Tensor} object on L{FunctionSpace} L{Function} or similar          :type stress: `Tensor` object on `FunctionSpace` `Function` or similar
         @note: All values needs to be set.  
   
416          """          """
417          self.eta=eta          self.setStokesEquation(f,fixed_u_mask, eta, surface_stress, stress, restoration_factor)
         A =self.__pde_u.createCoefficient("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.__pde_prec.setValue(D=1/self.eta)  
         self.__pde_u.setValue(A=A*self.eta,q=fixed_u_mask)  
         self.__f=f  
         self.__surface_stress=surface_stress  
         self.__stress=stress  
418    
419       def inner_pBv(self,p,v):       def Bv(self,v,tol):
420           """           """
421           returns inner product of element p and div(v)           returns inner product of element p and div(v)
422    
423           @param p: a pressure increment           :param v: a residual
424           @param v: a residual           :return: inner product of element p and div(v)
425           @return: inner product of element p and div(v)           :rtype: ``float``
426           @rtype: C{float}           """
427             self.__pde_proj.setValue(Y=-util.div(v))
428             self.getSolverOptionsDiv().setTolerance(tol)
429             self.getSolverOptionsDiv().setAbsoluteTolerance(0.)
430             out=self.__pde_proj.getSolution()
431             return out
432    
433         def inner_pBv(self,p,Bv):
434             """
435             returns inner product of element p and Bv=-div(v)
436    
437             :param p: a pressure increment
438             :param Bv: a residual
439             :return: inner product of element p and Bv=-div(v)
440             :rtype: ``float``
441           """           """
442           return util.integrate(-p*util.div(v))           return util.integrate(util.interpolate(p,escript.Function(self.domain))*util.interpolate(Bv, escript.Function(self.domain)))
443    
444       def inner_p(self,p0,p1):       def inner_p(self,p0,p1):
445           """           """
446           Returns inner product of p0 and p1           Returns inner product of p0 and p1
447    
448           @param p0: a pressure           :param p0: a pressure
449           @param p1: a pressure           :param p1: a pressure
450           @return: inner product of p0 and p1           :return: inner product of p0 and p1
451           @rtype: C{float}           :rtype: ``float``
452           """           """
453           s0=util.interpolate(p0/self.eta,Function(self.domain))           s0=util.interpolate(p0, escript.Function(self.domain))
454           s1=util.interpolate(p1/self.eta,Function(self.domain))           s1=util.interpolate(p1, escript.Function(self.domain))
455           return util.integrate(s0*s1)           return util.integrate(s0*s1)
456    
457       def norm_v(self,v):       def norm_v(self,v):
458           """           """
459           returns the norm of v           returns the norm of v
460    
461           @param v: a velovity           :param v: a velovity
462           @return: norm of v           :return: norm of v
463           @rtype: non-negative C{float}           :rtype: non-negative ``float``
464           """           """
465           return util.sqrt(util.integrate(util.length(util.grad(v))))           return util.sqrt(util.integrate(util.length(util.grad(v))**2))
466    
467       def getV(self, p, v0):  
468         def getDV(self, p, v, tol):
469           """           """
470           return the value for v for a given p (overwrite)           return the value for v for a given p
471    
472           @param p: a pressure           :param p: a pressure
473           @param v0: a initial guess for the value v to return.           :param v: a initial guess for the value v to return.
474           @return: v given as M{v= A^{-1} (f-B^*p)}           :return: dv given as *Adv=(f-Av-B^*p)*
475           """           """
476           self.__pde_u.setTolerance(self.getSubProblemTolerance())           self.updateStokesEquation(v,p)
477           self.__pde_u.setValue(Y=self.__f, y=self.__surface_stress, r=v0)           self.__pde_v.setValue(Y=self.__f, y=self.__surface_stress)
478             self.getSolverOptionsVelocity().setTolerance(tol)
479             self.getSolverOptionsVelocity().setAbsoluteTolerance(0.)
480           if self.__stress.isEmpty():           if self.__stress.isEmpty():
481              self.__pde_u.setValue(X=p*util.kronecker(self.domain))              self.__pde_v.setValue(X=p*util.kronecker(self.domain)-2*self.eta*util.symmetric(util.grad(v)))
482           else:           else:
483              self.__pde_u.setValue(X=self.__stress+p*util.kronecker(self.domain))              self.__pde_v.setValue(X=self.__stress+p*util.kronecker(self.domain)-2*self.eta*util.symmetric(util.grad(v)))
484           out=self.__pde_u.getSolution(verbose=self.show_details)           out=self.__pde_v.getSolution()
485           return  out           return  out
486    
487         def norm_Bv(self,Bv):
          raise NotImplementedError,"no v calculation implemented."  
   
   
      def norm_Bv(self,v):  
488          """          """
489          Returns Bv (overwrite).          Returns Bv (overwrite).
490    
491          @rtype: equal to the type of p          :rtype: equal to the type of p
492          @note: boundary conditions on p should be zero!          :note: boundary conditions on p should be zero!
493          """          """
494          return util.sqrt(util.integrate(util.div(v)**2))          return util.sqrt(util.integrate(util.interpolate(Bv, escript.Function(self.domain))**2))
495    
496       def solve_AinvBt(self,p):       def solve_AinvBt(self,p, tol):
497           """           """
498           Solves M{Av=B^*p} with accuracy L{self.getSubProblemTolerance()}           Solves *Av=B^*p* with accuracy `tol`
499    
500           @param p: a pressure increment           :param p: a pressure increment
501           @return: the solution of M{Av=B^*p}           :return: the solution of *Av=B^*p*
502           @note: boundary conditions on v should be zero!           :note: boundary conditions on v should be zero!
503           """           """
504           self.__pde_u.setTolerance(self.getSubProblemTolerance())           self.__pde_v.setValue(Y=escript.Data(), y=escript.Data(), X=-p*util.kronecker(self.domain))
505           self.__pde_u.setValue(Y=Data(), y=Data(), r=Data(),X=-p*util.kronecker(self.domain))           out=self.__pde_v.getSolution()
          out=self.__pde_u.getSolution(verbose=self.show_details)  
506           return  out           return  out
507    
508       def solve_precB(self,v):       def solve_prec(self,Bv, tol):
509           """           """
510           applies preconditioner for for M{BA^{-1}B^*} to M{Bv}           applies preconditioner for for *BA^{-1}B^** to *Bv*
511           with accuracy L{self.getSubProblemTolerance()} (overwrite).           with accuracy `self.getSubProblemTolerance()`
512    
513           @param v: velocity increment           :param Bv: velocity increment
514           @return: M{p=P(Bv)} where M{P^{-1}} is an approximation of M{BA^{-1}B^*}           :return: *p=P(Bv)* where *P^{-1}* is an approximation of *BA^{-1}B^ * )*
515           @note: boundary conditions on p are zero.           :note: boundary conditions on p are zero.
516           """           """
517           self.__pde_prec.setValue(Y=-util.div(v))           self.__pde_prec.setValue(Y=Bv)
518           self.__pde_prec.setTolerance(self.getSubProblemTolerance())           self.getSolverOptionsPressure().setTolerance(tol)
519           return self.__pde_prec.getSolution(verbose=self.show_details)           self.getSolverOptionsPressure().setAbsoluteTolerance(0.)
520             out=self.__pde_prec.getSolution()
521             return out

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