/[escript]/trunk/escriptcore/py_src/flows.py
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revision 2208 by gross, Mon Jan 12 06:37:07 2009 UTC revision 3569 by gross, Thu Sep 1 02:42:36 2011 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"""
18  __license__="""Licensed under the Open Software License version 3.0  __license__="""Licensed under the Open Software License version 3.0
19  http://www.opensource.org/licenses/osl-3.0.php"""  http://www.opensource.org/licenses/osl-3.0.php"""
20  __url__="http://www.uq.edu.au/esscc/escript-finley"  __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"
34    
35  from escript import *  import escript
36  import util  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  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
   
     M{u_i+k_{ij}*p_{,j} = g_i}  
     M{u_{i,i} = f}  
   
     where M{p} represents the pressure and M{u} the Darcy flux. M{k} represents the permeability,  
   
     @note: The problem is solved in a least squares formulation.  
     """  
   
     def __init__(self, domain,useReduced=False):  
         """  
         initializes the Darcy flux problem  
         @param domain: domain of the problem  
         @type domain: L{Domain}  
         """  
         self.domain=domain  
         self.__pde_v=LinearPDESystem(domain)  
         if useReduced: self.__pde_v.setReducedOrderOn()  
         self.__pde_v.setSymmetryOn()  
         self.__pde_v.setValue(D=util.kronecker(domain), A=util.outer(util.kronecker(domain),util.kronecker(domain)))  
         self.__pde_p=LinearSinglePDE(domain)  
         self.__pde_p.setSymmetryOn()  
         if useReduced: self.__pde_p.setReducedOrderOn()  
         self.__f=Scalar(0,self.__pde_v.getFunctionSpaceForCoefficient("X"))  
         self.__g=Vector(0,self.__pde_v.getFunctionSpaceForCoefficient("Y"))  
         self.__ATOL= None  
   
     def setValue(self,f=None, g=None, location_of_fixed_pressure=None, location_of_fixed_flux=None, permeability=None):  
         """  
         assigns values to model parameters  
   
         @param f: volumetic sources/sinks  
         @type f: scalar value on the domain (e.g. L{Data})  
         @param g: flux sources/sinks  
         @type g: vector values on the domain (e.g. L{Data})  
         @param location_of_fixed_pressure: mask for locations where pressure is fixed  
         @type location_of_fixed_pressure: scalar value on the domain (e.g. L{Data})  
         @param location_of_fixed_flux:  mask for locations where flux is fixed.  
         @type location_of_fixed_flux: vector values on the domain (e.g. L{Data})  
         @param permeability: permeability tensor. If scalar C{s} is given the tensor with  
                              C{s} on the main diagonal is used. If vector C{v} is given the tensor with  
                              C{v} on the main diagonal is used.  
         @type permeability: scalar, vector or tensor values on the domain (e.g. L{Data})  
   
         @note: the values of parameters which are not set by calling C{setValue} are not altered.  
         @note: at any point on the boundary of the domain the pressure (C{location_of_fixed_pressure} >0)  
                or the normal component of the flux (C{location_of_fixed_flux[i]>0} if direction of the normal  
                is along the M{x_i} axis.  
         """  
         if f !=None:  
            f=util.interpolate(f, self.__pde_v.getFunctionSpaceForCoefficient("X"))  
            if f.isEmpty():  
                f=Scalar(0,self.__pde_v.getFunctionSpaceForCoefficient("X"))  
            else:  
                if f.getRank()>0: raise ValueError,"illegal rank of f."  
            self.f=f  
         if g !=None:    
            g=util.interpolate(g, self.__pde_p.getFunctionSpaceForCoefficient("Y"))  
            if g.isEmpty():  
              g=Vector(0,self.__pde_v.getFunctionSpaceForCoefficient("Y"))  
            else:  
              if not g.getShape()==(self.domain.getDim(),):  
                raise ValueError,"illegal shape of g"  
            self.__g=g  
   
         if location_of_fixed_pressure!=None: self.__pde_p.setValue(q=location_of_fixed_pressure)  
         if location_of_fixed_flux!=None: self.__pde_v.setValue(q=location_of_fixed_flux)  
   
         if permeability!=None:  
            perm=util.interpolate(permeability,self.__pde_p.getFunctionSpaceForCoefficient("A"))  
            if perm.getRank()==0:  
                perm=perm*util.kronecker(self.domain.getDim())  
            elif perm.getRank()==1:  
                perm, perm2=Tensor(0.,self.__pde_p.getFunctionSpaceForCoefficient("A")), perm  
                for i in range(self.domain.getDim()): perm[i,i]=perm2[i]  
            elif perm.getRank()==2:  
               pass  
            else:  
               raise ValueError,"illegal rank of permeability."  
            self.__permeability=perm  
            self.__pde_p.setValue(A=util.transposed_tensor_mult(self.__permeability,self.__permeability))  
   
   
     def getFlux(self,p=None, fixed_flux=Data(),tol=1.e-8, 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(tol)  
         g=self.__g  
         f=self.__f  
         self.__pde_v.setValue(X=f*util.kronecker(self.domain), r=fixed_flux)  
         if p == None:  
            self.__pde_v.setValue(Y=g)  
         else:  
            self.__pde_v.setValue(Y=g-util.tensor_mult(self.__permeability,util.grad(p)))  
         return self.__pde_v.getSolution(verbose=show_details)  
   
     def getPressure(self,v=None, fixed_pressure=Data(),tol=1.e-8, show_details=False):  
         """  
         returns the pressure for a given flux C{v} where the pressure is equal to C{fixed_pressure}  
         on locations where C{location_of_fixed_pressure} is positive (see L{setValue}).  
         Note that C{g} is used, see L{setValue}.  
           
         @param v: flux.  
         @type v: vector-valued on the domain (e.g. L{Data}).  
         @param fixed_pressure: pressure on the locations of the domain marked be C{location_of_fixed_pressure}.  
         @type fixed_pressure: vector values on the domain (e.g. L{Data}).  
         @param tol: relative tolerance to be used.  
         @type tol: positive C{float}.  
         @return: pressure  
         @rtype: L{Data}  
         @note: the method uses the least squares solution M{p=(Q^*Q)^{-1}Q^*(g-u)} where and M{(Qp)_i=k_{ij}p_{,j}}  
                for the permeability M{k_{ij}}  
         """  
         self.__pde_v.setTolerance(tol)  
         g=self.__g  
         self.__pde_p.setValue(r=fixed_pressure)  
         if v == None:  
            self.__pde_p.setValue(X=util.transposed_tensor_mult(self.__permeability,g-v))  
         else:  
            self.__pde_p.setValue(X=util.transposed_tensor_mult(self.__permeability,g))  
         return self.__pde_p.getSolution(verbose=show_details)  
   
     def setTolerance(self,atol=0,rtol=1e-8,p_ref=None,v_ref=None):  
         """  
         set the tolerance C{ATOL} used to terminate the solution process. It is used  
   
         M{ATOL = atol + rtol * max( |g-v_ref|, |Qp_ref| )}  
   
         where M{|f|^2 = integrate(length(f)^2)} and M{(Qp)_i=k_{ij}p_{,j}} for the permeability M{k_{ij}}. If C{v_ref} or C{p_ref} is not present zero is assumed.  
   
         The iteration is terminated if for the current approximation C{p}, flux C{v=(I+D^*D)^{-1}(D^*f-g-Qp)} and their residual  
   
         M{r=Q^*(g-Qp-v)}  
   
         the condition  
   
         M{<(Q^*Q)^{-1} r,r> <= ATOL}  
   
         holds. M{D} is the M{div} operator and M{(Qp)_i=k_{ij}p_{,j}} for the permeability M{k_{ij}}  
   
         @param atol: absolute tolerance for the pressure  
         @type atol: non-negative C{float}  
         @param rtol: relative tolerance for the pressure  
         @type rtol: non-negative C{float}  
         @param p_ref: reference pressure. If not present zero is used. You may use physical arguments to set a resonable value for C{p_ref}, use the  
         L{getPressure} method or use  the value from a previous time step.  
         @type p_ref: scalar value on the domain (e.g. L{Data}).  
         @param v_ref: reference velocity.  If not present zero is used. You may use physical arguments to set a resonable value for C{v_ref}, use the  
         L{getFlux} method or use  the value from a previous time step.  
         @type v_ref: vector-valued on the domain (e.g. L{Data}).  
         @return: used absolute tolerance.  
         @rtype: positive C{float}  
         """  
         g=self.__g  
         if not v_ref == None:  
            f1=util.integrate(util.length(util.interpolate(g-v_ref,Function(self.domain)))**2)  
         else:  
            f1=util.integrate(util.length(util.interpolate(g))**2)  
         if not p_ref == None:  
            f2=util.integrate(util.length(util.tensor_mult(self.__permeability,util.grad(p_ref)))**2)  
         else:  
            f2=0  
         self.__ATOL= atol + rtol * util.sqrt(max(f1,f2))  
         if self.__ATOL<=0:  
            raise ValueError,"Positive tolerance (=%e) is expected."%self.__ATOL  
         return self.__ATOL  
           
     def getTolerance(self):  
         """  
         returns the current tolerance.  
43        
44          @return: used absolute tolerance.     *u_i+k_{ij}*p_{,j} = g_i*
45          @rtype: positive C{float}     *u_{i,i} = f*
46          """    
47          if self.__ATOL==None:     where *p* represents the pressure and *u* the Darcy flux. *k* represents the permeability,
48             raise ValueError,"no tolerance is defined."    
49          return self.__ATOL     :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      def solve(self,u0,p0, max_iter=100, verbose=False, show_details=False, sub_rtol=1.e-8):                 where *l* is the length scale, *K* is the inverse of the permeability tensor, and *w* is a positive weighting factor.
52           """     :cvar SMOOTH: global smoothing by solving the PDE *K_{ij} u_j= - p_{,j} + K_{ij} g_j*
53           solves the problem.     """
54       EVAL="EVAL"
55           The iteration is terminated if the residual norm is less then self.getTolerance().     SIMPLE="EVAL"
56       POST="POST"
57           @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.     SMOOTH="SMOOTH"
58           @type u0: vector value on the domain (e.g. L{Data}).     def __init__(self, domain, useReduced=False, solver="EVAL", verbose=False, w=1.):
59           @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.        """
60           @type p0: scalar value on the domain (e.g. L{Data}).        initializes the Darcy flux problem
61           @param sub_rtol: tolerance to be used in the sub iteration. It is recommended that M{sub_rtol<rtol*5.e-3}        :param domain: domain of the problem
62           @type sub_rtol: positive-negative C{float}        :type domain: `Domain`
63           @param verbose: if set some information on iteration progress are printed        :param useReduced: uses reduced oreder on flux and pressure
64           @type verbose: C{bool}        :type useReduced: ``bool``
65           @param show_details:  if set information on the subiteration process are printed.        :param solver: solver method
66           @type show_details: C{bool}        :type solver: in [`DarcyFlow.EVAL`, `DarcyFlow.POST',  `DarcyFlow.SMOOTH' ]
67           @return: flux and pressure        :param verbose: if ``True`` some information on the iteration progress are printed.
68           @rtype: C{tuple} of L{Data}.        :type verbose: ``bool``
69          :param w: weighting factor for `DarcyFlow.POST` solver
70           @note: The problem is solved as a least squares form        :type w: ``float``
71          
72           M{(I+D^*D)u+Qp=D^*f+g}        """
73           M{Q^*u+Q^*Qp=Q^*g}        if not solver in [DarcyFlow.EVAL, DarcyFlow.POST,  DarcyFlow.SMOOTH ] :
74              raise ValueError,"unknown solver %d."%solver
75    
76           where M{D} is the M{div} operator and M{(Qp)_i=k_{ij}p_{,j}} for the permeability M{k_{ij}}.        self.domain=domain
77           We eliminate the flux form the problem by setting        self.solver=solver
78          self.useReduced=useReduced
79          self.verbose=verbose
80          self.l=None
81          self.w=None
82        
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.EVAL:
88             self.__pde_v=None
89         if self.verbose: print "DarcyFlow: simple solver is used."
90    
91          elif self.solver  == self.POST:
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             self.__pde_v=LinearPDESystem(domain)
96             self.__pde_v.setSymmetryOn()
97             if self.useReduced: self.__pde_v.setReducedOrderOn()
98         self.w=w
99             self.l=util.vol(self.domain)**(1./self.domain.getDim()) # length scale
100    
101          elif self.solver  == self.SMOOTH:
102             self.__pde_v=LinearPDESystem(domain)
103             self.__pde_v.setSymmetryOn()
104             if self.useReduced: self.__pde_v.setReducedOrderOn()
105         if self.verbose: print "DarcyFlow: flux smoothing is used."
106         self.w=0
107    
108          self.__f=escript.Scalar(0,self.__pde_p.getFunctionSpaceForCoefficient("X"))
109          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          self.location_of_fixed_flux = escript.Vector(0, self.__pde_p.getFunctionSpaceForCoefficient("q"))
112        
113            
114       def setValue(self,f=None, g=None, location_of_fixed_pressure=None, location_of_fixed_flux=None, permeability=None):
115          """
116          assigns values to model parameters
117    
118           M{u=(I+D^*D)^{-1}(D^*f-g-Qp)} with u=u0 on location_of_fixed_flux        :param f: volumetic sources/sinks
119          :type f: scalar value on the domain (e.g. `escript.Data`)
120          :param g: flux sources/sinks
121          :type g: vector values on the domain (e.g. `escript.Data`)
122          :param location_of_fixed_pressure: mask for locations where pressure is fixed
123          :type location_of_fixed_pressure: scalar value on the domain (e.g. `escript.Data`)
124          :param location_of_fixed_flux:  mask for locations where flux is fixed.
125          :type location_of_fixed_flux: vector values on the domain (e.g. `escript.Data`)
126          :param permeability: permeability tensor. If scalar ``s`` is given the tensor with ``s`` on the main diagonal is used.
127          :type permeability: scalar or symmetric tensor values on the domain (e.g. `escript.Data`)
128    
129          :note: the values of parameters which are not set by calling ``setValue`` are not altered.
130          :note: at any point on the boundary of the domain the pressure
131                 (``location_of_fixed_pressure`` >0) or the normal component of the
132                 flux (``location_of_fixed_flux[i]>0``) if direction of the normal
133                 is along the *x_i* axis.
134    
135           form the first equation. Inserted into the second equation we get        """
136          if location_of_fixed_pressure!=None:
137               self.location_of_fixed_pressure=util.wherePositive(location_of_fixed_pressure)
138               self.__pde_p.setValue(q=self.location_of_fixed_pressure)
139          if location_of_fixed_flux!=None:
140              self.location_of_fixed_flux=util.wherePositive(location_of_fixed_flux)
141              if not self.__pde_v == None: self.__pde_v.setValue(q=self.location_of_fixed_flux)
142                
143          if permeability!=None:
144        
145         perm=util.interpolate(permeability,self.__pde_p.getFunctionSpaceForCoefficient("A"))
146            
147         if perm.getRank()==0:
148    
149           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          perm_inv=(1./perm)
150            perm_inv=perm_inv*util.kronecker(self.domain.getDim())
151            perm=perm*util.kronecker(self.domain.getDim())
152            
153            
154         elif perm.getRank()==2:
155            perm_inv=util.inverse(perm)
156         else:
157            raise ValueError,"illegal rank of permeability."
158                    
159           which is solved using the PCG method (precondition is M{Q^*Q}). In each iteration step       self.__permeability=perm
160           PDEs with operator M{I+D^*D} and with M{Q^*Q} needs to be solved using a sub iteration scheme.       self.__permeability_inv=perm_inv
161           """      
162           self.verbose=verbose           #====================
163           self.show_details= show_details and self.verbose       self.__pde_p.setValue(A=self.__permeability)
164           self.__pde_v.setTolerance(sub_rtol)           if self.solver  == self.EVAL:
165           self.__pde_p.setTolerance(sub_rtol)                pass # no extra work required
166           ATOL=self.getTolerance()           elif self.solver  == self.POST:
167           if self.verbose: print "DarcyFlux: absolute tolerance = %e"%ATOL          k=util.kronecker(self.domain.getDim())
168           #########################################################################################################################          self.omega = self.w*util.length(perm_inv)*self.l*self.domain.getSize()
169           #          self.__pde_v.setValue(D=self.__permeability_inv, A=self.omega*util.outer(k,k))
170           #   we solve:           elif self.solver  == self.SMOOTH:
171           #            self.__pde_v.setValue(D=self.__permeability_inv)
172           #      Q^*(I-(I+D^*D)^{-1})Q dp =  Q^* (g-u0-Qp0 - (I+D^*D)^{-1} ( D^*(f-Du0)+g-u0-Qp0) )  
173           #        if g != None:
174           #   residual is      g=util.interpolate(g, self.__pde_p.getFunctionSpaceForCoefficient("Y"))
175           #      if g.isEmpty():
176           #    r=  Q^* (g-u0-Qp0 - (I+D^*D)^{-1} ( D^*(f-Du0)+g-u0-Qp0) - Q dp +(I+D^*D)^{-1})Q dp ) = Q^* (g - Qp - v)            g=Vector(0,self.__pde_p.getFunctionSpaceForCoefficient("Y"))
177           #      else:
178           #        with v = (I+D^*D)^{-1} (D^*f+g-Qp) including BC          if not g.getShape()==(self.domain.getDim(),): raise ValueError,"illegal shape of g"
179           #      self.__g=g
180           #    we use (g - Qp, v) to represent the residual. not that        if f !=None:
181           #       f=util.interpolate(f, self.__pde_p.getFunctionSpaceForCoefficient("Y"))
182           #    dr(dp)=( -Q(dp), dv) with dv = - (I+D^*D)^{-1} Q(dp)       if f.isEmpty():      
183           #            f=Scalar(0,self.__pde_p.getFunctionSpaceForCoefficient("Y"))
184           #   while the initial residual is       else:
185           #           if f.getRank()>0: raise ValueError,"illegal rank of f."
186           #      r0=( g - Qp0, v00) with v00=(I+D^*D)^{-1} (D^*f+g-Qp0) including BC       self.__f=f
          #    
          d0=self.__g-util.tensor_mult(self.__permeability,util.grad(p0))  
          self.__pde_v.setValue(Y=d0, X=self.__f*util.kronecker(self.domain), r=u0)  
          v00=self.__pde_v.getSolution(verbose=show_details)  
          if self.verbose: print "DarcyFlux: range of initial flux = ",util.inf(v00), util.sup(v00)  
          self.__pde_v.setValue(r=Data())  
          # start CG  
          r=ArithmeticTuple(d0, v00)  
          p,r=PCG(r,self.__Aprod_PCG,p0,self.__Msolve_PCG,self.__inner_PCG,atol=ATOL, rtol=0.,iter_max=max_iter, verbose=self.verbose)  
          return r[1],p  
   
     def __Aprod_PCG(self,dp):  
           if self.show_details: print "DarcyFlux: Applying operator"  
           #  -dr(dp) = (Qdp,du) where du = (I+D^*D)^{-1} (Qdp)  
           mQdp=util.tensor_mult(self.__permeability,util.grad(dp))  
           self.__pde_v.setValue(Y=mQdp,X=Data(), r=Data())  
           du=self.__pde_v.getSolution(verbose=self.show_details)  
           return ArithmeticTuple(mQdp,du)  
   
     def __inner_PCG(self,p,r):  
          a=util.tensor_mult(self.__permeability,util.grad(p))  
          f0=util.integrate(util.inner(a,r[0]))  
          f1=util.integrate(util.inner(a,r[1]))  
          # print "__inner_PCG:",f0,f1,"->",f0-f1  
          return f0-f1  
   
     def __Msolve_PCG(self,r):  
           if self.show_details: print "DarcyFlux: Applying preconditioner"  
           self.__pde_p.setValue(X=util.transposed_tensor_mult(self.__permeability,r[0]-r[1]), r=Data())  
           return self.__pde_p.getSolution(verbose=self.show_details)  
187    
188  class StokesProblemCartesian(HomogeneousSaddlePointProblem):     def getSolverOptionsFlux(self):
189          """
190          Returns the solver options used to solve the flux problems
191          :return: `SolverOptions`
192          """
193          if self.__pde_v == None:
194              return None
195          else:
196              return self.__pde_v.getSolverOptions()
197          
198       def setSolverOptionsFlux(self, options=None):
199          """
200          Sets the solver options used to solve the flux problems
201          If ``options`` is not present, the options are reset to default
202          :param options: `SolverOptions`
203          """
204          if not self.__pde_v == None:
205              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        solves        return self.__pde_p.setSolverOptions(options)
223          
224       def solve(self, u0, p0):
225          """
226          solves the problem.
227          
228          :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.
229          :type u0: vector value on the domain (e.g. `escript.Data`).
230          :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.
231          :type p0: scalar value on the domain (e.g. `escript.Data`).
232          :return: flux and pressure
233          :rtype: ``tuple`` of `escript.Data`.
234    
235          """
236          self.__pde_p.setValue(X=self.__g ,
237                                Y=self.__f,
238                                y= - util.inner(self.domain.getNormal(),u0 * self.location_of_fixed_flux),
239                                r=p0)
240          p=self.__pde_p.getSolution()
241          u = self.getFlux(p, u0)
242          return u,p
243          
244       def getFlux(self,p, u0=None):
245            """
246            returns the flux for a given pressure ``p`` where the flux is equal to ``u0``
247            on locations where ``location_of_fixed_flux`` is positive (see `setValue`).
248            Notice that ``g`` is used, see `setValue`.
249    
250            :param p: pressure.
251            :type p: scalar value on the domain (e.g. `escript.Data`).
252            :param u0: flux on the locations of the domain marked be ``location_of_fixed_flux``.
253            :type u0: vector values on the domain (e.g. `escript.Data`) or ``None``
254            :return: flux
255            :rtype: `escript.Data`
256            """
257            u_eval=self.__g-util.tensor_mult(self.__permeability,util.grad(p))
258            if self.solver  == self.EVAL:
259               u = self.__g-util.tensor_mult(self.__permeability,util.grad(p))
260            elif self.solver  == self.POST or self.solver  == self.SMOOTH:
261                self.__pde_v.setValue(Y=util.tensor_mult(self.__permeability_inv,self.__g)-util.grad(p))
262                if u0 == None:
263               self.__pde_v.setValue(r=escript.Data())
264            else:
265               self.__pde_v.setValue(r=u0)
266                u= self.__pde_v.getSolution()
267        return u
268          
269    class StokesProblemCartesian(HomogeneousSaddlePointProblem):
270         """
271         solves
272    
273            -(eta*(u_{i,j}+u_{j,i}))_j + p_i = f_i-stress_{ij,j}            -(eta*(u_{i,j}+u_{j,i}))_j + p_i = f_i-stress_{ij,j}
274                  u_{i,i}=0                  u_{i,i}=0
# Line 333  class StokesProblemCartesian(Homogeneous Line 276  class StokesProblemCartesian(Homogeneous
276            u=0 where  fixed_u_mask>0            u=0 where  fixed_u_mask>0
277            eta*(u_{i,j}+u_{j,i})*n_j-p*n_i=surface_stress +stress_{ij}n_j            eta*(u_{i,j}+u_{j,i})*n_j-p*n_i=surface_stress +stress_{ij}n_j
278    
279        if surface_stress is not given 0 is assumed.       if surface_stress is not given 0 is assumed.
280    
281        typical usage:       typical usage:
282    
283              sp=StokesProblemCartesian(domain)              sp=StokesProblemCartesian(domain)
284              sp.setTolerance()              sp.setTolerance()
285              sp.initialize(...)              sp.initialize(...)
286              v,p=sp.solve(v0,p0)              v,p=sp.solve(v0,p0)
287        """       """
288        def __init__(self,domain,**kwargs):       def __init__(self,domain,**kwargs):
289           """           """
290           initialize the Stokes Problem           initialize the Stokes Problem
291    
292           @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
293           @type domain: L{Domain}           LBB complient, for instance using quadratic and linear approximation on the same element or using linear approximation
294           @warning: The apprximation order needs to be two otherwise you may see oscilations in the pressure.           with macro elements for the pressure.
295    
296             :param domain: domain of the problem.
297             :type domain: `Domain`
298           """           """
299           HomogeneousSaddlePointProblem.__init__(self,**kwargs)           HomogeneousSaddlePointProblem.__init__(self,**kwargs)
300           self.domain=domain           self.domain=domain
301           self.vol=util.integrate(1.,Function(self.domain))           self.__pde_v=LinearPDE(domain,numEquations=self.domain.getDim(),numSolutions=self.domain.getDim())
302           self.__pde_u=LinearPDE(domain,numEquations=self.domain.getDim(),numSolutions=self.domain.getDim())           self.__pde_v.setSymmetryOn()
303           self.__pde_u.setSymmetryOn()      
          # self.__pde_u.setSolverMethod(self.__pde_u.DIRECT)  
          # self.__pde_u.setSolverMethod(preconditioner=LinearPDE.RILU)  
               
304           self.__pde_prec=LinearPDE(domain)           self.__pde_prec=LinearPDE(domain)
305           self.__pde_prec.setReducedOrderOn()           self.__pde_prec.setReducedOrderOn()
          # self.__pde_prec.setSolverMethod(self.__pde_prec.LUMPING)  
306           self.__pde_prec.setSymmetryOn()           self.__pde_prec.setSymmetryOn()
307    
308           self.__pde_proj=LinearPDE(domain)           self.__pde_proj=LinearPDE(domain)
309           self.__pde_proj.setReducedOrderOn()           self.__pde_proj.setReducedOrderOn()
310         self.__pde_proj.setValue(D=1)
311           self.__pde_proj.setSymmetryOn()           self.__pde_proj.setSymmetryOn()
          self.__pde_proj.setValue(D=1.)  
312    
313        def initialize(self,f=Data(),fixed_u_mask=Data(),eta=1,surface_stress=Data(),stress=Data()):       def getSolverOptionsVelocity(self):
314             """
315         returns the solver options used  solve the equation for velocity.
316        
317         :rtype: `SolverOptions`
318         """
319         return self.__pde_v.getSolverOptions()
320         def setSolverOptionsVelocity(self, options=None):
321             """
322         set the solver options for solving the equation for velocity.
323        
324         :param options: new solver  options
325         :type options: `SolverOptions`
326         """
327             self.__pde_v.setSolverOptions(options)
328         def getSolverOptionsPressure(self):
329             """
330         returns the solver options used  solve the equation for pressure.
331         :rtype: `SolverOptions`
332         """
333         return self.__pde_prec.getSolverOptions()
334         def setSolverOptionsPressure(self, options=None):
335             """
336         set the solver options for solving the equation for pressure.
337         :param options: new solver  options
338         :type options: `SolverOptions`
339         """
340         self.__pde_prec.setSolverOptions(options)
341    
342         def setSolverOptionsDiv(self, options=None):
343             """
344         set the solver options for solving the equation to project the divergence of
345         the velocity onto the function space of presure.
346        
347         :param options: new solver options
348         :type options: `SolverOptions`
349         """
350         self.__pde_proj.setSolverOptions(options)
351         def getSolverOptionsDiv(self):
352             """
353         returns the solver options for solving the equation to project the divergence of
354         the velocity onto the function space of presure.
355        
356         :rtype: `SolverOptions`
357         """
358         return self.__pde_proj.getSolverOptions()
359    
360         def updateStokesEquation(self, v, p):
361             """
362             updates the Stokes equation to consider dependencies from ``v`` and ``p``
363             :note: This method can be overwritten by a subclass. Use `setStokesEquation` to set new values.
364             """
365             pass
366         def setStokesEquation(self, f=None,fixed_u_mask=None,eta=None,surface_stress=None,stress=None, restoration_factor=None):
367            """
368            assigns new values to the model parameters.
369    
370            :param f: external force
371            :type f: `Vector` object in `FunctionSpace` `Function` or similar
372            :param fixed_u_mask: mask of locations with fixed velocity.
373            :type fixed_u_mask: `Vector` object on `FunctionSpace` `Solution` or similar
374            :param eta: viscosity
375            :type eta: `Scalar` object on `FunctionSpace` `Function` or similar
376            :param surface_stress: normal surface stress
377            :type surface_stress: `Vector` object on `FunctionSpace` `FunctionOnBoundary` or similar
378            :param stress: initial stress
379        :type stress: `Tensor` object on `FunctionSpace` `Function` or similar
380            """
381            if eta !=None:
382                k=util.kronecker(self.domain.getDim())
383                kk=util.outer(k,k)
384                self.eta=util.interpolate(eta, escript.Function(self.domain))
385            self.__pde_prec.setValue(D=1/self.eta)
386                self.__pde_v.setValue(A=self.eta*(util.swap_axes(kk,0,3)+util.swap_axes(kk,1,3)))
387            if restoration_factor!=None:
388                n=self.domain.getNormal()
389                self.__pde_v.setValue(d=restoration_factor*util.outer(n,n))
390            if fixed_u_mask!=None:
391                self.__pde_v.setValue(q=fixed_u_mask)
392            if f!=None: self.__f=f
393            if surface_stress!=None: self.__surface_stress=surface_stress
394            if stress!=None: self.__stress=stress
395    
396         def initialize(self,f=escript.Data(),fixed_u_mask=escript.Data(),eta=1,surface_stress=escript.Data(),stress=escript.Data(), restoration_factor=0):
397          """          """
398          assigns values to the model parameters          assigns values to the model parameters
399    
400          @param f: external force          :param f: external force
401          @type f: L{Vector} object in L{FunctionSpace} L{Function} or similar          :type f: `Vector` object in `FunctionSpace` `Function` or similar
402          @param fixed_u_mask: mask of locations with fixed velocity.          :param fixed_u_mask: mask of locations with fixed velocity.
403          @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
404          @param eta: viscosity          :param eta: viscosity
405          @type eta: L{Scalar} object on L{FunctionSpace} L{Function} or similar          :type eta: `Scalar` object on `FunctionSpace` `Function` or similar
406          @param surface_stress: normal surface stress          :param surface_stress: normal surface stress
407          @type eta: L{Vector} object on L{FunctionSpace} L{FunctionOnBoundary} or similar          :type surface_stress: `Vector` object on `FunctionSpace` `FunctionOnBoundary` or similar
408          @param stress: initial stress          :param stress: initial stress
409      @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.  
   
410          """          """
411          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,Y=f,y=surface_stress)  
         self.__stress=stress  
412    
413        def B(self,v):       def Bv(self,v,tol):
414          """           """
415          returns div(v)           returns inner product of element p and div(v)
         @rtype: equal to the type of p  
416    
417          @note: boundary conditions on p should be zero!           :param v: a residual
418          """           :return: inner product of element p and div(v)
419          if self.show_details: print "apply divergence:"           :rtype: ``float``
420          self.__pde_proj.setValue(Y=-util.div(v))           """
421          self.__pde_proj.setTolerance(self.getSubProblemTolerance())           self.__pde_proj.setValue(Y=-util.div(v))
422          return self.__pde_proj.getSolution(verbose=self.show_details)       self.getSolverOptionsDiv().setTolerance(tol)
423         self.getSolverOptionsDiv().setAbsoluteTolerance(0.)
424             out=self.__pde_proj.getSolution()
425             return out
426    
427        def inner_pBv(self,p,Bv):       def inner_pBv(self,p,Bv):
428           """           """
429           returns inner product of element p and Bv  (overwrite)           returns inner product of element p and Bv=-div(v)
           
          @type p: equal to the type of p  
          @type Bv: equal to the type of result of operator B  
          @rtype: C{float}  
430    
431           @rtype: equal to the type of p           :param p: a pressure increment
432             :param Bv: a residual
433             :return: inner product of element p and Bv=-div(v)
434             :rtype: ``float``
435           """           """
436           s0=util.interpolate(p,Function(self.domain))           return util.integrate(util.interpolate(p,escript.Function(self.domain))*util.interpolate(Bv, escript.Function(self.domain)))
          s1=util.interpolate(Bv,Function(self.domain))  
          return util.integrate(s0*s1)  
437    
438        def inner_p(self,p0,p1):       def inner_p(self,p0,p1):
439           """           """
440           returns inner product of element p0 and p1  (overwrite)           Returns inner product of p0 and p1
           
          @type p0: equal to the type of p  
          @type p1: equal to the type of p  
          @rtype: C{float}  
441    
442           @rtype: equal to the type of p           :param p0: a pressure
443             :param p1: a pressure
444             :return: inner product of p0 and p1
445             :rtype: ``float``
446           """           """
447           s0=util.interpolate(p0/self.eta,Function(self.domain))           s0=util.interpolate(p0, escript.Function(self.domain))
448           s1=util.interpolate(p1/self.eta,Function(self.domain))           s1=util.interpolate(p1, escript.Function(self.domain))
449           return util.integrate(s0*s1)           return util.integrate(s0*s1)
450    
451        def inner_v(self,v0,v1):       def norm_v(self,v):
452           """           """
453           returns inner product of two element v0 and v1  (overwrite)           returns the norm of v
           
          @type v0: equal to the type of v  
          @type v1: equal to the type of v  
          @rtype: C{float}  
454    
455           @rtype: equal to the type of v           :param v: a velovity
456             :return: norm of v
457             :rtype: non-negative ``float``
458           """           """
459       gv0=util.grad(v0)           return util.sqrt(util.integrate(util.length(util.grad(v))**2))
460       gv1=util.grad(v1)  
          return util.integrate(util.inner(gv0,gv1))  
461    
462        def solve_A(self,u,p):       def getDV(self, p, v, tol):
463           """           """
464           solves Av=f-Au-B^*p (v=0 on fixed_u_mask)           return the value for v for a given p (overwrite)
465    
466             :param p: a pressure
467             :param v: a initial guess for the value v to return.
468             :return: dv given as *Adv=(f-Av-B^*p)*
469           """           """
470           if self.show_details: print "solve for velocity:"           self.updateStokesEquation(v,p)
471           self.__pde_u.setTolerance(self.getSubProblemTolerance())           self.__pde_v.setValue(Y=self.__f, y=self.__surface_stress)
472         self.getSolverOptionsVelocity().setTolerance(tol)
473         self.getSolverOptionsVelocity().setAbsoluteTolerance(0.)
474           if self.__stress.isEmpty():           if self.__stress.isEmpty():
475              self.__pde_u.setValue(X=-2*self.eta*util.symmetric(util.grad(u))+p*util.kronecker(self.domain))              self.__pde_v.setValue(X=p*util.kronecker(self.domain)-2*self.eta*util.symmetric(util.grad(v)))
476           else:           else:
477              self.__pde_u.setValue(X=self.__stress-2*self.eta*util.symmetric(util.grad(u))+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)))
478           out=self.__pde_u.getSolution(verbose=self.show_details)           out=self.__pde_v.getSolution()
479           return  out           return  out
480    
481        def solve_prec(self,p):       def norm_Bv(self,Bv):
482           if self.show_details: print "apply preconditioner:"          """
483           self.__pde_prec.setTolerance(self.getSubProblemTolerance())          Returns Bv (overwrite).
484           self.__pde_prec.setValue(Y=p)  
485           q=self.__pde_prec.getSolution(verbose=self.show_details)          :rtype: equal to the type of p
486           return q          :note: boundary conditions on p should be zero!
487            """
488            return util.sqrt(util.integrate(util.interpolate(Bv, escript.Function(self.domain))**2))
489    
490         def solve_AinvBt(self,p, tol):
491             """
492             Solves *Av=B^*p* with accuracy `tol`
493    
494             :param p: a pressure increment
495             :return: the solution of *Av=B^*p*
496             :note: boundary conditions on v should be zero!
497             """
498             self.__pde_v.setValue(Y=escript.Data(), y=escript.Data(), X=-p*util.kronecker(self.domain))
499             out=self.__pde_v.getSolution()
500             return  out
501    
502         def solve_prec(self,Bv, tol):
503             """
504             applies preconditioner for for *BA^{-1}B^** to *Bv*
505             with accuracy `self.getSubProblemTolerance()`
506    
507             :param Bv: velocity increment
508             :return: *p=P(Bv)* where *P^{-1}* is an approximation of *BA^{-1}B^ * )*
509             :note: boundary conditions on p are zero.
510             """
511             self.__pde_prec.setValue(Y=Bv)
512         self.getSolverOptionsPressure().setTolerance(tol)
513         self.getSolverOptionsPressure().setAbsoluteTolerance(0.)
514             out=self.__pde_prec.getSolution()
515             return out

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