/[escript]/trunk/escriptcore/py_src/flows.py
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revision 2208 by gross, Mon Jan 12 06:37:07 2009 UTC revision 3582 by gross, Tue Sep 6 02:02:17 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        """              sp.setStokesEquation(...) # new values for some parameters
288        def __init__(self,domain,**kwargs):              v1,p1=sp.solve(v,p)
289         """
290         def __init__(self,domain,**kwargs):
291           """           """
292           initialize the Stokes Problem           initialize the Stokes Problem
293    
294           @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
295           @type domain: L{Domain}           LBB complient, for instance using quadratic and linear approximation on the same element or using linear approximation
296           @warning: The apprximation order needs to be two otherwise you may see oscilations in the pressure.           with macro elements for the pressure.
297    
298             :param domain: domain of the problem.
299             :type domain: `Domain`
300           """           """
301           HomogeneousSaddlePointProblem.__init__(self,**kwargs)           HomogeneousSaddlePointProblem.__init__(self,**kwargs)
302           self.domain=domain           self.domain=domain
303           self.vol=util.integrate(1.,Function(self.domain))           self.__pde_v=LinearPDE(domain,numEquations=self.domain.getDim(),numSolutions=self.domain.getDim())
304           self.__pde_u=LinearPDE(domain,numEquations=self.domain.getDim(),numSolutions=self.domain.getDim())           self.__pde_v.setSymmetryOn()
305           self.__pde_u.setSymmetryOn()      
          # self.__pde_u.setSolverMethod(self.__pde_u.DIRECT)  
          # self.__pde_u.setSolverMethod(preconditioner=LinearPDE.RILU)  
               
306           self.__pde_prec=LinearPDE(domain)           self.__pde_prec=LinearPDE(domain)
307           self.__pde_prec.setReducedOrderOn()           self.__pde_prec.setReducedOrderOn()
          # self.__pde_prec.setSolverMethod(self.__pde_prec.LUMPING)  
308           self.__pde_prec.setSymmetryOn()           self.__pde_prec.setSymmetryOn()
309    
310           self.__pde_proj=LinearPDE(domain)           self.__pde_proj=LinearPDE(domain)
311           self.__pde_proj.setReducedOrderOn()           self.__pde_proj.setReducedOrderOn()
312         self.__pde_proj.setValue(D=1)
313           self.__pde_proj.setSymmetryOn()           self.__pde_proj.setSymmetryOn()
          self.__pde_proj.setValue(D=1.)  
314    
315        def initialize(self,f=Data(),fixed_u_mask=Data(),eta=1,surface_stress=Data(),stress=Data()):       def getSolverOptionsVelocity(self):
316             """
317         returns the solver options used  solve the equation for velocity.
318        
319         :rtype: `SolverOptions`
320         """
321         return self.__pde_v.getSolverOptions()
322         def setSolverOptionsVelocity(self, options=None):
323             """
324         set the solver options for solving the equation for velocity.
325        
326         :param options: new solver  options
327         :type options: `SolverOptions`
328         """
329             self.__pde_v.setSolverOptions(options)
330         def getSolverOptionsPressure(self):
331             """
332         returns the solver options used  solve the equation for pressure.
333         :rtype: `SolverOptions`
334         """
335         return self.__pde_prec.getSolverOptions()
336         def setSolverOptionsPressure(self, options=None):
337             """
338         set the solver options for solving the equation for pressure.
339         :param options: new solver  options
340         :type options: `SolverOptions`
341         """
342         self.__pde_prec.setSolverOptions(options)
343    
344         def setSolverOptionsDiv(self, options=None):
345             """
346         set the solver options for solving the equation to project the divergence of
347         the velocity onto the function space of presure.
348        
349         :param options: new solver options
350         :type options: `SolverOptions`
351         """
352         self.__pde_proj.setSolverOptions(options)
353         def getSolverOptionsDiv(self):
354             """
355         returns the solver options for solving the equation to project the divergence of
356         the velocity onto the function space of presure.
357        
358         :rtype: `SolverOptions`
359         """
360         return self.__pde_proj.getSolverOptions()
361    
362         def updateStokesEquation(self, v, p):
363             """
364             updates the Stokes equation to consider dependencies from ``v`` and ``p``
365             :note: This method can be overwritten by a subclass. Use `setStokesEquation` to set new values to model parameters.
366             """
367             pass
368         def setStokesEquation(self, f=None,fixed_u_mask=None,eta=None,surface_stress=None,stress=None, restoration_factor=None):
369            """
370            assigns new values to the model parameters.
371    
372            :param f: external force
373            :type f: `Vector` object in `FunctionSpace` `Function` or similar
374            :param fixed_u_mask: mask of locations with fixed velocity.
375            :type fixed_u_mask: `Vector` object on `FunctionSpace` `Solution` or similar
376            :param eta: viscosity
377            :type eta: `Scalar` object on `FunctionSpace` `Function` or similar
378            :param surface_stress: normal surface stress
379            :type surface_stress: `Vector` object on `FunctionSpace` `FunctionOnBoundary` or similar
380            :param stress: initial stress
381        :type stress: `Tensor` object on `FunctionSpace` `Function` or similar
382            """
383            if eta !=None:
384                k=util.kronecker(self.domain.getDim())
385                kk=util.outer(k,k)
386                self.eta=util.interpolate(eta, escript.Function(self.domain))
387            self.__pde_prec.setValue(D=1/self.eta)
388                self.__pde_v.setValue(A=self.eta*(util.swap_axes(kk,0,3)+util.swap_axes(kk,1,3)))
389            if restoration_factor!=None:
390                n=self.domain.getNormal()
391                self.__pde_v.setValue(d=restoration_factor*util.outer(n,n))
392            if fixed_u_mask!=None:
393                self.__pde_v.setValue(q=fixed_u_mask)
394            if f!=None: self.__f=f
395            if surface_stress!=None: self.__surface_stress=surface_stress
396            if stress!=None: self.__stress=stress
397    
398         def initialize(self,f=escript.Data(),fixed_u_mask=escript.Data(),eta=1,surface_stress=escript.Data(),stress=escript.Data(), restoration_factor=0):
399          """          """
400          assigns values to the model parameters          assigns values to the model parameters
401    
402          @param f: external force          :param f: external force
403          @type f: L{Vector} object in L{FunctionSpace} L{Function} or similar          :type f: `Vector` object in `FunctionSpace` `Function` or similar
404          @param fixed_u_mask: mask of locations with fixed velocity.          :param fixed_u_mask: mask of locations with fixed velocity.
405          @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
406          @param eta: viscosity          :param eta: viscosity
407          @type eta: L{Scalar} object on L{FunctionSpace} L{Function} or similar          :type eta: `Scalar` object on `FunctionSpace` `Function` or similar
408          @param surface_stress: normal surface stress          :param surface_stress: normal surface stress
409          @type eta: L{Vector} object on L{FunctionSpace} L{FunctionOnBoundary} or similar          :type surface_stress: `Vector` object on `FunctionSpace` `FunctionOnBoundary` or similar
410          @param stress: initial stress          :param stress: initial stress
411      @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.  
   
412          """          """
413          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  
414    
415        def B(self,v):       def Bv(self,v,tol):
416          """           """
417          returns div(v)           returns inner product of element p and div(v)
         @rtype: equal to the type of p  
418    
419          @note: boundary conditions on p should be zero!           :param v: a residual
420          """           :return: inner product of element p and div(v)
421          if self.show_details: print "apply divergence:"           :rtype: ``float``
422          self.__pde_proj.setValue(Y=-util.div(v))           """
423          self.__pde_proj.setTolerance(self.getSubProblemTolerance())           self.__pde_proj.setValue(Y=-util.div(v))
424          return self.__pde_proj.getSolution(verbose=self.show_details)       self.getSolverOptionsDiv().setTolerance(tol)
425         self.getSolverOptionsDiv().setAbsoluteTolerance(0.)
426             out=self.__pde_proj.getSolution()
427             return out
428    
429        def inner_pBv(self,p,Bv):       def inner_pBv(self,p,Bv):
430           """           """
431           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}  
432    
433           @rtype: equal to the type of p           :param p: a pressure increment
434             :param Bv: a residual
435             :return: inner product of element p and Bv=-div(v)
436             :rtype: ``float``
437           """           """
438           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)  
439    
440        def inner_p(self,p0,p1):       def inner_p(self,p0,p1):
441           """           """
442           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}  
443    
444           @rtype: equal to the type of p           :param p0: a pressure
445             :param p1: a pressure
446             :return: inner product of p0 and p1
447             :rtype: ``float``
448           """           """
449           s0=util.interpolate(p0/self.eta,Function(self.domain))           s0=util.interpolate(p0, escript.Function(self.domain))
450           s1=util.interpolate(p1/self.eta,Function(self.domain))           s1=util.interpolate(p1, escript.Function(self.domain))
451           return util.integrate(s0*s1)           return util.integrate(s0*s1)
452    
453        def inner_v(self,v0,v1):       def norm_v(self,v):
454           """           """
455           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}  
456    
457           @rtype: equal to the type of v           :param v: a velovity
458             :return: norm of v
459             :rtype: non-negative ``float``
460           """           """
461       gv0=util.grad(v0)           return util.sqrt(util.integrate(util.length(util.grad(v))**2))
462       gv1=util.grad(v1)  
          return util.integrate(util.inner(gv0,gv1))  
463    
464        def solve_A(self,u,p):       def getDV(self, p, v, tol):
465           """           """
466           solves Av=f-Au-B^*p (v=0 on fixed_u_mask)           return the value for v for a given p
467    
468             :param p: a pressure
469             :param v: a initial guess for the value v to return.
470             :return: dv given as *Adv=(f-Av-B^*p)*
471           """           """
472           if self.show_details: print "solve for velocity:"           self.updateStokesEquation(v,p)
473           self.__pde_u.setTolerance(self.getSubProblemTolerance())           self.__pde_v.setValue(Y=self.__f, y=self.__surface_stress)
474         self.getSolverOptionsVelocity().setTolerance(tol)
475         self.getSolverOptionsVelocity().setAbsoluteTolerance(0.)
476           if self.__stress.isEmpty():           if self.__stress.isEmpty():
477              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)))
478           else:           else:
479              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)))
480           out=self.__pde_u.getSolution(verbose=self.show_details)           out=self.__pde_v.getSolution()
481           return  out           return  out
482    
483        def solve_prec(self,p):       def norm_Bv(self,Bv):
484           if self.show_details: print "apply preconditioner:"          """
485           self.__pde_prec.setTolerance(self.getSubProblemTolerance())          Returns Bv (overwrite).
486           self.__pde_prec.setValue(Y=p)  
487           q=self.__pde_prec.getSolution(verbose=self.show_details)          :rtype: equal to the type of p
488           return q          :note: boundary conditions on p should be zero!
489            """
490            return util.sqrt(util.integrate(util.interpolate(Bv, escript.Function(self.domain))**2))
491    
492         def solve_AinvBt(self,p, tol):
493             """
494             Solves *Av=B^*p* with accuracy `tol`
495    
496             :param p: a pressure increment
497             :return: the solution of *Av=B^*p*
498             :note: boundary conditions on v should be zero!
499             """
500             self.__pde_v.setValue(Y=escript.Data(), y=escript.Data(), X=-p*util.kronecker(self.domain))
501             out=self.__pde_v.getSolution()
502             return  out
503    
504         def solve_prec(self,Bv, tol):
505             """
506             applies preconditioner for for *BA^{-1}B^** to *Bv*
507             with accuracy `self.getSubProblemTolerance()`
508    
509             :param Bv: velocity increment
510             :return: *p=P(Bv)* where *P^{-1}* is an approximation of *BA^{-1}B^ * )*
511             :note: boundary conditions on p are zero.
512             """
513             self.__pde_prec.setValue(Y=Bv)
514         self.getSolverOptionsPressure().setTolerance(tol)
515         self.getSolverOptionsPressure().setAbsoluteTolerance(0.)
516             out=self.__pde_prec.getSolution()
517             return out

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