/[escript]/trunk/escriptcore/py_src/flows.py
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revision 2208 by gross, Mon Jan 12 06:37:07 2009 UTC revision 3905 by gross, Tue Jun 5 08:33:41 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"""
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 *  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  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.__pde_v=LinearPDESystem(domain)        """
60          if useReduced: self.__pde_v.setReducedOrderOn()        initializes the Darcy flux problem
61          self.__pde_v.setSymmetryOn()        :param domain: domain of the problem
62          self.__pde_v.setValue(D=util.kronecker(domain), A=util.outer(util.kronecker(domain),util.kronecker(domain)))        :type domain: `Domain`
63          self.__pde_p=LinearSinglePDE(domain)        :param useReduced: uses reduced oreder on flux and pressure
64          self.__pde_p.setSymmetryOn()        :type useReduced: ``bool``
65          if useReduced: self.__pde_p.setReducedOrderOn()        :param solver: solver method
66          self.__f=Scalar(0,self.__pde_v.getFunctionSpaceForCoefficient("X"))        :type solver: in [`DarcyFlow.EVAL`, `DarcyFlow.POST',  `DarcyFlow.SMOOTH' ]
67          self.__g=Vector(0,self.__pde_v.getFunctionSpaceForCoefficient("Y"))        :param verbose: if ``True`` some information on the iteration progress are printed.
68          self.__ATOL= None        :type verbose: ``bool``
69          :param w: weighting factor for `DarcyFlow.POST` solver
70      def setValue(self,f=None, g=None, location_of_fixed_pressure=None, location_of_fixed_flux=None, permeability=None):        :type w: ``float``
71          """        
72          assigns values to model parameters        """
73          if not solver in [DarcyFlow.EVAL, DarcyFlow.POST,  DarcyFlow.SMOOTH ] :
74          @param f: volumetic sources/sinks            raise ValueError("unknown solver %d."%solver)
         @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)  
75    
76      def getPressure(self,v=None, fixed_pressure=Data(),tol=1.e-8, show_details=False):        self.domain=domain
77          """        self.solver=solver
78          returns the pressure for a given flux C{v} where the pressure is equal to C{fixed_pressure}        self.useReduced=useReduced
79          on locations where C{location_of_fixed_pressure} is positive (see L{setValue}).        self.verbose=verbose
80          Note that C{g} is used, see L{setValue}.        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             x=self.domain.getX()
100             self.l=min( [util.sup(x[i])-util.inf(x[i]) for i in xrange(self.domain.getDim()) ] )
101             #self.l=util.vol(self.domain)**(1./self.domain.getDim()) # length scale
102    
103          elif self.solver  == self.SMOOTH:
104             self.__pde_v=LinearPDESystem(domain)
105             self.__pde_v.setSymmetryOn()
106             if self.useReduced: self.__pde_v.setReducedOrderOn()
107             if self.verbose: print("DarcyFlow: flux smoothing is used.")
108             self.w=0
109    
110          self.__f=escript.Scalar(0,self.__pde_p.getFunctionSpaceForCoefficient("X"))
111          self.__g=escript.Vector(0,self.__pde_p.getFunctionSpaceForCoefficient("Y"))
112          self.location_of_fixed_pressure = escript.Scalar(0, self.__pde_p.getFunctionSpaceForCoefficient("q"))
113          self.location_of_fixed_flux = escript.Vector(0, self.__pde_p.getFunctionSpaceForCoefficient("q"))
114          self.perm_scale=1.
115        
116                    
117          @param v: flux.     def setValue(self,f=None, g=None, location_of_fixed_pressure=None, location_of_fixed_flux=None, permeability=None):
118          @type v: vector-valued on the domain (e.g. L{Data}).        """
119          @param fixed_pressure: pressure on the locations of the domain marked be C{location_of_fixed_pressure}.        assigns values to model parameters
         @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  
120    
121          M{<(Q^*Q)^{-1} r,r> <= ATOL}        :param f: volumetic sources/sinks
122          :type f: scalar value on the domain (e.g. `escript.Data`)
123          :param g: flux sources/sinks
124          :type g: vector values on the domain (e.g. `escript.Data`)
125          :param location_of_fixed_pressure: mask for locations where pressure is fixed
126          :type location_of_fixed_pressure: scalar value on the domain (e.g. `escript.Data`)
127          :param location_of_fixed_flux:  mask for locations where flux is fixed.
128          :type location_of_fixed_flux: vector values on the domain (e.g. `escript.Data`)
129          :param permeability: permeability tensor. If scalar ``s`` is given the tensor with ``s`` on the main diagonal is used.
130          :type permeability: scalar or symmetric tensor values on the domain (e.g. `escript.Data`)
131    
132          :note: the values of parameters which are not set by calling ``setValue`` are not altered.
133          :note: at any point on the boundary of the domain the pressure
134                 (``location_of_fixed_pressure`` >0) or the normal component of the
135                 flux (``location_of_fixed_flux[i]>0``) if direction of the normal
136                 is along the *x_i* axis.
137    
138          holds. M{D} is the M{div} operator and M{(Qp)_i=k_{ij}p_{,j}} for the permeability M{k_{ij}}        """
139          if location_of_fixed_pressure!=None:
140               self.location_of_fixed_pressure=util.wherePositive(location_of_fixed_pressure)
141               self.__pde_p.setValue(q=self.location_of_fixed_pressure)
142          if location_of_fixed_flux!=None:
143              self.location_of_fixed_flux=util.wherePositive(location_of_fixed_flux)
144              if not self.__pde_v == None: self.__pde_v.setValue(q=self.location_of_fixed_flux)
145                
146          if permeability!=None:
147        
148             perm=util.interpolate(permeability,self.__pde_p.getFunctionSpaceForCoefficient("A"))
149             self.perm_scale=util.Lsup(util.length(perm))
150             if self.verbose: print(("DarcyFlow: permeability scaling factor = %e."%self.perm_scale))
151             perm=perm*(1./self.perm_scale)
152            
153             if perm.getRank()==0:
154    
155          @param atol: absolute tolerance for the pressure              perm_inv=(1./perm)
156          @type atol: non-negative C{float}              perm_inv=perm_inv*util.kronecker(self.domain.getDim())
157          @param rtol: relative tolerance for the pressure              perm=perm*util.kronecker(self.domain.getDim())
158          @type rtol: non-negative C{float}          
159          @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          
160          L{getPressure} method or use  the value from a previous time step.           elif perm.getRank()==2:
161          @type p_ref: scalar value on the domain (e.g. L{Data}).              perm_inv=util.inverse(perm)
162          @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           else:
163          L{getFlux} method or use  the value from a previous time step.              raise ValueError("illegal rank of permeability.")
164          @type v_ref: vector-valued on the domain (e.g. L{Data}).          
165          @return: used absolute tolerance.           self.__permeability=perm
166          @rtype: positive C{float}           self.__permeability_inv=perm_inv
167          """      
168          g=self.__g           #====================
169          if not v_ref == None:           self.__pde_p.setValue(A=self.__permeability)
170             f1=util.integrate(util.length(util.interpolate(g-v_ref,Function(self.domain)))**2)           if self.solver  == self.EVAL:
171          else:                pass # no extra work required
172             f1=util.integrate(util.length(util.interpolate(g))**2)           elif self.solver  == self.POST:
173          if not p_ref == None:                k=util.kronecker(self.domain.getDim())
174             f2=util.integrate(util.length(util.tensor_mult(self.__permeability,util.grad(p_ref)))**2)                self.omega = self.w*util.length(perm_inv)*self.l*self.domain.getSize()
175                  #self.__pde_v.setValue(D=self.__permeability_inv, A=self.omega*util.outer(k,k))
176                  self.__pde_v.setValue(D=self.__permeability_inv, A_reduced=self.omega*util.outer(k,k))
177             elif self.solver  == self.SMOOTH:
178                self.__pde_v.setValue(D=self.__permeability_inv)
179    
180          if g != None:
181            g=util.interpolate(g, self.__pde_p.getFunctionSpaceForCoefficient("Y"))
182            if g.isEmpty():
183                 g=Vector(0,self.__pde_p.getFunctionSpaceForCoefficient("Y"))
184          else:          else:
185             f2=0               if not g.getShape()==(self.domain.getDim(),): raise ValueError("illegal shape of g")
186          self.__ATOL= atol + rtol * util.sqrt(max(f1,f2))          self.__g=g
187          if self.__ATOL<=0:        if f !=None:
188             raise ValueError,"Positive tolerance (=%e) is expected."%self.__ATOL           f=util.interpolate(f, self.__pde_p.getFunctionSpaceForCoefficient("Y"))
189          return self.__ATOL           if f.isEmpty():      
190                         f=Scalar(0,self.__pde_p.getFunctionSpaceForCoefficient("Y"))
191      def getTolerance(self):           else:
192          """               if f.getRank()>0: raise ValueError("illegal rank of f.")
193          returns the current tolerance.           self.__f=f
     
         @return: used absolute tolerance.  
         @rtype: positive C{float}  
         """  
         if self.__ATOL==None:  
            raise ValueError,"no tolerance is defined."  
         return self.__ATOL  
   
     def solve(self,u0,p0, max_iter=100, verbose=False, show_details=False, sub_rtol=1.e-8):  
          """  
          solves the problem.  
   
          The iteration is terminated if the residual norm is less then self.getTolerance().  
   
          @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.  
          @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 sub_rtol: tolerance to be used in the sub iteration. It is recommended that M{sub_rtol<rtol*5.e-3}  
          @type sub_rtol: positive-negative C{float}  
          @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  
194    
195           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     def getSolverOptionsFlux(self):
196                  """
197           which is solved using the PCG method (precondition is M{Q^*Q}). In each iteration step        Returns the solver options used to solve the flux problems
198           PDEs with operator M{I+D^*D} and with M{Q^*Q} needs to be solved using a sub iteration scheme.        :return: `SolverOptions`
199           """        """
200           self.verbose=verbose        if self.__pde_v == None:
201           self.show_details= show_details and self.verbose            return None
202           self.__pde_v.setTolerance(sub_rtol)        else:
203           self.__pde_p.setTolerance(sub_rtol)            return self.__pde_v.getSolverOptions()
204           ATOL=self.getTolerance()        
205           if self.verbose: print "DarcyFlux: absolute tolerance = %e"%ATOL     def setSolverOptionsFlux(self, options=None):
206           #########################################################################################################################        """
207           #        Sets the solver options used to solve the flux problems
208           #   we solve:        If ``options`` is not present, the options are reset to default
209           #          :param options: `SolverOptions`
210           #      Q^*(I-(I+D^*D)^{-1})Q dp =  Q^* (g-u0-Qp0 - (I+D^*D)^{-1} ( D^*(f-Du0)+g-u0-Qp0) )        """
211           #        if not self.__pde_v == None:
212           #   residual is            self.__pde_v.setSolverOptions(options)
213           #      
214           #    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)     def getSolverOptionsPressure(self):
215           #        """
216           #        with v = (I+D^*D)^{-1} (D^*f+g-Qp) including BC        Returns the solver options used to solve the pressure problems
217           #        :return: `SolverOptions`
218           #    we use (g - Qp, v) to represent the residual. not that        """
219           #        return self.__pde_p.getSolverOptions()
220           #    dr(dp)=( -Q(dp), dv) with dv = - (I+D^*D)^{-1} Q(dp)        
221           #     def setSolverOptionsPressure(self, options=None):
222           #   while the initial residual is        """
223           #        Sets the solver options used to solve the pressure problems
224           #      r0=( g - Qp0, v00) with v00=(I+D^*D)^{-1} (D^*f+g-Qp0) including BC        If ``options`` is not present, the options are reset to default
225           #          
226           d0=self.__g-util.tensor_mult(self.__permeability,util.grad(p0))        :param options: `SolverOptions`
227           self.__pde_v.setValue(Y=d0, X=self.__f*util.kronecker(self.domain), r=u0)        :note: if the adaption of subtolerance is choosen, the tolerance set by ``options`` will be overwritten before the solver is called.
228           v00=self.__pde_v.getSolution(verbose=show_details)        """
229           if self.verbose: print "DarcyFlux: range of initial flux = ",util.inf(v00), util.sup(v00)        return self.__pde_p.setSolverOptions(options)
230           self.__pde_v.setValue(r=Data())        
231           # start CG     def solve(self, u0, p0):
232           r=ArithmeticTuple(d0, v00)        """
233           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)        solves the problem.
234           return r[1],p        
235          :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.
236      def __Aprod_PCG(self,dp):        :type u0: vector value on the domain (e.g. `escript.Data`).
237            if self.show_details: print "DarcyFlux: Applying operator"        :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.
238            #  -dr(dp) = (Qdp,du) where du = (I+D^*D)^{-1} (Qdp)        :type p0: scalar value on the domain (e.g. `escript.Data`).
239            mQdp=util.tensor_mult(self.__permeability,util.grad(dp))        :return: flux and pressure
240            self.__pde_v.setValue(Y=mQdp,X=Data(), r=Data())        :rtype: ``tuple`` of `escript.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)  
241    
 class StokesProblemCartesian(HomogeneousSaddlePointProblem):  
242        """        """
243        solves        self.__pde_p.setValue(X=self.__g * 1./self.perm_scale,
244                                Y=self.__f * 1./self.perm_scale,
245                                y= - util.inner(self.domain.getNormal(),u0 * self.location_of_fixed_flux * 1./self.perm_scale ),
246                                r=p0)
247          p=self.__pde_p.getSolution()
248          u = self.getFlux(p, u0)
249          return u,p
250          
251       def getFlux(self,p, u0=None):
252            """
253            returns the flux for a given pressure ``p`` where the flux is equal to ``u0``
254            on locations where ``location_of_fixed_flux`` is positive (see `setValue`).
255            Notice that ``g`` is used, see `setValue`.
256    
257            :param p: pressure.
258            :type p: scalar value on the domain (e.g. `escript.Data`).
259            :param u0: flux on the locations of the domain marked be ``location_of_fixed_flux``.
260            :type u0: vector values on the domain (e.g. `escript.Data`) or ``None``
261            :return: flux
262            :rtype: `escript.Data`
263            """
264            if self.solver  == self.EVAL:
265               u = self.__g-self.perm_scale * util.tensor_mult(self.__permeability,util.grad(p))
266            elif self.solver  == self.POST or self.solver  == self.SMOOTH:
267                self.__pde_v.setValue(Y=util.tensor_mult(self.__permeability_inv,self.__g * 1./self.perm_scale)-util.grad(p))
268                if u0 == None:
269                   self.__pde_v.setValue(r=escript.Data())
270                else:
271                   if not isinstance(u0, escript.Data) : u0 = escript.Vector(u0, escript.Solution(self.domain))
272                   self.__pde_v.setValue(r=1./self.perm_scale * u0)
273                   u= self.__pde_v.getSolution() * self.perm_scale
274            return u
275          
276    class StokesProblemCartesian(HomogeneousSaddlePointProblem):
277         """
278         solves
279    
280            -(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}
281                  u_{i,i}=0                  u_{i,i}=0
# Line 333  class StokesProblemCartesian(Homogeneous Line 283  class StokesProblemCartesian(Homogeneous
283            u=0 where  fixed_u_mask>0            u=0 where  fixed_u_mask>0
284            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
285    
286        if surface_stress is not given 0 is assumed.       if surface_stress is not given 0 is assumed.
287    
288        typical usage:       typical usage:
289    
290              sp=StokesProblemCartesian(domain)              sp=StokesProblemCartesian(domain)
291              sp.setTolerance()              sp.setTolerance()
292              sp.initialize(...)              sp.initialize(...)
293              v,p=sp.solve(v0,p0)              v,p=sp.solve(v0,p0)
294        """              sp.setStokesEquation(...) # new values for some parameters
295        def __init__(self,domain,**kwargs):              v1,p1=sp.solve(v,p)
296         """
297         def __init__(self,domain,**kwargs):
298           """           """
299           initialize the Stokes Problem           initialize the Stokes Problem
300    
301           @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
302           @type domain: L{Domain}           LBB complient, for instance using quadratic and linear approximation on the same element or using linear approximation
303           @warning: The apprximation order needs to be two otherwise you may see oscilations in the pressure.           with macro elements for the pressure.
304    
305             :param domain: domain of the problem.
306             :type domain: `Domain`
307           """           """
308           HomogeneousSaddlePointProblem.__init__(self,**kwargs)           HomogeneousSaddlePointProblem.__init__(self,**kwargs)
309           self.domain=domain           self.domain=domain
310           self.vol=util.integrate(1.,Function(self.domain))           self.__pde_v=LinearPDE(domain,numEquations=self.domain.getDim(),numSolutions=self.domain.getDim())
311           self.__pde_u=LinearPDE(domain,numEquations=self.domain.getDim(),numSolutions=self.domain.getDim())           self.__pde_v.setSymmetryOn()
312           self.__pde_u.setSymmetryOn()      
          # self.__pde_u.setSolverMethod(self.__pde_u.DIRECT)  
          # self.__pde_u.setSolverMethod(preconditioner=LinearPDE.RILU)  
               
313           self.__pde_prec=LinearPDE(domain)           self.__pde_prec=LinearPDE(domain)
314           self.__pde_prec.setReducedOrderOn()           self.__pde_prec.setReducedOrderOn()
          # self.__pde_prec.setSolverMethod(self.__pde_prec.LUMPING)  
315           self.__pde_prec.setSymmetryOn()           self.__pde_prec.setSymmetryOn()
316    
317           self.__pde_proj=LinearPDE(domain)           self.__pde_proj=LinearPDE(domain)
318           self.__pde_proj.setReducedOrderOn()           self.__pde_proj.setReducedOrderOn()
319             self.__pde_proj.setValue(D=1)
320           self.__pde_proj.setSymmetryOn()           self.__pde_proj.setSymmetryOn()
          self.__pde_proj.setValue(D=1.)  
321    
322        def initialize(self,f=Data(),fixed_u_mask=Data(),eta=1,surface_stress=Data(),stress=Data()):       def getSolverOptionsVelocity(self):
323             """
324         returns the solver options used  solve the equation for velocity.
325        
326         :rtype: `SolverOptions`
327         """
328             return self.__pde_v.getSolverOptions()
329         def setSolverOptionsVelocity(self, options=None):
330             """
331         set the solver options for solving the equation for velocity.
332        
333         :param options: new solver  options
334         :type options: `SolverOptions`
335         """
336             self.__pde_v.setSolverOptions(options)
337         def getSolverOptionsPressure(self):
338             """
339         returns the solver options used  solve the equation for pressure.
340         :rtype: `SolverOptions`
341         """
342             return self.__pde_prec.getSolverOptions()
343         def setSolverOptionsPressure(self, options=None):
344             """
345         set the solver options for solving the equation for pressure.
346         :param options: new solver  options
347         :type options: `SolverOptions`
348         """
349             self.__pde_prec.setSolverOptions(options)
350    
351         def setSolverOptionsDiv(self, options=None):
352             """
353         set the solver options for solving the equation to project the divergence of
354         the velocity onto the function space of presure.
355        
356         :param options: new solver options
357         :type options: `SolverOptions`
358         """
359             self.__pde_proj.setSolverOptions(options)
360         def getSolverOptionsDiv(self):
361             """
362         returns the solver options for solving the equation to project the divergence of
363         the velocity onto the function space of presure.
364        
365         :rtype: `SolverOptions`
366         """
367             return self.__pde_proj.getSolverOptions()
368    
369         def updateStokesEquation(self, v, p):
370             """
371             updates the Stokes equation to consider dependencies from ``v`` and ``p``
372             :note: This method can be overwritten by a subclass. Use `setStokesEquation` to set new values to model parameters.
373             """
374             pass
375         def setStokesEquation(self, f=None,fixed_u_mask=None,eta=None,surface_stress=None,stress=None, restoration_factor=None):
376            """
377            assigns new values to the model parameters.
378    
379            :param f: external force
380            :type f: `Vector` object in `FunctionSpace` `Function` or similar
381            :param fixed_u_mask: mask of locations with fixed velocity.
382            :type fixed_u_mask: `Vector` object on `FunctionSpace` `Solution` or similar
383            :param eta: viscosity
384            :type eta: `Scalar` object on `FunctionSpace` `Function` or similar
385            :param surface_stress: normal surface stress
386            :type surface_stress: `Vector` object on `FunctionSpace` `FunctionOnBoundary` or similar
387            :param stress: initial stress
388        :type stress: `Tensor` object on `FunctionSpace` `Function` or similar
389            """
390            if eta !=None:
391                k=util.kronecker(self.domain.getDim())
392                kk=util.outer(k,k)
393                self.eta=util.interpolate(eta, escript.Function(self.domain))
394                self.__pde_prec.setValue(D=1/self.eta)
395                self.__pde_v.setValue(A=self.eta*(util.swap_axes(kk,0,3)+util.swap_axes(kk,1,3)))
396            if restoration_factor!=None:
397                n=self.domain.getNormal()
398                self.__pde_v.setValue(d=restoration_factor*util.outer(n,n))
399            if fixed_u_mask!=None:
400                self.__pde_v.setValue(q=fixed_u_mask)
401            if f!=None: self.__f=f
402            if surface_stress!=None: self.__surface_stress=surface_stress
403            if stress!=None: self.__stress=stress
404    
405         def initialize(self,f=escript.Data(),fixed_u_mask=escript.Data(),eta=1,surface_stress=escript.Data(),stress=escript.Data(), restoration_factor=0):
406          """          """
407          assigns values to the model parameters          assigns values to the model parameters
408    
409          @param f: external force          :param f: external force
410          @type f: L{Vector} object in L{FunctionSpace} L{Function} or similar          :type f: `Vector` object in `FunctionSpace` `Function` or similar
411          @param fixed_u_mask: mask of locations with fixed velocity.          :param fixed_u_mask: mask of locations with fixed velocity.
412          @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
413          @param eta: viscosity          :param eta: viscosity
414          @type eta: L{Scalar} object on L{FunctionSpace} L{Function} or similar          :type eta: `Scalar` object on `FunctionSpace` `Function` or similar
415          @param surface_stress: normal surface stress          :param surface_stress: normal surface stress
416          @type eta: L{Vector} object on L{FunctionSpace} L{FunctionOnBoundary} or similar          :type surface_stress: `Vector` object on `FunctionSpace` `FunctionOnBoundary` or similar
417          @param stress: initial stress          :param stress: initial stress
418      @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.  
   
419          """          """
420          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  
421    
422        def B(self,v):       def Bv(self,v,tol):
423          """           """
424          returns div(v)           returns inner product of element p and div(v)
         @rtype: equal to the type of p  
425    
426          @note: boundary conditions on p should be zero!           :param v: a residual
427          """           :return: inner product of element p and div(v)
428          if self.show_details: print "apply divergence:"           :rtype: ``float``
429          self.__pde_proj.setValue(Y=-util.div(v))           """
430          self.__pde_proj.setTolerance(self.getSubProblemTolerance())           self.__pde_proj.setValue(Y=-util.div(v))
431          return self.__pde_proj.getSolution(verbose=self.show_details)           self.getSolverOptionsDiv().setTolerance(tol)
432             self.getSolverOptionsDiv().setAbsoluteTolerance(0.)
433             out=self.__pde_proj.getSolution()
434             return out
435    
436        def inner_pBv(self,p,Bv):       def inner_pBv(self,p,Bv):
437           """           """
438           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}  
439    
440           @rtype: equal to the type of p           :param p: a pressure increment
441             :param Bv: a residual
442             :return: inner product of element p and Bv=-div(v)
443             :rtype: ``float``
444           """           """
445           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)  
446    
447        def inner_p(self,p0,p1):       def inner_p(self,p0,p1):
448           """           """
449           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}  
450    
451           @rtype: equal to the type of p           :param p0: a pressure
452             :param p1: a pressure
453             :return: inner product of p0 and p1
454             :rtype: ``float``
455           """           """
456           s0=util.interpolate(p0/self.eta,Function(self.domain))           s0=util.interpolate(p0, escript.Function(self.domain))
457           s1=util.interpolate(p1/self.eta,Function(self.domain))           s1=util.interpolate(p1, escript.Function(self.domain))
458           return util.integrate(s0*s1)           return util.integrate(s0*s1)
459    
460        def inner_v(self,v0,v1):       def norm_v(self,v):
461           """           """
462           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}  
463    
464           @rtype: equal to the type of v           :param v: a velovity
465             :return: norm of v
466             :rtype: non-negative ``float``
467           """           """
468       gv0=util.grad(v0)           return util.sqrt(util.integrate(util.length(util.grad(v))**2))
469       gv1=util.grad(v1)  
          return util.integrate(util.inner(gv0,gv1))  
470    
471        def solve_A(self,u,p):       def getDV(self, p, v, tol):
472           """           """
473           solves Av=f-Au-B^*p (v=0 on fixed_u_mask)           return the value for v for a given p
474    
475             :param p: a pressure
476             :param v: a initial guess for the value v to return.
477             :return: dv given as *Adv=(f-Av-B^*p)*
478           """           """
479           if self.show_details: print "solve for velocity:"           self.updateStokesEquation(v,p)
480           self.__pde_u.setTolerance(self.getSubProblemTolerance())           self.__pde_v.setValue(Y=self.__f, y=self.__surface_stress)
481             self.getSolverOptionsVelocity().setTolerance(tol)
482             self.getSolverOptionsVelocity().setAbsoluteTolerance(0.)
483           if self.__stress.isEmpty():           if self.__stress.isEmpty():
484              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)))
485           else:           else:
486              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)))
487           out=self.__pde_u.getSolution(verbose=self.show_details)           out=self.__pde_v.getSolution()
488           return  out           return  out
489    
490        def solve_prec(self,p):       def norm_Bv(self,Bv):
491           if self.show_details: print "apply preconditioner:"          """
492           self.__pde_prec.setTolerance(self.getSubProblemTolerance())          Returns Bv (overwrite).
493           self.__pde_prec.setValue(Y=p)  
494           q=self.__pde_prec.getSolution(verbose=self.show_details)          :rtype: equal to the type of p
495           return q          :note: boundary conditions on p should be zero!
496            """
497            return util.sqrt(util.integrate(util.interpolate(Bv, escript.Function(self.domain))**2))
498    
499         def solve_AinvBt(self,p, tol):
500             """
501             Solves *Av=B^*p* with accuracy `tol`
502    
503             :param p: a pressure increment
504             :return: the solution of *Av=B^*p*
505             :note: boundary conditions on v should be zero!
506             """
507             self.__pde_v.setValue(Y=escript.Data(), y=escript.Data(), X=-p*util.kronecker(self.domain))
508             out=self.__pde_v.getSolution()
509             return  out
510    
511         def solve_prec(self,Bv, tol):
512             """
513             applies preconditioner for for *BA^{-1}B^** to *Bv*
514             with accuracy `self.getSubProblemTolerance()`
515    
516             :param Bv: velocity increment
517             :return: *p=P(Bv)* where *P^{-1}* is an approximation of *BA^{-1}B^ * )*
518             :note: boundary conditions on p are zero.
519             """
520             self.__pde_prec.setValue(Y=Bv)
521             self.getSolverOptionsPressure().setTolerance(tol)
522             self.getSolverOptionsPressure().setAbsoluteTolerance(0.)
523             out=self.__pde_prec.getSolution()
524             return out

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