/[escript]/trunk/escriptcore/py_src/flows.py
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revision 2208 by gross, Mon Jan 12 06:37:07 2009 UTC revision 4446 by caltinay, Tue Jun 11 04:00:15 2013 UTC
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1  ########################################################  # -*- coding: utf-8 -*-
2    ##############################################################################
3  #  #
4  # Copyright (c) 2003-2008 by University of Queensland  # Copyright (c) 2003-2013 by University of Queensland
5  # Earth Systems Science Computational Center (ESSCC)  # http://www.uq.edu.au
 # http://www.uq.edu.au/esscc  
6  #  #
7  # Primary Business: Queensland, Australia  # Primary Business: Queensland, Australia
8  # Licensed under the Open Software License version 3.0  # Licensed under the Open Software License version 3.0
9  # http://www.opensource.org/licenses/osl-3.0.php  # http://www.opensource.org/licenses/osl-3.0.php
10  #  #
11  ########################################################  # Development until 2012 by Earth Systems Science Computational Center (ESSCC)
12    # Development since 2012 by School of Earth Sciences
13    #
14    ##############################################################################
15    
16  __copyright__="""Copyright (c) 2003-2008 by University of Queensland  __copyright__="""Copyright (c) 2003-2013 by University of Queensland
17  Earth Systems Science Computational Center (ESSCC)  http://www.uq.edu.au
 http://www.uq.edu.au/esscc  
18  Primary Business: Queensland, Australia"""  Primary Business: Queensland, Australia"""
19  __license__="""Licensed under the Open Software License version 3.0  __license__="""Licensed under the Open Software License version 3.0
20  http://www.opensource.org/licenses/osl-3.0.php"""  http://www.opensource.org/licenses/osl-3.0.php"""
21  __url__="http://www.uq.edu.au/esscc/escript-finley"  __url__="https://launchpad.net/escript-finley"
22    
23  """  """
24  Some models for flow  Some models for flow
25    
26  @var __author__: name of author  :var __author__: name of author
27  @var __copyright__: copyrights  :var __copyright__: copyrights
28  @var __license__: licence agreement  :var __license__: licence agreement
29  @var __url__: url entry point on documentation  :var __url__: url entry point on documentation
30  @var __version__: version  :var __version__: version
31  @var __date__: date of the version  :var __date__: date of the version
32  """  """
33    
34  __author__="Lutz Gross, l.gross@uq.edu.au"  __author__="Lutz Gross, l.gross@uq.edu.au"
35    
36  from escript import *  from . import escriptcpp as escore
37  import util  from . import util
38  from linearPDEs import LinearPDE, LinearPDESystem, LinearSinglePDE  from . import linearPDEs as lpe
39  from pdetools import HomogeneousSaddlePointProblem,Projector, ArithmeticTuple, PCG, NegativeNorm  from . import pdetools as pdt
40    
41  class DarcyFlow(object):  class DarcyFlow(object):
42      """     """
43      solves the problem     solves the problem
44      
45      M{u_i+k_{ij}*p_{,j} = g_i}     *u_i+k_{ij}*p_{,j} = g_i*
46      M{u_{i,i} = f}     *u_{i,i} = f*
47      
48      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,
49      
50      @note: The problem is solved in a least squares formulation.     :cvar EVAL: direct pressure gradient evaluation for flux
51      """     :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*
52                   where *l* is the length scale, *K* is the inverse of the permeability tensor, and *w* is a positive weighting factor.
53      def __init__(self, domain,useReduced=False):     :cvar SMOOTH: global smoothing by solving the PDE *K_{ij} u_j= - p_{,j} + K_{ij} g_j*
54          """     """
55          initializes the Darcy flux problem     EVAL="EVAL"
56          @param domain: domain of the problem     SIMPLE="EVAL"
57          @type domain: L{Domain}     POST="POST"
58          """     SMOOTH="SMOOTH"
59          self.domain=domain     def __init__(self, domain, useReduced=False, solver="POST", verbose=False, w=1.):
60          self.__pde_v=LinearPDESystem(domain)        """
61          if useReduced: self.__pde_v.setReducedOrderOn()        initializes the Darcy flux problem.
         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))  
   
62    
63      def getFlux(self,p=None, fixed_flux=Data(),tol=1.e-8, show_details=False):        :param domain: domain of the problem
64          """        :type domain: `Domain`
65          returns the flux for a given pressure C{p} where the flux is equal to C{fixed_flux}        :param useReduced: uses reduced oreder on flux and pressure
66          on locations where C{location_of_fixed_flux} is positive (see L{setValue}).        :type useReduced: ``bool``
67          Note that C{g} and C{f} are used, see L{setValue}.        :param solver: solver method
68                  :type solver: in [`DarcyFlow.EVAL`, `DarcyFlow.POST`, `DarcyFlow.SMOOTH` ]
69          @param p: pressure.        :param verbose: if ``True`` some information on the iteration progress are printed.
70          @type p: scalar value on the domain (e.g. L{Data}).        :type verbose: ``bool``
71          @param fixed_flux: flux on the locations of the domain marked be C{location_of_fixed_flux}.        :param w: weighting factor for `DarcyFlow.POST` solver
72          @type fixed_flux: vector values on the domain (e.g. L{Data}).        :type w: ``float``
73          @param tol: relative tolerance to be used.        
74          @type tol: positive C{float}.        """
75          @return: flux        if not solver in [DarcyFlow.EVAL, DarcyFlow.POST,  DarcyFlow.SMOOTH ] :
76          @rtype: L{Data}            raise ValueError("unknown solver %d."%solver)
         @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)  
77    
78      def getPressure(self,v=None, fixed_pressure=Data(),tol=1.e-8, show_details=False):        self.domain=domain
79          """        self.solver=solver
80          returns the pressure for a given flux C{v} where the pressure is equal to C{fixed_pressure}        self.useReduced=useReduced
81          on locations where C{location_of_fixed_pressure} is positive (see L{setValue}).        self.verbose=verbose
82          Note that C{g} is used, see L{setValue}.        self.l=None
83          self.w=None
84        
85          self.__pde_p=lpe.LinearSinglePDE(domain)
86          self.__pde_p.setSymmetryOn()
87          if self.useReduced: self.__pde_p.setReducedOrderOn()
88    
89          if self.solver  == self.EVAL:
90             self.__pde_v=None
91             if self.verbose: print("DarcyFlow: simple solver is used.")
92    
93          elif self.solver  == self.POST:
94             if util.inf(w)<0.:
95                raise ValueError("Weighting factor must be non-negative.")
96             if self.verbose: print("DarcyFlow: global postprocessing of flux is used.")
97             self.__pde_v=lpe.LinearPDESystem(domain)
98             self.__pde_v.setSymmetryOn()
99             if self.useReduced: self.__pde_v.setReducedOrderOn()
100             self.w=w
101             x=self.domain.getX()
102             self.l=min( [util.sup(x[i])-util.inf(x[i]) for i in range(self.domain.getDim()) ] )
103             #self.l=util.vol(self.domain)**(1./self.domain.getDim()) # length scale
104    
105          elif self.solver  == self.SMOOTH:
106             self.__pde_v=lpe.LinearPDESystem(domain)
107             self.__pde_v.setSymmetryOn()
108             if self.useReduced: self.__pde_v.setReducedOrderOn()
109             if self.verbose: print("DarcyFlow: flux smoothing is used.")
110             self.w=0
111    
112          self.__f=escore.Data(0,self.__pde_p.getFunctionSpaceForCoefficient("X"))
113          self.__g=escore.Vector(0,self.__pde_p.getFunctionSpaceForCoefficient("Y"))
114          self.__permeability_invXg=escore.Vector(0,self.__pde_p.getFunctionSpaceForCoefficient("Y"))
115          self.__permeability_invXg_ref=util.numpy.zeros((self.domain.getDim()),util.numpy.float64)
116          self.ref_point_id=None
117          self.ref_point=util.numpy.zeros((self.domain.getDim()),util.numpy.float64)
118          self.location_of_fixed_pressure = escore.Data(0, self.__pde_p.getFunctionSpaceForCoefficient("q"))
119          self.location_of_fixed_flux = escore.Vector(0, self.__pde_p.getFunctionSpaceForCoefficient("q"))
120          self.perm_scale=1.
121        
122                    
123          @param v: flux.     def setValue(self,f=None, g=None, location_of_fixed_pressure=None, location_of_fixed_flux=None, permeability=None):
124          @type v: vector-valued on the domain (e.g. L{Data}).        """
125          @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  
126    
127          M{<(Q^*Q)^{-1} r,r> <= ATOL}        :param f: volumetic sources/sinks
128          :type f: scalar value on the domain (e.g. `escript.Data`)
129          :param g: flux sources/sinks
130          :type g: vector values on the domain (e.g. `escript.Data`)
131          :param location_of_fixed_pressure: mask for locations where pressure is fixed
132          :type location_of_fixed_pressure: scalar value on the domain (e.g. `escript.Data`)
133          :param location_of_fixed_flux:  mask for locations where flux is fixed.
134          :type location_of_fixed_flux: vector values on the domain (e.g. `escript.Data`)
135          :param permeability: permeability tensor. If scalar ``s`` is given the tensor with ``s`` on the main diagonal is used.
136          :type permeability: scalar or symmetric tensor values on the domain (e.g. `escript.Data`)
137    
138          :note: the values of parameters which are not set by calling ``setValue`` are not altered.
139          :note: at any point on the boundary of the domain the pressure
140                 (``location_of_fixed_pressure`` >0) or the normal component of the
141                 flux (``location_of_fixed_flux[i]>0``) if direction of the normal
142                 is along the *x_i* axis.
143    
144          holds. M{D} is the M{div} operator and M{(Qp)_i=k_{ij}p_{,j}} for the permeability M{k_{ij}}        """
145          if location_of_fixed_pressure!=None:
146               self.location_of_fixed_pressure=util.wherePositive(util.interpolate(location_of_fixed_pressure, self.__pde_p.getFunctionSpaceForCoefficient("q")))
147               self.ref_point_id=self.location_of_fixed_pressure.maxGlobalDataPoint()
148               if not self.location_of_fixed_pressure.getTupleForGlobalDataPoint(*self.ref_point_id)[0] > 0: raise ValueError("pressure needs to be fixed at least one point.")
149               self.ref_point=self.__pde_p.getFunctionSpaceForCoefficient("q").getX().getTupleForGlobalDataPoint(*self.ref_point_id)
150               if self.verbose: print(("DarcyFlow: reference point at %s."%(self.ref_point,)))
151               self.__pde_p.setValue(q=self.location_of_fixed_pressure)
152          if location_of_fixed_flux!=None:
153              self.location_of_fixed_flux=util.wherePositive(location_of_fixed_flux)
154              if not self.__pde_v == None:
155                  self.__pde_v.setValue(q=self.location_of_fixed_flux)
156                
157          if permeability!=None:
158        
159             perm=util.interpolate(permeability,self.__pde_p.getFunctionSpaceForCoefficient("A"))
160             self.perm_scale=util.Lsup(util.length(perm))
161             if self.verbose: print(("DarcyFlow: permeability scaling factor = %e."%self.perm_scale))
162             perm=perm*(1./self.perm_scale)
163            
164             if perm.getRank()==0:
165    
166          @param atol: absolute tolerance for the pressure              perm_inv=(1./perm)
167          @type atol: non-negative C{float}              perm_inv=perm_inv*util.kronecker(self.domain.getDim())
168          @param rtol: relative tolerance for the pressure              perm=perm*util.kronecker(self.domain.getDim())
169          @type rtol: non-negative C{float}          
170          @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          
171          L{getPressure} method or use  the value from a previous time step.           elif perm.getRank()==2:
172          @type p_ref: scalar value on the domain (e.g. L{Data}).              perm_inv=util.inverse(perm)
173          @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:
174          L{getFlux} method or use  the value from a previous time step.              raise ValueError("illegal rank of permeability.")
175          @type v_ref: vector-valued on the domain (e.g. L{Data}).          
176          @return: used absolute tolerance.           self.__permeability=perm
177          @rtype: positive C{float}           self.__permeability_inv=perm_inv
178          """      
179          g=self.__g           #====================
180          if not v_ref == None:           self.__pde_p.setValue(A=self.__permeability)
181             f1=util.integrate(util.length(util.interpolate(g-v_ref,Function(self.domain)))**2)           if self.solver  == self.EVAL:
182          else:                pass # no extra work required
183             f1=util.integrate(util.length(util.interpolate(g))**2)           elif self.solver  == self.POST:
184          if not p_ref == None:                k=util.kronecker(self.domain.getDim())
185             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()
186                  #self.__pde_v.setValue(D=self.__permeability_inv, A=self.omega*util.outer(k,k))
187                  self.__pde_v.setValue(D=self.__permeability_inv, A_reduced=self.omega*util.outer(k,k))
188             elif self.solver  == self.SMOOTH:
189                self.__pde_v.setValue(D=self.__permeability_inv)
190    
191          if g != None:
192            g=util.interpolate(g, self.__pde_p.getFunctionSpaceForCoefficient("Y"))
193            if g.isEmpty():
194                 g=Vector(0,self.__pde_p.getFunctionSpaceForCoefficient("Y"))
195          else:          else:
196             f2=0               if not g.getShape()==(self.domain.getDim(),): raise ValueError("illegal shape of g")
197          self.__ATOL= atol + rtol * util.sqrt(max(f1,f2))          self.__g=g
198          if self.__ATOL<=0:          self.__permeability_invXg=util.tensor_mult(self.__permeability_inv,self.__g * (1./self.perm_scale ))
199             raise ValueError,"Positive tolerance (=%e) is expected."%self.__ATOL          self.__permeability_invXg_ref=util.integrate(self.__permeability_invXg)/util.vol(self.domain)
200          return self.__ATOL        if f !=None:
201                     f=util.interpolate(f, self.__pde_p.getFunctionSpaceForCoefficient("Y"))
202      def getTolerance(self):           if f.isEmpty():      
203          """               f=Scalar(0,self.__pde_p.getFunctionSpaceForCoefficient("Y"))
204          returns the current tolerance.           else:
205                   if f.getRank()>0: raise ValueError("illegal rank of f.")
206          @return: used absolute tolerance.           self.__f=f
         @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  
207    
208           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):
209                  """
210           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
211           PDEs with operator M{I+D^*D} and with M{Q^*Q} needs to be solved using a sub iteration scheme.        :return: `SolverOptions`
212           """        """
213           self.verbose=verbose        if self.__pde_v == None:
214           self.show_details= show_details and self.verbose            return None
215           self.__pde_v.setTolerance(sub_rtol)        else:
216           self.__pde_p.setTolerance(sub_rtol)            return self.__pde_v.getSolverOptions()
217           ATOL=self.getTolerance()        
218           if self.verbose: print "DarcyFlux: absolute tolerance = %e"%ATOL     def setSolverOptionsFlux(self, options=None):
219           #########################################################################################################################        """
220           #        Sets the solver options used to solve the flux problems
221           #   we solve:        If ``options`` is not present, the options are reset to default
222           #          :param options: `SolverOptions`
223           #      Q^*(I-(I+D^*D)^{-1})Q dp =  Q^* (g-u0-Qp0 - (I+D^*D)^{-1} ( D^*(f-Du0)+g-u0-Qp0) )        """
224           #        if not self.__pde_v == None:
225           #   residual is            self.__pde_v.setSolverOptions(options)
226           #      
227           #    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):
228           #        """
229           #        with v = (I+D^*D)^{-1} (D^*f+g-Qp) including BC        Returns the solver options used to solve the pressure problems
230           #        :return: `SolverOptions`
231           #    we use (g - Qp, v) to represent the residual. not that        """
232           #        return self.__pde_p.getSolverOptions()
233           #    dr(dp)=( -Q(dp), dv) with dv = - (I+D^*D)^{-1} Q(dp)        
234           #     def setSolverOptionsPressure(self, options=None):
235           #   while the initial residual is        """
236           #        Sets the solver options used to solve the pressure problems
237           #      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
238           #          
239           d0=self.__g-util.tensor_mult(self.__permeability,util.grad(p0))        :param options: `SolverOptions`
240           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.
241           v00=self.__pde_v.getSolution(verbose=show_details)        """
242           if self.verbose: print "DarcyFlux: range of initial flux = ",util.inf(v00), util.sup(v00)        return self.__pde_p.setSolverOptions(options)
243           self.__pde_v.setValue(r=Data())        
244           # start CG     def solve(self, u0, p0):
245           r=ArithmeticTuple(d0, v00)        """
246           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.
247           return r[1],p        
248          :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.
249      def __Aprod_PCG(self,dp):        :type u0: vector value on the domain (e.g. `escript.Data`).
250            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.
251            #  -dr(dp) = (Qdp,du) where du = (I+D^*D)^{-1} (Qdp)        :type p0: scalar value on the domain (e.g. `escript.Data`).
252            mQdp=util.tensor_mult(self.__permeability,util.grad(dp))        :return: flux and pressure
253            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)  
254    
 class StokesProblemCartesian(HomogeneousSaddlePointProblem):  
255        """        """
256        solves        p0=util.interpolate(p0, self.__pde_p.getFunctionSpaceForCoefficient("q"))
257          if self.ref_point_id == None:
258              p_ref=0
259          else:
260              p_ref=p0.getTupleForGlobalDataPoint(*self.ref_point_id)[0]
261          p0_hydrostatic=p_ref+util.inner(self.__permeability_invXg_ref, self.__pde_p.getFunctionSpaceForCoefficient("q").getX() - self.ref_point)
262          g_2=self.__g - util.tensor_mult(self.__permeability, self.__permeability_invXg_ref * self.perm_scale)
263          self.__pde_p.setValue(X=g_2 * 1./self.perm_scale,
264                                Y=self.__f * 1./self.perm_scale,
265                                y= - util.inner(self.domain.getNormal(),u0 * self.location_of_fixed_flux * 1./self.perm_scale ),
266                                r=p0 - p0_hydrostatic)
267          pp=self.__pde_p.getSolution()
268          u = self._getFlux(pp, u0)
269          return u,pp + p0_hydrostatic
270          
271       def getFlux(self,p, u0=None):
272            """
273            returns the flux for a given pressure ``p`` where the flux is equal to ``u0``
274            on locations where ``location_of_fixed_flux`` is positive (see `setValue`).
275            Notice that ``g`` is used, see `setValue`.
276    
277            :param p: pressure.
278            :type p: scalar value on the domain (e.g. `escript.Data`).
279            :param u0: flux on the locations of the domain marked be ``location_of_fixed_flux``.
280            :type u0: vector values on the domain (e.g. `escript.Data`) or ``None``
281            :return: flux
282            :rtype: `escript.Data`
283            """
284            p=util.interpolate(p, self.__pde_p.getFunctionSpaceForCoefficient("q"))
285            if self.ref_point_id == None:
286                p_ref=0
287            else:
288                p_ref=p.getTupleForGlobalDataPoint(*self.ref_point_id)[0]
289            p_hydrostatic=p_ref+util.inner(self.__permeability_invXg_ref, self.__pde_p.getFunctionSpaceForCoefficient("q").getX() - self.ref_point)
290            return self._getFlux(p-p_hydrostatic, u0)
291    
292       def _getFlux(self, pp, u0=None):
293            """
294            returns the flux for a given pressure ``pp`` where the flux is equal to
295            ``u0`` on locations where ``location_of_fixed_flux`` is positive (see
296            `setValue`). Notice that ``g`` is used, see `setValue`.
297    
298            :param pp: pressure.
299            :type pp: scalar value on the domain (i.e. `escript.Data`).
300            :param u0: flux on the locations of the domain marked in ``location_of_fixed_flux``.
301            :type u0: vector values on the domain (i.e. `escript.Data`) or ``None``
302            :return: flux
303            :rtype: `escript.Data`
304            """
305            if self.solver  == self.EVAL:
306               u = self.__g - util.tensor_mult(self.__permeability, self.perm_scale * (util.grad(pp) + self.__permeability_invXg_ref))
307            elif self.solver  == self.POST or self.solver  == self.SMOOTH:
308                self.__pde_v.setValue(Y= self.__permeability_invXg - (util.grad(pp) + self.__permeability_invXg_ref))
309    
310                if u0 == None:
311                   self.__pde_v.setValue(r=escore.Data())
312                else:
313                   if not isinstance(u0, escore.Data) : u0 = escore.Vector(u0, escore.Solution(self.domain))
314                   self.__pde_v.setValue(r=1./self.perm_scale * u0)
315                u= self.__pde_v.getSolution() * self.perm_scale
316            return u
317          
318    class StokesProblemCartesian(pdt.HomogeneousSaddlePointProblem):
319         """
320         solves
321    
322            -(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}
323                  u_{i,i}=0                  u_{i,i}=0
# Line 333  class StokesProblemCartesian(Homogeneous Line 325  class StokesProblemCartesian(Homogeneous
325            u=0 where  fixed_u_mask>0            u=0 where  fixed_u_mask>0
326            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
327    
328        if surface_stress is not given 0 is assumed.       if surface_stress is not given 0 is assumed.
329    
330        typical usage:       typical usage:
331    
332              sp=StokesProblemCartesian(domain)              sp=StokesProblemCartesian(domain)
333              sp.setTolerance()              sp.setTolerance()
334              sp.initialize(...)              sp.initialize(...)
335              v,p=sp.solve(v0,p0)              v,p=sp.solve(v0,p0)
336        """              sp.setStokesEquation(...) # new values for some parameters
337        def __init__(self,domain,**kwargs):              v1,p1=sp.solve(v,p)
338         """
339         def __init__(self,domain,**kwargs):
340           """           """
341           initialize the Stokes Problem           initialize the Stokes Problem
342    
343           @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
344           @type domain: L{Domain}           LBB complient, for instance using quadratic and linear approximation on the same element or using linear approximation
345           @warning: The apprximation order needs to be two otherwise you may see oscilations in the pressure.           with macro elements for the pressure.
346    
347             :param domain: domain of the problem.
348             :type domain: `Domain`
349           """           """
350           HomogeneousSaddlePointProblem.__init__(self,**kwargs)           pdt.HomogeneousSaddlePointProblem.__init__(self,**kwargs)
351           self.domain=domain           self.domain=domain
352           self.vol=util.integrate(1.,Function(self.domain))           self.__pde_v=lpe.LinearPDE(domain,numEquations=self.domain.getDim(),numSolutions=self.domain.getDim())
353           self.__pde_u=LinearPDE(domain,numEquations=self.domain.getDim(),numSolutions=self.domain.getDim())           self.__pde_v.setSymmetryOn()
354           self.__pde_u.setSymmetryOn()      
355           # self.__pde_u.setSolverMethod(self.__pde_u.DIRECT)           self.__pde_prec=lpe.LinearPDE(domain)
          # self.__pde_u.setSolverMethod(preconditioner=LinearPDE.RILU)  
               
          self.__pde_prec=LinearPDE(domain)  
356           self.__pde_prec.setReducedOrderOn()           self.__pde_prec.setReducedOrderOn()
          # self.__pde_prec.setSolverMethod(self.__pde_prec.LUMPING)  
357           self.__pde_prec.setSymmetryOn()           self.__pde_prec.setSymmetryOn()
358    
359           self.__pde_proj=LinearPDE(domain)           self.__pde_proj=lpe.LinearPDE(domain)
360           self.__pde_proj.setReducedOrderOn()           self.__pde_proj.setReducedOrderOn()
361             self.__pde_proj.setValue(D=1)
362           self.__pde_proj.setSymmetryOn()           self.__pde_proj.setSymmetryOn()
          self.__pde_proj.setValue(D=1.)  
363    
364        def initialize(self,f=Data(),fixed_u_mask=Data(),eta=1,surface_stress=Data(),stress=Data()):       def getSolverOptionsVelocity(self):
365             """
366         returns the solver options used  solve the equation for velocity.
367        
368         :rtype: `SolverOptions`
369         """
370             return self.__pde_v.getSolverOptions()
371         def setSolverOptionsVelocity(self, options=None):
372             """
373         set the solver options for solving the equation for velocity.
374        
375         :param options: new solver  options
376         :type options: `SolverOptions`
377         """
378             self.__pde_v.setSolverOptions(options)
379         def getSolverOptionsPressure(self):
380             """
381         returns the solver options used  solve the equation for pressure.
382         :rtype: `SolverOptions`
383         """
384             return self.__pde_prec.getSolverOptions()
385         def setSolverOptionsPressure(self, options=None):
386             """
387         set the solver options for solving the equation for pressure.
388         :param options: new solver  options
389         :type options: `SolverOptions`
390         """
391             self.__pde_prec.setSolverOptions(options)
392    
393         def setSolverOptionsDiv(self, options=None):
394             """
395         set the solver options for solving the equation to project the divergence of
396         the velocity onto the function space of presure.
397        
398         :param options: new solver options
399         :type options: `SolverOptions`
400         """
401             self.__pde_proj.setSolverOptions(options)
402         def getSolverOptionsDiv(self):
403             """
404         returns the solver options for solving the equation to project the divergence of
405         the velocity onto the function space of presure.
406        
407         :rtype: `SolverOptions`
408         """
409             return self.__pde_proj.getSolverOptions()
410    
411         def updateStokesEquation(self, v, p):
412             """
413             updates the Stokes equation to consider dependencies from ``v`` and ``p``
414             :note: This method can be overwritten by a subclass. Use `setStokesEquation` to set new values to model parameters.
415             """
416             pass
417         def setStokesEquation(self, f=None,fixed_u_mask=None,eta=None,surface_stress=None,stress=None, restoration_factor=None):
418            """
419            assigns new values to the model parameters.
420    
421            :param f: external force
422            :type f: `Vector` object in `FunctionSpace` `Function` or similar
423            :param fixed_u_mask: mask of locations with fixed velocity.
424            :type fixed_u_mask: `Vector` object on `FunctionSpace` `Solution` or similar
425            :param eta: viscosity
426            :type eta: `Scalar` object on `FunctionSpace` `Function` or similar
427            :param surface_stress: normal surface stress
428            :type surface_stress: `Vector` object on `FunctionSpace` `FunctionOnBoundary` or similar
429            :param stress: initial stress
430        :type stress: `Tensor` object on `FunctionSpace` `Function` or similar
431            """
432            if eta !=None:
433                k=util.kronecker(self.domain.getDim())
434                kk=util.outer(k,k)
435                self.eta=util.interpolate(eta, escore.Function(self.domain))
436                self.__pde_prec.setValue(D=1/self.eta)
437                self.__pde_v.setValue(A=self.eta*(util.swap_axes(kk,0,3)+util.swap_axes(kk,1,3)))
438            if restoration_factor!=None:
439                n=self.domain.getNormal()
440                self.__pde_v.setValue(d=restoration_factor*util.outer(n,n))
441            if fixed_u_mask!=None:
442                self.__pde_v.setValue(q=fixed_u_mask)
443            if f!=None: self.__f=f
444            if surface_stress!=None: self.__surface_stress=surface_stress
445            if stress!=None: self.__stress=stress
446    
447         def initialize(self,f=escore.Data(),fixed_u_mask=escore.Data(),eta=1,surface_stress=escore.Data(),stress=escore.Data(), restoration_factor=0):
448          """          """
449          assigns values to the model parameters          assigns values to the model parameters
450    
451          @param f: external force          :param f: external force
452          @type f: L{Vector} object in L{FunctionSpace} L{Function} or similar          :type f: `Vector` object in `FunctionSpace` `Function` or similar
453          @param fixed_u_mask: mask of locations with fixed velocity.          :param fixed_u_mask: mask of locations with fixed velocity.
454          @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
455          @param eta: viscosity          :param eta: viscosity
456          @type eta: L{Scalar} object on L{FunctionSpace} L{Function} or similar          :type eta: `Scalar` object on `FunctionSpace` `Function` or similar
457          @param surface_stress: normal surface stress          :param surface_stress: normal surface stress
458          @type eta: L{Vector} object on L{FunctionSpace} L{FunctionOnBoundary} or similar          :type surface_stress: `Vector` object on `FunctionSpace` `FunctionOnBoundary` or similar
459          @param stress: initial stress          :param stress: initial stress
460      @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.  
   
461          """          """
462          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  
463    
464        def B(self,v):       def Bv(self,v,tol):
465          """           """
466          returns div(v)           returns inner product of element p and div(v)
         @rtype: equal to the type of p  
467    
468          @note: boundary conditions on p should be zero!           :param v: a residual
469          """           :return: inner product of element p and div(v)
470          if self.show_details: print "apply divergence:"           :rtype: ``float``
471          self.__pde_proj.setValue(Y=-util.div(v))           """
472          self.__pde_proj.setTolerance(self.getSubProblemTolerance())           self.__pde_proj.setValue(Y=-util.div(v))
473          return self.__pde_proj.getSolution(verbose=self.show_details)           self.getSolverOptionsDiv().setTolerance(tol)
474             self.getSolverOptionsDiv().setAbsoluteTolerance(0.)
475             out=self.__pde_proj.getSolution()
476             return out
477    
478        def inner_pBv(self,p,Bv):       def inner_pBv(self,p,Bv):
479           """           """
480           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}  
481    
482           @rtype: equal to the type of p           :param p: a pressure increment
483             :param Bv: a residual
484             :return: inner product of element p and Bv=-div(v)
485             :rtype: ``float``
486           """           """
487           s0=util.interpolate(p,Function(self.domain))           return util.integrate(util.interpolate(p,escore.Function(self.domain))*util.interpolate(Bv, escore.Function(self.domain)))
          s1=util.interpolate(Bv,Function(self.domain))  
          return util.integrate(s0*s1)  
488    
489        def inner_p(self,p0,p1):       def inner_p(self,p0,p1):
490           """           """
491           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}  
492    
493           @rtype: equal to the type of p           :param p0: a pressure
494             :param p1: a pressure
495             :return: inner product of p0 and p1
496             :rtype: ``float``
497           """           """
498           s0=util.interpolate(p0/self.eta,Function(self.domain))           s0=util.interpolate(p0, escore.Function(self.domain))
499           s1=util.interpolate(p1/self.eta,Function(self.domain))           s1=util.interpolate(p1, escore.Function(self.domain))
500           return util.integrate(s0*s1)           return util.integrate(s0*s1)
501    
502        def inner_v(self,v0,v1):       def norm_v(self,v):
503           """           """
504           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}  
505    
506           @rtype: equal to the type of v           :param v: a velovity
507             :return: norm of v
508             :rtype: non-negative ``float``
509           """           """
510       gv0=util.grad(v0)           return util.sqrt(util.integrate(util.length(util.grad(v))**2))
511       gv1=util.grad(v1)  
          return util.integrate(util.inner(gv0,gv1))  
512    
513        def solve_A(self,u,p):       def getDV(self, p, v, tol):
514           """           """
515           solves Av=f-Au-B^*p (v=0 on fixed_u_mask)           return the value for v for a given p
516    
517             :param p: a pressure
518             :param v: a initial guess for the value v to return.
519             :return: dv given as *Adv=(f-Av-B^*p)*
520           """           """
521           if self.show_details: print "solve for velocity:"           self.updateStokesEquation(v,p)
522           self.__pde_u.setTolerance(self.getSubProblemTolerance())           self.__pde_v.setValue(Y=self.__f, y=self.__surface_stress)
523             self.getSolverOptionsVelocity().setTolerance(tol)
524             self.getSolverOptionsVelocity().setAbsoluteTolerance(0.)
525           if self.__stress.isEmpty():           if self.__stress.isEmpty():
526              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)))
527           else:           else:
528              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)))
529           out=self.__pde_u.getSolution(verbose=self.show_details)           out=self.__pde_v.getSolution()
530             return  out
531    
532         def norm_Bv(self,Bv):
533            """
534            Returns Bv (overwrite).
535    
536            :rtype: equal to the type of p
537            :note: boundary conditions on p should be zero!
538            """
539            return util.sqrt(util.integrate(util.interpolate(Bv, escore.Function(self.domain))**2))
540    
541         def solve_AinvBt(self,p, tol):
542             """
543             Solves *Av=B^*p* with accuracy `tol`
544    
545             :param p: a pressure increment
546             :return: the solution of *Av=B^*p*
547             :note: boundary conditions on v should be zero!
548             """
549             self.__pde_v.setValue(Y=escore.Data(), y=escore.Data(), X=-p*util.kronecker(self.domain))
550             out=self.__pde_v.getSolution()
551           return  out           return  out
552    
553        def solve_prec(self,p):       def solve_prec(self,Bv, tol):
554           if self.show_details: print "apply preconditioner:"           """
555           self.__pde_prec.setTolerance(self.getSubProblemTolerance())           applies preconditioner for for *BA^{-1}B^** to *Bv*
556           self.__pde_prec.setValue(Y=p)           with accuracy ``self.getSubProblemTolerance()``
557           q=self.__pde_prec.getSolution(verbose=self.show_details)  
558           return q           :param Bv: velocity increment
559             :return: *p=P(Bv)* where *P^{-1}* is an approximation of *BA^{-1}B^ * )*
560             :note: boundary conditions on p are zero.
561             """
562             self.__pde_prec.setValue(Y=Bv)
563             self.getSolverOptionsPressure().setTolerance(tol)
564             self.getSolverOptionsPressure().setAbsoluteTolerance(0.)
565             out=self.__pde_prec.getSolution()
566             return out

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