/[escript]/trunk/escriptcore/py_src/flows.py
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revision 2123 by gross, Wed Dec 3 03:26:02 2008 UTC revision 4154 by jfenwick, Tue Jan 22 09:30:23 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
37  import util  escore=escriptcpp
38  from linearPDEs import LinearPDE, LinearPDESystem, LinearSinglePDE  #from . import escript
39  from pdetools import HomogeneousSaddlePointProblem,Projector, ArithmeticTuple, PCG  from . import util
40    from .linearPDEs import LinearPDE, LinearPDESystem, LinearSinglePDE, SolverOptions
41    from .pdetools import HomogeneousSaddlePointProblem,Projector, ArithmeticTuple, PCG, NegativeNorm, GMRES
42    
43  class DarcyFlow(object):  class DarcyFlow(object):
44      """     """
45      solves the problem     solves the problem
46      
47      M{u_i+k_{ij}*p_{,j} = g_i}     *u_i+k_{ij}*p_{,j} = g_i*
48      M{u_{i,i} = f}     *u_{i,i} = f*
49      
50       where *p* represents the pressure and *u* the Darcy flux. *k* represents the permeability,
51      
52       :cvar EVAL: direct pressure gradient evaluation for flux
53       :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*
54                   where *l* is the length scale, *K* is the inverse of the permeability tensor, and *w* is a positive weighting factor.
55       :cvar SMOOTH: global smoothing by solving the PDE *K_{ij} u_j= - p_{,j} + K_{ij} g_j*
56       """
57       EVAL="EVAL"
58       SIMPLE="EVAL"
59       POST="POST"
60       SMOOTH="SMOOTH"
61       def __init__(self, domain, useReduced=False, solver="POST", verbose=False, w=1.):
62          """
63          initializes the Darcy flux problem.
64    
65      where M{p} represents the pressure and M{u} the Darcy flux. M{k} represents the permeability,        :param domain: domain of the problem
66          :type domain: `Domain`
67          :param useReduced: uses reduced oreder on flux and pressure
68          :type useReduced: ``bool``
69          :param solver: solver method
70          :type solver: in [`DarcyFlow.EVAL`, `DarcyFlow.POST`, `DarcyFlow.SMOOTH` ]
71          :param verbose: if ``True`` some information on the iteration progress are printed.
72          :type verbose: ``bool``
73          :param w: weighting factor for `DarcyFlow.POST` solver
74          :type w: ``float``
75          
76          """
77          if not solver in [DarcyFlow.EVAL, DarcyFlow.POST,  DarcyFlow.SMOOTH ] :
78              raise ValueError("unknown solver %d."%solver)
79    
80      @note: The problem is solved in a least squares formulation.        self.domain=domain
81      """        self.solver=solver
82          self.useReduced=useReduced
83          self.verbose=verbose
84          self.l=None
85          self.w=None
86        
87          self.__pde_p=LinearSinglePDE(domain)
88          self.__pde_p.setSymmetryOn()
89          if self.useReduced: self.__pde_p.setReducedOrderOn()
90    
91          if self.solver  == self.EVAL:
92             self.__pde_v=None
93             if self.verbose: print("DarcyFlow: simple solver is used.")
94    
95          elif self.solver  == self.POST:
96             if util.inf(w)<0.:
97                raise ValueError("Weighting factor must be non-negative.")
98             if self.verbose: print("DarcyFlow: global postprocessing of flux is used.")
99             self.__pde_v=LinearPDESystem(domain)
100             self.__pde_v.setSymmetryOn()
101             if self.useReduced: self.__pde_v.setReducedOrderOn()
102             self.w=w
103             x=self.domain.getX()
104             self.l=min( [util.sup(x[i])-util.inf(x[i]) for i in range(self.domain.getDim()) ] )
105             #self.l=util.vol(self.domain)**(1./self.domain.getDim()) # length scale
106    
107          elif self.solver  == self.SMOOTH:
108             self.__pde_v=LinearPDESystem(domain)
109             self.__pde_v.setSymmetryOn()
110             if self.useReduced: self.__pde_v.setReducedOrderOn()
111             if self.verbose: print("DarcyFlow: flux smoothing is used.")
112             self.w=0
113    
114          self.__f=escore.Data(0,self.__pde_p.getFunctionSpaceForCoefficient("X"))
115          self.__g=escore.Vector(0,self.__pde_p.getFunctionSpaceForCoefficient("Y"))
116          self.__permeability_invXg=escore.Vector(0,self.__pde_p.getFunctionSpaceForCoefficient("Y"))
117          self.__permeability_invXg_ref=util.numpy.zeros((self.domain.getDim()),util.numpy.float64)
118          self.ref_point_id=None
119          self.ref_point=util.numpy.zeros((self.domain.getDim()),util.numpy.float64)
120          self.location_of_fixed_pressure = escore.Data(0, self.__pde_p.getFunctionSpaceForCoefficient("q"))
121          self.location_of_fixed_flux = escore.Vector(0, self.__pde_p.getFunctionSpaceForCoefficient("q"))
122          self.perm_scale=1.
123        
124            
125       def setValue(self,f=None, g=None, location_of_fixed_pressure=None, location_of_fixed_flux=None, permeability=None):
126          """
127          assigns values to model parameters
128    
129      def __init__(self, domain):        :param f: volumetic sources/sinks
130          """        :type f: scalar value on the domain (e.g. `escript.Data`)
131          initializes the Darcy flux problem        :param g: flux sources/sinks
132          @param domain: domain of the problem        :type g: vector values on the domain (e.g. `escript.Data`)
133          @type domain: L{Domain}        :param location_of_fixed_pressure: mask for locations where pressure is fixed
134          """        :type location_of_fixed_pressure: scalar value on the domain (e.g. `escript.Data`)
135          self.domain=domain        :param location_of_fixed_flux:  mask for locations where flux is fixed.
136          self.__pde_v=LinearPDESystem(domain)        :type location_of_fixed_flux: vector values on the domain (e.g. `escript.Data`)
137          self.__pde_v.setValue(D=util.kronecker(domain), A=util.outer(util.kronecker(domain),util.kronecker(domain)))        :param permeability: permeability tensor. If scalar ``s`` is given the tensor with ``s`` on the main diagonal is used.
138          self.__pde_v.setSymmetryOn()        :type permeability: scalar or symmetric tensor values on the domain (e.g. `escript.Data`)
139          self.__pde_p=LinearSinglePDE(domain)  
140          self.__pde_p.setSymmetryOn()        :note: the values of parameters which are not set by calling ``setValue`` are not altered.
141          self.__f=Scalar(0,self.__pde_v.getFunctionSpaceForCoefficient("X"))        :note: at any point on the boundary of the domain the pressure
142          self.__g=Vector(0,self.__pde_v.getFunctionSpaceForCoefficient("Y"))               (``location_of_fixed_pressure`` >0) or the normal component of the
143                 flux (``location_of_fixed_flux[i]>0``) if direction of the normal
144                 is along the *x_i* axis.
145    
146      def setValue(self,f=None, g=None, location_of_fixed_pressure=None, location_of_fixed_flux=None, permeability=None):        """
147          """        if location_of_fixed_pressure!=None:
148          assigns values to model parameters             self.location_of_fixed_pressure=util.wherePositive(util.interpolate(location_of_fixed_pressure, self.__pde_p.getFunctionSpaceForCoefficient("q")))
149               self.ref_point_id=self.location_of_fixed_pressure.maxGlobalDataPoint()
150               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.")
151               self.ref_point=self.__pde_p.getFunctionSpaceForCoefficient("q").getX().getTupleForGlobalDataPoint(*self.ref_point_id)
152               if self.verbose: print(("DarcyFlow: reference point at %s."%(self.ref_point,)))
153               self.__pde_p.setValue(q=self.location_of_fixed_pressure)
154          if location_of_fixed_flux!=None:
155              self.location_of_fixed_flux=util.wherePositive(location_of_fixed_flux)
156              if not self.__pde_v == None:
157                  self.__pde_v.setValue(q=self.location_of_fixed_flux)
158                
159          if permeability!=None:
160        
161             perm=util.interpolate(permeability,self.__pde_p.getFunctionSpaceForCoefficient("A"))
162             self.perm_scale=util.Lsup(util.length(perm))
163             if self.verbose: print(("DarcyFlow: permeability scaling factor = %e."%self.perm_scale))
164             perm=perm*(1./self.perm_scale)
165            
166             if perm.getRank()==0:
167    
168          @param f: volumetic sources/sinks              perm_inv=(1./perm)
169          @type f: scalar value on the domain (e.g. L{Data})              perm_inv=perm_inv*util.kronecker(self.domain.getDim())
170          @param g: flux sources/sinks              perm=perm*util.kronecker(self.domain.getDim())
171          @type g: vector values on the domain (e.g. L{Data})          
172          @param location_of_fixed_pressure: mask for locations where pressure is fixed          
173          @type location_of_fixed_pressure: scalar value on the domain (e.g. L{Data})           elif perm.getRank()==2:
174          @param location_of_fixed_flux:  mask for locations where flux is fixed.              perm_inv=util.inverse(perm)
175          @type location_of_fixed_flux: vector values on the domain (e.g. L{Data})           else:
176          @param permeability: permeability tensor. If scalar C{s} is given the tensor with              raise ValueError("illegal rank of permeability.")
177                               C{s} on the main diagonal is used. If vector C{v} is given the tensor with          
178                               C{v} on the main diagonal is used.           self.__permeability=perm
179          @type permeability: scalar, vector or tensor values on the domain (e.g. L{Data})           self.__permeability_inv=perm_inv
180        
181          @note: the values of parameters which are not set by calling C{setValue} are not altered.           #====================
182          @note: at any point on the boundary of the domain the pressure (C{location_of_fixed_pressure} >0)           self.__pde_p.setValue(A=self.__permeability)
183                 or the normal component of the flux (C{location_of_fixed_flux[i]>0} if direction of the normal           if self.solver  == self.EVAL:
184                 is along the M{x_i} axis.                pass # no extra work required
185          """           elif self.solver  == self.POST:
186          if f !=None:                k=util.kronecker(self.domain.getDim())
187             f=util.interpolate(f, self.__pde_v.getFunctionSpaceForCoefficient("X"))                self.omega = self.w*util.length(perm_inv)*self.l*self.domain.getSize()
188             if f.isEmpty():                #self.__pde_v.setValue(D=self.__permeability_inv, A=self.omega*util.outer(k,k))
189                 f=Scalar(0,self.__pde_v.getFunctionSpaceForCoefficient("X"))                self.__pde_v.setValue(D=self.__permeability_inv, A_reduced=self.omega*util.outer(k,k))
190             else:           elif self.solver  == self.SMOOTH:
191                 if f.getRank()>0: raise ValueError,"illegal rank of f."              self.__pde_v.setValue(D=self.__permeability_inv)
192             self.f=f  
193          if g !=None:          if g != None:
194             g=util.interpolate(g, self.__pde_p.getFunctionSpaceForCoefficient("Y"))          g=util.interpolate(g, self.__pde_p.getFunctionSpaceForCoefficient("Y"))
195             if g.isEmpty():          if g.isEmpty():
196               g=Vector(0,self.__pde_v.getFunctionSpaceForCoefficient("Y"))               g=Vector(0,self.__pde_p.getFunctionSpaceForCoefficient("Y"))
197             else:          else:
198               if not g.getShape()==(self.domain.getDim(),):               if not g.getShape()==(self.domain.getDim(),): raise ValueError("illegal shape of g")
199                 raise ValueError,"illegal shape of g"          self.__g=g
200             self.__g=g          self.__permeability_invXg=util.tensor_mult(self.__permeability_inv,self.__g * (1./self.perm_scale ))
201            self.__permeability_invXg_ref=util.integrate(self.__permeability_invXg)/util.vol(self.domain)
202          if location_of_fixed_pressure!=None: self.__pde_p.setValue(q=location_of_fixed_pressure)        if f !=None:
203          if location_of_fixed_flux!=None: self.__pde_v.setValue(q=location_of_fixed_flux)           f=util.interpolate(f, self.__pde_p.getFunctionSpaceForCoefficient("Y"))
204             if f.isEmpty():      
205          if permeability!=None:               f=Scalar(0,self.__pde_p.getFunctionSpaceForCoefficient("Y"))
206             perm=util.interpolate(permeability,self.__pde_p.getFunctionSpaceForCoefficient("A"))           else:
207             if perm.getRank()==0:               if f.getRank()>0: raise ValueError("illegal rank of f.")
208                 perm=perm*util.kronecker(self.domain.getDim())           self.__f=f
            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))  
209    
210       def getSolverOptionsFlux(self):
211          """
212          Returns the solver options used to solve the flux problems
213          :return: `SolverOptions`
214          """
215          if self.__pde_v == None:
216              return None
217          else:
218              return self.__pde_v.getSolverOptions()
219          
220       def setSolverOptionsFlux(self, options=None):
221          """
222          Sets the solver options used to solve the flux problems
223          If ``options`` is not present, the options are reset to default
224          :param options: `SolverOptions`
225          """
226          if not self.__pde_v == None:
227              self.__pde_v.setSolverOptions(options)
228        
229       def getSolverOptionsPressure(self):
230          """
231          Returns the solver options used to solve the pressure problems
232          :return: `SolverOptions`
233          """
234          return self.__pde_p.getSolverOptions()
235          
236       def setSolverOptionsPressure(self, options=None):
237          """
238          Sets the solver options used to solve the pressure problems
239          If ``options`` is not present, the options are reset to default
240          
241          :param options: `SolverOptions`
242          :note: if the adaption of subtolerance is choosen, the tolerance set by ``options`` will be overwritten before the solver is called.
243          """
244          return self.__pde_p.setSolverOptions(options)
245          
246       def solve(self, u0, p0):
247          """
248          solves the problem.
249          
250          :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.
251          :type u0: vector value on the domain (e.g. `escript.Data`).
252          :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.
253          :type p0: scalar value on the domain (e.g. `escript.Data`).
254          :return: flux and pressure
255          :rtype: ``tuple`` of `escript.Data`.
256    
257      def getFlux(self,p, fixed_flux=Data(),tol=1.e-8):        """
258          p0=util.interpolate(p0, self.__pde_p.getFunctionSpaceForCoefficient("q"))
259          if self.ref_point_id == None:
260              p_ref=0
261          else:
262              p_ref=p0.getTupleForGlobalDataPoint(*self.ref_point_id)[0]
263          p0_hydrostatic=p_ref+util.inner(self.__permeability_invXg_ref, self.__pde_p.getFunctionSpaceForCoefficient("q").getX() - self.ref_point)
264          g_2=self.__g - util.tensor_mult(self.__permeability, self.__permeability_invXg_ref * self.perm_scale)
265          self.__pde_p.setValue(X=g_2 * 1./self.perm_scale,
266                                Y=self.__f * 1./self.perm_scale,
267                                y= - util.inner(self.domain.getNormal(),u0 * self.location_of_fixed_flux * 1./self.perm_scale ),
268                                r=p0 - p0_hydrostatic)
269          pp=self.__pde_p.getSolution()
270          u = self._getFlux(pp, u0)
271          return u,pp + p0_hydrostatic
272          
273       def getFlux(self,p, u0=None):
274          """          """
275          returns the flux for a given pressure C{p} where the flux is equal to C{fixed_flux}          returns the flux for a given pressure ``p`` where the flux is equal to ``u0``
276          on locations where C{location_of_fixed_flux} is positive (see L{setValue}).          on locations where ``location_of_fixed_flux`` is positive (see `setValue`).
277          Note that C{g} and C{f} are used, L{setValue}.          Notice that ``g`` is used, see `setValue`.
278            
279          @param p: pressure.          :param p: pressure.
280          @type p: scalar value on the domain (e.g. L{Data}).          :type p: scalar value on the domain (e.g. `escript.Data`).
281          @param fixed_flux: flux on the locations of the domain marked be C{location_of_fixed_flux}.          :param u0: flux on the locations of the domain marked be ``location_of_fixed_flux``.
282          @type fixed_flux: vector values on the domain (e.g. L{Data}).          :type u0: vector values on the domain (e.g. `escript.Data`) or ``None``
283          @param tol: relative tolerance to be used.          :return: flux
284          @type tol: positive float.          :rtype: `escript.Data`
         @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}}  
285          """          """
286          self.__pde_v.setTolerance(tol)          p=util.interpolate(p, self.__pde_p.getFunctionSpaceForCoefficient("q"))
287          self.__pde_v.setValue(Y=self.__g, X=self.__f*util.kronecker(self.domain), r=fixed_flux)          if self.ref_point_id == None:
288          return self.__pde_v.getSolution()              p_ref=0
289            else:
290      def solve(self,u0,p0,atol=0,rtol=1e-8, max_iter=100, verbose=False, show_details=False, sub_rtol=1.e-8):              p_ref=p.getTupleForGlobalDataPoint(*self.ref_point_id)[0]
291           """          p_hydrostatic=p_ref+util.inner(self.__permeability_invXg_ref, self.__pde_p.getFunctionSpaceForCoefficient("q").getX() - self.ref_point)
292           solves the problem.          return self._getFlux(p-p_hydrostatic, u0)
   
          The iteration is terminated if the error in the pressure is less then C{rtol * |q| + atol} where  
          C{|q|} denotes the norm of the right hand side (see escript user's guide for details).  
   
          @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 atol: absolute tolerance for the pressure  
          @type atol: non-negative C{float}  
          @param rtol: relative tolerance for the pressure  
          @type rtol: non-negative C{float}  
          @param 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  
   
          M{Q^*(I-(I+D^*D)^{-1})Qp= Q^*(g-(I+D^*D)^{-1}(D^*f+g))} with p=p0  on location_of_fixed_pressure  
           
          which is solved using the PCG method (precondition is M{Q^*Q}). In each iteration step  
          PDEs with operator M{I+D^*D} and with M{Q^*Q} needs to be solved using a sub iteration scheme.  
          """  
          self.verbose=verbose  
          self.show_details= show_details and self.verbose  
          self.__pde_v.setTolerance(sub_rtol)  
          self.__pde_p.setTolerance(sub_rtol)  
          p2=p0*self.__pde_p.getCoefficient("q")  
          u2=u0*self.__pde_v.getCoefficient("q")  
          g=self.__g-u2-util.tensor_mult(self.__permeability,util.grad(p2))  
          f=self.__f-util.div(u2)  
          self.__pde_v.setValue(Y=g, X=f*util.kronecker(self.domain), r=Data())  
          dv=self.__pde_v.getSolution(verbose=show_details)  
          self.__pde_p.setValue(X=util.transposed_tensor_mult(self.__permeability,g-dv))  
          self.__pde_p.setValue(r=Data())  
          dp=self.__pde_p.getSolution(verbose=self.show_details)  
          norm_rhs=self.__inner_PCG(dp,ArithmeticTuple(g,dv))  
          if norm_rhs<0:  
              raise NegativeNorm,"negative norm. Maybe the sub-tolerance is too large."  
          ATOL=util.sqrt(norm_rhs)*rtol +atol  
          if not ATOL>0:  
              raise ValueError,"Negative absolute tolerance (rtol = %e, norm right hand side =%, atol =%e)."%(rtol, util.sqrt(norm_rhs), atol)  
          rhs=ArithmeticTuple(g,dv)  
          dp,r=PCG(rhs,self.__Aprod_PCG,self.__Msolve_PCG,self.__inner_PCG,atol=ATOL, rtol=0.,iter_max=max_iter, x=p0-p2, verbose=self.verbose, initial_guess=True)  
          return u2+r[1],p2+dp  
           
     def __Aprod_PCG(self,p):  
           if self.show_details: print "DarcyFlux: Applying operator"  
           Qp=util.tensor_mult(self.__permeability,util.grad(p))  
           self.__pde_v.setValue(Y=Qp,X=Data())  
           w=self.__pde_v.getSolution(verbose=self.show_details)  
           return ArithmeticTuple(Qp,w)  
   
     def __inner_PCG(self,p,r):  
          a=util.tensor_mult(self.__permeability,util.grad(p))  
          return util.integrate(util.inner(a,r[0]-r[1]))  
   
     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]))  
           return self.__pde_p.getSolution(verbose=self.show_details)  
293    
294       def _getFlux(self, pp, u0=None):
295            """
296            returns the flux for a given pressure ``pp`` where the flux is equal to
297            ``u0`` on locations where ``location_of_fixed_flux`` is positive (see
298            `setValue`). Notice that ``g`` is used, see `setValue`.
299    
300            :param pp: pressure.
301            :type pp: scalar value on the domain (i.e. `escript.Data`).
302            :param u0: flux on the locations of the domain marked in ``location_of_fixed_flux``.
303            :type u0: vector values on the domain (i.e. `escript.Data`) or ``None``
304            :return: flux
305            :rtype: `escript.Data`
306            """
307            if self.solver  == self.EVAL:
308               u = self.__g - util.tensor_mult(self.__permeability, self.perm_scale * (util.grad(pp) + self.__permeability_invXg_ref))
309            elif self.solver  == self.POST or self.solver  == self.SMOOTH:
310                self.__pde_v.setValue(Y= self.__permeability_invXg - (util.grad(pp) + self.__permeability_invXg_ref))
311                print
312                if u0 == None:
313                   self.__pde_v.setValue(r=escore.Data())
314                else:
315                   if not isinstance(u0, escore.Data) : u0 = escore.Vector(u0, escore.Solution(self.domain))
316                   self.__pde_v.setValue(r=1./self.perm_scale * u0)
317                u= self.__pde_v.getSolution() * self.perm_scale
318            return u
319          
320  class StokesProblemCartesian(HomogeneousSaddlePointProblem):  class StokesProblemCartesian(HomogeneousSaddlePointProblem):
321        """       """
322        solves       solves
323    
324            -(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}
325                  u_{i,i}=0                  u_{i,i}=0
# Line 230  class StokesProblemCartesian(Homogeneous Line 327  class StokesProblemCartesian(Homogeneous
327            u=0 where  fixed_u_mask>0            u=0 where  fixed_u_mask>0
328            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
329    
330        if surface_stress is not given 0 is assumed.       if surface_stress is not given 0 is assumed.
331    
332        typical usage:       typical usage:
333    
334              sp=StokesProblemCartesian(domain)              sp=StokesProblemCartesian(domain)
335              sp.setTolerance()              sp.setTolerance()
336              sp.initialize(...)              sp.initialize(...)
337              v,p=sp.solve(v0,p0)              v,p=sp.solve(v0,p0)
338        """              sp.setStokesEquation(...) # new values for some parameters
339        def __init__(self,domain,**kwargs):              v1,p1=sp.solve(v,p)
340         """
341         def __init__(self,domain,**kwargs):
342           """           """
343           initialize the Stokes Problem           initialize the Stokes Problem
344    
345           @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
346           @type domain: L{Domain}           LBB complient, for instance using quadratic and linear approximation on the same element or using linear approximation
347           @warning: The apprximation order needs to be two otherwise you may see oscilations in the pressure.           with macro elements for the pressure.
348    
349             :param domain: domain of the problem.
350             :type domain: `Domain`
351           """           """
352           HomogeneousSaddlePointProblem.__init__(self,**kwargs)           HomogeneousSaddlePointProblem.__init__(self,**kwargs)
353           self.domain=domain           self.domain=domain
354           self.vol=util.integrate(1.,Function(self.domain))           self.__pde_v=LinearPDE(domain,numEquations=self.domain.getDim(),numSolutions=self.domain.getDim())
355           self.__pde_u=LinearPDE(domain,numEquations=self.domain.getDim(),numSolutions=self.domain.getDim())           self.__pde_v.setSymmetryOn()
356           self.__pde_u.setSymmetryOn()      
          # self.__pde_u.setSolverMethod(self.__pde_u.DIRECT)  
          # self.__pde_u.setSolverMethod(preconditioner=LinearPDE.RILU)  
               
357           self.__pde_prec=LinearPDE(domain)           self.__pde_prec=LinearPDE(domain)
358           self.__pde_prec.setReducedOrderOn()           self.__pde_prec.setReducedOrderOn()
          self.__pde_prec.setSolverMethod(self.__pde_prec.LUMPING)  
359           self.__pde_prec.setSymmetryOn()           self.__pde_prec.setSymmetryOn()
360    
361           self.__pde_proj=LinearPDE(domain)           self.__pde_proj=LinearPDE(domain)
362           self.__pde_proj.setReducedOrderOn()           self.__pde_proj.setReducedOrderOn()
363             self.__pde_proj.setValue(D=1)
364           self.__pde_proj.setSymmetryOn()           self.__pde_proj.setSymmetryOn()
          self.__pde_proj.setValue(D=1.)  
365    
366        def initialize(self,f=Data(),fixed_u_mask=Data(),eta=1,surface_stress=Data(),stress=Data()):       def getSolverOptionsVelocity(self):
367             """
368         returns the solver options used  solve the equation for velocity.
369        
370         :rtype: `SolverOptions`
371         """
372             return self.__pde_v.getSolverOptions()
373         def setSolverOptionsVelocity(self, options=None):
374             """
375         set the solver options for solving the equation for velocity.
376        
377         :param options: new solver  options
378         :type options: `SolverOptions`
379         """
380             self.__pde_v.setSolverOptions(options)
381         def getSolverOptionsPressure(self):
382             """
383         returns the solver options used  solve the equation for pressure.
384         :rtype: `SolverOptions`
385         """
386             return self.__pde_prec.getSolverOptions()
387         def setSolverOptionsPressure(self, options=None):
388             """
389         set the solver options for solving the equation for pressure.
390         :param options: new solver  options
391         :type options: `SolverOptions`
392         """
393             self.__pde_prec.setSolverOptions(options)
394    
395         def setSolverOptionsDiv(self, options=None):
396             """
397         set the solver options for solving the equation to project the divergence of
398         the velocity onto the function space of presure.
399        
400         :param options: new solver options
401         :type options: `SolverOptions`
402         """
403             self.__pde_proj.setSolverOptions(options)
404         def getSolverOptionsDiv(self):
405             """
406         returns the solver options for solving the equation to project the divergence of
407         the velocity onto the function space of presure.
408        
409         :rtype: `SolverOptions`
410         """
411             return self.__pde_proj.getSolverOptions()
412    
413         def updateStokesEquation(self, v, p):
414             """
415             updates the Stokes equation to consider dependencies from ``v`` and ``p``
416             :note: This method can be overwritten by a subclass. Use `setStokesEquation` to set new values to model parameters.
417             """
418             pass
419         def setStokesEquation(self, f=None,fixed_u_mask=None,eta=None,surface_stress=None,stress=None, restoration_factor=None):
420          """          """
421          assigns values to the model parameters          assigns new values to the model parameters.
422    
423          @param f: external force          :param f: external force
424          @type f: L{Vector} object in L{FunctionSpace} L{Function} or similar          :type f: `Vector` object in `FunctionSpace` `Function` or similar
425          @param fixed_u_mask: mask of locations with fixed velocity.          :param fixed_u_mask: mask of locations with fixed velocity.
426          @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
427          @param eta: viscosity          :param eta: viscosity
428          @type eta: L{Scalar} object on L{FunctionSpace} L{Function} or similar          :type eta: `Scalar` object on `FunctionSpace` `Function` or similar
429          @param surface_stress: normal surface stress          :param surface_stress: normal surface stress
430          @type eta: L{Vector} object on L{FunctionSpace} L{FunctionOnBoundary} or similar          :type surface_stress: `Vector` object on `FunctionSpace` `FunctionOnBoundary` or similar
431          @param stress: initial stress          :param stress: initial stress
432      @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.  
   
433          """          """
434          self.eta=eta          if eta !=None:
435          A =self.__pde_u.createCoefficient("A")              k=util.kronecker(self.domain.getDim())
436      self.__pde_u.setValue(A=Data())              kk=util.outer(k,k)
437          for i in range(self.domain.getDim()):              self.eta=util.interpolate(eta, escore.Function(self.domain))
438          for j in range(self.domain.getDim()):              self.__pde_prec.setValue(D=1/self.eta)
439              A[i,j,j,i] += 1.              self.__pde_v.setValue(A=self.eta*(util.swap_axes(kk,0,3)+util.swap_axes(kk,1,3)))
440              A[i,j,i,j] += 1.          if restoration_factor!=None:
441      self.__pde_prec.setValue(D=1/self.eta)              n=self.domain.getNormal()
442          self.__pde_u.setValue(A=A*self.eta,q=fixed_u_mask,Y=f,y=surface_stress)              self.__pde_v.setValue(d=restoration_factor*util.outer(n,n))
443          self.__stress=stress          if fixed_u_mask!=None:
444                self.__pde_v.setValue(q=fixed_u_mask)
445            if f!=None: self.__f=f
446            if surface_stress!=None: self.__surface_stress=surface_stress
447            if stress!=None: self.__stress=stress
448    
449        def B(self,v):       def initialize(self,f=escore.Data(),fixed_u_mask=escore.Data(),eta=1,surface_stress=escore.Data(),stress=escore.Data(), restoration_factor=0):
450          """          """
451          returns div(v)          assigns values to the model parameters
         @rtype: equal to the type of p  
452    
453          @note: boundary conditions on p should be zero!          :param f: external force
454            :type f: `Vector` object in `FunctionSpace` `Function` or similar
455            :param fixed_u_mask: mask of locations with fixed velocity.
456            :type fixed_u_mask: `Vector` object on `FunctionSpace` `Solution` or similar
457            :param eta: viscosity
458            :type eta: `Scalar` object on `FunctionSpace` `Function` or similar
459            :param surface_stress: normal surface stress
460            :type surface_stress: `Vector` object on `FunctionSpace` `FunctionOnBoundary` or similar
461            :param stress: initial stress
462            :type stress: `Tensor` object on `FunctionSpace` `Function` or similar
463          """          """
464          if self.show_details: print "apply divergence:"          self.setStokesEquation(f,fixed_u_mask, eta, surface_stress, stress, restoration_factor)
         self.__pde_proj.setValue(Y=-util.div(v))  
         self.__pde_proj.setTolerance(self.getSubProblemTolerance())  
         return self.__pde_proj.getSolution(verbose=self.show_details)  
465    
466        def inner_pBv(self,p,Bv):       def Bv(self,v,tol):
467           """           """
468           returns inner product of element p and Bv  (overwrite)           returns inner product of element p and div(v)
           
          @type p: equal to the type of p  
          @type Bv: equal to the type of result of operator B  
          @rtype: C{float}  
469    
470           @rtype: equal to the type of p           :param v: a residual
471             :return: inner product of element p and div(v)
472             :rtype: ``float``
473           """           """
474           s0=util.interpolate(p,Function(self.domain))           self.__pde_proj.setValue(Y=-util.div(v))
475           s1=util.interpolate(Bv,Function(self.domain))           self.getSolverOptionsDiv().setTolerance(tol)
476           return util.integrate(s0*s1)           self.getSolverOptionsDiv().setAbsoluteTolerance(0.)
477             out=self.__pde_proj.getSolution()
478             return out
479    
480        def inner_p(self,p0,p1):       def inner_pBv(self,p,Bv):
481           """           """
482           returns inner product of element p0 and p1  (overwrite)           returns inner product of element p and Bv=-div(v)
483            
484           @type p0: equal to the type of p           :param p: a pressure increment
485           @type p1: equal to the type of p           :param Bv: a residual
486           @rtype: C{float}           :return: inner product of element p and Bv=-div(v)
487             :rtype: ``float``
488             """
489             return util.integrate(util.interpolate(p,escore.Function(self.domain))*util.interpolate(Bv, escore.Function(self.domain)))
490    
491           @rtype: equal to the type of p       def inner_p(self,p0,p1):
492           """           """
493           s0=util.interpolate(p0/self.eta,Function(self.domain))           Returns inner product of p0 and p1
494           s1=util.interpolate(p1/self.eta,Function(self.domain))  
495             :param p0: a pressure
496             :param p1: a pressure
497             :return: inner product of p0 and p1
498             :rtype: ``float``
499             """
500             s0=util.interpolate(p0, escore.Function(self.domain))
501             s1=util.interpolate(p1, escore.Function(self.domain))
502           return util.integrate(s0*s1)           return util.integrate(s0*s1)
503    
504        def inner_v(self,v0,v1):       def norm_v(self,v):
505           """           """
506           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}  
507    
508           @rtype: equal to the type of v           :param v: a velovity
509             :return: norm of v
510             :rtype: non-negative ``float``
511           """           """
512       gv0=util.grad(v0)           return util.sqrt(util.integrate(util.length(util.grad(v))**2))
513       gv1=util.grad(v1)  
          return util.integrate(util.inner(gv0,gv1))  
514    
515        def solve_A(self,u,p):       def getDV(self, p, v, tol):
516           """           """
517           solves Av=f-Au-B^*p (v=0 on fixed_u_mask)           return the value for v for a given p
518    
519             :param p: a pressure
520             :param v: a initial guess for the value v to return.
521             :return: dv given as *Adv=(f-Av-B^*p)*
522           """           """
523           if self.show_details: print "solve for velocity:"           self.updateStokesEquation(v,p)
524           self.__pde_u.setTolerance(self.getSubProblemTolerance())           self.__pde_v.setValue(Y=self.__f, y=self.__surface_stress)
525             self.getSolverOptionsVelocity().setTolerance(tol)
526             self.getSolverOptionsVelocity().setAbsoluteTolerance(0.)
527           if self.__stress.isEmpty():           if self.__stress.isEmpty():
528              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)))
529           else:           else:
530              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)))
531           out=self.__pde_u.getSolution(verbose=self.show_details)           out=self.__pde_v.getSolution()
532             return  out
533    
534         def norm_Bv(self,Bv):
535            """
536            Returns Bv (overwrite).
537    
538            :rtype: equal to the type of p
539            :note: boundary conditions on p should be zero!
540            """
541            return util.sqrt(util.integrate(util.interpolate(Bv, escore.Function(self.domain))**2))
542    
543         def solve_AinvBt(self,p, tol):
544             """
545             Solves *Av=B^*p* with accuracy `tol`
546    
547             :param p: a pressure increment
548             :return: the solution of *Av=B^*p*
549             :note: boundary conditions on v should be zero!
550             """
551             self.__pde_v.setValue(Y=escore.Data(), y=escore.Data(), X=-p*util.kronecker(self.domain))
552             out=self.__pde_v.getSolution()
553           return  out           return  out
554    
555        def solve_prec(self,p):       def solve_prec(self,Bv, tol):
556           if self.show_details: print "apply preconditioner:"           """
557           self.__pde_prec.setTolerance(self.getSubProblemTolerance())           applies preconditioner for for *BA^{-1}B^** to *Bv*
558           self.__pde_prec.setValue(Y=p)           with accuracy ``self.getSubProblemTolerance()``
559           q=self.__pde_prec.getSolution(verbose=self.show_details)  
560           return q           :param Bv: velocity increment
561             :return: *p=P(Bv)* where *P^{-1}* is an approximation of *BA^{-1}B^ * )*
562             :note: boundary conditions on p are zero.
563             """
564             self.__pde_prec.setValue(Y=Bv)
565             self.getSolverOptionsPressure().setTolerance(tol)
566             self.getSolverOptionsPressure().setAbsoluteTolerance(0.)
567             out=self.__pde_prec.getSolution()
568             return out

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