/[escript]/trunk/escriptcore/py_src/flows.py
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revision 2208 by gross, Mon Jan 12 06:37:07 2009 UTC revision 3911 by jfenwick, Thu Jun 14 01:01:03 2012 UTC
# Line 1  Line 1 
1    # -*- coding: utf-8 -*-
2  ########################################################  ########################################################
3  #  #
4  # Copyright (c) 2003-2008 by University of Queensland  # Copyright (c) 2003-2012 by University of Queensland
5  # Earth Systems Science Computational Center (ESSCC)  # Earth Systems Science Computational Center (ESSCC)
6  # http://www.uq.edu.au/esscc  # http://www.uq.edu.au/esscc
7  #  #
# Line 10  Line 11 
11  #  #
12  ########################################################  ########################################################
13    
14  __copyright__="""Copyright (c) 2003-2008 by University of Queensland  __copyright__="""Copyright (c) 2003-2012 by University of Queensland
15  Earth Systems Science Computational Center (ESSCC)  Earth Systems Science Computational Center (ESSCC)
16  http://www.uq.edu.au/esscc  http://www.uq.edu.au/esscc
17  Primary Business: Queensland, Australia"""  Primary Business: Queensland, Australia"""
18  __license__="""Licensed under the Open Software License version 3.0  __license__="""Licensed under the Open Software License version 3.0
19  http://www.opensource.org/licenses/osl-3.0.php"""  http://www.opensource.org/licenses/osl-3.0.php"""
20  __url__="http://www.uq.edu.au/esscc/escript-finley"  __url__="https://launchpad.net/escript-finley"
21    
22  """  """
23  Some models for flow  Some models for flow
24    
25  @var __author__: name of author  :var __author__: name of author
26  @var __copyright__: copyrights  :var __copyright__: copyrights
27  @var __license__: licence agreement  :var __license__: licence agreement
28  @var __url__: url entry point on documentation  :var __url__: url entry point on documentation
29  @var __version__: version  :var __version__: version
30  @var __date__: date of the version  :var __date__: date of the version
31  """  """
32    
33  __author__="Lutz Gross, l.gross@uq.edu.au"  __author__="Lutz Gross, l.gross@uq.edu.au"
34    
35  from escript import *  from . import escript
36  import util  from . import util
37  from linearPDEs import LinearPDE, LinearPDESystem, LinearSinglePDE  from .linearPDEs import LinearPDE, LinearPDESystem, LinearSinglePDE, SolverOptions
38  from pdetools import HomogeneousSaddlePointProblem,Projector, ArithmeticTuple, PCG, NegativeNorm  from .pdetools import HomogeneousSaddlePointProblem,Projector, ArithmeticTuple, PCG, NegativeNorm, GMRES
39    
40  class DarcyFlow(object):  class DarcyFlow(object):
41      """     """
42      solves the problem     solves the problem
43      
44      M{u_i+k_{ij}*p_{,j} = g_i}     *u_i+k_{ij}*p_{,j} = g_i*
45      M{u_{i,i} = f}     *u_{i,i} = f*
46      
47      where M{p} represents the pressure and M{u} the Darcy flux. M{k} represents the permeability,     where *p* represents the pressure and *u* the Darcy flux. *k* represents the permeability,
48      
49      @note: The problem is solved in a least squares formulation.     :cvar EVAL: direct pressure gradient evaluation for flux
50      """     :cvar POST: global postprocessing of flux by solving the PDE *K_{ij} u_j + (w * K * l u_{k,k})_{,i}= - p_{,j} + K_{ij} g_j*
51                   where *l* is the length scale, *K* is the inverse of the permeability tensor, and *w* is a positive weighting factor.
52      def __init__(self, domain,useReduced=False):     :cvar SMOOTH: global smoothing by solving the PDE *K_{ij} u_j= - p_{,j} + K_{ij} g_j*
53          """     """
54          initializes the Darcy flux problem     EVAL="EVAL"
55          @param domain: domain of the problem     SIMPLE="EVAL"
56          @type domain: L{Domain}     POST="POST"
57          """     SMOOTH="SMOOTH"
58          self.domain=domain     def __init__(self, domain, useReduced=False, solver="POST", verbose=False, w=1.):
59          self.__pde_v=LinearPDESystem(domain)        """
60          if useReduced: self.__pde_v.setReducedOrderOn()        initializes the Darcy flux problem
61          self.__pde_v.setSymmetryOn()        :param domain: domain of the problem
62          self.__pde_v.setValue(D=util.kronecker(domain), A=util.outer(util.kronecker(domain),util.kronecker(domain)))        :type domain: `Domain`
63          self.__pde_p=LinearSinglePDE(domain)        :param useReduced: uses reduced oreder on flux and pressure
64          self.__pde_p.setSymmetryOn()        :type useReduced: ``bool``
65          if useReduced: self.__pde_p.setReducedOrderOn()        :param solver: solver method
66          self.__f=Scalar(0,self.__pde_v.getFunctionSpaceForCoefficient("X"))        :type solver: in [`DarcyFlow.EVAL`, `DarcyFlow.POST',  `DarcyFlow.SMOOTH' ]
67          self.__g=Vector(0,self.__pde_v.getFunctionSpaceForCoefficient("Y"))        :param verbose: if ``True`` some information on the iteration progress are printed.
68          self.__ATOL= None        :type verbose: ``bool``
69          :param w: weighting factor for `DarcyFlow.POST` solver
70      def setValue(self,f=None, g=None, location_of_fixed_pressure=None, location_of_fixed_flux=None, permeability=None):        :type w: ``float``
71          """        
72          assigns values to model parameters        """
73          if not solver in [DarcyFlow.EVAL, DarcyFlow.POST,  DarcyFlow.SMOOTH ] :
74          @param f: volumetic sources/sinks            raise ValueError("unknown solver %d."%solver)
         @type f: scalar value on the domain (e.g. L{Data})  
         @param g: flux sources/sinks  
         @type g: vector values on the domain (e.g. L{Data})  
         @param location_of_fixed_pressure: mask for locations where pressure is fixed  
         @type location_of_fixed_pressure: scalar value on the domain (e.g. L{Data})  
         @param location_of_fixed_flux:  mask for locations where flux is fixed.  
         @type location_of_fixed_flux: vector values on the domain (e.g. L{Data})  
         @param permeability: permeability tensor. If scalar C{s} is given the tensor with  
                              C{s} on the main diagonal is used. If vector C{v} is given the tensor with  
                              C{v} on the main diagonal is used.  
         @type permeability: scalar, vector or tensor values on the domain (e.g. L{Data})  
   
         @note: the values of parameters which are not set by calling C{setValue} are not altered.  
         @note: at any point on the boundary of the domain the pressure (C{location_of_fixed_pressure} >0)  
                or the normal component of the flux (C{location_of_fixed_flux[i]>0} if direction of the normal  
                is along the M{x_i} axis.  
         """  
         if f !=None:  
            f=util.interpolate(f, self.__pde_v.getFunctionSpaceForCoefficient("X"))  
            if f.isEmpty():  
                f=Scalar(0,self.__pde_v.getFunctionSpaceForCoefficient("X"))  
            else:  
                if f.getRank()>0: raise ValueError,"illegal rank of f."  
            self.f=f  
         if g !=None:    
            g=util.interpolate(g, self.__pde_p.getFunctionSpaceForCoefficient("Y"))  
            if g.isEmpty():  
              g=Vector(0,self.__pde_v.getFunctionSpaceForCoefficient("Y"))  
            else:  
              if not g.getShape()==(self.domain.getDim(),):  
                raise ValueError,"illegal shape of g"  
            self.__g=g  
   
         if location_of_fixed_pressure!=None: self.__pde_p.setValue(q=location_of_fixed_pressure)  
         if location_of_fixed_flux!=None: self.__pde_v.setValue(q=location_of_fixed_flux)  
   
         if permeability!=None:  
            perm=util.interpolate(permeability,self.__pde_p.getFunctionSpaceForCoefficient("A"))  
            if perm.getRank()==0:  
                perm=perm*util.kronecker(self.domain.getDim())  
            elif perm.getRank()==1:  
                perm, perm2=Tensor(0.,self.__pde_p.getFunctionSpaceForCoefficient("A")), perm  
                for i in range(self.domain.getDim()): perm[i,i]=perm2[i]  
            elif perm.getRank()==2:  
               pass  
            else:  
               raise ValueError,"illegal rank of permeability."  
            self.__permeability=perm  
            self.__pde_p.setValue(A=util.transposed_tensor_mult(self.__permeability,self.__permeability))  
   
   
     def getFlux(self,p=None, fixed_flux=Data(),tol=1.e-8, show_details=False):  
         """  
         returns the flux for a given pressure C{p} where the flux is equal to C{fixed_flux}  
         on locations where C{location_of_fixed_flux} is positive (see L{setValue}).  
         Note that C{g} and C{f} are used, see L{setValue}.  
           
         @param p: pressure.  
         @type p: scalar value on the domain (e.g. L{Data}).  
         @param fixed_flux: flux on the locations of the domain marked be C{location_of_fixed_flux}.  
         @type fixed_flux: vector values on the domain (e.g. L{Data}).  
         @param tol: relative tolerance to be used.  
         @type tol: positive C{float}.  
         @return: flux  
         @rtype: L{Data}  
         @note: the method uses the least squares solution M{u=(I+D^*D)^{-1}(D^*f-g-Qp)} where M{D} is the M{div} operator and M{(Qp)_i=k_{ij}p_{,j}}  
                for the permeability M{k_{ij}}  
         """  
         self.__pde_v.setTolerance(tol)  
         g=self.__g  
         f=self.__f  
         self.__pde_v.setValue(X=f*util.kronecker(self.domain), r=fixed_flux)  
         if p == None:  
            self.__pde_v.setValue(Y=g)  
         else:  
            self.__pde_v.setValue(Y=g-util.tensor_mult(self.__permeability,util.grad(p)))  
         return self.__pde_v.getSolution(verbose=show_details)  
75    
76      def getPressure(self,v=None, fixed_pressure=Data(),tol=1.e-8, show_details=False):        self.domain=domain
77          """        self.solver=solver
78          returns the pressure for a given flux C{v} where the pressure is equal to C{fixed_pressure}        self.useReduced=useReduced
79          on locations where C{location_of_fixed_pressure} is positive (see L{setValue}).        self.verbose=verbose
80          Note that C{g} is used, see L{setValue}.        self.l=None
81          self.w=None
82        
83          self.__pde_p=LinearSinglePDE(domain)
84          self.__pde_p.setSymmetryOn()
85          if self.useReduced: self.__pde_p.setReducedOrderOn()
86    
87          if self.solver  == self.EVAL:
88             self.__pde_v=None
89             if self.verbose: print("DarcyFlow: simple solver is used.")
90    
91          elif self.solver  == self.POST:
92             if util.inf(w)<0.:
93                raise ValueError("Weighting factor must be non-negative.")
94             if self.verbose: print("DarcyFlow: global postprocessing of flux is used.")
95             self.__pde_v=LinearPDESystem(domain)
96             self.__pde_v.setSymmetryOn()
97             if self.useReduced: self.__pde_v.setReducedOrderOn()
98             self.w=w
99             x=self.domain.getX()
100             self.l=min( [util.sup(x[i])-util.inf(x[i]) for i in xrange(self.domain.getDim()) ] )
101             #self.l=util.vol(self.domain)**(1./self.domain.getDim()) # length scale
102    
103          elif self.solver  == self.SMOOTH:
104             self.__pde_v=LinearPDESystem(domain)
105             self.__pde_v.setSymmetryOn()
106             if self.useReduced: self.__pde_v.setReducedOrderOn()
107             if self.verbose: print("DarcyFlow: flux smoothing is used.")
108             self.w=0
109    
110          self.__f=escript.Scalar(0,self.__pde_p.getFunctionSpaceForCoefficient("X"))
111          self.__g=escript.Vector(0,self.__pde_p.getFunctionSpaceForCoefficient("Y"))
112          self.__permeability_invXg=escript.Vector(0,self.__pde_p.getFunctionSpaceForCoefficient("Y"))
113          self.__permeability_invXg_ref=util.numpy.zeros((self.domain.getDim()),util.numpy.float64)
114          self.ref_point_id=None
115          self.ref_point=util.numpy.zeros((self.domain.getDim()),util.numpy.float64)
116          self.location_of_fixed_pressure = escript.Scalar(0, self.__pde_p.getFunctionSpaceForCoefficient("q"))
117          self.location_of_fixed_flux = escript.Vector(0, self.__pde_p.getFunctionSpaceForCoefficient("q"))
118          self.perm_scale=1.
119        
120                    
121          @param v: flux.     def setValue(self,f=None, g=None, location_of_fixed_pressure=None, location_of_fixed_flux=None, permeability=None):
122          @type v: vector-valued on the domain (e.g. L{Data}).        """
123          @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  
124    
125          M{<(Q^*Q)^{-1} r,r> <= ATOL}        :param f: volumetic sources/sinks
126          :type f: scalar value on the domain (e.g. `escript.Data`)
127          :param g: flux sources/sinks
128          :type g: vector values on the domain (e.g. `escript.Data`)
129          :param location_of_fixed_pressure: mask for locations where pressure is fixed
130          :type location_of_fixed_pressure: scalar value on the domain (e.g. `escript.Data`)
131          :param location_of_fixed_flux:  mask for locations where flux is fixed.
132          :type location_of_fixed_flux: vector values on the domain (e.g. `escript.Data`)
133          :param permeability: permeability tensor. If scalar ``s`` is given the tensor with ``s`` on the main diagonal is used.
134          :type permeability: scalar or symmetric tensor values on the domain (e.g. `escript.Data`)
135    
136          :note: the values of parameters which are not set by calling ``setValue`` are not altered.
137          :note: at any point on the boundary of the domain the pressure
138                 (``location_of_fixed_pressure`` >0) or the normal component of the
139                 flux (``location_of_fixed_flux[i]>0``) if direction of the normal
140                 is along the *x_i* axis.
141    
142          holds. M{D} is the M{div} operator and M{(Qp)_i=k_{ij}p_{,j}} for the permeability M{k_{ij}}        """
143          if location_of_fixed_pressure!=None:
144               self.location_of_fixed_pressure=util.wherePositive(util.interpolate(location_of_fixed_pressure, self.__pde_p.getFunctionSpaceForCoefficient("q")))
145               self.ref_point_id=self.location_of_fixed_pressure.maxGlobalDataPoint()
146               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.")
147               self.ref_point=self.__pde_p.getFunctionSpaceForCoefficient("q").getX().getTupleForGlobalDataPoint(*self.ref_point_id)
148               if self.verbose: print(("DarcyFlow: reference point at %s."%(self.ref_point,)))
149               self.__pde_p.setValue(q=self.location_of_fixed_pressure)
150          if location_of_fixed_flux!=None:
151              self.location_of_fixed_flux=util.wherePositive(location_of_fixed_flux)
152              if not self.__pde_v == None:
153                  self.__pde_v.setValue(q=self.location_of_fixed_flux)
154                
155          if permeability!=None:
156        
157             perm=util.interpolate(permeability,self.__pde_p.getFunctionSpaceForCoefficient("A"))
158             self.perm_scale=util.Lsup(util.length(perm))
159             if self.verbose: print(("DarcyFlow: permeability scaling factor = %e."%self.perm_scale))
160             perm=perm*(1./self.perm_scale)
161            
162             if perm.getRank()==0:
163    
164          @param atol: absolute tolerance for the pressure              perm_inv=(1./perm)
165          @type atol: non-negative C{float}              perm_inv=perm_inv*util.kronecker(self.domain.getDim())
166          @param rtol: relative tolerance for the pressure              perm=perm*util.kronecker(self.domain.getDim())
167          @type rtol: non-negative C{float}          
168          @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          
169          L{getPressure} method or use  the value from a previous time step.           elif perm.getRank()==2:
170          @type p_ref: scalar value on the domain (e.g. L{Data}).              perm_inv=util.inverse(perm)
171          @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:
172          L{getFlux} method or use  the value from a previous time step.              raise ValueError("illegal rank of permeability.")
173          @type v_ref: vector-valued on the domain (e.g. L{Data}).          
174          @return: used absolute tolerance.           self.__permeability=perm
175          @rtype: positive C{float}           self.__permeability_inv=perm_inv
176          """      
177          g=self.__g           #====================
178          if not v_ref == None:           self.__pde_p.setValue(A=self.__permeability)
179             f1=util.integrate(util.length(util.interpolate(g-v_ref,Function(self.domain)))**2)           if self.solver  == self.EVAL:
180          else:                pass # no extra work required
181             f1=util.integrate(util.length(util.interpolate(g))**2)           elif self.solver  == self.POST:
182          if not p_ref == None:                k=util.kronecker(self.domain.getDim())
183             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()
184                  #self.__pde_v.setValue(D=self.__permeability_inv, A=self.omega*util.outer(k,k))
185                  self.__pde_v.setValue(D=self.__permeability_inv, A_reduced=self.omega*util.outer(k,k))
186             elif self.solver  == self.SMOOTH:
187                self.__pde_v.setValue(D=self.__permeability_inv)
188    
189          if g != None:
190            g=util.interpolate(g, self.__pde_p.getFunctionSpaceForCoefficient("Y"))
191            if g.isEmpty():
192                 g=Vector(0,self.__pde_p.getFunctionSpaceForCoefficient("Y"))
193          else:          else:
194             f2=0               if not g.getShape()==(self.domain.getDim(),): raise ValueError("illegal shape of g")
195          self.__ATOL= atol + rtol * util.sqrt(max(f1,f2))          self.__g=g
196          if self.__ATOL<=0:          self.__permeability_invXg=util.tensor_mult(self.__permeability_inv,self.__g * (1./self.perm_scale ))
197             raise ValueError,"Positive tolerance (=%e) is expected."%self.__ATOL          self.__permeability_invXg_ref=util.integrate(self.__permeability_invXg)/util.vol(self.domain)
198          return self.__ATOL        if f !=None:
199                     f=util.interpolate(f, self.__pde_p.getFunctionSpaceForCoefficient("Y"))
200      def getTolerance(self):           if f.isEmpty():      
201          """               f=Scalar(0,self.__pde_p.getFunctionSpaceForCoefficient("Y"))
202          returns the current tolerance.           else:
203                   if f.getRank()>0: raise ValueError("illegal rank of f.")
204          @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  
205    
206           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):
207                  """
208           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
209           PDEs with operator M{I+D^*D} and with M{Q^*Q} needs to be solved using a sub iteration scheme.        :return: `SolverOptions`
210           """        """
211           self.verbose=verbose        if self.__pde_v == None:
212           self.show_details= show_details and self.verbose            return None
213           self.__pde_v.setTolerance(sub_rtol)        else:
214           self.__pde_p.setTolerance(sub_rtol)            return self.__pde_v.getSolverOptions()
215           ATOL=self.getTolerance()        
216           if self.verbose: print "DarcyFlux: absolute tolerance = %e"%ATOL     def setSolverOptionsFlux(self, options=None):
217           #########################################################################################################################        """
218           #        Sets the solver options used to solve the flux problems
219           #   we solve:        If ``options`` is not present, the options are reset to default
220           #          :param options: `SolverOptions`
221           #      Q^*(I-(I+D^*D)^{-1})Q dp =  Q^* (g-u0-Qp0 - (I+D^*D)^{-1} ( D^*(f-Du0)+g-u0-Qp0) )        """
222           #        if not self.__pde_v == None:
223           #   residual is            self.__pde_v.setSolverOptions(options)
224           #      
225           #    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):
226           #        """
227           #        with v = (I+D^*D)^{-1} (D^*f+g-Qp) including BC        Returns the solver options used to solve the pressure problems
228           #        :return: `SolverOptions`
229           #    we use (g - Qp, v) to represent the residual. not that        """
230           #        return self.__pde_p.getSolverOptions()
231           #    dr(dp)=( -Q(dp), dv) with dv = - (I+D^*D)^{-1} Q(dp)        
232           #     def setSolverOptionsPressure(self, options=None):
233           #   while the initial residual is        """
234           #        Sets the solver options used to solve the pressure problems
235           #      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
236           #          
237           d0=self.__g-util.tensor_mult(self.__permeability,util.grad(p0))        :param options: `SolverOptions`
238           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.
239           v00=self.__pde_v.getSolution(verbose=show_details)        """
240           if self.verbose: print "DarcyFlux: range of initial flux = ",util.inf(v00), util.sup(v00)        return self.__pde_p.setSolverOptions(options)
241           self.__pde_v.setValue(r=Data())        
242           # start CG     def solve(self, u0, p0):
243           r=ArithmeticTuple(d0, v00)        """
244           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.
245           return r[1],p        
246          :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.
247      def __Aprod_PCG(self,dp):        :type u0: vector value on the domain (e.g. `escript.Data`).
248            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.
249            #  -dr(dp) = (Qdp,du) where du = (I+D^*D)^{-1} (Qdp)        :type p0: scalar value on the domain (e.g. `escript.Data`).
250            mQdp=util.tensor_mult(self.__permeability,util.grad(dp))        :return: flux and pressure
251            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)  
252    
 class StokesProblemCartesian(HomogeneousSaddlePointProblem):  
253        """        """
254        solves        p0=util.interpolate(p0, self.__pde_p.getFunctionSpaceForCoefficient("q"))
255          if self.ref_point_id == None:
256              p_ref=0
257          else:
258              p_ref=p0.getTupleForGlobalDataPoint(*self.ref_point_id)[0]
259          p0_hydrostatic=p_ref+util.inner(self.__permeability_invXg_ref, self.__pde_p.getFunctionSpaceForCoefficient("q").getX() - self.ref_point)
260          g_2=self.__g - util.tensor_mult(self.__permeability, self.__permeability_invXg_ref * self.perm_scale)
261          self.__pde_p.setValue(X=g_2 * 1./self.perm_scale,
262                                Y=self.__f * 1./self.perm_scale,
263                                y= - util.inner(self.domain.getNormal(),u0 * self.location_of_fixed_flux * 1./self.perm_scale ),
264                                r=p0 - p0_hydrostatic)
265          pp=self.__pde_p.getSolution()
266          u = self._getFlux(pp, u0)
267          return u,pp + p0_hydrostatic
268          
269       def getFlux(self,p, u0=None):
270            """
271            returns the flux for a given pressure ``p`` where the flux is equal to ``u0``
272            on locations where ``location_of_fixed_flux`` is positive (see `setValue`).
273            Notice that ``g`` is used, see `setValue`.
274    
275            :param p: pressure.
276            :type p: scalar value on the domain (e.g. `escript.Data`).
277            :param u0: flux on the locations of the domain marked be ``location_of_fixed_flux``.
278            :type u0: vector values on the domain (e.g. `escript.Data`) or ``None``
279            :return: flux
280            :rtype: `escript.Data`
281            """
282            p=util.interpolate(p, self.__pde_p.getFunctionSpaceForCoefficient("q"))
283            if self.ref_point_id == None:
284                p_ref=0
285            else:
286                p_ref=p.getTupleForGlobalDataPoint(*self.ref_point_id)[0]
287            p_hydrostatic=p_ref+util.inner(self.__permeability_invXg_ref, self.__pde_p.getFunctionSpaceForCoefficient("q").getX() - self.ref_point)
288            return self._getFlux(p-p_hydrostatic, u0)
289    
290       def _getFlux(self,pp, u0=None):
291            """
292            returns the flux for a given pressure ``p`` where the flux is equal to ``u0``
293            on locations where ``location_of_fixed_flux`` is positive (see `setValue`).
294            Notice that ``g`` is used, see `setValue`.
295    
296            :param p: pressure.
297            :type p: scalar value on the domain (e.g. `escript.Data`).
298            :param u0: flux on the locations of the domain marked be ``location_of_fixed_flux``.
299            :type u0: vector values on the domain (e.g. `escript.Data`) or ``None``
300            :return: flux
301            :rtype: `escript.Data`
302            """
303            if self.solver  == self.EVAL:
304               u = self.__g - util.tensor_mult(self.__permeability, self.perm_scale * (util.grad(pp) + self.__permeability_invXg_ref))
305            elif self.solver  == self.POST or self.solver  == self.SMOOTH:
306                self.__pde_v.setValue(Y= self.__permeability_invXg - (util.grad(pp) + self.__permeability_invXg_ref))
307                print
308                if u0 == None:
309                   self.__pde_v.setValue(r=escript.Data())
310                else:
311                   if not isinstance(u0, escript.Data) : u0 = escript.Vector(u0, escript.Solution(self.domain))
312                   self.__pde_v.setValue(r=1./self.perm_scale * u0)
313                u= self.__pde_v.getSolution() * self.perm_scale
314            return u
315          
316    class StokesProblemCartesian(HomogeneousSaddlePointProblem):
317         """
318         solves
319    
320            -(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}
321                  u_{i,i}=0                  u_{i,i}=0
# Line 333  class StokesProblemCartesian(Homogeneous Line 323  class StokesProblemCartesian(Homogeneous
323            u=0 where  fixed_u_mask>0            u=0 where  fixed_u_mask>0
324            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
325    
326        if surface_stress is not given 0 is assumed.       if surface_stress is not given 0 is assumed.
327    
328        typical usage:       typical usage:
329    
330              sp=StokesProblemCartesian(domain)              sp=StokesProblemCartesian(domain)
331              sp.setTolerance()              sp.setTolerance()
332              sp.initialize(...)              sp.initialize(...)
333              v,p=sp.solve(v0,p0)              v,p=sp.solve(v0,p0)
334        """              sp.setStokesEquation(...) # new values for some parameters
335        def __init__(self,domain,**kwargs):              v1,p1=sp.solve(v,p)
336         """
337         def __init__(self,domain,**kwargs):
338           """           """
339           initialize the Stokes Problem           initialize the Stokes Problem
340    
341           @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
342           @type domain: L{Domain}           LBB complient, for instance using quadratic and linear approximation on the same element or using linear approximation
343           @warning: The apprximation order needs to be two otherwise you may see oscilations in the pressure.           with macro elements for the pressure.
344    
345             :param domain: domain of the problem.
346             :type domain: `Domain`
347           """           """
348           HomogeneousSaddlePointProblem.__init__(self,**kwargs)           HomogeneousSaddlePointProblem.__init__(self,**kwargs)
349           self.domain=domain           self.domain=domain
350           self.vol=util.integrate(1.,Function(self.domain))           self.__pde_v=LinearPDE(domain,numEquations=self.domain.getDim(),numSolutions=self.domain.getDim())
351           self.__pde_u=LinearPDE(domain,numEquations=self.domain.getDim(),numSolutions=self.domain.getDim())           self.__pde_v.setSymmetryOn()
352           self.__pde_u.setSymmetryOn()      
          # self.__pde_u.setSolverMethod(self.__pde_u.DIRECT)  
          # self.__pde_u.setSolverMethod(preconditioner=LinearPDE.RILU)  
               
353           self.__pde_prec=LinearPDE(domain)           self.__pde_prec=LinearPDE(domain)
354           self.__pde_prec.setReducedOrderOn()           self.__pde_prec.setReducedOrderOn()
          # self.__pde_prec.setSolverMethod(self.__pde_prec.LUMPING)  
355           self.__pde_prec.setSymmetryOn()           self.__pde_prec.setSymmetryOn()
356    
357           self.__pde_proj=LinearPDE(domain)           self.__pde_proj=LinearPDE(domain)
358           self.__pde_proj.setReducedOrderOn()           self.__pde_proj.setReducedOrderOn()
359             self.__pde_proj.setValue(D=1)
360           self.__pde_proj.setSymmetryOn()           self.__pde_proj.setSymmetryOn()
          self.__pde_proj.setValue(D=1.)  
361    
362        def initialize(self,f=Data(),fixed_u_mask=Data(),eta=1,surface_stress=Data(),stress=Data()):       def getSolverOptionsVelocity(self):
363             """
364         returns the solver options used  solve the equation for velocity.
365        
366         :rtype: `SolverOptions`
367         """
368             return self.__pde_v.getSolverOptions()
369         def setSolverOptionsVelocity(self, options=None):
370             """
371         set the solver options for solving the equation for velocity.
372        
373         :param options: new solver  options
374         :type options: `SolverOptions`
375         """
376             self.__pde_v.setSolverOptions(options)
377         def getSolverOptionsPressure(self):
378             """
379         returns the solver options used  solve the equation for pressure.
380         :rtype: `SolverOptions`
381         """
382             return self.__pde_prec.getSolverOptions()
383         def setSolverOptionsPressure(self, options=None):
384             """
385         set the solver options for solving the equation for pressure.
386         :param options: new solver  options
387         :type options: `SolverOptions`
388         """
389             self.__pde_prec.setSolverOptions(options)
390    
391         def setSolverOptionsDiv(self, options=None):
392             """
393         set the solver options for solving the equation to project the divergence of
394         the velocity onto the function space of presure.
395        
396         :param options: new solver options
397         :type options: `SolverOptions`
398         """
399             self.__pde_proj.setSolverOptions(options)
400         def getSolverOptionsDiv(self):
401             """
402         returns the solver options for solving the equation to project the divergence of
403         the velocity onto the function space of presure.
404        
405         :rtype: `SolverOptions`
406         """
407             return self.__pde_proj.getSolverOptions()
408    
409         def updateStokesEquation(self, v, p):
410             """
411             updates the Stokes equation to consider dependencies from ``v`` and ``p``
412             :note: This method can be overwritten by a subclass. Use `setStokesEquation` to set new values to model parameters.
413             """
414             pass
415         def setStokesEquation(self, f=None,fixed_u_mask=None,eta=None,surface_stress=None,stress=None, restoration_factor=None):
416            """
417            assigns new values to the model parameters.
418    
419            :param f: external force
420            :type f: `Vector` object in `FunctionSpace` `Function` or similar
421            :param fixed_u_mask: mask of locations with fixed velocity.
422            :type fixed_u_mask: `Vector` object on `FunctionSpace` `Solution` or similar
423            :param eta: viscosity
424            :type eta: `Scalar` object on `FunctionSpace` `Function` or similar
425            :param surface_stress: normal surface stress
426            :type surface_stress: `Vector` object on `FunctionSpace` `FunctionOnBoundary` or similar
427            :param stress: initial stress
428        :type stress: `Tensor` object on `FunctionSpace` `Function` or similar
429            """
430            if eta !=None:
431                k=util.kronecker(self.domain.getDim())
432                kk=util.outer(k,k)
433                self.eta=util.interpolate(eta, escript.Function(self.domain))
434                self.__pde_prec.setValue(D=1/self.eta)
435                self.__pde_v.setValue(A=self.eta*(util.swap_axes(kk,0,3)+util.swap_axes(kk,1,3)))
436            if restoration_factor!=None:
437                n=self.domain.getNormal()
438                self.__pde_v.setValue(d=restoration_factor*util.outer(n,n))
439            if fixed_u_mask!=None:
440                self.__pde_v.setValue(q=fixed_u_mask)
441            if f!=None: self.__f=f
442            if surface_stress!=None: self.__surface_stress=surface_stress
443            if stress!=None: self.__stress=stress
444    
445         def initialize(self,f=escript.Data(),fixed_u_mask=escript.Data(),eta=1,surface_stress=escript.Data(),stress=escript.Data(), restoration_factor=0):
446          """          """
447          assigns values to the model parameters          assigns values to the model parameters
448    
449          @param f: external force          :param f: external force
450          @type f: L{Vector} object in L{FunctionSpace} L{Function} or similar          :type f: `Vector` object in `FunctionSpace` `Function` or similar
451          @param fixed_u_mask: mask of locations with fixed velocity.          :param fixed_u_mask: mask of locations with fixed velocity.
452          @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
453          @param eta: viscosity          :param eta: viscosity
454          @type eta: L{Scalar} object on L{FunctionSpace} L{Function} or similar          :type eta: `Scalar` object on `FunctionSpace` `Function` or similar
455          @param surface_stress: normal surface stress          :param surface_stress: normal surface stress
456          @type eta: L{Vector} object on L{FunctionSpace} L{FunctionOnBoundary} or similar          :type surface_stress: `Vector` object on `FunctionSpace` `FunctionOnBoundary` or similar
457          @param stress: initial stress          :param stress: initial stress
458      @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.  
   
459          """          """
460          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  
461    
462        def B(self,v):       def Bv(self,v,tol):
463          """           """
464          returns div(v)           returns inner product of element p and div(v)
         @rtype: equal to the type of p  
465    
466          @note: boundary conditions on p should be zero!           :param v: a residual
467          """           :return: inner product of element p and div(v)
468          if self.show_details: print "apply divergence:"           :rtype: ``float``
469          self.__pde_proj.setValue(Y=-util.div(v))           """
470          self.__pde_proj.setTolerance(self.getSubProblemTolerance())           self.__pde_proj.setValue(Y=-util.div(v))
471          return self.__pde_proj.getSolution(verbose=self.show_details)           self.getSolverOptionsDiv().setTolerance(tol)
472             self.getSolverOptionsDiv().setAbsoluteTolerance(0.)
473             out=self.__pde_proj.getSolution()
474             return out
475    
476        def inner_pBv(self,p,Bv):       def inner_pBv(self,p,Bv):
477           """           """
478           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}  
479    
480           @rtype: equal to the type of p           :param p: a pressure increment
481             :param Bv: a residual
482             :return: inner product of element p and Bv=-div(v)
483             :rtype: ``float``
484           """           """
485           s0=util.interpolate(p,Function(self.domain))           return util.integrate(util.interpolate(p,escript.Function(self.domain))*util.interpolate(Bv, escript.Function(self.domain)))
          s1=util.interpolate(Bv,Function(self.domain))  
          return util.integrate(s0*s1)  
486    
487        def inner_p(self,p0,p1):       def inner_p(self,p0,p1):
488           """           """
489           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}  
490    
491           @rtype: equal to the type of p           :param p0: a pressure
492             :param p1: a pressure
493             :return: inner product of p0 and p1
494             :rtype: ``float``
495           """           """
496           s0=util.interpolate(p0/self.eta,Function(self.domain))           s0=util.interpolate(p0, escript.Function(self.domain))
497           s1=util.interpolate(p1/self.eta,Function(self.domain))           s1=util.interpolate(p1, escript.Function(self.domain))
498           return util.integrate(s0*s1)           return util.integrate(s0*s1)
499    
500        def inner_v(self,v0,v1):       def norm_v(self,v):
501           """           """
502           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}  
503    
504           @rtype: equal to the type of v           :param v: a velovity
505             :return: norm of v
506             :rtype: non-negative ``float``
507           """           """
508       gv0=util.grad(v0)           return util.sqrt(util.integrate(util.length(util.grad(v))**2))
509       gv1=util.grad(v1)  
          return util.integrate(util.inner(gv0,gv1))  
510    
511        def solve_A(self,u,p):       def getDV(self, p, v, tol):
512           """           """
513           solves Av=f-Au-B^*p (v=0 on fixed_u_mask)           return the value for v for a given p
514    
515             :param p: a pressure
516             :param v: a initial guess for the value v to return.
517             :return: dv given as *Adv=(f-Av-B^*p)*
518           """           """
519           if self.show_details: print "solve for velocity:"           self.updateStokesEquation(v,p)
520           self.__pde_u.setTolerance(self.getSubProblemTolerance())           self.__pde_v.setValue(Y=self.__f, y=self.__surface_stress)
521             self.getSolverOptionsVelocity().setTolerance(tol)
522             self.getSolverOptionsVelocity().setAbsoluteTolerance(0.)
523           if self.__stress.isEmpty():           if self.__stress.isEmpty():
524              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)))
525           else:           else:
526              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)))
527           out=self.__pde_u.getSolution(verbose=self.show_details)           out=self.__pde_v.getSolution()
528           return  out           return  out
529    
530        def solve_prec(self,p):       def norm_Bv(self,Bv):
531           if self.show_details: print "apply preconditioner:"          """
532           self.__pde_prec.setTolerance(self.getSubProblemTolerance())          Returns Bv (overwrite).
533           self.__pde_prec.setValue(Y=p)  
534           q=self.__pde_prec.getSolution(verbose=self.show_details)          :rtype: equal to the type of p
535           return q          :note: boundary conditions on p should be zero!
536            """
537            return util.sqrt(util.integrate(util.interpolate(Bv, escript.Function(self.domain))**2))
538    
539         def solve_AinvBt(self,p, tol):
540             """
541             Solves *Av=B^*p* with accuracy `tol`
542    
543             :param p: a pressure increment
544             :return: the solution of *Av=B^*p*
545             :note: boundary conditions on v should be zero!
546             """
547             self.__pde_v.setValue(Y=escript.Data(), y=escript.Data(), X=-p*util.kronecker(self.domain))
548             out=self.__pde_v.getSolution()
549             return  out
550    
551         def solve_prec(self,Bv, tol):
552             """
553             applies preconditioner for for *BA^{-1}B^** to *Bv*
554             with accuracy `self.getSubProblemTolerance()`
555    
556             :param Bv: velocity increment
557             :return: *p=P(Bv)* where *P^{-1}* is an approximation of *BA^{-1}B^ * )*
558             :note: boundary conditions on p are zero.
559             """
560             self.__pde_prec.setValue(Y=Bv)
561             self.getSolverOptionsPressure().setTolerance(tol)
562             self.getSolverOptionsPressure().setAbsoluteTolerance(0.)
563             out=self.__pde_prec.getSolution()
564             return out

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