# Diff of /trunk/escript/py_src/flows.py

revision 2208 by gross, Mon Jan 12 06:37:07 2009 UTC revision 2486 by gross, Tue Jun 23 03:38:54 2009 UTC
# Line 16  http://www.uq.edu.au/esscc Line 16  http://www.uq.edu.au/esscc
20
21  """  """
22  Some models for flow  Some models for flow
# Line 33  __author__="Lutz Gross, l.gross@uq.edu.a Line 33  __author__="Lutz Gross, l.gross@uq.edu.a
33
34  from escript import *  from escript import *
35  import util  import util
36  from linearPDEs import LinearPDE, LinearPDESystem, LinearSinglePDE  from linearPDEs import LinearPDE, LinearPDESystem, LinearSinglePDE, SolverOptions
37  from pdetools import HomogeneousSaddlePointProblem,Projector, ArithmeticTuple, PCG, NegativeNorm  from pdetools import HomogeneousSaddlePointProblem,Projector, ArithmeticTuple, PCG, NegativeNorm, GMRES
38
39  class DarcyFlow(object):  class DarcyFlow(object):
40      """      """
41      solves the problem      solves the problem
42
43      M{u_i+k_{ij}*p_{,j} = g_i}      M{u_i+k_{ij}*p_{,j} = g_i}
44      M{u_{i,i} = f}      M{u_{i,i} = f}
45
46      where M{p} represents the pressure and M{u} the Darcy flux. M{k} represents the permeability,      where M{p} represents the pressure and M{u} the Darcy flux. M{k} represents the permeability,
47
48      @note: The problem is solved in a least squares formulation.      @note: The problem is solved in a least squares formulation.
49      """      """
50
51      def __init__(self, domain,useReduced=False):      def __init__(self, domain, weight=None, useReduced=False, adaptSubTolerance=True):
52          """          """
53          initializes the Darcy flux problem          initializes the Darcy flux problem
54          @param domain: domain of the problem          @param domain: domain of the problem
55          @type domain: L{Domain}          @type domain: L{Domain}
56        @param useReduced: uses reduced oreder on flux and pressure
57        @type useReduced: C{bool}
58        @param adaptSubTolerance: switches on automatic subtolerance selection
60          """          """
61          self.domain=domain          self.domain=domain
62            if weight == None:
63               s=self.domain.getSize()
64               self.__l=(3.*util.longestEdge(self.domain)*s/util.sup(s))**2
65               # self.__l=(3.*util.longestEdge(self.domain))**2
66               # self.__l=(0.1*util.longestEdge(self.domain)*s/util.sup(s))**2
67            else:
68               self.__l=weight
69          self.__pde_v=LinearPDESystem(domain)          self.__pde_v=LinearPDESystem(domain)
70          if useReduced: self.__pde_v.setReducedOrderOn()          if useReduced: self.__pde_v.setReducedOrderOn()
71          self.__pde_v.setSymmetryOn()          self.__pde_v.setSymmetryOn()
72          self.__pde_v.setValue(D=util.kronecker(domain), A=util.outer(util.kronecker(domain),util.kronecker(domain)))          self.__pde_v.setValue(D=util.kronecker(domain), A=self.__l*util.outer(util.kronecker(domain),util.kronecker(domain)))
73          self.__pde_p=LinearSinglePDE(domain)          self.__pde_p=LinearSinglePDE(domain)
74          self.__pde_p.setSymmetryOn()          self.__pde_p.setSymmetryOn()
75          if useReduced: self.__pde_p.setReducedOrderOn()          if useReduced: self.__pde_p.setReducedOrderOn()
76          self.__f=Scalar(0,self.__pde_v.getFunctionSpaceForCoefficient("X"))          self.__f=Scalar(0,self.__pde_v.getFunctionSpaceForCoefficient("X"))
77          self.__g=Vector(0,self.__pde_v.getFunctionSpaceForCoefficient("Y"))          self.__g=Vector(0,self.__pde_v.getFunctionSpaceForCoefficient("Y"))
78          self.__ATOL= None          self.setTolerance()
79            self.setAbsoluteTolerance()
81        self.verbose=False
82        def getSolverOptionsFlux(self):
83        """
84        Returns the solver options used to solve the flux problems
85
86        M{(I+D^*D)u=F}
87
88        @return: L{SolverOptions}
89        """
90        return self.__pde_v.getSolverOptions()
91        def setSolverOptionsFlux(self, options=None):
92        """
93        Sets the solver options used to solve the flux problems
94
95        M{(I+D^*D)u=F}
96
97        If C{options} is not present, the options are reset to default
98        @param options: L{SolverOptions}
99        @note: if the adaption of subtolerance is choosen, the tolerance set by C{options} will be overwritten before the solver is called.
100        """
101        return self.__pde_v.setSolverOptions(options)
102        def getSolverOptionsPressure(self):
103        """
104        Returns the solver options used to solve the pressure problems
105
106        M{(Q^*Q)p=Q^*G}
107
108        @return: L{SolverOptions}
109        """
110        return self.__pde_p.getSolverOptions()
111        def setSolverOptionsPressure(self, options=None):
112        """
113        Sets the solver options used to solve the pressure problems
114
115        M{(Q^*Q)p=Q^*G}
116
117        If C{options} is not present, the options are reset to default
118        @param options: L{SolverOptions}
119        @note: if the adaption of subtolerance is choosen, the tolerance set by C{options} will be overwritten before the solver is called.
120        """
121        return self.__pde_p.setSolverOptions(options)
122
123      def setValue(self,f=None, g=None, location_of_fixed_pressure=None, location_of_fixed_flux=None, permeability=None):      def setValue(self,f=None, g=None, location_of_fixed_pressure=None, location_of_fixed_flux=None, permeability=None):
124          """          """
# Line 78  class DarcyFlow(object): Line 132  class DarcyFlow(object):
132          @type location_of_fixed_pressure: scalar value on the domain (e.g. L{Data})          @type location_of_fixed_pressure: scalar value on the domain (e.g. L{Data})
133          @param location_of_fixed_flux:  mask for locations where flux is fixed.          @param location_of_fixed_flux:  mask for locations where flux is fixed.
134          @type location_of_fixed_flux: vector values on the domain (e.g. L{Data})          @type location_of_fixed_flux: vector values on the domain (e.g. L{Data})
135          @param permeability: permeability tensor. If scalar C{s} is given the tensor with          @param permeability: permeability tensor. If scalar C{s} is given the tensor with
136                               C{s} on the main diagonal is used. If vector C{v} is given the tensor with                               C{s} on the main diagonal is used. If vector C{v} is given the tensor with
137                               C{v} on the main diagonal is used.                               C{v} on the main diagonal is used.
138          @type permeability: scalar, vector or tensor values on the domain (e.g. L{Data})          @type permeability: scalar, vector or tensor values on the domain (e.g. L{Data})
139
# Line 88  class DarcyFlow(object): Line 142  class DarcyFlow(object):
142                 or the normal component of the flux (C{location_of_fixed_flux[i]>0} if direction of the normal                 or the normal component of the flux (C{location_of_fixed_flux[i]>0} if direction of the normal
143                 is along the M{x_i} axis.                 is along the M{x_i} axis.
144          """          """
145          if f !=None:          if f !=None:
146             f=util.interpolate(f, self.__pde_v.getFunctionSpaceForCoefficient("X"))             f=util.interpolate(f, self.__pde_v.getFunctionSpaceForCoefficient("X"))
147             if f.isEmpty():             if f.isEmpty():
148                 f=Scalar(0,self.__pde_v.getFunctionSpaceForCoefficient("X"))                 f=Scalar(0,self.__pde_v.getFunctionSpaceForCoefficient("X"))
149             else:             else:
150                 if f.getRank()>0: raise ValueError,"illegal rank of f."                 if f.getRank()>0: raise ValueError,"illegal rank of f."
151             self.f=f             self.__f=f
152          if g !=None:            if g !=None:
153             g=util.interpolate(g, self.__pde_p.getFunctionSpaceForCoefficient("Y"))             g=util.interpolate(g, self.__pde_p.getFunctionSpaceForCoefficient("Y"))
154             if g.isEmpty():             if g.isEmpty():
155               g=Vector(0,self.__pde_v.getFunctionSpaceForCoefficient("Y"))               g=Vector(0,self.__pde_v.getFunctionSpaceForCoefficient("Y"))
# Line 121  class DarcyFlow(object): Line 175  class DarcyFlow(object):
175             self.__permeability=perm             self.__permeability=perm
176             self.__pde_p.setValue(A=util.transposed_tensor_mult(self.__permeability,self.__permeability))             self.__pde_p.setValue(A=util.transposed_tensor_mult(self.__permeability,self.__permeability))
177
178        def setTolerance(self,rtol=1e-4):
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}}
179          """          """
180          self.__pde_v.setTolerance(tol)          sets the relative tolerance C{rtol} used to terminate the solution process. The iteration is terminated if
181          g=self.__g
182          f=self.__f          M{|g-v-Qp| <= atol + rtol * min( max( |g-v|, |Qp| ), max( |v|, |g-Qp| ) ) }
self.__pde_v.setValue(X=f*util.kronecker(self.domain), r=fixed_flux)
if p == None:
self.__pde_v.setValue(Y=g)
else:
return self.__pde_v.getSolution(verbose=show_details)
183
184      def getPressure(self,v=None, fixed_pressure=Data(),tol=1.e-8, show_details=False):          where C{atol} is an absolut tolerance (see L{setAbsoluteTolerance}), M{|f|^2 = integrate(length(f)^2)} and M{(Qp)_i=k_{ij}p_{,j}} for the permeability M{k_{ij}}.
185
186            @param rtol: relative tolerance for the pressure
187            @type rtol: non-negative C{float}
188          """          """
189          returns the pressure for a given flux C{v} where the pressure is equal to C{fixed_pressure}          if rtol<0:
190          on locations where C{location_of_fixed_pressure} is positive (see L{setValue}).              raise ValueError,"Relative tolerance needs to be non-negative."
191          Note that C{g} is used, see L{setValue}.          self.__rtol=rtol
192                def getTolerance(self):
@param v: flux.
@type v: vector-valued on the domain (e.g. L{Data}).
@param fixed_pressure: pressure on the locations of the domain marked be C{location_of_fixed_pressure}.
@type fixed_pressure: vector values on the domain (e.g. L{Data}).
@param tol: relative tolerance to be used.
@type tol: positive C{float}.
@return: pressure
@rtype: L{Data}
@note: the method uses the least squares solution M{p=(Q^*Q)^{-1}Q^*(g-u)} where and M{(Qp)_i=k_{ij}p_{,j}}
for the permeability M{k_{ij}}
193          """          """
194          self.__pde_v.setTolerance(tol)          returns the relative tolerance
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)
195
196      def setTolerance(self,atol=0,rtol=1e-8,p_ref=None,v_ref=None):          @return: current relative tolerance
197            @rtype: C{float}
198          """          """
199          set the tolerance C{ATOL} used to terminate the solution process. It is used          return self.__rtol

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
200
201          M{r=Q^*(g-Qp-v)}      def setAbsoluteTolerance(self,atol=0.):
202            """
203          the condition          sets the absolute tolerance C{atol} used to terminate the solution process. The iteration is terminated if
204
205          M{<(Q^*Q)^{-1} r,r> <= ATOL}          M{|g-v-Qp| <= atol + rtol * min( max( |g-v|, |Qp| ), max( |v|, |g-Qp| ) ) }
206
207          holds. M{D} is the M{div} operator and M{(Qp)_i=k_{ij}p_{,j}} for the permeability M{k_{ij}}          where C{rtol} is an absolut tolerance (see L{setTolerance}), M{|f|^2 = integrate(length(f)^2)} and M{(Qp)_i=k_{ij}p_{,j}} for the permeability M{k_{ij}}.
208
209          @param atol: absolute tolerance for the pressure          @param atol: absolute tolerance for the pressure
210          @type atol: non-negative C{float}          @type atol: non-negative C{float}
@param rtol: relative tolerance for the pressure
@type rtol: non-negative C{float}
@param p_ref: reference pressure. If not present zero is used. You may use physical arguments to set a resonable value for C{p_ref}, use the
L{getPressure} method or use  the value from a previous time step.
@type p_ref: scalar value on the domain (e.g. L{Data}).
@param v_ref: reference velocity.  If not present zero is used. You may use physical arguments to set a resonable value for C{v_ref}, use the
L{getFlux} method or use  the value from a previous time step.
@type v_ref: vector-valued on the domain (e.g. L{Data}).
@return: used absolute tolerance.
@rtype: positive C{float}
211          """          """
212          g=self.__g          if atol<0:
213          if not v_ref == None:              raise ValueError,"Absolute tolerance needs to be non-negative."
214             f1=util.integrate(util.length(util.interpolate(g-v_ref,Function(self.domain)))**2)          self.__atol=atol
215          else:      def getAbsoluteTolerance(self):
216             f1=util.integrate(util.length(util.interpolate(g))**2)         """
217          if not p_ref == None:         returns the absolute tolerance
219          else:         @return: current absolute tolerance
220             f2=0         @rtype: C{float}
221          self.__ATOL= atol + rtol * util.sqrt(max(f1,f2))         """
222          if self.__ATOL<=0:         return self.__atol
223             raise ValueError,"Positive tolerance (=%e) is expected."%self.__ATOL      def getSubProblemTolerance(self):
224          return self.__ATOL      """
225                Returns a suitable subtolerance
226      def getTolerance(self):      @type: C{float}
227          """      """
228          returns the current tolerance.      return max(util.EPSILON**(0.75),self.getTolerance()**2)
229          def setSubProblemTolerance(self):
230          @return: used absolute tolerance.           """
231          @rtype: positive C{float}           Sets the relative tolerance to solve the subproblem(s) if subtolerance adaption is selected.
232          """           """
234             raise ValueError,"no tolerance is defined."           sub_tol=self.getSubProblemTolerance()
235          return self.__ATOL               self.getSolverOptionsFlux().setTolerance(sub_tol)
236             self.getSolverOptionsFlux().setAbsoluteTolerance(0.)
237             self.getSolverOptionsPressure().setTolerance(sub_tol)
238             self.getSolverOptionsPressure().setAbsoluteTolerance(0.)
239             if self.verbose: print "DarcyFlux: relative subtolerance is set to %e."%sub_tol
240
241      def solve(self,u0,p0, max_iter=100, verbose=False, show_details=False, sub_rtol=1.e-8):      def solve(self,u0,p0, max_iter=100, verbose=False, max_num_corrections=10):
242           """           """
243           solves the problem.           solves the problem.
244
245           The iteration is terminated if the residual norm is less then self.getTolerance().           The iteration is terminated if the residual norm is less then self.getTolerance().
246
247           @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.           @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.
248           @type u0: vector value on the domain (e.g. L{Data}).           @type u0: vector value on the domain (e.g. L{Data}).
249           @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.           @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.
250           @type p0: scalar value on the domain (e.g. L{Data}).           @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}
251           @param verbose: if set some information on iteration progress are printed           @param verbose: if set some information on iteration progress are printed
252           @type verbose: C{bool}           @type verbose: C{bool}
@param show_details:  if set information on the subiteration process are printed.
@type show_details: C{bool}
253           @return: flux and pressure           @return: flux and pressure
254           @rtype: C{tuple} of L{Data}.           @rtype: C{tuple} of L{Data}.
255
256           @note: The problem is solved as a least squares form           @note: The problem is solved as a least squares form
257
258           M{(I+D^*D)u+Qp=D^*f+g}           M{(I+D^*D)u+Qp=D^*f+g}
259           M{Q^*u+Q^*Qp=Q^*g}           M{Q^*u+Q^*Qp=Q^*g}
260
261           where M{D} is the M{div} operator and M{(Qp)_i=k_{ij}p_{,j}} for the permeability M{k_{ij}}.           where M{D} is the M{div} operator and M{(Qp)_i=k_{ij}p_{,j}} for the permeability M{k_{ij}}.
262           We eliminate the flux form the problem by setting           We eliminate the flux form the problem by setting
263
264           M{u=(I+D^*D)^{-1}(D^*f-g-Qp)} with u=u0 on location_of_fixed_flux           M{u=(I+D^*D)^{-1}(D^*f-g-Qp)} with u=u0 on location_of_fixed_flux
265
266           form the first equation. Inserted into the second equation we get           form the first equation. Inserted into the second equation we get
267
268           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           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
269
270           which is solved using the PCG method (precondition is M{Q^*Q}). In each iteration step           which is solved using the PCG method (precondition is M{Q^*Q}). In each iteration step
271           PDEs with operator M{I+D^*D} and with M{Q^*Q} needs to be solved using a sub iteration scheme.           PDEs with operator M{I+D^*D} and with M{Q^*Q} needs to be solved using a sub iteration scheme.
272           """           """
273           self.verbose=verbose           self.verbose=verbose
274           self.show_details= show_details and self.verbose           rtol=self.getTolerance()
275           self.__pde_v.setTolerance(sub_rtol)           atol=self.getAbsoluteTolerance()
276           self.__pde_p.setTolerance(sub_rtol)       self.setSubProblemTolerance()
277           ATOL=self.getTolerance()
278           if self.verbose: print "DarcyFlux: absolute tolerance = %e"%ATOL           num_corrections=0
279           #########################################################################################################################           converged=False
280           #           p=p0
281           #   we solve:           norm_r=None
282           #             while not converged:
283           #      Q^*(I-(I+D^*D)^{-1})Q dp =  Q^* (g-u0-Qp0 - (I+D^*D)^{-1} ( D^*(f-Du0)+g-u0-Qp0) )                 v=self.getFlux(p, fixed_flux=u0)
284           #                 Qp=self.__Q(p)
285           #   residual is                 norm_v=self.__L2(v)
286           #                 norm_Qp=self.__L2(Qp)
287           #    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)                 if norm_v == 0.:
288           #                    if norm_Qp == 0.:
289           #        with v = (I+D^*D)^{-1} (D^*f+g-Qp) including BC                       return v,p
290           #                    else:
291           #    we use (g - Qp, v) to represent the residual. not that                      fac=norm_Qp
292           #                 else:
293           #    dr(dp)=( -Q(dp), dv) with dv = - (I+D^*D)^{-1} Q(dp)                    if norm_Qp == 0.:
294           #                      fac=norm_v
295           #   while the initial residual is                    else:
296           #                      fac=2./(1./norm_v+1./norm_Qp)
297           #      r0=( g - Qp0, v00) with v00=(I+D^*D)^{-1} (D^*f+g-Qp0) including BC                 ATOL=(atol+rtol*fac)
298           #                   if self.verbose:
299           d0=self.__g-util.tensor_mult(self.__permeability,util.grad(p0))                      print "DarcyFlux: L2 norm of v = %e."%norm_v
300           self.__pde_v.setValue(Y=d0, X=self.__f*util.kronecker(self.domain), r=u0)                      print "DarcyFlux: L2 norm of k.grad(p) = %e."%norm_Qp
301           v00=self.__pde_v.getSolution(verbose=show_details)                      print "DarcyFlux: L2 defect u = %e."%(util.integrate(util.length(self.__g-util.interpolate(v,Function(self.domain))-Qp)**2)**(0.5),)
302           if self.verbose: print "DarcyFlux: range of initial flux = ",util.inf(v00), util.sup(v00)                      print "DarcyFlux: L2 defect div(v) = %e."%(util.integrate((self.__f-util.div(v))**2)**(0.5),)
303           self.__pde_v.setValue(r=Data())                      print "DarcyFlux: absolute tolerance ATOL = %e."%ATOL
304           # start CG                 if norm_r == None or norm_r>ATOL:
305           r=ArithmeticTuple(d0, v00)                     if num_corrections>max_num_corrections:
306           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)                           raise ValueError,"maximum number of correction steps reached."
307           return r[1],p                     p,r, norm_r=PCG(self.__g-util.interpolate(v,Function(self.domain))-Qp,self.__Aprod,p,self.__Msolve_PCG,self.__inner_PCG,atol=0.5*ATOL, rtol=0.,iter_max=max_iter, verbose=self.verbose)
308                       num_corrections+=1
309      def __Aprod_PCG(self,dp):                 else:
310            if self.show_details: print "DarcyFlux: Applying operator"                     converged=True
311            #  -dr(dp) = (Qdp,du) where du = (I+D^*D)^{-1} (Qdp)           return v,p
313            self.__pde_v.setValue(Y=mQdp,X=Data(), r=Data())           return util.sqrt(util.integrate(util.length(util.interpolate(v,Function(self.domain)))**2))
314            du=self.__pde_v.getSolution(verbose=self.show_details)
315            return ArithmeticTuple(mQdp,du)      def __Q(self,p):
317
318        def __Aprod(self,dp):
319              if self.getSolverOptionsFlux().isVerbose(): print "DarcyFlux: Applying operator"
320              Qdp=self.__Q(dp)
321              self.__pde_v.setValue(Y=-Qdp,X=Data(), r=Data())
322              du=self.__pde_v.getSolution()
323              # self.__pde_v.getOperator().saveMM("proj.mm")
324              return Qdp+du
325        def __inner_GMRES(self,r,s):
326             return util.integrate(util.inner(r,s))
327
328      def __inner_PCG(self,p,r):      def __inner_PCG(self,p,r):
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
330
331      def __Msolve_PCG(self,r):      def __Msolve_PCG(self,r):
332            if self.show_details: print "DarcyFlux: Applying preconditioner"        if self.getSolverOptionsPressure().isVerbose(): print "DarcyFlux: Applying preconditioner"
333            self.__pde_p.setValue(X=util.transposed_tensor_mult(self.__permeability,r[0]-r[1]), r=Data())            self.__pde_p.setValue(X=util.transposed_tensor_mult(self.__permeability,r), Y=Data(), r=Data())
334            return self.__pde_p.getSolution(verbose=self.show_details)            # self.__pde_p.getOperator().saveMM("prec.mm")
335              return self.__pde_p.getSolution()
336
337        def getFlux(self,p=None, fixed_flux=Data()):
338            """
339            returns the flux for a given pressure C{p} where the flux is equal to C{fixed_flux}
340            on locations where C{location_of_fixed_flux} is positive (see L{setValue}).
341            Note that C{g} and C{f} are used, see L{setValue}.
342
343            @param p: pressure.
344            @type p: scalar value on the domain (e.g. L{Data}).
345            @param fixed_flux: flux on the locations of the domain marked be C{location_of_fixed_flux}.
346            @type fixed_flux: vector values on the domain (e.g. L{Data}).
347            @param tol: relative tolerance to be used.
348            @type tol: positive C{float}.
349            @return: flux
350            @rtype: L{Data}
351            @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}}
352                   for the permeability M{k_{ij}}
353            """
354        self.setSubProblemTolerance()
355            g=self.__g
356            f=self.__f
357            self.__pde_v.setValue(X=self.__l*f*util.kronecker(self.domain), r=fixed_flux)
358            if p == None:
359               self.__pde_v.setValue(Y=g)
360            else:
361               self.__pde_v.setValue(Y=g-self.__Q(p))
362            return self.__pde_v.getSolution()
363
365        """       """
366        solves       solves
367
368            -(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}
369                  u_{i,i}=0                  u_{i,i}=0
# Line 333  class StokesProblemCartesian(Homogeneous Line 371  class StokesProblemCartesian(Homogeneous
372            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
373
374        if surface_stress is not given 0 is assumed.       if surface_stress is not given 0 is assumed.
375
376        typical usage:       typical usage:
377
378              sp=StokesProblemCartesian(domain)              sp=StokesProblemCartesian(domain)
379              sp.setTolerance()              sp.setTolerance()
380              sp.initialize(...)              sp.initialize(...)
381              v,p=sp.solve(v0,p0)              v,p=sp.solve(v0,p0)
382        """       """
383        def __init__(self,domain,**kwargs):       def __init__(self,domain,adaptSubTolerance=True, **kwargs):
384           """           """
385           initialize the Stokes Problem           initialize the Stokes Problem
386
387           @param domain: domain of the problem. The approximation order needs to be two.           @param domain: domain of the problem. The approximation order needs to be two.
388           @type domain: L{Domain}           @type domain: L{Domain}
389         @param adaptSubTolerance: If True the tolerance for subproblem is set automatically.
391           @warning: The apprximation order needs to be two otherwise you may see oscilations in the pressure.           @warning: The apprximation order needs to be two otherwise you may see oscilations in the pressure.
392           """           """
394           self.domain=domain           self.domain=domain
395           self.vol=util.integrate(1.,Function(self.domain))           self.vol=util.integrate(1.,Function(self.domain))
396           self.__pde_u=LinearPDE(domain,numEquations=self.domain.getDim(),numSolutions=self.domain.getDim())           self.__pde_u=LinearPDE(domain,numEquations=self.domain.getDim(),numSolutions=self.domain.getDim())
397           self.__pde_u.setSymmetryOn()           self.__pde_u.setSymmetryOn()
398           # self.__pde_u.setSolverMethod(self.__pde_u.DIRECT)
# self.__pde_u.setSolverMethod(preconditioner=LinearPDE.RILU)

399           self.__pde_prec=LinearPDE(domain)           self.__pde_prec=LinearPDE(domain)
400           self.__pde_prec.setReducedOrderOn()           self.__pde_prec.setReducedOrderOn()
# self.__pde_prec.setSolverMethod(self.__pde_prec.LUMPING)
401           self.__pde_prec.setSymmetryOn()           self.__pde_prec.setSymmetryOn()
402
403           self.__pde_proj=LinearPDE(domain)           self.__pde_proj=LinearPDE(domain)
404           self.__pde_proj.setReducedOrderOn()           self.__pde_proj.setReducedOrderOn()
405         self.__pde_proj.setValue(D=1)
406           self.__pde_proj.setSymmetryOn()           self.__pde_proj.setSymmetryOn()
self.__pde_proj.setValue(D=1.)
407
409             """
410         returns the solver options used  solve the equation for velocity.
411
412         @rtype: L{SolverOptions}
413         """
414         return self.__pde_u.getSolverOptions()
415         def setSolverOptionsVelocity(self, options=None):
416             """
417         set the solver options for solving the equation for velocity.
418
419         @param options: new solver  options
420         @type options: L{SolverOptions}
421         """
422             self.__pde_u.setSolverOptions(options)
423         def getSolverOptionsPressure(self):
424             """
425         returns the solver options used  solve the equation for pressure.
426         @rtype: L{SolverOptions}
427         """
428         return self.__pde_prec.getSolverOptions()
429         def setSolverOptionsPressure(self, options=None):
430             """
431         set the solver options for solving the equation for pressure.
432         @param options: new solver  options
433         @type options: L{SolverOptions}
434         """
435         self.__pde_prec.setSolverOptions(options)
436
437         def setSolverOptionsDiv(self, options=None):
438             """
439         set the solver options for solving the equation to project the divergence of
440         the velocity onto the function space of presure.
441
442         @param options: new solver options
443         @type options: L{SolverOptions}
444         """
445         self.__pde_prec.setSolverOptions(options)
446         def getSolverOptionsDiv(self):
447             """
448         returns the solver options for solving the equation to project the divergence of
449         the velocity onto the function space of presure.
450
451         @rtype: L{SolverOptions}
452         """
453         return self.__pde_prec.getSolverOptions()
454         def setSubProblemTolerance(self):
455             """
456         Updates the tolerance for subproblems
457             """
459                 sub_tol=self.getSubProblemTolerance()
460             self.getSolverOptionsDiv().setTolerance(sub_tol)
461             self.getSolverOptionsDiv().setAbsoluteTolerance(0.)
462             self.getSolverOptionsPressure().setTolerance(sub_tol)
463             self.getSolverOptionsPressure().setAbsoluteTolerance(0.)
464             self.getSolverOptionsVelocity().setTolerance(sub_tol)
465             self.getSolverOptionsVelocity().setAbsoluteTolerance(0.)
466
467
469          """          """
470          assigns values to the model parameters          assigns values to the model parameters
471
# Line 383  class StokesProblemCartesian(Homogeneous Line 480  class StokesProblemCartesian(Homogeneous
480          @param stress: initial stress          @param stress: initial stress
481      @type stress: L{Tensor} object on L{FunctionSpace} L{Function} or similar      @type stress: L{Tensor} object on L{FunctionSpace} L{Function} or similar
482          @note: All values needs to be set.          @note: All values needs to be set.

483          """          """
484          self.eta=eta          self.eta=eta
485          A =self.__pde_u.createCoefficient("A")          A =self.__pde_u.createCoefficient("A")
486      self.__pde_u.setValue(A=Data())      self.__pde_u.setValue(A=Data())
487          for i in range(self.domain.getDim()):          for i in range(self.domain.getDim()):
488          for j in range(self.domain.getDim()):          for j in range(self.domain.getDim()):
489              A[i,j,j,i] += 1.              A[i,j,j,i] += 1.
490              A[i,j,i,j] += 1.              A[i,j,i,j] += 1.
491      self.__pde_prec.setValue(D=1/self.eta)      self.__pde_prec.setValue(D=1/self.eta)
493            self.__f=f
494            self.__surface_stress=surface_stress
495          self.__stress=stress          self.__stress=stress
496
497        def B(self,v):       def Bv(self,v):
498          """           """
499          returns div(v)           returns inner product of element p and div(v)
@rtype: equal to the type of p
500
501          @note: boundary conditions on p should be zero!           @param p: a pressure increment
502          """           @param v: a residual
503          if self.show_details: print "apply divergence:"           @return: inner product of element p and div(v)
self.__pde_proj.setValue(Y=-util.div(v))
self.__pde_proj.setTolerance(self.getSubProblemTolerance())
return self.__pde_proj.getSolution(verbose=self.show_details)

def inner_pBv(self,p,Bv):
"""
returns inner product of element p and Bv  (overwrite)

@type p: equal to the type of p
@type Bv: equal to the type of result of operator B
504           @rtype: C{float}           @rtype: C{float}

@rtype: equal to the type of p
505           """           """
506           s0=util.interpolate(p,Function(self.domain))           self.__pde_proj.setValue(Y=-util.div(v))
507           s1=util.interpolate(Bv,Function(self.domain))           return self.__pde_proj.getSolution()
return util.integrate(s0*s1)
508
509        def inner_p(self,p0,p1):       def inner_pBv(self,p,Bv):
510           """           """
511           returns inner product of element p0 and p1  (overwrite)           returns inner product of element p and Bv=-div(v)
512
513           @type p0: equal to the type of p           @param p: a pressure increment
514           @type p1: equal to the type of p           @param v: a residual
515             @return: inner product of element p and Bv=-div(v)
516           @rtype: C{float}           @rtype: C{float}
517             """
518             return util.integrate(util.interpolate(p,Function(self.domain))*util.interpolate(Bv,Function(self.domain)))
519
520           @rtype: equal to the type of p       def inner_p(self,p0,p1):
521             """
522             Returns inner product of p0 and p1
523
524             @param p0: a pressure
525             @param p1: a pressure
526             @return: inner product of p0 and p1
527             @rtype: C{float}
528           """           """
529           s0=util.interpolate(p0/self.eta,Function(self.domain))           s0=util.interpolate(p0/self.eta,Function(self.domain))
530           s1=util.interpolate(p1/self.eta,Function(self.domain))           s1=util.interpolate(p1/self.eta,Function(self.domain))
531           return util.integrate(s0*s1)           return util.integrate(s0*s1)
532
533        def inner_v(self,v0,v1):       def norm_v(self,v):
534           """           """
535           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}
536
537           @rtype: equal to the type of v           @param v: a velovity
538             @return: norm of v
539             @rtype: non-negative C{float}
540           """           """
return util.integrate(util.inner(gv0,gv1))
542
543        def solve_A(self,u,p):       def getV(self, p, v0):
544           """           """
545           solves Av=f-Au-B^*p (v=0 on fixed_u_mask)           return the value for v for a given p (overwrite)
546
547             @param p: a pressure
548             @param v0: a initial guess for the value v to return.
549             @return: v given as M{v= A^{-1} (f-B^*p)}
550           """           """
551           if self.show_details: print "solve for velocity:"           self.__pde_u.setValue(Y=self.__f, y=self.__surface_stress, r=v0)
self.__pde_u.setTolerance(self.getSubProblemTolerance())
552           if self.__stress.isEmpty():           if self.__stress.isEmpty():
554           else:           else:
556           out=self.__pde_u.getSolution(verbose=self.show_details)           out=self.__pde_u.getSolution()
557           return  out           return  out
558
559        def solve_prec(self,p):       def norm_Bv(self,Bv):
560           if self.show_details: print "apply preconditioner:"          """
561           self.__pde_prec.setTolerance(self.getSubProblemTolerance())          Returns Bv (overwrite).
562           self.__pde_prec.setValue(Y=p)
563           q=self.__pde_prec.getSolution(verbose=self.show_details)          @rtype: equal to the type of p
564           return q          @note: boundary conditions on p should be zero!
565            """
566            return util.sqrt(util.integrate(util.interpolate(Bv,Function(self.domain))**2))
567
568         def solve_AinvBt(self,p):
569             """
570             Solves M{Av=B^*p} with accuracy L{self.getSubProblemTolerance()}
571
572             @param p: a pressure increment
573             @return: the solution of M{Av=B^*p}
574             @note: boundary conditions on v should be zero!
575             """
576             self.__pde_u.setValue(Y=Data(), y=Data(), r=Data(),X=-p*util.kronecker(self.domain))
577             out=self.__pde_u.getSolution()
578             return  out
579
580         def solve_prec(self,Bv):
581             """
582             applies preconditioner for for M{BA^{-1}B^*} to M{Bv}
583             with accuracy L{self.getSubProblemTolerance()}
584
585             @param v: velocity increment
586             @return: M{p=P(Bv)} where M{P^{-1}} is an approximation of M{BA^{-1}B^*}
587             @note: boundary conditions on p are zero.
588             """
589             self.__pde_prec.setValue(Y=Bv)
590             return self.__pde_prec.getSolution()

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