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

revision 2156 by gross, Mon Dec 15 05:09:02 2008 UTC revision 2288 by gross, Tue Feb 24 06:11:48 2009 UTC
# Line 34  __author__="Lutz Gross, l.gross@uq.edu.a Line 34  __author__="Lutz Gross, l.gross@uq.edu.a
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
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):      def __init__(self, domain,useReduced=False):
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          """          """
57          self.domain=domain          self.domain=domain
58            self.__l=util.longestEdge(self.domain)**2
59          self.__pde_v=LinearPDESystem(domain)          self.__pde_v=LinearPDESystem(domain)
60          self.__pde_v.setValue(D=util.kronecker(domain), A=util.outer(util.kronecker(domain),util.kronecker(domain)))          if useReduced: self.__pde_v.setReducedOrderOn()
61          self.__pde_v.setSymmetryOn()          self.__pde_v.setSymmetryOn()
62            self.__pde_v.setValue(D=util.kronecker(domain), A=self.__l*util.outer(util.kronecker(domain),util.kronecker(domain)))
63          self.__pde_p=LinearSinglePDE(domain)          self.__pde_p=LinearSinglePDE(domain)
64          self.__pde_p.setSymmetryOn()          self.__pde_p.setSymmetryOn()
65            if useReduced: self.__pde_p.setReducedOrderOn()
66          self.__f=Scalar(0,self.__pde_v.getFunctionSpaceForCoefficient("X"))          self.__f=Scalar(0,self.__pde_v.getFunctionSpaceForCoefficient("X"))
67          self.__g=Vector(0,self.__pde_v.getFunctionSpaceForCoefficient("Y"))          self.__g=Vector(0,self.__pde_v.getFunctionSpaceForCoefficient("Y"))
68            self.setTolerance()
69            self.setAbsoluteTolerance()
70            self.setSubProblemTolerance()
71
72      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):
73          """          """
# Line 75  class DarcyFlow(object): Line 81  class DarcyFlow(object):
81          @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})
82          @param location_of_fixed_flux:  mask for locations where flux is fixed.          @param location_of_fixed_flux:  mask for locations where flux is fixed.
83          @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})
84          @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
85                               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
86                               C{v} on the main diagonal is used.                               C{v} on the main diagonal is used.
87          @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})
88
# Line 85  class DarcyFlow(object): Line 91  class DarcyFlow(object):
91                 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
92                 is along the M{x_i} axis.                 is along the M{x_i} axis.
93          """          """
94          if f !=None:          if f !=None:
95             f=util.interpolate(f, self.__pde_v.getFunctionSpaceForCoefficient("X"))             f=util.interpolate(f, self.__pde_v.getFunctionSpaceForCoefficient("X"))
96             if f.isEmpty():             if f.isEmpty():
97                 f=Scalar(0,self.__pde_v.getFunctionSpaceForCoefficient("X"))                 f=Scalar(0,self.__pde_v.getFunctionSpaceForCoefficient("X"))
98             else:             else:
99                 if f.getRank()>0: raise ValueError,"illegal rank of f."                 if f.getRank()>0: raise ValueError,"illegal rank of f."
100             self.f=f             self.__f=f
101          if g !=None:            if g !=None:
102             g=util.interpolate(g, self.__pde_p.getFunctionSpaceForCoefficient("Y"))             g=util.interpolate(g, self.__pde_p.getFunctionSpaceForCoefficient("Y"))
103             if g.isEmpty():             if g.isEmpty():
104               g=Vector(0,self.__pde_v.getFunctionSpaceForCoefficient("Y"))               g=Vector(0,self.__pde_v.getFunctionSpaceForCoefficient("Y"))
# Line 118  class DarcyFlow(object): Line 124  class DarcyFlow(object):
124             self.__permeability=perm             self.__permeability=perm
125             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))
126
127        def setTolerance(self,rtol=1e-4):
128            """
129            sets the relative tolerance C{rtol} used to terminate the solution process. The iteration is terminated if
130
131            M{|g-v-Qp| <= atol + rtol * min( max( |g-v|, |Qp| ), max( |v|, |g-Qp| ) ) }
132
133            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}}.
134
135      def getFlux(self,p, fixed_flux=Data(),tol=1.e-8, show_details=False):          @param rtol: relative tolerance for the pressure
136            @type rtol: non-negative C{float}
137          """          """
138          returns the flux for a given pressure C{p} where the flux is equal to C{fixed_flux}          if rtol<0:
139          on locations where C{location_of_fixed_flux} is positive (see L{setValue}).              raise ValueError,"Relative tolerance needs to be non-negative."
140          Note that C{g} and C{f} are used, L{setValue}.          self.__rtol=rtol
141                def getTolerance(self):
@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 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}}
142          """          """
143          self.__pde_v.setTolerance(tol)          returns the relative tolerance
144          self.__pde_v.setValue(Y=self.__g, X=self.__f*util.kronecker(self.domain), r=fixed_flux)
145          return self.__pde_v.getSolution(verbose=show_details)          @return: current relative tolerance
146            @rtype: C{float}
147            """
148            return self.__rtol
149
150        def setAbsoluteTolerance(self,atol=0.):
151            """
152            sets the absolute tolerance C{atol} used to terminate the solution process. The iteration is terminated if
153
154            M{|g-v-Qp| <= atol + rtol * min( max( |g-v|, |Qp| ), max( |v|, |g-Qp| ) ) }
155
156            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}}.
157
158            @param atol: absolute tolerance for the pressure
159            @type atol: non-negative C{float}
160            """
161            if atol<0:
162                raise ValueError,"Absolute tolerance needs to be non-negative."
163            self.__atol=atol
164        def getAbsoluteTolerance(self):
165           """
166           returns the absolute tolerance
167
168           @return: current absolute tolerance
169           @rtype: C{float}
170           """
171           return self.__atol
172
173        def setSubProblemTolerance(self,rtol=None):
174             """
175             Sets the relative tolerance to solve the subproblem(s). If C{rtol} is not present
176             C{self.getTolerance()**2} is used.
177
178             @param rtol: relative tolerence
179             @type rtol: positive C{float}
180             """
181             if rtol == None:
182                  if self.getTolerance()<=0.:
183                      raise ValueError,"A positive relative tolerance must be set."
184                  self.__sub_tol=max(util.EPSILON**(0.75),self.getTolerance()**2)
185             else:
186                 if rtol<=0:
187                     raise ValueError,"sub-problem tolerance must be positive."
188                 self.__sub_tol=max(util.EPSILON**(0.75),rtol)
189
190      def solve(self,u0,p0,atol=0,rtol=1e-8, max_iter=100, verbose=False, show_details=False, sub_rtol=1.e-8):      def getSubProblemTolerance(self):
191           """           """
192             Returns the subproblem reduction factor.
193
194             @return: subproblem reduction factor
195             @rtype: C{float}
196             """
197             return self.__sub_tol
198
199        def solve(self,u0,p0, max_iter=100, verbose=False, show_details=False, max_num_corrections=10):
200             """
201           solves the problem.           solves the problem.
202
203           The iteration is terminated if the error in the pressure is less then C{rtol * |q| + atol} where           The iteration is terminated if the residual norm is less then self.getTolerance().
C{|q|} denotes the norm of the right hand side (see escript user's guide for details).
204
205           @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.
206           @type u0: vector value on the domain (e.g. L{Data}).           @type u0: vector value on the domain (e.g. L{Data}).
207           @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.
208           @type p0: scalar value on the domain (e.g. L{Data}).           @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}
209           @param verbose: if set some information on iteration progress are printed           @param verbose: if set some information on iteration progress are printed
210           @type verbose: C{bool}           @type verbose: C{bool}
211           @param show_details:  if set information on the subiteration process are printed.           @param show_details:  if set information on the subiteration process are printed.
212           @type show_details: C{bool}           @type show_details: C{bool}
213           @return: flux and pressure           @return: flux and pressure
214           @rtype: C{tuple} of L{Data}.           @rtype: C{tuple} of L{Data}.
215
216           @note: The problem is solved as a least squares form           @note: The problem is solved as a least squares form
217
218           M{(I+D^*D)u+Qp=D^*f+g}           M{(I+D^*D)u+Qp=D^*f+g}
219           M{Q^*u+Q^*Qp=Q^*g}           M{Q^*u+Q^*Qp=Q^*g}
220
221           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}}.
222           We eliminate the flux form the problem by setting           We eliminate the flux form the problem by setting
223
224           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
225
226           form the first equation. Inserted into the second equation we get           form the first equation. Inserted into the second equation we get
227
228           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
229
230           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
231           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.
232           """           """
233           self.verbose=verbose           self.verbose=verbose or True
234           self.show_details= show_details and self.verbose           self.show_details= show_details and self.verbose
235           self.__pde_v.setTolerance(sub_rtol)           rtol=self.getTolerance()
236           self.__pde_p.setTolerance(sub_rtol)           atol=self.getAbsoluteTolerance()
237           u2=u0*self.__pde_v.getCoefficient("q")           if self.verbose: print "DarcyFlux: initial sub tolerance = %e"%self.getSubProblemTolerance()
238           #
239           # first the reference velocity is calculated from           num_corrections=0
240           #           converged=False
241           #   (I+D^*D)u_ref=D^*f+g (including bundray conditions for u)           p=p0
242           #           norm_r=None
243           self.__pde_v.setValue(Y=self.__g, X=self.__f*util.kronecker(self.domain), r=u0)           while not converged:
244           u_ref=self.__pde_v.getSolution(verbose=show_details)                 v=self.getFlux(p, fixed_flux=u0, show_details=self.show_details)
245           if self.verbose: print "DarcyFlux: maximum reference flux = ",util.Lsup(u_ref)                 Qp=self.__Q(p)
246           self.__pde_v.setValue(r=Data())                 norm_v=self.__L2(v)
247           #                 norm_Qp=self.__L2(Qp)
248           #   and then we calculate a reference pressure                 if norm_v == 0.:
249           #                    if norm_Qp == 0.:
250           #       Q^*Qp_ref=Q^*g-Q^*u_ref ((including bundray conditions for p)                       return v,p
251           #                    else:
252           self.__pde_p.setValue(X=util.transposed_tensor_mult(self.__permeability,(self.__g-u_ref)), r=p0)                      fac=norm_Qp
253           p_ref=self.__pde_p.getSolution(verbose=self.show_details)                 else:
254           if self.verbose: print "DarcyFlux: maximum reference pressure = ",util.Lsup(p_ref)                    if norm_Qp == 0.:
255           self.__pde_p.setValue(r=Data())                      fac=norm_v
256           #                    else:
257           #   (I+D^*D)du + Qdp = - Qp_ref                       u=du+u_ref                      fac=2./(1./norm_v+1./norm_Qp)
258           #   Q^*du + Q^*Qdp = Q^*g-Q^*u_ref-Q^*Qp_ref=0        p=dp+pref                 ATOL=(atol+rtol*fac)
259           #                 if self.verbose:
260           #      du= -(I+D^*D)^(-1} Q(p_ref+dp)  u = u_ref+du                      print "DarcyFlux: L2 norm of v = %e."%norm_v
261           #                      print "DarcyFlux: L2 norm of k.grad(p) = %e."%norm_Qp
262           #  => Q^*(I-(I+D^*D)^(-1})Q dp = Q^*(I+D^*D)^(-1} Qp_ref                      print "DarcyFlux: absolute tolerance ATOL = %e."%ATOL
263           #  or Q^*(I-(I+D^*D)^(-1})Q p = Q^*Qp_ref                 if norm_r == None or norm_r>ATOL:
264           #                     if num_corrections>max_num_corrections:
265           #   r= Q^*( (I+D^*D)^(-1} Qp_ref - Q dp + (I+D^*D)^(-1})Q dp) = Q^*(-du-Q dp)                           raise ValueError,"maximum number of correction steps reached."
266           #            with du=-(I+D^*D)^(-1} Q(p_ref+dp)                     p,r, norm_r=PCG(self.__g-util.interpolate(v,Function(self.domain))-Qp,self.__Aprod,p,self.__Msolve_PCG,self.__inner_PCG,atol=0.1*ATOL, rtol=0.,iter_max=max_iter, verbose=self.verbose)
267           #                     num_corrections+=1
268           #  we use the (du,Qdp) to represent the resudual                 else:
269           #  Q^*Q is a preconditioner                     converged=True
270           #             return v,p
271           #  <(Q^*Q)^{-1}r,r> -> right hand side norm is <Qp_ref,Qp_ref>  #
272           #  #
273           Qp_ref=util.tensor_mult(self.__permeability,util.grad(p_ref))  #               r_hat=g-util.interpolate(v,Function(self.domain))-Qp
274           norm_rhs=util.sqrt(util.integrate(util.inner(Qp_ref,Qp_ref)))  #               #===========================================================================
275           ATOL=max(norm_rhs*rtol +atol, 200. * util.EPSILON * norm_rhs)  #               norm_r_hat=self.__L2(r_hat)
276           if not ATOL>0:  #               norm_v=self.__L2(v)
277               raise ValueError,"Negative absolute tolerance (rtol = %e, norm right hand side =%, atol =%e)."%(rtol, norm_rhs, atol)  #               norm_g=self.__L2(g)
278           if self.verbose: print "DarcyFlux: norm of right hand side = %e (absolute tolerance = %e)"%(norm_rhs,ATOL)  #               norm_gv=self.__L2(g-v)
279           #  #               norm_Qp=self.__L2(Qp)
280           #   caclulate the initial residual  #               norm_gQp=self.__L2(g-Qp)
281           #  #               fac=min(max(norm_v,norm_gQp),max(norm_Qp,norm_gv))
282           self.__pde_v.setValue(X=Data(), Y=-util.tensor_mult(self.__permeability,util.grad(p0)), r=Data())  #               fac=min(norm_v,norm_Qp,norm_gv)
283           du=self.__pde_v.getSolution(verbose=show_details)  #               norm_r_hat_PCG=util.sqrt(self.__inner_PCG(self.__Msolve_PCG(r_hat),r_hat))
284           r=ArithmeticTuple(util.tensor_mult(self.__permeability,util.grad(p0-p_ref)), du)  #               print "norm_r_hat = ",norm_r_hat,norm_r_hat_PCG, norm_r_hat_PCG/norm_r_hat
285           dp,r=PCG(r,self.__Aprod_PCG,p0,self.__Msolve_PCG,self.__inner_PCG,atol=ATOL, rtol=0.,iter_max=max_iter, verbose=self.verbose)  #               if r!=None:
286           util.saveVTK("d.vtu",p=dp,p_ref=p_ref)  #                   print "diff = ",self.__L2(r-r_hat)/norm_r_hat
287           return u_ref+r[1],dp  #                   sub_tol=min(rtol/self.__L2(r-r_hat)*norm_r_hat,1.)*self.getSubProblemTolerance()
288            #                   self.setSubProblemTolerance(sub_tol)
289      def __Aprod_PCG(self,p):  #                   print "subtol_new=",self.getSubProblemTolerance()
290            if self.show_details: print "DarcyFlux: Applying operator"  #               print "norm_v = ",norm_v
291            Qp=util.tensor_mult(self.__permeability,util.grad(p))  #               print "norm_gv = ",norm_gv
292            self.__pde_v.setValue(Y=Qp,X=Data())  #               print "norm_Qp = ",norm_Qp
293            w=self.__pde_v.getSolution(verbose=self.show_details)  #               print "norm_gQp = ",norm_gQp
294            return ArithmeticTuple(-Qp,w)  #               print "norm_g = ",norm_g
295    #               print "max(norm_v,norm_gQp)=",max(norm_v,norm_gQp)
296    #               print "max(norm_Qp,norm_gv)=",max(norm_Qp,norm_gv)
297    #               if fac == 0:
298    #                   if self.verbose: print "DarcyFlux: trivial case!"
299    #                   return v,p
300    #               #===============================================================================
301    #               # norm_v=util.sqrt(self.__inner_PCG(self.__Msolve_PCG(v),v))
302    #               # norm_Qp=self.__L2(Qp)
303    #               norm_r_hat=util.sqrt(self.__inner_PCG(self.__Msolve_PCG(r_hat),r_hat))
304    #               # print "**** norm_v, norm_Qp :",norm_v,norm_Qp
305    #
306    #               ATOL=(atol+rtol*2./(1./norm_v+1./norm_Qp))
307    #               if self.verbose:
308    #                   print "DarcyFlux: residual = %e"%norm_r_hat
309    #                   print "DarcyFlux: absolute tolerance ATOL = %e."%ATOL
310    #               if norm_r_hat <= ATOL:
311    #                   print "DarcyFlux: iteration finalized."
312    #                   converged=True
313    #               else:
314    #                   # p=GMRES(r_hat,self.__Aprod, p, self.__inner_GMRES, atol=ATOL, rtol=0., iter_max=max_iter, iter_restart=20, verbose=self.verbose,P_R=self.__Msolve_PCG)
315    #                   # p,r=PCG(r_hat,self.__Aprod,p,self.__Msolve_PCG,self.__inner_PCG,atol=ATOL*min(0.1,norm_r_hat_PCG/norm_r_hat), rtol=0.,iter_max=max_iter, verbose=self.verbose)
316    #                   p,r, norm_r=PCG(r_hat,self.__Aprod,p,self.__Msolve_PCG,self.__inner_PCG,atol=0.1*ATOL, rtol=0.,iter_max=max_iter, verbose=self.verbose)
317    #               print "norm_r =",norm_r
318    #         return v,p
319        def __L2(self,v):
320             return util.sqrt(util.integrate(util.length(util.interpolate(v,Function(self.domain)))**2))
321
322        def __Q(self,p):
324
325        def __Aprod(self,dp):
326              self.__pde_v.setTolerance(self.getSubProblemTolerance())
327              if self.show_details: print "DarcyFlux: Applying operator"
328              Qdp=self.__Q(dp)
329              self.__pde_v.setValue(Y=-Qdp,X=Data(), r=Data())
330              du=self.__pde_v.getSolution(verbose=self.show_details)
331              return Qdp+du
332        def __inner_GMRES(self,r,s):
333             return util.integrate(util.inner(r,s))
334
335      def __inner_PCG(self,p,r):      def __inner_PCG(self,p,r):
336           a=util.tensor_mult(self.__permeability,util.grad(p))           return util.integrate(util.inner(self.__Q(p), r))
out=-util.integrate(util.inner(a,r[0]+r[1]))
return out
337
338      def __Msolve_PCG(self,r):      def __Msolve_PCG(self,r):
339              self.__pde_p.setTolerance(self.getSubProblemTolerance())
340            if self.show_details: print "DarcyFlux: Applying preconditioner"            if self.show_details: print "DarcyFlux: Applying preconditioner"
341            self.__pde_p.setValue(X=-util.transposed_tensor_mult(self.__permeability,r[0]+r[1]))            self.__pde_p.setValue(X=util.transposed_tensor_mult(self.__permeability,r), Y=Data(), r=Data())
342            return self.__pde_p.getSolution(verbose=self.show_details)            return self.__pde_p.getSolution(verbose=self.show_details)
343
344        def getFlux(self,p=None, fixed_flux=Data(), show_details=False):
345            """
346            returns the flux for a given pressure C{p} where the flux is equal to C{fixed_flux}
347            on locations where C{location_of_fixed_flux} is positive (see L{setValue}).
348            Note that C{g} and C{f} are used, see L{setValue}.
349
350            @param p: pressure.
351            @type p: scalar value on the domain (e.g. L{Data}).
352            @param fixed_flux: flux on the locations of the domain marked be C{location_of_fixed_flux}.
353            @type fixed_flux: vector values on the domain (e.g. L{Data}).
354            @param tol: relative tolerance to be used.
355            @type tol: positive C{float}.
356            @return: flux
357            @rtype: L{Data}
358            @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}}
359                   for the permeability M{k_{ij}}
360            """
361            self.__pde_v.setTolerance(self.getSubProblemTolerance())
362            g=self.__g
363            f=self.__f
364            self.__pde_v.setValue(X=self.__l*f*util.kronecker(self.domain), r=fixed_flux)
365            if p == None:
366               self.__pde_v.setValue(Y=g)
367            else:
368               self.__pde_v.setValue(Y=g-self.__Q(p))
369            return self.__pde_v.getSolution(verbose=show_details)
370
372        """       """
373        solves       solves
374
375            -(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}
376                  u_{i,i}=0                  u_{i,i}=0
# Line 264  class StokesProblemCartesian(Homogeneous Line 378  class StokesProblemCartesian(Homogeneous
378            u=0 where  fixed_u_mask>0            u=0 where  fixed_u_mask>0
379            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
380
381        if surface_stress is not given 0 is assumed.       if surface_stress is not given 0 is assumed.
382
383        typical usage:       typical usage:
384
385              sp=StokesProblemCartesian(domain)              sp=StokesProblemCartesian(domain)
386              sp.setTolerance()              sp.setTolerance()
387              sp.initialize(...)              sp.initialize(...)
388              v,p=sp.solve(v0,p0)              v,p=sp.solve(v0,p0)
389        """       """
390        def __init__(self,domain,**kwargs):       def __init__(self,domain,**kwargs):
391           """           """
392           initialize the Stokes Problem           initialize the Stokes Problem
393
# Line 288  class StokesProblemCartesian(Homogeneous Line 402  class StokesProblemCartesian(Homogeneous
402           self.__pde_u.setSymmetryOn()           self.__pde_u.setSymmetryOn()
403           # self.__pde_u.setSolverMethod(self.__pde_u.DIRECT)           # self.__pde_u.setSolverMethod(self.__pde_u.DIRECT)
404           # self.__pde_u.setSolverMethod(preconditioner=LinearPDE.RILU)           # self.__pde_u.setSolverMethod(preconditioner=LinearPDE.RILU)
405
406           self.__pde_prec=LinearPDE(domain)           self.__pde_prec=LinearPDE(domain)
407           self.__pde_prec.setReducedOrderOn()           self.__pde_prec.setReducedOrderOn()
408           # self.__pde_prec.setSolverMethod(self.__pde_prec.LUMPING)           # self.__pde_prec.setSolverMethod(self.__pde_prec.LUMPING)
409           self.__pde_prec.setSymmetryOn()           self.__pde_prec.setSymmetryOn()
410
411           self.__pde_proj=LinearPDE(domain)       def initialize(self,f=Data(),fixed_u_mask=Data(),eta=1,surface_stress=Data(),stress=Data()):
self.__pde_proj.setReducedOrderOn()
self.__pde_proj.setSymmetryOn()
self.__pde_proj.setValue(D=1.)

412          """          """
413          assigns values to the model parameters          assigns values to the model parameters
414
# Line 314  class StokesProblemCartesian(Homogeneous Line 423  class StokesProblemCartesian(Homogeneous
423          @param stress: initial stress          @param stress: initial stress
424      @type stress: L{Tensor} object on L{FunctionSpace} L{Function} or similar      @type stress: L{Tensor} object on L{FunctionSpace} L{Function} or similar
425          @note: All values needs to be set.          @note: All values needs to be set.
426
427          """          """
428          self.eta=eta          self.eta=eta
429          A =self.__pde_u.createCoefficient("A")          A =self.__pde_u.createCoefficient("A")
430      self.__pde_u.setValue(A=Data())      self.__pde_u.setValue(A=Data())
431          for i in range(self.domain.getDim()):          for i in range(self.domain.getDim()):
432          for j in range(self.domain.getDim()):          for j in range(self.domain.getDim()):
433              A[i,j,j,i] += 1.              A[i,j,j,i] += 1.
434              A[i,j,i,j] += 1.              A[i,j,i,j] += 1.
435      self.__pde_prec.setValue(D=1/self.eta)      self.__pde_prec.setValue(D=1/self.eta)
437            self.__f=f
438            self.__surface_stress=surface_stress
439          self.__stress=stress          self.__stress=stress
440
441        def B(self,v):       def inner_pBv(self,p,v):
442          """           """
443          returns div(v)           returns inner product of element p and div(v)
@rtype: equal to the type of p
444
445          @note: boundary conditions on p should be zero!           @param p: a pressure increment
446          """           @param v: a residual
447          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
448           @rtype: C{float}           @rtype: C{float}

@rtype: equal to the type of p
449           """           """
450           s0=util.interpolate(p,Function(self.domain))           return util.integrate(-p*util.div(v))
s1=util.interpolate(Bv,Function(self.domain))
return util.integrate(s0*s1)
451
452        def inner_p(self,p0,p1):       def inner_p(self,p0,p1):
453           """           """
454           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}
455
456           @rtype: equal to the type of p           @param p0: a pressure
457             @param p1: a pressure
458             @return: inner product of p0 and p1
459             @rtype: C{float}
460           """           """
461           s0=util.interpolate(p0/self.eta,Function(self.domain))           s0=util.interpolate(p0/self.eta,Function(self.domain))
462           s1=util.interpolate(p1/self.eta,Function(self.domain))           s1=util.interpolate(p1/self.eta,Function(self.domain))
463           return util.integrate(s0*s1)           return util.integrate(s0*s1)
464
465        def inner_v(self,v0,v1):       def norm_v(self,v):
466           """           """
467           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}
468
469           @rtype: equal to the type of v           @param v: a velovity
470             @return: norm of v
471             @rtype: non-negative C{float}
472           """           """
return util.integrate(util.inner(gv0,gv1))
474
475        def solve_A(self,u,p):       def getV(self, p, v0):
476           """           """
477           solves Av=f-Au-B^*p (v=0 on fixed_u_mask)           return the value for v for a given p (overwrite)
478
479             @param p: a pressure
480             @param v0: a initial guess for the value v to return.
481             @return: v given as M{v= A^{-1} (f-B^*p)}
482           """           """
if self.show_details: print "solve for velocity:"
483           self.__pde_u.setTolerance(self.getSubProblemTolerance())           self.__pde_u.setTolerance(self.getSubProblemTolerance())
484             self.__pde_u.setValue(Y=self.__f, y=self.__surface_stress, r=v0)
485           if self.__stress.isEmpty():           if self.__stress.isEmpty():
487           else:           else:
489           out=self.__pde_u.getSolution(verbose=self.show_details)           out=self.__pde_u.getSolution(verbose=self.show_details)
490           return  out           return  out
491
492        def solve_prec(self,p):
493           if self.show_details: print "apply preconditioner:"           raise NotImplementedError,"no v calculation implemented."
494
495
496         def norm_Bv(self,v):
497            """
498            Returns Bv (overwrite).
499
500            @rtype: equal to the type of p
501            @note: boundary conditions on p should be zero!
502            """
503            return util.sqrt(util.integrate(util.div(v)**2))
504
505         def solve_AinvBt(self,p):
506             """
507             Solves M{Av=B^*p} with accuracy L{self.getSubProblemTolerance()}
508
509             @param p: a pressure increment
510             @return: the solution of M{Av=B^*p}
511             @note: boundary conditions on v should be zero!
512             """
513             self.__pde_u.setTolerance(self.getSubProblemTolerance())
514             self.__pde_u.setValue(Y=Data(), y=Data(), r=Data(),X=-p*util.kronecker(self.domain))
515             out=self.__pde_u.getSolution(verbose=self.show_details)
516             return  out
517
518         def solve_precB(self,v):
519             """
520             applies preconditioner for for M{BA^{-1}B^*} to M{Bv}
521             with accuracy L{self.getSubProblemTolerance()} (overwrite).
522
523             @param v: velocity increment
524             @return: M{p=P(Bv)} where M{P^{-1}} is an approximation of M{BA^{-1}B^*}
525             @note: boundary conditions on p are zero.
526             """
527             self.__pde_prec.setValue(Y=-util.div(v))
528           self.__pde_prec.setTolerance(self.getSubProblemTolerance())           self.__pde_prec.setTolerance(self.getSubProblemTolerance())
529           self.__pde_prec.setValue(Y=p)           return self.__pde_prec.getSolution(verbose=self.show_details)
q=self.__pde_prec.getSolution(verbose=self.show_details)
return q

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