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

revision 2169 by caltinay, Wed Dec 17 03:08:58 2008 UTC revision 2349 by gross, Mon Mar 30 08:14:23 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 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      Represents and 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      where M{p} represents the pressure and M{u} the Darcy flux. M{k} represents the permeability,
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          """          """
74          Assigns values to model parameters.          assigns values to model parameters
75
76          @param f: volumetric sources/sinks          @param f: volumetic sources/sinks
77          @type f: scalar value on the domain, e.g. L{Data}          @type f: scalar value on the domain (e.g. L{Data})
78          @param g: flux sources/sinks          @param g: flux sources/sinks
79          @type g: vector value on the domain, e.g. L{Data}          @type g: vector values on the domain (e.g. L{Data})
80          @param location_of_fixed_pressure: mask for locations where pressure is fixed          @param location_of_fixed_pressure: mask for locations where pressure is fixed
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 value 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          @param permeability: permeability tensor. If scalar C{s} is given the tensor with
85                               tensor with C{s} on the main diagonal is used. If                               C{s} on the main diagonal is used. If vector C{v} is given the tensor with
86                               vector C{v} is given the tensor with C{v} on the                               C{v} on the main diagonal is used.
87                               main diagonal is used.          @type permeability: scalar, vector or tensor values on the domain (e.g. L{Data})
88          @type permeability: scalar, vector or tensor values on the domain, e.g.
89                              L{Data}          @note: the values of parameters which are not set by calling C{setValue} are not altered.
90            @note: at any point on the boundary of the domain the pressure (C{location_of_fixed_pressure} >0)
91          @note: the values of parameters which are not set by calling                 or the normal component of the flux (C{location_of_fixed_flux[i]>0} if direction of the normal
92                 C{setValue} are not altered                 is along the M{x_i} axis.
@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 the direction of
the normal 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"))
# Line 98  class DarcyFlow(object): Line 97  class DarcyFlow(object):
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():
# Line 125  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):
def getFlux(self,p, fixed_flux=Data(),tol=1.e-8, show_details=False):
128          """          """
129          Returns the flux for a given pressure C{p}.          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
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.

@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 by
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}}
"""
self.__pde_v.setTolerance(tol)
self.__pde_v.setValue(Y=self.__g, X=self.__f*util.kronecker(self.domain), r=fixed_flux)
return self.__pde_v.getSolution(verbose=show_details)

def solve(self, u0, p0, atol=0, rtol=1e-8, max_iter=100, verbose=False, show_details=False, sub_rtol=1.e-8):
"""
Solves the problem.

The iteration is terminated if the error in the pressure is less than
M{rtol * |q| + atol} where M{|q|} denotes the norm of the right hand
side (see escript user's guide for details).

@param u0: initial guess for the flux. At locations in the domain
marked by C{location_of_fixed_flux} the value of C{u0} is
kept unchanged.
@type u0: vector value on the domain, e.g. L{Data}
@param p0: initial guess for the pressure. At locations in the domain
marked by C{location_of_fixed_pressure} the value of C{p0}
is kept unchanged.
@type p0: scalar value on the domain, e.g. L{Data}
@param atol: absolute tolerance for the pressure
@type atol: non-negative C{float}
135          @param rtol: relative tolerance for the pressure          @param rtol: relative tolerance for the pressure
136          @type rtol: non-negative C{float}          @type rtol: non-negative C{float}
137          @param sub_rtol: tolerance to be used in the sub iteration. It is          """
138                           recommended that M{sub_rtol<rtol*5.e-3}          if rtol<0:
139          @type sub_rtol: positive-negative C{float}              raise ValueError,"Relative tolerance needs to be non-negative."
140          @param verbose: if True information on iteration progress is printed          self.__rtol=rtol
141          @type verbose: C{bool}      def getTolerance(self):
142          @param show_details: if True information on the sub-iteration process          """
143                               is printed          returns the relative tolerance
144          @type show_details: C{bool}
145          @return: flux and pressure          @return: current relative tolerance
146          @rtype: C{tuple} of L{Data}          @rtype: C{float}
147            """
148          @note: the problem is solved in a least squares formulation:          return self.__rtol
149
150          M{(I+D^*D)u+Qp=D^*f+g}      def setAbsoluteTolerance(self,atol=0.):
151            """
152          M{Q^*u+Q^*Qp=Q^*g}          sets the absolute tolerance C{atol} used to terminate the solution process. The iteration is terminated if

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 from the problem by
setting

M{u=(I+D^*D)^{-1}(D^*f-g-Qp)} with M{u=u0} on C{location_of_fixed_flux}

from the first equation. Inserted into the second equation we get

M{Q^*(I-(I+D^*D)^{-1})Qp= Q^*(g-(I+D^*D)^{-1}(D^*f+g))} with M{p=p0}
on C{location_of_fixed_pressure}

which is solved using the PCG method (precondition is M{Q^*Q}).
In each iteration step PDEs with operator M{I+D^*D} and with M{Q^*Q}
need to be solved using a sub-iteration scheme.
"""
self.verbose=verbose
self.show_details= show_details and self.verbose
self.__pde_v.setTolerance(sub_rtol)
self.__pde_p.setTolerance(sub_rtol)
u2=u0*self.__pde_v.getCoefficient("q")
#
# first the reference velocity is calculated from
#
#   (I+D^*D)u_ref=D^*f+g (including bundray conditions for u)
#
self.__pde_v.setValue(Y=self.__g, X=self.__f*util.kronecker(self.domain), r=u0)
u_ref=self.__pde_v.getSolution(verbose=show_details)
if self.verbose: print "DarcyFlux: maximum reference flux = ",util.Lsup(u_ref)
self.__pde_v.setValue(r=Data())
#
#   and then we calculate a reference pressure
#
#       Q^*Qp_ref=Q^*g-Q^*u_ref ((including bundray conditions for p)
#
self.__pde_p.setValue(X=util.transposed_tensor_mult(self.__permeability,(self.__g-u_ref)), r=p0)
p_ref=self.__pde_p.getSolution(verbose=self.show_details)
if self.verbose: print "DarcyFlux: maximum reference pressure = ",util.Lsup(p_ref)
self.__pde_p.setValue(r=Data())
#
#   (I+D^*D)du + Qdp = - Qp_ref                       u=du+u_ref
#   Q^*du + Q^*Qdp = Q^*g-Q^*u_ref-Q^*Qp_ref=0        p=dp+pref
#
#      du= -(I+D^*D)^(-1} Q(p_ref+dp)  u = u_ref+du
#
#  => Q^*(I-(I+D^*D)^(-1})Q dp = Q^*(I+D^*D)^(-1} Qp_ref
#  or Q^*(I-(I+D^*D)^(-1})Q p = Q^*Qp_ref
#
#   r= Q^*( (I+D^*D)^(-1} Qp_ref - Q dp + (I+D^*D)^(-1})Q dp) = Q^*(-du-Q dp)
#            with du=-(I+D^*D)^(-1} Q(p_ref+dp)
#
#  we use the (du,Qdp) to represent the resudual
#  Q^*Q is a preconditioner
#
#  <(Q^*Q)^{-1}r,r> -> right hand side norm is <Qp_ref,Qp_ref>
#
norm_rhs=util.sqrt(util.integrate(util.inner(Qp_ref,Qp_ref)))
ATOL=max(norm_rhs*rtol +atol, 200. * util.EPSILON * norm_rhs)
if not ATOL>0:
raise ValueError,"Negative absolute tolerance (rtol = %e, norm right hand side = %e, atol =%e)."%(rtol, norm_rhs, atol)
if self.verbose: print "DarcyFlux: norm of right hand side = %e (absolute tolerance = %e)"%(norm_rhs,ATOL)
#
#   caclulate the initial residual
#
du=self.__pde_v.getSolution(verbose=show_details)
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)
util.saveVTK("d.vtu",p=dp,p_ref=p_ref)
return u_ref+r[1],dp

def __Aprod_PCG(self,p):
if self.show_details: print "DarcyFlux: Applying operator"
self.__pde_v.setValue(Y=Qp,X=Data())
w=self.__pde_v.getSolution(verbose=self.show_details)
return ArithmeticTuple(-Qp,w)
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 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.
202
203             The iteration is terminated if the residual norm is less then self.getTolerance().
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.
206             @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.
208             @type p0: scalar value on the domain (e.g. L{Data}).
209             @param verbose: if set some information on iteration progress are printed
210             @type verbose: C{bool}
211             @param show_details:  if set information on the subiteration process are printed.
212             @type show_details: C{bool}
213             @return: flux and pressure
214             @rtype: C{tuple} of L{Data}.
215
216             @note: The problem is solved as a least squares form
217
218             M{(I+D^*D)u+Qp=D^*f+g}
219             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}}.
222             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
225
226             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
229
230             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.
232             """
233             self.verbose=verbose or True
234             self.show_details= show_details and self.verbose
235             rtol=self.getTolerance()
236             atol=self.getAbsoluteTolerance()
237             if self.verbose: print "DarcyFlux: initial sub tolerance = %e"%self.getSubProblemTolerance()
238
239             num_corrections=0
240             converged=False
241             p=p0
242             norm_r=None
243             while not converged:
244                   v=self.getFlux(p, fixed_flux=u0, show_details=self.show_details)
245                   Qp=self.__Q(p)
246                   norm_v=self.__L2(v)
247                   norm_Qp=self.__L2(Qp)
248                   if norm_v == 0.:
249                      if norm_Qp == 0.:
250                         return v,p
251                      else:
252                        fac=norm_Qp
253                   else:
254                      if norm_Qp == 0.:
255                        fac=norm_v
256                      else:
257                        fac=2./(1./norm_v+1./norm_Qp)
258                   ATOL=(atol+rtol*fac)
259                   if self.verbose:
260                        print "DarcyFlux: L2 norm of v = %e."%norm_v
261                        print "DarcyFlux: L2 norm of k.grad(p) = %e."%norm_Qp
262                        print "DarcyFlux: absolute tolerance ATOL = %e."%ATOL
263                   if norm_r == None or norm_r>ATOL:
264                       if num_corrections>max_num_corrections:
265                             raise ValueError,"maximum number of correction steps reached."
266                       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                   else:
269                       converged=True
270             return v,p
271    #
272    #
273    #               r_hat=g-util.interpolate(v,Function(self.domain))-Qp
274    #               #===========================================================================
275    #               norm_r_hat=self.__L2(r_hat)
276    #               norm_v=self.__L2(v)
277    #               norm_g=self.__L2(g)
278    #               norm_gv=self.__L2(g-v)
279    #               norm_Qp=self.__L2(Qp)
280    #               norm_gQp=self.__L2(g-Qp)
281    #               fac=min(max(norm_v,norm_gQp),max(norm_Qp,norm_gv))
282    #               fac=min(norm_v,norm_Qp,norm_gv)
283    #               norm_r_hat_PCG=util.sqrt(self.__inner_PCG(self.__Msolve_PCG(r_hat),r_hat))
284    #               print "norm_r_hat = ",norm_r_hat,norm_r_hat_PCG, norm_r_hat_PCG/norm_r_hat
285    #               if r!=None:
286    #                   print "diff = ",self.__L2(r-r_hat)/norm_r_hat
287    #                   sub_tol=min(rtol/self.__L2(r-r_hat)*norm_r_hat,1.)*self.getSubProblemTolerance()
288    #                   self.setSubProblemTolerance(sub_tol)
289    #                   print "subtol_new=",self.getSubProblemTolerance()
290    #               print "norm_v = ",norm_v
291    #               print "norm_gv = ",norm_gv
292    #               print "norm_Qp = ",norm_Qp
293    #               print "norm_gQp = ",norm_gQp
294    #               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, iter_max = 100000)
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):
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          if self.show_details: print "DarcyFlux: Applying preconditioner"            self.__pde_p.setTolerance(self.getSubProblemTolerance())
340          self.__pde_p.setValue(X=-util.transposed_tensor_mult(self.__permeability,r[0]+r[1]))            if self.show_details: print "DarcyFlux: Applying preconditioner"
341          return self.__pde_p.getSolution(verbose=self.show_details)            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, iter_max = 100000)
343
344  class StokesProblemCartesian(HomogeneousSaddlePointProblem):      def getFlux(self,p=None, fixed_flux=Data(), show_details=False):
345        """          """
346        Represents and solves the problem          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, iter_max=100000)
370
371        M{-(eta*(u_{i,j}+u_{j,i}))_j + p_i = f_i-stress_{ij,j}}  class StokesProblemCartesian(HomogeneousSaddlePointProblem):
372         """
373         solves
374
375        M{u_{i,i}=0} and M{u=0} where C{fixed_u_mask}>0            -(eta*(u_{i,j}+u_{j,i}))_j + p_i = f_i-stress_{ij,j}
376                    u_{i,i}=0
377
378        M{eta*(u_{i,j}+u_{j,i})*n_j-p*n_i=surface_stress +stress_{ij}n_j}            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
380
381        If C{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           Initializes the Stokes Problem.           initialize the Stokes Problem
393
394           @param domain: domain of the problem. The approximation order needs           @param domain: domain of the problem. The approximation order needs to be two.
to be two.
395           @type domain: L{Domain}           @type domain: L{Domain}
396           @warning: The approximation order needs to be two otherwise you may           @warning: The apprximation order needs to be two otherwise you may see oscilations in the pressure.
see oscillations in the pressure.
397           """           """
399           self.domain=domain           self.domain=domain
# Line 318  class StokesProblemCartesian(Homogeneous Line 408  class StokesProblemCartesian(Homogeneous
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
412           self.__pde_proj.setReducedOrderOn()          """
413           self.__pde_proj.setSymmetryOn()          assigns values to the model parameters
414           self.__pde_proj.setValue(D=1.)
415            @param f: external force
416        def initialize(self,f=Data(),fixed_u_mask=Data(),eta=1,surface_stress=Data(),stress=Data()):          @type f: L{Vector} object in L{FunctionSpace} L{Function} or similar
418           Assigns values to the model parameters.          @type fixed_u_mask: L{Vector} object on L{FunctionSpace} L{Solution} or similar
419            @param eta: viscosity
420           @param f: external force          @type eta: L{Scalar} object on L{FunctionSpace} L{Function} or similar
421           @type f: L{Vector} object in L{FunctionSpace} L{Function} or similar          @param surface_stress: normal surface stress
422           @param fixed_u_mask: mask of locations with fixed velocity          @type eta: L{Vector} object on L{FunctionSpace} L{FunctionOnBoundary} or similar
423           @type fixed_u_mask: L{Vector} object on L{FunctionSpace}, L{Solution}          @param stress: initial stress
424                               or similar      @type stress: L{Tensor} object on L{FunctionSpace} L{Function} or similar
425           @param eta: viscosity          @note: All values needs to be set.
426           @type eta: L{Scalar} object on L{FunctionSpace}, L{Function} or similar
427           @param surface_stress: normal surface stress          """
428           @type surface_stress: L{Vector} object on L{FunctionSpace},          self.eta=eta
429                                 L{FunctionOnBoundary} or similar          A =self.__pde_u.createCoefficient("A")
430           @param stress: initial stress      self.__pde_u.setValue(A=Data())
431           @type stress: L{Tensor} object on L{FunctionSpace}, L{Function} or          for i in range(self.domain.getDim()):
432                         similar          for j in range(self.domain.getDim()):
433           @note: All values need to be set.              A[i,j,j,i] += 1.
434           """              A[i,j,i,j] += 1.
435           self.eta=eta      self.__pde_prec.setValue(D=1/self.eta)
437           self.__pde_u.setValue(A=Data())          self.__f=f
438           for i in range(self.domain.getDim()):          self.__surface_stress=surface_stress
439               for j in range(self.domain.getDim()):          self.__stress=stress
440                   A[i,j,j,i] += 1.
441                   A[i,j,i,j] += 1.       def inner_pBv(self,p,v):
442           self.__pde_prec.setValue(D=1/self.eta)           """
443           self.__pde_u.setValue(A=A*self.eta,q=fixed_u_mask,Y=f,y=surface_stress)           returns inner product of element p and div(v)
444           self.__stress=stress
445             @param p: a pressure increment
446        def B(self,v):           @param v: a residual
447           """           @return: inner product of element p and div(v)
448           Returns M{div(v)}.           @rtype: C{float}
@return: M{div(v)}
@rtype: equal to the type of p

@note: Boundary conditions on p should be zero!
"""
if self.show_details: print "apply divergence:"
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
@return: inner product of p and Bv
@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
455
456           @type p0: equal to the type of p           @param p0: a pressure
457           @type p1: equal to the type of p           @param p1: a pressure
458           @return: inner product of p0 and p1           @return: inner product of p0 and p1
459           @rtype: equal to the type of p           @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 elements v0 and v1 (overwrite).           returns the norm of v
468
469           @type v0: equal to the type of v           @param v: a velovity
470           @type v1: equal to the type of v           @return: norm of v
471           @return: inner product of v0 and v1           @rtype: non-negative C{float}
@rtype: equal to the type of v
472           """           """
return util.integrate(util.inner(gv0,gv1))
474
475        def solve_A(self,u,p):       def getV(self, p, v0):
476           """           """
477           Solves M{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)
490             return  out
491
492
493             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)           out=self.__pde_u.getSolution(verbose=self.show_details)
516           return out           return  out
517
518        def solve_prec(self,p):       def solve_precB(self,v):
519           """           """
520           Applies the preconditioner.           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           if self.show_details: print "apply preconditioner:"           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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