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

revision 2208 by gross, Mon Jan 12 06:37:07 2009 UTC revision 2386 by gross, Wed Apr 15 03:54:25 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      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):
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            if weight == None:
59               s=self.domain.getSize()
60               self.__l=(3.*util.longestEdge(self.domain)*s/util.sup(s))**2
61            else:
62               self.__l=weight
63          self.__pde_v=LinearPDESystem(domain)          self.__pde_v=LinearPDESystem(domain)
64          if useReduced: self.__pde_v.setReducedOrderOn()          if useReduced: self.__pde_v.setReducedOrderOn()
65          self.__pde_v.setSymmetryOn()          self.__pde_v.setSymmetryOn()
66          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)))
67            # self.__pde_v.setSolverMethod(preconditioner=self.__pde_v.ILU0)
68          self.__pde_p=LinearSinglePDE(domain)          self.__pde_p=LinearSinglePDE(domain)
69          self.__pde_p.setSymmetryOn()          self.__pde_p.setSymmetryOn()
70          if useReduced: self.__pde_p.setReducedOrderOn()          if useReduced: self.__pde_p.setReducedOrderOn()
71          self.__f=Scalar(0,self.__pde_v.getFunctionSpaceForCoefficient("X"))          self.__f=Scalar(0,self.__pde_v.getFunctionSpaceForCoefficient("X"))
72          self.__g=Vector(0,self.__pde_v.getFunctionSpaceForCoefficient("Y"))          self.__g=Vector(0,self.__pde_v.getFunctionSpaceForCoefficient("Y"))
73          self.__ATOL= None          self.setTolerance()
74            self.setAbsoluteTolerance()
75            self.setSubProblemTolerance()
76
77      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):
78          """          """
# Line 78  class DarcyFlow(object): Line 86  class DarcyFlow(object):
86          @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})
87          @param location_of_fixed_flux:  mask for locations where flux is fixed.          @param location_of_fixed_flux:  mask for locations where flux is fixed.
88          @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})
89          @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
90                               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
91                               C{v} on the main diagonal is used.                               C{v} on the main diagonal is used.
92          @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})
93
# Line 88  class DarcyFlow(object): Line 96  class DarcyFlow(object):
96                 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
97                 is along the M{x_i} axis.                 is along the M{x_i} axis.
98          """          """
99          if f !=None:          if f !=None:
100             f=util.interpolate(f, self.__pde_v.getFunctionSpaceForCoefficient("X"))             f=util.interpolate(f, self.__pde_v.getFunctionSpaceForCoefficient("X"))
101             if f.isEmpty():             if f.isEmpty():
102                 f=Scalar(0,self.__pde_v.getFunctionSpaceForCoefficient("X"))                 f=Scalar(0,self.__pde_v.getFunctionSpaceForCoefficient("X"))
103             else:             else:
104                 if f.getRank()>0: raise ValueError,"illegal rank of f."                 if f.getRank()>0: raise ValueError,"illegal rank of f."
105             self.f=f             self.__f=f
106          if g !=None:            if g !=None:
107             g=util.interpolate(g, self.__pde_p.getFunctionSpaceForCoefficient("Y"))             g=util.interpolate(g, self.__pde_p.getFunctionSpaceForCoefficient("Y"))
108             if g.isEmpty():             if g.isEmpty():
109               g=Vector(0,self.__pde_v.getFunctionSpaceForCoefficient("Y"))               g=Vector(0,self.__pde_v.getFunctionSpaceForCoefficient("Y"))
# Line 121  class DarcyFlow(object): Line 129  class DarcyFlow(object):
129             self.__permeability=perm             self.__permeability=perm
130             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))
131
132        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}}
133          """          """
134          self.__pde_v.setTolerance(tol)          sets the relative tolerance C{rtol} used to terminate the solution process. The iteration is terminated if
135          g=self.__g
136          f=self.__f          M{|g-v-Qp| <= atol + rtol * min( max( |g-v|, |Qp| ), max( |v|, |g-Qp| ) ) }
137          self.__pde_v.setValue(X=f*util.kronecker(self.domain), r=fixed_flux)
138          if p == None:          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}}.
self.__pde_v.setValue(Y=g)
else:
return self.__pde_v.getSolution(verbose=show_details)
139
140      def getPressure(self,v=None, fixed_pressure=Data(),tol=1.e-8, show_details=False):          @param rtol: relative tolerance for the pressure
141            @type rtol: non-negative C{float}
142          """          """
143          returns the pressure for a given flux C{v} where the pressure is equal to C{fixed_pressure}          if rtol<0:
144          on locations where C{location_of_fixed_pressure} is positive (see L{setValue}).              raise ValueError,"Relative tolerance needs to be non-negative."
145          Note that C{g} is used, see L{setValue}.          self.__rtol=rtol
146                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}}
147          """          """
148          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)
149
150      def setTolerance(self,atol=0,rtol=1e-8,p_ref=None,v_ref=None):          @return: current relative tolerance
151            @rtype: C{float}
152          """          """
153          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.
154
155          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      def setAbsoluteTolerance(self,atol=0.):
156            """
157          M{r=Q^*(g-Qp-v)}          sets the absolute tolerance C{atol} used to terminate the solution process. The iteration is terminated if

the condition
158
159          M{<(Q^*Q)^{-1} r,r> <= ATOL}          M{|g-v-Qp| <= atol + rtol * min( max( |g-v|, |Qp| ), max( |v|, |g-Qp| ) ) }
160
161          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}}.
162
163          @param atol: absolute tolerance for the pressure          @param atol: absolute tolerance for the pressure
164          @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}
"""
g=self.__g
if not v_ref == None:
f1=util.integrate(util.length(util.interpolate(g-v_ref,Function(self.domain)))**2)
else:
f1=util.integrate(util.length(util.interpolate(g))**2)
if not p_ref == None:
else:
f2=0
self.__ATOL= atol + rtol * util.sqrt(max(f1,f2))
if self.__ATOL<=0:
raise ValueError,"Positive tolerance (=%e) is expected."%self.__ATOL
return self.__ATOL

def getTolerance(self):
165          """          """
166          returns the current tolerance.          if atol<0:
167                  raise ValueError,"Absolute tolerance needs to be non-negative."
168          @return: used absolute tolerance.          self.__atol=atol
169          @rtype: positive C{float}      def getAbsoluteTolerance(self):
170          """         """
171          if self.__ATOL==None:         returns the absolute tolerance
172             raise ValueError,"no tolerance is defined."
173          return self.__ATOL         @return: current absolute tolerance
174           @rtype: C{float}
175           """
176           return self.__atol
177
178        def setSubProblemTolerance(self,rtol=None):
179             """
180             Sets the relative tolerance to solve the subproblem(s). If C{rtol} is not present
181             C{self.getTolerance()**2} is used.
182
183             @param rtol: relative tolerence
184             @type rtol: positive C{float}
185             """
186             if rtol == None:
187                  if self.getTolerance()<=0.:
188                      raise ValueError,"A positive relative tolerance must be set."
189                  self.__sub_tol=max(util.EPSILON**(0.75),self.getTolerance()**2)
190             else:
191                 if rtol<=0:
192                     raise ValueError,"sub-problem tolerance must be positive."
193                 self.__sub_tol=max(util.EPSILON**(0.75),rtol)
194
195        def getSubProblemTolerance(self):
196             """
197             Returns the subproblem reduction factor.
198
199      def solve(self,u0,p0, max_iter=100, verbose=False, show_details=False, sub_rtol=1.e-8):           @return: subproblem reduction factor
200           """           @rtype: C{float}
201             """
202             return self.__sub_tol
203
204        def solve(self,u0,p0, max_iter=100, verbose=False, show_details=False, max_num_corrections=10):
205             """
206           solves the problem.           solves the problem.
207
208           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().
209
210           @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.
211           @type u0: vector value on the domain (e.g. L{Data}).           @type u0: vector value on the domain (e.g. L{Data}).
212           @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.
213           @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}
214           @param verbose: if set some information on iteration progress are printed           @param verbose: if set some information on iteration progress are printed
215           @type verbose: C{bool}           @type verbose: C{bool}
216           @param show_details:  if set information on the subiteration process are printed.           @param show_details:  if set information on the subiteration process are printed.
217           @type show_details: C{bool}           @type show_details: C{bool}
218           @return: flux and pressure           @return: flux and pressure
219           @rtype: C{tuple} of L{Data}.           @rtype: C{tuple} of L{Data}.
220
221           @note: The problem is solved as a least squares form           @note: The problem is solved as a least squares form
222
223           M{(I+D^*D)u+Qp=D^*f+g}           M{(I+D^*D)u+Qp=D^*f+g}
224           M{Q^*u+Q^*Qp=Q^*g}           M{Q^*u+Q^*Qp=Q^*g}
225
226           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}}.
227           We eliminate the flux form the problem by setting           We eliminate the flux form the problem by setting
228
229           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
230
231           form the first equation. Inserted into the second equation we get           form the first equation. Inserted into the second equation we get
232
233           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
234
235           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
236           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.
237           """           """
238           self.verbose=verbose           self.verbose=verbose
239           self.show_details= show_details and self.verbose           self.show_details= show_details and self.verbose
240           self.__pde_v.setTolerance(sub_rtol)           rtol=self.getTolerance()
241           self.__pde_p.setTolerance(sub_rtol)           atol=self.getAbsoluteTolerance()
242           ATOL=self.getTolerance()           if self.verbose: print "DarcyFlux: initial sub tolerance = %e"%self.getSubProblemTolerance()
243           if self.verbose: print "DarcyFlux: absolute tolerance = %e"%ATOL
244           #########################################################################################################################           num_corrections=0
245           #           converged=False
246           #   we solve:           p=p0
247           #             norm_r=None
248           #      Q^*(I-(I+D^*D)^{-1})Q dp =  Q^* (g-u0-Qp0 - (I+D^*D)^{-1} ( D^*(f-Du0)+g-u0-Qp0) )           while not converged:
249           #                 v=self.getFlux(p, fixed_flux=u0, show_details=self.show_details)
250           #   residual is                 Qp=self.__Q(p)
251           #                 norm_v=self.__L2(v)
252           #    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)                 norm_Qp=self.__L2(Qp)
253           #                 if norm_v == 0.:
254           #        with v = (I+D^*D)^{-1} (D^*f+g-Qp) including BC                    if norm_Qp == 0.:
255           #                       return v,p
256           #    we use (g - Qp, v) to represent the residual. not that                    else:
257           #                      fac=norm_Qp
258           #    dr(dp)=( -Q(dp), dv) with dv = - (I+D^*D)^{-1} Q(dp)                 else:
259           #                    if norm_Qp == 0.:
260           #   while the initial residual is                      fac=norm_v
261           #                    else:
262           #      r0=( g - Qp0, v00) with v00=(I+D^*D)^{-1} (D^*f+g-Qp0) including BC                      fac=2./(1./norm_v+1./norm_Qp)
263           #                   ATOL=(atol+rtol*fac)
265           self.__pde_v.setValue(Y=d0, X=self.__f*util.kronecker(self.domain), r=u0)                      print "DarcyFlux: L2 norm of v = %e."%norm_v
266           v00=self.__pde_v.getSolution(verbose=show_details)                      print "DarcyFlux: L2 norm of k.grad(p) = %e."%norm_Qp
267           if self.verbose: print "DarcyFlux: range of initial flux = ",util.inf(v00), util.sup(v00)                      print "DarcyFlux: absolute tolerance ATOL = %e."%ATOL
268           self.__pde_v.setValue(r=Data())                 if norm_r == None or norm_r>ATOL:
269           # start CG                     if num_corrections>max_num_corrections:
270           r=ArithmeticTuple(d0, v00)                           raise ValueError,"maximum number of correction steps reached."
271           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)                     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)
272           return r[1],p                     num_corrections+=1
273                   else:
274                       converged=True
275             return v,p
276        def __L2(self,v):
277             return util.sqrt(util.integrate(util.length(util.interpolate(v,Function(self.domain)))**2))
278
279      def __Aprod_PCG(self,dp):      def __Q(self,p):
280            if self.show_details: print "DarcyFlux: Applying operator"            return util.tensor_mult(self.__permeability,util.grad(p))
#  -dr(dp) = (Qdp,du) where du = (I+D^*D)^{-1} (Qdp)
self.__pde_v.setValue(Y=mQdp,X=Data(), r=Data())
du=self.__pde_v.getSolution(verbose=self.show_details)
return ArithmeticTuple(mQdp,du)
281
282        def __Aprod(self,dp):
283              self.__pde_v.setTolerance(self.getSubProblemTolerance())
284              if self.show_details: print "DarcyFlux: Applying operator"
285              Qdp=self.__Q(dp)
286              self.__pde_v.setValue(Y=-Qdp,X=Data(), r=Data())
287              du=self.__pde_v.getSolution(verbose=self.show_details, iter_max = 100000)
288              # self.__pde_v.getOperator().saveMM("proj.mm")
289              return Qdp+du
290        def __inner_GMRES(self,r,s):
291             return util.integrate(util.inner(r,s))
292
293      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
295
296      def __Msolve_PCG(self,r):      def __Msolve_PCG(self,r):
297              self.__pde_p.setTolerance(self.getSubProblemTolerance())
298            if self.show_details: print "DarcyFlux: Applying preconditioner"            if self.show_details: print "DarcyFlux: Applying preconditioner"
299            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())
300            return self.__pde_p.getSolution(verbose=self.show_details)            # self.__pde_p.getOperator().saveMM("prec.mm")
301              return self.__pde_p.getSolution(verbose=self.show_details, iter_max = 100000)
302
303        def getFlux(self,p=None, fixed_flux=Data(), show_details=False):
304            """
305            returns the flux for a given pressure C{p} where the flux is equal to C{fixed_flux}
306            on locations where C{location_of_fixed_flux} is positive (see L{setValue}).
307            Note that C{g} and C{f} are used, see L{setValue}.
308
309            @param p: pressure.
310            @type p: scalar value on the domain (e.g. L{Data}).
311            @param fixed_flux: flux on the locations of the domain marked be C{location_of_fixed_flux}.
312            @type fixed_flux: vector values on the domain (e.g. L{Data}).
313            @param tol: relative tolerance to be used.
314            @type tol: positive C{float}.
315            @return: flux
316            @rtype: L{Data}
317            @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}}
318                   for the permeability M{k_{ij}}
319            """
320            self.__pde_v.setTolerance(self.getSubProblemTolerance())
321            g=self.__g
322            f=self.__f
323            self.__pde_v.setValue(X=self.__l*f*util.kronecker(self.domain), r=fixed_flux)
324            if p == None:
325               self.__pde_v.setValue(Y=g)
326            else:
327               self.__pde_v.setValue(Y=g-self.__Q(p))
328            return self.__pde_v.getSolution(verbose=show_details, iter_max=100000)
329
331        """       """
332        solves       solves
333
334            -(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}
335                  u_{i,i}=0                  u_{i,i}=0
# Line 333  class StokesProblemCartesian(Homogeneous Line 337  class StokesProblemCartesian(Homogeneous
338            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
339
340        if surface_stress is not given 0 is assumed.       if surface_stress is not given 0 is assumed.
341
342        typical usage:       typical usage:
343
344              sp=StokesProblemCartesian(domain)              sp=StokesProblemCartesian(domain)
345              sp.setTolerance()              sp.setTolerance()
346              sp.initialize(...)              sp.initialize(...)
347              v,p=sp.solve(v0,p0)              v,p=sp.solve(v0,p0)
348        """       """
349        def __init__(self,domain,**kwargs):       def __init__(self,domain,**kwargs):
350           """           """
351           initialize the Stokes Problem           initialize the Stokes Problem
352
# Line 356  class StokesProblemCartesian(Homogeneous Line 360  class StokesProblemCartesian(Homogeneous
360           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())
361           self.__pde_u.setSymmetryOn()           self.__pde_u.setSymmetryOn()
362           # self.__pde_u.setSolverMethod(self.__pde_u.DIRECT)           # self.__pde_u.setSolverMethod(self.__pde_u.DIRECT)
363           # self.__pde_u.setSolverMethod(preconditioner=LinearPDE.RILU)           # self.__pde_u.setSolverMethod(preconditioner=LinearPDE.ILU0)
364
365           self.__pde_prec=LinearPDE(domain)           self.__pde_prec=LinearPDE(domain)
366           self.__pde_prec.setReducedOrderOn()           self.__pde_prec.setReducedOrderOn()
367           # self.__pde_prec.setSolverMethod(self.__pde_prec.LUMPING)           # self.__pde_prec.setSolverMethod(self.__pde_prec.LUMPING)
368           self.__pde_prec.setSymmetryOn()           self.__pde_prec.setSymmetryOn()
369
self.__pde_proj.setReducedOrderOn()
self.__pde_proj.setSymmetryOn()
self.__pde_proj.setValue(D=1.)

371          """          """
372          assigns values to the model parameters          assigns values to the model parameters
373
# Line 383  class StokesProblemCartesian(Homogeneous Line 382  class StokesProblemCartesian(Homogeneous
382          @param stress: initial stress          @param stress: initial stress
383      @type stress: L{Tensor} object on L{FunctionSpace} L{Function} or similar      @type stress: L{Tensor} object on L{FunctionSpace} L{Function} or similar
384          @note: All values needs to be set.          @note: All values needs to be set.
385
386          """          """
387          self.eta=eta          self.eta=eta
388          A =self.__pde_u.createCoefficient("A")          A =self.__pde_u.createCoefficient("A")
389      self.__pde_u.setValue(A=Data())      self.__pde_u.setValue(A=Data())
390          for i in range(self.domain.getDim()):          for i in range(self.domain.getDim()):
391          for j in range(self.domain.getDim()):          for j in range(self.domain.getDim()):
392              A[i,j,j,i] += 1.              A[i,j,j,i] += 1.
393              A[i,j,i,j] += 1.              A[i,j,i,j] += 1.
394      self.__pde_prec.setValue(D=1/self.eta)      self.__pde_prec.setValue(D=1/self.eta)
396            self.__f=f
397            self.__surface_stress=surface_stress
398          self.__stress=stress          self.__stress=stress
399
400        def B(self,v):       def inner_pBv(self,p,v):
401          """           """
402          returns div(v)           returns inner product of element p and div(v)
@rtype: equal to the type of p
403
404          @note: boundary conditions on p should be zero!           @param p: a pressure increment
405          """           @param v: a residual
406          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
407           @rtype: C{float}           @rtype: C{float}

@rtype: equal to the type of p
408           """           """
409           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)
410
411        def inner_p(self,p0,p1):       def inner_p(self,p0,p1):
412           """           """
413           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}
414
415           @rtype: equal to the type of p           @param p0: a pressure
416             @param p1: a pressure
417             @return: inner product of p0 and p1
418             @rtype: C{float}
419           """           """
420           s0=util.interpolate(p0/self.eta,Function(self.domain))           s0=util.interpolate(p0/self.eta,Function(self.domain))
421           s1=util.interpolate(p1/self.eta,Function(self.domain))           s1=util.interpolate(p1/self.eta,Function(self.domain))
422           return util.integrate(s0*s1)           return util.integrate(s0*s1)
423
424        def inner_v(self,v0,v1):       def norm_v(self,v):
425           """           """
426           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}
427
428           @rtype: equal to the type of v           @param v: a velovity
429             @return: norm of v
430             @rtype: non-negative C{float}
431           """           """
return util.integrate(util.inner(gv0,gv1))
433
434        def solve_A(self,u,p):       def getV(self, p, v0):
435           """           """
436           solves Av=f-Au-B^*p (v=0 on fixed_u_mask)           return the value for v for a given p (overwrite)
437
438             @param p: a pressure
439             @param v0: a initial guess for the value v to return.
440             @return: v given as M{v= A^{-1} (f-B^*p)}
441           """           """
if self.show_details: print "solve for velocity:"
442           self.__pde_u.setTolerance(self.getSubProblemTolerance())           self.__pde_u.setTolerance(self.getSubProblemTolerance())
443             self.__pde_u.setValue(Y=self.__f, y=self.__surface_stress, r=v0)
444           if self.__stress.isEmpty():           if self.__stress.isEmpty():
446           else:           else:
448           out=self.__pde_u.getSolution(verbose=self.show_details)           out=self.__pde_u.getSolution(verbose=self.show_details)
449           return  out           return  out
450
451        def solve_prec(self,p):
452           if self.show_details: print "apply preconditioner:"           raise NotImplementedError,"no v calculation implemented."
453
454
455         def norm_Bv(self,v):
456            """
457            Returns Bv (overwrite).
458
459            @rtype: equal to the type of p
460            @note: boundary conditions on p should be zero!
461            """
462            return util.sqrt(util.integrate(util.div(v)**2))
463
464         def solve_AinvBt(self,p):
465             """
466             Solves M{Av=B^*p} with accuracy L{self.getSubProblemTolerance()}
467
468             @param p: a pressure increment
469             @return: the solution of M{Av=B^*p}
470             @note: boundary conditions on v should be zero!
471             """
472             self.__pde_u.setTolerance(self.getSubProblemTolerance())
473             self.__pde_u.setValue(Y=Data(), y=Data(), r=Data(),X=-p*util.kronecker(self.domain))
474             out=self.__pde_u.getSolution(verbose=self.show_details)
475             return  out
476
477         def solve_precB(self,v):
478             """
479             applies preconditioner for for M{BA^{-1}B^*} to M{Bv}
480             with accuracy L{self.getSubProblemTolerance()} (overwrite).
481
482             @param v: velocity increment
483             @return: M{p=P(Bv)} where M{P^{-1}} is an approximation of M{BA^{-1}B^*}
484             @note: boundary conditions on p are zero.
485             """
486             self.__pde_prec.setValue(Y=-util.div(v))
487           self.__pde_prec.setTolerance(self.getSubProblemTolerance())           self.__pde_prec.setTolerance(self.getSubProblemTolerance())
488           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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