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# $Id:$ |
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# |
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####################################################### |
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# |
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# Copyright 2008 by University of Queensland |
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# |
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# http://esscc.uq.edu.au |
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# Primary Business: Queensland, Australia |
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# Licensed under the Open Software License version 3.0 |
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# http://www.opensource.org/licenses/osl-3.0.php |
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# |
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####################################################### |
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# |
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|
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""" |
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Some models for flow |
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|
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@var __author__: name of author |
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@var __copyright__: copyrights |
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@var __license__: licence agreement |
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@var __url__: url entry point on documentation |
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@var __version__: version |
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@var __date__: date of the version |
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""" |
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|
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__author__="Lutz Gross, l.gross@uq.edu.au" |
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__copyright__=""" Copyright (c) 2008 by ACcESS MNRF |
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http://www.access.edu.au |
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Primary Business: Queensland, Australia""" |
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__license__="""Licensed under the Open Software License version 3.0 |
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http://www.opensource.org/licenses/osl-3.0.php""" |
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__url__="http://www.iservo.edu.au/esys" |
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__version__="$Revision:$" |
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__date__="$Date:$" |
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|
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from escript import * |
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import util |
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from linearPDEs import LinearPDE |
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from pdetools import HomogeneousSaddlePointProblem,Projector |
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|
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class PlateMantelModel(HomogeneousSaddlePointProblem): |
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""" |
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This class implements the reology of a self-consistent plate-mantel model proposed in |
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U{Hans-Bernd Muhlhaus<emailto:h.muhlhaus@uq.edu.au>} and U{Klaus Regenauer-Lieb<mailto:klaus.regenauer-lieb@csiro.au>}: |
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I{Towards a self-consistent plate mantle model that includes elasticity: simple benchmarks and application to basic modes of convection}, |
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see U{doi: 10.1111/j.1365-246X.2005.02742.x<http://www3.interscience.wiley.com/journal/118661486/abstract?CRETRY=1&SRETRY=0>}. |
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|
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typical usage: |
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|
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sp=PlateMantelModel(domain,stress=0,v=0) |
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sp.setTolerance() |
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sp.initialize(...) |
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v,p=sp.solve(v0,p0) |
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""" |
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def __init__(self, domain, stress=0, v=0, p=0, t=0, useJaumannStress=True, **kwargs): |
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""" |
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initializes PlateMantelModel |
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|
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@param domain: problem domain |
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@type domain: L{domain} |
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@param stress: initial stress |
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@param v: initial velocity field |
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@param t: initial time |
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@param useJaumannStress: C{True} if Jaumann stress is used |
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""" |
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HomogeneousSaddlePointProblem.__init__(self,**kwargs) |
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self.__domain=domain |
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self.__t=t |
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self.setSmall() |
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self.__vol=util.integrate(1.,Function(self.__domain)) |
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self.__useJaumannStress=useJaumannStress |
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#======================= |
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# state variables: |
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# |
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if isinstance(stress,Data): |
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self.__stress=util.interpolate(stress,Function(domain)) |
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else: |
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self.__stress=Data(stress,(domain.getDim(),domain.getDim()),Function(domain)) |
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self.__stress-=util.trace(self.__stress)*(util.kronecker(domain)/domain.getDim()) |
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if isinstance(v,Data): |
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self.__v=util.interpolate(v,Solution(domain)) |
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else: |
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self.__v=Data(v,(domain.getDim(),),Solution(domain)) |
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if isinstance(p,Data): |
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self.__p=util.interpolate(p,ReducedSolution(domain)) |
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else: |
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self.__p=Data(p,(),ReducedSolution(domain)) |
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self.__tau=util.sqrt(0.5)*util.length(self.__stress) |
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self.__D=util.symmetric(util.grad(self.__v)) |
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#======================= |
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# parameters |
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# |
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self.__mu=None |
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self.__eta_N=None |
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self.__eta_0=None |
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self.__tau_0=None |
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self.__n=None |
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self.__eta_Y=None |
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self.__tau_Y=None |
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self.__n_Y=None |
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#======================= |
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# solme value to track: |
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# |
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self.__mu_eff=None |
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self.__h_eff=None |
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self.__eta_eff=None |
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#======================= |
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# PDE related stuff |
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self.__pde_u=LinearPDE(domain,numEquations=self.getDomain().getDim(),numSolutions=self.getDomain().getDim()) |
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self.__pde_u.setSymmetryOn() |
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# self.__pde_u.setSolverMethod(preconditioner=LinearPDE.ILU0) |
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|
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self.__pde_prec=LinearPDE(domain) |
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self.__pde_prec.setReducedOrderOn() |
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self.__pde_prec.setSymmetryOn() |
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|
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self.__pde_proj=LinearPDE(domain) |
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self.__pde_proj.setReducedOrderOn() |
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self.__pde_proj.setSymmetryOn() |
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self.__pde_proj.setValue(D=1.) |
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|
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def useJaumannStress(self): |
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""" |
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return C{True} if Jaumann stress is included. |
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""" |
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return self.__useJaumannStress |
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def setSmall(self,small=util.sqrt(util.EPSILON)): |
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""" |
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sets small value |
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|
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@param small: positive small value |
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""" |
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self.__small=small |
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def getSmall(self): |
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""" |
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returns small value |
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@rtype: positive float |
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""" |
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return self.__small |
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def getDomain(self): |
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""" |
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returns the domain |
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""" |
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return self.__domain |
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def getStress(self): |
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""" |
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returns current stress |
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""" |
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return self.__stress |
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def getStretching(self): |
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""" |
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return stretching |
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""" |
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return self.__D |
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def getTau(self): |
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""" |
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returns current second stress deviatoric invariant |
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""" |
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return self.__tau |
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def getPressure(self): |
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""" |
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returns current pressure |
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""" |
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return self.__p |
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def getVelocity(self): |
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""" |
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returns current velocity |
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""" |
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return self.__v |
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def getTime(self): |
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""" |
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returns current time |
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""" |
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return self.__t |
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def getVolume(self): |
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""" |
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returns domain volume |
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""" |
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return self.__vol |
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|
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def getEtaEffective(self): |
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""" |
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returns effective shear viscocity |
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""" |
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return self.__eta_eff |
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def getMuEffective(self): |
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""" |
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returns effective shear modulus |
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""" |
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return self.__mu_eff |
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def getHEffective(self): |
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""" |
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returns effective h |
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""" |
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return self.__h_eff |
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def getMechanicalPower(self): |
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""" |
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returns the locl mechanical power M{D_ij Stress_ij} |
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""" |
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return util.inner(self.getStress(),self.getStretching()) |
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|
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def initialize(self, mu=None, eta_N=None, eta_0=None, tau_0=None, n=None, eta_Y=None, tau_Y=None, n_Y=None, F=None, q=None): |
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""" |
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sets the model parameters. |
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|
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@param mu: shear modulus. If C{mu==None} ... |
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@param eta_N: Newtonian viscosity. |
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@param eta_0: power law viscovity for tau=tau0 |
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@param tau_0: reference tau for power low. If C{tau_0==None} constant viscosity is assumed. |
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@param n: power of power law. if C{n=None}, constant viscosity is assumed. |
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@param eta_Y: =None |
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@param tau_Y: =None |
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@param n_Y: =None |
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@param F: external force. |
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@param q: location of constraints |
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""" |
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if mu != None: self.__mu=mu |
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if eta_N != None: self.__eta_N=eta_N |
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if eta_0 != None: self.__eta_0=eta_0 |
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if tau_0 != None: self.__tau_0=tau_0 |
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if n != None: self.__n=n |
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if eta_Y != None: self.__eta_Y=eta_Y |
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if tau_Y != None: self.__tau_Y=tau_Y |
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if n_Y != None: self.__n_Y=n_Y |
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if F != None: self.__pde_u.setValue(Y=F) |
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if q != None: self.__pde_u.setValue(q=q) |
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|
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def updateEffectiveCoefficients(self, dt, tau): |
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""" |
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updates the effective coefficints depending on tau and time step size dt. |
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""" |
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h_vis=None |
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h_yie=None |
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# calculate eta_eff = ( 1/eta_N + 1/eta_vis + 1/eta_yie)^{-1} |
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# with eta_vis = eta_0 * (tau/tau_0) ^ (1-n) |
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# eta_yie = eta_Y * (tau/tau_Y) ^ (1-n_Y) |
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s=Scalar(0.,Function(self.getDomain())) |
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if self.__eta_N!=None: |
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s+=1/self.__eta_N |
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if self.__eta_0!=None and self.__tau_0 != None and self.__n!=None: |
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if self.__tau_0 !=None and self.__n!=None: |
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eta_vis=self.__eta_0*(tau/self.__tau_0)**(1-self.__n) |
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s+=1./(eta_vis+self.__eta_0*self.getSmall()) |
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h_vis=eta_vis/(self.__n-1) |
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if self.__tau_Y!=None and self.__eta_Y!=None and self.__n_Y!=None: |
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eta_yie=self.__eta_Y*(tau/self.__tau_Y)**(1-self.__n_Y) |
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s+=1/(eta_yie+self.getSmall()*self.__eta_Y) |
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h_yie=self.__eta_Y/(self.__n_Y-1) |
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self.__eta_eff=1/s |
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# calculate eta_eff = ( 1/h_vis + 1/h_yie)^{-1} |
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# with h_vis = eta_vis/(n-1) |
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# h_yie = eta_yie/(n_Y-1) |
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if h_vis == None: |
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if h_yie==None: |
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self__h_eff=None |
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else: |
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self__h_eff=h_yie |
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else: |
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if h_yie==None: |
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self__h_eff=h_vis |
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else: |
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self__h_eff=1/((1./h_vis)+(1./h_yie)) |
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# calculate mu_eff = ( 1/mu + dt/eta_eff)^{-1} = mu*eta_eff/(mu*dt+eta_eff) |
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if self.__mu == None: |
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self.__mu_eff=self.__eta_eff/dt |
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else: |
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self.__mu_eff=1./(1./self.__mu+dt/self.__eta_eff) |
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|
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def update(self,dt,max_inner_iter=20, verbose=False, show_details=False, tol=10., solver="PCG"): |
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""" |
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updates stress, velocity and pressure for time increment dt |
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|
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@param dt: time increment |
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@param max_inner_iter: maximum number of iteration steps in the incompressible solver |
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@param verbose: prints some infos in the incompressible solve |
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@param show_details: prints some infos while solving PDEs |
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@param tol: tolerance for the time step |
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""" |
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stress_last=self.getStress() |
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# we should use something like FGMRES to merge the two iterations! |
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e=10.*tol |
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# get values from last time step and use them as initial guess: |
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v=self.__v |
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stress=self.__stress |
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p=self.__p |
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tau=self.__tau |
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while e > tol: # revise stress calculation if this is used. |
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# |
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# update the effective coefficients: |
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# |
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self.updateEffectiveCoefficients(dt,tau) |
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eta_eff=self.getEtaEffective() |
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mu_eff=self.getMuEffective() |
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h_eff=self.getHEffective() |
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# |
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# create some temporary variables: |
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# |
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self.__pde_u.setValue(A=Data()) # save memory! |
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k3=util.kronecker(Function(self.getDomain())) |
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k3Xk3=util.outer(k3,k3) |
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self.__f3=mu_eff*dt |
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A=self.__f3*(util.swap_axes(k3Xk3,0,3)+util.swap_axes(k3Xk3,1,3)) |
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print "mueff=",util.inf(mu_eff),util.sup(mu_eff) |
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if h_eff == None: |
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s=0 |
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self.__f0=0 |
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self.__f1=mu_eff/eta_eff*dt |
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else: |
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s=mu_eff*dt/(h_eff+mu_eff*dt) |
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Lsup_tau=Lsup(tau) |
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if Lsup>0: |
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self.__f0=mu_eff*s*dt/(tau+Lsup_tau*self.getSmall())**2 |
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A+=util.outer((-self.__f0)*stress,stress) |
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else: |
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self.__f0=0. |
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self.__f1=mu_eff/eta_eff*(1-s)*dt |
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|
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if self.useJaumannStress(): |
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self.__f2=mu_eff/eta_eff*dt**2 |
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sXk3=util.outer(stress,self.__f2*(-k3/2)) |
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|
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A+=util.swap_axes(sXk3,0,3) |
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A+=util.swap_axes(sXk3,1,3) |
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A-=util.swap_axes(sXk3,0,2) |
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A-=util.swap_axes(sXk3,1,2) |
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else: |
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self.__f2=0 |
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self.__pde_u.setValue(A=A) |
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self.__pde_prec.setValue(D=1/mu_eff) |
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|
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print "X f0:",util.inf(self.__f0), util.sup(self.__f0) |
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print "X f1:",util.inf(self.__f1), util.sup(self.__f1) |
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print "X f2:",util.inf(self.__f2), util.sup(self.__f2) |
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print "X f3:",util.inf(self.__f3), util.sup(self.__f3) |
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|
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v_old=v |
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v,p=self.solve(v,p,max_iter=max_inner_iter, verbose=verbose, show_details=show_details, solver=solver) |
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# update stress |
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stress=stress_last+dt*self.getStressChange(v) |
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stress-=util.trace(stress)*(util.kronecker(self.getDomain())/self.getDomain().getDim()) |
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tau=util.sqrt(0.5)*util.length(stress) |
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# calculate error: |
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e=util.Lsup(v_old-v)/util.Lsup(v) |
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# state variables: |
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self.__t+=dt |
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self.__v=v |
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self.__p=p |
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self.__stress=stress |
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self.__tau=tau |
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self.__D=util.symmetric(util.grad(v)) |
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|
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def getStressChange(self,v): |
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""" |
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returns the stress change due to a given velocity field v |
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""" |
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stress=self.getStress() |
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g=util.grad(v) |
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D=util.symmetric(g) |
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W=util.nonsymmetric(g) |
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U=util.nonsymmetric(util.tensor_mult(W,stress)) |
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dstress=2*self.__f3*D-self.__f0*util.inner(stress,D)*stress-self.__f1*stress-2*self.__f2*U |
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return dstress |
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|
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def B(self,arg): |
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""" |
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div operator |
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""" |
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d=util.div(arg) |
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self.__pde_proj.setValue(Y=d) |
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self.__pde_proj.setTolerance(self.getSubProblemTolerance()) |
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return self.__pde_proj.getSolution(verbose=self.show_details) |
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|
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def solve_prec(self,p): |
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""" |
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preconditioner |
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""" |
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#proj=Projector(domain=self.getDomain(), reduce = True, fast=False) |
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self.__pde_prec.setTolerance(self.getSubProblemTolerance()) |
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self.__pde_prec.setValue(Y=p) |
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q=self.__pde_prec.getSolution(verbose=self.show_details) |
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return q |
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|
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def inner(self,p0,p1): |
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""" |
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inner product for pressure |
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""" |
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s0=util.interpolate(p0,Function(self.getDomain())) |
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s1=util.interpolate(p1,Function(self.getDomain())) |
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return util.integrate(s0*s1) |
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|
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def solve_A(self,u,p): |
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""" |
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solves Av=f-Au-B^*p (v=0 on fixed_u_mask) |
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""" |
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self.__pde_u.setTolerance(self.getSubProblemTolerance()) |
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self.__pde_u.setValue(X=-self.getStressChange(u)-p*util.kronecker(self.getDomain())) |
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return self.__pde_u.getSolution(verbose=self.show_details) |
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|
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##############Added by Artak ########################## |
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def inner_a(self,a0,a1): |
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p0=util.interpolate(a0[1],Function(self.domain)) |
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p1=util.interpolate(a1[1],Function(self.domain)) |
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alfa=(1/self.vol)*util.integrate(p0) |
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beta=(1/self.vol)*util.integrate(p1) |
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v0=util.grad(a0[0]) |
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v1=util.grad(a1[0]) |
| 407 |
return util.integrate((p0-alfa)*(p1-beta)+((1/self.eta)**2)*util.inner(v0,v1)) |
| 408 |
|
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def stoppingcriterium(self,Bv,v,p): |
| 410 |
n_r=util.sqrt(self.inner(Bv,Bv)) |
| 411 |
n_v=util.sqrt(util.integrate(util.length(util.grad(v))**2)) |
| 412 |
if self.verbose: print "PCG step %s: L2(div(v)) = %s, L2(grad(v))=%s"%(self.iter,n_r,n_v) |
| 413 |
if self.iter == 0: self.__n_v=n_v; |
| 414 |
self.__n_v, n_v_old =n_v, self.__n_v |
| 415 |
self.iter+=1 |
| 416 |
if self.iter>1 and n_r <= n_v*self.getTolerance() and abs(n_v_old-self.__n_v) <= n_v * self.getTolerance(): |
| 417 |
if self.verbose: print "PCG terminated after %s steps."%self.iter |
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return True |
| 419 |
else: |
| 420 |
return False |
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|
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def stoppingcriterium2(self,norm_r,norm_b,solver='GMRES',TOL=None): |
| 423 |
if TOL==None: |
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TOL=self.getTolerance() |
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if self.verbose: print "%s step %s: L2(r) = %s, L2(b)*TOL=%s"%(solver,self.iter,norm_r,norm_b*TOL) |
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self.iter+=1 |
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|
| 428 |
if norm_r <= norm_b*TOL: |
| 429 |
if self.verbose: print "%s terminated after %s steps."%(solver,self.iter) |
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return True |
| 431 |
else: |
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return False |
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|
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###################################################### |
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