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######################################################## |
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# |
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# Copyright (c) 2003-2008 by University of Queensland |
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# Earth Systems Science Computational Center (ESSCC) |
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# http://www.uq.edu.au/esscc |
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# |
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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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__copyright__="""Copyright (c) 2003-2008 by University of Queensland |
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Earth Systems Science Computational Center (ESSCC) |
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http://www.uq.edu.au/esscc |
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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.uq.edu.au/esscc/escript-finley" |
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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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|
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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 Defect, NewtonGMRES, ArithmeticTuple |
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|
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class PowerLaw(object): |
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""" |
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this implements the power law for a composition of a set of materials where the viscosity eta of each material is given by a |
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power law relationship of the form |
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|
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M{eta=eta_N*(tau/tau_t)**(1.-1./power)} |
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|
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where tau is equivalent stress and eta_N, tau_t and power are given constant. Moreover an elastic component can be considered. |
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Moreover tau meets the Drucker-Prager type yield condition |
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|
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M{tau <= tau_Y + friction * pressure} |
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|
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where gamma_dot is the equivalent. |
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""" |
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def __init__(self, numMaterials=1,verbose=False): |
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""" |
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initializes a power law |
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|
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@param numMaterials: number of materials |
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@type numMaterials: C{int} |
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@param verbose: if C{True} some informations are printed. |
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@type verbose: C{bool} |
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""" |
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if numMaterials<1: |
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raise ValueError,"at least one material must be defined." |
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self.__numMaterials=numMaterials |
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self.__eta_N=[None for i in xrange(self.__numMaterials)] |
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self.__tau_t=[1. for i in xrange(self.__numMaterials)] |
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self.__power=[1. for i in xrange(self.__numMaterials)] |
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self.__tau_Y=None |
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self.__friction=None |
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self.__mu=None |
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self.__verbose=verbose |
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self.setEtaTolerance() |
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#=========================================================================== |
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def getNumMaterials(self): |
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""" |
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returns the numebr of materials |
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|
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@return: number of materials |
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@rtype: C{int} |
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""" |
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return self.__numMaterials |
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def validMaterialId(self,id=0): |
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""" |
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checks if a given material id is valid |
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|
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@param id: a material id |
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@type id: C{int} |
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@return: C{True} is the id is valid |
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@rtype: C{bool} |
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""" |
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return 0<=id and id<self.getNumMaterials() |
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def setEtaTolerance(self,rtol=util.sqrt(util.EPSILON)): |
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""" |
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sets the relative tolerance for the effectice viscosity. |
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|
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@param rtol: relative tolerance |
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@type rtol: positive C{float} |
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""" |
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if rtol<=0: |
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raise ValueError,"rtol needs to positive." |
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self.__rtol=rtol |
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def getEtaTolerance(self): |
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""" |
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returns the relative tolerance for the effectice viscosity. |
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|
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@return: relative tolerance |
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@rtype rtol: positive C{float} |
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""" |
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return self.__rtol |
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#=========================================================================== |
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def setDruckerPragerLaw(self,tau_Y=None,friction=None): |
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""" |
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Sets the parameters for the Drucker-Prager model. |
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|
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@param tau_Y: yield stress |
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@param friction: friction coefficient |
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""" |
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self.__tau_Y=tau_Y |
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self.__friction=friction |
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def getFriction(self): |
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""" |
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returns the friction coefficient |
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|
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@return: friction coefficient |
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""" |
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return self.__friction |
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def getTauY(self): |
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""" |
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returns the yield stress |
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|
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@return: the yield stress |
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""" |
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return self.__tau_Y |
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#=========================================================================== |
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def getElasticShearModulus(self): |
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""" |
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returns the elastic shear modulus. |
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|
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@return: elastic shear modulus |
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""" |
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return self.__mu |
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def setElasticShearModulus(self,mu=None): |
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""" |
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Sets the elastic shear modulus. |
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|
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@param mu: elastic shear modulus |
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""" |
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self.__mu=mu |
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#=========================================================================== |
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def getPower(self, id=None): |
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""" |
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returns the power in the power law |
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|
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@param id: if present, the power for material C{id} is returned. |
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@type id: C{int} |
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@return: the list of the powers for all matrials is returned. If C{id} is present only the power for material C{id} is returned. |
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""" |
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if id == None: |
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return self.__power |
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else: |
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if self.validMaterialId(id): |
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return self.__power[id] |
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else: |
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raise ValueError,"Illegal material id %s."%id |
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def getEtaN(self, id=None): |
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""" |
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returns the viscosity |
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|
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@param id: if present, the viscosity for material C{id} is returned. |
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@type id: C{int} |
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@return: the list of the viscosities for all matrials is returned. If C{id} is present only the viscosity for material C{id} is returned. |
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""" |
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if id == None: |
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return self.__eta_N |
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else: |
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if self.validMaterialId(id): |
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return self.__eta_N[id] |
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else: |
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raise ValueError,"Illegal material id %s."%id |
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def getTauT(self, id=None): |
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""" |
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returns the transition stress |
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|
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@param id: if present, the transition stress for material C{id} is returned. |
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@type id: C{int} |
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@return: the list of the transition stresses for all matrials is returned. If C{id} is present only the transition stress for material C{id} is returned. |
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""" |
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if id == None: |
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return self.__tau_t |
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else: |
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if self.validMaterialId(id): |
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return self.__tau_t[id] |
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else: |
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raise ValueError,"Illegal material id %s."%id |
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|
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def setPowerLaw(self,eta_N, id=0, tau_t=1, power=1): |
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""" |
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Sets the power-law parameters for material id |
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|
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@param id: material id |
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@type id: C{int} |
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@param eta_N: viscosity for tau=tau_t |
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@param tau_t: transition stress |
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@param power: power law coefficient |
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""" |
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if self.validMaterialId(id): |
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self.__eta_N[id]=eta_N |
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self.__power[id]=power |
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self.__tau_t[id]=tau_t |
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else: |
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raise ValueError,"Illegal material id %s."%id |
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|
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def setPowerLaws(self,eta_N, tau_t, power): |
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""" |
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Sets the parameters of the power-law for all materials. |
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|
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@param eta_N: list of viscosities for tau=tau_t |
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@param tau_t: list of transition stresses |
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@param power: list of power law coefficient |
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""" |
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if len(eta_N)!=self.__numMaterials or len(tau_t)!=self.__numMaterials or len(power)!=self.__numMaterials: |
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raise ValueError,"%s materials are expected."%self.__numMaterials |
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for i in xrange(self.__numMaterials): |
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self.setPowerLaw(id=i, eta_N=eta_N[i],tau_t=tau_t[i],power=power[i]) |
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|
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#=========================================================================== |
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def getEtaEff(self,gamma_dot, eta0=None, pressure=None,dt=None, iter_max=10): |
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""" |
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returns the effective viscosity eta_eff such that |
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|
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M{tau=eta_eff * gamma_dot} |
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|
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by solving a non-linear problem for tau. |
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|
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@param gamma_dot: equivalent strain gamma_dot |
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@param eta0: initial guess for the effective viscosity (e.g from a previous time step). If not present, an initial guess is calculated. |
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@param pressure: pressure used to calculate yield condition |
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@param dt: time step size. only needed if elastic component is considered. |
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@type dt: positive C{float} if present |
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@param iter_max: maximum number of iteration steps. |
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@type iter_max: C{int} |
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@return: effective viscosity. |
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""" |
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SMALL=1./(util.DBLE_MAX/100.) |
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numMaterial=self.getNumMaterials() |
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s=[1.-1./p for p in self.getPower() ] |
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eta_N=self.getEtaN() |
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tau_t=self.getTauT() |
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mu=self.getElasticShearModulus() |
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fric=self.getFriction() |
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tau_Y=self.getTauY() |
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if eta0==None: |
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theta=0. |
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for i in xrange(numMaterial): |
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inv_eta_i=0**s[i]/eta_N[i] |
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theta=theta+inv_eta_i |
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if util.inf(theta)<=0: |
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raise ValueError,"unable to set positive initial guess for eta_eff. Most likely no power law with power 1 set." |
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eta_eff=1./theta |
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else: |
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if util.inf(eta0)<=0: |
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raise ValueError,"initial guess for eta_eff is not positive." |
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eta_eff=eta0 |
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|
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if mu !=None and dt == None: |
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raise ValueError,"Time stepsize dt must be given." |
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if dt !=None: |
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if dt<=0: raise ValueError,"time step size must be positive." |
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if tau_Y==None and fric==None: |
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eta_max=None |
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else: |
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if fric == None: |
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eta_max=tau_Y/(gamma_dot+SMALL*util.whereZero(gamma_dot)) |
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else: |
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if tau_Y==None: tau_Y==0 |
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if util.inf(fric)<=0: |
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raise ValueError,"if friction present it needs to be positive." |
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eta_max=fric*util.clip(tau_Y/fric+pressure,minval=0)/(gamma_dot+SMALL*util.whereZero(gamma_dot)) |
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rtol=self.getEtaTolerance() |
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iter =0 |
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converged=False |
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tau=eta_eff*gamma_dot |
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if self.__verbose: print "Start calculation of eta_eff (tolerance = %s)\ninitial max eta_eff = %s, tau = %s."%(rtol,util.Lsup(eta_eff),util.Lsup(tau)) |
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while not converged: |
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if iter>max(iter_max,1): |
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raise RuntimeError,"tolerance not reached after %s steps."%max(iter_max,1) |
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#=========================================== |
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theta=0. # =1/eta |
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omega=0. # = tau*theta'= eta'*tau/eta**2 |
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if mu !=None: theta=1./(dt*mu) |
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for i in xrange(numMaterial): |
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inv_eta_i=(tau/tau_t[i])**s[i]/eta_N[i] |
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theta=theta+inv_eta_i |
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omega=omega+s[i]*inv_eta_i |
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#=========================================== |
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eta_eff, eta_eff_old=util.clip(eta_eff*(theta+omega)/(eta_eff*theta**2+omega),maxval=eta_max), eta_eff |
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tau=eta_eff*gamma_dot |
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d=util.Lsup(eta_eff-eta_eff_old) |
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l=util.Lsup(eta_eff) |
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iter+=1 |
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if self.__verbose: print "step %s: correction = %s, max eta_eff = %s, max tau= %s"%(iter, d, l,util.Lsup(tau)) |
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converged= d<= rtol* l |
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return eta_eff |
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|
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#==================================================================================================================================== |
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|
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class IncompressibleIsotropicKelvinFlow(Defect): |
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""" |
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This class implements the rheology of an isotropic Kelvin material. |
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|
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Typical usage:: |
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|
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sp = IncompressibleIsotropicKelvinFlow(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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@note: This model has been used in the self-consistent plate-mantle model |
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proposed in U{Hans-Bernd Muhlhaus<emailto:h.muhlhaus@uq.edu.au>} |
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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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""" |
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def __init__(self, domain, stress=0, v=0, p=0, t=0, numMaterials=1, useJaumannStress=True, **kwargs): |
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""" |
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Initializes the model. |
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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 deviatoric stress |
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@param v: initial velocity field |
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@param p: initial pressure |
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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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(not supported yet) |
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""" |
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super(IncompressibleIsotropicKelvinFlow, self).__init__(**kwargs) |
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self.__domain=domain |
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self.__t=t |
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self.__vol=util.integrate(1.,Function(self.__domain)) |
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self.__useJaumannStress=useJaumannStress |
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self.__v_boundary=Vector(0,Solution(self.__domain)) |
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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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#======================= |
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# PDE related stuff |
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self.__pde_v=LinearPDE(domain,numEquations=self.getDomain().getDim(),numSolutions=self.getDomain().getDim()) |
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self.__pde_v.setSymmetryOn() |
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self.__pde_v.setSolverMethod(preconditioner=LinearPDE.RILU) |
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|
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self.__pde_p=LinearPDE(domain) |
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self.__pde_p.setReducedOrderOn() |
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self.__pde_p.setSymmetryOn() |
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|
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self.setTolerance() |
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self.setSmall() |
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|
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def useJaumannStress(self): |
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""" |
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Returns C{True} if Jaumann stress is included. |
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""" |
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return self.__useJaumannStress |
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|
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def setSmall(self,small=util.sqrt(util.EPSILON)): |
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""" |
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Sets a small value to be used. |
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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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|
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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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|
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def setTolerance(self,tol=1.e-4): |
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""" |
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Sets the tolerance. |
| 397 |
""" |
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self.__pde_v.setTolerance(tol**2) |
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self.__pde_p.setTolerance(tol**2) |
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self.__tol=tol |
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|
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def getTolerance(self): |
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""" |
| 404 |
Returns the set tolerance. |
| 405 |
@rtype: positive float |
| 406 |
""" |
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return self.__tol |
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|
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def getDomain(self): |
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""" |
| 411 |
Returns the domain. |
| 412 |
""" |
| 413 |
return self.__domain |
| 414 |
|
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def getTime(self): |
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""" |
| 417 |
Returns current time. |
| 418 |
""" |
| 419 |
return self.__t |
| 420 |
|
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def setExternals(self, F=None, f=None, q=None, v_boundary=None): |
| 422 |
""" |
| 423 |
Sets externals. |
| 424 |
|
| 425 |
@param F: external force |
| 426 |
@param f: surface force |
| 427 |
@param q: location of constraints |
| 428 |
""" |
| 429 |
if F != None: self.__pde_v.setValue(Y=F) |
| 430 |
if f != None: self.__pde_v.setValue(y=f) |
| 431 |
if q != None: self.__pde_v.setValue(q=q) |
| 432 |
if v_boundary != None: self.__v_boundary=v_boundary |
| 433 |
|
| 434 |
def bilinearform(self,arg0,arg1): |
| 435 |
s0=util.deviatoric(util.symmetric(util.grad(arg0[0]))) |
| 436 |
s1=util.deviatoric(util.symmetric(util.grad(arg1[0]))) |
| 437 |
# s0=util.interpolate(arg0[0],Function(self.getDomain())) |
| 438 |
# s1=util.interpolate(arg1[0],Function(self.getDomain())) |
| 439 |
p0=util.interpolate(arg0[1],Function(self.getDomain())) |
| 440 |
p1=util.interpolate(arg1[1],Function(self.getDomain())) |
| 441 |
a=util.integrate(self.__p_weight**2*util.inner(s0,s1))+util.integrate(p0*p1) |
| 442 |
return a |
| 443 |
|
| 444 |
def getEtaEff(self,strain, pressure): |
| 445 |
if self.__mu==None: |
| 446 |
eps=util.length(strain)*util.sqrt(2) |
| 447 |
else: |
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eps=util.length(strain+self.__stress/((2*self.__dt)*self.__mu))*util.sqrt(2) |
| 449 |
p=util.interpolate(pressure,eps.getFunctionSpace()) |
| 450 |
if self.__tau_Y!= None: |
| 451 |
tmp=self.__tau_Y+self.__friction*p |
| 452 |
m=util.wherePositive(eps)*util.wherePositive(tmp) |
| 453 |
eta_max=m*tmp/(eps+(1-m)*util.EPSILON)+(1-m)*util.DBLE_MAX |
| 454 |
else: |
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eta_max=util.DBLE_MAX |
| 456 |
# initial guess: |
| 457 |
tau=util.length(self.__stress)/util.sqrt(2) |
| 458 |
# start the iteration: |
| 459 |
cc=0 |
| 460 |
TOL=1e-7 |
| 461 |
dtau=util.DBLE_MAX |
| 462 |
print "tau = ", tau, "eps =",eps |
| 463 |
while cc<10 and dtau>TOL*util.Lsup(tau): |
| 464 |
eta_eff2,eta_eff_dash=self.evalEtaEff(tau,return_dash=True) |
| 465 |
eta_eff=util.clip(eta_eff2-eta_eff_dash*tau/(1-eta_eff_dash*eps),maxval=eta_max) |
| 466 |
tau, tau2=eta_eff*eps, tau |
| 467 |
dtau=util.Lsup(tau2-tau) |
| 468 |
print "step ",cc,dtau, util.Lsup(tau) |
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cc+=1 |
| 470 |
return eta_eff |
| 471 |
|
| 472 |
def getEtaCharacteristic(self): |
| 473 |
a=0 |
| 474 |
for i in xrange(self.__numMaterials): |
| 475 |
a=a+1./self.__eta_N[i] |
| 476 |
return 1/a |
| 477 |
|
| 478 |
def evalEtaEff(self, tau, return_dash=False): |
| 479 |
a=Scalar(0,tau.getFunctionSpace()) # =1/eta |
| 480 |
if return_dash: a_dash=Scalar(0,tau.getFunctionSpace()) # =(1/eta)' |
| 481 |
s=util.Lsup(tau) |
| 482 |
if s>0: |
| 483 |
m=util.wherePositive(tau) |
| 484 |
tau2=s*util.EPSILON*(1.-m)+m*tau |
| 485 |
for i in xrange(self.__numMaterials): |
| 486 |
eta_N=self.__eta_N[i] |
| 487 |
tau_t=self.__tau_t[i] |
| 488 |
if tau_t==None: |
| 489 |
a+=1./eta_N |
| 490 |
else: |
| 491 |
power=1.-1./self.__power[i] |
| 492 |
c=1./(tau_t**power*eta_N) |
| 493 |
a+=c*tau2**power |
| 494 |
if return_dash: a_dash+=power*c*tau2**(power-1.) |
| 495 |
else: |
| 496 |
for i in xrange(self.__numMaterials): |
| 497 |
eta_N=self.__eta_N[i] |
| 498 |
power=1.-1./self.__power[i] |
| 499 |
a+=util.whereZero(power)/eta_N |
| 500 |
if self.__mu!=None: a+=1./(self.__dt*self.__mu) |
| 501 |
out=1/a |
| 502 |
if return_dash: |
| 503 |
return out,-out**2*a_dash |
| 504 |
else: |
| 505 |
return out |
| 506 |
|
| 507 |
def eval(self,arg): |
| 508 |
v=arg[0] |
| 509 |
p=arg[1] |
| 510 |
D=self.getDeviatoricStrain(v) |
| 511 |
eta_eff=self.getEtaEff(D,p) |
| 512 |
print "eta_eff=",eta_eff |
| 513 |
# solve for dv |
| 514 |
self.__pde_v.setValue(A=Data()) # save memory! |
| 515 |
k3=util.kronecker(Function(self.getDomain())) |
| 516 |
k3Xk3=util.outer(k3,k3) |
| 517 |
self.__pde_v.setValue(A=eta_eff*(util.swap_axes(k3Xk3,0,3)+util.swap_axes(k3Xk3,1,3)),X=-eta_eff*D+p*util.kronecker(self.getDomain())) |
| 518 |
dv=self.__pde_v.getSolution(verbose=self.__verbose) |
| 519 |
print "resistep dv =",dv |
| 520 |
# solve for dp |
| 521 |
v2=v+dv |
| 522 |
self.__pde_p.setValue(D=1/eta_eff,Y=util.div(v2)) |
| 523 |
dp=self.__pde_p.getSolution(verbose=self.__verbose) |
| 524 |
print "resistep dp =",dp |
| 525 |
return ArithmeticTuple(dv,dp) |
| 526 |
|
| 527 |
def update(self, dt, iter_max=100, inner_iter_max=20, verbose=False): |
| 528 |
""" |
| 529 |
Updates stress, velocity and pressure for time increment dt. |
| 530 |
|
| 531 |
@param dt: time increment |
| 532 |
@param inner_iter_max: maximum number of iteration steps in the |
| 533 |
incompressible solver |
| 534 |
@param verbose: prints some infos in the incompressible solver |
| 535 |
""" |
| 536 |
self.__verbose=verbose |
| 537 |
self.__dt=dt |
| 538 |
tol=self.getTolerance() |
| 539 |
# set the initial velocity: |
| 540 |
m=util.wherePositive(self.__pde_v.getCoefficient("q")) |
| 541 |
v_new=self.__v*(1-m)+self.__v_boundary*m |
| 542 |
# and off we go: |
| 543 |
x=ArithmeticTuple(v_new, self.__p) |
| 544 |
# self.__p_weight=util.interpolate(1./self.getEtaCharacteristic(),Function(self.__domain))**2 |
| 545 |
self.__p_weight=self.getEtaCharacteristic() |
| 546 |
# self.__p_weight=util.interpolate(1./self.getEtaCharacteristic()**2,self.__p.getFunctionSpace()) |
| 547 |
atol=self.norm(x)*self.__tol |
| 548 |
x_new=NewtonGMRES(self, x, iter_max=iter_max,sub_iter_max=inner_iter_max, atol=atol,rtol=0., verbose=verbose) |
| 549 |
self.__v=x_new[0] |
| 550 |
self.__p=x_new[1] |
| 551 |
1/0 |
| 552 |
# self.__stress=self.getUpdatedStress(...) |
| 553 |
self.__t+=dt |
| 554 |
return self.__v, self.__p |
| 555 |
|
| 556 |
#======================================================================== |
| 557 |
|
| 558 |
def getNewDeviatoricStress(self,D,eta_eff=None): |
| 559 |
if eta_eff==None: eta_eff=self.evalEtaEff(self.__stress,D,self.__p) |
| 560 |
s=(2*eta_eff)*D |
| 561 |
if self.__mu!=None: s+=eta_eff/(self.__dt*self.__mu)*self.__last_stress |
| 562 |
return s |
| 563 |
|
| 564 |
def getDeviatoricStress(self): |
| 565 |
""" |
| 566 |
Returns current stress. |
| 567 |
""" |
| 568 |
return self.__stress |
| 569 |
|
| 570 |
def getDeviatoricStrain(self,velocity=None): |
| 571 |
""" |
| 572 |
Returns strain. |
| 573 |
""" |
| 574 |
if velocity==None: velocity=self.getVelocity() |
| 575 |
return util.deviatoric(util.symmetric(util.grad(velocity))) |
| 576 |
|
| 577 |
def getPressure(self): |
| 578 |
""" |
| 579 |
Returns current pressure. |
| 580 |
""" |
| 581 |
return self.__p |
| 582 |
|
| 583 |
def getVelocity(self): |
| 584 |
""" |
| 585 |
Returns current velocity. |
| 586 |
""" |
| 587 |
return self.__v |
| 588 |
|
| 589 |
def getTau(self,stress=None): |
| 590 |
""" |
| 591 |
Returns current second stress deviatoric invariant. |
| 592 |
""" |
| 593 |
if stress==None: stress=self.getDeviatoricStress() |
| 594 |
return util.sqrt(0.5)*util.length(stress) |
| 595 |
|
| 596 |
def getGammaDot(self,strain=None): |
| 597 |
""" |
| 598 |
Returns current second stress deviatoric invariant. |
| 599 |
""" |
| 600 |
if strain==None: strain=self.getDeviatoricStrain() |
| 601 |
return util.sqrt(2)*util.length(strain) |
| 602 |
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