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#
# Copyright (c) 2003-2017 by The University of Queensland
# http://www.uq.edu.au
#
# Primary Business: Queensland, Australia
# Licensed under the Apache License, version 2.0
# http://www.apache.org/licenses/LICENSE-2.0
#
# Development until 2012 by Earth Systems Science Computational Center (ESSCC)
# Development 2012-2013 by School of Earth Sciences
# Development from 2014 by Centre for Geoscience Computing (GeoComp)
#
##############################################################################
from __future__ import print_function, division
__copyright__="""Copyright (c) 2003-2017 by The University of Queensland
http://www.uq.edu.au
Primary Business: Queensland, Australia"""
__license__="""Licensed under the Apache License, version 2.0
http://www.apache.org/licenses/LICENSE-2.0"""
__url__="https://launchpad.net/escript-finley"
"""
Some models for flow
:var __author__: name of author
:var __copyright__: copyrights
:var __license__: licence agreement
:var __url__: url entry point on documentation
:var __version__: version
:var __date__: date of the version
"""
__author__="Lutz Gross, l.gross@uq.edu.au"
from . import escriptcpp as escore
from . import util
from .flows import StokesProblemCartesian
from .pdetools import MaxIterReached
class PowerLaw(object):
"""
this implements the power law for a composition of a set of materials where the viscosity eta of each material is given by a
power law relationship of the form
*eta=eta_N*(tau/tau_t)**(1./power-1.)*
where tau is equivalent stress and eta_N, tau_t and power are given constant. Moreover an elastic component can be considered.
Moreover tau meets the Drucker-Prager type yield condition
*tau <= tau_Y + friction * pressure*
where gamma_dot is the equivalent.
"""
def __init__(self, numMaterials=1,verbose=False):
"""
initializes a power law
:param numMaterials: number of materials
:type numMaterials: ``int``
:param verbose: if ``True`` some information is printed.
:type verbose: ``bool``
"""
if numMaterials<1:
raise ValueError("at least one material must be defined.")
self.__numMaterials=numMaterials
self.__eta_N=[None for i in range(self.__numMaterials)]
self.__tau_t=[1. for i in range(self.__numMaterials)]
self.__power=[1. for i in range(self.__numMaterials)]
self.__tau_Y=None
self.__friction=None
self.__mu=None
self.__verbose=verbose
self.setEtaTolerance()
#===========================================================================
def getNumMaterials(self):
"""
returns the numebr of materials
:return: number of materials
:rtype: ``int``
"""
return self.__numMaterials
def validMaterialId(self,id=0):
"""
checks if a given material id is valid
:param id: a material id
:type id: ``int``
:return: ``True`` is the id is valid
:rtype: ``bool``
"""
return 0<=id and id<self.getNumMaterials()
def setEtaTolerance(self,rtol=1.e-4):
"""
sets the relative tolerance for the effectice viscosity.
:param rtol: relative tolerance
:type rtol: positive ``float``
"""
if rtol<=0:
raise ValueError("rtol needs to positive.")
self.__rtol=rtol
def getEtaTolerance(self):
"""
returns the relative tolerance for the effectice viscosity.
:return: relative tolerance
:rtype: positive ``float``
"""
return self.__rtol
#===========================================================================
def setDruckerPragerLaw(self,tau_Y=None,friction=None):
"""
Sets the parameters for the Drucker-Prager model.
:param tau_Y: yield stress
:param friction: friction coefficient
"""
self.__tau_Y=tau_Y
self.__friction=friction
def getFriction(self):
"""
returns the friction coefficient
:return: friction coefficient
"""
return self.__friction
def getTauY(self):
"""
returns the yield stress
:return: the yield stress
"""
return self.__tau_Y
#===========================================================================
def getElasticShearModulus(self):
"""
returns the elastic shear modulus.
:return: elastic shear modulus
"""
return self.__mu
def setElasticShearModulus(self,mu=None):
"""
Sets the elastic shear modulus.
:param mu: elastic shear modulus
"""
self.__mu=mu
#===========================================================================
def getPower(self, id=None):
"""
returns the power in the power law
:param id: if present, the power for material ``id`` is returned.
:type id: ``int``
:return: the list of the powers for all matrials is returned. If ``id`` is present only the power for material ``id`` is returned.
"""
if id is None:
return self.__power
else:
if self.validMaterialId(id):
return self.__power[id]
else:
raise ValueError("Illegal material id %s."%id)
def getEtaN(self, id=None):
"""
returns the viscosity
:param id: if present, the viscosity for material ``id`` is returned.
:type id: ``int``
:return: the list of the viscosities for all matrials is returned. If ``id`` is present only the viscosity for material ``id`` is returned.
"""
if id is None:
return self.__eta_N
else:
if self.validMaterialId(id):
return self.__eta_N[id]
else:
raise ValueError("Illegal material id %s."%id)
def getTauT(self, id=None):
"""
returns the transition stress
:param id: if present, the transition stress for material ``id`` is returned.
:type id: ``int``
:return: the list of the transition stresses for all matrials is returned. If ``id`` is present only the transition stress for material ``id`` is returned.
"""
if id is None:
return self.__tau_t
else:
if self.validMaterialId(id):
return self.__tau_t[id]
else:
raise ValueError("Illegal material id %s."%id)
def setPowerLaw(self,eta_N, id=0, tau_t=1, power=1):
"""
Sets the power-law parameters for material id
:param id: material id
:type id: ``int``
:param eta_N: viscosity for tau=tau_t
:param tau_t: transition stress
:param power: power law coefficient
"""
if self.validMaterialId(id):
self.__eta_N[id]=eta_N
self.__power[id]=power
self.__tau_t[id]=tau_t
else:
raise ValueError("Illegal material id %s."%id)
def setPowerLaws(self,eta_N, tau_t, power):
"""
Sets the parameters of the power-law for all materials.
:param eta_N: list of viscosities for tau=tau_t
:param tau_t: list of transition stresses
:param power: list of power law coefficient
"""
if len(eta_N)!=self.__numMaterials or len(tau_t)!=self.__numMaterials or len(power)!=self.__numMaterials:
raise ValueError("%s materials are expected."%self.__numMaterials)
for i in range(self.__numMaterials):
self.setPowerLaw(id=i, eta_N=eta_N[i],tau_t=tau_t[i],power=power[i])
#===========================================================================
def getEtaEff(self,gamma_dot, eta0=None, pressure=None,dt=None, iter_max=30):
"""
returns the effective viscosity eta_eff such that
*tau=eta_eff * gamma_dot*
by solving a non-linear problem for tau.
:param gamma_dot: equivalent strain gamma_dot
:param eta0: initial guess for the effective viscosity (e.g from a previous time step). If not present, an initial guess is calculated.
:param pressure: pressure used to calculate yield condition
:param dt: time step size. only needed if elastic component is considered.
:type dt: positive ``float`` if present
:param iter_max: maximum number of iteration steps.
:type iter_max: ``int``
:return: effective viscosity.
"""
if pressure is None:
p2 = None
else:
p2=(abs(pressure)+pressure)/2.
SMALL=1./(util.DBLE_MAX/100.)
numMaterial=self.getNumMaterials()
s=[p-1. for p in self.getPower() ]
eta_N=self.getEtaN()
tau_t=self.getTauT()
mu=self.getElasticShearModulus()
fric=self.getFriction()
tau_Y=self.getTauY()
if eta0 is None:
theta=0.
for i in range(numMaterial):
inv_eta_i=0**s[i]/eta_N[i]
theta=theta+inv_eta_i
if util.inf(theta)<=0:
raise ValueError("unable to set positive initial guess for eta_eff. Most likely no power law with power 1 set.")
eta_eff=1./theta
else:
if util.inf(eta0)<=0:
raise ValueError("initial guess for eta_eff is not positive.")
eta_eff=eta0
if mu !=None:
if dt is None: raise ValueError("Time stepsize dt must be given.")
if dt<=0: raise ValueError("Time step size must be positive.")
if tau_Y is None and fric is None:
eta_max=None
else:
if fric is None or p2 is None:
eta_max=tau_Y/(gamma_dot+SMALL*util.whereZero(gamma_dot))
else:
if tau_Y is None: tau_Y==0
if util.inf(fric)<=0:
raise ValueError("if friction present it needs to be positive.")
eta_max=fric*util.clip(tau_Y/fric+p2,minval=0)/(gamma_dot+SMALL*util.whereZero(gamma_dot))
rtol=self.getEtaTolerance()
iter =0
converged=False
tau=eta_eff*gamma_dot
if self.__verbose: print(("PowerLaw: Start calculation of eta_eff (tolerance = %s)\nPowerLaw: initial max eta_eff = %s, tau = %s."%(rtol,util.Lsup(eta_eff),util.Lsup(tau))))
while not converged:
if iter>max(iter_max,1):
raise RuntimeError("tolerance not reached after %s steps."%max(iter_max,1))
#===========================================
theta=0. # =1/eta
omega=0. # = tau*theta'= eta'*tau/eta**2
if mu !=None: theta=1./(dt*mu)
for i in range(numMaterial):
inv_eta_i=(tau/tau_t[i])**s[i]/eta_N[i]
theta=theta+inv_eta_i
omega=omega+s[i]*inv_eta_i
#===========================================
eta_eff, eta_eff_old=util.clip(eta_eff*(theta+omega)/(eta_eff*theta**2+omega),maxval=eta_max), eta_eff
tau=eta_eff*gamma_dot
d=util.Lsup(eta_eff-eta_eff_old)
l=util.Lsup(eta_eff)
iter+=1
if self.__verbose: print(("PowerLaw: step %s: correction = %s, max eta_eff = %s, max tau= %s"%(iter, d, l,util.Lsup(tau))))
converged= d<= rtol* l
if self.__verbose: print(("PowerLaw: Start calculation of eta_eff finalized after %s steps."%iter))
return eta_eff
#====================================================================================================================================
class Rheology(object):
"""
General framework to implement a rheology
"""
def __init__(self, domain, stress=None, v=None, p=None, t=0, verbose=True):
"""
Initializes the rheology
:param domain: problem domain
:type domain: `Domain`
:param stress: initial (deviatoric) stress
:type stress: a tensor value/field of order 2
:param v: initial velocity field
:type v: a vector value/field
:param p: initial pressure
:type p: a scalar value/field
:param t: initial time
:type t: ``float``
"""
self.__domain=domain
self.__t=t
self.__verbose=verbose
#=======================
#
# state variables:
#
if stress is None: stress=Tensor(0.,escore.Function(self.__domain))
if v is None: v=Vector(0.,escore.Solution(self.__domain))
if p is None: p=Vector(0.,escore.ReducedSolution(self.__domain))
self.setStatus(t, v, p, stress)
self.setExternals(F=escore.Data(), f=escore.Data(), fixed_v_mask=escore.Data(), v_boundary=escore.Data(), restoration_factor=0)
def getDomain(self):
"""
returns the domain.
:return: the domain
:rtype: `Domain`
"""
return self.__domain
def getTime(self):
"""
Returns current time.
:return: current time
:rtype: ``float``
"""
return self.__t
def setExternals(self, F=None, f=None, fixed_v_mask=None, v_boundary=None, restoration_factor=None):
"""
sets external forces and velocity constraints
:param F: external force
:type F: vector value/field
:param f: surface force
:type f: vector value/field on boundary
:param fixed_v_mask: location of constraints maked by positive values
:type fixed_v_mask: vector value/field
:param v_boundary: value of velocity at location of constraints
:type v_boundary: vector value/field
:param restoration_factor: factor for normal restoration force
:type restoration_factor: scalar values/field
:note: Only changing parameters need to be specified.
"""
if F is not None: self.__F=F
if f is not None: self.__f=f
if fixed_v_mask is not None: self.__fixed_v_mask=fixed_v_mask
if v_boundary is not None: self.__v_boundary=v_boundary
if restoration_factor is not None: self.__restoration_factor=restoration_factor
def getForce(self):
"""
Returns the external force
:return: external force
:rtype: `Data`
"""
return self.__F
def getSurfaceForce(self):
"""
Returns the surface force
:return: surface force
:rtype: `Data`
"""
return self.__f
def getVelocityConstraint(self):
"""
Returns the constraint for the velocity as a pair of the
mask of the location of the constraint and the values.
:return: the locations of fixed velocity and value of velocities at these locations
:rtype: ``tuple`` of `Data` s
"""
return self.__fixed_v_mask, self.__v_boundary
def getRestorationFactor(self):
"""
Returns the restoring force factor
:return: restoring force factor
:rtype: `float` or `Data`
"""
return self.__restoration_factor
def checkVerbose(self):
"""
Returns True if verbose is switched on
:return: value of verbosity flag
:rtype: ``bool``
"""
return self.__verbose
def setTime(self,t=0.):
"""
Updates current time.
:param t: new time mark
:type t: ``float``
"""
self.__t=t
#=======================================================================================
def getStress(self):
"""
Returns current stress.
:return: current stress
:rtype: `Data` of rank 2
"""
s=self.getDeviatoricStress()
p=self.getPressure()
k=util.kronecker(self.getDomain())
return s-p*(k/trace(k))
def getDeviatoricStress(self):
"""
Returns current deviatoric stress.
:return: current deviatoric stress
:rtype: `Data` of rank 2
"""
return self.__stress
def setDeviatoricStress(self, stress):
"""
Sets the current deviatoric stress
:param stress: new deviatoric stress
:type stress: `Data` of rank 2
"""
dom=self.getDomain()
s=util.interpolate(stress,escore.Function(dom))
self.__stress=util.deviatoric(s)
def getPressure(self):
"""
Returns current pressure.
:return: current stress
:rtype: scalar `Data`
"""
return self.__p
def setPressure(self, p):
"""
Sets current pressure.
:param p: new deviatoric stress
:type p: scalar `Data`
"""
self.__p=util.interpolate(p,escore.ReducedSolution(self.getDomain()))
def getVelocity(self):
"""
Returns current velocity.
:return: current velocity
:rtype: vector `Data`
"""
return self.__v
def setVelocity(self, v):
"""
Sets current velocity.
:param v: new current velocity
:type v: vector `Data`
"""
self.__v=util.interpolate(v,escore.Solution(self.getDomain()))
def setStatus(self,t, v, p, stress):
"""
Resets the current status given by pressure p and velocity v.
:param t: new time mark
:type t: `float`
:param v: new current velocity
:type v: vector `Data`
:param p: new deviatoric stress
:type p: scalar `Data`
:param stress: new deviatoric stress
:type stress: `Data` of rank 2
"""
self.setDeviatoricStress(stress)
self.setVelocity(v)
self.setPressure(p)
self.setDeviatoricStrain()
self.setGammaDot()
self.setTime(t)
def setDeviatoricStrain(self, D=None):
"""
set deviatoric strain
:param D: new deviatoric strain. If ``D`` is not present the current velocity is used.
:type D: `Data` of rank 2
"""
if D is None:
self.__D=self.getDeviatoricStrain(self.getVelocity())
else:
self.__D=util.deviatoric(util.interpolate(D,escore.Function(self.getDomain())))
def getDeviatoricStrain(self, v=None):
"""
Returns deviatoric strain of current velocity or if ``v`` is present the
deviatoric strain of velocity ``v``:
:param v: a velocity field
:type v: `Data` of rank 1
:return: deviatoric strain of the current velocity field or if ``v`` is present the deviatoric strain of velocity ``v``
:rtype: `Data` of rank 2
"""
if v is None:
return self.__D
else:
return util.deviatoric(util.symmetric(util.grad(v)))
def getTau(self):
"""
Returns current second invariant of deviatoric stress
:return: second invariant of deviatoric stress
:rtype: scalar `Data`
"""
s=self.getDeviatoricStress()
return util.sqrt(0.5)*util.length(s)
def setGammaDot(self, gammadot=None):
"""
set the second invariant of deviatoric strain rate. If ``gammadot`` is not present zero is used.
:param gammadot: second invariant of deviatoric strain rate.
:type gammadot: `Data` of rank 1
"""
if gammadot is None:
self.__gammadot = escore.Scalar(0.,escore.Function(self.getDomain()))
else:
self.__gammadot=gammadot
def getGammaDot(self, D=None):
"""
Returns current second invariant of deviatoric strain rate or if ``D`` is present the second invariant of ``D``.
:param D: deviatoric strain rate tensor
:type D: `Data` of rank 0
:return: second invariant of deviatoric strain
:rtype: scalar `Data`
"""
if D is None:
return self.__gammadot
else:
return util.sqrt(2.)*util.length(D)
#====================================================================================================================================
class IncompressibleIsotropicFlowCartesian(PowerLaw,Rheology, StokesProblemCartesian):
"""
This class implements the rheology of an isotropic Kelvin material.
Typical usage::
sp = IncompressibleIsotropicFlowCartesian(domain, stress=0, v=0)
sp.initialize(...)
v,p = sp.solve()
:note: This model has been used in the self-consistent plate-mantle model
proposed in `Hans-Bernd Muhlhaus <mailto:h.muhlhaus@uq.edu.au>`_
and `Klaus Regenauer-Lieb <mailto:klaus.regenauer-lieb@csiro.au>`_:
"Towards a self-consistent plate mantle model that includes elasticity: simple benchmarks and application to basic modes of convection",
see `doi: 10.1111/j.1365-246X.2005.02742.x <http://www3.interscience.wiley.com/journal/118661486/abstract>`_
"""
def __init__(self, domain, stress=0, v=0, p=0, t=0, numMaterials=1, verbose=True):
"""
Initializes the model.
:param domain: problem domain
:type domain: `Domain`
:param stress: initial (deviatoric) stress
:type stress: a tensor value/field of order 2
:param v: initial velocity field
:type v: a vector value/field
:param p: initial pressure
:type p: a scalar value/field
:param t: initial time
:type t: ``float``
:param numMaterials: number of materials
:type numMaterials: ``int``
:param verbose: if ``True`` some information is printed.
:type verbose: ``bool``
"""
PowerLaw. __init__(self, numMaterials,verbose=verbose)
Rheology. __init__(self, domain, stress, v, p, t,verbose=verbose)
StokesProblemCartesian.__init__(self,domain,verbose=verbose)
self.__eta_eff=None
def getCurrentEtaEff(self):
"""
returns the effective viscosity used in the last iteration step of the last time step.
"""
return self.__eta_eff
def updateStokesEquation(self, v, p):
"""
updates the underlying Stokes equation to consider dependencies from ``v`` and ``p``
"""
dt=self.__dt
mu=self.getElasticShearModulus()
F=self.getForce()
f=self.getSurfaceForce()
mask_v,v_b=self.getVelocityConstraint()
s_last=self.getDeviatoricStress()
#
# calculate eta_eff if we don't have one or elasticity is present.
#
if mu is None:
gamma=self.getGammaDot(self.getDeviatoricStrain(v))
else:
gamma=self.getGammaDot(self.getDeviatoricStrain(v)+s_last/(2*dt*mu))
self.__eta_eff_save=self.getEtaEff(gamma, pressure=p,dt=dt, eta0=self.__eta_eff_save, iter_max=self.__eta_iter_max)
if self.checkVerbose(): print("IncompressibleIsotropicFlowCartesian: eta_eff has been updated.")
if mu is None:
stress0=escore.Data()
else:
stress0=-(self.__eta_eff_save/(dt*mu))*s_last
self.setStokesEquation(eta=self.__eta_eff_save,stress=stress0)
def initialize(self, F=None, f=None, fixed_v_mask=None, v_boundary=None, restoration_factor=None):
"""
sets external forces and velocity constraints
:param F: external force
:type F: vector value/field
:param f: surface force
:type f: vector value/field on boundary
:param fixed_v_mask: location of constraints maked by positive values
:type fixed_v_mask: vector value/field
:param v_boundary: value of velocity at location of constraints
:type v_boundary: vector value/field
:param restoration_factor: factor for normal restoration force
:type restoration_factor: scalar values/field
:note: Only changing parameters need to be specified.
"""
self.setExternals(F, f, fixed_v_mask, v_boundary, restoration_factor)
def update(self, dt, iter_max=10, eta_iter_max=20, verbose=False, usePCG=True, max_correction_steps=50):
"""
Updates stress, velocity and pressure for time increment dt.
:param dt: time increment
:param iter_max: maximum number of iteration steps in the incompressible solver
:param eta_iter_max: maximum number of iteration steps in the incompressible solver
:param verbose: prints some infos in the incompressible solver
"""
mu=self.getElasticShearModulus()
if mu is not None:
if not dt > 0.:
raise ValueError("dt must be positive.")
else:
dt=max(0,dt)
self.__dt=dt
self.__eta_iter_max=max(eta_iter_max,1)
v_last=self.getVelocity()
s_last=self.getDeviatoricStress()
mask_v,v_b=self.getVelocityConstraint()
p_last=self.getPressure()
self.__eta_eff_save=self.getCurrentEtaEff()
self.setStokesEquation(f=self.getForce(),fixed_u_mask=mask_v,surface_stress=self.getSurfaceForce(), restoration_factor=self.getRestorationFactor())
if self.checkVerbose(): print(("IncompressibleIsotropicFlowCartesian: start iteration for t = %s."%(self.getTime()+dt,)))
#
# get a new velcocity and pressure:
#
if mask_v.isEmpty():
v0=v_last
else:
if v_b.isEmpty():
v0=v_last*(1.-mask_v)
else:
v0=v_b*mask_v+v_last*(1.-mask_v)
v,p=self._solve(v0,p_last,verbose=self.checkVerbose(),max_iter=iter_max,usePCG=usePCG, max_correction_steps=max_correction_steps)
#
# finally we can update the return values:
#
self.setPressure(p)
self.setVelocity(v)
self.setDeviatoricStrain(self.getDeviatoricStrain(v))
if mu is None:
D=self.getDeviatoricStrain(v)
else:
D=self.getDeviatoricStrain(v)+s_last/(2*dt*mu)
gamma=self.getGammaDot(D)
self.setGammaDot(gamma)
self.__eta_eff = self.getEtaEff(self.getGammaDot(), pressure=p,dt=dt, eta0=self.__eta_eff_save, iter_max=self.__eta_iter_max)
self.setDeviatoricStress(2.*self.__eta_eff*D)
self.setTime(self.getTime()+dt)
if self.checkVerbose(): print(("IncompressibleIsotropicFlowCartesian: iteration on time step %s completed."%(self.getTime(),)))
return self.getVelocity(), self.getPressure()
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