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# -*- coding: utf-8 -*- |
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######################################################## |
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# |
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# Copyright (c) 2003-2010 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) 2003-2010 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__="https://launchpad.net/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, LinearPDESystem, LinearSinglePDE, SolverOptions |
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from pdetools import HomogeneousSaddlePointProblem,Projector, ArithmeticTuple, PCG, NegativeNorm, GMRES |
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|
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class DarcyFlow(object): |
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""" |
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solves the problem |
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|
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*u_i+k_{ij}*p_{,j} = g_i* |
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*u_{i,i} = f* |
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|
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where *p* represents the pressure and *u* the Darcy flux. *k* represents the permeability, |
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|
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:note: The problem is solved in a least squares formulation. |
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""" |
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|
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def __init__(self, domain, useReduced=False, adaptSubTolerance=True, solveForFlux=False): |
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""" |
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initializes the Darcy flux problem |
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:param domain: domain of the problem |
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:type domain: `Domain` |
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:param useReduced: uses reduced oreder on flux and pressure |
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:type useReduced: ``bool`` |
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:param adaptSubTolerance: switches on automatic subtolerance selection |
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:type adaptSubTolerance: ``bool`` |
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:param solveForFlux: if True the solver solves for the flux (do not use!) |
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:type solveForFlux: ``bool`` |
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""" |
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self.domain=domain |
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self.solveForFlux=solveForFlux |
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self.useReduced=useReduced |
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self.__adaptSubTolerance=adaptSubTolerance |
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self.verbose=False |
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|
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self.__pde_k=LinearPDESystem(domain) |
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self.__pde_k.setSymmetryOn() |
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if self.useReduced: self.__pde_k.setReducedOrderOn() |
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|
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self.__pde_p=LinearSinglePDE(domain) |
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self.__pde_p.setSymmetryOn() |
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if self.useReduced: self.__pde_p.setReducedOrderOn() |
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|
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self.__pde_l=LinearSinglePDE(domain) # this is here for getSolverOptionsWeighting |
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# self.__pde_l.setSymmetryOn() |
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# if self.useReduced: self.__pde_l.setReducedOrderOn() |
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self.setTolerance() |
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self.setAbsoluteTolerance() |
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self.__f=Scalar(0,self.__pde_p.getFunctionSpaceForCoefficient("X")) |
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self.__g=Vector(0,self.__pde_p.getFunctionSpaceForCoefficient("Y")) |
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|
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def getSolverOptionsFlux(self): |
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""" |
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Returns the solver options used to solve the flux problems |
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|
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*K^{-1} u=F* |
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|
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:return: `SolverOptions` |
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""" |
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return self.__pde_k.getSolverOptions() |
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|
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def setSolverOptionsFlux(self, options=None): |
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""" |
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Sets the solver options used to solve the flux problems |
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|
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*K^{-1}u=F* |
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|
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If ``options`` is not present, the options are reset to default |
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|
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:param options: `SolverOptions` |
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:note: if the adaption of subtolerance is choosen, the tolerance set by ``options`` will be overwritten before the solver is called. |
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""" |
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return self.__pde_v.setSolverOptions(options) |
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|
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def getSolverOptionsPressure(self): |
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""" |
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Returns the solver options used to solve the pressure problems |
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|
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*(Q^* K Q)p=-Q^*G* |
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|
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:return: `SolverOptions` |
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""" |
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return self.__pde_p.getSolverOptions() |
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|
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def setSolverOptionsPressure(self, options=None): |
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""" |
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Sets the solver options used to solve the pressure problems |
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|
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*(Q^* K Q)p=-Q^*G* |
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|
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If ``options`` is not present, the options are reset to default |
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|
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:param options: `SolverOptions` |
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:note: if the adaption of subtolerance is choosen, the tolerance set by ``options`` will be overwritten before the solver is called. |
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""" |
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return self.__pde_p.setSolverOptions(options) |
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|
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def getSolverOptionsWeighting(self): |
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""" |
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Returns the solver options used to solve the pressure problems |
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|
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*(D K D^*)p=-D F* |
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|
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:return: `SolverOptions` |
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""" |
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return self.__pde_l.getSolverOptions() |
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|
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def setSolverOptionsWeighting(self, options=None): |
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""" |
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Sets the solver options used to solve the pressure problems |
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|
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*(D K D^*)p=-D F* |
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|
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If ``options`` is not present, the options are reset to default |
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|
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:param options: `SolverOptions` |
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:note: if the adaption of subtolerance is choosen, the tolerance set by ``options`` will be overwritten before the solver is called. |
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""" |
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return self.__pde_l.setSolverOptions(options) |
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|
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|
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def setValue(self,f=None, g=None, location_of_fixed_pressure=None, location_of_fixed_flux=None, permeability=None): |
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""" |
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assigns values to model parameters |
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|
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:param f: volumetic sources/sinks |
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:type f: scalar value on the domain (e.g. `Data`) |
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:param g: flux sources/sinks |
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:type g: vector values on the domain (e.g. `Data`) |
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:param location_of_fixed_pressure: mask for locations where pressure is fixed |
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:type location_of_fixed_pressure: scalar value on the domain (e.g. `Data`) |
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:param location_of_fixed_flux: mask for locations where flux is fixed. |
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:type location_of_fixed_flux: vector values on the domain (e.g. `Data`) |
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:param permeability: permeability tensor. If scalar ``s`` is given the tensor with |
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``s`` on the main diagonal is used. |
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:type permeability: scalar or tensor values on the domain (e.g. `Data`) |
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:note: the values of parameters which are not set by calling ``setValue`` are not altered. |
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:note: at any point on the boundary of the domain the pressure (``location_of_fixed_pressure`` >0) |
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or the normal component of the flux (``location_of_fixed_flux[i]>0`` if direction of the normal |
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is along the *x_i* axis. |
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""" |
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if f !=None: |
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f=util.interpolate(f, self.__pde_p.getFunctionSpaceForCoefficient("X")) |
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if f.isEmpty(): |
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f=Scalar(0,self.__pde_p.getFunctionSpaceForCoefficient("X")) |
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else: |
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if f.getRank()>0: raise ValueError,"illegal rank of f." |
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self.__f=f |
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if g !=None: |
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g=util.interpolate(g, self.__pde_p.getFunctionSpaceForCoefficient("Y")) |
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if g.isEmpty(): |
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g=Vector(0,self.__pde_p.getFunctionSpaceForCoefficient("Y")) |
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else: |
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if not g.getShape()==(self.domain.getDim(),): |
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raise ValueError,"illegal shape of g" |
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self.__g=g |
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if location_of_fixed_pressure!=None: |
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self.__pde_p.setValue(q=location_of_fixed_pressure) |
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#self.__pde_l.setValue(q=location_of_fixed_pressure) |
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if location_of_fixed_flux!=None: |
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self.__pde_k.setValue(q=location_of_fixed_flux) |
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|
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if permeability!=None: |
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perm=util.interpolate(permeability,self.__pde_p.getFunctionSpaceForCoefficient("A")) |
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if perm.getRank()==0: |
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perm_inv=(1./perm)*util.kronecker(self.domain.getDim()) |
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perm=perm*util.kronecker(self.domain.getDim()) |
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elif perm.getRank()==2: |
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perm_inv=util.inverse(perm) |
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else: |
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raise ValueError,"illegal rank of permeability." |
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|
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self.__permeability=perm |
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self.__permeability_inv=perm_inv |
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self.__l =(util.longestEdge(self.domain)**2*util.length(self.__permeability_inv))/10 |
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#self.__l =(self.domain.getSize()**2*util.length(self.__permeability_inv))/10 |
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if self.solveForFlux: |
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self.__pde_k.setValue(D=self.__permeability_inv) |
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else: |
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self.__pde_k.setValue(D=self.__permeability_inv, A=self.__l*util.outer(util.kronecker(self.domain),util.kronecker(self.domain))) |
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self.__pde_p.setValue(A=self.__permeability) |
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#self.__pde_l.setValue(D=1/self.__l) |
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#self.__pde_l.setValue(A=self.__permeability) |
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|
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def __applWeight(self, v, f=None): |
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# solves L p = f-Dv with p = 0 |
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if self.getSolverOptionsWeighting().isVerbose() or self.verbose: print "DarcyFlux: Applying weighting operator" |
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if f == None: |
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return -util.div(v)*self.__l |
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else: |
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return (f-util.div(v))*self.__l |
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# if f == None: |
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# self.__pde_l.setValue(Y=-util.div(v)) |
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# else: |
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# return (f-util.div(v))/self.__l |
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# return self.__pde_l.getSolution() |
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|
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def __getPressure(self, v, p0, g=None): |
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# solves (G*KG)p = G^(g-v) with p = p0 where location_of_fixed_pressure>0 |
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if self.getSolverOptionsPressure().isVerbose() or self.verbose: print "DarcyFlux: Pressure update" |
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if g == None: |
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self.__pde_p.setValue(X=-v, r=p0) |
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else: |
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self.__pde_p.setValue(X=g-v, r=p0) |
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p=self.__pde_p.getSolution() |
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return p |
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|
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def __Aprod_v(self,dv): |
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# calculates: (a,b,c) = (K^{-1}(dv + KG * dp), L^{-1}Ddv, dp) with (G*KG)dp = - G^*dv |
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dp=self.__getPressure(dv, p0=Data()) # dp = (G*KG)^{-1} (0-G^*dv) |
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a=util.tensor_mult(self.__permeability_inv,dv)+util.grad(dp) # a= K^{-1}u+G*dp |
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b= - self.__applWeight(dv) # b = - (D K D^*)^{-1} (0-Dv) |
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return ArithmeticTuple(a,b,-dp) |
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|
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def __Msolve_PCG_v(self,r): |
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# K^{-1} u = r[0] + D^*r[1] = K^{-1}(dv + KG * dp) + D^*L^{-1}Ddv |
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if self.getSolverOptionsFlux().isVerbose() or self.verbose: print "DarcyFlux: Applying preconditioner" |
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self.__pde_k.setValue(X=r[1]*util.kronecker(self.domain), Y=r[0], r=Data()) |
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# self.__pde_p.getOperator().saveMM("prec.mm") |
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return self.__pde_k.getSolution() |
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|
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def __inner_PCG_v(self,v,r): |
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return util.integrate(util.inner(v,r[0])+util.div(v)*r[1]) |
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|
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def __Aprod_p(self,dp): |
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if self.getSolverOptionsFlux().isVerbose(): print "DarcyFlux: Applying operator" |
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Gdp=util.grad(dp) |
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self.__pde_k.setValue(Y=-Gdp,X=Data(), r=Data()) |
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du=self.__pde_k.getSolution() |
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# self.__pde_v.getOperator().saveMM("proj.mm") |
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return ArithmeticTuple(util.tensor_mult(self.__permeability,Gdp),-du) |
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|
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def __getFlux(self,p, v0, f=None, g=None): |
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# solves (K^{-1}+D^*L^{-1} D) v = D^*L^{-1}f + K^{-1}g - Gp |
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if f!=None: |
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self.__pde_k.setValue(X=self.__applWeight(v0*0,self.__f)*util.kronecker(self.domain)) |
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self.__pde_k.setValue(r=v0) |
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g2=util.tensor_mult(self.__permeability_inv,g) |
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if p == None: |
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self.__pde_k.setValue(Y=g2) |
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else: |
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self.__pde_k.setValue(Y=g2-util.grad(p)) |
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return self.__pde_k.getSolution() |
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|
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#v=self.__getFlux(p, u0, f=self.__f, g=g2) |
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def __Msolve_PCG_p(self,r): |
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if self.getSolverOptionsPressure().isVerbose(): print "DarcyFlux: Applying preconditioner" |
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self.__pde_p.setValue(X=r[0]-r[1], Y=Data(), r=Data(), y=Data()) |
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# self.__pde_p.getOperator().saveMM("prec.mm") |
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return self.__pde_p.getSolution() |
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|
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def __inner_PCG_p(self,p,r): |
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return util.integrate(util.inner(util.grad(p), r[0]-r[1])) |
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|
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def __L2(self,v): |
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return util.sqrt(util.integrate(util.length(util.interpolate(v,Function(self.domain)))**2)) |
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|
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def solve(self,u0,p0, max_iter=100, verbose=False, max_num_corrections=10): |
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""" |
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solves the problem. |
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|
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The iteration is terminated if the residual norm is less then self.getTolerance(). |
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|
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:param u0: initial guess for the flux. At locations in the domain marked by ``location_of_fixed_flux`` the value of ``u0`` is kept unchanged. |
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:type u0: vector value on the domain (e.g. `Data`). |
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:param p0: initial guess for the pressure. At locations in the domain marked by ``location_of_fixed_pressure`` the value of ``p0`` is kept unchanged. |
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:type p0: scalar value on the domain (e.g. `Data`). |
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:param verbose: if set some information on iteration progress are printed |
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:type verbose: ``bool`` |
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:return: flux and pressure |
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:rtype: ``tuple`` of `Data`. |
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|
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:note: The problem is solved as a least squares form |
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*(K^[-1]+D^* (DKD^*)^[-1] D)u+G p=D^* (DKD^*)^[-1] f + K^[-1]g* |
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*G^*u+G^* K Gp=G^*g* |
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|
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where *D* is the *div* operator and *(Gp)_i=p_{,i}* for the permeability *K=k_{ij}*. |
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""" |
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self.verbose=verbose |
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rtol=self.getTolerance() |
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atol=self.getAbsoluteTolerance() |
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self.setSubProblemTolerance() |
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num_corrections=0 |
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converged=False |
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norm_r=None |
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|
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# Eliminate the hydrostatic pressure: |
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if self.verbose: print "DarcyFlux: calculate hydrostatic pressure component." |
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self.__pde_p.setValue(X=self.__g, r=p0, y=-util.inner(self.domain.getNormal(),u0)) |
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p0=self.__pde_p.getSolution() |
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g2=self.__g - util.tensor_mult(self.__permeability, util.grad(p0)) |
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norm_g2=util.integrate(util.inner(g2,util.tensor_mult(self.__permeability_inv,g2)))**0.5 |
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|
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p=p0*0 |
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if self.solveForFlux: |
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v=u0.copy() |
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else: |
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v=self.__getFlux(p, u0, f=self.__f, g=g2) |
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|
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while not converged and norm_g2 > 0: |
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Gp=util.grad(p) |
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KGp=util.tensor_mult(self.__permeability,Gp) |
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if self.verbose: |
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def_p=g2-(v+KGp) |
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def_v=self.__f-util.div(v) |
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print "DarcyFlux: L2: g-v-K*grad(p) = %e (v = %e)."%(self.__L2(def_p),self.__L2(v)) |
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print "DarcyFlux: L2: f-div(v) = %e (grad(v) = %e)."%(self.__L2(def_v),self.__L2(util.grad(v))) |
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print "DarcyFlux: K^{-1}-norm of v = %e."%util.integrate(util.inner(v,util.tensor_mult(self.__permeability_inv,v)))**0.5 |
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print "DarcyFlux: K^{-1}-norm of g2 = %e."%norm_g2 |
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print "DarcyFlux: K-norm of grad(dp) = %e."%util.integrate(util.inner(Gp,KGp))**0.5 |
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ATOL=atol+rtol*norm_g2 |
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if self.verbose: print "DarcyFlux: absolute tolerance ATOL = %e."%(ATOL,) |
| 347 |
if norm_r == None or norm_r>ATOL: |
| 348 |
if num_corrections>max_num_corrections: |
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raise ValueError,"maximum number of correction steps reached." |
| 350 |
|
| 351 |
if self.solveForFlux: |
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# initial residual is r=K^{-1}*(g-v-K*Gp)+D^*L^{-1}(f-Du) |
| 353 |
v,r, norm_r=PCG(ArithmeticTuple(util.tensor_mult(self.__permeability_inv,g2-v)-Gp,self.__applWeight(v,self.__f),p), |
| 354 |
self.__Aprod_v, |
| 355 |
v, |
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self.__Msolve_PCG_v, |
| 357 |
self.__inner_PCG_v, |
| 358 |
atol=ATOL, rtol=0.,iter_max=max_iter, verbose=self.verbose) |
| 359 |
p=r[2] |
| 360 |
else: |
| 361 |
# initial residual is r=G^*(g2-KGp - v) |
| 362 |
p,r, norm_r=PCG(ArithmeticTuple(g2-KGp,v), |
| 363 |
self.__Aprod_p, |
| 364 |
p, |
| 365 |
self.__Msolve_PCG_p, |
| 366 |
self.__inner_PCG_p, |
| 367 |
atol=ATOL, rtol=0.,iter_max=max_iter, verbose=self.verbose) |
| 368 |
v=r[1] |
| 369 |
if self.verbose: print "DarcyFlux: residual norm = %e."%norm_r |
| 370 |
num_corrections+=1 |
| 371 |
else: |
| 372 |
if self.verbose: print "DarcyFlux: stopping criterium reached." |
| 373 |
converged=True |
| 374 |
return v,p+p0 |
| 375 |
def setTolerance(self,rtol=1e-4): |
| 376 |
""" |
| 377 |
sets the relative tolerance ``rtol`` used to terminate the solution process. The iteration is terminated if |
| 378 |
|
| 379 |
*|g-v-K gard(p)|_PCG <= atol + rtol * |K^{1/2}g2|_0* |
| 380 |
|
| 381 |
where ``atol`` is an absolut tolerance (see `setAbsoluteTolerance`). |
| 382 |
|
| 383 |
:param rtol: relative tolerance for the pressure |
| 384 |
:type rtol: non-negative ``float`` |
| 385 |
""" |
| 386 |
if rtol<0: |
| 387 |
raise ValueError,"Relative tolerance needs to be non-negative." |
| 388 |
self.__rtol=rtol |
| 389 |
def getTolerance(self): |
| 390 |
""" |
| 391 |
returns the relative tolerance |
| 392 |
:return: current relative tolerance |
| 393 |
:rtype: ``float`` |
| 394 |
""" |
| 395 |
return self.__rtol |
| 396 |
|
| 397 |
def setAbsoluteTolerance(self,atol=0.): |
| 398 |
""" |
| 399 |
sets the absolute tolerance ``atol`` used to terminate the solution process. The iteration is terminated if |
| 400 |
|
| 401 |
*|g-v-K gard(p)|_PCG <= atol + rtol * |K^{1/2}g2|_0* |
| 402 |
|
| 403 |
|
| 404 |
where ``rtol`` is an absolut tolerance (see `setTolerance`), *|f|^2 = integrate(length(f)^2)* and *(Qp)_i=k_{ij}p_{,j}* for the permeability *k_{ij}*. |
| 405 |
|
| 406 |
:param atol: absolute tolerance for the pressure |
| 407 |
:type atol: non-negative ``float`` |
| 408 |
""" |
| 409 |
if atol<0: |
| 410 |
raise ValueError,"Absolute tolerance needs to be non-negative." |
| 411 |
self.__atol=atol |
| 412 |
def getAbsoluteTolerance(self): |
| 413 |
""" |
| 414 |
returns the absolute tolerance |
| 415 |
:return: current absolute tolerance |
| 416 |
:rtype: ``float`` |
| 417 |
""" |
| 418 |
return self.__atol |
| 419 |
def getSubProblemTolerance(self): |
| 420 |
""" |
| 421 |
Returns a suitable subtolerance |
| 422 |
:type: ``float`` |
| 423 |
""" |
| 424 |
return max(util.EPSILON**(0.5),self.getTolerance()**2) |
| 425 |
|
| 426 |
def setSubProblemTolerance(self): |
| 427 |
""" |
| 428 |
Sets the relative tolerance to solve the subproblem(s) if subtolerance adaption is selected. |
| 429 |
""" |
| 430 |
if self.__adaptSubTolerance: |
| 431 |
sub_tol=self.getSubProblemTolerance() |
| 432 |
self.getSolverOptionsFlux().setTolerance(sub_tol) |
| 433 |
self.getSolverOptionsFlux().setAbsoluteTolerance(0.) |
| 434 |
self.getSolverOptionsPressure().setTolerance(sub_tol) |
| 435 |
self.getSolverOptionsPressure().setAbsoluteTolerance(0.) |
| 436 |
self.getSolverOptionsWeighting().setTolerance(sub_tol) |
| 437 |
self.getSolverOptionsWeighting().setAbsoluteTolerance(0.) |
| 438 |
if self.verbose: print "DarcyFlux: relative subtolerance is set to %e."%sub_tol |
| 439 |
|
| 440 |
|
| 441 |
class DarcyFlowOld(object): |
| 442 |
""" |
| 443 |
solves the problem |
| 444 |
|
| 445 |
*u_i+k_{ij}*p_{,j} = g_i* |
| 446 |
*u_{i,i} = f* |
| 447 |
|
| 448 |
where *p* represents the pressure and *u* the Darcy flux. *k* represents the permeability, |
| 449 |
|
| 450 |
:note: The problem is solved in a least squares formulation. |
| 451 |
""" |
| 452 |
|
| 453 |
def __init__(self, domain, weight=None, useReduced=False, adaptSubTolerance=True): |
| 454 |
""" |
| 455 |
initializes the Darcy flux problem |
| 456 |
:param domain: domain of the problem |
| 457 |
:type domain: `Domain` |
| 458 |
:param useReduced: uses reduced oreder on flux and pressure |
| 459 |
:type useReduced: ``bool`` |
| 460 |
:param adaptSubTolerance: switches on automatic subtolerance selection |
| 461 |
:type adaptSubTolerance: ``bool`` |
| 462 |
""" |
| 463 |
self.domain=domain |
| 464 |
if weight == None: |
| 465 |
s=self.domain.getSize() |
| 466 |
self.__l=(3.*util.longestEdge(self.domain)*s/util.sup(s))**2 |
| 467 |
# self.__l=(3.*util.longestEdge(self.domain))**2 |
| 468 |
#self.__l=(0.1*util.longestEdge(self.domain)*s/util.sup(s))**2 |
| 469 |
else: |
| 470 |
self.__l=weight |
| 471 |
self.__pde_v=LinearPDESystem(domain) |
| 472 |
if useReduced: self.__pde_v.setReducedOrderOn() |
| 473 |
self.__pde_v.setSymmetryOn() |
| 474 |
self.__pde_v.setValue(D=util.kronecker(domain), A=self.__l*util.outer(util.kronecker(domain),util.kronecker(domain))) |
| 475 |
self.__pde_p=LinearSinglePDE(domain) |
| 476 |
self.__pde_p.setSymmetryOn() |
| 477 |
if useReduced: self.__pde_p.setReducedOrderOn() |
| 478 |
self.__f=Scalar(0,self.__pde_v.getFunctionSpaceForCoefficient("X")) |
| 479 |
self.__g=Vector(0,self.__pde_v.getFunctionSpaceForCoefficient("Y")) |
| 480 |
self.setTolerance() |
| 481 |
self.setAbsoluteTolerance() |
| 482 |
self.__adaptSubTolerance=adaptSubTolerance |
| 483 |
self.verbose=False |
| 484 |
def getSolverOptionsFlux(self): |
| 485 |
""" |
| 486 |
Returns the solver options used to solve the flux problems |
| 487 |
|
| 488 |
*(I+D^*D)u=F* |
| 489 |
|
| 490 |
:return: `SolverOptions` |
| 491 |
""" |
| 492 |
return self.__pde_v.getSolverOptions() |
| 493 |
def setSolverOptionsFlux(self, options=None): |
| 494 |
""" |
| 495 |
Sets the solver options used to solve the flux problems |
| 496 |
|
| 497 |
*(I+D^*D)u=F* |
| 498 |
|
| 499 |
If ``options`` is not present, the options are reset to default |
| 500 |
:param options: `SolverOptions` |
| 501 |
:note: if the adaption of subtolerance is choosen, the tolerance set by ``options`` will be overwritten before the solver is called. |
| 502 |
""" |
| 503 |
return self.__pde_v.setSolverOptions(options) |
| 504 |
def getSolverOptionsPressure(self): |
| 505 |
""" |
| 506 |
Returns the solver options used to solve the pressure problems |
| 507 |
|
| 508 |
*(Q^*Q)p=Q^*G* |
| 509 |
|
| 510 |
:return: `SolverOptions` |
| 511 |
""" |
| 512 |
return self.__pde_p.getSolverOptions() |
| 513 |
def setSolverOptionsPressure(self, options=None): |
| 514 |
""" |
| 515 |
Sets the solver options used to solve the pressure problems |
| 516 |
|
| 517 |
*(Q^*Q)p=Q^*G* |
| 518 |
|
| 519 |
If ``options`` is not present, the options are reset to default |
| 520 |
:param options: `SolverOptions` |
| 521 |
:note: if the adaption of subtolerance is choosen, the tolerance set by ``options`` will be overwritten before the solver is called. |
| 522 |
""" |
| 523 |
return self.__pde_p.setSolverOptions(options) |
| 524 |
|
| 525 |
def setValue(self,f=None, g=None, location_of_fixed_pressure=None, location_of_fixed_flux=None, permeability=None): |
| 526 |
""" |
| 527 |
assigns values to model parameters |
| 528 |
|
| 529 |
:param f: volumetic sources/sinks |
| 530 |
:type f: scalar value on the domain (e.g. `Data`) |
| 531 |
:param g: flux sources/sinks |
| 532 |
:type g: vector values on the domain (e.g. `Data`) |
| 533 |
:param location_of_fixed_pressure: mask for locations where pressure is fixed |
| 534 |
:type location_of_fixed_pressure: scalar value on the domain (e.g. `Data`) |
| 535 |
:param location_of_fixed_flux: mask for locations where flux is fixed. |
| 536 |
:type location_of_fixed_flux: vector values on the domain (e.g. `Data`) |
| 537 |
:param permeability: permeability tensor. If scalar ``s`` is given the tensor with |
| 538 |
``s`` on the main diagonal is used. If vector ``v`` is given the tensor with |
| 539 |
``v`` on the main diagonal is used. |
| 540 |
:type permeability: scalar, vector or tensor values on the domain (e.g. `Data`) |
| 541 |
|
| 542 |
:note: the values of parameters which are not set by calling ``setValue`` are not altered. |
| 543 |
:note: at any point on the boundary of the domain the pressure (``location_of_fixed_pressure`` >0) |
| 544 |
or the normal component of the flux (``location_of_fixed_flux[i]>0`` if direction of the normal |
| 545 |
is along the *x_i* axis. |
| 546 |
""" |
| 547 |
if f !=None: |
| 548 |
f=util.interpolate(f, self.__pde_v.getFunctionSpaceForCoefficient("X")) |
| 549 |
if f.isEmpty(): |
| 550 |
f=Scalar(0,self.__pde_v.getFunctionSpaceForCoefficient("X")) |
| 551 |
else: |
| 552 |
if f.getRank()>0: raise ValueError,"illegal rank of f." |
| 553 |
self.__f=f |
| 554 |
if g !=None: |
| 555 |
g=util.interpolate(g, self.__pde_p.getFunctionSpaceForCoefficient("Y")) |
| 556 |
if g.isEmpty(): |
| 557 |
g=Vector(0,self.__pde_v.getFunctionSpaceForCoefficient("Y")) |
| 558 |
else: |
| 559 |
if not g.getShape()==(self.domain.getDim(),): |
| 560 |
raise ValueError,"illegal shape of g" |
| 561 |
self.__g=g |
| 562 |
|
| 563 |
if location_of_fixed_pressure!=None: self.__pde_p.setValue(q=location_of_fixed_pressure) |
| 564 |
if location_of_fixed_flux!=None: self.__pde_v.setValue(q=location_of_fixed_flux) |
| 565 |
|
| 566 |
if permeability!=None: |
| 567 |
perm=util.interpolate(permeability,self.__pde_p.getFunctionSpaceForCoefficient("A")) |
| 568 |
if perm.getRank()==0: |
| 569 |
perm=perm*util.kronecker(self.domain.getDim()) |
| 570 |
elif perm.getRank()==1: |
| 571 |
perm, perm2=Tensor(0.,self.__pde_p.getFunctionSpaceForCoefficient("A")), perm |
| 572 |
for i in range(self.domain.getDim()): perm[i,i]=perm2[i] |
| 573 |
elif perm.getRank()==2: |
| 574 |
pass |
| 575 |
else: |
| 576 |
raise ValueError,"illegal rank of permeability." |
| 577 |
self.__permeability=perm |
| 578 |
self.__pde_p.setValue(A=util.transposed_tensor_mult(self.__permeability,self.__permeability)) |
| 579 |
|
| 580 |
def setTolerance(self,rtol=1e-4): |
| 581 |
""" |
| 582 |
sets the relative tolerance ``rtol`` used to terminate the solution process. The iteration is terminated if |
| 583 |
|
| 584 |
*|g-v-Qp| <= atol + rtol * min( max( |g-v|, |Qp| ), max( |v|, |g-Qp| ) )* |
| 585 |
|
| 586 |
where ``atol`` is an absolut tolerance (see `setAbsoluteTolerance`), *|f|^2 = integrate(length(f)^2)* and *(Qp)_i=k_{ij}p_{,j}* for the permeability *k_{ij}*. |
| 587 |
|
| 588 |
:param rtol: relative tolerance for the pressure |
| 589 |
:type rtol: non-negative ``float`` |
| 590 |
""" |
| 591 |
if rtol<0: |
| 592 |
raise ValueError,"Relative tolerance needs to be non-negative." |
| 593 |
self.__rtol=rtol |
| 594 |
def getTolerance(self): |
| 595 |
""" |
| 596 |
returns the relative tolerance |
| 597 |
|
| 598 |
:return: current relative tolerance |
| 599 |
:rtype: ``float`` |
| 600 |
""" |
| 601 |
return self.__rtol |
| 602 |
|
| 603 |
def setAbsoluteTolerance(self,atol=0.): |
| 604 |
""" |
| 605 |
sets the absolute tolerance ``atol`` used to terminate the solution process. The iteration is terminated if |
| 606 |
|
| 607 |
*|g-v-Qp| <= atol + rtol * min( max( |g-v|, |Qp| ), max( |v|, |g-Qp| ) )* |
| 608 |
|
| 609 |
where ``rtol`` is an absolut tolerance (see `setTolerance`), *|f|^2 = integrate(length(f)^2)* and *(Qp)_i=k_{ij}p_{,j}* for the permeability *k_{ij}*. |
| 610 |
|
| 611 |
:param atol: absolute tolerance for the pressure |
| 612 |
:type atol: non-negative ``float`` |
| 613 |
""" |
| 614 |
if atol<0: |
| 615 |
raise ValueError,"Absolute tolerance needs to be non-negative." |
| 616 |
self.__atol=atol |
| 617 |
def getAbsoluteTolerance(self): |
| 618 |
""" |
| 619 |
returns the absolute tolerance |
| 620 |
|
| 621 |
:return: current absolute tolerance |
| 622 |
:rtype: ``float`` |
| 623 |
""" |
| 624 |
return self.__atol |
| 625 |
def getSubProblemTolerance(self): |
| 626 |
""" |
| 627 |
Returns a suitable subtolerance |
| 628 |
@type: ``float`` |
| 629 |
""" |
| 630 |
return max(util.EPSILON**(0.75),self.getTolerance()**2) |
| 631 |
def setSubProblemTolerance(self): |
| 632 |
""" |
| 633 |
Sets the relative tolerance to solve the subproblem(s) if subtolerance adaption is selected. |
| 634 |
""" |
| 635 |
if self.__adaptSubTolerance: |
| 636 |
sub_tol=self.getSubProblemTolerance() |
| 637 |
self.getSolverOptionsFlux().setTolerance(sub_tol) |
| 638 |
self.getSolverOptionsFlux().setAbsoluteTolerance(0.) |
| 639 |
self.getSolverOptionsPressure().setTolerance(sub_tol) |
| 640 |
self.getSolverOptionsPressure().setAbsoluteTolerance(0.) |
| 641 |
if self.verbose: print "DarcyFlux: relative subtolerance is set to %e."%sub_tol |
| 642 |
|
| 643 |
def solve(self,u0,p0, max_iter=100, verbose=False, max_num_corrections=10): |
| 644 |
""" |
| 645 |
solves the problem. |
| 646 |
|
| 647 |
The iteration is terminated if the residual norm is less then self.getTolerance(). |
| 648 |
|
| 649 |
:param u0: initial guess for the flux. At locations in the domain marked by ``location_of_fixed_flux`` the value of ``u0`` is kept unchanged. |
| 650 |
:type u0: vector value on the domain (e.g. `Data`). |
| 651 |
:param p0: initial guess for the pressure. At locations in the domain marked by ``location_of_fixed_pressure`` the value of ``p0`` is kept unchanged. |
| 652 |
:type p0: scalar value on the domain (e.g. `Data`). |
| 653 |
:param verbose: if set some information on iteration progress are printed |
| 654 |
:type verbose: ``bool`` |
| 655 |
:return: flux and pressure |
| 656 |
:rtype: ``tuple`` of `Data`. |
| 657 |
|
| 658 |
:note: The problem is solved as a least squares form |
| 659 |
|
| 660 |
*(I+D^*D)u+Qp=D^*f+g* |
| 661 |
*Q^*u+Q^*Qp=Q^*g* |
| 662 |
|
| 663 |
where *D* is the *div* operator and *(Qp)_i=k_{ij}p_{,j}* for the permeability *k_{ij}*. |
| 664 |
We eliminate the flux form the problem by setting |
| 665 |
|
| 666 |
*u=(I+D^*D)^{-1}(D^*f-g-Qp)* with u=u0 on location_of_fixed_flux |
| 667 |
|
| 668 |
form the first equation. Inserted into the second equation we get |
| 669 |
|
| 670 |
*Q^*(I-(I+D^*D)^{-1})Qp= Q^*(g-(I+D^*D)^{-1}(D^*f+g))* with p=p0 on location_of_fixed_pressure |
| 671 |
|
| 672 |
which is solved using the PCG method (precondition is *Q^*Q*). In each iteration step |
| 673 |
PDEs with operator *I+D^*D* and with *Q^*Q* needs to be solved using a sub iteration scheme. |
| 674 |
""" |
| 675 |
self.verbose=verbose |
| 676 |
rtol=self.getTolerance() |
| 677 |
atol=self.getAbsoluteTolerance() |
| 678 |
self.setSubProblemTolerance() |
| 679 |
num_corrections=0 |
| 680 |
converged=False |
| 681 |
p=p0 |
| 682 |
norm_r=None |
| 683 |
while not converged: |
| 684 |
v=self.getFlux(p, fixed_flux=u0) |
| 685 |
Qp=self.__Q(p) |
| 686 |
norm_v=self.__L2(v) |
| 687 |
norm_Qp=self.__L2(Qp) |
| 688 |
if norm_v == 0.: |
| 689 |
if norm_Qp == 0.: |
| 690 |
return v,p |
| 691 |
else: |
| 692 |
fac=norm_Qp |
| 693 |
else: |
| 694 |
if norm_Qp == 0.: |
| 695 |
fac=norm_v |
| 696 |
else: |
| 697 |
fac=2./(1./norm_v+1./norm_Qp) |
| 698 |
ATOL=(atol+rtol*fac) |
| 699 |
if self.verbose: |
| 700 |
print "DarcyFlux: L2 norm of v = %e."%norm_v |
| 701 |
print "DarcyFlux: L2 norm of k.util.grad(p) = %e."%norm_Qp |
| 702 |
print "DarcyFlux: L2 defect u = %e."%(util.integrate(util.length(self.__g-util.interpolate(v,Function(self.domain))-Qp)**2)**(0.5),) |
| 703 |
print "DarcyFlux: L2 defect div(v) = %e."%(util.integrate((self.__f-util.div(v))**2)**(0.5),) |
| 704 |
print "DarcyFlux: absolute tolerance ATOL = %e."%ATOL |
| 705 |
if norm_r == None or norm_r>ATOL: |
| 706 |
if num_corrections>max_num_corrections: |
| 707 |
raise ValueError,"maximum number of correction steps reached." |
| 708 |
p,r, norm_r=PCG(self.__g-util.interpolate(v,Function(self.domain))-Qp,self.__Aprod,p,self.__Msolve_PCG,self.__inner_PCG,atol=0.5*ATOL, rtol=0.,iter_max=max_iter, verbose=self.verbose) |
| 709 |
num_corrections+=1 |
| 710 |
else: |
| 711 |
converged=True |
| 712 |
return v,p |
| 713 |
def __L2(self,v): |
| 714 |
return util.sqrt(util.integrate(util.length(util.interpolate(v,Function(self.domain)))**2)) |
| 715 |
|
| 716 |
def __Q(self,p): |
| 717 |
return util.tensor_mult(self.__permeability,util.grad(p)) |
| 718 |
|
| 719 |
def __Aprod(self,dp): |
| 720 |
if self.getSolverOptionsFlux().isVerbose(): print "DarcyFlux: Applying operator" |
| 721 |
Qdp=self.__Q(dp) |
| 722 |
self.__pde_v.setValue(Y=-Qdp,X=Data(), r=Data()) |
| 723 |
du=self.__pde_v.getSolution() |
| 724 |
# self.__pde_v.getOperator().saveMM("proj.mm") |
| 725 |
return Qdp+du |
| 726 |
def __inner_GMRES(self,r,s): |
| 727 |
return util.integrate(util.inner(r,s)) |
| 728 |
|
| 729 |
def __inner_PCG(self,p,r): |
| 730 |
return util.integrate(util.inner(self.__Q(p), r)) |
| 731 |
|
| 732 |
def __Msolve_PCG(self,r): |
| 733 |
if self.getSolverOptionsPressure().isVerbose(): print "DarcyFlux: Applying preconditioner" |
| 734 |
self.__pde_p.setValue(X=util.transposed_tensor_mult(self.__permeability,r), Y=Data(), r=Data()) |
| 735 |
# self.__pde_p.getOperator().saveMM("prec.mm") |
| 736 |
return self.__pde_p.getSolution() |
| 737 |
|
| 738 |
def getFlux(self,p=None, fixed_flux=Data()): |
| 739 |
""" |
| 740 |
returns the flux for a given pressure ``p`` where the flux is equal to ``fixed_flux`` |
| 741 |
on locations where ``location_of_fixed_flux`` is positive (see `setValue`). |
| 742 |
Note that ``g`` and ``f`` are used, see `setValue`. |
| 743 |
|
| 744 |
:param p: pressure. |
| 745 |
:type p: scalar value on the domain (e.g. `Data`). |
| 746 |
:param fixed_flux: flux on the locations of the domain marked be ``location_of_fixed_flux``. |
| 747 |
:type fixed_flux: vector values on the domain (e.g. `Data`). |
| 748 |
:return: flux |
| 749 |
:rtype: `Data` |
| 750 |
:note: the method uses the least squares solution *u=(I+D^*D)^{-1}(D^*f-g-Qp)* where *D* is the *div* operator and *(Qp)_i=k_{ij}p_{,j}* |
| 751 |
for the permeability *k_{ij}* |
| 752 |
""" |
| 753 |
self.setSubProblemTolerance() |
| 754 |
g=self.__g |
| 755 |
f=self.__f |
| 756 |
self.__pde_v.setValue(X=self.__l*f*util.kronecker(self.domain), r=fixed_flux) |
| 757 |
if p == None: |
| 758 |
self.__pde_v.setValue(Y=g) |
| 759 |
else: |
| 760 |
self.__pde_v.setValue(Y=g-self.__Q(p)) |
| 761 |
return self.__pde_v.getSolution() |
| 762 |
|
| 763 |
class StokesProblemCartesian(HomogeneousSaddlePointProblem): |
| 764 |
""" |
| 765 |
solves |
| 766 |
|
| 767 |
-(eta*(u_{i,j}+u_{j,i}))_j + p_i = f_i-stress_{ij,j} |
| 768 |
u_{i,i}=0 |
| 769 |
|
| 770 |
u=0 where fixed_u_mask>0 |
| 771 |
eta*(u_{i,j}+u_{j,i})*n_j-p*n_i=surface_stress +stress_{ij}n_j |
| 772 |
|
| 773 |
if surface_stress is not given 0 is assumed. |
| 774 |
|
| 775 |
typical usage: |
| 776 |
|
| 777 |
sp=StokesProblemCartesian(domain) |
| 778 |
sp.setTolerance() |
| 779 |
sp.initialize(...) |
| 780 |
v,p=sp.solve(v0,p0) |
| 781 |
""" |
| 782 |
def __init__(self,domain,**kwargs): |
| 783 |
""" |
| 784 |
initialize the Stokes Problem |
| 785 |
|
| 786 |
The approximation spaces used for velocity (=Solution(domain)) and pressure (=ReducedSolution(domain)) must be |
| 787 |
LBB complient, for instance using quadratic and linear approximation on the same element or using linear approximation |
| 788 |
with macro elements for the pressure. |
| 789 |
|
| 790 |
:param domain: domain of the problem. |
| 791 |
:type domain: `Domain` |
| 792 |
""" |
| 793 |
HomogeneousSaddlePointProblem.__init__(self,**kwargs) |
| 794 |
self.domain=domain |
| 795 |
self.__pde_u=LinearPDE(domain,numEquations=self.domain.getDim(),numSolutions=self.domain.getDim()) |
| 796 |
self.__pde_u.setSymmetryOn() |
| 797 |
|
| 798 |
self.__pde_prec=LinearPDE(domain) |
| 799 |
self.__pde_prec.setReducedOrderOn() |
| 800 |
self.__pde_prec.setSymmetryOn() |
| 801 |
|
| 802 |
self.__pde_proj=LinearPDE(domain) |
| 803 |
self.__pde_proj.setReducedOrderOn() |
| 804 |
self.__pde_proj.setValue(D=1) |
| 805 |
self.__pde_proj.setSymmetryOn() |
| 806 |
|
| 807 |
def getSolverOptionsVelocity(self): |
| 808 |
""" |
| 809 |
returns the solver options used solve the equation for velocity. |
| 810 |
|
| 811 |
:rtype: `SolverOptions` |
| 812 |
""" |
| 813 |
return self.__pde_u.getSolverOptions() |
| 814 |
def setSolverOptionsVelocity(self, options=None): |
| 815 |
""" |
| 816 |
set the solver options for solving the equation for velocity. |
| 817 |
|
| 818 |
:param options: new solver options |
| 819 |
:type options: `SolverOptions` |
| 820 |
""" |
| 821 |
self.__pde_u.setSolverOptions(options) |
| 822 |
def getSolverOptionsPressure(self): |
| 823 |
""" |
| 824 |
returns the solver options used solve the equation for pressure. |
| 825 |
:rtype: `SolverOptions` |
| 826 |
""" |
| 827 |
return self.__pde_prec.getSolverOptions() |
| 828 |
def setSolverOptionsPressure(self, options=None): |
| 829 |
""" |
| 830 |
set the solver options for solving the equation for pressure. |
| 831 |
:param options: new solver options |
| 832 |
:type options: `SolverOptions` |
| 833 |
""" |
| 834 |
self.__pde_prec.setSolverOptions(options) |
| 835 |
|
| 836 |
def setSolverOptionsDiv(self, options=None): |
| 837 |
""" |
| 838 |
set the solver options for solving the equation to project the divergence of |
| 839 |
the velocity onto the function space of presure. |
| 840 |
|
| 841 |
:param options: new solver options |
| 842 |
:type options: `SolverOptions` |
| 843 |
""" |
| 844 |
self.__pde_proj.setSolverOptions(options) |
| 845 |
def getSolverOptionsDiv(self): |
| 846 |
""" |
| 847 |
returns the solver options for solving the equation to project the divergence of |
| 848 |
the velocity onto the function space of presure. |
| 849 |
|
| 850 |
:rtype: `SolverOptions` |
| 851 |
""" |
| 852 |
return self.__pde_proj.getSolverOptions() |
| 853 |
|
| 854 |
def updateStokesEquation(self, v, p): |
| 855 |
""" |
| 856 |
updates the Stokes equation to consider dependencies from ``v`` and ``p`` |
| 857 |
:note: This method can be overwritten by a subclass. Use `setStokesEquation` to set new values. |
| 858 |
""" |
| 859 |
pass |
| 860 |
def setStokesEquation(self, f=None,fixed_u_mask=None,eta=None,surface_stress=None,stress=None, restoration_factor=None): |
| 861 |
""" |
| 862 |
assigns new values to the model parameters. |
| 863 |
|
| 864 |
:param f: external force |
| 865 |
:type f: `Vector` object in `FunctionSpace` `Function` or similar |
| 866 |
:param fixed_u_mask: mask of locations with fixed velocity. |
| 867 |
:type fixed_u_mask: `Vector` object on `FunctionSpace` `Solution` or similar |
| 868 |
:param eta: viscosity |
| 869 |
:type eta: `Scalar` object on `FunctionSpace` `Function` or similar |
| 870 |
:param surface_stress: normal surface stress |
| 871 |
:type surface_stress: `Vector` object on `FunctionSpace` `FunctionOnBoundary` or similar |
| 872 |
:param stress: initial stress |
| 873 |
:type stress: `Tensor` object on `FunctionSpace` `Function` or similar |
| 874 |
""" |
| 875 |
if eta !=None: |
| 876 |
k=util.kronecker(self.domain.getDim()) |
| 877 |
kk=util.outer(k,k) |
| 878 |
self.eta=util.interpolate(eta, Function(self.domain)) |
| 879 |
self.__pde_prec.setValue(D=1/self.eta) |
| 880 |
self.__pde_u.setValue(A=self.eta*(util.swap_axes(kk,0,3)+util.swap_axes(kk,1,3))) |
| 881 |
if restoration_factor!=None: |
| 882 |
n=self.domain.getNormal() |
| 883 |
self.__pde_u.setValue(d=restoration_factor*util.outer(n,n)) |
| 884 |
if fixed_u_mask!=None: |
| 885 |
self.__pde_u.setValue(q=fixed_u_mask) |
| 886 |
if f!=None: self.__f=f |
| 887 |
if surface_stress!=None: self.__surface_stress=surface_stress |
| 888 |
if stress!=None: self.__stress=stress |
| 889 |
|
| 890 |
def initialize(self,f=Data(),fixed_u_mask=Data(),eta=1,surface_stress=Data(),stress=Data(), restoration_factor=0): |
| 891 |
""" |
| 892 |
assigns values to the model parameters |
| 893 |
|
| 894 |
:param f: external force |
| 895 |
:type f: `Vector` object in `FunctionSpace` `Function` or similar |
| 896 |
:param fixed_u_mask: mask of locations with fixed velocity. |
| 897 |
:type fixed_u_mask: `Vector` object on `FunctionSpace` `Solution` or similar |
| 898 |
:param eta: viscosity |
| 899 |
:type eta: `Scalar` object on `FunctionSpace` `Function` or similar |
| 900 |
:param surface_stress: normal surface stress |
| 901 |
:type surface_stress: `Vector` object on `FunctionSpace` `FunctionOnBoundary` or similar |
| 902 |
:param stress: initial stress |
| 903 |
:type stress: `Tensor` object on `FunctionSpace` `Function` or similar |
| 904 |
""" |
| 905 |
self.setStokesEquation(f,fixed_u_mask, eta, surface_stress, stress, restoration_factor) |
| 906 |
|
| 907 |
def Bv(self,v,tol): |
| 908 |
""" |
| 909 |
returns inner product of element p and div(v) |
| 910 |
|
| 911 |
:param v: a residual |
| 912 |
:return: inner product of element p and div(v) |
| 913 |
:rtype: ``float`` |
| 914 |
""" |
| 915 |
self.__pde_proj.setValue(Y=-util.div(v)) |
| 916 |
self.getSolverOptionsDiv().setTolerance(tol) |
| 917 |
self.getSolverOptionsDiv().setAbsoluteTolerance(0.) |
| 918 |
out=self.__pde_proj.getSolution() |
| 919 |
return out |
| 920 |
|
| 921 |
def inner_pBv(self,p,Bv): |
| 922 |
""" |
| 923 |
returns inner product of element p and Bv=-div(v) |
| 924 |
|
| 925 |
:param p: a pressure increment |
| 926 |
:param Bv: a residual |
| 927 |
:return: inner product of element p and Bv=-div(v) |
| 928 |
:rtype: ``float`` |
| 929 |
""" |
| 930 |
return util.integrate(util.interpolate(p,Function(self.domain))*util.interpolate(Bv,Function(self.domain))) |
| 931 |
|
| 932 |
def inner_p(self,p0,p1): |
| 933 |
""" |
| 934 |
Returns inner product of p0 and p1 |
| 935 |
|
| 936 |
:param p0: a pressure |
| 937 |
:param p1: a pressure |
| 938 |
:return: inner product of p0 and p1 |
| 939 |
:rtype: ``float`` |
| 940 |
""" |
| 941 |
s0=util.interpolate(p0,Function(self.domain)) |
| 942 |
s1=util.interpolate(p1,Function(self.domain)) |
| 943 |
return util.integrate(s0*s1) |
| 944 |
|
| 945 |
def norm_v(self,v): |
| 946 |
""" |
| 947 |
returns the norm of v |
| 948 |
|
| 949 |
:param v: a velovity |
| 950 |
:return: norm of v |
| 951 |
:rtype: non-negative ``float`` |
| 952 |
""" |
| 953 |
return util.sqrt(util.integrate(util.length(util.grad(v))**2)) |
| 954 |
|
| 955 |
|
| 956 |
def getDV(self, p, v, tol): |
| 957 |
""" |
| 958 |
return the value for v for a given p (overwrite) |
| 959 |
|
| 960 |
:param p: a pressure |
| 961 |
:param v: a initial guess for the value v to return. |
| 962 |
:return: dv given as *Adv=(f-Av-B^*p)* |
| 963 |
""" |
| 964 |
self.updateStokesEquation(v,p) |
| 965 |
self.__pde_u.setValue(Y=self.__f, y=self.__surface_stress) |
| 966 |
self.getSolverOptionsVelocity().setTolerance(tol) |
| 967 |
self.getSolverOptionsVelocity().setAbsoluteTolerance(0.) |
| 968 |
if self.__stress.isEmpty(): |
| 969 |
self.__pde_u.setValue(X=p*util.kronecker(self.domain)-2*self.eta*util.symmetric(util.grad(v))) |
| 970 |
else: |
| 971 |
self.__pde_u.setValue(X=self.__stress+p*util.kronecker(self.domain)-2*self.eta*util.symmetric(util.grad(v))) |
| 972 |
out=self.__pde_u.getSolution() |
| 973 |
return out |
| 974 |
|
| 975 |
def norm_Bv(self,Bv): |
| 976 |
""" |
| 977 |
Returns Bv (overwrite). |
| 978 |
|
| 979 |
:rtype: equal to the type of p |
| 980 |
:note: boundary conditions on p should be zero! |
| 981 |
""" |
| 982 |
return util.sqrt(util.integrate(util.interpolate(Bv,Function(self.domain))**2)) |
| 983 |
|
| 984 |
def solve_AinvBt(self,p, tol): |
| 985 |
""" |
| 986 |
Solves *Av=B^*p* with accuracy `tol` |
| 987 |
|
| 988 |
:param p: a pressure increment |
| 989 |
:return: the solution of *Av=B^*p* |
| 990 |
:note: boundary conditions on v should be zero! |
| 991 |
""" |
| 992 |
self.__pde_u.setValue(Y=Data(), y=Data(), X=-p*util.kronecker(self.domain)) |
| 993 |
out=self.__pde_u.getSolution() |
| 994 |
return out |
| 995 |
|
| 996 |
def solve_prec(self,Bv, tol): |
| 997 |
""" |
| 998 |
applies preconditioner for for *BA^{-1}B^** to *Bv* |
| 999 |
with accuracy `self.getSubProblemTolerance()` |
| 1000 |
|
| 1001 |
:param Bv: velocity increment |
| 1002 |
:return: *p=P(Bv)* where *P^{-1}* is an approximation of *BA^{-1}B^ * )* |
| 1003 |
:note: boundary conditions on p are zero. |
| 1004 |
""" |
| 1005 |
self.__pde_prec.setValue(Y=Bv) |
| 1006 |
self.getSolverOptionsPressure().setTolerance(tol) |
| 1007 |
self.getSolverOptionsPressure().setAbsoluteTolerance(0.) |
| 1008 |
out=self.__pde_prec.getSolution() |
| 1009 |
return out |
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