On 9/21/07, joelvuillon <[EMAIL PROTECTED]> wrote:
>
>
> Hi,
> I am working with Florian De Vuyst who is in touch with Daniel Wheeler for
> solving Elasticity system with Fipy.
>
> Daniel provides the following formulation
>
> > mesh = ...
> >
> > u = CellVariable(mesh=mesh)
> > v = CellVariable(mesh=mesh)
> >
> > The equations will be,
> >
> > Xhat = (1, 0)
> > Yhat = (0, 1)
> >
> > u_eqn = DiffusionTerm(mu) \
> > + mu * (u.getFaceGrad().dot(Xhat) * Xhat \
> > + v.getFaceGrad().dot(Xhat) *
> Yhat).getDivergence() \
> > + ConvectionTerm(lambda * Xhat) \
> > + lambda * v.getGrad().dot(Yhat) \
> > + f
> >
> > v_eqn = DiffusionTerm(mu) \
> > + mu * (u.getFaceGrad().dot(Yhat) * Xhat \
> > + v.getFaceGrad().dot(Yhat) *
> Yhat).getDivergence() \
> > + ConvectionTerm(lambda * Yhat) \
> > + lambda * u.getGrad().dot(Xhat) \
> > + g
>
>
> for solving the system for U=(u,v) :
>
> div.[ mu grad(U) + mu [grad(U)]^T + lambda I tr(grad(U)) ] + F = (0,0).
>
> However, I do not agree with that formulation for the particular term div.[
> lambda I tr(grad(U)) ] which seems has been written as div.[ lambda (U) ].
My apologies, my colleague just pointed out what the equation represents and
your formulation below is correct.
Although your representation is correct, the terms that are not defined
by the "DiffusionTerm" will be solved explicitly and I imagine that this
could cause some convergence issues related to grid size. You may want
to make the following alterations. First define a diffusion
coefficient as a face
variable (assuming you are using fipy version 1.2)
uDiffCoeffVector = VectorFaceVariable(mesh=mesh, value=(2 * mu + lambda, mu))
uDiffCoeff = uDiffCoeffVector.dot(abs(mesh._getFaceNormals()))
u_Eqn = DiffusionTerm(uDiffCoeff) \
+ (mu * v.getFaceGrad().dot(Xhat) * Yhat \
+ lambda * v.getFaceGrad().dot(Yhat) * Xhat).getDivergence()
You can do the same for the v_eqn. This formulation will only work for
a regular grid.
I hope this helps.
>
> Instead, I propose the following formulation :
>
> u_eqn = DiffusionTerm(mu) \
> + mu * (u.getFaceGrad().dot(Xhat) * Xhat \
> + v.getFaceGrad().dot(Xhat) *
> Yhat).getDivergence() \
> + lambda * (u.getFaceGrad().dot(Xhat) \
> +
> v.getFaceGrad().dot(Yhat))* Xhat).getDivergence() \
> + f
>
> v_eqn = DiffusionTerm(mu) \
> + mu * (u.getFaceGrad().dot(Yhat) * Xhat \
> + v.getFaceGrad().dot(Yhat) *
> Yhat).getDivergence() \
> + lambda * (u.getFaceGrad().dot(Xhat) \
> +
> v.getFaceGrad().dot(Yhat))*Yhat).getDivergence() \
> + g
>
> Best Regards,
>
> Joel
>
--
Daniel Wheeler
