Re: [Getfem-users] Mooney-Rivlin and negative material constants

[Igor Peterlik]

Dear Yves,


sorry for the delay, but I was waiting for another experimental verification
of the parameters done by my colleagues...

First, the definition of two-constant Mooney-Rivlin material given by you 
bellow is exactly what I need. Before I start using getfem, I briefly studied the 
code (getfem_nonlinear_elasticity.h and cc), so I was sure this is the correct implementation.

Now, something about the constants. New laboratory experiments and parameter fitting 
has shown, that the constants are really negative (more precisely 
C10=-2752, C01=720). Moreover, my colleagues modelled the simulation
using these constants in ANSYS getting the same result as in experimental
setting. Therefore, the constants are acceptable (I also found some other
literature considering negative constants when modelling silicon and 
soft-tissues materials).

So the problem is that the classical Newton method is not suitable as
the tangent stiffness matrix A(u) is indefinite (in fact the more than half
of eigenvalues is negative for A(0)). I tried some preconditioning, namely
adding a multiple of identity matrix to make the tangent stiffness positive-
definite as well as Murphy-Golub-Wathen preconditioning, however 
it did not show any improvement and the Newton still diverged (
for very small step, it converges but the "minimum" is apparently
wrong).

The other two possibilities I considered (without trying) are Truncated Newton 
method,
when the direction of Newton is given also by the negative curvature
detected in conjug. gradients when solving the linear system in each step
and the other one -- doing some prediction first (for example by incremental
loading) and using the Newton as correction.

Therefore, I would like to ask for an advice -- what kind of method should
help in this case? Do you think some of those considered above are suitable?
Aren't there also some other users coping with indefinite Hessians?


        Thank you very much in advance


                        Best regards

                                Igor Peterlik







On Mon, 4 Jun 2007, Yves Renard wrote:

> Le jeudi 31 mai 2007 11:30, Igor Peterlik a écrit :
> > Dear getfem experts,
> >
> > First, thanks for providing such a great FEM code. I found
> > it really useful for my research.
> >
> > And now, the problem. I want to model
> > a deformation of a silicon cylindric object. According to
> > real lab measurements, Mooney-Rivlin material with constants
> > C_10=2.94e+03 and C_01=-1.11e+03 should be suitable for
> > realistic modelling of the object.
>
> Indeed, i am not an expert in  Mooney-Rivlin material  constants.
> It may be understandable that if the constant C_01 is to much negative, the
> poly-convexity of the law could be lost (but really, I did not study this at
> all). Are you sure the values you obtained are in an acceptable range ?
>
> The constants in the Getfem brick are defined as follows (strain energy) :
>
> W = C_01 ( I_1(C) - 3) + C_10 (I_2(C) - 3)
>
> where $C  = (\nabla \phi)^T \nabla \phi$ is the right Cauchy Green tensor and
> I_1, I_2 the two first usual invariants. Is it the definition you expected ?
>
>
> Yves.
>
> -------------------------------------------------------------------------
>  Yves Renard (address@hidden)       tel : (33) 04.72.43.80.11
>  Pole de Mathematiques, INSA de Lyon          fax : (33) 04.72.43.85.29
>  Institut Camille Jordan - CNRS UMR 5208
>  20, rue Albert Einstein
>  69621 Villeurbanne Cedex, FRANCE
>  http://math.univ-lyon1.fr/~renard
> -------------------------------------------------------------------------