[Getfem-users] Questions about user-defined strain energy

[David Danan]

Dear Getfem-users,

to check whether i understand correctly the implementation (and because it's actually much clearer in my code that way), i am trying to replace some predefined bricks for strain energy by the equivalent version using GWFL.

For that, i have several questions

First question

For instance

lawname = 'SaintVenant Kirchhoff'

clambda,cmu = params["clambda"],params["cmu"]

model.add_initialized_data('paramsSVK', [clambda, cmu])

idBrick=model.add_finite_strain_elasticity_brick(mim, lawname, 'u', 'paramsSVK')

becomes

clambda,cmu = params["clambda"],params["cmu"]

model.add_initialized_data('paramsSVK', [clambda, cmu])

idBrick=model.add_nonlinear_generic_assembly_brick(mim,"(paramsSVK(1)/2)*pow(Trace(Green_Lagrange_E),2)+paramsSVK(2)*Trace(Green_Lagrange_E'*Green_Lagrange_E)")

And the results seems to be the same.

However, for the incompressible Mooney Rivlin strain energy case, it does not behave as i expected

Using the example there as a basis in a 3D case

https://github.com/getfem-doc/getfem/blob/master/interface/tests/python/demo_nonlinear_elasticity.py

and after some simplifications

I have tried to replace the last line in

lawname = 'Incompressible Mooney Rivlin'

model.add_initialized_data('paramsIMR', [1,1])

model.add_finite_strain_elasticity_brick(mim, lawname, 'u', 'paramsIMR')

By this

model.add_nonlinear_generic_assembly_brick(mim, "Incompressible_Mooney_Rivlin_potential(Grad_u, [paramsIMR(1);paramsIMR(2)])")

Which was exactly the same, of course. Next, i have tried to replace it by these lines

model.add_macro("F", "Grad_u+Id(3)")

model.add_nonlinear_generic_assembly_brick(mim, "paramsIMR(1)* ( Matrix_j1(Right_Cauchy_Green(F)) - 3 )+ paramsIMR(2)* ( Matrix_j2(Right_Cauchy_Green(F)) - 3 )")

Which failed to converge

I thought this _expression_ was consistent with the implementation of Mooney_Rivlin_hyperelastic_law::strain_energy and the compute_invariants here

http://download-mirror.savannah.gnu.org/releases/getfem/doc/getfem_reference/getfem__nonlinear__elasticity_8cc_source.html

and the documentation

https://getfem.org/userdoc/model_nonlinear_elasticity.html?highlight=finite%20strain

But, clearly, i am missing something. Could you explain what I am doing wrong?

Second question

In a 2D case, i would like to be able to use either a plane strain approximation based on a given strain energy _expression_ in 3D.

In the implementation, it is nicely done in  scalar_type plane_strain_hyperelastic_law::strain_energy there

http://download-mirror.savannah.gnu.org/releases/getfem/doc/getfem_reference/getfem__nonlinear__elasticity_8cc_source.html

Is it possible to do the same using the GWFL?

Third question

I would like to be able to compute the von-mises field for any strain energy function.

If It is an existing brick, the method

md.compute_finite_strain_elasticity_Von_Mises(lawname, 'u', 'params', mfdu)

will do the trick just fine.

If it's not the case, i guess i can use something akin to the actual implementation of compute_finite_strain_elasticity_Von_Mises

 

  std::string expr = "sqrt(3/2)*Norm(Deviator(Cauchy_stress_from_PK2("

  + adapted_lawname + "_PK2(Grad_" + varname + "," + params + "),Grad_"

  + varname + ")))";

  ga_interpolation_Lagrange_fem(md, expr, mf_vm, VM, rg);

  }

combined with local_projection to get the value at the elements.

The question is: is it possible to compute Piola Kirchhoff 2 from the strain energy within the GWFL _expression_ given to local_projection?

I have the impression it's the only thing i need to be able to do what i want.

Let W be a strain energy function and E be the Green-Lagrange tensor (defined as macros, let's say), is Diff(W, E) doing exactly what i am expecting for this purpose?

If it's not the case, is there another way to do it?

Thank you in advance,

kind regards,

David.