[deal.II] Question about how to get the Stiffness matrix partial derivative with respect to density

[Lance Zhang]

Hello ,

Thanks for your time to see this question.

The given conditions are stiffness matrix,destination function and x from 
Ax=b,x is movement ,b is force,A is global stiffness matrix.
J(x)=||x||norm-1
There is one equation:

dJ(x,θ)/dθ=∂J(x,θ)/∂θ+∂J(x,θ)/∂x 
*∂x/∂θ=∂J(x,θ)/∂θ+∂J(x,θ)/∂x*[A^-1(θ))(-∂A(θ)/∂θ)x],because b has no 
relation with density.

I don't know how to get [A^-1(θ))(-∂A(θ)/∂θ)x],A is matrix with nxn,if θ is 
mx1 vector,∂A(θ)/∂θ is nxnxm matrix. 

But I don't know how to calculate (A^-1(θ))(-∂A(θ)/∂θ),I can simply get the 
value of A^-1.
I use cook_membrane.cc this file for dJ(x,θ)/dθ.

I found the density was just included from the mu invoked from 
parameters.prm.

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>


public:
    Material_Compressible_Neo_Hook_One_Field(const double mu,
                                             const double nu)
      :
      kappa((2.0 * mu * (1.0 + nu)) / (3.0 * (1.0 - 2.0 * nu))),
      c_1(mu / 2.0)
    {
      Assert(kappa > 0, ExcInternalError());
    }

    const double kappa;
    const double c_1;

    // Value of the volumetric free energy
    NumberType
    get_Psi_vol(const NumberType &det_F) const
    {
      return (kappa / 4.0) * (det_F*det_F - 1.0 - 2.0*std::log(det_F));
    }

    NumberType
    get_dPsi_vol_dJ(const NumberType &det_F) const
    {
      return (kappa / 2.0) * (det_F - 1.0 / det_F);
    }

    get_d2Psi_vol_dJ2(const NumberType &det_F) const
    {
      return ( (kappa / 2.0) * (1.0 + 1.0 / (det_F * det_F)));
    }    NumberType
    get_Psi_iso(const SymmetricTensor<2,dim,NumberType> &b_bar) const
    {
      return c_1 * (trace(b_bar) - dim);
    }
    SymmetricTensor<2,dim,NumberType>
    get_tau_bar(const SymmetricTensor<2,dim,NumberType> &b_bar) const
    {
      return 2.0 * c_1 * b_bar;
    }

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 assemble_system_one_cell(const typename 
DoFHandler<dim>::active_cell_iterator &cell,
                             ScratchData_ASM &scratch,
                             PerTaskData_ASM &data)
    {
      // Due to the C++ specialization rules, we need one more
      // level of indirection in order to define the assembly
      // routine for all different number. The next function call
      // is specialized for each NumberType, but to prevent having
      // to specialize the whole class along with it we have inlined
      // the definition of the other functions that are common to
      // all implementations.
      assemble_system_tangent_residual_one_cell(cell, scratch, data);
      assemble_neumann_contribution_one_cell(cell, scratch, data);
    }



<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<



Unfortunately ,I did not find density vector in this file  to calculate 
-∂A(θ)/∂θ.

Could anyone provide any hint or suggestions?

Thanks in advance!
Best regards
Lance

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