Imbricated finite element method (FE\(^2\) model)#

This model has been developed during the past few years [Feyel] in order to directly replace the constitutive equations (at the structural scale). At each Gauss point of the structural scale, the constitutive equations are thus replaced by an other finite element computation at the composite scale. The figure bellow shows the schematic principle of the FE\(^2\) approach :

  • finite element computation at the structural scale which delivers the macroscopic strain \(\boldsymbol{E}\),

  • localization which determines the local strain field at the fiber/matrix scale \(\boldsymbol{\epsilon}\),

  • a finite element computation which delivers the local stress field \(\boldsymbol{\sigma}\),

  • homogenization of \(\boldsymbol{\sigma}\) to compute the macroscopic stress \(\boldsymbol{\Sigma}\).

../_images/schema_ef2.png

Any localization/homogenization scheme can be used, but in the specific SiC/Ti case, the regular position of the fiber into the matrix allows to use the classical periodic homogenization.

Localization/homogenisation scheme#

Description#

Two files are necessary, the first one periodic.mat is defined in the **material section of the macroscopic problem and the second one file_local is defined in the **local section of the periodic.mat file.

Syntax#

This is the syntax of the periodic.mat file.

***behavior mmc_fe2 **local periodic [ **file file_name ] [ **update_tg_matrix tg_matrix_option ] [ **import name name_dest ]

where file_name is the name of the file where the localization options are indicated. The import option specifies which macroscopic parameters must imported in the microscopic problem. name is the name of the parameter in the macroscopic problem and name_dest is its name in the microscopic problem.

Syntax#

This is the syntax of the **file file_name file.

**convergence ctype **reinit itype **maxiter maxiter **ratio ratio **local_problem local_name **localization ltype \(~\,\) *impose_elset_dof elset_name \(~\,\) *macro_E_size Esize \(~\,\) **homogeneisation htype \(~\,\) *macro_flux_size Fsize

**convergence

give the convergence type : absolu, automatic

**reinit

(see ***sequence)

**maxiter

maximum number of iteration to solve the microproblem.

**ratio

specify the maximum global ratio for the convergence. method of localisation form the macroscopic scale to the microscopic scale. Two types : to_periodic_plane_strain or to_general_periodic_plane_strain. *macro_E_size size of the macroscopic strain tensor (4 in 2D or 6 in 3D).

**homogeneisation

method of homogenisation. Two types : classic_2d or classic_3d *macro_flux_size size of the macroscopic stress tensor.

Example of a FE\(^2\) test#

Macroscopic problem#

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% This is the macroscopic problem          %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

****calcul
***mesh **predefined rve2d
***resolution newton
**sequence 1
*time 0.5
*increment 1O
*ratio .1
*algorithm p1p2p3
*iteration 100
***bc
**impose_element_dof  1 E11 1. time
***material    *file periodic.mat
***output
    **test
      *precision 5
      *small 1.e-8
      *gauss_var 1 1 eto11 eto22 eto33 eto12 sig11 sig22 sig33 sig12
****return

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% This is the periodic.mat file            %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
***behavior mmc_fe2
  **local periodic
  **file file_local
  **update_tg_matrix  ask_localization
***return

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% This is the file_local file              %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
**convergence absolu
**reinit p1p2p3
**maxiter 1000
**ratio .005
**local_problem micro
**localisation to_general_periodic_plane_strain
    *impose_elset_dof ALL_ELEMENT
    *macro_E_size 4
**homogeneisation classic_2d
    *macro_flux_size 4
***return

Microscopic problem#

Description#

This part is the description of the microscopic problem. No comment, see ****calcul.

****calcul
  ***mesh
    **elset ALL_ELEMENT periodic_general_plane_strain
            periodic_info total_displacement_field
***linear_solver frontal
***resolution
  **sequence
    *ratio absolu 0.001
    *time  1.
    *increment 30
    *algorithm   eeeeee
  ***bc
    **impose_nodal_dof
      centre  U1 0.
      centre  U2 0.
      ALL_NODE U3 0.
  ***equation
    **mpc2 gauche U1 droite U1
    **mpc2 gauche U2 droite U2

    **mpc2 haut U1 bas U1
    **mpc2 haut U2 bas U2

    **mpc2 bg U1 bd U1
    **mpc2 bg U2 bd U2
    **mpc2 bg U1 hg U1
    **mpc2 bg U2 hg U2
    **mpc2 bg U1 hd U1
    **mpc2 bg U2 hd U2
***material
  **elset fibre
    *file  fibre
  **elset matrice
    *file matrix
***output
    **test
      *precision 5
      *small 1.e-8
      *gauss_var 1 1 eto11 eto22 eto33 eto12 sig11 sig22 sig33 sig12
****return