Interpolation of arbitary quantities

Once a solution has been computed, it is quite easy to extract any quantity of interest on it with the interpolation functions for instance for post-treatment.

Basic interpolation

The file getfem/getfem_interpolation.h defines the function getfem::interpolation(...) to interpolate a solution from a given mesh/finite element method on another mesh and/or another Lagrange finite element method:

getfem::interpolation(mf1, mf2, U, V, extrapolation = 0);

where mf1 is a variable of type getfem::mesh_fem and describes the finite element method on which the source field U is defined, mf2 is the finite element method on which U will be interpolated. extrapolation is an optional parameter. The values are 0 not to allow the extrapolation, 1 for an extrapolation of the exterior points near the boundary and 2 for the extrapolation of all exterior points (could be expensive).

The dimension of U should be a multiple of mf1.nb_dof(), and the interpolated data V should be correctly sized (multiple of mf2.nb_dof()).

... important:

``mf2`` should be of Lagrange type for the interpolation to make sense but the
meshes linked to ``mf1`` and ``mf2`` may be different (and this is the
interest of this function). There is no restriction for the dimension of the
domain (you can interpolate a 2D mesh on a line etc.).

If you need to perform more than one interpolation between the same finite element methods, it might be more efficient to use the function:

getfem::interpolation(mf1, mf2, M, extrapolation = 0);

where M is a row matrix which will be filled with the linear map representing the interpolation (i.e. such that V = MU). The matrix should have the correct dimensions (i.e. mf2.nb_dof()``x``mf1.nb_dof()). Once this matrix is built, the interpolation is done with a simple matrix multiplication:

gmm::mult(M, U, V);

Interpolation based on the high-level weak form language

It is possible to extract some arbitrary expressions on possibly several fields thanks to the weak form language and the interpolation functions.

This is specially dedicated to the model object (but it can also be used with a ga_workspace object). For instance if md is a valid object containing some defined variables u (vectorial) and p (scalar), one can interpolate on a Lagrange finite element method an expression such as p*Trace(Grad_u). The resulting expression can be scalar, vectorial or tensorial. The size of the resulting vector is automatically adapted.

The high-level generic interpolation functions are defined in the file getfem/getfem_generic_assembly.h.

There is different interpolation functions corresponding to the interpolation on a Lagrange fem on the same mesh, the interpolation on a cloud on points or on a getfem::im_data object.

Interpolation on a Lagrange fem:

void getfem::ga_interpolation_Lagrange_fem(workspace, mf, result);

where workspace is a getfem::ga_workspace object which aims to store the different variables and data (see Compute arbitrary terms - high-level generic assembly procedures - Weak-form language), mf is the getfem::mesh_fem object reresenting the Lagrange fem on which the interpolation is to be done and result is a beot::base_vector which store the interpolatin. Note that the workspace should contain the epression to be interpolated.

void getfem::ga_interpolation_Lagrange_fem(md, expr, mf, result, rg=mesh_region::all_convexes());

where md is a getfem::model object (containing the variables and data), expr (std::string object) is the expression to be interpolated, mf is the getfem::mesh_fem object reresenting the Lagrange fem on which the interpolation is to be done, result is the vector in which the interpolation is stored and rg is the optional mesh region.

Interpolation on a cloud of points:

void getfem::ga_interpolation_mti(md, expr, mti, result, extrapolation = 0, rg=mesh_region::all_convexes(), nbpoints = size_type(-1));

where md is a getfem::model object (containing the variables and data), expr (std::string object) is the expression to be interpolated, mti is a getfem::mesh_trans_inv object which stores the cloud of points (see getfem/getfem_interpolation.h), result is the vector in which the interpolation is stored, extrapolation is an option for extrapolating the field outside the mesh for outside points, rg is the optional mesh region and nbpoints is the optional maximal number of points.

Interpolation on an im_data object (on the Gauss points of an integration method):

void getfem::ga_interpolation_im_data(md, expr, im_data &imd,
 base_vector &result, const mesh_region &rg=mesh_region::all_convexes());

where md is a getfem::model object (containing the variables and data), expr (std::string object) is the expression to be interpolated, imd is a getfem::im_data object which refers to a integration method (see getfem/getfem_im_data.h), result is the vector in which the interpolation is stored and rg is the optional mesh region.