Mesh function Application Object

A Mesh_function is a Maths_function that:

has a domain that consists of a table of real tuple spaces;

The table is a one dimensional array for an unstructured mesh, but can be a rectangualar array of any dimension for a structured mesh.

The real tuples spaces can have different dimensions, if the mesh contains cells of different dimension.

The real tuple spaces can have different bounds, if the mesh contains cells of different shape, such as tetrahedrons and hexahedrons.
has a set of control values, such that the function within each cell is determined by a subset of the control values and the basis for that cell; and

The assignment of control values to cells is determined by the topology of the mesh. A control value is usually, but not necessarily, the value of the function at a position within the mesh.
interpolates or extrapolates separately within each cell from the control values assigned to that cell.

EXAMPLE - The Mesh_function 'my_function_from_gauss_points' is defined with respect to the mesh shown in Figure 1. This function has 16 control values. The function uses bi-linear interpolation/extrapolation within each cell from the control values that are at Gauss points within the cell.

Figure 1: Mesh function with interpolation from Gauss points

EXAMPLE - The Mesh_function 'my_function_from_corners' is defined with respect to the mesh shown in Figure 2. This function has 9 control values. The function uses bi-linear interpolation/extrapolation within each cell from the control values that are at corners of the cell.

Figure 2: Mesh function with interpolation from corner vertices

EXPRESS specification:

*)
ENTITY Mesh_function
  SUBTYPE OF (Maths_function);
  basis :  OPTIONAL LIST[1,?] OF Mesh_function_basis;
  control_values :  Table;
  mesh :  Mesh;
  uniform_basis :  OPTIONAL BOOLEAN;
  values_at_vertices :  BOOLEAN;
  WHERE
    necessary_basis :  values_at_vertices OR EXISTS (basis);
    uniform_specified :  EXISTS (basis) = EXISTS (uniform_basis);
    one_basis_if_uniform :  IF EXISTS (basis) AND 
                               EXISTS (uniform_basis) THEN
                              IF uniform_basis THEN
                                SIZEOF (basis) = 1;
                              ELSE
                                TRUE;
                              ENDIF;
                            ELSE
                              TRUE;
                            ENDIF;   
END_ENTITY;
(*

Attribute definitions:

basis: the Mesh_function_basis that specifies the method of interpolation or extrapolation to be used within each cell of the mesh.

If uniform_basis is true, then a single Mesh_function_basis is specified for use within each cell. If uniform_basis is false, then a Mesh_function_basis is specified separately for each cell of the mesh.

control_values: the Table that contains the control values at the vertices of the cells of the mesh or at the discretisation points within each cell. The Table shall be a structured according to the following rule.

The Table is a function with:

domain:

an integer tuple space that identifies the discretisation points of the Mesh_function;

range:

the range of the Mesh_function.

The components of the integer tuple space domain are shown in Table 1.

Table 1: Domain of the control values Table for a Mesh_function

unstructured mesh values at vertices i
where:

i is the position in the vertex array specified for the unstructured mesh.

unstructured mesh discretisation points for each cell j₁, j₂, ...j_n, i
where:

j₁, j₂, ...j_n is the position in the pattern of discretisation points, for the cell basis;

i is the position in the cell array specified for the unstructured mesh.

structured mesh values at vertices i₁, i₂, ...i_n
where:

i₁, i₂, ...i_n is the position of the vertex in the structured mesh.

structured mesh discretisation points for each cell j₁, j₂, ...j_n, i₁, i₂, ...i_n
where:

j₁, j₂, ...j_n is the position in the pattern of discretisation points, for the cell basis;

i₁, i₂, ...i_n is the position of the vertex in the structured mesh.

mesh: the Mesh for the Mesh_function.

uniform_basis: a flag that indicate whether or not the Mesh_function a uniform basis, as follows:

if uniform_basis is true, then each cell of the mesh has the same Mesh_function_basis;
if uniform_basis is false, then a Mesh_function_basis is specified separately for each cell of the mesh.

values_at_vertices: a flag that indicate whether or not the control values are specified for the vertices or the cells of Mesh, as follows:

if values_at_vertices is true, then the control values are specified for the vertices of the Mesh;
if values_at_vertices is false, then the control values are specified for a separate pattern of discretisation points for each cell of the Mesh.

Formal propositions:

necessary_basis: if values_at_vertices is false, then a basis shall be supplied.

NOTE - If the control values are specified for a separate pattern of discretisation points for each cell, then the basis attribute is required to specify the pattern.

uniform_specified: a uniform_basis flag shall be supplied if, and only if, a basis is supplied.

one_basis_if_uniform: the length of the basis list shall be 1, if the a uniform_basis flag is true.

NOTE - each cell has the same basis if the uniform_basis flag is true. Therefore, only one basis shall be supplied.

unstructured mesh	values at vertices	i where: i is the position in the vertex array specified for the unstructured mesh.
unstructured mesh	discretisation points for each cell	j₁, j₂, ...j_n, i where: j₁, j₂, ...j_n is the position in the pattern of discretisation points, for the cell basis; i is the position in the cell array specified for the unstructured mesh.
structured mesh	values at vertices	i₁, i₂, ...i_n where: i₁, i₂, ...i_n is the position of the vertex in the structured mesh.
structured mesh	discretisation points for each cell	j₁, j₂, ...j_n, i₁, i₂, ...i_n where: j₁, j₂, ...j_n is the position in the pattern of discretisation points, for the cell basis; i₁, i₂, ...i_n is the position of the vertex in the structured mesh.