Mesh function basis Application Object

A Mesh_function_basis is a Maths_function that has:

• a domain that is a real tuple space; and
• a range that is a space of table functions.

NOTE - A Mesh_function_basis is used such that the domain is a parametric space that identifies points within a cell of a mesh. Hence the dimension of the real tuple space shall be the same as the topological dimension of the cell.

Each table function in the range of a Mesh_function_basis has:

• a domain that is a subscript space; and
• a range that is the reals.

NOTE - A Mesh_function_basis is used such that

• the domain of each table function in its range is a parametric space that identifies the control values for a cell;
• the range of each table function is its range is a weighting for a control value.

Each table function specifies the weightings for control values that define the value of a Mesh_function at a point within a cell.

EXAMPLE - There is bi-linear interpolation/extrapolation from Gauss points for each cell of the mesh shown in Figure 1. The Mesh_function_basis these cells has:

• cell_topological_dimension=2;
• cell_control_value_array_dimesion=2;
• cell_control_value_array_order=(1, 1);
• control_value_positions given by a table as follows:

  term(1,1): (-0.577, -0.577)	term(1,2): (-0.577, +0.577)
  term(2,1): (+0.577, -0.577)	term(2,2): (+0.577, +0.577)

  These positions are consistent with each position within a cell being identified by a point in the tuple space that is the Cartesian product [-1.0, 1.0]´ [-1.0, 1.0].

EXAMPLE - There is bi-linear interpolation from vertices for each cell of the mesh shown in Figure 2. The Mesh_function_basis these cells has:

• cell_topological_dimension=2;
• cell_control_value_array_dimesion=2;
• cell_control_value_array_order=(1, 1);
• control_value_positions given by a table as follows:

  term(1,1): (0.0, 0.0)	term(1,2): (0.0, 1.0)
  term(2,1): (1.0, 0.0)	term(2,2): (1.0, 1.0)

  These positions are consistent with each position within a cell being identified by a point in the tuple space that is the Cartesian product [0.0, 1.0]´ [0.0, 1.0].

*)
ENTITY Mesh_function_basis;
  SUBTYPE OF (Maths_functon);
  cell_topological_dimension :  INTEGER;
  control_value_array_dimension :  INTEGER;
  control_value_array_order :  ARRAY[1,control_value_array_dimension] OF INTEGER;
  control_value_positions :  Table;
END_ENTITY;
(*

Attribute definitions:

cell_topology_dimension: the dimension of the real tuple space that is the domain of the Mesh_function_basis.

control_value_array_dimension: the dimension of the subscript space for each table function in the range of the Mesh_function_basis.

control_value_array_order: the number of control values - 1 for each control value array direction.

control_value_positions: the table that specifies the 'positions' of the control values within the domain of the Mesh_function_basis. The 'position' of a control value is the point within the domain that has a weighting of 1.0 for that control value and a weighting of 0.0 for all other control values.

The control_value_positions table shall be a function that has:

• a domain that is an integer tuple space that identifies the discretisation points for a cell; and
• a range that is a real tuple space that identifies positions within a cell.

A control value for a Mesh_function_basis need not have a position. Whether or not a control value has a position is determined by the nature Mesh_function_basis. If a control value does not have a position, then the corresponding value of the control_value_positions function is of no significance.

Informal propositions:

consistent_cell_topology_dimension: The cell_topology_dimension shall be the same at the topological_dimension of any topological_region to which the Mesh_function_basis is assigned.

consistent_control_value_position_table_domain: The domain of the table of control value positions shall be consistent with the control_value_array_order. It shall be the tuple space that is the Cartesian product of integer intervals [1, n₁]´[1, n₂]´ ...[1, n_m], where:

• m is the control_value_array_dimension; and
• n_i is control_value_array_order[i] + 1.

consistent_control_value_position_table_range: The dimension of the tuple space that is the range of the table of control value positions shall be equal to the cell_topology_dimension.

valid_weighting_at_control_value_position: If a control value is assigned to a position, then the value of the Mesh_function_basis at that position shall be:

• 1 for the control value assigned to the position;
• 0 for all other control values