Product of mesh Application Object

A Product_of_mesh is a relationship that is between:

• operands that are a 1 dimensional Mesh and a n dimensional Mesh; and
• a product that is an n+1 dimensional Mesh,

and that indicates the n+1 dimensional Mesh is the Cartesian product of the operands.

The ordering of cells and vertices of the product Mesh is as follows:

• cell i + n(j-1) of the product mesh corresponds to cell i of the first operand and cell j of the second operand, where n is the total number of cells of the first operand;
• vertex i + m(j-1) of the product mesh corresponds to vertex i of the first operand and vertex j of the second operand, where m is the total number of vertices of the first operand;

EXAMPLE - A product of a 2 dimensional mesh consisting of 4 cells and a 1 dimensional mesh consisting of 2 cells is shown in Figure 1.

Figure 1: A Product_of_mesh between a 2D mesh and a 1D mesh

EXPRESS specification:

*)
ENTITY product_of_mesh;
  operands  :  LIST[2,2] OF Mesh
  product  :  Mesh;
WHERE
  consistent_topological_dimension : 
    IF ('MESH_ARM.REGION' IN TYPEOF(operands[1])) AND
       ('MESH_ARM.REGION' IN TYPEOF(operands[2])) AND
       ('MESH_ARM.REGION' IN TYPEOF(product)) THEN
      operands[1]\Region.topological_dimension + 
        operands[2]\Region.topological_dimension =  
          product\Region.topological_dimension;
    ELSE
      FALSE;
    ENDIF;
  single_dimensional_operand : 
    IF ('MESH_ARM.REGION' IN TYPEOF(operands[1])) AND
       ('MESH_ARM.REGION' IN TYPEOF(operands[2])) THEN
      operands[1]\Region.topological_dimension = 1 OR 
        operands[2]\Region.topological_dimension = 1;  
    ELSE
      FALSE;
    ENDIF;
END_ENTITY;
(*

Attribute definitions:

operands: the instances of Mesh that define the product.

product: the Mesh that is the Cartesian product of the operands.

Formal propositions:

consistent_dimensions: the sum of the topological_dimension for each of the operands shall be equal to the topological_dimension of the product.

single_dimensional_operand: one of the operands shall have a topological_dimension equal to 1.