ࡱ> OQNY bjbjWW 0==-!]8$"4z$$p A 4u z||||||$k_x  |  $pz $Lpz z  <r@zpV$i@Cs (X" Topology and meshes An explanation of the approach in the EACM AuthorDavid LealDate10th February 2000Version3.0CirculationRay Goult, NAFEMS CAD-FE Working Group, PDES Inc. EA team, Boeing Fluid Dynamics AP team, Keith Hunten, Phil Kraushar The production of this document was funded by the BSI under the Consultancy Drafting Scheme Introduction At present, there are two completely independent approaches to topology in STEP: generic topology in part 42, which is a generalisation of CAD practice; and mesh topology in part 104, which is derived from FEA practice. The Engineering Analysis Core Model (EACM) seeks to develop the topology in part 104 in the following ways: the definition of a mesh will be separated from the definition of a finite element model, so that: the same mesh can be used for more than one model; a mesh can be used for models that are not finite elements, such as finite difference or finite volume; a mesh can be used for the interpolation of test results, where there is no model; structured as well as unstructured meshes will be supported. Part 42 defines topological objects by their boundaries, but analysis requires internal topological structures. The EACM will integrate the definition of a mesh with the existing boundary topology in part 42. This document outlines how it is proposed to do this for initial review, prior to the CD draft of the EACM. A Mesh The high level ARM model for a mesh is shown in  REF _Ref474813358 \h Figure 1. Figure  SEQ Figure \* ARABIC 1: High level model for a mesh Notes: The ARM entity topological_object maps to topological_representation_item. The ARM entity topological_definition maps to product_definition_shape, but this could be mis-using the concept of 'shape'. What about a new entity product_definition_topology? The ARM entity physical_space maps to the thing that is meshed - usually a product_definition. A topological_object can be a vertex with a topological dimension of zero. A topological_object can be a region with a topological dimension of 1 (edge), 2 (face), 3 (volume) or higher. A topological_object can be an assembly of parts with different topological dimensions. A region is a topolgical_object of 1 or higher dimension. A topological_object of dimension 2 or 3 can have a shape, such as quadrilateral or hexahedron. This is purely a topological shape, such as specified for a finite element, and is not the geometric shape. The entity mesh_product enables a 4D mesh of space-time to be created from a 3D mesh of space and a 1D mesh of time (i.e. a division of time into time steps). It would be legitimate to create an information model in which physical objects such as pipe, beam and line (electrical) were subtypes of 'edge'. This is not convenient because: it would result in a plethora of AND supertype relationships; and this is merely one view of a pipe (say), which can equally be regarded as a subtype of 'topological volume' if we look in more detail. However, we see that the relationship between a pipe and 'edge' is a classification, and one of many possible classifications of a pipe. This leads to the conclusion that a topological_object is a class. Structured mesh The ARM model for a structured mesh is shown in  REF _Ref473958026 \h  \* MERGEFORMAT Figure 2. Figure  SEQ Figure \* ARABIC 2: Structured mesh Notes: The AIM model will look much the same, with structured_mesh being a new subtype of topological_representation_item (from a new part 5x?). A structured_mesh is a region with an internal structure. The region shall be quadrilateral for a 2D structured_mesh, or hexahedral for a 3D structured_mesh. The number of cells of the mesh in each direction is specified. The entity component_of_structured_mesh allows a component of the mesh to be extracted, which is either: of the same dimension - i.e. a sub-region; or of a lower dimension - such as a face or edge. Unstructured mesh Unstructured meshes are already supported by part 104. The proposed model below is a simplification of part 104, in which all the cells of the mesh are referenced as a single list. This will enable a field to be described with respect to a mesh in a concise way, whether the mesh is structured or unstructured. The ARM model is shown in  REF _Ref473958809 \h  \* MERGEFORMAT Figure 3. Figure  SEQ Figure \* ARABIC 3: An unstructured mesh Notes: The AIM model will look much the same, with unstructured_mesh being a new subtype of topological_representation_item (from a new part 5x?). A list of cells shall be specified for an unstructured mesh. A list of vertices can be specified for an unstructured mesh if it is intended to take advantage of this vertex order to defined values at mesh vertices in a concise way using a linearized_table_function from STEP part 50. A cell can have a different order in different directions. This is an extension to part 104, required by p-refinement. Support for a high order elements (such as 4 by 4 by 2 say) requires an algorithm for the definition of vertex positions - replacing the diagrams in part 104. The entity unstructured_mesh is an extension of connected_face_set in part 42, into n dimensions. For an unstructured_mesh, the component cells are ordered, so that a mesh_function (see below) has a concise form. A mesh function A mesh function is a subtype of maths_function from part 50. The control values of a mesh function are defined either: at the vertices of the mesh; or at a pattern of discretisation points for each cell of the mesh. The function is evaluated by interpolating or extrapolating within each cell of the mesh separately. The ARM model is shown in  REF _Ref473959378 \h  \* MERGEFORMAT Figure 4. Figure  SEQ Figure \* ARABIC 4: A mesh function Notes: The table contains values for either the vertices of the mesh or for the discretisation points within each cell. The ARM entity table maps to linearized_table_function in STEP part 50, which encodes the control values as a single packed array. Usually the subtype standard_table_function will be used, because there is unlikely to be any symmetry or banded structure to the table of control values. The mesh_function_basis specified the pattern of discretisation points within each cell, and the order of interpolation or extrapolation that is used. If uniform_bases is true, then a single basis is specified for all cells. If uniform_bases is false, then a list of bases is specified - one for each cell. An entity maths_representation_by_function is proposed as a subtype of representation. This will allow a property_definition to be represented by a maths_function. The entity mesh_function_basis is a class or family of mathematical function. The interpolation/extrapolation from the discretisation points within a cell is a particular mathematical function that is a member of the class or family. 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