Toward a Meta-Model for Computational Engineering

Geometric modeling and finite element analysis have matured in recent decades. Both methods are used extensively in engineering design. However, the link between geometric modeling, physical modeling and finite element analysis is currently cumbersome, error-prone, and ad-hoc. Topological domain modeling provides the missing link. In this paper, we propose a combined topological modeling and finite element modeling method that allows not only topological modeling, but also promotes geometric and physical modeling, by providing a topological base space for the definition of finite element meshes, fields, and the definition and solution of boundary value problems. We call the method the Constructive Topological Domain Method (CTDM). In this method, Primitive Topological Domains (PTDs), each possessing a natural coordinate space, are combined in multiple n-dimensional Cartesian coordinate spaces, called charts, using generalizations of Boolean set operations, to create Constructed Topological Domains (CTDs) capable of acting as the base spaces of fiber bundles. The charts are glued together to create an atlas, within which the CTD is defined. The fiber of the bundle may describe, in addition to geometry, physical fields like density, stress, and temperature. Finite element meshes may be defined upon each of the PTDs from which the CTD is constructed, enabling the definition and solution of boundary value problems, thus avoiding the difficult and messy problem of creating a single finite element mesh to represent the entire CTD. A modified finite element method, to handle the individually meshed PTDs, is described. The boundary conditions may be specified as analytical or as finite element-based fields upon each of the PTDs. The CTDM appears to be a promising approach to robust mathematical and computational modeling of physical objects. Simple examples are presented.