Constructive topological domain modeling for solution of problems with moving boundaries

摘要

Both geometric modeling and finite element analysis are employed in fracture propagation modeling in solids. However, the link between geometric modeling, physical modeling, and finite element analysis is currently ad hoc. Constructive topological domain modeling (CTDM) provides an integrated solution. In CTDM, 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 PTDs embedded in charts create an atlas, within which the CTD is defined. The fiber of the bundle may describe, in addition to geometry, physical fields like density, displacement, stress, and temperature. Mapped 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 mathematically messy problem of creating a single finite element mesh to represent the entire CTD. A modified finite element method, to handle the individually meshed, but overlapping and connected, hypercubical PTDs, is described. The boundary conditions may be specified as analytical, computational, or 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. Examples of linear elastic fracture analysis and fracture propagation modeling are presented.