1Chain-Based Representations for Solid and Physical Modeling

Abstract — In this paper we show that the (co)chain complex associated with a decomposition of the computational domain, commonly called a mesh in computational science and engineer-ing, can be represented by a block-bidiagonal matrix that we call the Hasse matrix. Moreover, we show that topology-preserving mesh refinements, produced by the action of (the simplest) Euler operators, can be reduced to multilinear transformations of the Hasse matrix representing the complex. Our main result is a new representation of the (co)chain complex underlying field computations, a representation that provides new insights into the transformations induced by local mesh refinements. Our approach is based on first principles and is general in that it applies to most representational domains that can be characterized as cell complexes, without any restrictions on their type, dimension, codimension, orientability, manifoldness, connectedness. This paper is a further contribution towards bridging the subject of computer representations for solid and physical modeling—which flourished border-line between computer graphics, engineering mechanics and computer science with its own methods and data structures—under the general cover of linear algebra and algebraic topology. The main advantage of such an approach is that topology, geometry and physics may coexist in one and the same formalized framework, concurring together to define, represent and simulate the behavior of the model. Last but not least, our tensor-based approach is a significant step forward in achieving close integration of ge-ometrical representations and physics-based simulations, i.e., in the concurrent modeling of shape and behavior. Contrary to what might appear at first sight, the theoretical complexity of the present approach is not greater than that of current methods, provided that sparse-matrix techniques with double element access (by rows and by columns) are exploited. Index Terms — Computational geometry, geometric modeling, mesh generation, topology, spatial data structures, algorithms