Name
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 engineering, 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 multi-linear 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. 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. 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. 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 employed. Last but not least, our tensorbased approach is a significant step forward in achieving close integration of geometrical representations and physics-based simulations, i.e., in the concurrent modeling of shape and behavior. |
Year | DOI | Venue |
|---|---|---|
2007 | 10.1145/1236246.1236259 | Symposium on Solid and Physical Modeling |
Keywords | Field | DocType |
cell complex,block-bidiagonal matrix,computer representation,tensorbased approach,present approach,chain complex underlying field,chain complex,computer science,computer graphics,physical modeling,hasse matrix,data structure,computational science and engineering,first principle,linear transformation,physical model,sparse matrix | Linear algebra,Topology,Algebraic topology,Computer science,Matrix (mathematics),Orientability,Euler's formula,Computer graphics,Computational topology,Computation | Conference |
Citations | PageRank | References |
7 | 0.60 | 7 |
Authors | ||
4 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Antonio Di Carlo | 1 | 24 | 3.51 |
| Franco Milicchio | 2 | 27 | 6.61 |
| Alberto Paoluzzi | 3 | 92 | 14.74 |
| Vadim Shapiro | 4 | 7 | 0.60 |