Combinatorial Laws for Physically Meaningful Design

Abstract

A typical computer representation of a design includes geometric and physical information organized in a suitable combinatorial data structure. Queries and transformations of these design representations are used to formulate most algorithms in computational design, including analysis, optimization, evolution, generation, and synthesis. Formal properties, and in particular existence and validity of the computed solutions, must be assured and preserved by all such algorithms. Using tools from algebraic topology, we show that a small set of the usual combinatorial operators: boundary (∂), coboundary (δ), and dualization (*)–are sufficient to represent a variety of physical laws and invariants. Specific examples include geometric integrity, balance and equilibrium, and surface smoothing. Our findings point a way toward systematic development of data structures and algorithms for design in a common formal computational framework.