On the Topology of Sheet Metal Parts

Abstract

This paper analyzes the topological properties of sheet metal parts represented schematically (zero thickness, zero bend radii). Although such parts are usually non-manifold objects, the paper establishes a general topological invariant f = s + b + e + w − v − gnm + m regarding the number of facets, components, bends, free edges, welds, vertices holes and volumes, respectively. Corresponding Euler operators are derived, providing a basis for a modeling system for sheet metal parts. With this invariant, it is possible to reason about manufacturing processes, such as number of components and arrangement of bend lines and weld lines, using only a single qualitative model of the product. This capability is particularly useful in the preliminary stage of conceptual design. A corresponding topological invariant v − e + f = s + m − gnm is also proposed for general sheet models and thin walled objects.