A Unified Simplicial Model for Mixed-Dimensional and Non-Manifold Deformable Elastic Objects

Abstract

We present a unified method to simulate deformable elastic bodies consisting of mixed-dimensional components represented with potentially non-manifold simplicial meshes. Building on well-known simplicial rod, shell, and solid models for elastic continua, we categorize and define a comprehensive palette expressing all possible constraints and elastic energies for stiff and flexible connections between the 1D, 2D, and 3D components of a single conforming simplicial mesh. This palette consists of three categories: point connections, in which simplices meet at a single vertex around which they may twist and bend; curve connections in which simplices share an edge around which they may rotate (bend) relative to one another; and surface connections, in which a shell is embedded on or into a solid. To define elastic behaviors across non-manifold point connections, we adapt and apply parallel transport concepts from elastic rods. To address discontinuous forces that would otherwise arise when large accumulated relative rotations wrap around in the space of angles, we develop an incremental angle-update strategy. Our method provides a conceptually simple, flexible, and highly expressive framework for designing complex elastic objects, by modeling the geometry with a single simplicial mesh and decorating its elements with appropriate physical models (rod, shell, solid) and connection types (point, curve, surface). We demonstrate a diverse set of possible interactions achievable with our method, through technical and application examples, including scenes featuring complex aquatic creatures, children's toys, and umbrellas.