Integrating Physics-Based Modeling with PDE Solids for Geometric Design

PDE techniques, which use Partial Differential Equations (PDEs) to model the shapes of various real-world objects, can unify their geometric attributes and functional constraints in geometric computing and graphics. This paper presents a unified dynamic approach that allows modelers to define the solid geometry of sculptured objects using the second-order or fourth-order elliptic PDEs subject to flexible boundary conditions. Founded upon the previous work on PDE solids by Bloor and Wilson, as well as our recent research on the interactive sculpting of physics-based PDE surfaces, our new formulation and its associated dynamic principle permit designers to directly deform PDE solids whose behaviors are natural and intuitive subject to imposed constraints. Users can easily model and interact with solids of complicated geometry and/or arbitrary topology from locally-defined PDE primitives through trimming operations. We employ the finite-difference discretization and the multi-grid subdivision to solve the PDEs numerically. Our PDE-based modeling software offers users various sculpting toolkits for solid design, allowing them to interactively modify the physical and geometric properties of arbitrary points, curve spans, regions of interest (either in the isoparametric or nonisoparametric form) on boundary surfaces, as well as any interior parts of modeled objects.