Dynamic PDE-based surface design using geometric and physical constraints

PDE surfaces, which are defined as solutions of partial differential equations (PDEs), offer many modeling advantages in surface blending, free-form surface modeling, and specifying surface's aesthetic or functional requirements. Despite the earlier advances of PDE surfaces, previous PDE-based techniques exhibit certain difficulties such as lack of interactive sculpting capabilities and restrained topological structure of modeled objects. This paper presents an integrated approach that can incorporate PDE surfaces into the powerful physics-based modeling framework, to realize the full potential of PDE methodology. We have developed a prototype system that allows interactive design of flexible topological surfaces as PDE surfaces and displacements using generalized boundary conditions as well as a variety of geometric and physical constraints, hence supporting various interactive techniques beyond the conventional boundary control. The system offers a set of sculpting toolkits that allow users to interactively modify arbitrary points, curve spans, and/or regions of interest across the entire PDE surfaces and displacements in an intuitive and physically meaningful way. To achieve real-time performance, we employ several simple, yet efficient numerical techniques, including the finite-difference discretization, the multigrid-like subdivision, and the mass-spring approximation of elastic PDE surfaces and displacements. In addition, we present the standard bivariant B-spline finite element approximations of dynamic PDEs, which can subsequently be sculpted and deformed directly in real-time subject to the intrinsic PDE constraints. Our experiments demonstrate many attractive advantages of the physics-based PDE formulation such as intuitive control, real-time feedback, and usability to both professional and common users.