Dynamic PDE surfaces with flexible and general geometric constraints

Abstract

PDE surfaces, whose behavior is governed by partial differential equations (PDEs), have demonstrated many modeling advantages in surface blending, free-form surface modeling, and surface aesthetic or functional specifications. Although PDE surfaces can potentially unify geometric attributes and functional constraints for surface design, current PDE based techniques exhibit certain difficulties such as the restrained topological structure of modeled objects and the lack of interactive editing functionalities. We propose an integrated approach and develop a set of algorithms that augment conventional PDE surfaces with material properties and dynamic behavior. The authors incorporate PDE surfaces into the powerful physics based framework, aiming to realize the full potential of the PDE methodology. We have implemented a prototype software environment that can offer users a wide array of PDE surfaces with flexible topology (through trimming and joining operations) as well as generalized boundary constraints. Using our system, designers can dynamically manipulate PDE surfaces at arbitrary location with applied forces. Our sculpting toolkits allow users to interactively modify arbitrary point, curve span, and/or region of interest throughout the entire PDE surface in an intuitive and predictable way. To achieve real time sculpting, we employ several simple, yet efficient numerical techniques such as finite difference discretization, multi-grid subdivision, and FEM approximation. Our experiments demonstrate many advantages of physics based PDE formulation such as intuitive control, real time feedback, and usability to both professional and non-expert users.