Parametric PDE techniques, which use partial differential equations (PDEs) defined over a 2D or 3D parametric domain to model graphical objects and processes, can unify geometric attributes and functional constraints of the models. PDEs can also model implicit shapes defined by level sets of scalar intensity fields. In this paper, we present an approach that integrates parametric and implicit trivariate PDEs to define geometric solid models containing both geometric information and intensity distribution subject to flexible boundary conditions. The integrated formulation of second-order or fourth-order elliptic PDEs permits designers to manipulate PDE objects of complex geometry and/or arbitrary topology through direct sculpting and free-form modeling. We developed a PDE-based geometric modeling system for shape design and manipulation of PDE objects. The integration of implicit PDEs with parametric geometry offers more general and arbitrary shape blending and free-form modeling for objects with intensity attributes than pure geometric models.

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 (1989, 1990, 1993), 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.

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.

This paper presents an integrated approach and a unified algorithm that combine the benefits of PDE surfaces and powerful physics‐based modeling techniques within one single modeling framework, in order to realize the full potential of PDE surfaces. We have developed a novel system that allows direct manipulation and interactive sculpting of PDE surfaces at arbitrary location, hence supporting various interactive techniques beyond the conventional boundary control. Our prototype software affords users to interactively modify point, normal, curvature, and arbitrary region of PDE surfaces in a predictable way. We employ several simple, yet effective numerical techniques including the finite‐difference discretization of the PDE surface, the multigrid‐like subdivision on the PDE surface, the mass‐spring approximation of the elastic PDE surface, etc. to achieve real‐time performance. In addition, our dynamic PDE surfaces can also be approximated using standard bivariate B‐spline finite elements, which can subsequently be sculpted and deformed directly in real‐time subject to intrinsic PDE constraints. Our experiments demonstrate many attractive advantages of our dynamic PDE formulation such as intuitive control, real‐time feedback, and usability to the general public.