A shape design system using volumetric implicit PDEs

Abstract

Solid modeling based on partial differential equations (PDEs) can potentially unify both geometric constraints and functional requirements within a single design framework to model real-world objects via its explicit, direct integration with parametric geometry. In contrast, implicit functions indirectly define geometric objects as the level-set of underlying scalar fields. To maximize the modeling potential of PDEbased methodology, in this paper we tightly couple PDEs with volumetric implicit functions in order to achieve interactive, intuitive shape representation, manipulation, and deformation. In particular, the unified approach can reconstruct the PDE geometry of arbitrary topology from scattered data points or a set of sketch curves. We make use of elliptic PDEs for boundary value problems to define the volumetric implicit function. The proposed implicit PDE model has the capability to reconstruct a complete solid model from partial information and facilitates the direct manipulation of underlying volumetric datasets via sketch curves and iso-surface sculpting, deformation of arbitrary interior regions, as well as a set of CSG operations inside the working space. The prototype system that we have developed allows designers to interactively sketch the curve outlines of the object, define intensity values and gradient directions, and specify interpolatory points in the 3D working space. The governing implicit PDE treats these constraints as generalized boundary conditions to determine the unknown scalar intensity values over the entire working space. The implicit shape is reconstructed with specified intensity value accordingly and can be deformed using a set of sculpting toolkits. We use the finite-difference discretization and variational interpolating approach with the localized iterative solver for the numerical integration of our PDEs in order to accommodate the diversity of generalized boundary and additional constraints. q

Method of trimming PDE surfaces

“…The PDE method has been discussed before in a number of different references, e.g. [3,22,7]. It has been shown how surfaces satisfying a wide range of functional requirements can be created by a suitable choice of the boundary conditions and appropriate values for the various design parameters associated with the method [23,24].…”

“…Since this initial application of the method on shape blending, in recent years, PDE based shapes have broadened their uses in shape description, e.g. [16,7,22]. This is due to the fact that PDE shapes in principle have the advantage that most of the information defining a shape is based on a boundary representation making it possible for complex geometry to be generated and controlled by very few design parameters [24].…”

Editing level-set models with sketched curves

“…Baerentzen and Christensen [BC02] used the Level Set Method in their volume sculpting system, which included such capabilities as add/remove material, smooth/unsmooth and morphological operations. Another PDE-based volumetric sculpting system [DQ04,DQ05] defined smooth surfaces as a solution to a fourth order elliptic PDE subject to geometric and physical constraints, such as curvature and normals. The surface is deformed by solving the PDE with the given boundary conditions.…”

A survey of partial differential equations in geometric design

“…However, implicit PDE surfaces can also be obtained from elliptic PDEs [20]. A brief description on how elliptic PDEs give rise to implicit PDE surfaces is given below, where the most relevant mathematical details are outlined.…”