Method of trimming PDE surfaces

Abstract

A method for trimming surfaces generated as solutions to Partial Differential Equations (PDEs) is presented. The work we present here utilises the 2D parameter space on which the trim curves are defined whose projection on the parametrically represented PDE surface is then trimmed out. To do this we define the trim curves to be a set of boundary conditions which enable us to solve a low order elliptic PDE on the parameter space. The chosen elliptic PDE is solved analytically, even in the case of a very general complex trim, allowing the design process to be carried out interactively in real time. To demonstrate the capability for this technique we discuss a series of examples where trimmed PDE surfaces may be applicable.

Modelling the Mechanical Behaviour of a Pharmaceutical Tablet Using PDEs

“…where χ(u, v) is the function defining a surface in 3D space while α is a smoothing parameter [7] and r defines the order of the PDE. Equation (2) is transformed to the Biharmonic Equation by taking r = 2 and α = 1.…”

“…The overall shape of the PDE surface depends on the derivative conditions which are defined by the derivative vector along the boundary curves [7]. The analytic solution of the fourth order PDE can be written as,…”

“…Many studies have been carried out to exploit the full potential of the PDE method in visual computing since the PDE surfaces offer many advantages over other type of surfaces. Most of all, this technique is capable of blending surfaces [1] and offers modelling tools to manipulate the shape of the PDE surface [7]. Moreover, smooth surfaces with high-order continuity requirements can be defined through PDEs since the formulation is well-conditioned and technically sound.…”

“…However, we have not found in the literature any work related to geometric modelling of pharmaceutical tablets based on the use of parametric surface representation. Therefore, the objective of this work is to model a solid cylindrical pharmaceutical tablet by utilising the PDE method [7] and to study the mechanical behaviour of the axially compressed PDE-based representation of a tablet.…”

Boundary Constrained Swept Surfaces for Modelling and Animation

“…The first influencing publication was made by Bloor and Wilson [BW89]. After that, many more researchers have made an effort for further investigations and developments [UBW99, ZY04, DHQ04, DQ05, BW05, Uga06]. Instead of using a partial differential equation, in this paper, we use the solution to an ordinary differential equation to formulate the profile curves of a swept surface.…”