Partial differential equation-based surface modelling and applications.

Abstract

Due to its advantages in creating complicated 3D models with small data, good continuities, and physics-based deformations for better realism etc., partial differential equation (PDE)-based modelling provides a powerful technique of creating, manipulating, and animating 3D models, and has been attracting considerable attention in the community of computer graphics in the last three decades. Various PDE-based approaches have been proposed for surface modelling. However, the following challenges have not been addressed. First, since PDE is a non-industry standard in CAD, CAM and CAE systems and lacks effective boundary control methods, the numerical solution for PDE-based surface modelling has few engineering applications. Second, there is no unified framework for solving different modelling problems. Existing research studies use special PDE-based mathematical models for specific applications, which cannot be applied on other occasions. Third, previous surface manipulation methods cannot deform shapes within arbitrary boundaries and usually involve heavy numerical calculations due to the use of numerical solutions. This thesis aims to address these challenges. For the first challenge, a numerical solution using the finite difference method to a fourth-order PDE was presented. Based on this solution, an optimal conversion of PDE surfaces representing high-speed train heads into NURBS surfaces was developed, and a novel multi-objective aerodynamic optimization method of high-speed train heads was proposed, which is the first pipeline of using the PDE-based approach to optimize shapes in the CFD simulation. For the second challenge, a unified PDE mathematical model for surface modelling using analytical 2-, 3- and 4- sided PDE patches with C n continuity was proposed. Based on the analytical solution, a PDE-based surface reconstruction method was proposed to generate optimal surfaces under the constraints of the feature curves in automotive styling design, and a PDE-based 3D modelling plug- in was developed for Blender. For the third challenge, a physics-based method was presented to interactively manipulate surface shapes of 3D models using the approximate analytical solution of a fourth-order PDE with C 1 continuity, and an interactive user interface was developed as a plug-in of Maya to facilitate surface manipulation.