Parameterization Of Triangular Domains Based On Quasi-Conformal Mapping

Constructing a parameter spline representation to interpolate a given boundary of a two-dimensional or three-dimensional area is a fundamental challenge in CAD.The Coons surface is a classic tool for addressing this problem.In recent years,the rise of isogeometric analysis(IGA)has brought renewed attention to this challenge,which is now known as domain parameterization.Domain parameterization is a crucial step for isogeometric analysis methods,as it has a significant impact on the accuracy and stability of the subsequent analysis.In IGA,it is generally required that parameterization be non-self-intersecting,meaning the mapping from the parameter domain to the computational domain should be bijective.Moreover,the distortion of the mapping should be minimized,such that the area and angle of the mapping remain as unchanged as possible.Many scholars have made significant progress in tackling this problem in recent years.However,most existing parameterization methods are limited to square or volume parameter domains.These parameterization methods are often difficult to accurately represent triangle-like domains,and cannot achieve high-quality mappings.In this paper,we propose an effective parameterization method for planar domains,using triangular Bezier surfaces and triangular rational Bezier surfaces as representations and quasi-conformal mapping as the calculation framework.Chapter 2 of this paper provides a brief overview of the definition of triangular Bezier surfaces and triangular rational Bezier surfaces on a triangular domain,as well as the relevant theoretical knowledge on directional derivatives and quasi-conformal mappings.This provides theoretical support for the subsequent research work.In Chapter 3,we propose an effective parameterization method for planar domains using triangular Bezier surfaces as the representation form and quasi-conformal mapping as the calculation framework.The problem addressed in this chapter is how to construct a quasi-conformal mapping with bijectivity and low distortion,given three corresponding boundaries between the parameter domain(unit right triangle)and the computational domain.To solve this problem,we formulate it as an optimization model,where the objective function aims to minimize the conformal distortion of the mapping and improve its smoothness,while the constraints ensure the injectivity of the mapping.To compute the mapping,we adopt an approach that alternately solves two quadratic optimization problems.We also compare this approach with a parametrization method based on tensor-product B-spline representation,and experimental results demonstrate the superiority of the proposed method in obtaining higher-quality mappings.In Chapter 4,the domain optimization algorithm based on triangular Bezier representation is extended to triangular rational Bezier,with the addition of weight factor optimization to the original model.By adjusting control points and weight factors,a high-quality triangular rational Bézier parameterization is constructed.Experimental results demonstrate that this method yields improved parameterization.In conclusion,the fifth chapter presents a concise summary of the research conducted in this thesis and suggests potential directions for future research.