Optimization of Geometrically Trimmed B-spline Surfaces

Unlike the visual trimming of B-spline surfaces, which hides unwanted portions in rendering, the geometric trimming approach provides a mathematically clean representation without redundancy. However, the process may lead to significant deviation from the corresponding portion on the original surface. Optimization is required to minimize approximation errors and obtain higher accuracy. In this paper, we describe the application of a novel global optimization method, so-called hybrid Parallel Tempering (PT) and Simulated Annealing (SA) method, for the minimization of B-spline surface representation errors. The high degree of freedom within the configuration of B-spline surfaces as well as the “rugged” landscapes of objective functions complicate the error minimization process. The hybrid PT/SA method, which is an effective algorithm to overcome the slow convergence, waiting dilemma, and initial value sensitivity, is a good candidate for optimizing geometrically trimmed B-spline surfaces. Examples of application to geometrically trimmed wing components are presented and discussed. Our preliminary results confirm our expectation. INTRODUCTION In most of the literature, the term “trimming” of B-spline surfaces refers to the visual trimming, which defines a trimmed surface by using the original surface and a trimming curve. When the trimmed surface is rendered, the original surface is tessellated so that it shows only the wanted portions. In contrast, geometric trimming creates new surfaces. Intersections are obtained to restrain the sampling points [1]. Then surface points are sampled on the retained part and reinterpolated into new surfaces [2, 3, 4]. In this paper, trimming refers to geometric trimming for the sake of brevity. An exact trimming is precluded due to the remarkably high degree of their intersections [5]. Depending on the selection of the interpolation points on the original surface, the trimming errors may vary greatly. The B-spline curve/surface shape modification problem has been addressed through optimization. Ferguson and Jones proposed methods to control curvature by formulating a constrained nonlinear optimization problem using the control points of B-splines in nonrational form as variables [6]. Moreton and Sequin studied the application of nonlinear optimization techniques to minimize a fairness functional based on the variation of curvature [7]. Hohenberger and Reuding investigated the possibilities of entering the weights in an automatic fairing process [8]. Laurent-Gengoux and Mekhilef investigated the optimization of a NURBS representation by using Polak-Ribiere technique with some improvement [9]. The Quasi-Newton BFGS method has been applied to optimize NURBS represented wing profiles by Trepanier, etc [10]. Most prior work focuses on curve shape optimization and local 1 Copyright © 2005 by ASME