IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 4, APRIL 2007 1541 Remeshing Driven by Smooth-Surface Approximation of Mesh Nodes Cássia R. S. Nunes , Renato C. Mesquita , and David A. Lowther Departamento de Engenharia Elétrica, Universidade Federal de Minas Gerais, 31270-010, Belo Horizonte, MG, Brazil Department of Electrical and Computer Engineering, McGill University, Montreal QC H3A 2A7, Canada This paper presents improvements in surface mesh of models generated by Boolean and assembly operations or surface reconstruction methods. The general concept consists of applying local mesh modification operators on the surface mesh to improve the shape quality of the elements without losing geometric information. To guarantee the model geometric characteristics, a smooth surface approximation of the model is evaluated and coupled to the operators of local mesh modifications. The approximation uses mesh information to generate the B-splines patches. Index Terms—Finite-element method (FEM), mesh generation, solid modelers. I. INTRODUCTION A UTOMATIC mesh generators can produce surface meshes with a specified quality degree at the beginning. But after some Boolean and assembly operations application, the quality can decrease drastically [1]. This problem also arises in models generated by a 3-D model acquisition process, such as from a scanning device. The reconstruction algorithms do not care about the element shape quality or their interconnections. On the other hand, highly regular meshes are necessary for engineers to perform effective finite-element (FE) analysis. A high-quality mesh results in a well-conditioned system, and minimizes numerical errors and singularities that might otherwise arise. The surface-mesh quality directly affects the quality of the volumetric mesh, which is the starting point for solving electro- magnetic problems by the FE method (FEM). In this context, remeshing is very important to reduce the number of sharp angles, and improve the nodes distribution and their nodal interconnections. The remeshing and adaptive mesh-refinement methods [2] improve the accuracy of the FE analysis in different ways: the former improves the mesh element quality considering the model geometric characteristics; the latter uses a posteriori error estimation to concentrate elements in a particular region of interest, without changing the geometric approximation or removing sharp angles. The remeshing process consists of applying a series of local mesh operators that can move, remove, or introduce nodes in the surface mesh. To avoid losing model geometric characteristics, it is necessary to know the model surface geometry, which will drive the nodes’ movements and ensure that they stay on top of the original model surface. Unfortunately, much of the time, the surface information is lost, and only the mesh configuration is available. To overcome this, we suggest using a smooth-surface representation to approximate the model geometry. This is our main contribution to the family of local mesh modification tech- niques. Digital Object Identifier 10.1109/TMAG.2006.892092 In a few words, our remeshing approach consists of eval- uating a set of surface approximations for the model geom- etry, choosing the edges that will participate in the optimization process, and applying the local mesh modifications to the edges set, aiming to improve the elements mesh quality with the geo- metric approximation guaranteed. II. SMOOTH-SURFACE APPROXIMATION B-spline surfaces are often used in CAD systems to approxi- mate scattered points. They give a high continuity degree, local control, and good approximation for a large variety of solids [3]. In our approach, the model surface is approximated by a set of B-spline surface patches. Each mesh face has a B-spline patch associated with it. Approximating the model surface by pieces decreases the processing time and approximation errors, be- cause the global parameterization is avoided. The patches are evaluated through a least-squares (LS) for- mulation [4], using face nodes and its neighbors as input. The problem consists of looking for a B-spline patch which approximates a model surface region, with a representation of the form (1) where are the control points and are the basis functions, usually polynomial, piecewise polynomial, or piecewise ra- tional. Let , be the mesh face vertices and the vertices of the faces around them. The points should be close to the corresponding data points . Then, the approximant is computed to minimize (2) Assuming that are given or precomputed from the input, the function is quadratic in the unknown control points, . This classical LS fitting always has a solution; however, the solution is not necessarily unique. Also, the resulting surface may not be sufficiently smooth. One may augment (2) with a regu- larization term, also called the smoothing term or penalty term, 0018-9464/$25.00 © 2007 IEEE