Automatic finite element mesh generation techniques have become commonly used tools for the analysis of complex, real-world models. All of these methods can, however, create distorted and even unusable elements. Fortunately, several techniques exist which can take an existing mesh and improve its quality. Smoothing (also referred to as mesh relaxation) is one such method, which repositions nodal locations, so as to minimize element distortion. In this paper, an overall mesh smoothing scheme is presented for meshes consisting of triangular, quadrilateral, or mixed triangular and quadrilateral elements. This paper describes an efficient and robust combination of constrained Laplacian smoothing together with an optimization-based smoothing algorithm. The smoothing algorithms have been implemented in ANSYS and performance times are presented along with several example models.

. We present an optimization-based approach for mesh untangling that maximizes the minimum area or volume of simplicial elements in a local submesh. These functions are linear with respect to the free vertex position; thus the problem can be formulated as a linear program that is solved by using the computationally inexpensive simplex method. We prove that the function level sets are convex regardless of the position of the free vertex, and hence the local subproblem is guaranteed to converge. Maximizing the minimum area or volume of mesh elements, although well-suited for mesh untangling, is not ideal for mesh improvement, and its use often results in poor quality meshes. We therefore combine the mesh untangling technique with optimization-based mesh improvement techniques and expand previous results to show that a commonly used two-dimensional mesh quality criterion can be guaranteed to converge when starting with a valid mesh. Typical results showing the effectiveness of the combine...

. We present a new shape measure for tetrahedral elements that is optimal in the sense that it gives the distance of a tetrahedron from the set of inverted elements. This measure is constructed from the condition number of the linear transformation between a unit equilateral tetrahedron and any tetrahedron with positive volume. We use this shape measure to formulate two optimization objective functions that are differentiated by their goal: the first seeks to improve the average quality of the tetrahedral mesh; the second aims to improve the worst-quality element in the mesh. Because the element condition number is not defined for tetrahedra with negative volume, these objective functions can be used only when the initial mesh is valid. Therefore, we formulate a third objective function using the determinant of the element Jacobian that is suitable for mesh untangling. We review the optimization techniques used with each objective function and present experimental results tha...

We present an effective and easy-to-implement angle-based smoothing scheme for triangular, quadrilateral and tri-quad mixed meshes. For each mesh node our algorithm compares all the pairs of adjacent angles incident to the node and adjusts these angles so that they become equal in the case of a triangular mesh and a quadrilateral mesh, or they form the ideal ratio in the case of a tri-quad mixed mesh. The size and shape quality of the mesh after this smoothing algorithm is much better than that after Laplacian smoothing. The proposed method is superior to Laplacian smoothing by reducing the risk of generating inverted elements and increasing the uniformity of element sizes. The computational cost of our smoothing method is yet much lower than optimization-based smoothing. To prove the effectiveness of this algorithm, we compared errors in approximating a given analytical surface by a set of bi-linear patches corresponding to a mesh with Laplacian smoothing and a mesh with the proposed smoothing method. The experiments show that a mesh smoothed with our method has roughly 20 % less approximation error.

This paper describes a comprehensive approach to construct quality meshes for implicit solvation models of biomolecular structures starting from atomic resolution data in the Protein Data Bank (PDB). First, a smooth volumetric electron density map is constructed from atomic data using weighted Gaussian isotropic kernel functions and a two-level clustering technique. This enables the selection of a smooth implicit solvation surface approximation to the Lee-Richards molecular surface. Next, a modified dual contouring method is used to extract triangular meshes for the surface, and tetrahedral meshes for the volume inside or outside the molecule within a bounding sphere/box of influence. Finally, geometric flow techniques are used to improve the surface and volume mesh quality. Several examples are presented, including generated meshes for biomolecules that have been successfully used in finite element simulations involving solvation energetics and rate binding constants. Key words: quality mesh, biomolecule, implicit solvation model, finite element simulation. 1

this paper for smoothing and untangling are local techniques; a globally optimal solution is not guaranteed although empirical evidence suggests Copyright c fl 2000 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2000; 0:0--0 Prepared using nmeauth.cls

this paper, we provide a deeper analysis of the tradeoffs associated with the cost of mesh improvement in terms of solution efficiency. We consider both finite element and finite volume discretization techniques, a number of different solvers, and a variety of application problems. The issue of solution accuracy will be addressed in a later paper. We focus initially on problems discretized using the finite element method. Such discretizations lead to large, sparse linear systems, which are often solved by using either conjugate gradient (CG) [16] or GMRES [31] iterative techniques. Several theoretical results that relate the convergence behavior of these algorithms to matrix characteristics such as condition number and spectral distribution. In turn, for simple applications such as Poisson's equation, these matrix characteristics can be theoretically related to the size and quality of the underlying finite element mesh. Theoretical results are not available for more complicated applications. To obtain insight into the convergence rates of CG and GMRES, we must empirically establish the relationship between mesh size and quality to convergence behavior. We do so by performing a series of experiments in which the parameters of mesh size and quality are varied both individually

In this paper we summarize our experiences with 3D constrained Delaunay triangulation algorithms for industrial applications. In addition, we report a robust implementation process for constructing 3D constrained triangulations from initial unconstrained triangulations, based on a minimalist approach, in which we minimize the use of geometrical operations such as intersections. This is achieved by inserting Steiner points on missing constraining edges and faces in the initial unconstrained triangulations. This approach allowed the generation of tetrahedral meshes for arbitrarily complex 3D domains.

This paper describes an algorithm to extract adaptive and quality quadrilateral/hexahedral meshes directly from volumetric imaging data. First, a bottom-up surface topology preserving octree-based algorithm is applied to select a starting octree level. Then the dual contouring method is used to extract a preliminary uniform quad/hex mesh, which is decomposed into finer quads/hexes adaptively without introducing any hanging nodes. The positions of all boundary vertices are recalculated to approximate the boundary surface more accurately. Mesh adaptivity can be controlled by a feature sensitive error function, the regions that users are interested in, or finite element calculation results. Finally, the relaxation based technique is deployed to improve mesh quality. Several demonstration examples are provided from a wide variety of application domains. Some extracted meshes have been extensively used in finite element simulations.

Abstract: This paper describes an approach to smooth the surface and improve the quality of quadrilateral/hexahedral meshes with feature preserved using geometric flow. For quadrilateral surface meshes, the surface diffusion flow is selected to remove noise by relocating vertices in the normal direction, and the aspect ratio is improved with feature preserved by adjusting vertex positions in the tangent direction. For hexahedral meshes, besides the surface vertex movement in the normal and tangent directions, interior vertices are relocated to improve the aspect ratio. Our method has the properties of noise removal, feature preservation and quality improvement of quadrilateral/hexahedral meshes, and it is especially suitable for biomolecular meshes because the surface diffusion flow preserves sphere accurately if the initial surface is close to a sphere. Several demonstration examples are provided from a wide variety of application domains. Some extracted meshes have been extensively used in finite element simulations. Key words: quadrilateral/hexahedral mesh, surface smoothing, feature preservation, quality improvement, geometric flow. 1