S.H. Lo

Volume discretization into tetrahedra-II. 3D triangulation by advancing front approach

Abstract Existing methods of automatic mesh generation for 3D solid objects are reviewed. Although the 3D Delaunay triangulation recently aroused much attention, its suitability as a finite element mesh generator is questioned. Although in 2D Delaunay triangulation, the ‘max-min’ angle criterion can be verified over the entire domain, no equivalent or similar criterion can be defined for its extension to 3D situations to ensure that tetrahedron elements so generated are well proportioned for numerical calculations. In this paper, a simple but versatile 3D triangulation scheme based on the advancing front technique for the discretization of arbitrary volumes is presented. To ensure that the tetrahedron elements generated are as equilateral as possible, the ratio of volume of the element to the sum of squares of edges put into a dimensionless form is adopted to judge the quality of a tetrahedron element. The quality of the finite element mesh can thus be ensured if the shape of each tetrahedron element is carefully controlled in the mesh construction process. Through the study of numerous examples of various characteristics, it is found that high-quality tetrahedron element meshes are obtained by the proposed algorithm.

A new scheme for the generation of a graded quadrilateral mesh

Abstract A new scheme for the generation of a quadrilateral element mesh is presented. The algorithm makes use of the fact that a triangular element mesh bounded by an even number of line segments can always be converted into quadrilaterals. By using the advancing front technique for schematically merging of triangular elements, high-quality well-graded quadrilateral meshes can be formed without any tedious treatment for isolated triangles. Unlike many other methods, no cut-lines or manual division of the problem domain into simpler subregions is required before conversion. As the number and the position of the boundary nodes are not altered during the mesh generation process, different material regions sharing common boundary lines can be treated individually. Since a background triangular mesh is all that is needed, the process can be applied to any arbitrary 2D domain with or without internal openings. In fact, the method has an equal area of application as a general triangulator. This also implies that the proposed scheme can be used to generate strongly graded quadrilateral meshes of the same gradation effect as the background triangular meshes for adaptive finite element analysis.

Fast Delaunay triangulation in three dimensions

Abstract An efficient algorithm for Delaunay triangulation of a given set of points in three dimensions based on the point insertion technique is presented. Various steps of the triangulation algorithm are reviewed and many acceleration procedures are devised to speed up the triangulation process. New features include the search of a neighbouring point by the layering scheme, locating the containing tetrahedron by random walk, formulas of important geometrical quantities of a new tetrahedron based on those of an existing one, a novel approach in establishing the adjacency relationship, the use of adjacency table and the management of memory. The resulting scheme is one of the fastest triangulation algorithms known to the authors, which is able to generate tetrahedra generation rate of 15 000 tetrahedra per second for randomly generated points on a HP 735 machine.

3D vibration analysis of solid and hollow circular cylinders via Chebyshev-Ritz method

A general approach is presented for solving the free vibration of solid and hollow circular cylinders. The analysis procedure is based on the small-strain, linear and exact elasticity theory. By taking the Chebyshev polynomial series multiplied by a boundary function to satisfy the geometric boundary conditions as the admissible functions, the Ritz method is applied to derive the frequency equation of the cylinder. According to the axisymmetric geometrical property of a circular cylinder, the vibration modes are divided into three distinct categories: axisymmetric vibration, torsional vibration and circumferential vibration. Moreover, for a cylinder with the same boundary conditions at the two ends, the vibration modes can be further divided into antisymmetric and symmetric ones in the length direction. Convergence and comparison studies demonstrate the high accuracy and small computational cost of the present method. A significant advantage over other Ritz solutions is that the present method can guarantee stable numerical operation even when a large number of terms of admissible functions are used. Not only the lower-order but also the higher-order frequencies can be obtained by using a few terms of the Chebyshev polynomials. Finally, the first several frequencies of circular cylinders with different boundary conditions, with respect to various parameters such as the length–radius ratio and the inside–outside radius ratio, are given. 2003 Elsevier Science B.V. All rights reserved.

OPTIMAL DELAUNAY POINT INSERTION

SUMMARY An efficient algorithm for Delaunay triangulation of a given set of points in d dimensions is presented. Various steps of the point insertion algorithm are reviewed and many acceleration procedures are implemented to speed up the triangulation process. New features include the search for a neighbouring point by a layering scheme, locating the containing simplex by a random walk, formulas of important geometrical quantities of a new simplex based on those of an old one, a novel approach in establishing the adjacency relationship using connection matrices. The resulting scheme seems to be one of the fastest triangulation algorithms known, which enables us to generate tetrahedra in R3 with a linear generation rate of 15 OOO tetrahedra per second for randomly generated points on an HP 735 workstation.

Finite element mesh generation over analytical curved surfaces

A scheme is proposed for the automatic generation of unstructured triangular meshes of arbitrary density distribution over curved surfaces. Instead of using the widely employed plane to surface mapping method, elements are generated directly on the curved surfaces by the advancing front technique. Given a segment on the boundary, the possibility of forming a new triangle with a node on the boundary or an interior node is examined. The elements formed are optimized in terms of a number of parameters including surface curvature, element quality and the given element density distribution. Two algorithms to bring a spatial point back on the curved surface without affecting the shape and size of the element are described. Some examples are given to illustrate the application of the method to various kinds of surfaces.

Generating quadrilateral elements on plane and over curved surfaces

Abstract An algorithm is proposed to generate quadrilateral elements over a triangular element mesh by selectively removing diagonals between triangles. The quality (shape) of a triangular element can be measured by the dimensionless α value (area/sum of squares of sides). As for quadrilateral elements, a new distortion coefficient β has to be introduced, with which the quality of a quadrilateral can be compared to those of the two triangles arising from a cut along either diagonal. A parameter γ can be specified by the user to give higher preference either to triangles or to quadrilaterals. Hence a careful selection of γ would lead to an optimized hybrid mesh consisting of both triangular and quadrilateral elements. Since a quadrilateral can always be divided into two triangles, the method can be considered as a general approach for the generation of quadrilateral elements both on plane and over curved surfaces. As shown by the examples, regular quadrilateral finite element meshes and quadrilateral elements at the interior part of a region are easily recovered by the algorithm.

Volume discretization into tetrahedra—I. Verification and orientation of boundary surfaces

Abstract Three-dimensional triangulation of an arbitrary volume is a time-consuming process, and the boundary surfaces of the object have to be checked very carefully to ensure no error is made. The data verification and preparation work for a general 3D mesh generation problem is best done more or less automatically by another program which not only detects any error in the data but also generates from the given triangular facets all the boundary surfaces with the correct orientations for the next step of volume discretization into tetrahedra, using the advancing front approach. This paper describes such a boundary data pre-processor for discretization of arbitrary objects into tetrahedron elements. In this data-checking and preparation process, first the size and quality of the given randomly numbered triangular facets will be examined, then the closure and orientability of the surfaces that can be constructed from the triangular elements will be verified. Each valid closed surfaces will be given a correct orientation according to its type (interior or exterior surface). The volumes so defined by these surfaces are identified and labelled for 3D mesh generation.

Optimization of tetrahedral meshes based on element shape measures

Abstract An element shape optimization procedure is presented, which can be considered as a general post-treatment process for three-dimensional tetrahedral meshes generated by Delaunay triangulation or refinement based on the subdivision of elements. The tetrahedral mesh is optimized with respect to a given element shape measure through a combined iterative scheme of local transformations and node relaxation. From the examples studied, a substantial gain in quality could be achieved in two cycles of iterations based on any valid shape measures, minimum solid angle θ, radius ratio ϱ and gamma coefficient γ. Although further research and evidence are required for a more definite conclusion, the γ-coefficient which is more economical to compute, seems to give better results than the other two shape measures. The largest mesh processed which, consists of 67,326 nodes and 360,824 elements, required a CPU time of a little more than 3 min on a IBM Power Station 3BT.

Stress analysis of inclusion problems of various shapes in an infinite anisotropic elastic medium

Abstract A boundary integral equation approach is used to solve an infinite anisotropic elastic inclusion problem subjected to remote loading. Continuous and discontinuous quadratic isoparametric boundary elements are employed to model the interfaces between the inclusions and the matrix. The inclusion–matrix interfaces are assumed to be perfectly bonded. Inclusions of various shapes and their interaction are investigated. Numerical examples are compared with existing analytical solutions. Relative to the finite element method and the volume integral method, the present method is more efficient and accurate in the analysis of elastic inclusion problems.