Generation of tetrahedral mesh of variable element size by sphere packing over an unbounded 3D domain

This paper describes an algorithm for the generation of tetrahedral mesh of specified element size over an unbounded three-dimensional domain. Starting from an arbitrary point in space (defined as the origin) and guided by the concept of advancing front, spheres of size compatible with the specified element size are packed tightly together one by one to form a cluster of spheres of different sizes. The compactness of the cluster of spheres is achieved by packing spheres at a site closest to the origin in a densest manner with tangent but no overlapping with as many other spheres as possible. In view of these criteria, a rotational mechanism between spheres is innovated, which allows the newly inserted sphere to follow the path by rotation between existing spheres until the lowest point is reached. The centres of the packing spheres provide ideal locations for Delaunay point insertion to form a triangulation of tetrahedral elements of size compatible with the specified value. Spheres of random size distribution or of size specified by a node spacing function are packed by the proposed algorithm. In all test examples, high-quality tetrahedral elements of size consistent with the specified node spacing are generated. The process is fast and robust, and the time complexity for mesh generation is expected to be almost linear; however, in the present implementation, a quasi-linear time relationship is observed as a search for the nearest node on the front is proposed in the sphere packing process. The memory requirement has been kept to a minimum as no additional data structure other than the adjacency relationship of the tetrahedral elements as required by the Delaunay triangulation is stored. The structure of the advancing frontal surface needs not be explicitly constructed nor updated as it is simply not required in the sphere packing process.     

Generation of anisotropic mesh by ellipse packing over an unbounded domain

With the advance of the finite element method, general fluid dynamic and traffic flow problems with arbitrary boundary definition over an unbounded domain are tackled. This paper describes an algorithm for the generation of anisotropic mesh of variable element size over an unbounded 2D domain by using the advancing front ellipse packing technique. Unlike the conventional frontal method, the procedure does not start from the object boundary but starts from a convenient point within an open domain. The sequence of construction of the packing ellipses is determined by the shortest distance from the fictitious centre in such a way that the generation front is more or less a circular loop with occasional minor concave parts due to element size variation. As soon as an ellipse is added to the generation front, finite elements are directly generated by properly connecting frontal segments with the centre of the new ellipse. Ellipses are packed closely and in contact with the existing ellipses by an iterative procedure according to the specified anisotropic metric tensor. The anisotropic meshes generated by ellipse packing can also be used through a mapping process to produce parametric surface meshes of various characteristics. The size and the orientation of the ellipses in the pack are controlled by the metric tensor as derived from the principal surface curvatures. In contrast to other mesh generation schemes, the domain boundary is not considered in the process of ellipse packing, this reduces a lot of geometrical checks for intersection between frontal segments. Five examples are given to show the effectiveness and robustness of anisotropic mesh generation and the application of ellipse packing to mesh generation over various curved surfaces.     

2D finite element mesh generation by medial axis subdivision

An algorithm for the automatic generation of two-dimensional finite element meshes using quadrilateral elements has been demonstrated. The technique uses information derived from the medial axis of a 2D region, the locus of the centre of an inscribed disc of maximal diameter as it rolls around the region interior. Using this information, an arbitrarily complex object can be subdivided into a series of meshable subregions. Within these subregions relatively conventional meshing patterns are then generated. The resulting meshes are well structured and flow smoothly round the object boundary with minimum mesh irregularity.     

An algorithm for automatic 2D finite element mesh generation with line constraints

In this study, an algorithm is designed specifically for automatic finite element (FE) mesh generation on the transverse structure of hulls reinforced by stiffeners. Stiffeners attached to the transverse structure are considered as line constraints in the geometry boundary. For the FE mesh generation used in this study, the line constraints are treated as boundaries and by that means the geometry domain attached to the line constraints is decomposed into sub-domains, constrained only by the closed boundaries. Then, the mesh can be generated directly on those sub-domains by the traditional approach. The performance of the proposed algorithm is evaluated and the quality of the generated mesh meets expectations.     

Development of an object-oriented finite element program with adaptive mesh refinement for multi-physics applications

In this paper, an object-oriented framework for numerical analysis of multi-physics applications is presented. The framework is divided into several basic sets of classes that enable the code segments to be built according to the type of problem to be solved. Fortran 2003 was used in the development of this finite element program due to its advantages for scientific and engineering programming and its new object-oriented features. The program was developed with h-type adaptive mesh refinement, and it was tested for several classical cases involving heat transfer, fluid mechanics and structural mechanics. The test cases show that the adaptive mesh is refined only in the localization region where the feature gradient is relatively high. The overall mesh refinement and the h-adaptive mesh refinement were justified with respect to the computational accuracy and the CPU time cost. Both methods can improve the computational accuracy with the refinement of mesh. The overall mesh refinement causes the CPU time cost to greatly increase as the mesh is refined. However, the CPU time cost does not increase very much with the increase of the level of h-adaptive mesh refinement. The CPU time cost can be saved by up to 90%, especially for the simulated system with a large number of elements and nodes.     

有限元网格划分相关问题分析研究

以有限元分析软件HyperMesh为计算平台,分析两个简单模型的静态受力产生的变形和应力,阐述有限元网格不同划分方法对计算精度的影响。     

Controlled cost of adaptive mesh refinement in practical 3D finite element analysis

In this paper, attention is restricted to mesh adaptivity. Traditionally, the most common mesh adaptive strategies for linear problems are used to reach a prescribed accuracy. This goal is best met with an h-adaptive scheme in combination with an error estimator. In an industrial context, the aim of the mechanical simulations in engineering design is not only to obtain greatest quality but more often a compromise between the desired quality and the computation cost (CPU time, storage, software, competence, human cost, computer used). In this paper we propose the use of alternative mesh refinement with an h-adaptive procedure for 3D elastic problems. The alternative mesh refinement criteria allow to obtain the maximum of accuracy for a prescribed cost. These adaptive strategies are based on a technique of error in constitutive relation (the process could be used with other error estimators) and an efficient adaptive technique which automatically takes into account the steep gradient areas. This work proposes a 3D method of adaptivity with the latest version of the INRIA automatic mesh generator GAMHIC3D.     

A hybrid method for variable-order fractional 2D optimal control problems on an unbounded domain

Engineering with Computers - In this study, the Caputo-Fabrizio fractional derivative is applied to generate a new category of variable-order fractional 2D optimization problems in an unbounded...     

Generation of triangular mesh with specified size by circle packing

This paper describes an algorithm for the generation of a finite element mesh with a specified element size over an unbound 2D domain using the advancing front circle packing technique. Unlike the conventional frontal method, the procedure does not start from the object boundary but starts from a convenient point within the open domain. As soon as a circle is added to the generation front, triangular elements are directly generated by properly connecting frontal segments with the centre of the new circle. Circles are packed closely and in contact with the existing circles by an iterative procedure according to the specified size control function. In contrast to other mesh generation schemes, the domain boundary is not considered in the process of circle packing, this reduces a lot of geometrical checks for intersection between frontal segments. If the mesh generation of a physical object is required, the object boundary can be introduced. The boundary recovery procedure is fast and robust by tracing neighbours of triangular elements. The finite element mesh generated by circle packing can also be used through a mapping process to produce parametric surface meshes of the required characteristics. The sizes of circles in the pack are controlled by the principal surface curvatures. Five examples are given to show the effectiveness and robustness of mesh generation and the application of circle packing to mesh generation over curved surfaces.     

3D finite element mesh generation of complicated tooth model based on CT slices

An interactive three-dimensional finite element generation method is presented for modelling a multi-connected teeth and mandible structure. The tetrahedron is chosen as the basic element type due to its rigorous adaptability to structures with geometric complexities. The mesh generation is implemented by allocating two quadrangles in adjacent CT image slices to form a set of tetrahedrons. By examining all the possible allocations and their degradations, an algorithm is developed for interactive mesh generation, resulting in a series of tetrahedrons consistent with all the others without overlapping and spacing. The developed system was applied to a tooth-mandibular structure, generating a complicated 3D FEM model consisting of 4762 nodes and 18,534 tetrahedral elements with nine different materials. This 3D model was successfully used to evaluate different tooth restoration strategies, which proved the viability and effectiveness of the proposed method.     

A finite element solution of diffraction problems in unbounded domains

In this paper, the method introduced in [3] is extended and applied to diffraction problems of acoustics and hydrodynamics. The problems dealt with are linear elliptic and may involve non-constant coefficients; they are set in unbounded domains. The method uses an integral representation formula with a regular kernel which allows an equivalent problem to be set in an interior annular closed region; it is shown how irregular frequencies are avoided. A variational formulation and its finite element discretization are detailed. Some numerical results are shown which support the validity of the technique and corroborate the theoretical analysis.     

An improved interface element with variable nodes for non-matching finite element meshes

Extensive improvements of the interface element method (IEM) are proposed for the efficient treatment of non-matching finite element meshes. Our approach enables us to establish the master element via the moving least-square (MLS) approximation, and so to remove the cumbersome process of constructing interface elements. The values of shape functions and their derivatives are therefore mapped from the master element, as in the conventional finite element method. For the assurance of convergence and compatibility condition, a patch test is demonstrated. Through several examples of 2D linear elasticity, the convergence rate is compared between the present interface element and the previous version.     

Mesh optimization: How to obtain good unstructured 3D finite element meshes with not-so-good mesh generators

We propose a new idea, based upon an optimization procedure, for improving three-dimensional finite element meshes. Our method can effectively be coupled with any automatic mesh generator in order to obtain a triangulation without badly distorted elements. In this way, we are able to weaken the requirements on the generator, allowing it to produce singular elements, and post-process the result to obtain a valid mesh. We report some experiments showing how this combination makes the meshing of arbitrary domains much more reliable, as compared to the usual approach of precluding distorted elements during the initial generation stage.     

Skeleton-based computational method for the generation of a 3D finite element mesh sizing function

This paper focuses on the generation of a three-dimensional (3D) mesh sizing function for geometry-adaptive finite element (FE) meshing. The mesh size at a point in the domain of a solid depends on the geometric complexity of the solid. This paper proposes a set of tools that are sufficient to measure the geometric complexity of a solid. Discrete skeletons of the input solid and its surfaces are generated, which are used as tools to measure the proximity between geometric entities and feature size. The discrete skeleton and other tools, which are used to measure the geometric complexity, generate source points that determine the size and local sizing function at certain points in the domain of the solid. An octree lattice is used to store the sizing function as it reduces the meshing time. The size at every lattice-node is calculated by interpolating the size of the source points. The algorithm has been tested on many industrial models, and it can be extended to consider other non-geometric factors that influence the mesh size, such as physics, boundary conditions, etc.Sandia National Laboratory is a multiprogram laboratory operated by the Sandia Corporation, a Lockheed Martin Company, for the US Department of Energy under contract DE-AC04-94AL85000.     

Three-dimensional mesh generation using principles of finite element method

The present work deals with the development of a three-dimensional mesh generation algorithm using the principles of FEM with special emphasis on the computational efficiency and the memory requirement. The algorithm makes use of a basic mesh that defines the total number of elements and nodes. Wavefront technique is used to renumber the nodes in order to reduce the bandwidth. By elastic distortion of the basic mesh, it is redefined to map onto actual geometry to be discretized. Later a finer distribution of mesh is done in the zones of interest to suit the nature of the problem. The same Finite Element code meant for stress analysis is adopted with necessary modifications. The algorithm has been extended to three-dimensional geometries. The current methodology is used to discretize a straight bevel gear and an hourglass worm to study their stress patterns.     

曲面的有限元网格顶点法矢算法

分析计算有限元三角形网格顶点法矢的各种算法原理,比较各种算法的结果精度,指出Max方法考虑了三角形网格的形状,且本质上是一种通过对四面体进行外接球面拟合的计算方法,结果精度很高.在此基础上,针对曲面在有限元网格划分后可能同时存在三角形网格和四边形网格,提出适应于单独的三角形网格和四边形网格与两者并存的混合网格的顶点法矢求取算法,计算结果表明了算法的适应性和有效性.     

The use of 2D enriched elements with bubble functions for finite element analysis

In this paper, the concept that adds the interior nodes of the Lagrange elements to the serendipity elements is described and a family of enriched elements is presented to improve the accuracy of finite element analysis. By the use of the static condensation technique at the element level, the extra computation time in using these elements can be ignored. Plane stress problems are used as examples in this paper. The numerical results show that these enriched elements are more accurate than the traditional serendipity elements. The convergence rate of the proposed elements is the same as the traditional serendipity elements. The error norm of the second and third order proposed elements can be reduced from 40% to 60% when compared with the use of the traditional serendipity elements. In the numerical examples, the use of the second and third order proposed elements not only give an improvement in element accuracy but also save computation time, when the precondition conjugate gradient method is used to solve the system of equations. The saving of computation time is due to the decrease of iteration number in iteration method.     

面向有限元网格生成的单元尺寸场光滑化算法

针对单元尺寸场的合适与否会直接影响到后续有限元网格质量的问题,提出一种尺寸修正算法来优化单元尺寸场。在Borouchaki等提出的H变化量（BOROUCHAKI H, HECHT F, FREY P J. Mesh gradation control. International Journal for Numerical Methods in Engineering, 1998,43(6):1143-1165）的基础上,引入尺寸梯度概念,进行一系列公式推导,得到二维的单元尺寸场的合理过渡要求,从而以定义在非结构背景网格的单元尺寸场为例,改进Borouchaki修正算法,提出了一种最少量地重置尺寸场中节点单元尺寸值,最大化地全局光滑单元尺寸场的新算法。最后给出若干实例的网格生成效果图,证明算法能帮助工程应用的模型生成更高质量的网格,跟其他修正算法相比,网格尺寸过渡明显更均匀。