Review of techniques for efficient network generation for finite element analysis

Triangle and tetrahedron elements possess special characteristics for automatic network generation in comparison with other element types. They always admit the application of only one element type within a whole arbitrary network structure, which leads to a simplification of the generation process and therefore to a reduction of the work involved. Different techniques for automatic two- and three-dimensional network generation with the application of these two element types will be analysed in this paper. It will be seen that the techniques differ considerably not only in the generation strategy, but also in the quality of the generated network structures for the finite element computations.     

RECOVERY BY EQUILIBRIUM IN PATCHES (REP)

A new recovery technique is developed in this paper. It is shown that, for many elements, the behaviour of the method is very similar to that of SPR. Because it does not need to identify super-convergent points, it is applicable for any form of element in which these points are not defined. The formulation is very simple and is based on equilibrating the recovered stresses, in the patch, in the same way that the standard FEM does. This procedure leads to a weak form of equilibrium equations of new stresses on the patch and consequently to answers satisfying the discrete equilibrium conditions. The formulation is consistent with non-linear formulations which iteratively equilibrate the problem. Therefore, this method can be used to project the Gauss points values to nodal points, with minimum disturbance of the global equilibrium. © 1997 by John Wiley & Sons, Ltd.     

A hierarchical approach to automatic finite element mesh generation

A point-based two-stage hierarchical method for automatic finite element mesh generation from a solid model is presented. Given the solid model of a component and the required nodal density distribution, nodes are generated according to the hierarchy—vertex, edge, face and solid. At the vertices, nodes are established naturally. Nodes on the edges, faces and inside the solid model are generated by recursive subdivision. The nodes are then connected to form a valid and well-conditioned finite element mesh of tetrahedron elements using modified Delaunay Triangulation. Checks are conducted to ensure the compatibility of geometry and topology between the solid model and the mesh.     

A normal offsetting technique for automatic mesh generation in three dimensions

A technique, based on a normal offsetting procedure, for the fully automatic generation of meshes suitable for finite element analysis in three dimensions is presented. The method is completely automatic, requiring no user intervention in the process and no special modelling procedures. The method is applied to three-dimensional solid geometries. The procedure positions nodes in the interior domain of an object by offsetting an initial set of nodes on the object boundary along vectors normal to the boundary to define a layer of new interior point locations. The offset points are processed to ensure good nodal spacing appropriate for generating well-shaped elements. Following processing, the offset points become a new boundary surrounding the remaining unmeshed region in the interior of the geometric domain. The offsetting procedure is applied again to this new boundary layer to form another offset layer farther into the domain interior. The offset-process-offset cycle is repeated until the entire region is filled with nodes. Tetrahedral elements are then formed by triangulation of the nodes. The boundary-based technique ensures good quality element shapes for analysis in critical boundary regions and facilitates applications involving integration of mesh generation with design geometry databases. Calculation of nodal locations are based on local parameters avoiding the higher-order time complexities associated with global calculations.     

Analytical stiffness matrices with Green–Lagrange strain measure

Separating the dependence on material and stress/strain state from the dependence on initial geometry, we obtain analytical secant and tangent stiffness matrices. For the case of a linear displacement triangle with uniform thickness and uniform constitutive behaviour closed‐form results are listed, directly suited for coding in a finite element program. The nodal positions of an element and the displacement assumption give three basic matrices of order three. These matrices do not depend on material and stress/strain state, and thus are unchanged during the necessary iterations for obtaining a solution based on Green–Lagrange strain measure. The approach is especially useful in design optimization, because analytical sensitivity analysis then can be performed. The case of a three node triangular ring element for axisymmetric analysis involves small modifications and extension to four node tetrahedron elements should be straight forward. Copyright © 2004 John Wiley & Sons, Ltd.     

Fully tensorial nodal and modal shape functions for triangles and tetrahedra

This paper presents nodal and modal shape functions for triangle and tetrahedron finite elements. The functions are constructed based on the fully tensorial expansions of one‐dimensional polynomials expressed in barycentric co‐ordinates. The nodal functions obtained from the application of the tensorial procedure are the standard h‐Lagrange shape functions presented in the literature. The modal shape functions use Jacobi polynomials and have a natural global C⁰ inter‐element continuity. An efficient Gauss–Jacobi numerical integration procedure is also presented to decrease the number of points for the consistent integration of the element matrices. An example illustrates the approximation properties of the modal functions. Copyright © 2005 John Wiley & Sons, Ltd.     

Fully automatic two dimensional mesh generation using normal offsetting

A technique, based on a normal offsetting procedure, for the fully automatic generation of two dimensional meshes suitable for finite element analysis is presented. The method positions nodes by first meshing the geometric entities that compose the object boundary, then offsetting those nodal locations along vectors normal to the boundary geometry. The offset row of nodes is processed to ensure a good nodal spacing appropriate for generating well shaped elements. Following processing, the new row is offset again and the cycle is repeated until the entire area is filled with nodes. The boundary based technique ensures good quality element shapes for analysis in critical boundary regions and facilitates applications involving integration of mesh generation with design geometry databases. Nodal locations are calculated based on local parameters avoiding the higher order time complexities associated with global calculations. A technique for controlling mesh density by overlaying an independent mesh density function on the geometry is also presented as part of the method. This approach allows mesh density to be automatically controlled by a variety of factors, such as previous analysis results, that are external to the actual mesh generation process. The independent nature of the function method allows different sources of density information to be used interchangeably without modification to the mesh generation procedure.     

A moving superimposed finite element method for structural topology optimization

Level set methods are becoming an attractive design tool in shape and topology optimization for obtaining efficient and lighter structures. In this paper, a dynamic implicit boundary‐based moving superimposed finite element method (s‐version FEM or S‐FEM) is developed for structural topology optimization using the level set methods, in which the variational interior and exterior boundaries are represented by the zero level set. Both a global mesh and an overlaying local mesh are integrated into the moving S‐FEM analysis model. A relatively coarse fixed Eulerian mesh consisting of bilinear rectangular elements is used as a global mesh. The local mesh consisting of flexible linear triangular elements is constructed to match the dynamic implicit boundary captured from nodal values of the implicit level set function. In numerical integration using the Gauss quadrature rule, the practical difficulty due to the discontinuities is overcome by the coincidence of the global and local meshes. A double mapping technique is developed to perform the numerical integration for the global and coupling matrices of the overlapped elements with two different co‐ordinate systems. An element killing strategy is presented to reduce the total number of degrees of freedom to improve the computational efficiency. A simple constraint handling approach is proposed to perform minimum compliance design with a volume constraint. A physically meaningful and numerically efficient velocity extension method is developed to avoid the complicated PDE solving procedure. The proposed moving S‐FEM is applied to structural topology optimization using the level set methods as an effective tool for the numerical analysis of the linear elasticity topology optimization problems. For the classical elasticity problems in the literature, the present S‐FEM can achieve numerical results in good agreement with those from the theoretical solutions and/or numerical results from the standard FEM. For the minimum compliance topology optimization problems in structural optimization, the present approach significantly outperforms the well‐recognized ‘ersatz material’ approach as expected in the accuracy of the strain field, numerical stability, and representation fidelity at the expense of increased computational time. It is also shown that the present approach is able to produce structures near the theoretical optimum. It is suggested that the present S‐FEM can be a promising tool for shape and topology optimization using the level set methods. Copyright © 2005 John Wiley & Sons, Ltd.     

Finite elements on the basis of continuum mechanics

The formulation of finite element models on the basis of different variational principles is reviewed. The degrees-of-freedom of the elements are defined in an abstract way without the help of nodal points. In this manner it is possible to describe elements of arbitrary shape and accuracy. The formulation is confined to linear elasto-statics. For two-dimensional structures two hybrid element models are developed using Legendre polynomials on the element boundaries. Examples of plane stress problems are used to test the generation of convergence by increasing the accuracy of the elements vs. by increasing the number of elements.     

High‐accuracy differential quadrature finite element method and its application to free vibrations of thin plate with curvilinear domain

Based on the differential quadrature (DQ) rule, the Gauss Lobatto quadrature rule and the variational principle, a DQ finite element method (DQFEM) is proposed for the free vibration analysis of thin plates. The DQFEM is a highly accurate and rapidly converging approach, and is distinct from the differential quadrature element method (DQEM) and the quadrature element method (QEM) by employing the function values themselves in the trial function for the title problem. The DQFEM, without using shape functions, essentially combines the high accuracy of the differential quadrature method (DQM) with the generality of the standard finite element formulation, and has superior accuracy to the standard FEM and FDM, and superior efficiency to the p‐version FEM and QEM in calculating the stiffness and mass matrices. By incorporating the reformulated DQ rules for general curvilinear quadrilaterals domains into the DQFEM, a curvilinear quadrilateral DQ finite plate element is also proposed. The inter‐element compatibility conditions as well as multiple boundary conditions can be implemented, simply and conveniently as in FEM, through modifying the nodal parameters when required at boundary grid points using the DQ rules. Thus, the DQFEM is capable of constructing curvilinear quadrilateral elements with any degree of freedom and any order of inter‐element compatibilities. A series of frequency comparisons of thin isotropic plates with irregular and regular planforms validate the performance of the DQFEM. Copyright © 2009 John Wiley & Sons, Ltd.     

Modification to the Suhara-Fukuda method of network generation

This note describes improvements to the Suhara-Fukuda method for generating a triangular finite element network. These are based on the use of a node generation procedure which distributes nodes is a regular pattern across the network area and is sufficiently flexible to permit any variation in element density. Because of the regular node distribution, poorly shaped elements will not arise and so the procedure for connecting the nodes to form elements can be simplified. Examples of networks generaed in this way are given.     

一维C~1有限元超收敛解答计算的EEP法

将新近提出的C0有限元后处理中超收敛解答计算的单元能量投影(Element Energy Projection,简称EEP)法推广到一维C1类有限元。根据单元投影定理具体推导了一般梁单元的计算公式,并对两个有代表性的单元给出了数值算例。分析和算例表明,EEP法在一维C1类有限元中再次获得令人满意的效果,即对任一单元中的任一点,从位移一直到三阶导数(如梁的挠度、转角、弯矩、剪力),匀可获得与结点位移精度相当的超收敛结果,而且可精确满足自然边界条件。     

Automatic two- and three-dimensional mesh generation based on fuzzy knowledge processing

This paper describes the development of a novel automatic FEM mesh generation algorithm based on the fuzzy knowledge processing technique.A number of local nodal patterns are stored in a nodal pattern database of the mesh generation system. These nodal patterns are determined a priori based on certain theories or past experience of experts of FEM analyses. For example, such human experts can determine certain nodal patterns suitable for stress concentration analyses of cracks, corners, holes and so on. Each nodal pattern possesses a membership function and a procedure of node placement according to this function. In the cases of the nodal patterns for stress concentration regions, the membership function which is utilized in the fuzzy knowledge processing has two meanings, i.e. the closeness of nodal location to each stress concentration field as well as nodal density. This is attributed to the fact that a denser nodal pattern is required near a stress concentration field. What a user has to do in a practical mesh generation process are to choose several local nodal patterns properly and to designate the maximum nodal density of each pattern. After those simple operations by the user, the system places the chosen nodal patterns automatically in an analysis domain and on its boundary, and connects them smoothly by the fuzzy knowledge processing technique. Then triangular or tetrahedral elements are generated by means of the advancing front method. The key issue of the present algorithm is an easy control of complex two- or three-dimensional nodal density distribution by means of the fuzzy knowledge processing technique.To demonstrate fundamental performances of the present algorithm, a prototype system was constructed with one of object-oriented languages, Smalltalk-80 on a 32-bit microcomputer, Macintosh II. The mesh generation of several two- and three-dimensional domains with cracks, holes and junctions was presented as examples.     

Automatic element reordering for finite element analysis with frontal solution schemes

This paper describes an element reordering algorithm which is suitable for use with a frontal solution package. The procedure is shown to generate efficient element numberings for a wide variety of test examples. In an effort to obtain an optimum elimination order, the algorithm first renumbers the nodes, and then uses this result to resequence the elements. This intermediate step is necessary because of the nature of the frontal solution procedure, which assembles variables on an element-by-element basis but eliminates them node by node. To renumber the nodes, a modified version of the King¹ algorithm is used. In order to minimize the number of nodal numbering schemes that need to be considered, the starting nodes are selected automatically by using some concepts from graph theory. Once the optimum numbering sequence has been ascertained, the elements are then reordered in an ascending sequence of their lowest-numbered nodes. This ensures that the new elimination order is preserved as closely as possible. For meshes that are composed of a single type of high-order element, it is only necessary to consider the vertex nodes in the renumbering process. This follows from the fact that mesh numberings which are optimal for low-order elements are also optimal for high-order elements. Significant economies in the reordering strategy may thus be achieved. A computer implementation of the algorithm, written in FORTRAN IV, is given.     

悬索桥结构分析中索鞍的精确模拟

为在悬索桥结构分析中精确模拟索鞍,建立了索段一端固定于鞍座上的两节点“左鞍座单元”和“右鞍座单元”,以及索段中一点固定于鞍座上的三节点“鞍座单元”,此固定点为新单元的一个节点。它们通过自动调整索与鞍座的切点而处于平衡状态,从而简化了计算。单元算法的推导基于有限元分析的基本原理和弹性悬链线的精确解,并利用了处于平衡状态时索与鞍座之间的内力关系。新单元可以考虑鞍座重量的影响,鞍槽纵向曲线可为复合圆曲线。新单元可以同常规单元一样直接用于索结构的有限元分析,设计的算例验证了其正确性,工程算例显示了其在悬索桥结构分析中的应用。     

An algorithm for automatically tracking ablating boundaries

An algorithm has been developed which automatically calculates the time-dependent positions of points on the ablating boundaries of two-dimensional continuum structures with geometrially complex shapes. The structural boundary may consist of ablating and non-ablating parts. The initial ablating and non-ablating boundaries are defined by the disjoint union of piecewise linear arcs passing through a finite set of nodal points on the boundary, each defined by a pair of rectangular Cartesian co-ordinates. For a specified ablation rate, the algorithm calculates successive positions of the boundary points at times specifiec by the user. The algorithm is designed such that it may be easily incorporated, along with an automated mesh generation procedure, into existing finite element codes for transient thermal or stress analysis of structures with ablating boundaries. Two examples are presented from the field of solid rocket analysis.     

A least-squares approach for uniform strain triangular and tetrahedral finite elements

A least-squares approach is presented for implementing uniform strain triangular and tetrahedral finite elements. The basis for the method is a weighted least-squares formulation in which a linear displacement field is fit to an element's nodal displacements. By including a greater number of nodes on the element boundary than is required to define the linear displacement field, it is possible to eliminate volumetric locking common to fully integrated lower-order elements. Such results can also be obtained using selective or reduced integration schemes, but the present approach is fundamentally different from those. The method is computationally efficient and can be used to distribute surface loads on an element edge or face in a continuously varying manner between vertex, mid-edge and mid-face nodes. Example problems in two- and three-dimensional linear elasticity are presented. Element types considered in the examples include a six-node triangle, eight-node tetrahedron, and ten-node tetrahedron. © 1998 John Wiley & Sons, Ltd.     

Boundary edge elements and spanning tree technique in three-dimensional electromagnetic field computation

The boundary discretization of the curl-free vector variable such as the magnetic field h and the divergence-free variable such as the surface current density k is discussed. Instead of introducing the scalar variables, the vector variables h and k are discretized by the curl-conform and the div-conform triangular edge elements, respectively. The degrees of freedom are associated with the boundary edges. In order to ensure the null curl of h and the null divergence of k, a spanning tree technique is used to identify the independent edges. The triangular edge elements contain the first-order nodal elements when expressing h or k by the scalar variables. The use of edge elements permits one to solve multiply connected problems if the independent edges are well identified, i.e. the necessary cuts are introduced in multiply connected domains. An automatic tree generation algorithm is presented. It permits one to determine automatically the additional edges on the necessary loops (cuts) of a multiply connected region. Some tree generation examples are illustrated. A numerical application to a three-dimensional multiply connected eddy current problem is reported at the end of the paper.     

Structural collapse analysis of framed structures under impact loads using ASI-Gauss finite element method

The main objective of this study is to devise a technique, which, when implemented into finite-element codes, is efficiently applicable to impact collapse analyses of framed structures. In this study, the formerly developed adaptively shifted integration (ASI) technique for the linear Timoshenko beam element is modified into the ASI-Gauss technique by placing the numerical integration points of the two consecutive elements forming an elastically deformed member in such a way that stresses and strains are evaluated at the Gaussian integration points of the two-element member. On comparison with the ASI technique, the ASI-Gauss technique proves its higher accuracy and efficiency in elastic range. Moreover, instead of applying impact loads in the form of nodal forces, we consider the impact phenomenon by means of contacts between the elements involved and the elemental contact algorithm is verified from the point of conservation of energy. Impact analyses considering member fracture with different sets of parameters are performed using a high-rise framed structure and a small aircraft. From the results obtained, we can observe propagation phenomena of impact loads and shock waves. Also, a proper difference in impact damage is obtained by different sets of parameters. The results also indicate that the mass of the aircraft has a stronger influence on impact damage than its velocity. Moreover, soon after impact, tensile stresses are observed in the columns that were compressed by dead loads before impact.     

A SUPERCONVERGENT STRESS RECOVERY TECHNIQUE WITH EQUILIBRIUM CONSTRAINT

A stress recovery technique is developed to extract more accurate nodal stress values from the raw stress values obtained directly from the finite element analysis. In the present method a stress field is assumed over a patch of elements, and a least-squares functional is formed using the discrete stress errors at the superconvergent stress points and the residual of the equilibrium equation expressed in the virtual work form. The results of numerical tests conducted on one-dimensional and two-dimensional example problems demonstrate the validity and effectiveness of the present method. The introduction of an equilibrium constraint allows a patch stress field of higher order than is possible without the equilibrium constraint and this leads to a recovered stress field of higher accuracy. Because the residual of equilibrium is expressed in the virtual work form, the proposed method can easily be applied to arbitrarily curved shell structures. © 1997 by John Wiley & Sons, Ltd.