Automatic generation of anisotropic quadrilateral meshes on three‐dimensional surfaces using metric specifications

A new algorithm for constructing full quadrilateral anisotropic meshes on 3D surfaces is proposed in this paper. The proposed method is based on the advancing front and the systemic merging techniques. Full quadrilateral meshes are constructed by systemically converting triangular elements in the background meshes into quadrilateral elements.By using the metric specifications to describe the element characteristics, the proposed algorithm is applicable to convert both isotropic and anisotropic triangular meshes into full quadrilateral meshes. Special techniques for generating anisotropic quadrilaterals such as new selection criteria of base segment for merging, new approaches for the modifications of the background mesh and construction of quadrilateral elements, are investigated and proposed in this study. Since the final quadrilateral mesh is constructed from a background triangular mesh and the merging procedure is carried out in the parametric space, the mesh generator is robust and no expensive geometrical computation that is commonly associated with direct quadrilateral mesh generation schemes is needed. Copyright © 2002 John Wiley & Sons, Ltd.     

Quadrilateral approaches for accurate free mesh method

This paper presents the formulation of free mesh method and two approaches of the method by incorporating quadrilateral elements. The approaches do not have any difference with the original free mesh method in their fundamental algorithms, wherein local system equations with triangular elements are created nodewise to create the global system equations, and their implementation is therefore very easy. The first approach creates quadrilateral elements inside every triangular element, whilst quadrilateral elements are generated outside every triangular element in the second approach. The results of numerical examples indicate that the approaches improve the accuracy of free mesh method, further opening the possibility for more improvement using an accurate quadrilateral element. Copyright © 2000 John Wiley & Sons, Ltd.     

A new eight-node quadrilateral shear-bending plate finite element

A new eight-node quadrilateral shear-bending Reissner–Mindlin plate finite element for the very thin and thick plates without locking and spurious zero-energy modes is presented. The element has very good convergence characteristics both for thin and thick plates, is hardly insensitive to mesh distortions, and passes the patch tests. The formulation of the element is derived from a displacement variational principle and some general criteria to compute inconsistent transverse shear strains. These criteria have been applied with success to four- and eight-node quadrilateral plate finite elements and could be applied to construct triangular elements. The eight-node quadrilateral shear-bending plate finite element proposed has been found to be very efficient.     

A ‘FE-Meshfree’ TRIA3 element based on partition of unity for linear and geometry nonlinear analyses

A new 3-node triangular element is developed on the basis of partition of unity (PU) concept. The formulation employs the parametric shape functions of classical triangular element (TRIA3) to construct the PU and the least square point interpolation method to construct the local displacement approximation. The proposed element synergizes the individual merits of finite element method and meshfree method. Moreover, the usual linear dependence problem associated with PU finite elements is eliminated in the present element. Application of the element to several linear and geometric nonlinear problems shows that the proposed element gives a performance better than that of classical linear triangular as well as linear quadrilateral elements, and comparable to that of quadratic quadrilateral element. The proposed element does not necessitate a new mesh or additional nodes in the mesh. It uses the same mesh as the classical TRIA3 element and is able to give more accurate solution than the TRIA3 element.     

Dispersion analysis of finite element semidiscretizations of the two-dimensional wave equation

The dispersive properties of finite element semidiscretizations of the two-dimensional wave equation are examined. Both bilinear quadrilateral elements and linear triangular elements are considered with diagonal and nondiagonal mass matrices in uniform meshes. It is shown that mass diagonalization and underintegration of the stiffness matrix of the quadrilateral element markedly increases dispersive errors. The dispersive properties of triangular meshes depends on the mesh layout; certain layouts introduce optical modes which amplify numerically induced oscillations and dispersive errors. Compared to the five-point Laplacian finite difference operator, rectangular finite element semidiscretizations with consistent mass matrices provide superior fidelity regardless of the wave direction.     

On the density of finite element matrices

Explicit formulae for the density of a class of finite element matrices are presented. The formulae are functions of the characteristics of the mesh and the element (piecewise polynomial) employed. Results are given for two-dimensional triangular and quadrilateral elements and three-dimensional tetrahedral and cuboid elements. These results are useful in the management of storage in finite element computer programmes.     

A family of simple and robust finite elements for linear and geometrically nonlinear analysis of laminated composite plates

A family of simple, displacement-based and shear-flexible triangular and quadrilateral flat plate/shell elements for linear and geometrically nonlinear analysis of thin to moderately thick laminate composite plates are introduced and summarized in this paper.

The developed elements are based on the first-order shear deformation theory (FSDT) and von-Karman’s large deflection theory, and total Lagrangian approach is employed to formulate the element for geometrically nonlinear analysis. The deflection and rotation functions of the element boundary are obtained from Timoshenko’s laminated composite beam functions, thus convergence can be ensured theoretically for very thin laminates and shear-locking problem is avoided naturally.

The flat triangular plate/shell element is of 3-node, 18-degree-of-freedom, and the plane displacement interpolation functions of the Allman’s triangular membrane element with drilling degrees of freedom are taken as the in-plane displacements of the element. The flat quadrilateral plate/shell element is of 4-node, 24-degree-of-freedom, and the linear displacement interpolation functions of a quadrilateral plane element with drilling degrees of freedom are taken as the in-plane displacements.

The developed elements are simple in formulation, free from shear-locking, and include conventional engineering degrees of freedom. Numerical examples demonstrate that the elements are convergent, not sensitive to mesh distortion, accurate and efficient for linear and geometric nonlinear analysis of thin to moderately thick laminates.     

An analysis of some higher-order triangular elements and their susceptibility to hourglassing in Lagrangian fluid simulations

In Lagrangian codes two of the main problems are hourglassing with bilinear quadrilateral elements and mesh locking with linear triangular elements. Higher-order triangular elements, with their greater number of degrees of freedom, should not suffer from mesh locking, Hourglassing, though, is caused by the interaction of the element and the quadrature used to evaluate the integrals of the derivatives on it. Higher-order triangular elements, as we shall see, unlike their linear counterparts, also suffer from this spurious mode. A higher-order element/quadrature combination is found that does not suffer from hourglassing. The main fault of this combination is the extra cost involved over the more usual linear or bilinear element with centroid quadrature combination. A short discussion is given regarding this point.     

A class of data structures for 2-D and 3-D adaptive mesh refinement

A ‘family’ of tree data structures for adaptive mesh refinement is described and details concerning the associated logic are provided. The data structures encompass triangular elements and quadrilateral elements in two dimensions and quadrilateral bricks in three dimensions. Furthermore, both linear (bilinear) and quadratic (biquadratic) element types, respectively, are developed. Representative refinement results are given for the bilinear, trilinear and biquadratic types and associated performance studies made for the refinement procedure.     

A two-level mesh repartitioning scheme for the displacement-based lower-order finite element methods in volumetric locking-free analyses

We present an approach for repartitioning existing lower-order finite element mesh based on quadrilateral or triangular elements for the linear and nonlinear volumetric locking-free analysis. This approach contains two levels of mesh repartitioning. The first-level mesh re-partitioning is an h-adaptive mesh refinement for the generation of a refined mesh needed in the second-level mesh coarsening. The second-level mesh coarsening involves a gradient smoothing scheme performed on each pair of adjacent elements selected based on the first-level refined mesh. With the repartitioned mesh and smoothed gradient, the equivalence between the mixed finite element formulation and the displacement-based finite element formulation is established. The extension to nonlinear finite element formulation is also considered. Several linear and non-linear numerical benchmarks are solved and numerical inf-sup tests are conducted to demonstrate the accuracy and stability of the proposed formulation in the nearly incompressible applications.     

Assessment of discretized errors and adaptive refinement with quadrilateral finite elements

This paper studies discretized errors, and their estimation in conjunction with quadrilateral finite element meshes which are generated by the intelligent mesh generator XFORMQ.¹ The exact energy error is used to evaluate the distortion effect of the quadrilateral mesh. The Zienkiewicz–Zhu² error estimate and actaptive procedure are applied to the short cantilever and the square plate problems using the quadrilateral mesh generator XFORMQ. It is shown that the multistage quadrilateral element refinement produces results superior to the triangular element refinement in the test cases.     

Analyzing static three-dimensional elastic domains with a new infinite boundary element formulation

A new two-dimensionally mapped infinite boundary element (IBE) is presented. The formulation is based on a triangular boundary element (BE) with linear shape functions instead of the quadrilateral IBEs usually found in the literature. The infinite solids analyzed are assumed to be three-dimensional, linear-elastic and isotropic, and Kelvin fundamental solutions are employed. One advantage of the proposed formulation over quadratic or higher order elements is that no additional degrees of freedom are added to the original BE mesh by the presence of the IBEs. Thus, the IBEs allow the mesh to be reduced without compromising the accuracy of the result. Two examples are presented, in which the numerical results show good agreement with authors using quadrilateral IBEs and analytical solutions.     

A generalized automatic mesh generation scheme for finite element method

This paper presents a generalized method which generates linear, triangular, quadrilateral and pentahedral elements for the finite element method. Depending on geometrical and material variations, the region to be discretized is manually divided into blocks such as lines, triangles, quadrilaterals, pentahedrons and hexahedrons in several appropriate co-ordinate systems. However, no connectivity information of the adjacent blocks is required by the user as input. The continuity of the generated nodal co-ordinates and element configurations at the block interface are automatically maintained to describe the geometry of structures, no matter how these five types of blocks are connected. Furthermore, a mesh grading algorithm which generates reliable mesh grade distributions in the interior of the triangular and quadrilateral blocks is established corresponding to the arbitrarily defined subdivision numbers for each edge line of blocks. This algorithm is extended to the mesh grading in the interior of the hexahedral and pentahedral blocks. Element numbers are also renumbered in this scheme, in addition to node numbers, in order to increase the computational efficiency of the global matrix assembly. Additional facilities, i.e. loading data generation, boundary condition data generation and so on, are also discussed. An illustrative and a practical example are given to demonstrate the capabilities of this scheme.     

The application of node-element rules for forecasting problems in the generation of finite element meshes

The number of equations which relate the mesh parameters (number of nodes, number of elements etc.) to each other is smaller than the number of unknowns contained therein. These rules are only applicable when several parameters of a defined mesh are known, but are unsuitable for use in the particularly interesting field of mesh projection unless additional inter-relationships can be found. Such relationships are given for uniform meshes in which besides the characteristics of the element type the ‘solidity value’ of the structure plays an important role. The equations are derived for two-dimensional triangular and quadrilateral elements and also for three-dimensional tetrahedral and hexahedral elements. The fact that these relationships are also suitable for estimating the parameters of any non-uniform mesh, discloses a wide field of application in the management of storage and in the control of programs for automatic mesh generation.     

Multiphase hybrid stress finite element analysis of heterogeneous media by simple mesh: One element with one interface

In this paper, a finite element method is proposed to analyze the microscopic and macroscopic mechanical behaviors of heterogeneous media with randomly distributed inclusions. A simple mesh partitioned the domain into regular quadrilateral or triangular elements, where one element may contain two phases. An assumed stress hybrid formulation is implemented in the finite element model and the functional is derived for an element containing two phases. Numerical examples were used to study the microscopic and macroscopic properties of the composites, such as the effective modulus, to validate of the proposed model. The results show that the proposed multiphase hybrid stress finite element model can accurately measure the stress fields of materials with arbitrary microstructural distributions and improve computational efficiency by about 30 to 1500 times in comparison with the traditional displacement based finite element method.     

A methodology for adaptive finite element analysis: Towards an integrated computational environment

This work introduces a methodology for self-adaptive numerical procedures, which relies on the various components of an integrated, object-oriented, computational environment involving pre-, analysis, and post-processing modules. A basic platform for numerical experiments and further development is provided, which allows implementation of new elements/error estimators and sensitivity analysis. A general implementation of the Superconvergent Patch Recovery (SPR) and the recently proposed Recovery by Equilibrium in Patches (REP) is presented. Both SPR and REP are compared and used for error estimation and for guiding the adaptive remeshing process. Moreover, the SPR is extended for calculating sensitivity quantities of first and higher orders. The mesh (re-)generation process is accomplished by means of modern methods combining quadtree and Delaunay triangulation techniques. Surface mesh generation in arbitrary domains is performed automatically (i.e. with no user intervention) during the self-adaptive analysis using either quadrilateral or triangular elements. These ideas are implemented in the Finite Element System Technology in Adaptivity (FESTA) software. The effectiveness and versatility of FESTA are demonstrated by representative numerical examples illustrating the interconnections among finite element analysis, recovery procedures, error estimation/adaptivity and automatic mesh generation.     

Unsymmetric extensions of Wilson's incompatible four-node quadrilateral and eight-node hexahedral elements

The unsymmetric finite element is based on the virtual work principle with different sets of test and trial functions. In this article, the incompatible four-node quadrilateral element and eight-node hexahedral element originated by Wilson et al. are extended to their unsymmetric forms. The isoparametric shape functions together with Wilson's incompatible functions are chosen as the test functions, while internal nodes at the middle of element sides/edges are added to generate the trial functions with quadratic completeness in the Cartesian coordinate system. A local area/volume coordinate frame is established so that the trial shape functions can be explicitly obtained. The key idea which avoids the matrix inversion is that the trial nodal shape functions are constructed by standard quadratic triangular/tetrahedral elements and then transformed in consistent with the quadrilateral/hexahedral elements. Numerical examples show that the present elements keep the merits of both incompatible and unsymmetric elements, that is, high numerical accuracy, insensitivity to mesh distortion, free of trapezoidal and volumetric locking, and easy implementation.     

Triangular checkerboard control using a wavelet‐based method in topology optimization

Topology optimization has been carried out mainly with structured quadrilateral finite elements. However, triangular elements facilitate mesh generation especially for problems having complex geometries that often appear in practical industrial problems. The use of triangular elements, especially low‐order triangular elements, causes a serious numerical trouble that is equivalent to the rectangular checkerboard pattern formation. The objective of this investigation is to develop a triangular checkerboard‐freeing method that directly restricts the design space. To this end, we use the multiscale design space that is mapped from the standard single‐scale density space. To facilitate the mapping, we employ the triangular mesh subdivision and propose a bi‐orthogonal wavelet transform suitable for a triangulated domain. For checkerboard‐free designs, a shrinkage method based on the wavelet frame appropriate for triangular mesh is proposed. Typical benchmark problems and a simplified roof‐reinforcing problem in an automobile body are considered to check the effectiveness of the proposed method. Copyright © 2005 John Wiley & Sons, Ltd.     

任意平面区域的变密度四边形网格生成方法

：基于三角形网格的合并方法,首先生成光滑过渡的变密度三角形网格,然后进行三角形的合并,接着将合并后残留的三角形细分成三个较小的四边形,相应把合并生成的四边形进一步细分成四个较小的四边形,最终生成了光滑过渡的四边形网格。     

On a finite element model for the analysis of through cracks in laminated anisotropic cylindrical shells

A finite element model for the analysis of laminated composite cylindrical shells with through cracks is presented. The analysis takes into account anisotropic elastic behaviour, bending-extensional coupling and transverse shear deformation effects. The proposed finite element model is based on the approach of dividing a cracked configuration into triangular shaped singular elements around the crack tip with adjoining quadrilateral shaped regular elements. The parabolic isoparametric cylindrical shell elements (both singular and regular) used in this model employ independent displacement and rotation interpolation in the shell middle surface. The numerical comparisons show the evidence to the conclusion that the proposed model will yield accurate stress intensity factors from a relatively coarse mesh. Through the analysis of a pressurised fibre composite cylindrical shell with an axial crack, the effect of material orthotropy on the crack tip stress intensity factors is shown to be quite significant.