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.     

Optimization of the finite element mesh for the solution of fracture problems

A study is presented concerning the effect of mesh geometry upon the accuracy of finite element solutions for fracture problems. Suggestions are given for the determination of the optimum mesh for several types of problems using linear displacement and quadratic displacement quadrilateral demonstrated that merely using smaller elements will not necessarily improve solution accuracy.     

Modified and Trefftz unsymmetric finite element models

The unsymmetric finite element method employs compatible test functions but incompatible trial functions. The pertinent 8-node quadrilateral and 20-node hexahedron unsymmetric elements possess exceptional immunity to mesh distortion. It was noted later that they are not invariant and the proposed remedy is to formulate the element stiffness matrix in a local frame and then transform the matrix back to the global frame. In this paper, a more efficient approach will be proposed to secure the invariance. To our best knowledge, unsymmetric 4-node quadrilateral and 8-node hexahedron do not exist. They will be devised by using the Trefftz functions as the trial function. Numerical examples show that the two elements also possess exceptional immunity to mesh distortion with respect to other advanced elements of the same nodal configurations.     

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.     

Application of infinite elements to phase change situations on deforming meshes

In recent years progress has been made in applying moving and deforming mesh systems to phase change problems. This allows the numerical attention where it is needed, near the migrating phase change zone. In spatially unbounded problems one hopes that numerically finite outer boundaries either escape significant activity or are automatically pushed further away as activity nears. Not infrequently this approach fails. Temperature activity often spreads more rapidly than phase change, thereby reaching far boundaries; stretching of the mesh by movement of far boundaries can challenge mesh control and cause ill-conditioning. In this paper the advantages of time dependent mesh adaption are enhanced by the joining of a new formulation for infinite elements to far boundaries. This is accomplished through a co-ordinate transformation within the framework of conventional 2-D quadratic, biquadratic, and linear–quadratic elements. Standard 2 by 2 Gauss–Legendre quadrature suffices throughout and normal Galerkin finite element features are undisturbed, including strict conservation of energy. The formulation is independent of global co-ordinates, entails no restrictions on the unknown function and should be applicable to other problem types. All test cases on quadrilateral and triangular grids show very significant improvements with infinite elements relative to comparable solution systems using strictly finite grids.     

Generalized nodes and high‐performance elements

The paper concerns the development of robust and high accuracy finite elements with only corner nodes using a partition‐of‐unity‐based finite‐element approximation. Construction of the partition‐of‐unity‐based approximation is accomplished by a physically defined local function of displacements. A 4‐node quadratic tetrahedral element and a 3‐node quadratic triangular element are developed. Eigenvalue analysis shows that linear dependencies in the partition‐of‐unity‐based finite‐element approximation constructed for the new elements are eliminable. Numerical calculations demonstrate that the new elements are robust, insensitive to mesh distortion, and offer quadratic accuracy, while also keeping mesh generation extremely simple. Copyright © 2005 John Wiley & Sons, Ltd.     

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.     

Automatic conversion of triangular finite element meshes to quadrilateral elements

A method is presented for the fully automatic conversion of a general finite element mesh containing triangular elements into a mesh composed of exclusively quadrilateral elements. The initial mesh may be constructed of entirely triangular elements or may consist of a mixture of triangular and quadrilateral elements. The technique used employs heuristic procedures and criteria to selectively combine adjacent triangular elements into quadrilaterals based on preestablished criteria for element quality. Additional procedures are included to eliminate isolated triangles. The methods operates completely without user intervention once the nodal co-ordinates and element connectivity of the original mesh are supplied.     

Anomalous behavior of bilinear quadrilateral finite elements for modeling cohesive cracks with XFEM/GFEM

The performance of partition‐of‐unity based methods such as the generalized finite element method or the extended finite element method is studied for the simulation of cohesive cracking. The focus of investigation is on the performance of bilinear quadrilateral finite elements using these methods. In particular, the approximation of the displacement jump field, representing cohesive cracks, by extended finite element method/generalized finite element method and its effect on the overall behavior at element and structural level is investigated. A single element test is performed with two different integration schemes, namely the Newton‐Cotes/Lobatto and the Gauss integration schemes, for the cracked interface contribution. It was found that cohesive crack segments subjected to a nonuniform opening in unstructured meshes (or an inclined crack in a structured finite element mesh) result in an unrealistic crack opening. The reasons for such behavior and its effect on the response at element level are discussed. Furthermore, a mesh refinement study is performed to analyze the overall response of a cohesively cracked body in a finite element analysis. Copyright © 2013 John Wiley & Sons, Ltd.     

Automatic generation of transitional meshes

Advances in commercial computer‐aided design software have made finite element analysis with three‐dimensional solid finite elements routinely available. Since these analyses usually confine themselves to those geometrical objects for which particular CAD systems can produce finite element meshes, expanding the capability of analyses becomes an issue of expanding the capability of generating meshes. This paper presents a method for stitching together two three‐dimensional meshes with diverse elements that can include tetrahedral, pentahedral and hexahedral solid finite elements. The stitching produces a mesh that coincides with the edges which already exist on the portion of boundaries that will be joined. Moreover, the transitional mesh does not introduce new edges on these boundaries. Since the boundaries of the regions to be stitched together can have a mixture of triangles and quadrilaterals, tetrahedral and pyramidal elements provide the transitional elements required to honor these constraints. On these boundaries a pyramidal element shares its base face with the quadrilateral faces of hexahedra and pentahedra. Tetrahedral elements share a face with the triangles on the boundary. Tetrahedra populate the remaining interior of the transitional region. Copyright © 2001 John Wiley & Sons, Ltd.     

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.     

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.     

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 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.     

A fractal patch test

Element consistency is generally checked using the patch test on an element patch of finite size. This condition may in certain cases be too restrictive, and disqualifies elements that appear to be convergent. A method termed ‘fractal patch test’ is presented, in which the patch size is maintained constant while the distorted mesh is refined. Examples are given for four-node quadrilateral elements used in plane stress and strain analysis, and for plate bending elements.     

An accurate and robust contact detection algorithm for particle‐solid interaction in combined finite‐discrete element analysis

When applying the combined finite‐discrete element method for analysis of dynamic problems, contact is often encountered between the finite elements and discrete elements, and thus an effective contact treatment is essential. In this paper, an accurate and robust contact detection algorithm is proposed to resolve contact problems between spherical particles, which represent rigid discrete elements, and convex quadrilateral mesh facets, which represent finite element boundaries of structural components. Different contact scenarios between particles and mesh facets, or edges, or vertices have been taken into account. For each potential contact pair, the contact search is performed in an hierarchical way starting from mesh facets, possibly going to edges and even further to vertices. The invalid contact pairs can be removed by means of two reasonable priorities defined in terms of geometric primitives and facet identifications. This hierarchical contact searching scheme is effective, and its implementation is straightforward. Numerical examples demonstrated the accuracy and robustness of the proposed algorithm. Copyright © 2015 John Wiley & Sons, Ltd.     

Finite element dispersion analysis for the three-dimensional second-order scalar wave equation

The dispersive properties of finite element semidiscretizations of the three-dimensional second-order scalar wave equation are examined for both plane and spherical waves. This analysis throws light on the performance and limitations of the finite element approximation over the entire spectrum of wavenumbers and provides guidance for optimal mesh discretization as well as mass representation. The 8-node trilinear element, 20-node serendipity element, 27-node triquadratic element and the linear and quadratic spherically symmetric elements are considered.     

An Imbricate Finite Element Method (I-FEM) using full, reduced, and smoothed integration

A method to design finite elements that imbricate with each other while being assembled, denoted as imbricate finite element method, is proposed to improve the smoothness and the accuracy of the approximation based upon low order elements. Although these imbricate elements rely on triangular meshes, the approximation stems from the shape functions of bilinear quadrilateral elements. These elements satisfy the standard requirements of the finite element method: continuity, delta function property, and partition of unity. The convergence of the proposed approximation is investigated by means of two numerical benchmark problems comparing three different schemes for the numerical integration including a cell-based smoothed FEM based on a quadratic shape of the elements edges. The method is compared to related existing methods.     

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.     

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.