Variable kinematic beam elements coupled via Arlequin method

In this work, beam elements based on different kinematic assumptions are combined through the Arlequin method. Computational costs are reduced assuming refined models only in those zones with a quasi-three-dimensional stress field. Variable kinematics beam elements are formulated on the basis of a unified formulation (UF). This formulation is extended to the Arlequin method to derive matrices related to the coupling zones between high- and low-order kinematic beam theories. According to UF, a N-order polynomials approximation is assumed on the beam cross-section for the unknown displacements, being N a free parameter of the formulation. Several hierarchical finite elements can be formulated. Part of the structure can be accurately modelled with computationally cheap low-order elements, part calls for computationally demanding high-order elements. Slender, moderately deep and deep beams are investigated. Square and I-shaped cross-sections are accounted for. A cross-ply laminated composite beam is considered as well. Results are assessed towards Navier-type analytical models and three-dimensional finite element solutions. The numerical investigation has shown that Arlequin method in the context of a hierarchical formulation effectively couples sub-domains having different order finite elements without loss of accuracy and reducing the computational cost.     

Stress intensity factor analyses of three-dimensional interface cracks using tetrahedral finite elements

A numerical method is proposed for evaluating the stress intensity factors of a three-dimensional bimaterial interfacial crack using tetrahedral finite elements. This technique is based on the M₁-integral method and employs the moving least-squares approximation. Stress or strain in the M₁-integral equation is automatically approximated from the nodal displacements obtained by the finite element analysis using the moving least-squares method. Therefore, the presented method needs no elemental information from the finite element analysis. In this study, stress intensity factor analyses of some typical three-dimensional interface crack problems using the tetrahedral finite elements are demonstrated.     

Extended finite element method and fast marching method for three-dimensional fatigue crack propagation

A numerical technique for planar three-dimensional fatigue crack growth simulations is proposed. The new technique couples the extended finite element method (X-FEM) to the fast marching method (FMM). In the X-FEM, a discontinuous function and the two-dimensional asymptotic crack-tip displacement fields are added to the finite element approximation to account for the crack using the notion of partition of unity. This enables the domain to be modeled by finite elements with no explicit meshing of the crack surfaces. The initial crack geometry is represented by level set functions, and subsequently signed distance functions are used to compute the enrichment functions that appear in the displacement-based finite element approximation. The FMM in conjunction with the Paris crack growth law is used to advance the crack front. Stress intensity factors for planar three-dimensional cracks are computed, and fatigue crack growth simulations for planar cracks are presented. Good agreement between the numerical results and theory is realized.     

Finite element calculation of three-dimensional hot forging

A three-dimensional finite, element model for the simulation of isothermal hot forging is presented. The material behaviour is assumed to be incompressible visco-plastic (Norton–Hoff law), with the associated friction law. The velocity field is calculated with a finite element approximation using eight node hexahedral or six node prismatic elements. An explicit Euler scheme is used for time integration. Simulations of compression of a cubic block and horizontal cylinder are performed. The computed results are compared with experimental measurements made on special Plasticine.     

Three-dimensional finite element analysis for through-wall crack in thick plate

A superposition method, in which analytical and finite element solutions are combined through the variational principle, is presented for determining the three-dimensional stress intensity factors of the through-wall crack. Cumbersome volume integral due to the singular terms at the crack tip is carried on by applying the least squares methods to the results obtained by the Legendre-Gauss quadrature. Several numerical calculations are made based on the present method, showing that the three-dimensional effects cannot be neglected especially for the cases of larger Poisson's raito. The comparison of the present method with the other finite element methods proposed for determining the three-dimensional stress intensity factors indicates that the former has the obvious superiority to the latters in respect to the numbers of elements and nodal points necessary to obtain reasonable results.     

Dual-mixed hp finite element methods using first-order stress functions and rotations

 A two-field dual-mixed variational formulation of three-dimensional elasticity in terms of the non-symmetric stress tensor and the skew-symmetric rotation tensor is considered in this paper. The translational equilibrium equations are satisfied a priori by introducing the tensor of first-order stress functions. It is pointed out that the use of six properly chosen first-order stress function components leads to a (three-dimensional) weak formulation which is analogous to the displacement-pressure formulation of elasticity and the velocity-pressure formulation of Stokes flow. Selection of stable mixed hp finite element spaces is based on this analogy. Basic issues of constructing curvilinear dual-mixed p finite elements with higher-order stress approximation and continuous surface tractions are discussed in the two-dimensional case where the number of independent variables reduces to three, namely two components of a first-order stress function vector and a scalar rotation. Numerical performance of three quadrilateral dual-mixed hp finite elements is presented and compared to displacement-based hp finite elements when the Poisson's ratio converges to the incompressible limit of 0.5. It is shown that the dual-mixed elements developed in this paper are free from locking in the energy norm as well as in the stress computations, both for h- and p-extensions. Received 22 October 1999     

An Alternative Positional FEM Formulation for Geometrically Non-linear Analysis of Shells: Curved Triangular Isoparametric Elements

This work presents an alternative finite element shell formulation based on non-conventional nodal parameters. The considered parameters are nodal positions (not displacements) and generalized vector components that comprise both director cossines and shell thickness variation at the same time. Although any objective strain measure could be adopted to develop the proposed formulation, non-linear engineering strain is chosen in order to take advantage of well know linear engineering stress–strain relations and to complement the easy geometrical appeal of positional formulation. The resulting formulation presents six degrees of freedom by each node and considers constant thickness variation. Consequently, the formulation fulfills a three dimensional compatible mapping and requires a relaxed three-dimensional constitutive relation to avoid thickness locking. Curved triangular elements with cubic approximation are adopted following a very simple notation. Several numerical simulations illustrate and confirm the accuracy and applicability of the proposed formulation.     

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.     

Three-dimensional finite element for analysing thin plate/shell structures

A three-dimensional (3-D) hexahedron finite element is presented for the analysis of thin plate/shell structures. The element employs an explicit algebraic definition of six uniform (continuum) strains, six rigid body modes and classical Lagrange-Germain-Kirchhoff thin plate bending modes. Nine additional stiffness factors are used to control higher-order hourglass modes. The element may be used for plate/shell analyses where the flat plate assumptions are appropriate. Also it can easily be adapted to form transition elements to lower order 2-D elements, or to higher-order 3-D continuum elements. The stiffness matrix satisfies the geometric isotropy requirement, passes the patch test, and gives essentially identical response to either applied transverse corner forces or to twisting moments applied on the corner, a requirement of Kirchhoff's corner conditions for a classical thin plate. Several examples are presented to demonstrate the performance of this finite element.     

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.     

Finite elements for the dynamic analysis of fluid-solid systems

Several new finite elements are presented for the idealization of two- and three-dimensional coupled fluid-solid systems subjected to static and dynamic loading. The elements are based on a displacement formulation in terms of the displacement degrees-of-freedom at the nodes of the element. The formulation includes the effects of compressible wave propagation and surface sloshing motion. The use of reduced integration techniques and the introduction of rotational constraints in the formulation of the element stiffness eliminates all unnecessary zero-energy modes. A simple method is given which allows the stability of a finite element mesh of fluid elements to be investigated prior to analysis. Hence, the previously encountered problems of ‘element locking’ and ‘hour glass’ modes have been eliminated and a condition of optimum constraint is obtained. Numerical examples are presented which illustrate the accuracy of the element. It is shown that the element behaves very well for non-rectangular geometry. The optimum constraint condition is clearly illustrated by the static solution of a rigid block floating on a mesh of fluid elements.     

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.     

复杂三维曲梁结构的无闭锁等几何分析算法研究

梁结构在工程中应用广泛,梁结构的仿真分析是计算力学的一个重要研究内容。该文研究了复杂三维曲梁结构的等几何分析方法,首次应用拟协调有限元中的多套函数技术,使用降阶基函数逼近梁内应变项,解决复杂三维曲梁结构仿真中的闭锁问题。利用全局坐标系列式方法,避免了单元刚度阵组装时的复杂坐标变换过程,提高计算效率。使用多片NURBS （非均匀有理B样条）数据表示复杂三维梁结构,可精确描述曲梁结构的几何形状,与有限元方法等仿真技术相比避免了网格生成过程,减少了几何误差。数值结果表明该文算法可有效解决闭锁问题,适于复杂三维曲梁结构的仿真分析。     

Transition finite elements for three-dimensional stress analysis

This paper presents isoparametric formulation for the three-dimensional transition finite elements. The transition finite elements are necessary for applications requiring the use of three-dimensional isoparametric solid elements and the curved shell elements. These elements provide proper connections between the two portions of the structure modelled with three-dimensional solids and the curved shell elements. The element properties are derived and presented in detail. Numerical examples are also presented to demonstrate the accuracy and the applications of such elements in three-dimensional stress analysis.     

Alternative vector potential formulations of 3-D magnetostatic field problems

A unified theory of three-dimensional vector potential formulations of magnetostatic field problems is presented. It is shown that existing formulations are based on one of two equivalent boundary value problems. A new formulation is derived, using the concept of projection between the space of arbitrary vector finite elements and the space of non-divergent vector finite elements. The merits of the different approaches are examined.     

Enrichment and coupling of the finite element and meshless methods

A mixed hierarchical approximation based on finite elements and meshless methods is presented. Two cases are considered. The first one couples regions where finite elements or meshless methods are used to interpolate: continuity and consistency is preserved. The second one enriches a finite element mesh with particles. Thus, there is no need to remesh in adaptive refinement processes. In both cases the same formulation is used, convergence is studied and examples are shown. Copyright © 2000 John Wiley & Sons, Ltd.     

On completeness of shape functions for finite element analysis

Some elements commonly used for analysis are examined for examined for completeness of polynomial interpolation and computational efficiency. Extensions to n-dimensional space are shown to be natural consequences of the interpolation, thus all elements considered here allow for finite element approximation in higher than three-dimensional spaces (e.g. space–time interpolations). From the study it is concluded that ‘serendipity’ class elements from the most efficient elements up to third-degree polynomial approximations. The method used here to develop the serendipity shape functions allows for different orders of interpolation along each edge. Thus, in zones where high accuracy is required meshes can now be easily changed from linear to quadratic or higher-order elements. Computations on some simple problems have demonstrated this to be a superior method than using large numbers of low ordered elements.     

Families of consistent conforming elements with singular derivative fields

Families of two-and three-dimensional finite elements are constructed for use in modelling fields with singular derivatives. The elements are complete over linear fields, conform with regular elements, and are easily programmed. A low order exact quadrature rule is also derived for element stiffness calculations.     

A corrected XFEM approximation without problems in blending elements

The extended finite element method (XFEM) enables local enrichments of approximation spaces. Standard finite elements are used in the major part of the domain and enriched elements are employed where special solution properties such as discontinuities and singularities shall be captured. In elements that blend the enriched areas with the rest of the domain problems arise in general. These blending elements often require a special treatment in order to avoid a decrease in the overall convergence rate. A modification of the XFEM approximation is proposed in this work. The enrichment functions are modified such that they are zero in the standard elements, unchanged in the elements with all their nodes being enriched, and varying continuously in the blending elements. All nodes in the blending elements are enriched. The modified enrichment function can be reproduced exactly everywhere in the domain and no problems arise in the blending elements. The corrected XFEM is applied to problems in linear elasticity and optimal convergence rates are achieved. Copyright © 2007 John Wiley & Sons, Ltd.     

Multiple quadrature underintegrated finite elements

New multiple-quadrature-point underintegrated finite elements with hourglass control are developed. The elements are selectively underintegrated to avoid volumetric and shear locking and save computational time. An approach for hourglass control is proposed such that the stabilization operators are obtained simply by taking the partial derivatives of the generalized strain rate vector with respect to the natural co-ordinates so that the elements require no stabilization parameter. To improve accuracy over the traditional one-point-quadrature elements, several quadrature points are used to integrate the internal forces, especially for tracing the plastic fronts in the mesh during loading and unloading in elastic–plastic analysis. Two- and four-point-quadrature elements are proposed for use in the two- and three-dimensional elements, respectively. Other multiple-quadrature points can also be employed. Several numerical examples such as thin beam, plate and shell problems are presented to demonstrate the applicability of the proposed elements.