A general mass lumping scheme for the variants of the extended finite element method

A new strategy for the mass matrix lumping of enriched elements for explicit transient analysis is presented. It is shown that to satisfy the kinetic energy conservation, the use of zero or negative masses for enriched degrees of freedom of lumped mass matrix may be necessary. For a completely cracked element, by lumping the mass of each side of the interface into the finite element nodes located at the same side and assigning zero masses to the enriched degrees of freedom, the kinetic energy for rigid body translations is conserved without transferring spurious energy across the interface. The time integration is performed by adopting an explicit-implicit technique, where the regular and enriched degrees of freedom are treated explicitly and implicitly, respectively. The proposed method can be viewed as a general mass lumping scheme for the variants of the extended finite element methods because it can be used irrespective of the enrichment method. It also preserves the optimal critical time step of an intact finite element by treating the enriched degrees of freedom implicitly. The accuracy and efficiency of the proposed mass matrix are validated with several benchmark examples.     

A pseudo-complementary finite element approach for the vibration analysis of thin plates in bending

The present paper is concerned with the numerical analysis of the steady-state and transient response of thin elastic plates. Based on a modification of the variational principle due to Hamilton wherein in contrast to the classical formulation not only displacements but also stress resultants represent independent (primal) variables, a new mixed hybrid finite element model is proposed. Introducing separate approximations for the displacement and stress field, stiffness and consistent mass matrix of a triangular plate element with three kinematic degrees of freedom per nodal point are obtained. The performance of the new element scheme is evaluated on the basis of several test examples representing a broad range of circumstances encountered in linear elastokinetic thin plate analysis. The obtained numerical results demonstrate that in terms of efficiency, reliability and accuracy the new element scheme competes most favorably with a series of well-established plate elements. This work was partially supported by the Deutsche Forschungsgemeinschaft.     

The gradient of the finite element variational indicator with respect to nodal and applications in fracture mechanics and mesh optimization

We derive a closed-form expression for the change in the variational indicator of a finite element mesh with respect to perturbations in nodal point co-ordinates. The expression is evaluated very effectively from standard finite element data obtained in one solution, and may be easily programmed as part of a general finite element code. We present the derivation for two- and three-dimensional isoparametric elements used in linear and nonlincar elasticity. The expression has practical applications in the computation of stress intensity factors in fracture mechanics and in the determination of the ‘optimal’ mesh with a given element-node connectivity. We demonstrate both applications by accurately determining the stress intensity factor of a Mode I crack using a finite element mesh which was improved using mesh optimization.     

关于单元能量投影法的两点注记

最近,袁驷等基于力学原理提出了一种一维有限元超收敛后处理计算格式,称为单元能量投影(EEP)法。大量数值例子显示:若真解充分光滑,对m次有限元解,EEP法后处理节点恢复导数具有h2m阶精度。首先利用限元超收敛理论中的一个基本估计式证明了线性元(m=1)节点恢复导数具有h2阶精度。另外,对EEP法高次元的内点计算公式提出了一点简化。     

A higher-order triangular finite element for the solution of field problems in orthotropic media

This paper presents a triangular finite element for the solution of two-dimensional field problems in orthotropic media. The element has nine degrees of freedom, these being the potential and its two derivatives at each node. The ‘stiffness’ matrix is derived analytically so that no further integration is required when computations are performed using the element. The results obtained using the element are compared with the exact mathematical solution of both a temperature distribution and a torsion problem.     

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.     

A solid shell element for the shell structured eyeball with application to radial keratotomy

In order to obtain the curvature changes of the exterior surface of a shell subjected to an internal pressure, it is necessary to evaluate the displacement derivatives up to the second order. To this end, a hexahedronal element is developed with three-dimensional elastic properties utilized. At a nodal point on the surface of the shell, the first-order derivatives of the three displacement components are used as degrees of freedom (d.o.f.) in addition to the components themselves. However, at a nodal point not on the surface, only the three displacement components are used. Therefore, an element with aii exterior surface has a total of 48 d.o.f. and other elements have 24 d.o.f. The stiffness matrix of the 8-node subparametric hexahedronal high-order element is derived from the strain energy consideration. The cubic shape functions in the surface co-ordinates and linear shape functions through the shell thickness are used. The second derivatives of displacement components are continuous at any point on the exterior surface of the element, although they may not be continuous across the nodal line of two adjacent elements.     

A general finite element for beams or beam-columns with or without an elastic foundation

A general finite element is derived for beams or beam-columns with or without a continuous Winkler type elastic foundation. The need to discretize members into shorter elements for convergence towards an ‘exact’ solution is eliminated by employing in the derivation of the element exact shape functions obtained from the equation of the elastic line. Inter-nodal values of deflections, bending moments and shear forces are obtained using the exact shape functions and trigonometric series. The effect of heavy compressive or tensile axial forces on bending stiffness is treated as a linear problem by considering the axial force as a constant parameter affecting the stiffness. FORTRAN subroutines to compute the stiffness matrix, equivalent nodal forces, deflected shape, bending moments and shear forces are provided and verified by an example.     

A finite element for edge-cracked beam columns

The stiffness matrix is derived for a finite element representing a beam column with rectangular cross section and a single edge crack. The element has zero length, and the standard nodal degrees of freedom associated with beam-column elements. To illustrate its capabilities, the element is used to model propagation of multiple cracks in a self-loaded fixed beam.     

Stress analysis of flat plates with shear using explicit stiffness matrix

An explicit expression for the stiffness matrix is worked out for a triangular plate bending element considering the effect of transverse shear deformation. The element has twelve nodes on the sides and four nodes internal to it. The formulation is displacement type and the use of area co-ordinates makes it possible to obtain the shape functions explicitly. Separate polynomials are assumed for transverse displacement and rotations. To obtain the element stiffness matrix no matrix inversion or numerical integration need be carried out and only a few matrix multiplications of low order are necessary. The element, which is initially of thirty five degrees of freedom, can be reduced to a thirty degrees of freedom one by condensation of the internal nodes. An interesting feature of the element developed is that the values of nodal moments computed at a node point, considering different elements surrounding the node, do not vary significantly. Thus the nodal moments can be obtained directly at node points. Also, the element does not give rise to any inconvenience like locking, even for very thin plates. The straightforward approach in formation of the element stiffness will cut down the storage space considerably and will also call for less CPU time, thus making the use of the element well suited to low capacity computers. A number of plate bending problems have been worked out using the present element for different thickness to side ratios and a comparison has been made with the available results. Good accuracy has been observed in all cases, even for a small number of elements.     

Smoothing techniques for certain primitive variable solutions of the Navier–Stokes equations

Several types of smoothing technique are considered which generate continuous approximation (i.e. nodal values) for vorticity and pressure from finite element solutions of the Navier–Stokes equations using quadrilateral elements. The simpler schemes are based on combinations of linear extrapolation and/or averaging algorithms which convert elementwise. Gauss point values to nodal point values. More complicated schemes, based on a global smoothing technique which employ the mass matrix (consistent or lumped), are also presented. An initial assessment of the accuracy of the several schemes is obtained by comparing the approximate vorticities with an analytical function. Next, qualitative vorticity comparisons are made from numerical solutions of the steady-state driven cavity problem. Finally, applications of smoothing techniques to discontinuous pressure fields are demonstrated.     

An engineers' defence of the patch test

This convergence criterion is proved for any ‘legalizable’ finite element formulation, i.e. any model for which a conforming displacement version is possible, using the same nodal variables. Engineers may therefore take control of (and responsibility for) convergence, for most of the elements in service today. Stummel's counter-example is explained as a technical misunderstanding, due to inadequate documentation of the patch test. It is suggested, without proof, that the test (properly applied) is universal, provided that it is always combined with an adequate test of stability.     

Free vibration analysis of arches using curved beam elements

The natural frequencies and mode shapes for the radial (in‐plane) bending vibrations of the uniform circular arches were investigated by means of the finite arch (curved beam) elements. Instead of the complicated explicit shape functions of the arch element given by the existing literature, the simple implicit shape functions associated with the tangential, radial (or normal) and rotational displacements of the arch element were derived and presented in matrix form. Based on the relationship between the nodal forces and the nodal displacements of a two‐node six‐degree‐of‐freedom arch element, the elemental stiffness matrix was derived, and based on the equation of kinetic energy and the implicit shape functions of an arch element the elemental consistent mass matrix with rotary inertia effect considered was obtained. Assembly of the foregoing elemental property matrices yields the overall stiffness and mass matrices of the complete curved beam. The standard techniques were used to determine the natural frequencies and mode shapes for the curved beam with various boundary conditions and subtended angles. In addition to the typical circular arches with constant curvatures, a hybrid beam constructed by using an arch segment connected with a straight beam segment at each of its two ends was also studied. For simplicity, a lumped mass model for the arch element was also presented. All numerical results were compared with the existing literature or those obtained from the finite element method based on the conventional straight beam element and good agreements were achieved. Copyright © 2003 John Wiley & Sons, Ltd.     

Hybrid-Trefftz six-node triangular finite element models for Helmholtz problem

In this paper, six-node hybrid-Trefftz triangular finite element models which can readily be incorporated into the standard finite element program framework in the form of additional element subroutines are devised via a hybrid variational principle for Helmholtz problem. In these elements, domain and boundary variables are independently assumed. The former is truncated from the Trefftz solution sets and the latter is obtained by the standard polynomial-based nodal interpolation. The equality of the two variables are enforced along the element boundary. Both the plane-wave solutions and Bessel solutions are employed to construct the domain variable. For full rankness of the element matrix, a minimal of six domain modes are required. By using local coordinates and directions, rank sufficient and invariant elements with six plane-wave modes, six Bessel solution modes and seven Bessel solution modes are devised. Numerical studies indicate that the hybrid-Trefftz elements are typically 50% less erroneous than their continuous Galerkin element counterpart.     

A method for dynamic crack and shear band propagation with phantom nodes

A new method for modelling of arbitrary dynamic crack and shear band propagation is presented. We show that by a rearrangement of the extended finite element basis and the nodal degrees of freedom, the discontinuity can be described by superposed elements and phantom nodes. Cracks are treated by adding phantom nodes and superposing elements on the original mesh. Shear bands are treated by adding phantom degrees of freedom. The proposed method simplifies the treatment of element‐by‐element crack and shear band propagation in explicit methods. A quadrature method for 4‐node quadrilaterals is proposed based on a single quadrature point and hourglass control. The proposed method provides consistent history variables because it does not use a subdomain integration scheme for the discontinuous integrand. Numerical examples for dynamic crack and shear band propagation are provided to demonstrate the effectiveness and robustness of the proposed method. Copyright © 2006 John Wiley & Sons, Ltd.     

On the eigenvalues of a uniform cantilever beam carrying any number of spring–damper–mass systems

The eigenvalues of a uniform cantilever beam carrying any number of spring–damper–mass systems with arbitrary magnitudes and locations were determined by means of the analytical‐and‐numerical‐combined method (ANCM). First of all, each spring–damper–mass system was replaced by a massless effective spring with spring constant k_eff, which is the main point that the ANCM is available for the present problem. Next, the equation of motion for the ‘constrained’ beam (with spring–damper–mass systems attached) was derived by using the natural frequencies and normal mode shapes of the ‘unconstrained’ beam (without carrying any attachments) incorporated with the expansion theorem. Finally, the equation of motion for the ‘constrained’ beam in ‘complex form’ is separated into the real and the imaginary parts. From either part, a set of simultaneous equations were obtained. Since the simultaneous equations are in ‘real form’, the eigenvalues of the ‘constrained’ beam were determined with the conventional numerical methods. To confirm the reliability of the presented theory, all the numerical results obtained from the ANCM were compared with the corresponding ones obtained from the conventional finite element method (FEM) and good agreement was achieved. Because the order of the property matrices for the equation of motion derived by using the ANCM is much lower than that by using the conventional FEM, the storing memory and the CPU time required by the ANCM are much less than those required by the FEM. Besides, the solution of the equation of motion derived from the ANCM can always be obtained with the general personal computers, but that from the FEM can sometimes be obtained only with the computers of workstations or main frames when the total degrees of freedom exceeding a certain limit. Copyright © 1999 John Wiley & Sons, Ltd.     

Improvement of PUFEM for the numerical solution of high‐frequency elastic wave scattering on unstructured triangular mesh grids

The purpose of this paper, which builds on previous work (Int. J. Numer. Meth. Engng 2009; 77 :1646–1669), is to improve a numerical scheme based on the partition of unity finite element method (PUFEM) for the solution of the time harmonic elastic wave equations. The approach consists to approximate the displacement field by the standard finite element shape functions, enriched locally by superimposing pressure (P) and shear (S) plane waves. The aim is to accurately model two‐dimensional elastic wave problems on relatively coarse mesh grids, capable of containing many wavelengths per nodal spacing, for wide ranges of frequencies. This allows us to relax the traditional requirement of about 10 nodal points per S wavelength. In this work, an exact integration scheme for the linear triangular finite element is developed to evaluate the oscillatory integrals arising from the use of the PUFEM. The main contribution here consists in developing an explicit closed‐form solution for two‐dimensional wave‐based integrals, when the phase variation is linear in the local coordinate element system. The evaluation of the element mass matrix is performed from appropriate edge integrals. All other element matrices, obtained by adequate splitting of the element stress tensor matrix, are simply deduced from the element mass matrix entries. The results show clearly that the proposed integration scheme evaluates accurately the entries of the global matrix with drastic reduction of the computational time. Numerical tests dealing with the scattering of S elastic plane waves by a circular rigid body show that, for the same discretization level, it is possible to improve the accuracy by using large elements associated with high numbers of approximating plane waves rather than using small elements with less plane waves. However, this increases the conditioning and the fill‐in of the global matrix. At high frequency, it is even possible to push the number of degrees of freedom per S wavelength under 2 and still achieve good accuracy. Finally, some remarks on the choice of the numbers of P and S plane waves leading to better accuracy and conditioning are discussed. Copyright © 2010 John Wiley & Sons, Ltd.     

The geometric stiffness of thick shell triangular finite elements for large rotations

This paper is concerned with the development of the geometric stiffness matrix of thick shell finite elements for geometrically nonlinear analysis of the Newton type. A linear shell element that is comprised of the constant stress triangular membrane element and the triangular discrete Kirchhoff Mindlin theory (DKMT) plate element is ‘upgraded’ to become a geometrically nonlinear thick shell finite element. Perturbation methods are used to derive the geometric stiffness matrix from the gradient, in global coordinates, of the nodal force vector when stresses are kept fixed. The present approach follows earlier works associated with trusses, space frames and thin shells. It has the advantage of explicitness and clear physical insight. A special procedure, tailored to triangular elements is used to isolate pure rotations to enable stress recovery via linear elastic constitutive relations. Several examples are solved. The results compare well with those available in the literature. Copyright © 2005 John Wiley & Sons, Ltd.     

A note on universal matrices for triangular finite elements for the quasi-harmonic equation

An algorithm to generate universal matrices for plane triangular finite elements for the general ‘quasi-harmonic’ equation is presented. For every member of the triangle family three numerical universal matrices are obtained which are independent of the size, shape and ‘material’ properties of the element. Of these, two are basic and the third can be generated from one of these two. The element ‘stiffness’ matrix is conveniently generated by manipulating these two basic matrices taking into account the size, shape and material properties of the element in a simple manner.