A spline wavelet finite element formulation of thin plate bending

The wavelet scaling functions of spline wavelets are used to construct the displacement interpolation functions of triangular and rectangular thin plate elements. The displacement shape functions are then expressed by spline wavelet functions. A spline wavelet finite element formulation of thin plate bending is developed by using the virtual work principle. Two numerical examples have shown that the bending deflections and moments of thin plates agree well with those obtained by the differential equations and conventional elements. It is demonstrated that the current spline wavelet finite element method (FEM) can achieve a high numerical accuracy and converges fast. The proposed spline wavelet finite element formulation has a wide range of applicability since it is developed in the same way like conventional displacement-based FEM.     

A simple finite element model for elastic-plastic plate bending

A consistent finite element model for a plate is developed based on triangular elements and a piecewise-linear displacement field. The resulting generalized stresses are the average normal moments across the element interfaces. Equilibrium equations are derived for each node, and a simple constitutive equation is obtained for each generalized stress. Applications are made to some square plate problems.     

A stable bilinear element for the Reissner-Mindlin plate model

In this paper, we introduce a new quadrilateral element using isoparametric bilinear basis functions for both components of the rotation vector and the deflection. The element is a stable modification of the MITC4 element. We report on calculations with this new element, the original MITC4 and also the bilinear element, with selective reduced integration. The numerical results are in accordance with the results of the numerical analysis and they show that (i) the method with reduced integration is highly unreliable and cannot be recommended; (ii) the MITC4 performs rather well, but its instability can lead to a decrease in accuracy for the deflection and especially to an inaccurate and oscillating shear force; (iii) the drawbacks of the MITC4 are not present in the modified method.     

A powerful finite element for plate bending

A powerful finite element formulation for plate bending has been developed using a modified version of the variational method of Trefftz. The notion of a boundary has been generalized to include the interelement boundary. All boundary conditions and the interelement continuity requirements (displacements, slopes, internal forces) have been obtained as natural conditions on the generalized boundary. Coordinate functions have been constructed to satisfy the nonhomogeneous Lagrange equation locally within the elements. Singularities due to isolated loads have been properly taken into account. For practical use a general quadrilateral element has been developed and its accuracy illustrated on several numerical examples. Work is in progress to extend the formulation to anisotropic and moderately thick plates and to vibration analysis.     

Aspects of a simple triangular plate bending finite element

This paper discusses an early phase of a study whose aim is the development of economical, yet still accurate, finite elements for dynamic analyses. Transverse shear deformation is utilized to construct a nine-degree-of-freedom plate bending element. Numerical difficulties, not unexpected, occur. A device for overcoming these difficulties is applied.     

Mixed finite element models for plate bending analysis theory

The theoretical background of mixed finite element models, in general for nonlinear problems, is briefly reexamined. In the first part of the paper, several alternative “mixed” formulations for 3-D continua undergoing large elastic deformations under the action of time dependent external loading are outlined and are examined incisively. It is concluded that mixed finite element formulations, wherein the interpolants for the stress field satisfy only a part of the domain equilibrium equations, are not only consistent from a theoretical standpoint but are also preferable from an implementation point of view. In the second part of the paper, alternative variational bases for the development of thin-plate elements are presented and discussed in detail. In light of this discussion, it is concluded that the “bad press” generated in the past concerning the practical relevance of the so-called assumed stress hybrid finite element model is not justified. Moreover, the advantages of this type of elements as compared with the “assumed displacement” or alternative mixed elements are outlined.     

The “heterosis” finite element for plate bending

A new. high-accuracy, finite element for thick and thin plate bending is developed, based upon Mindlin plate theory. The element is a 9-node quadrilateral, which exhibits improved characteristics in comparison with the 8-node serendipity, or the 9-node Lagrange elements. In particular, the element stiffness possesses correct rank, and high accuracy is attainable for extremely thin plates. Due to the consistently good performance of the element, it is proposed as a candidate for inclusion in finite element programs available to the general user.     

Dynamic analysis of structures using a Reissner-Mindlin finite strip formulation

In this paper a finite strip formulation based on Reissner-Mindlin plate theory for dynamic analysis of prismatic shell type structure is presented. Detailed expressions of the relevant strip matrices for a variety of structures using the simple two node linear strip element are given. Examples of the good performance of the linear strip element for free and forced vibration analysis of plates, bridges and axisymmetric shells are presented.     

On an equilibrium finite element method for plate bending problems

A mathematical analysis of the so called «equilibrium» finite element method is carried out. Convergence of the scheme and optimal error bounds are proved.     

Experience with finite element modeling of thin plate bending

A finite element modeling technique which utilizes different elements for the boundary region and the interior region is tested for plate bending analysis. Numerical results indicate that model types designated as TYPE 1, TYPE 2 and TYPE 3 do not exhibit detrimental locking effect even for very thin plate situations. For extremely weakly constrained problems, TYPE 2A and TYPE 3A can be used for additional stabilization against possible occurrence of kinematic modes.     

Explicit formulation for a high precision triangular laminated anisotropic thin plate finite element

An efficient formulation of the stiffness matrix is presented for a high precision triangular laminated anisotropic thin plate finite element. The formulation is based on the classical lamination theory which is reviewed briefly. The stiffness matrix is obtained simply by pre and post multiplication of a few basic matrices, which are presented explicitly. It is believed that this formulation is almost an order of magnitude faster than those available for similar order elements. In addition, the present element formulation is readily applicable as a thin flat shell element. A complete listing of FORTRAN subroutines is presented for the users, to ease implementation of the algorithm.     

Polygon-circle paradox in the finite element analysis of bending of a simply supported plate

In this paper, it is shown that the so-called polygon-circle paradox in the analysis of simply supported plates is not a paradox but a consequence of the alteration of boundary conditions due to a polygonal domain-approximation. Simple ways of avoiding potential problems in the analysis of bending of simply supported curved plates, using a polygonal domain approximation, in the context of the finite element method, are presented and demonstrated to be valid.     

Mixed finite element models for plate bending analysis: A new element and its applications

With a few exceptions, finite element packages available in today's commercial software environment contain in their libraries displacement-type elements only. The present paper aims to demonstrate the feasibility that properly formulated mixed-type elements compete most favorably with displacement-type elements and should, therefore, be considered as potential candidates for inclusion in general purpose finite element packages. In doing so, the development of a new triangular doubly—curved mixed-hybrid shallow shell element and its extensive testing in carefully chosen example problems are reported on.     

Continuous and discontinuous finite element methods for a peridynamics model of mechanics

In contrast to classical partial differential equation models, the recently developed peridynamic nonlocal continuum model for solid mechanics is an integro-differential equation that does not involve spatial derivatives of the displacement field. As a result, the peridynamic model admits solutions having jump discontinuities so that it has been successfully applied to fracture problems. The peridynamic model features a horizon which is a length scale that determines the extent of the nonlocal interactions. Based on a variational formulation, continuous and discontinuous Galerkin finite element methods are developed for the peridynamic model. Discontinuous discretizations are conforming for the model without the need to account for fluxes across element edges. Through a series of simple, one-dimensional computational experiments, we investigate the convergence behavior of the finite element approximations and compare the results with theoretical estimates. One issue addressed is the effect of the relative sizes of the horizon and the grid. For problems with smooth solutions, we find that continuous and discontinuous piecewise-linear approximations result in the same accuracy as that obtained by continuous piecewise-linear approximations for classical models. Piecewise-constant approximations are less robust and require the grid size to be small with respect to the horizon. We then study problems having solutions containing jump discontinuities for which we find that continuous approximations are not appropriate whereas discontinuous approximations can result in the same convergence behavior as that seen for smooth solutions. In case a grid point is placed at the locations of the jump discontinuities, such results are directly obtained. In the general case, we show that such results can be obtained through a simple, automated, abrupt, local refinement of elements containing the discontinuity. In order to reduce the number of degrees of freedom while preserving accuracy, we also briefly consider a hybrid discretization which combines continuous discretizations in regions where the solution is smooth with discontinuous discretizations in small regions surrounding the jump discontinuities.     

Least-squares finite element formulation for shear-deformable shells

We present a least-squares based finite element formulation for the numerical analysis of shear-deformable shell structures. The variational problem is obtained by minimizing the least-squares functional, defined as the sum of the squares of the shell equilibrium equations residuals measured in suitable norms of Hilbert spaces. The use of least-squares principles leads to a variational unconstrained minimization problem where compatibility conditions between approximation spaces never arise, i.e. stability requirements such as inf–sup conditions never arise. The proposed formulation retains the generalized displacements and stress resultants as independent variables and, in view of the nature of the variational setting upon which the finite element model is built, allows for equal-order interpolation. A p-type hierarchical basis is used to construct the discrete finite element model based on the least-squares formulation. Exponentially fast decay of the least-squares functional is verified for increasing order of the modal expansions. Several well established benchmark problems are solved to demonstrate the predictive capability of the least-squares based shell elements. Shell elements based on this formulation are shown to be effective in both membrane- and bending-dominated states.     

An improved rectangular element for plate bending analysis

This paper presents a new, simple, rectangular finite element with twelve degrees of freedom for the bending analysis of thin plates. Three interpolation functions corresponding to the normal deflections and tangential slopes at the nodal points are written in parametric form. Convergence requirements are then used to find relationships among the parameters included in these functions. To identify the optimal values of the still undetermined parameters extensive comparisons are carried out using plate problems with different loading and boundary conditions. Certain values of the unknown parameters are found to produce displacement results with faster rate of convergence than those of other simple elements. When comparisons are based on a measure representing the actual computational effort rather than the mesh size the proposed element is found to excell higher-order elements as well. Stress results are also calculated for the proposed element and found to be fairly close to the exact values.     

Multivariable spline element analysis for plate bending problems

A multivariable spline element analysis for a plate bending problem is presented. The bicubic spline functions are used to construct the bending moments and transverse displacements field. The spline element equations with multiple variables are derived based on Hellinger-Reissner principle. Some numerical results are given and compared with other methods.     

Stabilized finite element formulation for elastic–plastic finite deformations

This paper presents a stabilized finite element formulation for nearly incompressible finite deformations in hyperelastic–plastic solids, such as metals. An updated Lagrangian finite element formulation is developed where mesh dependent terms are added to enhance the stability of the mixed finite element formulation. This formulation circumvents the restriction on the displacement and pressure fields due to the Babuška–Brezzi condition and provides freedom in choosing interpolation functions in the incompressible or nearly incompressible limit, typical in metal forming applications. Moreover, it facilitates the use of low order simplex elements (i.e. P1/P1), reducing the degrees of freedom required for the solution in the incompressible limit when stable elements are necessary. Linearization of the weak form is derived for implementation into a finite element code. Numerical experiments with P1/P1 elements show that the method is effective in incompressible conditions and can be advantageous in metal forming analysis.     

On a physically stabilized one point finite element formulation for three-dimensional finite elasto-plasticity

In the case of linear elasticity, a direct connection between the concept of reduced integration with hourglass stabilization and a mixed method can usually be established. In the non-linear case, this is in general not possible. To overcome this difficulty we suggest in this paper a new concept based on a Taylor expansion of the constitutively dependent quantities with respect to the centre of the element. The push-forward of the second (linear) term of the Taylor series for the first Piola–Kirchhoff stress tensor to the current configuration determines the so-called hourglass stabilization part of the residual force vector. Due to the fact that the element uses only one Gauss point and the hourglass stabilization part is computed by means of a simple functional evaluation, the present element technology is very efficient from the computational point of view.In contrast to the 2D case the computation of the Jacobi determinant only in the centre of the 3D element does not yield the correct volume, if the element shape deviates from being a parallelipiped. It is shown in the paper that the error becomes negligibly small for a relatively coarse discretization. The formulation is free of volumetric locking and can compete with shell formulations up to an aspect ratio of about hundred. For bending-dominated problems, at least two elements over the thickness are needed in order to compute the onset of plastification correctly. The element behaves very robustly in finite elasticity and inelasticity, also when large element distortions occur.