A new mixed finite element based on different approximations of the minors of deformation tensors

Finite element formulations for arbitrary hyperelastic strain energy functions that are characterized by a locking-free behavior for incompressible materials, a good bending performance and accurate solutions for coarse meshes need still attention. Therefore, the main goal of this contribution is to provide an improved mixed finite element for quasi-incompressible finite elasticity. Based on the knowledge that the minors of the deformation gradient play a major role for the transformation of infinitesimal line-, area- and volume elements, as well as in the formulation of polyconvex strain energy functions a mixed finite element with different interpolation orders of the terms related to the minors is developed. Due to the formulation it is possible to condensate the mixed element formulation at element level to a pure displacement form. Examples show the performance and robustness of the element.     

FEM considering local bending effects for flat sliding isolators

Presented in this paper are an advanced analytical model and finite element formulations including the local bending moment effect for the flat sliding system. This study reveals that the local bending effect is a very important factor. The results presented in this paper show that the design of the isolation without considering this effect may endanger structures during earthquakes. The new formulations taking into consideration the local effect provide an effective tool for the more realistic simulation of the behavior of flat-sliding-isolated structures subjected to earthquake loadings. The proposed finite element formulations can be applied directly for both two- and three-dimensional analyses without any further assumption, and can generally be placed at any location in the structure. The implementation of the proposed techniques in existing finite element computer codes is a simple task. Examples of structures equipped with flat-sliding isolators is also given to demonstrate the paramount findings through the proposed formulations. Results obtained from the numerical analyses suggest that the local bending moment effects are extremely important, and should be taken into account to assure the safety of isolated-structures during earthquakes.     

On various formulations of large amplitude free vibrations of beams

Various finite element formulations of large amplitude free vibrations of beams with immovably supported ends are discussed in this paper. Analytical formulation based on the Rayleigh-Ritz method is also presented. Numerical results of the analytical approach are seen to be in good agreement with some of these finite element formulations. Mixed finite element formulations based on two methods are derived to study the large amplitude free vibrations of beams. The mixed finite element methods also show good agreement with the analytical and the above finite element formulations. Various points of view raised from time to time on the applicability of these formulations can now be clarified through these formulations and the numerical results. The weakness of the so-called improved Ritz-type finite element model in predicting the nonlinear frequency ratio is highlighted through various results of the above formulations. As a typical example, a hinged-hinged beam on immovable ends is considered for all the above formulations and the nonlinear frequencies show a good agreement amongst themselves at all amplitude levels.     

Stabilised finite element methods for a bending moment formulation of the Reissner-Mindlin plate model

This work presents new stabilised finite element methods for a bending moments formulation of the Reissner-Mindlin plate model. The introduction of the bending moment as an extra unknown leads to a new weak formulation, where the symmetry of this variable is imposed strongly in the space. This weak problem is proved to be well-posed, and stabilised Galerkin schemes for its discretisation are presented and analysed. The finite element methods are such that the bending moment tensor is sought in a finite element space constituted of piecewise linear continuos and symmetric tensors. Optimal error estimates are proved, and these findings are illustrated by representative numerical experiments.     

A geometrical nonlinear eccentric 3D-beam element with arbitrary cross-sections

In this paper a finite element formulation of eccentric space curved beams with arbitrary cross-sections is derived. Based on a Timoshenko beam kinematic, the strain measures are derived by exploitation of the Green-Lagrangean strain tensor. Thus, the formulation is conformed with existing nonlinear shell theories. Finite rotations are described by orthogonal transformations of the basis systems from the initial to the current configuration. Since for arbitrary cross-sections the centroid and shear center do not coincide, torsion bending coupling occurs in the linear as well as in the finite deformation case. The linearization of the boundary value formulation leads to a symmetric bilinear form for conservative loads. The resulting finite element model is characterized by 6 degrees of freedom at the nodes and therefore is fully compatible with existing shell elements. Since the reference curve lies arbitrarily to the line of centroids, the element can be used to model eccentric stiffener of shells with arbitrary cross-sections.     

Finite element formulations for transient dynamic problems in solids using explicit time integration

This paper presents the displacement, mixed and stress formulations of the finite element method when applied to transient dynamic problems of solids. The formulations are chosen so that explicit time integration may be used. Large deformations are considered for these formulations, and infinitesimal strain assumptions are employed with the stress formulation. Displacement formulations are well-known, but the mixed formulations presented provide a viable alternative. The stress formulation has not proven successful for the large deformation problem, but when infinitesimal strains are assumed, the formulation is attractive. A problem of an internally pressurized ring is solved in order to evaluate the different proposed formulations.     

Finite element analysis of lateral buckling for beam structures

The application of finite element analysis to lateral buckling problems, locating the critical points and tracing the postbifurcation path, is treated on the basis of a geometrically nonlinear formulation for a beam with small elastic strain but with possibly large rotations. The existing finite element formulations for thin beams are examined in the aspect of application to bifurcation problems, such as lateral buckling, and the choice of an appropriate rotation parameter for representing incremental or variational rotations in finite element formulations is discussed in relation to locating bifurcation points. This is illustrated through several numerical examples and followed by appropriate discussion.     

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.     

Finite element formulation for the large amplitude free vibrations of beams and orthotropic circular plates

A finite element formulation for the large amplitude free oscillations of beams and orthotropic circular plates is presented in this paper. The present formulation does not need the knowledge of longitudinal/inplane forces developed due to large displacements and thus avoids the use of corresponding geometric stiffness matrices, which were used in earlier finite element formulations. The convergence of the results obtained by using the present formulation is very good. Comparison of the present results with the earlier work wherever possible confirms the reliability and effectiveness of the present finite element formulation.     

Gershgorin theory for stiffness and stability of evolution systems and convection-diffusion

An analysis of stiffness and stability based on Gershgorin theorems for eigenvalues is developed for initial value systems. In particular, semidiscrete formulations for evolution problems are analysed. Common techniques such as semidiscrete finite difference and finite element methods are examined using eigenvalue bounds to characterize stiffness and stability of the associated systems. The analysis is applied to a prototype convection-diffusion problem to demonstrate the arguments and clarify several current questions concerning the qualitative nature of the solution and errors, including effects of: “lumped” versus “consistent” finite element formulations; high- or low-degree bases; mesh refinement, dimensionality and differing material properties. To study general initial value systems such as those arising in consistent finite element formulations, a generalized Gershgorin theory and computable bounds in the chordal metric are utilized.     

A computer algebra based finite element development environment

A finite element development environment based on the technical computing program Mathematica is described. The environment is used to automatically program standard element formulations and develop new elements with novel features. Source code can also be exported in a format compatible with commercial finite element program user-element facilities. The development environment is demonstrated for three mixed Petrov–Galerkin plane stress elements: a standard formulation, an advanced formulation incorporating rotational degrees of freedom and a standard formulation in which the stiffness matrix is integrated analytically, before being exported as ANSYS user elements. The results presented illustrate the accuracy of the standard mixed formulation element and the enhancement of performance when rotational degrees of freedom are added. Further, the analytically integrated element shows that computational requirements can be greatly reduced when analytical integration schemes are used in the formation.     

High order finite elements for shells

In this article the p-version finite element method is applied to thin-walled structures. Two different hierarchic element formulations are compared, a shell approach as well as a shell-like, solid formulation. Both approaches are compared for linear elastic and elastoplastic problems. Special emphasis is placed on the efficiency as well as on determining the area of application for both formulations.     

Frame element with lateral deformable supports: Formulations and numerical validation

This paper presents the theory and the numerical validation of three different formulations of nonlinear frame elements with nonlinear lateral deformable supports. The governing differential equations of the problem are derived first and the three different finite element formulations are then presented. The first model follows a displacement-based formulation, which is based on the virtual displacement principle. The second one follows the force-based formulation, which is based on the virtual force principle. The third model follows the Hellinger–Reissner mixed formulation, which is based on the two-field mixed variational principle. The selection of the displacement and force interpolation functions for the different formulations is discussed. Tonti’s diagrams are used to conveniently represent the equations governing both the strong and the weak forms of the problem. The general matrix equations of the three formulations are presented, with some details on the issues regarding the elements’ implementations in a general-purpose finite element program. The convergence, accuracy, and computational times of the three elements are studied through a numerical example. The distinctive element characteristics in terms of force and deformation discontinuities between adjacent elements are discussed. The capability of the proposed frame models to trace the softening response due to softening of the foundation is also investigated. Overall, the force-based and the mixed models are much more accurate than the displacement-based model and require very few elements to reach the converged solution. The force-based element is slightly more accurate than the mixed model, but it is more prone to numerical instabilities as it involves inverting the element flexibility matrix.     

Discussion of finite element formulations of nonlinear beam vibrations

The Lagrange-type, Galerkin, and Ritz-type finite element formulations for large amplitude vibrations of immovably supported slender beams are reexamined. Inconsistency in the definition of frequency or criterion of defining nonlinearity is discussed, and validity of the frequency solution is examined. Improved finite element results by including both longitudinal displacement and inertia in the formulation are presented and compared with available Rayleigh-Ritz continuum solutions.     

A boundary element analysis of symmetric laminated composite shallow shells

This paper presents a boundary element formulation for the analysis of symmetric laminated composite shallow shells where only the boundary is discretized. Classical plate bending and plane elasticity formulations are coupled and effects of curvature are treated as body forces. Fundamental solutions for elastostatic formulations are used and body forces are written as a sum of approximation functions multiplied by unknown coefficients. Two approximation functions are used. Domain integrals which arise in the formulation are transformed into boundary integrals by the radial integration method. Results for the approximation functions are compared and the accuracy of the proposed formulation is assessed by results from literature. It was shown that results obtained with the approximation function called augmented thin plate spline present very good agreement with literature even for shells that are not so shallow.     

On the Coupling of Local Discontinuous Galerkin and Conforming Finite Element Methods

The finite element formulation resulting from coupling the local discontinuous Galerkin method with a standard conforming finite element method for elliptic problems is analyzed. The transmission conditions across the interface separating the subdomains where the different formulations are applied are taken into account by a suitable definition of the so-called numerical fluxes. An error analysis leading to optimal a priori error estimates is presented for arbitrary meshes with possible hanging nodes. Numerical experiments validating the theoretical results are reported.     

Vibration of thin cylindrical shells based on a mixed finite element formulation

A Hellinger-Reissner functional for thin circular cylindrical shells is presented. A mixed finite element formulation is developed from this functional, which is free from line integrals and relaxed continuity terms. This element is applied to the problem of vibration of rectangular cylindrical shells. Bilinear trial functions are used for all field variables. The element satisfies the compatibility and completeness requirements. The numerical results obtained in this work show that convergence is quite rapid and monotonic for a much smaller number of degrees of freedom than other finite element formulations.     

Finite element analysis of shell structures

Summary A survey of effective finite element formulations for the analysis of shell structures is presented. First, the basic requirements for shell elements are discussed, in which it is emphasized that generality and reliability are most important items. A general displacement-based formulation is then briefly reviewed. This formulation is not effective, but it is used as a starting point for developing a general and effective approach using the mixed interpolation of the tensorial components. The formulation of various MITC elements (that is, elements based on Mixed Interpolation of Tensorial Components) are presented. Theoretical results (applicable to plate analysis) and various numerical results of analyses of plates and shells are summarized. These illustrate some current capabilities and the potential for further finite element developments.