Solving boundary-value problems using hp-version finite elements in time

An hp-version finite element method for one-dimensional boundary value problems is presented. The method is based on a similar approach developed by the authors for solution of optimal control problems. The primary applications for the methodology include two-point- and multi-point-boundary-value problems, for example, in the time domain. Results presented for a 7-state/3-phase missile problem show that the method is very efficient for time-marching applications. Furthermore, it easily solves time-domain problems with discontinuities in the system equations and/or in the states, where the time at which these jumps (i.e. ‘events’) take place is determined by equations that govern the states. An example involving friction with intermittent sticking is presented to illustrate the power of the method. © 1998 John Wiley & Sons, Ltd.     

Triangular and tetrahedral space-time finite elements in vibration analysis

Recently, several methods of time integration of the equations of motion have been proposed. Many of them result in square mass, damping and stiffness matrices. The space–time finite element method is an extension of the FEM, familiar to most engineers, over the time domain. A special approach enables the use of triangular, tetrahedral and hyper-tetrahedral elements in time and space. By special division of the space–time layer the triangular matrix of coefficients in the system of equations can be obtained. A simple algorithm enables the storage of non-zero coefficients only. Dynamic solution requires a small amount of the memory compared to other methods, and ensures considerable reduction of arithmetic operations. The method presented is also efficient in solving both linear and non-linear problems. Matrices for a beam and plane stress/strain element are derived. Exemplary problems solved by the method described have proved the effectiveness of the application of triangular and tetrahedral space–time elements in vibration analysis.     

On the use of space-time finite elements in the solution of elasto-dynamic fracture problems

The use of a discontinuous Galerkin (DG) formulation for the solution of dynamic fracture problems in linear elastic media with and without cohesive zones is explored. The results are compared with closed-form as well as numerical solutions available from the literature. The effectiveness of the space-time finite element method in the study of dynamic fracture problems is demonstrated, especially in those cases in which dynamic fracture occurs along with time discontinuous loading.     

An implementation of the hp-version of the finite element method for Reissner-Mindlin plate problems

Reissner-Mindlin plate theory is still a topic of research in finite element analysis. One reason for the continuous development of new plate elements is that it is still difficult to construct elements which are accurate and stable against the well-known shear locking effect. In this paper we suggest an approach which allows high order polynomial degrees of the shape functions for deflection and rotations. A balanced adaptive mesh-refinement and increase of the polynomial degree in an hp-version finite element program is presented and it is shown in numerical examples that the results are highly accurate and that high order elements show virtually no shear locking even for very small plate thickness.     

An adaptive hp-version of the finite element method applied to flame propagation problems

This paper describes an adaptive hp-version mesh refinement strategy and its application to the finite element solution of one-dimensional flame propagation problems. The aim is to control the spatial and time discretization errors below a prescribed error tolerance at all time levels. In the algorithm, the optimal time step is first determined in an adaptive manner by considering the variation of the computable error in the reaction zone. Later, the method uses a p-version refinement till the computable a posteriori error is brought down below the tolerance. During the p-version, if the maximum allowable degree of approximation is reached in some elements of the mesh without satisfying the global error tolerance criterion, then conversion from p- to h-version is performed. In the conversion procedure, a gradient based non-uniform h-version refinement has been introduced in the elements of higher degree approximation. In this way, p-version and h-version approaches are used alternately till the a posteriori error criteria are satisfied. The mesh refinement is based on the element error indicators, according to a statistical error equi-distribution procedure. Numerical simulations have been carried out for a linear parabolic problem and premixed flame propagation in one-space dimension. © 1997 John Wiley & Sons, Ltd.     

An expert system for the optimal mesh design in the hp-version of the finite element method

This paper suggests a simple expert system frame and provides the domain knowledge for the optimal mesh design and the prediction of the error in the energy norm for the problem of plane elasticity using the hp-extension in the finite element method. The expert system monitors the progress of the analysis, guides the user through the various steps and is able to reason about its own advice. In an example the user–expert communication is shown and the superiority of the results is demonstrated.     

Polygonal finite elements for finite elasticity

Nonlinear elastic materials are of great engineering interest, but challenging to model with standard finite elements. The challenges arise because nonlinear elastic materials are characterized by non‐convex stored‐energy functions as a result of their ability to undergo large reversible deformations, are incompressible or nearly incompressible, and often times possess complex microstructures. In this work, we propose and explore an alternative approach to model finite elasticity problems in two dimensions by using polygonal discretizations. We present both lower order displacement‐based and mixed polygonal finite element approximations, the latter of which consist of a piecewise constant pressure field and a linearly‐complete displacement field at the element level. Through numerical studies, the mixed polygonal finite elements are shown to be stable and convergent. For demonstration purposes, we deploy the proposed polygonal discretization to study the nonlinear elastic response of rubber filled with random and periodic distributions of rigid particles, as well as the development of cavitation instabilities in elastomers containing vacuous defects. These physically‐based examples illustrate the potential of polygonal finite elements in studying and modeling nonlinear elastic materials with complex microstructures under finite deformations. Copyright © 2014 John Wiley & Sons, Ltd.     

Dynamic response in the time domain by coupled boundary and finite elements

This paper reports on the development of a finite element — boundary element coupling procedure for the analysis of arbitrary shaped elastic bodies subjected to dynamic loads. The coupling is accomplished through equilibrium and compatibility considerations along the boundary element — finite element interface.Several numerical studies are performed where one part of a uniform body is treated by finite elements, whereas the remaining region is descretized by boundary elements. The examples demonstrate the influence of different finite element approaches and the applicability and the accuracy of the proposed procedure.     

Tetrahedral polynomial finite elements for the Helmholtz equation

It is shown that the Helmholtz equation in three dimension leads to finite element approximations on tetrahedral elements that closely resemble the corresponding two-dimensional treatment on triangle. For each polynomial order, there exist two numeric universal matrices independent to tetrahedron size and shape; the element matrices are always given as linear combinations of row and column permutations of these. Numeric matrices are given up to third-order, and the permutation schemes are shown in detail. Experimental computer programs using these elements have shown fast matrix assembly times; convergence rates are essentially similar to those obtained with the corresponding triangular elements.     

Permutation rules for simplex finite elements

Simplicial finite elements have many symmetries, which are commonly exploited in programs to save computing time and storage. All possible permutations of the interpolation functions of an N-simplex can be expressed in terms of N basic permutation operations. The relevant permutation matrices are given for triangles up to order 20 and tetrahedra up to order 10.     

Forbidden nodes for curved finite elements

It is shown here that certain interpolating polynomials of degrees four and five may not always be uniquely defined on triangular-shaped elements which have one curved side. Conditions which indicate non-uniqueness are given, together with some geometrical interpretations concerning the location of the node on the curved side. A numerical example is given to demonstrate that there ar curves for which every point is unsuitable to be chosen as a node.     

Anisotropic mixed finite elements for elasticity

In this paper, we present a family of mixed finite elements, which are suitable for the discretization of slim domains. The displacement space is chosen as Nédélec's space of tangential continuous elements, whereas the stress is approximated by normal–normal continuous symmetric tensor‐valued finite elements. We show stability of the system on a slim domain discretized by a tensor product mesh, where the constant of stability does not depend on the aspect ratio of the discretization. We give interpolation operators for the finite element spaces, and thereby obtain optimal order a priori error estimates for the approximate solution. All estimates are independent of the aspect ratio of the finite elements. Copyright © 2011 John Wiley & Sons, Ltd.     

Mixing finite elements and finite differences for the analysis ofsilicon on sapphire

The analysis of electron devices involves the solution of the Poisson equation, made nonlinear as a result of the charge cloud density being a function of the unknown potential. Because of the nonlinear term, the finite-element scheme becomes time consuming. However, the finite difference method, which is shown to model the term Δ² more accurately, cannot handle line charge densities and the rapid variation in the source term near such charge accumulations. It is shown that there is much to be gained by mixing the two methods in analyzing electron devices, and the method is demonstrated for a silicon-on-sapphire device     

Crypto-DOF finite elements

New finite elements, called crypto-DOF elements, are introduced, utilizing degrees of freedom that satisfy the differential equation but do not appear explicitly in the discrete equations. The crypto-DOF elements can be advantageous in adaptive finite element techniques, and to approximate singular-type eigensolutions (e.g. in convective diffusion or stress concentration problems) without altering the original mesh or introducing new degrees of freedom. In this paper the crypto-DOF elements, the construction of crypto-DOF element families and selected numerical results are presented.     

Polytope finite elements

In this paper, finite elements based on arbitrary convex and non‐convex polytopes are introduced. Polytopes in combination with natural element coordinates (NECs) permit a uniform element formulation of interpolation functions that are independent of the dimension of space, localization and the number of vertices. NECs based on the natural neighbor interpolation are restricted to the polytope and can be understood as an extension of the barycentric coordinates on simplexes. The differentiation and integration of these interpolation functions on the basis of NECs is essential for finite element approximations. The accuracy of the finite element interpolation or approximation can be controlled by either applying the h‐version or by utilizing the p‐version of the finite element method (FEM). Advantages in the handling of hanging nodes are discussed. Furthermore, we present construction methods for Lagrangian as well as for hierarchical interpolation functions based on NECs. Numerical experiments on different convex and non‐convex decompositions will show the usability, accuracy and convergence of the developed polytope FEM. Copyright © 2007 John Wiley & Sons, Ltd.     

Hybrid finite elements for the computation of sandwich plates

In this article, new hybrid finite elements are developed on the basis of the displacements and the Pian and Tong functionals using Lagrange multipliers in order to compute correctly and efficiently interlaminar stresses in sandwich structures. These elements represent the mechanical behaviour of sandwich structures in an accurate way, especially at interfaces, where the force equilibrium state must be ensured. They permit to obtain the values of interlaminar stresses using a coarse mesh through the thickness of the sandwich structure. These hybrid elements are assessed and compared through several examples of static linear problems with solutions found in the literature. Copyright © 2001 John Wiley & Sons, Ltd.     

Least-squares finite elements for first-order hyperbolic systems

A class of least-squares finite element methods has been developed for first-order systems and here we study this approach for hyperbolic problems. The formulation of the least-squares method is developed in detail and compared with the Petrov-Galerkin and Taylor-Galerkin procedures. A stability analysis is carried out and the extension to the non-linear problem described. Numerical comparison studies demonstrate the performance of the method and suggest that it is a promising alternative to existing schemes. Applications considered include the convection equation, inviscid Burger's equation and shallow-water equations.     

A drafting program for isoparametric finite elements

Considerable effort has been invested lately in the application of isoparametric finite elements for numerical solution of a wide range of applied mechanics problems. In fact, several general purpose computer programs are now available which are based upon such finite elements. In the present paper, a new application of the isoparametric finite element concept is introduced which significantly extends its usefulness for many practical structural configurations. In this application, final working or architectural drawings of the structure are made from the same (or similar) finite element model as was utilized in a structural integrity analysis. The hardware necessary to produce such drawing, a computer driven plotter or automated drafting machine, is available commercially or through most data centres, and the software concepts required are described herein.     

MITC finite elements for laminated composite plates

Within the framework of the first‐order shear deformation theory, 4‐ and 9‐node elements for the analysis of laminated composite plates are derived from the MITC family developed by Bathe and coworkers. To this end the bases of the MITC formulation are illustrated and suitably extended to incorporate the laminate theory. The proposed elements are locking‐free, they do not have zero‐energy modes and provide accurate in‐plane deformations. Two consecutive regularizations of the extensional and flexural strain fields and the correction of the resulting out‐of‐plane stress profiles necessary to enforce exact fulfillment of the boundary conditions are shown to yield very satisfactory results in terms of transverse and normal stresses. The features of the proposed elements are assessed through several numerical examples, either for regular and highly distorted meshes. Comparisons with analytical solutions are also shown. Copyright © 2001 John Wiley & Sons, Ltd.