Buckling of laminated plates - A simple finite element based on higher-order theory

A four-noded rectangular element with seven degrees of freedom at each node is developed for buckling analysis of laminated plate structures having any number of layers with a constant thickness of individual layers. The displacement model is so chosen that it can explain adequately the parabolic distribution of transverse shear stresses and the non-linearity of in-plane displacements across the thickness. A geometrical stiffness matrix is developed using in-plane stresses. A wide range of plates from thick to thin are examined under uniaxial loading conditions. The results are compared with the existing analytical and numerical solutions. The present formulations confirm its applicability for buckling analysis of a wide range of plates.     

Estimation of temperature in rubber-like materials using non-linear finite element analysis based on strain history

A finite element procedure for hyper-elastic materials such as rubber has been developed to estimate the temperature rise during cyclic loading. The irreversible mechanical work developed in rubber has been used to determine the heat generation rate for carrying out thermal analysis. The evaluation of the heat energy is dependent on the strains. The finite element analysis assumes Green–Lagrangian strain displacement relations, Mooney–Rivlin strain energy density function for constitutive relationship, incremental equilibrium equations, and Total Lagrangian approach and the stress and strain of the rubber-like materials are evaluated using a degenerated shell element with assumed strain field technique, considering both material and geometric non-linearities. A transient heat conduction analysis has been carried out to estimate the temperature rise for different time steps in rubber-like materials using Galerkin's formulations. A numerical example is presented and the computed temperature values for various load steps agree closely with the experimental results reported in the literature.     

A finite element method for the solution of a potential theory integral equation

This paper discusses a finite element approximation for an integral equation of the second kind deduced from a potential theory boundary value problem in two variables. The equation is shown to admit a unique solution, to be variational and coercive in the Hilbert space of functions σ ε H^1/2(Γ), f_rd γ = 0. The Galerkin method with finite elements as trial functions is shown to lead to an optimal rate of convergence.     

An extended beam theory for smart materials applications part I: Extended beam models,duality theory,and finite element simulations

An extended theory for elastic and plastic beam problems is studied. By introducing new dependent and independent variables, the standard Timoshenko beam model is extended to take account of shear variation in the lateral direction. The dynamic governing equations are established via Hamilton's principle, and existence and uniqueness results for the solution of the static problem are proved. Using the theory of convex analysis, the duality theory for the extended beam model is developed. Moreover, the extended theory for rigid-perfectly plastic beams is also established. Based on the extended model, a finite-element method is proposed and numerical results are obtained indicating the usefulness of the extended theory in applications.The work of the first author was supported in part by National Science Foundation under Grant DMS9400565.     

A finite element model of temperature distribution in the human torso

A finite element model is developed for the calculation of steady state temperature distribution throughout the human torso. The torso is considered as a cylinder and the differential equations of the model are expressed in their equivalent variational form. The solution is approximated using a rational finite element basis which fully exploits symmetry.     

Modeling the nonlinear deformation of composite laminates based on plasticity theory

The plasticity theory has been successfully used for describing the nonlinear deformation of laminated composite materials under a monotonically increasing loading. Generally, several tests are needed to determine the parameters of the plastic potential for a laminate. We explore an alternative approach and obtain the plastic potential by using theoretical considerations based on a laminate analysis. The model is shown to provide an accurate prediction for the response of a cross-ply glass/epoxy laminate under uniaxial tensile loading at different angles to the material orthotropy axes. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 43, No. 3, pp. 309–318, May–June, 2007.     

A stabilized mixed finite element method for the biharmonic equation based on biorthogonal systems

We propose a stabilized finite element method for the approximation of the biharmonic equation with a clamped boundary condition. The mixed formulation of the biharmonic equation is obtained by introducing the gradient of the solution and a Lagrange multiplier as new unknowns. Working with a pair of bases forming a biorthogonal system, we can easily eliminate the gradient of the solution and the Lagrange multiplier from the saddle point system leading to a positive definite formulation. Using a superconvergence property of a gradient recovery operator, we prove an optimal a priori estimate for the finite element discretization for a class of meshes.     

A monotonicity analysis theory for the dependent eigenvalues of the vibrating systems

This paper presents and discusses the monotonicity analysis theory for the generalized eigenvalues of the nonlinear structural eigensystems. The analysis is based on investigating the mass and stiffness matrices which are associated with both the mixed and exact finite element models. These models can be distinguished by the shape functions derived from the choice of displacement field which plays a crucial role in the accuracy and efficiency of the solution. The main emphasis of this contribution is the derivation and the investigation of this analysis for large scale eigenproblems.     

A comparative analysis of adaptive algorithms in the finite element method for solving the boundary value problem for a stationary reaction-diffusion equation

A new adaptive algorithm is proposed for constructing grids in the hp-version of the finite element method with piecewise polynomial basis functions. This algorithm allows us to find a solution (with local singularities) to the boundary value problem for a one-dimensional reaction-diffusion equation and smooth the grid solution via the adaptive elimination and addition of grid nodes. This algorithm is compared to one proposed earlier that adaptively refines the grid and deletes nodes with the help of an estimate for the local effect of trial addition of new basis functions and the removal of old ones. Results are presented from numerical experiments aimed at assessing the performance of the proposed algorithm on a singularly perturbed model problem with a smooth solution.     

A new finite element model for the analysis of arbitrary stiffened shells

A new stiffened shallow shell finite element has been introduced for the static analysis of stiffened plates and shells. This approach has been presented to cater for stiffeners in which the positions and properties remain undisturbed in the formulation and the element can accomodate the stiffener anywhere within the shell element and in any direction, which introduces a considerable flexibility in the analysis. This is a distinct improvement over the existing models. Stiffened shells having various disposition of stiffeners as available in the literature, have been analysed by the proposed approach. Comparison obtained with the existing theoretical and/or experimental values have indicated good accuracy with relatively coarser mesh sizes and less CPU time.     

A novel far-boundary condition for the finite element analysis of infinite reservoir

The present paper deals with the finite element analysis of the reservoir of infinite extent using a novel far-boundary condition. The equations of motion are expressed in terms of the pressure only assuming water as inviscid and incompressible. The truncation boundary condition is developed numerically from the classical wave equation. Comparative studies show that the proposed far-boundary condition is numerically efficient and accurate over the existing ones, available in the literature. The effect of the geometry of the reservoir bed and the adjacent structure on the development hydrodynamic pressure has been studied. The results show that the geometry of the reservoir bed and as well as the adjacent structure has considerable effect on the development of hydrodynamic pressure at the dam–reservoir interface.     

A unifying theory of a posteriori error control for nonconforming finite element methods

Residual-based a posteriori error estimates were derived within one unifying framework for lowest-order conforming, nonconforming, and mixed finite element schemes in Carstensen [Numer Math 100:617–637, 2005]. Therein, the key assumption is that the conforming first-order finite element space annulates the linear and bounded residual ℓ written . That excludes particular nonconforming finite element methods (NCFEMs) on parallelograms in that . The present paper generalises the aforementioned theory to more general situations to deduce new a posteriori error estimates, also for mortar and discontinuous Galerkin methods. The key assumption is the existence of some bounded linear operator with some elementary properties. It is conjectured that the more general hypothesis (H1)–(H3) can be established for all known NCFEMs. Applications on various nonstandard finite element schemes for the Laplace, Stokes, and Navier–Lamé equations illustrate the presented unifying theory of a posteriori error control for NCFEM. Supported by DFG Research Center MATHEON “Mathematics for key technologies” in Berlin and the German Indian Project DST-DAAD (PPP-05). J. Hu was partially supported by National Science Foundation of China under Grant No.10601003.     

A partitioned finite element scheme based on Gauge-Uzawa method for time-dependent MHD equations

In this paper, we mainly introduce a partitioned scheme based on Gauge-Uzawa finite element method for the 2D time-dependent incompressible magnetohydrodynamics (MHD) equations. It is a fully decoupled projection method which combines the Gauge and Uzawa methods within a variational formulation. Firstly, the temporal discretization is based on backward Euler technique for the linear term and semi-implicit scheme for the nonlinear term. Secondly, the spatial approximation of fluid velocity, hydrodynamic pressure, and magnetic field apply the mixed element method. Finally, the validity, reliability, and accuracy of the proposed algorithms are supported by numerical experiments.     

A finite element method for a biharmonic equation based on gradient recovery operators

A new non-conforming finite element method is proposed for the approximation of the biharmonic equation with clamped boundary condition. The new formulation is based on a gradient recovery operator. Optimal a priori error estimates are proved for the proposed approach. The approach is also extended to cover a singularly perturbed problem.     

A time-stepping procedure based on convolution for the mixed finite element approximation for porous media flow

A time-stepping procedure is established and analyzed for the problem of miscible displacement in porous medium. The fluid velocity based on mixed element method is post-processed by the convolution with Bramble-Schatz kernel. Convergence analysis shows that the property of superconvergence, originally held for the approximation of velocity, can be retained for the approximation of concentration variable. Compared with the traditional results in references, the constraint conditions between parameters in this paper are improved essentially.     

A stabilized finite element method based on two local Gauss integrations for the Stokes equations

This paper considers a stabilized method based on the difference between a consistent and an under-integrated mass matrix of the pressure for the Stokes equations approximated by the lowest equal-order finite element pairs (i.e., the _P1$P_1$–_P1$P_1$ and _Q1$Q_1$–_Q1$Q_1$ pairs). This method only offsets the discrete pressure space by the residual of the simple and symmetry term at element level in order to circumvent the inf–sup condition. Optimal error estimates are obtained by applying the standard Galerkin technique. Finally, the numerical illustrations agree completely with the theoretical expectations.     

A comparison of finite element based solution schemes for depicting overland flow

Three distinct finite element based schemes are presented and used to solve the direct run-off hydrodynamic equations associated with overland flow. One scheme is shown to have distinct advantages over the other two in that the computer time required for a solution of comparable accuracy is considerably reduced.     

A performance study of tetrahedral and hexahedral elements in 3-D finite element structural analysis

This study compares the performance of linear and quadratic tetrahedral elements and hexahedral elements in various structural problems. The problems selected demonstrate different types of behavior, namely, bending, shear, torsional and axial deformations. It was observed that the results obtained with quadratic tetrahedral elements and hexahedral elements were equivalent in terms of both accuracy and CPU time.     

A reduced finite volume element formulation and numerical simulations based on POD for parabolic problems

A proper orthogonal decomposition (POD) method is applied to a usual finite volume element (FVE) formulation for parabolic equations such that it is reduced to a POD FVE formulation with lower dimensions and high enough accuracy. The error estimates between the reduced POD FVE solution and the usual FVE solution are analyzed. It is shown by numerical examples that the results of numerical computation are consistent with theoretical conclusions. Moreover, it is also shown that the reduced POD FVE formulation based on POD method is both feasible and highly efficient.     

Mixed finite element methods based on Riesz-representing operators for the shell problem

1. IntroductionAs far as the shell problem is concerned, [l] established a mixed formulation in the clas-sical W that the K-ellipticity and the lnfSup condition are introduced. Unfortunately it isvery dndcult to construct ndxed elemellts simultaneously satisfying the K-ellipticity and theInfSup conditionI2], which are indeed prerequisites Of the stability and the convergence. Inaddition, the indefiniteness of the resulting linear algebraic system complicates the solutionalgorithIn.ffecentl…