2D elastic analysis of the sandwich panel buckling problem: benchmark solutions and accurate finite element formulations

This paper is concerned with the elastic stability of a sandwich beam panel using classical elasticity. An exact solution for the buckling problem of a sandwich panel (wide beam) in uniaxial compression is presented. Various formulations that correspond to the use of different pairs of energetically conjugate stress and strain measures for the infinitesimal elastic stability of the sandwich panel are discussed. Results from the present two-dimensional analyses to predict the global and local buckling of a sandwich panel are compared with previous theoretical and experimental results. A new finite element formulation for the bifurcation buckling problem is also introduced. In this new formulation, terms that influence the buckling load, which have been omitted in popular commercial codes are pointed out and their significance in influencing the buckling load is identified. The formulation and results presented here can be used as a benchmark solution to establish the accuracy of numerical methods for computing the buckling behavior of thick, orthotropic solids.     

Methods of finite element analysis of arbitrary buckling forms of sandwich plates and shells

Two statements of the problem of arbitrary buckling forms (BFs) (including synphasic, antiphasic, mixed flexural, flexural-shear, and shear forms in the tangential directions) of general-form sandwich shells and two schemes of its solution by the FEM are given. The first of the schemes is based on the use of refined linear equations for determination of the precritical stress-strain state and linearized equations of neutral equilibrium with all parametric addends necessary to determine the critical loads and reveal the possible BFs. The second one uses the general geometrically nonlinear relations of elasticity theory for investigation of the whole deformation process up to buckling in terms of a modified incremental (stepwise) statement of the problem. Examples of solution of particular problems are given.Center for Study of Dynamics and Stability, Tupolev Kazan' State Technical University, Kazan', Tatarstan, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. 4, pp. 473–486, July–August, 2000.     

Dynamic crack analysis in 2D elastic solids with the singular edge-based smoothed finite element method

The singular edge-based smoothed finite element method (sES-FEM) is developed for stationary dynamic crack analysis in two-dimensional (2D) elastic solids. The paper aims at providing a better understanding of the dynamic fracture behaviors in linear elastic solids by means of the strain smoothing technique. The strains are smoothed and the system stiffness matrix is performed using the strain smoothing over the smoothing domains associated with the element edges. A two-layer singular five-node crack-tip element is employed while the standard implicit time integration scheme is used for solving the discrete sES-FEM equation system. Dynamic stress intensity factors (DSIFs) are extracted using the domain-form of interaction integrals in terms of the smoothing technique. The normalized DSIFs are compared with reference solutions showing a high accuracy of the sES-FEM. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Shape sensitivity analysis of an elastic contact problem: Convergence of the Nitsche based finite element approximation

In a recent work, we introduced a finite element approximation for the shape optimization of an elastic structure in sliding contact with a rigid foundation where the contact condition (Signorini’s condition) is approximated by Nitsche’s method and the shape gradient is obtained via the adjoint state method. The motivation of this work is to propose an a priori convergence analysis of the numerical approximation of the variables of the shape gradient (displacement and adjoint state) and to show some numerical results in agreement with the theoretical ones. The main difficulty comes from the non-differentiability of the contact condition in the classical sense which requires the notion of conical differentiability.     

A mixed differential quadrature and finite element free vibration and buckling analysis of thick beams on two-parameter elastic foundations

As a first endeavor, a mixed differential quadrature (DQ) and finite element (FE) method for boundary value structural problems in the context of free vibration and buckling analysis of thick beams supported on two-parameter elastic foundations is presented. The formulations are based on the two-dimensional theory of elasticity. The problem domain along axial direction is discretized using finite elements. The resulting system of equations and the related boundary conditions are discretized in the thickness direction and in strong-form using DQM. The method benefits from low computational efforts of the DQ in conjunction with the effectiveness of the FE method in general geometry and systematic boundary treatment resulting in highly accurate and fast convergence behavior solution. The boundary conditions at the top and bottom surface of the beams are implemented accurately. The presented formulations provide an effective analysis tool for beams free of shear locking. Comparisons are made with results from elasticity solutions as well as higher-order beam theory.     

An optimal C $^0$ finite element algorithm for the 2D biharmonic problem: theoretical analysis and numerical results

Summary. The aim of this paper is to give a new method for the numerical approximation of the biharmonic problem. This method is based on the mixed method given by Ciarlet-Raviart and have the same numerical properties of the Glowinski-Pironneau method. The error estimate associated to these methods are of order O(h) for k The algorithm proposed in this paper converges even for k, without any regularity condition on or . We have an error estimate of order O(h) in case of regularity. Received February 5, 1999 / Revised version received February 23, 2000 / Published online May 4, 2001     

Plane finite element for static and free vibration analysis of sandwich plates

Conclusions Triangular sandwich finite elements (MPLW30) with 30 DOF have been developed for the static and free vibration analysis of sandwich plates with thin faces and low core shear modules. The results of numerical examples presented here demonstrate the accuracy and suitability of the formulations for the analysis of sandwich plates with k_H > 25 and k_E > 200.Published in Mekhanika Kompozitnykh Materialov, Vol. 30, No. 2, pp 238–248, March–April, 1994.     

Homogeneous solutions in the equilibrium problem of an elastic cylinder of finite length

The problem of the equilibrium of a finite elastic cylinder, under the action of axisymmetric normal and tangential loads on its surface, is considered. Within the framework of the method of homogeneous solutions, one establishes the connection between the representations of the displacement vector in the cylinder in the form of series in layer and cylindrical homogeneous solutions. The expansion coefficients in both representations are expressed in terms of the solution of an infinite regular system of linear algebraic equations.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 20, pp. 3–9, 1989.     

An analysis of the Eddy current problem by the least-squares finite element in 2-D

This report discusses an analysis of least-squares finite element for a steady electromagnetic field in 2-D. The Maxwell equations for the magnetic field strength H are written into a first-order linear system of PDE. The analysis shows that the regular finite element spaces with piecewise linear polynomials can be chosen to represent the H and the conducted electric density J. The error of the numerical results in H-1 norm should be bounded by Ch.     

DFT modal analysis of spectral element methods for the 2D elastic wave equation

The DFT modal analysis is a dispersion analysis technique that transforms the equations of a numerical scheme to the discrete Fourier transform domain sampled in the mesh nodes. This technique provides a natural matching of exact and approximate modes of propagation. We extend this technique to spectral element methods for the 2D isotropic elastic wave equation, by using a Rayleigh quotient approximation of the eigenvalue problem that characterizes the dispersion relation, taking full advantage of the tensor product representation of the spectral element matrices. Numerical experiments illustrate the dependence of dispersion errors on the grid resolution, polynomial degree, and discretization in time. We consider spectral element methods with Chebyshev and Legendre collocation points.     

A finite element method for the double-layer potential solutions of the Neumann exterior problem

A variational formulation for the integral equation used for the double layer potential solution of the Neumann exterior problem in the Laplace equation was proposed in [4]. This formulation allows the use of a finite element method which we describe and experiment here.     

Multiscale finite element discretizations based on local defect correction for the biharmonic eigenvalue problem of plate buckling

In this paper, we study the multiscale finite element discretizations about the biharmonic eigenvalue problem of plate buckling. On the basis of the work of Dai and Zhou (SIAM J. Numer. Anal. 46[1] [2008] 295‐324), we establish a three‐scale scheme, a multiscale discretization scheme, and the associated parallel version based on local defect correction. We first prove a local priori error estimate of finite element approximations, then give the error estimates of multiscale discretization schemes. Theoretical analysis and numerical experiments indicate that our schemes are suitable and efficient for eigenfunctions with local low smoothness.     

Nonlinear finite element solutions of thermoelastic deflection and stress responses of internally damaged curved panel structure

Thermoelastic deflection and corresponding stresses of the pre-damaged layered panel structure are investigated numerically in this article including the large deformation kinematics under the linearly varying temperature field. The composite structural deformation kinematics is derived using two different polynomial type of kinematic theories including the through-thickness stretching effect. The inter-laminar separation between the adjacent layers is incurred via the sub-laminate approach and Green–Lagrange strain to count the total structural deformation. Also, the intermittent displacement continuity conditions are imposed in the current mathematical model to establish the displacement continuity between the separated layers. The variational principle is adopted for the evaluation of the nonlinear structural equilibrium equations and solved via total Lagrangian approach. The convergence and the corresponding validity of the currently derived nonlinear finite element solutions are checked by solving different sets of numerical examples. Additionally, the comprehensive inferences are drawn from various numerical examples for the well-defined important input parameter including the size, position, and location of delamination.     

On the convergence of finite element solutions to the interface problem for the Stokes system

The Stokes system with a discontinuous coefficient (Stokes interface problem) and its finite element approximations are considered. We firstly show a general error estimate. To derive explicit convergence rates, we introduce some appropriate assumptions on the regularity of exact solutions and on a geometric condition for the triangulation. We mainly deal with the MINI element approximation and then consider P1-iso-P2/P1 element approximation. Results are expected to give an instructive remark in numerical analysis for two-phase flow problems.     

Mixed finite element analysis of a thermally nonlinear coupled problem

In Loula and Zhou [Comput Appl Math 20 (2001), 321–339], a thermally coupled nonlinear elliptic system modeling a large class of engineering problems was considered, and some mathematical and numerical analyses (C⁰ Lagrangian finite elements combined with a fixed point algorithm) were given. To continue our work, we propose in this article a mixed method for the potential equation and present the corresponding analyses and numerical implementations. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

On the mixed finite element approximation for the buckling of plates

We consider the mixed finite element method for the buckling problem of the thin plate by using piecewise linear polynomials. We give error estimates for the approximate eigenvalues and the eigenfunctions.     

3D finite element analysis of evaporative laser cutting

A three-dimensional computational model of evaporative laser-cutting process has been developed using a finite element method. Steady heat transfer equation is used to model the laser-cutting process with a moving laser. The laser is assumed continuous wave Gaussian beam. The finite element surfaces on evaporation side are nonplanar and approximated by bilinear polynomial surfaces. Semi-infinite elements are introduced to approximate the semi-infinite domain. An iterative scheme is used to handle the geometric nonlinearity due to the unknown groove shape. The convergence studies are performed for various meshes. Numerical results about groove shapes and temperature distributions are presented and also compared with those by semi-analytical methods.     

Quadratic finite element approximation of the Signorini problem

Applying high order finite elements to unilateral contact variational inequalities may provide more accurate computed solutions, compared with linear finite elements. Up to now, there was no significant progress in the mathematical study of their performances. The main question is involved with the modeling of the nonpenetration Signorini condition on the discrete solution along the contact region. In this work we describe two nonconforming quadratic finite element approximations of the Poisson-Signorini problem, responding to the crucial practical concern of easy implementation, and we present the numerical analysis of their efficiency. By means of Falk's Lemma we prove optimal and quasi-optimal convergence rates according to the regularity of the exact solution.

     

On finite element schemes of the dirichlet problem

In this paper, we consider finite element schemes applied to the Dirichlet problem for the system of nonlinear elliptic equations, based on piecewise linear polynomials, and present iterative methods for solving algebraic nonlinear equations, which construct monotone sequences. Furthermore, we derive error estimates which imply uniform convergence. Our results are based on the discrete maximum principle. Finally, some typical numerical examples are given to demonstrate the usefulness of convergence results.