Showing papers on "Finite element method published in 1998"

A new Meshless Local Petrov-Galerkin (MLPG) approach in computational mechanics

Abstract: A local symmetric weak form (LSWF) for linear potential problems is developed, and a truly meshless method, based on the LSWF and the moving least squares approximation, is presented for solving potential problems with high accuracy. The essential boundary conditions in the present formulation are imposed by a penalty method. The present method does not need a “finite element mesh”, either for purposes of interpolation of the solution variables, or for the integration of the “energy”. All integrals can be easily evaluated over regularly shaped domains (in general, spheres in three-dimensional problems) and their boundaries. No post-smoothing technique is required for computing the derivatives of the unknown variable, since the original solution, using the moving least squares approximation, is already smooth enough. Several numerical examples are presented in the paper. In the example problems dealing with Laplace & Poisson's equations, high rates of convergence with mesh refinement for the Sobolev norms ||·||0 and ||·||1 have been found, and the values of the unknown variable and its derivatives are quite accurate. In essence, the present meshless method based on the LSWF is found to be a simple, efficient, and attractive method with a great potential in engineering applications.

The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media

Abstract: Keywords: Methode des elements finis ; Sols non satures ; Consolidation Reference Record created on 2004-09-07, modified on 2016-08-08

Thermomechanical analysis of functionally graded cylinders and plates

Abstract: The dynamic thermoelastic response of functionally graded cylinders and plates is studied. Thermomechanical coupling is included in the formulation, and a finite element model of the formulation is developed. The heat conduction and the thermoelastic equations are solved for a functionally graded axisymmetric cylinder subjected to thermal loading. In addition, a thermoelastic boundary value problem using the first-order shear deformation plate theory (FSDT) that accounts for the transverse shear strains and the rotations, coupled with a three-dimensional heat conduction equation, is formulated for a functionally graded plate. Both problems are studied by varying the volume fraction of a ceramic and a metal using a power law distribution.

Finite Element Analysis of Acoustic Scattering

Abstract: A cognitive journey towards the reliable simulation of scattering problems using finite element methods, with the pre-asymptotic analysis of Galerkin FEM for the Helmholtz equation with moderate and large wave number forming the core of this book. Starting from the basic physical assumptions, the author methodically develops both the strong and weak forms of the governing equations, while the main chapter on finite element analysis is preceded by a systematic treatment of Galerkin methods for indefinite sesquilinear forms. In the final chapter, three dimensional computational simulations are presented and compared with experimental data. The author also includes broad reference material on numerical methods for the Helmholtz equation in unbounded domains, including Dirichlet-to-Neumann methods, absorbing boundary conditions, infinite elements and the perfectly matched layer. A self-contained and easily readable work.

Progressive Delamination Using Interface Elements

Abstract: The paper describes a method for modelling progressive mixed-mode delamination in fibre composites. The procedure, which is incorporated within the non-linear finite element method, is based on the use of interface elements in conjunction with softening relationships between the stresses and the relative displacements. Fracture mechanics is indirectly introduced by relating the areas under the stress/displacement curves to the critical fracture energies.

Impact-induced delamination of composites: A 2D simulation

Abstract: The delamination process in thin composite plates subjected to low-velocity impact is simulated using a specially developed 2D cohesive/volumetric finite element scheme. Cohesive elements are introduced along the boundaries of the inner layers and inside the transverse plies to simulate the spontaneous initiation and propagation of transverse matrix cracks and delamination fronts. The analysis is performed within the framework of the finite deformation theory of elasticity to account for the nonlinear stiffening of the thin composite plate and the large rotations which accompany the fracture process. The simulation is dynamic and uses an explicit time stepping scheme. Comparison with existing experiments performed on graphite/epoxy laminates indicates that the cohesive/volumetric finite element scheme is able to capture the complex mechanisms leading to the delamination, including the initial micro-cracking of the matrix, the appearance of critical transverse matrix cracks and the rapid propagation of delamination cracks initiated at the intersections between the critical matrix cracks and the adjacent plies.

Finite element methods and their convergence for elliptic and parabolic interface problems

Abstract: In this paper, we consider the finite element methods for solving second order elliptic and parabolic interface problems in two-dimensional convex polygonal domains. Nearly the same optimal $L^2$ -norm and energy-norm error estimates as for regular problems are obtained when the interfaces are of arbitrary shape but are smooth, though the regularities of the solutions are low on the whole domain. The assumptions on the finite element triangulation are reasonable and practical.

A local boundary integral equation (LBIE) method in computational mechanics, and a meshless discretization approach

Abstract: The Galerkin finite element method (GFEM) owes its popularity to the local nature of nodal basis functions, i.e., the nodal basis function, when viewed globally, is non-zero only over a patch of elements connecting the node in question to its immediately neighboring nodes. The boundary element method (BEM), on the other hand, reduces the dimensionality of the problem by one, through involving the trial functions and their derivatives, only in the integrals over the global boundary of the domain; whereas, the GFEM involves the integration of the “energy” corresponding to the trial function over a patch of elements immediately surrounding the node. The GFEM leads to banded, sparse and symmetric matrices; the BEM based on the global boundary integral equation (GBIE) leads to full and unsymmetrical matrices. Because of the seemingly insurmountable difficulties associated with the automatic generation of element-meshes in GFEM, especially for 3-D problems, there has been a considerable interest in element free Galerkin methods (EFGM) in recent literature. However, the EFGMs still involve domain integrals over shadow elements and lead to difficulties in enforcing essential boundary conditions and in treating nonlinear problems. The object of the present paper is to present a new method that combines the advantageous features of all the three methods: GFEM, BEM and EFGM. It is a meshless method. It involves only boundary integration, however, over a local boundary centered at the node in question; it poses no difficulties in satisfying essential boundary conditions; it leads to banded and sparse system matrices; it uses the moving least squares (MLS) approximations. The method is based on a Local Boundary Integral Equation (LBIE) approach, which is quite general and easily applicable to nonlinear problems, and non-homogeneous domains. The concept of a “companion solution” is introduced so that the LBIE for the value of trial solution at the source point, inside the domain Ω of the given problem, involves only the trial function in the integral over the local boundary Ω s of a sub-domain Ω s centered at the node in question. This is in contrast to the traditional GBIE which involves the trial function as well as its gradient over the global boundary Γ of Ω. For source points that lie on Γ, the integrals over Ω s involve, on the other hand, both the trial function and its gradient. It is shown that the satisfaction of the essential as well as natural boundary conditions is quite simple and algorithmically very efficient in the present LBIE approach. In the example problems dealing with Laplace and Poisson's equations, high rates of convergence for the Sobolev norms ||·||0 and ||·||1 have been found. In essence, the present EF-LBIE (Element Free-Local Boundary Integral Equation) approach is found to be a simple, efficient, and attractive alternative to the EFG methods that have been extensively popularized in recent literature.

New applications of integral equations methods for solvation continuum models: ionic solutions and liquid crystals

Abstract: We present a new method for solving numerically the equations associated with solvation continuum models, which also works when the solvent is an anisotropic dielectric or an ionic solution This method is based on the integral equation formalism Its theoretical background is set up and some numerical results for simple systems are given This method is much more effective than three‐dimensional methods used so far, like finite elements or finite differences, in terms of both numerical accuracy and computational costs

A modified collocation method and a penalty formulation for enforcing the essential boundary conditions in the element free Galerkin method

Abstract: The Element free Galerkin method, which is based on the Moving Least Squares approximation, requires only nodal data and no element connectivity, and therefore is more flexible than the conventional finite element method. Direct imposition of essential boundary conditions for the element free Galerkin (EFG) method is always difficult because the shape functions from the Moving Least Squares approximation do not have the delta function property. In the prior literature, a direct collocation of the fictitious nodal values u used as undetermined coefficients in the MLS approximation, uh(x) [uh(x)=Φ·u], was used to enforce the essential boundary conditions. A modified collocation method using the actual nodal values of the trial function uh(x) is presented here, to enforce the essential boundary conditions. This modified collocation method is more consistent with the variational basis of the EFG method. Alternatively, a penalty formulation for easily imposing the essential boundary conditions in the EFG method with the MLS approximation is also presented. The present penalty formulation yields a symmetric positive definite system stiffness matrix. Numerical examples show that the present penalty method does not exhibit any volumetric locking and retains high rates of convergence for both displacements and strain energy. The penalty method is easy to implement as compared to the Lagrange multiplier method, which increases the number of degrees of freedom and yields a non-positive definite system matrix.

Application of an Ultra Weak Variational Formulation of Elliptic PDEs to the Two-Dimensional Helmholtz Problem

Abstract: A new technique to solve elliptic linear PDEs, called ultra weak variational formulation (UWVF) in this paper, is introduced in [B. Despres, C. R. Acad. Sci. Paris, 318 (1994), pp. 939--944]. This paper is devoted to an evaluation of the potentialities of this technique. It is applied to a model wave problem, the two-dimensional Helmholtz problem. The new method is presented in three parts following the same style of presentation as the classical one of the finite elements method, even though they are definitely conceptually different methods. The first part is committed to the variational formulation and to the continuous problem. The second part defines the discretization process using a Galerkin procedure. The third part actually studies the efficiency of the technique from the order of convergence point of view. This is achieved using theoretical proofs and a series of numerical experiments. In particular, it is proven and shown the order of convergence is lower bounded by a linear function of the number of degrees of freedom. An application to scattering problems is presented in a fourth part.

Schwarz Alternating Method

Abstract: A discrete technique of the Schwarz alternating method is presented in this last chapter, to combine the Ritz-Galerkin and finite element methods. This technique is well suited for solving singularity problems in parallel, and requires a little more computation for large overlap of subdomains. The convergence rate of the iterative procedure, which depends upon overlap of subdomains, will be studied. Also a balance strategy will be proposed to couple the iteration number with the element size used in the FEM. For the crack-infinity problem of singularity the total CPU time by the technique in this chapter is much less than that by the nonconforming combination in Chapter 12.

An introduction to programming the meshless Element F reeGalerkin method

Abstract: A detailed description of the Element Free Galerkin (EFG) method and its numerical implementation is presented with the goal of familiarizing scientists and engineers with the new computational technique. In this spirit, an in-depth explanation of the essential concepts which comprise the method is given with specific emphasis on the one-dimensional formulation. First, the EFG algorithm for a one-dimensional problem in linear elastostatics is given; the results are compared to those achievable with standard finite element techniques. A step by step explanation of the MATLAB program used to solve the problem is given with specific references to the EFG method in one-dimension. Next, a simplified two-dimensional implementation to linear elastostatics is described. Results are calculated with the method and the aid of a two-dimensional MATLAB EFG program, and conclusions are drawn about the method and its capabilities. The source programs used to solve both the one-dimensional and two-dimensional problems are provided in the Appendices and are available on the web.

An Arbitrary Lagrangian-Eulerian finite element method

Abstract: This paper describes an Arbitrary Lagrangian- Eulerian (ALE) finite element method for the simulation of fluid domains with moving structures. The fluid is viscous, incompressible and unsteady and the fluid motion is solved by a fractional step discretization of the Navier-Stokes equations. The emphasis is on convection dominated flows, and a three-step method is used for the convection term. The moving structure causes the mesh of the fluid domain to move, and a new algorithm is proposed to solve the important and crucial problem of the calculation of the mesh velocities. Numerical calculations of the added mass and added damping of a vibrating two-dimensional circular cylinder in the frequency Reynolds number range Re w =20−2000 are performed to evaluate the proposed ALE finite element method. The numerically calculated added mass and added damping are compared to both analytical and numerical results. To further demonstrate the generality of the method, a numerical simulation of flow past an oscillating schematic sports car is presented.

A practical numerical approach for large deformation problems in soil

Abstract: A practical method is presented for numerical analysis of problems in solid (in particular soil) mechanics which involve large strains or deformations. The method is similar to what is referred to as ‘arbitrary Lagrangian–Eulerian’, with simple infinitesimal strain incremental analysis combined with regular updating of co-ordinates, remeshing of the domain and interpolation of material and stress parameters. The technique thus differs from the Lagrangian or Eulerian methods more commonly used. Remeshing is accomplished using a fully automatic remeshing technique based on normal offsetting, Delaunay triangulation and Laplacian smoothing. This technique is efficient and robust. It ensures good quality shape and distribution of elements for boundary regions of irregular shape, and is very quick computationally. With remeshing and interpolation, small fluctuations appeared initially in the load-deformation results. In order to minimize these, different increment sizes and remeshing frequencies were explored. Also, various planar linear interpolation techniques were compared, and the unique element method found to work best. Application of the technique is focused on the widespread problem of penetration of surface foundations into soft soil, including deep penetration of foundations where soil flows back over the upper surface of the foundation. Numerical results are presented for a plane strain footing and an axisymmetric jack-up (spudcan) foundation, penetrating deeply into soil which has been modelled as a simple Tresca or Von Mises material, but allowing for increase of the soil strength with depth. The computed results are compared with plasticity solutions for bearing capacity. The numerical method is shown to work extremely well, with potential application to a wide range of soil–structure interaction problems. © 1998 John Wiley & Sons, Ltd.

Finite Element Methods of Least-Squares Type

Abstract: We consider the application of least-squares variational principles to the numerical solution of partial differential equations. Our main focus is on the development of least-squares finite element methods for elliptic boundary value problems arising in fields such as fluid flows, linear elasticity, and convection-diffusion. For many of these problems, least-squares principles offer numerous theoretical and computational advantages in the algorithmic design and implementation of corresponding finite element methods that are not present in standard Galerkin discretizations. Most notably, the use of least-squares principles leads to symmetric and positive definite algebraic problems and allows us to circumvent stability conditions such as the inf-sup condition arising in mixed methods for the Stokes and Navier--Stokes equations. As a result, application of least-squares principles has led to the development of robust and efficient finite element methods for a large class of problems of practical importance.

Incompressible Flow and the Finite Element Method: Advection-Diffusion and Isothermal Laminar Flow

Abstract: The advection-diffusion equation the Navier-Stokes equation derived quantities. Appendices: weak operators some element matrices and projection methods.

Mixed finite element methods for viscoelastic flow analysis : a review

Abstract: The progress made during the past decade in the application of mixed finite element methods to solve viscoelastic flow problems using differential constitutive equations is reviewed. The algorithmic developments are discussed in detail. Starting with the classical mixed formulation, the elastic viscous stress splitting (EVSS) method as well as the related discrete EVSS and the so-called EVSS-G method are discussed among others. Furthermore, stabilization techniques such as the streamline upwind Petrov‐Galerkin (SUPG) and the discontinuous Galerkin (DG) are reviewed. The performance of the numerical schemes for both smooth and non-smooth benchmark problems is discussed. Finally, the capabilities of viscoelastic flow solvers to predict experimental observations are reviewed. # 1998 Elsevier Science B.V. All rights reserved.

A mixed displacement-pressure formulation for poroelastic materials

Abstract: Recently, finite element models based on Biot’s displacement (u¯,U¯) formulation for poroelastic materials have been extensively used to predict the acoustical and structural behavior of multilayer structures. These models while accurate lead to large frequency dependent matrices for three-dimensional problems necessitating important setup time, computer storage and solution time. In this paper, a novel exact mixed displacement pressure (u¯,p) formulation is presented. The formulation derives directly from Biot’s poroelasticity equations. It has the form of a classical coupled fluid-structure problem involving the dynamic equations of the skeleton in vacuo and the equivalent fluid in the rigid skeleton limit. The governing (u¯,p) equations and their weak integral form are given together with the coupling conditions with acoustic media. The numerical implementation of the presented approach in a finite element code is discussed. Examples are presented to show the accuracy and effectiveness of the presented formulation.

A general environment for the treatment of discrete problems and its application to the finite element method

Abstract: A general computer aided description environment for the treatment of discrete problems, based on a concise structure for both development and application levels, is described and applied to the finite element method. Its characteristics reveal its great ability to welcome in an evolutive way a wide range of physical models and numerical methods, with any coupling of them. It is therefore adapted to various activities, such as research, collaboration, education, training and industrial studies.