Showing papers on "Extended finite element method published in 1989"

An efficient, accurate, simple ALE method for nonlinear finite element programs

Abstract: A simple arbitrary Lagrangian-Eulerian formulation is presented. An operator split separates the Lagrangian and Eulerian processes, allowing a Lagrangian finite element program to be extended to this formulation with little difficulty. Example problems illustrate the strengths and weaknesses of the formulation.

A uniformly accurate finite element method for the Reissner-Mindlin plate

Abstract: A simple finite element method for the Reissner–Mindlin plate model in the prim-itive variables is presented and analyzed The method uses nonconforming linear finite elements for the transverse displacement and conforming linear finite elements enriched by bubbles for the rotation, with the computation of the element stiffness matrix modified by the inclusion of a simple elementwise averaging It is proved that the method converges with optimal order uniformly with respect to thickness

Concepts of an adaptive hierarchical finite element code

Abstract: This paper presents the mathematical concepts underlying the new adaptive finite element code KASKADE, which, in its present form, applies to linear scalar second-order 2D elliptic problems on general domains. The starting point for this new development is the recent work on hierarchical finite element bases by H. Yserentant ( Numer. Math. 49 , 379–412 (1986)). It is shown that this approach permits a flexible balance among iterative solver, local error estimator, and local mesh refinement device—the main components of an adaptive PDE code. Without use of standard multigrid techniques, the same kind of computational complexity is achieved—independent of any uniformity restrictions on the applied meshes. In addition, the method is extremely simple and all computations are purely local, making the method particularly attractive in view of parallel computing. The algorithmic approach is illustrated by a well-known critical test problem.

Nonuniform deformations in polycrystals and aspects of the validity of the Taylor model

Abstract: F ull solutions to mixed rate boundary value problems over polycrystalline domains are performed via the finite element method. In order to make these finite element calculations feasible, an idealized two-dimensional crystal structure is studied. These boundary value problems rigorously satisfy the averaging theorems of Hill (Proc. R. Soc.A326, 131, 1972) so that well defined Taylor model analogue problems may be identified and solved. Comparisons between the finite element solutions and their corresponding Taylor model analogues yield a quantitative assessment of the Taylor model's validity with respect to its predictions of texture development and global stress-strain response. The finite element calculations also provide physical insight into the mechanisms contributing to the development of nonuniform and localized deformations in polycrystals.

Finite Element Model for Curved Embedded Reinforcement

Abstract: The geometric relations required for an embedded finite element representation of generally curved reinforcing bars or prestressing tendons are developed. For practical reasons, the reinforcing layers are described in global coordinates, independently of the finite element mesh. An inverse mapping procedure is developed to transform global coordinates of points on the reinforcement layer into local natural coordinates in the parent element. The strain field in the layer is discussed, including a bond slip model. The principle of virtual work is used to derive the various element matrices. The procedure is successfully tested, using both regular and irregular meshes, on three test problems: a uniform strain field, and two versions of a quarter ring under external pressure.

The local projection $P^0-P^1$ -discontinuous-Galerkin finite element method for scalar conservation laws

Abstract: In this paper we introducé the Local Projection P° P-Discontinuous Galerkin finite elemente method (ÂI1P ° P -scheme) for solving numencally scalar conservation laws This is an exphcit method obtained by modifying the expltcit Discontinuous Galerkin method introduced by G Chavent and G Salzano [3], via a simple local projection based on the monotomcity-preserving projections introduced by van Leer [13] The resulting scheme is an extension o f Godunov scheme that vérifies a local maximum pnnciple, and is TV DM (total variation diminishing in the means) Convergence to a weak solution is proven We display numencal évidence that the scheme is an entropy scheme of order one even when discontinuities are present Resumé —Nous proposons une méthode d'éléments finis discontinus P° P avec projection locale pour le calcul des lois de conservation scalaires C'est un schéma explicite obtenu en modifiant la méthode de Galerkin discontinue explicite, introduite par G Chavent et G Salzano [3], a l'aide d'une simple projection locale basée sur les projections introduites par Van Leer [13] qui garde ses propriétés de conservation de la monotomcité Le schéma correspondant est une extension du schéma de Godunov qui vérifie un principe du maximum localy et est DVTM (diminue la variation totale sur les moyennes) Nous démontrons la convergence vers une solution faible, et fournissons des résultats numériques montrant que le schema est entropique d'ordre un même en présence de discontinuité 1. INRODUCTION In this paper we introducé and analyze a new finite element method, the local projection P° P^Discontinuous Galerkin method (AILP°P^scheme), devised to solve numerically the scalar conservation law 3,w + dj(«) = 0 , on (0, r ) x R , u(t = 0) = u0, inR, K ' } where the nonhnear function/ is assumed to be C, and the initial data w0 is assumed to belong to the space L^R) n BV (R). This finite element (*) Received in December 1987 (0 INRIA, Domaine de Voluceau, Rocquencourt, B.P 105, 78153 Le Chesnay Cedex, France , and CEREMADE, Université Faris-Dauphme, 75775 Pans Cedex 16, France () IMA, University of Minnesota, 514 Vincent Hall, Minneapolis, Minnesota 55455, USA M AN Modélisation mathématique et Analyse numérique 0764-583X/89/04/565/28/$ 4 10 Mathematical Modellmg and Numencal Analysis © AFCET Gauthier-Villars 566 G CHAVENT, B COCKBURN method is a predictor-corrector method whose prédiction is given by the explicit P°F-Discontinuous-Galerkin method introduced by G Chavent and G Salzano in [3], and whose correction is obtained by means of a very simple local projection, that we shall call AH, based on the monotonicitypreservmg projection introduced by Van Leer in [13] The basic idea of this method is to write the approximate solution uh as the sum of a piecewise-constant function Uh, and a function uh whose restriction to each element has zero-mean, and to consider the method as a finite différence scheme for the means Uh The function üh is considered as a parameter The local projection All acts on the parameter üh, and is constructed in order to preserve the conservativity, and enforce the stabüity of the scheme for the means Uh In the extreme case m which the parameter üh is set identically equal to zero by the local projection Au, our scheme reduces to the well known Godunov scheme In the gênerai case, the scheme for the means keeps the local maximum pnnciple venfied by Godunov scheme, and is TVD (total variation dimimshing) Thus, the AHP ° P ̂ scheme is conservative, positive, and TVDM, i e total variation dimmishmg in the means We show that these properties, together with some properties of the local projection Au, imply the existence of a subsequence converging to a weak solution of (1 1) Our numencal results mdicate that if the cfl-numbei is mildly small enough, the scheme converges to the entropy solution with a rate of convergence equal to 1 m the L(0, T, L^J-norm even in the présence of discontinuities In 74 Le Samt and Ra^iart [9] introduced the Discontinuous-Galerkin method for solving the neutron transport équation |x dtu -f v dxu + cru = g They choose their approximate function to be piecewise a polynomial of at most degree k >: 0 in each of the variables t, and x In this way they obtained an ïmphcit scheme, but they did not had to solve ït globally Indeed, they proved that ït is possible to solve ït locally due to the fact that the direction of the propagation of the information, (|x, v), is always the same In the gênerai case, this is no longer true, for the local direction of propagation, (1, f'(u)), dépends on values that have not been calculated yet ' To overcome this difficulty, m 1978 G Chavent and G Salzano [3] modified this method and obtamed an explicit scheme that we shall call the P°P-Discontinuous-Galerkin method In this method the tand x-directions are treated in a different way the approximate solution is taken to be piecewise constant in time, and piecewise linear in space The two main advantages of the method are that ït is explicit, and that ït is very easy to generahze to the case of several space dimensions However, the scheme has a very restrictive stabüity condition — as we shall prove later —, and ït Modélisation mathématique et Analyse numérique Mathematical Modelhng and Numencal Analysis P°P-DISCONTINUOUS-GALERKIN FINITE ELEMENT 567 may not converge to the entropy solution in the case in which the nonlinearity ƒ is nonconvex — as the numerical évidence we shall display indicates. In 1984 one of the authors [4] modified the scheme and obtained a scheme called the G-l/2 scheme, for which the convergence to the entropy solution was proven in the gênerai case. A further development of the ideas involved in the construction of this scheme lead to the theory of quasimonotone schemes for which L°°(0, T ; L(R))-error estimâtes have been obtained ; see [5]. The scheme we now introducé can be considered as a simplification of the initial G-l/2 scheme. This simplification leads to a very simple, and much cheaper algorithm, but complicates enormously the proof of its convergence. At each time step the AILP ° P ̂ scheme consists of two phases : in the first, a prédiction is obtained by using the unchanged P°P ^method ; in the second, a correction is obtained by applying the local projection All to it. This projection dépends on a parameter, 0 e [0, 1], (0 may vary from element to element, but we have performed our numerical experiments with G = constant) and is based on the monotonicity-preserving local projections introduced by Van Leer in [13] : for 0 == 1 the All projection coincides with the one defined in [13, (66)] (thus, the AILP ° P ̂ scheme can be considered as a Discontinuous-Galerkin finite element version of the schemes introduced in [13]). One of the main contributions of this work is that we have proved that in fact the use of the local projection All — originally devised in order to produce positive and monotonicity-preserving schemes — renders the scheme under considération a TVDM scheme whose approximate solution vérifies a local maximum principle ; see Proposition 3.2. These two properties allow us to conclude that the scheme is indeed total variation bounded (TVB) and that it générâtes a subsequence converging in L°°(0, T ; L1 1 0C(IR)) to a weak solution of (1.1) ; see Theorem 3.3. The problem of pro ving that the weak solution is indeed the entropy solution is still open. A resuit in this direction is the proof of the convergence of MUSCL-type semidiscrete schemes in the case of a convex (or concave) nonlinearity by Osher in [10]. Also, Johnson and Pitkaranta [7] have analized the Discontinuous-Galerkin method in the linear case. An outline of the paper follows. In Section 2 we define the P° PDiscontinuous-Galerkin method, we obtain the L c/Z-stabüity condition for the linear case, and display some numerical expériences that show the typical behavior of the method. In Section 3 we define the local-projection P° P ^Discontinuous-Galerkin method, we obtain some stability properties, prove the convergence to a weak solution, and test it in the same examples the P° P ̂ Discontinuous-Galerkin method was tested. We end with some concluding remarks in Section 4. In what follows, the P° P ^DiscontinuousGalerkin method will be referred to simply by the P° P ^scheme, and the local-projection P° P ^Discontinuous-Galerkin by the AILPP ^scheme.

A further introduction to finite element analysis of electromagnetic problems

Abstract: An elementary tutorial introduction in finite-element numerical analysis is presented. The finite-element method is applied to Laplacian electrostatic field problems. Suggestions are offered on how the basic concepts developed can be extended to finite-element analysis of problems involving Poisson's or the wave equation. A step-by-step procedure for coding the numerical method is presented; a useful, working FORTRAN program is also included. >

New finite element techniques for modeling deformation histories of continents with stratified temperature‐dependent rheology

Abstract: Previous techniques for modeling anelastic deformation of the lithosphere have included plane strain models, restricted to two-dimensional problems, and quasi-three-dimensional “plane stress” or “thin plate” models, that did not accurately include the effects of the shallow frictional layer, or of kinematic detachment of crust from mantle This paper presents techniques to remedy these deficiencies of thin plate models An iteration strategy in which the rheology is linearized using artificial prestress and a particular effective viscosity tensor causes the calculation of horizontal velocities to converge monotonically, even with a frictional layer at the top of the lithosphere A technique using two planar grids allows deformation and displacement to be different at the crust and mantle levels, at far less cost than that of a three-dimensional grid A finite element technique is developed for computing the changes in thickness of these layers caused by pure shear, simple shear, and pressure gradients A technique based on relaxation of perturbation eigenfunctions solves the heat equation in the lithosphere during deformation Accuracy of component numerical methods is good for simple test problems, but in realistic nonlinear problems utilizing all components, only the precision can be determined because of the lack of analytic solutions Precision of the combined program is tested with a realistic sample problem and presented as a function of the number of iterations in each velocity solution (convergence factor 073 to 088), size of time step in the predictor/corrector time integration (convergence as Δt08), and number of degrees of freedom in the finite element grid (convergence as N−05 to −08 for most variables) Overall cost of a simulation varies with the fractional precision Π as Π−33 A new consequence of kinematic detachment, a moving wave of crustal thickness, is found; unfortunately, the form of the wave depends on the finite element size This means that element size must be chosen to approximate the smoothing by flexural rigidity effects that were neglected because of cost

Finite element/finite difference propagation algorithm for integrated optical device

Abstract: A propagation algorithm based on finite elements and a finite difference discretisation of the scalar wave equation is investigated as an alternative to the beam propagation method. The new approach overcomes the assumption of low contrast media in the BPM and allows the propagation of arbitrary input fields in strongly guiding structures.

A finite element dynamic analysis of spatial mechanisms with flexible links

Abstract: A finite element based method is presented for dynamic analysis of spatial mechanisms with flexible links. In this analysis the geometrically nonlinear relations for the element deformations in terms of the nodal position and orientation coordinates play a central role. The equations of motion are formulated in terms of mixed sets of generalized coordinates of the mechanism with rigid links and deformation mode coordinates that characterize deformation of the elements. A dynamic simulation of a spatial slider crank mechanism with flexible connecting rod and a simulation of a rotating shaft passing through the major critical speed is presented.

The h‐p version of the finite element method for parabolic equations. Part I. The p‐version in time

Abstract: The paper is the first in the series addressing the h-p version of the finite element method for parabolic equations. The h-p version is applied to both time and space variables. The present paper addresses the case when in time the p-version with one single time element is used. Error estimation is given and numerical computations are presented.

A mixed finite element formulation for the boundary controllability of the wave equation

Abstract: : This paper introduces the mixed finite element method as a viable numerical procedure for the boundary controllability of the linear wave equation. Another numerical implementation using Galerkin finite elements has been investigated. However, due to approximation problems of the normal derivative on the boundary, the method becomes unstable as the mesh is refined. To correct for the ill-posedness of the approximate problem, a Tychonoff regularization method was implemented. The aforementioned paper also presents other possible remedies; among them is the mixed finite element method. The fixed finite element approximation is a plausible procedure to overcome these difficulties since the derivative at certain nodal values arises naturally from the formulation.

A sparse matrix open boundary method for finite element analysis

Abstract: A novel implicit mapping method for handling exterior- and far-field problems is described. The method provides a way of maintaining sparsity. Several boundary methods are compared, and the implicit mapping approach is described in detail. In terms of complexity the proposed method is more expensive than a hybrid or ballooning method, but this cost is offset for many problems by its simplicity and the fact that far-field values are readily available. In addition, the method can be used to extend the range of boundary conditions available in a standard finite-element package to include inhomogeneous Neumann conditions. >

Original Formulation of the Finite Element Method

Abstract: Following a brief summary of the 1952 state of the art of structural analysis, the paper describes the circumstances that led to the formulation of the finite element method by members of the Structural Dyanmics Unit at the Boeing Airplane Company. It is noted that the central feature of the procedure that was developed is the evaluation of the stiffness properties of structural elements based on assumed sets of displacement interpolation functions.

A decoupled finite element streamline-upwind scheme for viscoelastic flow problems

Abstract: Parallel to a recently reported mixed finite element method by Marchal and Crochet, a new decoupled finite element streamline-upwinding scheme is developed for viscoelastic flows. While relatively cheap Picard iteration used previously in our streamline element method is maintained, the streamline integration of the constitutive equation is replaced by a Galerkin discretization with streamline-upwinding applied to the convective terms. Using a Maxwell or an Oldroyd-B model and a simple bilinear interpolation for the extra-stress without element subdivision, our calculation of the flow through a four-to-one circular contraction has reached a Weissenberg number limit slightly higher than that reached by the corresponding mixed method with a two by two element subdivision for the extra-stress. Numerical experiments also show that both the conventional Galerkin and the consistent streamline-upwind/Petrov-Galerkin methods give poorer performance than the non-consistent streamline-upwinding approach, confirming results found by the mixed method. The validity of the non-consistent upwinding approach is discussed, and we show that it leads to effectively changed constitutive models which yield smoothed solutions and increased stability.

Mixed formulations for finite element analysis of magnetostatic and electrostatic problems

Abstract: This paper presents some mixed formulations for finite element analysis of magnetostatic and electrostatic problems We employ the magnetic and electric fields as fundamental unknowns instead of the vector potential and the scalar potential, and the proposed approach appears to be desirable for three-dimensional finite element analyses We also give brief comments on the use of the vector potential for the magnetostatic problem

A modified method of characteristics technique and mixed finite elements method for simulation of groundwater solute transport

Abstract: A comprehensive groundwater solute transport simulator is developed based on the modified method of characteristics (MMOC) combined with the Galerkin finite element method for the transport equation and the mixed finite element (MFE) method for the groundwater flow equation. The preconditioned conjugate gradient algorithm is used to solve the two large sparse algebraic system of equations arising from the MMOC and MFE discretizations. The MMOC takes time steps in the direction of flow, along the characteristics of the velocity field of the total fluid. The physical diffusion and dispersion terms are treated by a standard finite element scheme. The crucial aspect of the MMOC technique is that it looks backward in time, along an approximate flow path, instead of forward in time as in many method of characteristics or moving mesh techniques. The MFE procedure involves solving for both the hydraulic head and the specific discharge simultaneously. One order of convergence is gained by the MFE method, as compared with other standard finite element methods, and therefore more accurate velocity fields are simulated. The overall advantages of the MMOC-MFE method include minimum numerical oscillation or grid orientation problems under steep concentration gradient simulations, and material balance errors are greatly reduced due to a very accurate velocity simulation by the MFE method. In addition, much larger time steps with Courant number well in excess of 1, as compared with the standard Galerkin finite element method, can be taken on a fixed spatial grid system without significant loss of accuracy.

A singular finite element for Stokes flow: The stick–slip problem

Abstract: SUMMARY Abrupt changes in boundary conditions in viscous flow problems give rise to stress singularities. Ordinary finite element methods account effectively for the global solution but perform poorly near the singularity. In this paper we develop singular finite elements, similar in principle to the crack tip elements used in fracture mechanics, to improve the solution accuracy in the vicinity of the singular point and to speed up the rate of convergence. These special elements surround the singular point, and the corresponding field shape functions embody the form of the singularity. Because the pressure is singular, there is no pressure node at the singular point. The method performs well when applied to the stick-slip problem and gives more accurate results than those from refined ordinary finite element meshes.

A nonconforming finite element method of upstream type applied to the stationary Navier-Stokes equation

Abstract: — We present a nonconforming finite element method with an upstream discretizatwn of the convective term for solving the stationary Navier-Stokes équations. The existence of at least one solution of the discrete problem and the convergence of subsequences of such solutions to a solution of the Navier-Stokes équations are estabhshed In addition, under certain assumptions on the data, uniqueness of the solution can be guarenteed and error estimâtes of the approximate solution are given Moreover, some favourable properties of the discrete algebraic system are discussed Resumé — Nous présentons une méthode non conforme d'éléments finis avec une discrétisation décentrée amont du terme de convection pour la résolution des équations de NavierStokes statwnnaires On prouve l'existence d'une solution au moins du problème discret et la convergence des sous-suites de telles solutions vers une solution des équations de Navier-Stokes statwnnaires En outre on peut sous certaines hypothèses sur les données garantir l'unicité et on donne alors des estimations d'erreur de la solution approximative En outre on discute quelques propriétés importantes du système algébrique discret

A finite element method for analyzing localization in rate dependent solids at finite strains

Abstract: The finite element method for localization analysis of Ortiz et al. [Comp. Methods Appl. Mech. Engrg. 61] is generalized to account for finite deformations and for material rate dependence. Special shape functions are added to the finite element basis to reproduce band-like localized deformation modes. The amplitudes of these additional modes are eliminated locally by static condensation. The performance of the enhanced element is illustrated in a problem involving shear localization in a plane strain tensile bar. Solutions based on the enhanced element are compared with corresponding results obtained from the underlying compatible isoparametric quadrilateral element and from crossedtriangular and uniformly reduced integration elements. In the finite deformation context, the enhanced element solution is not very sensitive to the precise specification of initial orientation of the additional band-like modes. The enhanced element formulation described here can be used for a broad range of rate independent and rate dependent material behaviors in two dimensional and three dimensional problems.

A quadrilateral 2‐D finite element based on mixed interpolation of tensorial components

Abstract: A quadrilateral 2‐D finite element for linear and non‐linear analysis of solids is presented. The element is based on the technique of mixed interpolation of tensorial components. It is shown that the new element is reliable and efficient, being apt, therefore, to be used in routine engineering applications.

Solution of atomic Hartree–Fock equations with the P version of the finite element method

Abstract: The Hartree–Fock equations for atoms are solved with the p version of the finite element method, which differs from the traditional finite element method in using high order, hierarchic polynomials as basis functions. Recursion formulas are developed for the analytical evaluation of integrals, which are crucial in reducing the computation time and maintaining the accuracy of the solution. A hierarchic computational approach is used where the solution at a certain level is used to start the calculation at the next level. Results are presented for closed and open shell atoms taken from various columns of the periodic table that show excellent agreement with accurate numerical calculations.