Showing papers on "Smoothed finite element method published in 1981"

An optimal order process for solving finite element equations

Abstract: A k-level iterative procedure for solving the algebraic equations which arise from the finite element approximation of elliptic boundary value problems is presented and analyzed. The work estimate for this procedure is proportional to the number of unknowns, an optimal order result. General geometry is permitted for the underlying domain, but the shape plays a role in the rate of convergence through elliptic regularity. Finally, a short discussion of the use of this method for parabolic problems is presented.

Discretization by finite elements of a model parameter dependent problem

Abstract: The discretization by finite elements of a model variational problem for a clamped loaded beam is studied with emphasis on the effect of the beam thickness, which appears as a parameter in the problem, on the accuracy. It is shown that the approximation achieved by a standard finite element method degenerates for thin beams. In contrast a large family of mixed finite element methods are shown to yield quasioptimal approximation independent of the thickness parameter. The most useful of these methods may be realized by replacing the integrals appearing in the stiffness matrix of the standard method by Gauss quadratures.

Hierarchical Finite Element Approaches Error Estimates and Adaptive Refinement

Abstract: : This paper is concerned with the identification of the discretization error in finite element solution and the definition of optimal refinement processes. The advantages and limitations of the hierarchical approach will be discussed and it will be shown how the intelligent enrichment of the finite element grid can be left to the computer if a capacity for a-posteriori error estimation exists within the finite element code.

The application of quasi‐Newton methods in fluid mechanics

Abstract: The use of quasi-Newton methods is studied for the solution of the nonlinear finite element equations that arise in the analysis of incompressible fluid flow. An effective procedure for the use of Broyden’s method in finite element analysis is presented. The quasi-Newton method is compared with the commonly employed successive substitution and Newton-Raphson procedures, and it is concluded that the use of Broyden‘s method can constitute an effective solution strategy.

Finite element analysis of earthquake‐induced sloshing in axisymmetric tanks

Abstract: A finite element anaylysis to predict the sloshing dislacements and hydrodynamic pressures in liquid-filled tanks subjected to earthquake ground motions is presented. Finite element equations were derived using the Galerkin formulation, and the predicted results were checked against the test data, showing a good agreement between the test and finite element results.

Improved finite element forms for the shallow-water wave equations

Abstract: SUMMARY This paper presents new finite element formulations of the shallow-water wave equations which use different basis functions for the velocity and height fields. These arrangements are analysed with the Fourier transform technique which was developed by Schoenstadt,' and they are also compared with other finite difference and finite element schemes. The new schemes are integrated in time for two initial states and compared with analytic solutions and numerical solutions from other schemes. The behaviour of the new forms is excellent and they are also convenient to apply in two dimensions with triangular elements.

A simple method to generate high-order accurate convection operators for explicit schemes based on linear finite elements

Abstract: SUMMARY A simple method is proposed to generate high-order accurate convection operators for lumped-explicit schemes based on linear or multilinear finite elements. The basic idea is to reduce the truncation error on the first-order spatial derivatives by exploiting the consistent mass matrix of the finite element method in a purely explicit multistep procedure. The effectiveness of the method is demonstrated on pure convection problems in one and two dimensions. KEY woms Heat Transfer Convection Finite Element Method

Finite element analysis of a scattering problem

Abstract: A finite element method for the solution of a scattering problem for the reduced wave equation is formulated and analyzed. The method involves a reformulation of the problem on a bounded domain with a nonlocal boundary condition. The space of trial functions includes piecewise polynomial functions and functions arising from spherical harmonics.

A comparison of localized finite element formulations for two‐dimensional wave diffraction and radiation problems

Abstract: Two of the most promising localized finite element methods are compared: the boundary series element method, in which a series of eigenfunctions is used to represent the far field solution; and the boundary integral element method, in which an integral equation is satisfied at the boundary between localized finite element and outer regions. The methods are applied to water of arbitrary depth. The theory of the two methods is summarized, and typical numerical results are discussed. Consideration is given to the well-known hydrodynamical reciprocal relations, and to the phenomenon of ‘irregular’ frequencies. The relative merits of the two methods are established.

The vortex method with finite elements

Abstract: This work shows that the method of charcteristics is well suited for the numerical solution of first order hyperbolic partial differential equations whose coefficients are approximated by functions piecewise constant on a finite element triangulation of the domain of integration. We apply this method to the numerical solution of Euler's equation and prove convergence when the time step and the mesh size tend to zero. The proof is based upon the results of regularity given by Kato and Wolibner and on L°° estimates for the solution of the Dirichlet problem given by Nitsche. The method obtained belongs to the family of vortex methods usually studied in a finite difference context. Introduction. The vortex method is based on an old concept of fluid mechanics which says that for two-dimensional nonviscous flows the vorticity in the fluid is transported by the flow; thus, if the initial distribution of vorticity consists of a finite number of point vortices, the flow at later times can be found by transport of these point vortices along the streamlines of the flow that they create. In mathematical terms this means that the two-dimensional stream functionvorticity formulation of Euler's equations -^+«Vw = 0, -A* = w, m = VA* in fi X 10, 7T, (1) J 3/ J ' L' u(t = 0) = 2 to,°5(x x,), *|r = *0, where 8 is the Dirac function, T, the boundary of fi, is integrated by (2) ■ This method was first implemented by Christiansen [6] and Chorin [5] and thoroughly tested by Baker [2] on the roll up of vortex sheets. From the theoretical point of view if fi = R2 Hald [10] showed that when (2) is discretized explicitly in time, when the Dirichlet problem is approximated by a suitable discretization of the corresponding Green's function and when the Dirac functions are smoothed by appropriate convolutions then the method converges. In Baker [2] the Dirichlet problem is discretized with finite differences, and as far as we know the convergence is not established in such a case. The present work is based on the rather straightforward observation that the system (2) is perhaps easier to analyze when it is discretized by the finite element Received May 1, 1980; revised June 30, 1980. 1980 Mathematics Subject Classification. Primary 65N30, 65N35; Secondary 76C05. © 1981 American Mathematical Society 0025-571 8/8 l/0000-OO09/$O5.50 119 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 120 CLAUDE BARDOS, MICHEL BERCOVIER AND OLIVIER PIRONNEAU method than by the two previously mentioned methods because the equation for the characteristics x,(r) can be integrated exactly if VA* is piecewise constant on a triangulation of fi X ]0, T[. However, the error analysis shows that in the finite element context it is no longer feasible to work with Dirac functions; it is better to use a piecewise constant discretization of w°(x). Therefore, we shall not work with point vortices but with a piecewise constant approximation of the vortex field (3) «(*, 0 = 2 «?(,)/(* *,(')) 1 = 1 where I(x x¡(t)) equals 1 if x and x,(r) belong to the same element of the triangulation and zero otherwise, where TV is the total number of elements and where j(i) is the index of the element to which x,(0) belongs. Therefore we will have to compute certain characteristics backward in t in order to define w(x, t) by (3). Thus, although in spirit identical, in practice the present method is substantially different from the point vortex method of [2] and [5]. Both have the advantage of being nondissipative; ours is conservative in a statistical sense only in terms of «°. On the other hand, we do not have to insert new vortices in some regions of the flow as in [6] and the method is more appropriate to smooth flows. But most of all an error analysis will be given and the method is unconditionally stable in time. This, by the way, may also be true of the cloud and cell vortex method [5], [2] as was observed by Baker. The proofs are involved and difficult in their details but the guidelines are simple: we assume that the regularity obtained by Wolibner [14] and Kato [11] for the solution holds. Thus, to measure the error between the exact solution and the appropriate solution we measure the distance between two particles, one transported by the exact flow and the other by the approximate flow. In the process L°° estimates of the finite element solution of the Dirichlet problem for -A will have to be established following the arguments of Nitsche [12]. For the sake of clarity and also because the method of characteristics in the finite element context can be useful for other hyperbolic systems, we begin with a presentation of the method for the transport equation. Then, in Section 2 the method and the error analysis is explained for the two-dimensional Euler equation. Finally, the numerical implementation and some numerical tests are presented in Section 3. 1. Finite Elements and Characteristics for the Transport Equation. 1. a. Statement of the Problem. Let fi be a bounded open set of R\", let Q = Q X ]0, T[, and u a divergence free (V-m = 0) vector of (H\\Q) n L°°(Q))n. Let/and p0 be two functions of LX(Q) and consider the problem , v ^+ mVp = / in fi x 10, 7T = Q, (1.1) 9r H J J ' L *' p(x, t) = p0(x, t) for all (x, r} e 5 = fi X (0} u 2\", where T is the boundary of fi, v is outward normal and 2~ = {{x, t): u(x, t) ■ v < 0, x e T}. 2\" represents the part of the boundary of Q where the flow enters into fi; thus License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use THE VORTEX METHOD WITH FINITE ELEMENTS 121 the boundary conditions for p are given at initial time and when the velocity u enters into fi. Several physical phenomena are governed by this equation, known as the transport equation. On rectangular domains fi (1.1) is easily discretized by any upwind finite difference scheme, but if fi is complicated there is no simple nondissipative finite element scheme, and the method of characteristics is usually considered as an expensive numerical method. Let us show that for first order accurate discretizations this is not so costly. The method of characteristics is based upon the following observation: given {x, /)EÖ define [Xx''(t), t} by dX \\u(X(r),r) if*(r)efi, (1.2) ^=l0 ifX^fi, Vre]0,r[, X(t) = x. If u is uniformly Lipschitz continuous with respect to x, (1.2) has a unique solution on ]0, r[; then we define (1.3) p(x, t) = Po(X*'(0), 0) + f7(X*-'(t),t) dr and claim that p is a solution of (1.1). If the data u, p0 and/ are not smooth the proof is difficult [15], but if p0,/and u are in C'(£?)> men p is differentiable with respect to {x, r} and p(Xx''(r), t) p(x, 0 \"§(t 0 + VxP(X*''(t) x) (1.4) + o(t t) + o(X x) dp and also -(^ + «vj^t o + (X*>'(r), t) p(x, i) = Cf(Xx'(a), a) da. Jt Therefore we have the following result: Proposition 1. When u is uniformly Lipschitz continuous in x and divergence free and p0, m and f are in LX(Q), then the solution o/ (1.1) is given by (1.2), (1.3). Remark. Equation (1.2) includes the possibility that a characteristic leaves fi before t = 0 with the convention that/|r = 0. l.b. Discretization. To discretize (1.1) we choose a triangulation <5h of fi made of nonoverlapping triangles if n = 2 or tetrahedra if n = 3 with the usual properties [4]: \\ = kir-, (1 5) ri n ry = 0 or one vertex, or one side (or face), (or one edge), N U t, = Q,h c fi, distance (fiA, fi) = 0(h2), h = size of largest side of %. i License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 122 CLAUDE BARDOS, MICHEL BERCOVIER AND OLIVIER PIRONNEAU If Ar denotes a time step then Q is approximated by (1.6) Q„ = U Pv; Py = T, x ]jAt, (j + l)Ar[, j 1.M £(7/Ar). Instead of u we shall approximate the stream function * of u, i.e. the function such that _/9*3 9*2 9*, 9*3 9*2 _ 9*, \\ y 9x2 9x3 ' 9x3 9x, ' 9x, 9x2 / (L7) =[— — ) if« = 2 \\ 9x2' 9X| / Let *A be an approximation of * in the space Hh: (1.8) Hh = [rjpA: