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Showing papers on "Smoothed finite element method published in 1978"


Book
01 Jan 1978
TL;DR: The finite element method has been applied to a variety of nonlinear problems, e.g., Elliptic boundary value problems as discussed by the authors, plate problems, and second-order problems.
Abstract: Preface 1. Elliptic boundary value problems 2. Introduction to the finite element method 3. Conforming finite element methods for second-order problems 4. Other finite element methods for second-order problems 5. Application of the finite element method to some nonlinear problems 6. Finite element methods for the plate problem 7. A mixed finite element method 8. Finite element methods for shells Epilogue Bibliography Glossary of symbols Index.

8,407 citations


Journal ArticleDOI
TL;DR: In this article, a-posteriori error estimates for finite element solutions are derived in an asymptotic form for h 0 where h measures the size of the elements.
Abstract: Computable a-posteriori error estimates for finite element solutions are derived in an asymptotic form for h 0 where h measures the size of the elements. The approach has similarity to the residual method but differs from it in the use of norms of negative Sobolev spaces corresponding to the given bilinear (energy) form. For clarity the presentation is restricted to one-dimensional model problems. More specifically, the source, eigenvalue, and parabolic problems are considered involving a linear, self-adjoint operator of the second order. Generalizations to more general one-dimensional problems are straightforward, and the results also extend to higher space dimensions; but this involves some additional considerations. The estimates can be used for a practical a-posteriori assessment of the accuracy of a computed finite element solution, and they provide a basis for the design of adaptive finite element solvers.

1,211 citations



Book
01 Jan 1978

255 citations



Journal ArticleDOI
TL;DR: In this article, the authors considered the model Dirichlet problem on a plane polygonal domain and derived the rate of convergence estimates in the maximum norm, up to the boundary, are given locally.
Abstract: The finite element method is considered when applied to a model Dirichlet problem on a plane polygonal domain. Rate of convergence estimates in the maximum norm, up to the boundary, are given locally. The rate of convergence may vary from point to point and is shown to depend on the local smoothness of the solution and on a possible pollution effect. In one of the applications given, a method is proposed for calculating the first few coefficients (stress intensity factors) in an expansion of the solution in singular functions at a corner from the finite element solution. In a second application the location of the maximum error is determined. A rather general class of non-quasi-uniform meshes is allowed in our present investigations. In a subsequent paper, Part 2 of this work, we shall consider meshes that are refined in a systematic fashion near a corner and derive sharper results for that case. 0. Introduction. Let Q2 be a bounded simply connected domain in the plane with boundary U2 consisting of a finite number of straight line segments meeting at vertices v;, j = 1, . . . , M, of interior angles 0 < oil S < aM < 27r (in a suitable ordering). We shall consider the Dirichlet problem -Au = f in Q2, (0.1) u= 0 on Q, where f is a given function, which for simplicity we assume to be smooth. To solve the problem (0.1) numerically, let Sh = Sh(2), 0 < h < 1, denote a one-parameter family of finite dimensional subspaces of H1 (2) n w2 (Q). We have in mind piecewise polynomials of a fixed degree on a sequence of partitions of Q2. In our considerations the partitions do not have to be quasi-uniform, not even locally (cf. examples in Section 9). Let uh E Sh be the approximate solution of (0.1) defined by the relation (0.2) A(uh, X) = (f X) for all X Esh. Here A(v, w) = fI Vv VW dx, and (v, w) = fvw dx. We wish to obtain local estimates up to the boundary in the maximum norm for the error u uh. Although our present assumptions allow meshes that are refined near a corner, in the subsequent paper, Part 2, we shall investigate the error in more detail in that case, and obtain sharper results. The general results derived in the present papei will be essential in those investigations. Received April 25, 1977. AMS (MOS) subject classifications (1970). Primary 65N30, 65N15. *This work was supported in part by the National Science Foundation. Copyright i 1978, American Mathematical Society

173 citations


Journal ArticleDOI
TL;DR: In this paper, a detailed analysis of a superconvergence phenomenon for the gradient of approximate solutions to second-order elliptic boundary value problems is given for the Serendipity family of curved isoparametric elements.
Abstract: The finite elements considered in this paper are those of the Serendipity family of curved isoparametric elements. There is given a detailed analysis of a superconvergence phenomenon for the gradient of approximate solutions to second order elliptic boundary value problems. An approach is proposed how to use the superconvergence in practical computations.

168 citations


Journal ArticleDOI
TL;DR: In this article, a modified quadrature formula is used to numerically integrate discontinuous functions in the expression of the stiffness matrix, which makes it possible to match one element with two elements side-by-side.
Abstract: Formulation of a transition element is presented. Such an element makes it possible to match one element with two elements side-by-side. A modified quadrature formula is used to numerically integrate discontinuous functions in the expression of the stiffness matrix. Numerical examples show the applicability of the element.

139 citations


Journal ArticleDOI
TL;DR: The strain energy release rate (G) converges rapidly in finite element approximations in which the finite element mesh is fixed and the order of polynomial displacement interpolations (p) is increased.
Abstract: The strain energy release rate (G) converges rapidly in finite element approximations in which the finite element mesh is fixed and the order of polynomial displacement interpolations (p) is increased. Numerical experiments indicate that the error inG is very closely estimated, even for small pand very coarse finite element meshes, by an expression of the form k (NDF)-1 in which k is a mesh dependent constant and NDF is the number of degrees-of-freedom. The method provides for very efficient and accurate computation of G without the use of special techniques.

114 citations


01 Mar 1978
TL;DR: Comprehensive survey of past and current literature on geometrically-nonlinear finite elements is organized into handbook form and serves as valuable reference when solving problems in nonlinear structural mechanics.
Abstract: Comprehensive survey of past and current literature on geometrically-nonlinear finite elements is organized into handbook form and serves as valuable reference when solving problems in nonlinear structural mechanics. Handbook provides rapid access to wide variety of element types and facilitates evaluation of different elements as to their features, probable accuracy, and complexity.

80 citations


Journal ArticleDOI
TL;DR: In this article, the Lax-Wendroff finite difference method was used for the analysis of the Tokachi-oki Earthquake tsunami problem and compared with the tide gauge records.
Abstract: Numerical analysis of tsunamis applying the finite element method is presented based on the shallow water wave equation. To discretize time, a two step explicit method is used. The scheme is the extension of the Lax-Wendroff finite difference method. The present finite element method is used for the analysis of the Tokachi-oki Earthquake tsunami problem and compared with the tide gauge records. The conclusion of this paper is that the present method is suitable for the prediction of the tsunami wave propagation problem.

Journal ArticleDOI
TL;DR: In this paper, a method of numerical calculation is developed for predicting two-dimensional shape changes at a cathode during electrodeposition using finite element methods to obtain the secondary potential field distribution in an electrolysis cell.
Abstract: A method of numerical calculation is developed for predicting two‐dimensional shape changes at a cathode during electrodeposition. The calculation uses finite element methods to obtain the secondary potential field distribution in an electrolysis cell. The cathode shape initially consists of parallel metal strips which are separated by, and coplanar with, insulating strips; the anode is at a fixed distance from the cathode. Transient numerical calculations provide a complete time history of cathode shape during deposition. Results are obtained in order to compile dimensionless shape change dependence on coulombs passed, polarization parameter, applied potential, and initial cathode shape.

Journal ArticleDOI
TL;DR: The design of wire-coating dies is described using finite element numerical analysis as a guide and two basic die geometries are examined and a new design which eliminates recirculation within the die is proposed.
Abstract: The design of wire-coating dies is described using finite element numerical analysis as a guide. Finite elements are able to accommodate awkward geometries and non-Newtonian fluid properties in a realistic manner, and produce streamline and stress patterns within the die. Two basic die geometries are examined and a new design which eliminates recirculation within the die is proposed.

Journal ArticleDOI
TL;DR: Several competitive numerical integration techniques for nonlinear dynamic analysis of structures by the finite element method are examined and compared for a plane stress problem in this paper, and it is concluded that the central difference predictor is the best, whereas the performances of the other two methods are about equal.
Abstract: Several competitive numerical integration techniques for nonlinear dynamic analysis of structures by the finite element method are examined and compared for a plane stress problem. Both material and geometric nonlinearities are included in the finite element formulation. Three explicit methods are investigated: (1)Central difference predictor; (2)two-cycle iteration with the trapezoidal rule; and (3)fourth-order Runge-Kutta method. A nodewise solution technique is generated at any state in the analysis. Three implicit methods also are studied: Newmark-Beta method, Houbolt method, and Park's stiffly-stable method. Among the three explicit methods, it is concluded that the central difference predictor is the best, whereas the performances of the other two methods are about equal. The three implicit approaches are nearly equal, but Park's stiffly-stable method is somewhat better than the Newmark-Beta method, and Houbolt's procedure must be rated third.

Journal ArticleDOI
15 Sep 1978
TL;DR: In this article, the finite element method is applied to collinear reactive scattering problems, where no basis set expansion of the wave function is required and a direct solution of the two-dimensional partial differential equation is achieved.
Abstract: The finite element method is applied to collinear reactive scattering problems. In this way no basis set expansion of the wave function is required and a direct solution of the two-dimensional partial differential equation is achieved. It is shown how to generally formulate this approach and achieve fast and accurate results. As a test calculation the method was applied to H + H2, yielding excellent agreement with close coupling results. Since no basis sets are used in the finite element calculation, no question of basis set convergence or closed channel behavior arises. Some discussion on applications to higher dimensions is also included.


Journal ArticleDOI
TL;DR: In this article, an incremental hybrid stress finite-element model, based on an incremental complementary energy principle involving both the incremental Piola-Lagrange stress and an incremental rotation tensor which leads to discretization of rotational equilibrium equations, is presented.
Abstract: The possibility of deriving a complementary energy principle, for the incremental analysis of finite deformations of nonlinear-elastic solids, in terms of incremental PiolaLagrange (unsymmetric) stress alone, is examined. A new incremental hybrid stress finite-element model, based on an incremental complementary energy principle involving both the incremental Piola-Lagrange stress, and an incremental rotation tensor which leads to discretization of rotational equilibrium equations, is presented. An application of this new method to the finite strain analysis of a compressible nonlinear-elastic solid is included, and the numerical results are discussed.

Journal ArticleDOI
TL;DR: In this paper, a sparse positive definite system of equations arising from the use of the finite element method to solve a two-dimensional boundary value problem is defined, and a common method of solving these equations is to use a finite element approach.
Abstract: Let $Ax = b$ be a sparse positive definite system of equations arising from the use of the finite element method to solve a two dimensional boundary value problem. A common method of solving these ...


Journal ArticleDOI
TL;DR: In this article, a finite element method is presented for predicting the coupled structural-acoustic response of a flexible cylinder, which contains an acoustic medium and is excited by mechanical forces.
Abstract: A finite element method is presented for predicting the coupled structural-acoustic response of a flexible cylinder, which contains an acoustic medium and is excited by mechanical forces The cylinder is represented by an existing axisymmetric, cylindrical shell element The acoustic space inside the cylinder is modelled using a new axisymmetric, acoustic ring element The cross-secton of the ring takes the form of an eight node, isoparametric element The coupled equations of motion for the cylinder and acoustic field are solved using modal analysis techniques The numerical results obtained are compared with results from an experimental investigation which is also described


Journal ArticleDOI
TL;DR: In this article, a finite element package is presented that is able to treat two-dimensional Schroedinger equation problems over a finite region with an arbitrary potential and homogeneous boundary conditions.


Journal ArticleDOI
TL;DR: In this paper, sufficient conditions under which a map whose domain is a compact set is a bijection onto a given set are given, and an algorithm based on elimination is given for the numerical inversion of these maps.
Abstract: This paper contains sufficient conditions under which a map whose domain is a compact set is a bijection onto a given set. Relative to certain isoparametric flnite element maps, one set of conditions involves the nonvanishing of the Jacobian; another the notion of overspill. An algorithm based on elimination is given for the numerical inversion of these maps.


01 Oct 1978
TL;DR: In this article, a posteriori estimators of error in strain energy were examined on the basis of a typical problem in linear elastic fracture mechanics, and two estimators were found to give close upper and lower bounds for the strain energy error.
Abstract: The results of numerical experiments are presented in which a posteriori estimators of error in strain energy were examined on the basis of a typical problem in linear elastic fracture mechanics. Two estimators were found to give close upper and lower bounds for the strain energy error. The potential significance of this is that the same estimators may provide a suitable basis for adaptive redistribution of the degrees of freedom in finite element models.

Journal ArticleDOI
TL;DR: In this paper, a variable domain finite element method is presented for the solution of problems in gravity flow with free surfaces, which is applied to flow over spillway and gives results that agree extremely well with experimental data.


Journal ArticleDOI
TL;DR: A finite element recursion technique is described for the solution of open-boundary electric and magnetic field problems by an annular "super-element" in which all the nodes except those on the inside edge have been eliminated.
Abstract: A finite element recursion technique is described for the solution of open-boundary electric and magnetic field problems. The method is to model the exterior region by an annular "super-element" in which all the nodes except those on the inside edge have been eliminated. The routine to generate the annulus and its element division automatically is detailed. This algorithm calculates the boundary coefficients representing the exterior Laplacian region from a knowledge of the interior region boundary node distribution. The method has been tested by comparing calculated solutions with results obtained from both analytical and experimental models and good agreement achieved.

Journal ArticleDOI
TL;DR: In this paper, the application of the finite element method to a variety of acoustics problems with arbitrary boundary shapes and impedances is examined, and the governing equations and boundary conditions are established in a variational format to include permeable membranes and boundary forcing functions.