Showing papers on "Extended finite element method published in 1978"
Book•
01 Jan 1978TL;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
TL;DR: The main theorem gives an error estimate in terms of localized quantities which can be computed approximately, and the estimate is optimal in the sense that, up to multiplicative constants which are independent of the mesh and solution, the upper and lower error bounds are the same.
Abstract: A mathematical theory is developed for a class of a-posteriors error estimates of finite element solutions. It is based on a general formulation of the finite element method in terms of certain bilinear forms on suitable Hilbert spaces. The main theorem gives an error estimate in terms of localized quantities which can be computed approximately. The estimate is optimal in the sense that, up to multiplicative constants which are independent of the mesh and solution, the upper and lower error bounds are the same. The theoretical results also lead to a heuristic characterization of optimal meshes, which in turn suggests a strategy for adaptive mesh refinement. Some numerical examples show the approach to be very effective.
1,431 citations
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
TL;DR: In this article, a discontinuous collocation-finite element method with interior penalties was proposed and analyzed for elliptic equations, motivated by the interior penalty L 2-Galerkin procedure of Douglas and Dupont.
Abstract: A discontinuous collocation-finite element method with interior penalties is proposed and analyzed for elliptic equations. The integral orthogonalities are motivated by the interior penalty L2- Galerkin procedure of Douglas and Dupont.
787 citations
343 citations
TL;DR: In this paper, a classification of mathematical commonly encountered in connection with solution of non-linear finite element problems is presented, and the principal methods for numerical solution of the nonlinear equations are surveyed and discussed.
Abstract: The paper presents a classification of mathematical commonly encountered in connection with solution of non−linear finite element problems. The principal methods for numerical solution of the non−linear equations are surveyed and discussed. Special emphasis is placed upon the description of an automatic load incrementation procedure with equilibrium iterations. It is shown how this algorithm can be adapted for solving problems involving instabilities, snap−through and snap−back. A simple scalar quantity denoted the current stiffness parameter is suggested; this parameter is used to characterize the overall behaviour of non−linear problems. It can also be used as a steering parameter in the solution process. The use of the present technique is illustrated by several examples.
267 citations
257 citations
Book•
01 Jan 1978255 citations
TL;DR: In this paper, a new high-accuracy finite element for thin and thick plate bending is developed, based upon Mindlin plate theory, which exhibits improved characteristics in comparison with the 8-node serendipity, or the 9-node Lagrange elements.
219 citations
213 citations
TL;DR: In this paper, the mixed finite element approximation of variational inequalities is studied, taking as model problems the so-called "obstacle problem" and "unilateral problem".
Abstract: We study the mixed finite element approximation of variational inequalities, taking as model problems the so called "obstacle problem" and "unilateral problem" Optimal error bounds are obtained in both cases
TL;DR: In this article, an economical method is derived to remove this singularity and which also produces accurate flexural response for rectilinear element geometry, which is equivalent to the incompatible model element of Wilson et al.
Abstract: Numerical codes which use a one-point quadrature integration rule to calculate stiffness matrices for the 2-D quadrilateral element and the 3-D hexahedral element, produce matrices which are singular with respect to a number of displacement patterns, other than the rigid body patterns. In this paper an economical method is derived to remove this singularity and which also produces accurate flexural response. For rectilinear element geometry the method is equivalent to the incompatible model element of Wilson et al.7 For non-rectilinear element geometry a slight modification of the scheme is required in order to assure that it passes the patch test. The method of this paper can also be used in finite difference codes which experience similar difficulties.
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.
TL;DR: In this article, a modified version of the variational method of Trefftz was presented for plate bending in which the coordinate functions satisfying the nonhomogeneous Lagrange equation were defined on subregions.
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.
TL;DR: In this paper, the dynamics of a viscous layer of fluid (salt) moving bouyantly through another viscous fluid of another viscosity and density (sedimentary strata) is modeled by means of a special finite element model.
TL;DR: In this paper, the authors extended the finite element method to direct calculation of combined modes I and II stress intensity factors for axisymmetric and planar structures of arbitrary geometry and loading.
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.
TL;DR: In this article, a finite strip method is presented for the post-locally-buckled analysis of prismatic thin walled structures under end compression, and sample problems are analyzed and the economy of the method compared with the finite element method is demonstrated.
TL;DR: In this article, a numerical solution of the Navier-Stokes equations in the form of a single, fourth order differential equation for stream function and the boundary conditions are replaced by an equivalent variational principle.
TL;DR: In this paper, an eight node isoparametric finite element is used to represent a rigid porous absorbing material and tests on an assembly of these elements for a one dimensional model gave good agreement with an exact solution for the input impedance.
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.
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.
TL;DR: A viscous approximation for steady creeping flow is extended to include the effect of elastic strains to illustrate both the method and the need for including elastic strains in the analysis of steady-state visco-plastic flows.
Abstract: A viscous approximation for steady creeping flow is extended to include the effect of elastic strains. Two examples are presented which illustrate both the method and the need for including elastic strains in the analysis of steady-state visco-plastic flows.
TL;DR: In this article, a finite element method was developed for the study of transmission of sound in non-uniform ducts without flow, and the formulation is based on a weighted residual approach and eight noded isoparametric elements are used.
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.
TL;DR: In this paper, the authors extended the finite element method using a singular element near the crack tip to the elastodynamic problems of cracks where the displacement function of the singular element is taken from the solution of a propagating crack.
Abstract: The finite element method using a singular element near the crack tip is extended to the elastodynamic problems of cracks where the displacement function of the singular element is taken from the solution of a propagating crack. The dynamic stress intensity factor for cracks of mode III or mode I deformations in a finite plate is determined. The results of computation for stationary cracks or propagating cracks under dynamic loadings are compared with the analytical solutions of other authors. It is shown that the present method satisfactorily describes the time variation of the stress intensity factor in dynamic crack problems.
TL;DR: In this paper, a finite strip method is presented for the stability analysis of prismatic thin walled structures under arbitrary loading, and sample problems are analyzed and the economy of the method compared with the finite element method is demonstrated.