Showing papers on "Biharmonic equation published in 1991"

Postprocessing schemes for some mixed finite elements

Abstract: — We consider some mixed finiie element methods for scalar second and fourth order elïiptic équations. For these methods we introducé and analyze some new postprocessing schemes. It is shown that by a simple postprocessing, performed separately on each element, one can obtain a considerably better approximation for the scalar variable than the original one. Resumé. — Nous considérons quelques méthodes d'éléments finis mixtes pour des équations aux dérivées partielles scalaires, elliptiques, du second ou du quatrième ordre* Pour ces méthodes, nous introduisons et analysons quelques techniques nouvelles de postraitement. On montre qu'un postraitement simple, effectué séparément sur chaque élément, permet d'obtenir une approximation bien meilleure sur la variable scalaire.

Area integral estimates for the biharmonic operator in Lipschitz domains

Abstract: Let DCR\" be a Lipschitz domain and let « be a function biharmonic in D , i.e., AAu = 0 in D . We prove that the nontangential maximal function and the square function of the gradient of u have equivalent Lp(dp) norms, where dp e A°°(do) and da is surface measure on 3D. Introduction In this paper we prove area integral and nontangential maximal function inequalities for functions biharmonic in a bounded Lipschitz domain D in R\" . Specifically, if u satisfies A2« = 0 in D CRn , then these functionals applied to Vu (the gradient of u) have comparable Lp(dp) norm (0

Pointwise and spectral control of plate vibrations.

Abstract: We consider the problem of controlling pointwise (by means of a time dependent Dirac measure supported by a given point) the motion of a vibrating plate O. Under general boundary conditions, including the special cases of simply supported or clamped plates, but of course excluding the cases where multiple eigenvalues exist for the biharmonic operator, we show the controlability of finite linear combinations of the eigenfunctions at any point of O where no eigenfunction vanishes at any time greater than half of the plate area. This result is optimal since no finite linear combination of the functions other than 0 is pointwise controllable at a time smaller than the plate's area. Under the same condition on the time, but for any domain O in R2, we solve the problem of internal spectral control, which means that for any open disk ? I O, any finite linear combination of eigenfunctions can be set to equilibrium by means of a control function h I D((0,T) x O) supported in (0,T) x ?.

A stabilized treatment of the biharmonic operator with spectral methods

Abstract: For the discretization of the biharmonic operator with spectral methods a high condition number like $O(N^8 )$, where N is the maximal degree of polynomials, occurs. Two different techniques are proposed for reducing this condition number to $O(N^4 )$. The best technique seems to be a splitting of the equation with the biharmonic operator into a system of two equations with the Laplace operator. An effective way of preconditioning is introduced. An effective iterative solver is presented.

Stabilization techniques for spectral methods

Abstract: We introduce a stabilized treatment of spectral methods. The condition number of the spectral systems is highly improved. Elliptic and biharmonic problems are considered. Suitable interpolants in the case of inhomogeneous Dirichlet boundary conditions are presented. For a direct solver the improvements with respect to rounding error propagation are numerically demonstrated.

Special Methods for Plate Analysis

Abstract: Special boundary element methods for the solution of plate bending problems are presented in this chapter. Some of these methods have been developed during the first efforts to apply the boundary integral equation method to the plate bending problem and before the appearance of general direct BEM formulations in the literature. Other special BEM’s have been developed later in order to overcome certain shortcomings and computational difficulties in the general direct BEM formulations or in order to solve problems for which the fundamental solution can not be established or is difficult to treat numerically. On the basis of common characteristics the special methods are classified in five groups: (a) Methods based on the biharmonic analysis, (b) Indirect boundary element methods (IBEM’s) (c) Boundary differential — integral equation methods (BDIEM’s) (d) Green’s function methods (GFM’s), (e) Other methods which lack common characteristics. For each group of special methods a separate section is devoted. The special methods are described by the most representative methods of each group. The description of each special method is supplemented by several numerical results which are compared with those obtained by analytical or other numerical methods in order to illustrate the versatility, the effectiveness and the accuracy of the special BEM’s. Concluding remarks are presented in the last section.

An approximate inverse based multigrid approach to the biharmonic problem

Abstract: The approximate inverse based multigrid algorithm FAPIN for the solution of large sparse linear systems of equations is examined. This algorithm has proven successful in the numerical solution of several second order boundary value problems. Here we are concerned with its application to fourth order problems. In particular, we demonstrate good multigrid performance with discrete problems arising from the biharmonic (plate) equation. The work presented also represents new experience with FAPIN using bicubic Hermite basis functions. Central to our development is the concept of an approximate inverse of a matrix. In particular, we use a least squares approximate inverse found by solving a Frobenius matrix norm minimization problem. This approximate inverse is used in the multigrid smoothers of our algorithm FAPIN. The algorithms presented are well suited for implementation on hypercube multiprocessors.

The simple layer potential for the biharmonic equation in \( n \) variables

Abstract: — A theory of the «simple layer potential» for the classical biharmonic problem in R" is worked out. This hinges on the study of a new class of singular integral operators, each of them trasforming a vector with n scalar components into a vector whose components are n differential forms of degree

An H o m interpolation result

Abstract: This paper presents a proof of an interpolation result related to the approximation theory for higher-order finite element or spectral methods when $C^1$ (or higher) regularity is convenient for the finite-dimensional subspaces. This can be a natural choice, for example, for the Stokes problem, the biharmonic problem, or higher-order plate and shell models. It is shown that the same intermediate spaces are obtained whether one (1) interpolates between two Sobolev spaces defined on a domain with nonsmooth boundary first and then enforces the homogeneous boundary conditions afterwards or (2) interpolates between two Sobolev spaces where the homogeneous boundary conditions are enforced throughout the interpolation process.

Solution of the plane stress problems of strain-hardening materials described by power-law using the complex pseudo-stress function

Abstract: In the present paper, the compatibility equation for the plane stress problems of power-law materials is transformed into a biharmonic equation by introducing the so-called complex pseudo-stress function, which makes it possible to solve the elastic-plastic plane stress problems of strain hardening materials described by power-law using the complex variable function method like that in the linear elasticity theory. By using this general method, the close-formed analytical solutions for the stress, strain and displacement components of the plane stress problems of power-law materials is deduced in the paper, which can also be used to solve the elasto-plastic plane stress problems of strain-hardening materials other than that described by power-law. As an example, the problem of a power-law material infinite plate containing a circular hole under uniaxial tension is solved by using this method, the results of which are compared with those of a known asymptotic analytical solution obtained by the perturbation method.

A boundary approximation method for fourth order problems

Abstract: We study approximate solutions of the biharmonic problem ∆2u=0, by a boundary approximation method for a class of given boundary conditions. We prove an O(n−r) error bound (in the space L2(Ω)) for the solution u belonging to Hr(Ω).

An analytical method for a mixed boundary value problem of circular plates under arbitrary lateral loads

Abstract: In this paper by using the concept of mixed boundary functions, an analytical method is proposed for a mixed boundary value problem of circular plates. The trial functions are constructed by using the series of particular solutions of the biharmonic equations in the polar coordinate system. Three examples are presented to show the stability and high convergence rate of the method.

ANALYTIC SOLUTION TO THE SCHRDINGER EQUATION WITH A HARMONIC OSCILLATOR POTENTIAL PLUS δ-POTENTIAL

Abstract: The analytic solution to the Schrodinger equation with a harmonic oscillator potentialplus δ-potential is presented in the coordinate representation. Its algebraic expression in theenergy representation is also given. The degeneracy of the energy levels is discussed and acomparison between the energy spectra of a harmonic oscillator and a particle in a δ-poten-tial is made.

Boundary Elements and Perturbation Theory for Vibrating Plates

Abstract: In this work, classical techniques from perturbation theory will be applied to develop a boundary integral formulation for low frequency forced vibrations of elastic plates The functional form of the applied load and the plate deflection are assumed to be products of a temporal function and corresponding spatial functions The separation of variables approach removes the difficulties associated with transient analysis The resulting boundary element formulation requires consecutive solutions to a set of coupled non-homogeneous biharmonic equations The domain integral normally associated with each non-homogeneous equation is transformed to a set of boundary integrals using the Rayleigh-Green identity Numerical solutions of the perturbation-based expansion equations of forced vibrations using the boundary element method (BEM) are presented and compared with analytical analysis

A Stabilized Treatment of the Biharmonic Operator with Spectral Methods.

Abstract: For the discretization of the biharmonic operator with spectral methods a high condition number like $O(N^8 )$, where N is the maximal degree of polynomials, occurs. Two different techniques are proposed for reducing this condition number to $O(N^4 )$. The best technique seems to be a splitting of the equation with the biharmonic operator into a system of two equations with the Laplace operator. An effective way of preconditioning is introduced. An effective iterative solver is presented.

On the Character of the Continuity, on the Boundary of a Nonsmooth Region, of a Generalized Solution of the Dirichlet Problem for the Biharmonic Equation

Abstract: This article is devoted to a study of the boundary behavior of solutions of the Dirichlet problem for the biharmonic equation in arbitrary plane regions.

Moving Boundary Free Flow with Viscosity by BEM

Abstract: Various moving boundary flows (with undetermined boundaries a priori) are of great interest in recent decades Both steady & unsteady cases for potential flows with one or two free surfaces have been solved using BEM by author as well as the interaction between the flow and structures In this paper, the flow is considered to be viscous or rotational so that the governing equation becomes biharmonic one It is shown that the biharmonic equation could be formulated as the combination of a Laplace equation and a Poisson one In such a way, the BEM discretized formulation will be similar to that of potential flow except the complicated boundary conditions which become difficult even to fit in an iterative scheme Nevertheless, the numerical implementation of its BEM equation in this paper could still be compiled on a PC-computer with lower cost