Showing papers on "Biharmonic equation published in 1973"
TL;DR: In this article, it was shown that the stiffness matrix produced by thin elastic plates becomes violently ill-conditioned as the thickness of the structure is reduced, and that the factor 1/t 2 causing this illconditioning can be removed from the stiffness matrices and consequently from its condition number by relating the thickness t to the diameter of the element h, without losing the rate of convergence provided by the degree of the shape functions inside the element.
73 citations
TL;DR: In this article, it was shown that the solution of the biharmonic variational inequality has bounded second derivatives provided that the obstacle and the data are smooth, and the second derivative was shown to be bounded for the case where the data and the obstacle are smooth.
Abstract: It is shown that the solution of the biharmonic variational inequality has bounded second derivatives provided that the obstacle and the data are smooth.
67 citations
TL;DR: In this paper, the authors considered the Green's function for the biharmonic equation in an infinite angular wedge and showed that if the angle a is less than ai Ä 0.81277, then the green's function does not remain positive; in fact it oscillates an infinite number of times near zero and near °°.
Abstract: The Green's function for the biharmonic equation in an infinite angular wedge is considered. The main result is that if the angle a is less than ai Ä 0.81277, then the Green's function does not remain positive; in fact it oscillates an infinite number of times near zero and near °°. The method uses a number of transformations of the problem including the Fourier transform. The inversion of the Fourier transform is accomplished by means of the calculus of residues and depends on the zeros of a certain transcendental function. The distribution of these zeros in the complex plane gives rise to the determination of the angle a. A general expression for the asymptotic behavior of the solution near zero and near infinity is obtained. This result has the physical interpretation that if a thin elastic plate is deflected downward at a point, the resulting shape taken by the plate will have ripples which protrude above the initial plane of the plate. blem
28 citations
TL;DR: Two methods for solving the biharmonic equation are compared, one method is direct, using eigenvalue-eigenvector decomposition, and the other method is iterative, solving a Poisson equation directly at each iteration.
Abstract: Two methods for solving the biharmonic equation are compared. One method is direct, using eigenvalue-eigenvector decomposition. The other method is iterative, solving a Poisson equation directly at each iteration.
27 citations
TL;DR: In this paper, the problem of completeness of a system of elementary solutions in the space of biharmonic functions with finite energy is investigated and the necessary and sufficient conditions are formulated for the boundary values which ensure that the solution belongs to the energy space.
21 citations
TL;DR: In this paper, it was shown that the Green's function has no nodal lines for the annular region with inner radius Y and outer radius 1, and that the principal eigenfunction has a diametric nodal line if Y is small enough (y. < l/71 5).
18 citations
15 citations
Journal Article•
TL;DR: In this article, the existence or nonexistence of bounded nonharmonie biharmonic functions on a non-compact Riemannian manifold was studied and conditions for the existence of such functions with given boundedness properties were compared.
Abstract: Albhough the existence of nonharmonie biharmonic functions on a noncompact Riemannian manifold is always assured, the same is not necessarily true if various boundedness properties are imposed upon them. For this reason it may be interesting to find conditions for the existence of nonharmonie biharmonic functions with given boundedness properties and to compare these conditions for various such properties. As a first step of this biharmonic classification problem we studied in Nakai-Sario [6] (see also O'Malla [8], Nakai [4]) the existence of Dirichlet finite biharmonic functions and showed, among other things, that this existence has no relation to the parabolicity or hyperbolicity of the manifold. In the present paper we will discuss the existence of bounded nonharmonie biharmonic functions. As in the case of Dirichlet finite biharmonic functions, the existence or nonexistence of bounded nonharmonie biharmonic functions turns out to be unrelated to the parabolicity or hyperbolicity of the base manifold. We also show that the existence of Dirichlet finite biharmonic functions has no relation to that of bounded nonharmonie biharmonic functions. In particular the former does not necessarily imply the existence of Dirichlet finite bounded nonharmonie biharmonic functions. Therefore, s is to be expected, there is no analogy with the case of harmonic functions.
14 citations
13 citations
01 Jan 1973
TL;DR: In this article, the authors discuss finite elements with harmonic interpolation functions and the possibility of solving bi-harmonic problems and also a nonlinear system as in the elasto-plastic problem.
Abstract: Publisher Summary This chapter discusses finite elements with harmonic interpolation functions. The basis of all finite element solutions of partial differential equations must be the satisfaction of three types of condition for best fit, namely, in the element interior on the interfaces, and on the external boundary of the entire region considered. It is pointed out that these conditions can be satisfied exactly, or in the mean, and that they are independent. Convergence of a solution may be obtained either by using the same type of element and increasing the number of elements toward infinity, while the size of each element tends to zero, or by keeping the division into elements unchanged and letting the number of degrees of freedom in each individual element increase toward infinity. Many engineering problems involve the solution of Laplace's equation. For this, if polynomial solutions are selected as interpolation functions in the elements, the condition in the interior is satisfied exactly. The energy integral method can no longer be used, but an alternative functional is proposed that consists solely of surface integrals. The method offers a possibility of satisfying both natural and essential boundary conditions for both the unknown function and its derivative. The chapter discusses the possibilities of solving biharmonic problems and also a nonlinear system as in the elasto-plastic problem.
12 citations
TL;DR: In this paper, a strongly implicit procedure is described which solves the system of 13 point finite difference equations associated with the biharmonic and similar fourth order elliptic equations, and for the majority of problems, a universal set of iteration parameters provide rapid rates of convergence.
TL;DR: In this paper, the parameter differentiation method of Joseph is applied to the title problem and general forms of the operator derivatives suitable to the bi-harmonic eigenvalue problem are found.
Abstract: The parameter differentiation method of Joseph is applied to the title problem. General forms of the operator derivatives suitable to the biharmonic eigenvalue problem are found. Applications to the simply-supported parallelogram plate and to the clamped elliptical plate are given, and comparisons are made with other exact and approximate calculations.
TL;DR: The class 0HP of Riemann surfaces without bounded or Dirichlet finite nonharmonic biharmonic functions is the smallest harmonically or analytically degenerate class as mentioned in this paper.
Abstract: The class 0HP of Riemann surfaces or Riemannian manifolds which do not carry (nonconstant) positive harmonic functions is the smallest harmonically or analytically degenerate class. In particular, it is strictly contained in the classes 0HB and 0HD of Riemann surfaces or Riemannian manifolds without bounded or Dirichlet finite harmonic functions, and in the classes 0AB and 0AD of Riemann surfaces without bounded or Dirichlet finite analytic functions. In the present paper we ask: Are there any relations between 0HP and the classes 0H2B and 0HzD of Riemannian manifolds without bounded or Dirichlet finite nonharmonic biharmonic functions? We shall show that the answer is in the negative. Explicitly, if 0 is a null class of AT-dimensional manifolds, and 0 its complement, then all four classes
TL;DR: In this paper, a method of establishing the eigenvalues and tables of them for various types of hollow cylinders is presented for the calculation of the effect of end loading on a semi-infinite length cylinder.
TL;DR: In this article, the formulation of a Monte Carlo method for the numerical solution of boundary value problems involving the homogeneous biharmonic equation is dealt with, where isotropic random walk and a mean value relation have been used.
Abstract: This paper deals with the formulation of a Monte Carlo method for the numerical solution of a class of boundary value problems involving the homogeneous biharmonic equation. Isotropic random walk and a mean value relation have been used.
In der vorliegenden Arbeit wird ein Monte-Carlo-Verfahren fur die numerische Losung zur homogenen biharmonischen Gleichung gehoriger Randwertaufgaben formuliert. Dabei werden das Modell einer isotropen Irrfahrt und eine Mittelwertrelation benutzt.
TL;DR: In this article, the authors presented a method of solution for an infinite wedge containing a symmetrically located circular hole, which is formulated separately according to the given in-plane edge tractions being even or odd with respect to the axis of the wedge.
Abstract: This paper presents a method of solution for an infinite wedge containing a symmetrically located circular hole. The solution is formulated separately according to the given in-plane edge tractions being even or odd with respect to the axis of the wedge. In either case, the stress function is constructed as the sum of four parts of biharmonic functions, two in the form of integrals and the other two in the form of series, in addition to a basic stress function for an otherwise unperforated wedge. The four parts as a whole give no traction along the edges and no stress at infinity of the wedge. Together with the basic stress function, the boundary conditions of no traction at the rim of hole are adjusted. Complex expressions are used in adjusting the boundary conditions. Finally, numerical examples are given for illustration.
TL;DR: In this paper, the second boundary value problem for the biharmonic equation is equivalent to the Dirichlet problems for two Poisson equations, and several finite difference approximations are defined to solve these Dirichelet problems and discretization error estimates are obtained.
Abstract: The second boundary value problem for the biharmonic equation is equivalent to the Dirichlet problems for two Poisson equations. Several finite difference approximations are defined to solve these Dirichlet problems and discretization error estimates are obtained. It is shown that the splitting of the biharmonic equation produces a numerically efficient procedure.
TL;DR: For coordinate systems with rotational symmetry, this paper showed that particular solutions of the biharmonic Poisson and first order Stokes equations exist and can be expressed in terms of simple derivatives and algebraic functions of the corresponding solutions.
Abstract: For coordinate systems with rotational symmetry it is shown that particular solutions of the biharmonic Poisson and first order Stokes equations exist and can be expressed in terms of simple derivatives and algebraic functions of the corresponding solutions of the Laplace and Stokes equations.
TL;DR: In this paper, the maximum and minimum principles for plate bending problems where the boundary is part clamped and part simply supported are derived in a unified manner from the canonical theory of complementary variational principles.
Abstract: Maximum and minimum principles for plate bending problems where the boundary is part clamped and part simply supported are derived in a unified manner from the canonical theory of complementary variational principles. Two new error bounds for approximate variational solutions are also presented.
TL;DR: In this article, the flux of lower state molecules emerging from the cavity of an almost phase locked biharmonic ammonia maser oscillator was found to be amplitude modulated at the bi-harmonic beat frequency.
TL;DR: In this paper, a complete sequence of orthogonal harmonic functions on a domain is constructed and the boundary values of these harmonic functions are found to be the eigenfunctions of a certain integral operator.
Abstract: A complete sequence of orthogonal harmonic functions on a domain is constructed. The boundary values of these harmonic functions are found to be the eigenfunctions of a certain integral operator.
TL;DR: In this paper, a unified approach for solving boundary value problems with a hole was suggested with special reference to problems relating thin infinite plates with holes, and closed form expressions for the stresses due to a uniform magnetic field present in the plane of deformation of a thin infinite conducting plate with a circular hole, the plate being deformed by a tension acting parallel to the direction of the magnetic field.
Abstract: Under certain specific assumption it has been observed that the basic equations of magneto-elasticity in the case of plane deformation lead to a biharmonic equation, as in the case of the classical plane theory of elasticity. The method of solving boundary value problems has been properly modified and a unified approach in solving such problems has been suggested with special reference to problems relating thin infinite plates with a hole. Closed form expressions have been obtained for the stresses due to a uniform magnetic field present in the plane of deformation of a thin infinite conducting plate with a circular hole, the plate being deformed by a tension acting parallel to the direction of the magnetic field.