Vibration of a Simply Supported L-Shaped Plate

1997 ◽  
Vol 119 (3) ◽  
pp. 464-467 ◽  
Author(s):  
R. Solecki
Keyword(s):  
Boundary Conditions ◽  
Integral Equations ◽  
Natural Vibration ◽  
Boundary Integral Equations ◽  
Rectangular Domain ◽  
Boundary Integral ◽  
Theoretical Development ◽  
Natural Vibrations ◽  
Simply Supported ◽  
Discontinuous Functions

Recently Solecki (1996) has shown that a differential equation for vibration of a rectangular plate with a cutout can be reduced to boundary integral equations. This was accomplished by filling the cutout with a “patch” made of the same material as the rest of the plate and separated from it by an infinitesimal gap. Thanks to this procedure it was possible to apply finite Fourier transformation of discontinuous functions in a rectangular domain. Subsequent application of the available boundary conditions led to a system of boundary integral equations. A plate simply supported along the perimeter, and fixed along the cutout (an L-shaped plate), was analyzed as an example. The general solution obtained by Solecki (1996) serves here to determine the frequencies of natural vibration of a L-shaped plate simply supported all around its perimeter. This problem is, however, more complicated than the previous example: to satisfy the boundary conditions an infinite series depending on discontinuous functions must be differentiated. The theoretical development is illustrated by numerical values of the frequencies of the natural vibrations of a square plate with a square cutout. The results are compared with the results obtained using finite elements method.

A New Boundary Equation Solution to the Plate Problem

1989 ◽  
Vol 56 (2) ◽  
pp. 364-374 ◽  
Author(s):  
J. T. Katsikadelis ◽  
A. E. Armena`kas
Keyword(s):  
Boundary Element Method ◽  
Boundary Conditions ◽  
Integral Equations ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Boundary Equation ◽  
Arbitrary Geometry ◽  
Equation Solution ◽  
The Finite Difference Method ◽  
Boundary Equations

A new boundary equation method is presented for analyzing plates of arbitrary geometry. The plates may have holes and may be subjected to any type of boundary conditions. The boundary value problem for the plate is formulated in terms of two differential and two integral coupled boundary equations which are solved numerically by discretizing the boundary. The differential equations are solved using the finite difference method while the integral equations are solved using the boundary element method. The main advantages of this new method are that the kernels of the boundary integral equations are simple and do not have hyper-singularities. Moreover, the same set of equations is employed for all types of boundary conditions. Furthermore, the use of intrinsic coordinates facilitates the modeling of plates with curvilinear boundaries. The numerical results demonstrate the accuracy and the efficiency of the method.

scholarly journals A generalized Calderón formula for open-arc diffraction problems: theoretical considerations

2015 ◽  
Vol 145 (2) ◽  
pp. 331-364 ◽  
Author(s):  
Stéphane K. Lintner ◽  
Oscar P. Bruno
Keyword(s):  
Boundary Conditions ◽  
Integral Equations ◽  
Krylov Subspace ◽  
Dimensional Space ◽  
Scattering Problem ◽  
Single Layer ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Iterative Solvers ◽  
Open Set

We deal with the general problem of scattering by open arcs in two-dimensional space. We show that this problem can be solved by means of certain second-kind integral equations of the form , where and are first-kind integral operators whose composition gives rise to a generalized Calderón formula of the form in a weighted, periodized Sobolev space. (Here is a continuous and continuously invertible operator and is a compact operator.) The formulation provides, for the first time, a second-kind integral equation for the open-arc scattering problem with Neumann boundary conditions. Numerical experiments show that, for both the Dirichlet and Neumann boundary conditions, our second-kind integral equations have spectra that are bounded away from zero and infinity as k → ∞; to the authors’ knowledge these are the first integral equations for these problems that possess this desirable property. This situation is in stark contrast with that arising from the related classical open-surface hypersingular and single-layer operators N and S, whose composition NS maps, for example, the function ϕ = 1 into a function that is not even square integrable. Our proofs rely on three main elements: algebraic manipulations enabled by the presence of integral weights; use of the classical result of continuity of the Cesàro operator; and explicit characterization of the point spectrum of , which, interestingly, can be decomposed into the union of a countable set and an open set, both of which are tightly clustered around . As shown in a separate contribution, the new approach can be used to construct simple, spectrally accurate numerical solvers and, when used in conjunction with Krylov-subspace iterative solvers such as the generalized minimal residual method, it gives rise to a dramatic reduction in the number of iterations compared with those required by other approaches.

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