Degenerate scale for the analysis of circular thin plate using the boundary integral equation method and boundary element methods

Abstract

In this paper, the degenerate scale for plate problem is studied. For the continuous model, we use the null-field integral equation, Fourier series and the series expansion in terms of degenerate kernel for fundamental solutions to examine the solvability of BIEM for circular thin plates. Any two of the four boundary integral equations in the plate formulation may be chosen. For the discrete model, the circulant is employed to determine the rank deficiency of the influence matrix. Both approaches, continuous and discrete models, lead to the same result of degenerate scale. We study the nonunique solution analytically for the circular plate and find degenerate scales. The similar properties of solvability condition between the membrane (Laplace) and plate (biharmonic) problems are also examined. The number of degenerate scales for the six boundary integral formulations is also determined.

A new method for Stokes problems with circular boundaries using degenerate kernel and Fourier series

“…To deal with the Stokes problem, two ways are used in the literature [3,14]. First, the biharmonic equation of Equation (6) is treated [3].…”

“…The direct BIEs for the domain point can be derived from the Rayleigh-Green identity [2,3] as follows:…”

“…However, u(x) is defined as the stream function in this paper instead of displacement for plate problem. The kernel functions U, (9)- (12) are expanded to degenerate kernels by using the separation of source and field points [3,17]. The kernel function U (s, x) in Equation (9) is the fundamental solution that satisfies…”

An Analytical Approach for the Green's Functions of Biharmonic Problems with Circular and Annular Domains

“…In the recent years, the Green's function of an isotropic annular plate clamped along one edge and free at the other was solved by Adewale [8]. Following the success of null-field integral formulation for typical BVPs [9][10][11], the present paper provides an analytical approach by using null-field integral equations in conjunction with Fourier series and degenerate kernels to derive the Green's functions of circular and annular plates.…”

On the solution of exterior plane problems by the boundary element method: A physical point of view

“…Other methods are used to perform a "regularization" of the problem and to recover the uniqueness [15,16,17,18,19,20]. This problem is recurrent within the literature [21,22,7,23,24]. The link between the condition number of the BEM matrices and the scaling of the problem has been also investigated [25].…”