Null-Field Integral Equation Approach for Plate Problems With Circular Boundaries

Abstract

In this paper, a semi-analytical approach for circular plate problems with multiple circular holes is presented. Null-field integral equation is employed to solve the plate problems while the kernel functions in the null-field integral equation are expanded to degenerate kernels based on the separation of field and source points in the fundamental solution. The unknown boundary densities of the circular plates are expressed in terms of Fourier series. It is noted that all the improper integrals are transformed to series sum and are easily calculated when the degenerate kernels and Fourier series are used. By matching the boundary conditions at the collocation points, a linear algebraic system is obtained. After determining the unknown Fourier coefficients, the displacement, slope, normal moment, and effective shear force of the plate can be obtained by using the boundary integral equations. Finally, two numerical examples are proposed to demonstrate the validity of the present method and the results are compared with the available exact solution, the finite element solution using ABAQUS software and the data of Bird and Steele.

Null-field integral equations and their applications

“…where the A(x) and B(x) can be found for the Laplace [4][5][6][7][8], Helmholtz [9], biharmonic [5] and biHelmholtz operators and the superscripts " i " and " e " denote the interior ( s x ≥ ) and exterior ( s x < ) cases, respectively. To classify the interior and exterior regions, Figure 2 shows for one-, two-and three-dimensional cases.…”

“…where s and x are the source and field points, respectively, n t u = ∂ ∂ , B is the boundary, x n denotes the outward normal vector at the field point x and the kernel function U(s,x), is the fundamental solution, and the other kernel functions, T(s,x), L(s,x) and M(s,x), are defined in the dual boundary integral method (BIEM) [9]. It is noted that more potentials are needed in eqns (3) and (4) for biharmonic and biHelmholtz cases.…”

“…In this paper, we review the recent development of the null-field integral equation approach [4][5][6][7][8][9] for boundary value problems (BVPs) with circular boundaries. The key idea is the expansion of kernel functions and boundary densities in the null-field integral equations.…”

Regularization of nearly singular integrals in the boundary element analysis for interior anisotropic thermal field near the boundary

“…A dual integral formulation that has regularized the singularities for the Laplace potential problem with a corner was derived by Chen and Hong (1994) using the contour approach surrounding the singularity. Very recently, the problem of the boundary layer effect has also been investigated by Chen et al (2006aChen et al ( , 2006b, who have derived null-field integral equations for a medium containing circular cavities. By introducing the concept of degenerate kernels, Chen et al (2006a) transformed the singular integrals into series sum when the null-field point was moved to the boundary.…”

“…Very recently, the problem of the boundary layer effect has also been investigated by Chen et al (2006aChen et al ( , 2006b, who have derived null-field integral equations for a medium containing circular cavities. By introducing the concept of degenerate kernels, Chen et al (2006a) transformed the singular integrals into series sum when the null-field point was moved to the boundary. In this article, the scheme of integration by parts is applied to treat the weakly singular integral, for which a general high-order interpolation of any families, such as the Serendipity, Lagrange, and Hermite, is considered.…”