Recent development of the null-field integral equation approach for engineering problems with circular boundaries

Abstract

In this paper, a systematic approach is proposed to deal with engineering problems containing circular boundaries. The mathematical tools, degenerate kernels and Fourier series, are utilized in the null-field integral formulation. The kernel function is expanded to the degenerate form and the boundary density is expressed in terms of Fourier series. By moving the null-field point to the boundary, the singularity is novelly eliminated. Three gains of singularity free, boundary-layer effect free and exponential convergence are achieved. By matching the boundary condition, a linear algebraic system is obtained. After obtaining the unknown Fourier coefficients, the solution can be obtained by using the integral representation. This systematic approach can be applied to solve the Laplace, Helmholtz, biharmonic and biHelmholtz problems. Besides, the circular inclusion as well as the electro-mechanical coupling of piezoelectricity are addressed. Finally, several examples, including Stokes flow and piezoelectricity, are demonstrated to show the validity of present formulation.