Some Recent Results 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 into Fourier series. By collocating the null-field point on the real boundary, the singularity is novelly avoided. Five gains of well-posed model, singularity free, boundary-layer effect free, exponential convergence and mesh-free approach 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, bi-Helmholtz and biharmonic problems. Besides, the circular inclusions as well as the electro-elastic coupling of piezoelectricity are addressed. Finally, several examples, including Stokes’ flow, elasticity and piezoelectricity, are demonstrated to show the validity of present formulation.