边界子波谱方法中的一种奇异积分方法

More Accurate and Efficient Singular Integrals for the Boundary Spectral Method Based on Wavelet Expansions

提出了一种处理三维轴对称H elm ho ltz积分方程子波谱方法的奇异积分方法。对于三维轴对称和三维问题的H elm ho ltz积分方程,在积分奇异点构造一个小球面,然后减去三维Lap lace方程核函数在该球面上的积分,消除奇异积分。算例表明该方法有较高的精度,计算方便,原理上可用于任意三维问题。

Purpose. Although existing singular integral methods, such as Refs. 2, 3 and 4, are already pretty accurate and efficient, we are of the opinion that their precision and efficiency can be further improved. In the full paper we explain in detail our method, believed to be of better precision and more efficient~ in this abstract, we just sketch an outline of our explanation. In order to make our sketchy explanation better understood, we refer often to the equations in the full paper. In the three-dimensional and axisymmetric Helmholtz integral equations eqs. （4） and （5） in the full paper , the auxiliary interior spherical surfaces are selected at the singular points and the singular integrals are removed by subtracting the three dimensional Laplace kernel on corresponding spherical surfaces from eqs. （4） and （5）. Then singular integrals can be turned into the elliptic integrals eq. （18） in the full paper which can be computed by analysis or the standard numerical methods. In the mathematical derivation, the equations preceding eq. （18） are of course necessary; for one reason or another, eqs. （19） through （22） are also necessary for our method. The radiation of a pulsating sphere is very efficiently solved using our new method, and the results, as compared with the analytical solution, show that the average deviation from the analytical solution is only 1.5%, thus confirming that our new method is of high precision.