Havelock型格林函数振荡项数值积分的稳定性研究

Numerical integration stability control strategies for wave-like part of the havelock form green function

Havelock型格林函数的传播项被积函数是高频振荡且奇异的复变函数,文献[4]引入变量代换获得了一种兼具积分效率和精度的积分方法,本文研究了该方法的积分稳定性,发现该方法仍存在如下的积分困难:(1)θ=γ时复函数中分母为零引起的计算溢出;(2)θ=π/2是复函数在y及z方向偏导数的无穷间断点;(3)场点与源点横坐标相同时伪奇异点变为真奇点。针对这些积分困难,采用极限公式计算θ=γ处复函数的值避免计算溢出;在保证积分精度的前提下采用截断方法略去θ=π/2邻近区域的积分消除无穷间断处的奇异;针对(3)采用分区法处理以避开原被积函数的高频振荡,并消除奇异性。伪奇异性存在的条件是场点必须在点源传播波的传播范围内,伪奇异点最多为2个。

The wave-like part of the Havelock form translating-pulsating source Green function is highly oscillatory and discussed. A method based on variable substitution and local refinement of integral steps technique is introduced to calculate the integration by Xu and Dong ） which makes calculation efficiency and accuracy possible. But there are still some difficulties in calculation about this method as follows. firstly, the complex function after variable substitution is still discussed when θ =γ; secondly, the y and z direction partial derivatives of the complex function are infinite when θ =γ/2 ; in addition, false singularities （which found by Xu and Dong） become true singularities when Y=0 （where Y=y-p,y and p are the abscissas of the field and source point） and the integration method may be failed. In order to improve numerical integration stability about this method, some auxiliary techniques are introduced in present study as follows.. （1） a limit formula which can remove the singularity of the complex function when is used and can avoid calculation overflow. （2） A truncation method is introduced to remove the infinite singularity of the y and z direction partial derivatives when θ =γ/2 （3） A region dividing integral method is performed to deal with the singularities and avoid high oscillation when Y=0. False singularities are also studied and it is interest to find that the false singularities should exist when the field point is in the propagation range of the waves produced by the source.