改进的无奇异局部边界积分方程方法

在局部边界积分方程方法中,当源节点位于分析域的整体边界上时,局部边界积分将出现奇异积分问题,这些奇异积分需要做特别的处理.为此,提出了对域内节点采用局部积分方程,而对边界节点直接采用移动最小二乘近似函数引入边界条件来解决奇异积分问题,这同时也解决了对积分边界进行插值引入近似误差的问题.作为应用和数值实验,对Laplace方程和Helmholtz方程问题进行了分析,取得了很好的数值结果.进而,在Helmholtz方程求解中,采用了含波解信息的修正基函数来代替单项式基函数进行近似.数值结果显示,这样处理是简单高效的,在高波数声传播问题的求解中非常具有前景.

Improved Non-Singular Local Boundary Integral Equation Method

When the source nodes are on the global boundary in the implementation of local boundary integral equation method(LBIEM),singularities in the local boundary integrals need to be treated specially.Local integral equations were adopted for the nodes inside the domain and moving least square approximation(MLSA) for the nodes on the global boundary,thus singularities will not occur in the new algorithm.At the same time,approximation errors of boundary integrals reduce significantly.As applications and numerical tests,Laplace equation and Helmholtz equation problems were considered and excellent numerical results were obtained.Furthermore,when solving the Helmholtz problems,the modified basis functions with wave solutions were adopted to replace the usually-used monomial basis functions.Numerical results show that this treatment is simple and effective and its application is promising in solutions for the wave propagation problem with high wave number.