层状介质中计算体积分方程的弱化BCGS-FFT算法

Weak-form BCGS-FFT algorithm for volume integral equations in stratified medium

采用弱化稳定型双共轭梯度快速Fourier变换(BCGS-FFT)算法精确计算了层状介质中的体积分方程。采用递推矩阵方法计算层状介质中的并矢Green函数,可以很方便地与体积分方程结合。将“屋顶”函数作为基函数和试探函数对体积分方程进行弱化离散,从而有效地避免了体积分方程的奇异性。离散后的体积分方程采用稳定型双共轭梯度迭代方法进行求解,从而得到异常体内电场的分布。假设异常体只分布在层状介质中的某一层介质内,则体积分方程内并矢Green函数与对比源之间的乘积可表示为褶积或相关形式,从而在每一次迭代过程中可以同时在x,y,z方向采用快速Fourier变换技术加快运算速度。数值算例说明了该算法的精确性和有效性。

The volume integral equations in stratified medium were accurately calculated by weak-form stabilized biconjugategradient fast Fourier transform （BCGS-FFT） algorithm. The dyadic Green＇s functions in stratified medium were calculated by recursive matrix method, which can be easily combined with the volume integral equations. The rooftop functions were chosen as basis function and testing function to get the weak-form discretization of the volume integral equations, thus the singularity can be circumvented. The discrete form of the volume integral equations was solved via the stabilized biconjugate-gradient iteration method, so that the distribution of the electric field within the inhomogeneous objects can be obtained. It is assumed that the inhomogeneous objects are only in one layer of the stratified medium, so that the product between the dyadic Green＇ s function and the contrast source within the volume integral equations can be expressed in the form of convolution or correlation, which can be accelerated by adopting the fast Fourier transform in x ,y and z directions during each iteration of BCGS. Numerical examples show the accuracy and efficiency of the algorithm.