各向异性介质中波动方程有限元法模拟及其稳定性

MODELING OF WAVE EQUATIONS IN ANISOTROPIC MEDIA USING FINITE ELEMENT METHOD AND ITS STABILITY C0NDITION

利用六面体单元和三线性插值函数，推导出在任意弹性各向异性介质中三维三分量波动方程所满足的有限元方程．给出用时域有限差分求解该有限元方程时，时间、空间采样率和弹性参数应满足的稳定性条件．提出避免每一时间步都求解一大型线性代数方程组的方法．在开始求波场前，先求质量矩阵的逆的因子形式，以后每时间步只需作刚度矩阵的逆的因子形式与向量的乘积运算，可大大提高有限元素法在微机上进行波动方程数值模拟的运算速度．结合两个数值模拟算例，说明该方法的可行性，并与伪谱法作了计算速度的比较，有限元法的计算速度高于伪谱法的计算速度．

The finite element equati0n for arbitrary anisotropic elastic medium is derived when the domain is divided with hexahedron elements and linear interpolation funchons are used in three direchons. The stability condih0ns of the numerical method, which relthes the sampling rates of hme and space and elashc constalltS, are given in thes paper. A method for solving the finite element equahons is also discussed. The method avoids solving a linear system at each hme step, and only needs the product of the faCtor form of the inverse of the mass matrix and a displacement vector. So the computational rate of the method is very high. Two examples of mode1ing show that the method proposed in this paper is feasible.