稀疏存储的显式有限元三角网格地震波数值模拟及其PML吸收边界条件

Explicit finite element method with triangle meshes stored by sparse format and its perfectly matched layers absorbing boundary condition

有限元法是复杂介质地震模拟的有力工具,它能比较客观地反映地震波的传播,比较细致地再现地震图像.但是,为了获得较精确的结果,有限元法模拟地震波的传播需要的网格点数多,具有计算量大和消耗内存多的缺点.针对上述缺点,本文对刚度矩阵采用压缩存储行（CSR）格式,以减少计算量并节省内存;采用集中质量矩阵得到对角的质量矩阵以提高有限元法（显式有限元）的计算效率;时间离散采用保能量的Newmark算法以提高有限元法的计算精度;采用变分形式（弱形式）的PML吸收边界条件对人工截断边界进行处理.通过与高精度的数值方法——谱元法的数值试验的对比表明,上述方法的引入可使有限元法在计算精度和计算效率方面均可取得比较显著的改进.为了获得相当的计算精度,相比于7阶谱元法,显式有限元法需要更精细的网格.然而,显式有限元法的计算速度比前者快近2倍,而内存需求仅为谱元法的1/4～1/6.

Finite element method is a useful tool for realistically modeling seismic wave in complex medium and displaying seismic wave fields in a quite detailed way.However,finite element method needs a large number of mesh grids for the high accuracy simulation of seismic wave,and suffers from large computational amount and computer memory resource.To surmount the mentioned shortages above,in this paper,the so called compressed spare row （CSR） method is adopted to reduce the computational amount and memory occupation of FEM; a diagonal mass matrix is obtained by using the lumped mass matrix rather the consistent mass matrix （explicit finite element method） to improve the computational efficiency; Newmark algorithm which has the property of energy-momentum conserving is used to improve the computational accuracy; and the artificial truncated boundaries is dealt with by the perfectly matched layer （PML） in variationai form （weak form）.The numerical tests demonstrate that the techniques mentioned above introduced to FEM make a significant improvement compared with spectral element method （SEM） which has a high computational accuracy.To obtain a comparable accuracy,the meshes of EFEM must be finer than that of SEM.The computational speed of EFEM is about 2 times faster than that of SEM with 7-order interpolation whereas the computer memory occupation amount of the former is about 4～6 times less than that of the latter.