A Non-Splitting PML for Transient Analysis of Poroelastic Media and Its Finite Element Implementation
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摘要: 在波场的数值模拟中,完全匹配层(perfectly matched layer, PML)已经被证明是一种十分有效的吸收技术,并得到了广泛的应用.为了解决具有无限域的多孔介质中2阶弹性波动方程数值模拟中的吸收边界问题,提出了一种非分裂格式的PML(nonsplitting perfectly matched layer, NPML).首先,基于Biot多孔介质波动理论,建立了以固相和流相位移表示的2阶动力控制方程,其中考虑了固体颗粒和孔隙流体的可压缩性、惯性以及孔隙流体的粘性.其次,根据复伸展坐标变换的定义,通过Laplace变换获得了非分裂格式PML的频域表达式.然后,借助辅助函数将该方程变换到时间域内,得到了一种有效的非分裂PML.最后,基于Galerkin近似方法,给出了其时域有限元计算格式.通过数值算例分析了该非分裂格式的PML在饱和介质动力响应分析中的有效性.Abstract: The perfectly matched layer (PML) absorbing boundary condition had been proved to be a highly effective absorption technique for the numerical simulation of wave propagation and therefore widely used. In order to solve the problems of absorbing boundary conditions in the numerical modeling of 2nd-order elastic wave equations for the infinite domain poroelastic media, a non-splitting perfectly matched layer (NPML) was proposed. Firstly, based on the theory of Biot’s wave equations and in view of the compressibility of solid particles and pore fluid, the inertia and the pore fluid viscosity, the 2nd-order dynamic governing equations were established in the form of solid and fluid displacements. Secondly, according to the complex coordinate stretching technique, the frequency domain formulations of the NPML were obtained by means of the Laplace transform. Afterwards, with the aid of auxiliary functions in the absorption layer, an effective NPML was built through the transform of the frequency domain formulations back to the time domain. Finally, the time domain finite element scheme of the NPML on the basis of Galerkin approximate method was provided. The effectiveness of the NPML in the dynamic response analysis of saturated poroelastic media is demonstrated with several numerical examples.
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