弹性波散射问题优化PML方法的收敛性
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摘要
基于频域复坐标拉伸,给出一种解时谐弹性波散射问题的单轴优化完美匹配层(PML)方法.该方法通过在吸收函数中引入一个小参数ε_0,使散射问题优化PML方法的计算不再依赖PML层的厚度.结果表明,当参数ε_0充分小时,优化的PML解指数收敛于原弹性波散射问题的解.
We gave a uniaxial optimal perfectly matched layer(PML)method for solving the scattering problem of time harmonic elastic wave based on the complex coordinate stretching in the frequency domain.By introducing a small parameter ε_0 in the absorption function,the optimal PML method for scattering problem was no longer dependent on the thickness of the PML layer.The results show that the optimal PML solution converges exponentially to the solution of the original elastic wave scattering problem if the parameter ε_0 is small sufficiently.
We gave a uniaxial optimal perfectly matched layer(PML)method for solving the scattering problem of time harmonic elastic wave based on the complex coordinate stretching in the frequency domain.By introducing a small parameter ε_0 in the absorption function,the optimal PML method for scattering problem was no longer dependent on the thickness of the PML layer.The results show that the optimal PML solution converges exponentially to the solution of the original elastic wave scattering problem if the parameter ε_0 is small sufficiently.
引文
[1]CHEN Zhiming,XIANG Xueshuang,ZHANG Xiaohui.Convergence of the PML Method for Elastic Wave Scattering Problems[J].Mathematics of Computation,2016,85:2687-2714.
[2]Bramble J H,Pasciak J E,Prieto A.A Note on the Existence and Uniqueness of Solutions of the Frequency Domain Elastic Wave Problems:A Priori Estimates in H1[J].J Math Anal Appl,2009,345(1):396-404.
[3]张德悦,马富明,方明.求解洞穴散射问题的带有吸收边界层的有限元方法[J].计算物理,2008,25(3):301-308.(ZHANG Deyue,MA Fuming,FANG Ming.Finite Element Method with Perfectly Matched Absorbing Layers for Wave Scattering from aCavity[J].Chinese Journal of Computational Physics,2008,25(3):301-308.)
[4]杨孝英,马富明,杜新伟.计算开洞穴电磁散射问题的一种优化PML方法[J].吉林大学学报(理学版),2009,47(2):185-190.(YANG Xiaoying,MA Fuming,DU Xinwei.An Optimal PML Method for Electromagnetic Scattering from Cavities[J].Journal of Jilin University(Science Edition),2009,47(2):185-190.)
[5]CHEN Zhiming,WU Xinming.Long-Time Stability and Convergence of the Uniaxial Perfectly Matched Layer Method for Time-Domain Acoustic Scattering Problems[J].SIAM J Numer Anal,2012,50(5):2632-2655.
[6]杨孝英,王旖旎.解声波散射问题的一种优化PML方法[J].吉林大学学报(理学版),2013,51(1):89-91.(YANG Xiaoying,WANG Yini.An Optimal Perfectly Matched Layer Technique for Uniaxial Scattering Problems[J].Journal of Jilin University(Science Edition),2013,51(1):89-91.)
[7]杨孝英.双层介质散射问题单轴优化PML方法的收敛性[J].吉林大学学报(理学版),2016,54(5):983-988.(YANG Xiaoying.Convergence of Single Axis Optimal Perfectly Matched Layer Method for Scattering Problems in Two-Layered Media[J].Journal of Jilin University(Science Edition),2016,54(5):983-988.)
[8]杨孝英.散射问题优化PML方法的收敛性[J].吉林大学学报(理学版),2016,54(4):709-714.(YANG Xiaoying.Convergence of Optimal Perfectly Matched Layer Method for Scattering Problems[J].Journal of Jilin University(Science Edition),2016,54(4):709-714.)
[9]CHEN Zhiming,CUI Tao,ZHANG Linbo.An Adaptive Anisotropic Perfectly Matched Layer Method for 3-D Time Harmonic Electromagnetic Scattering Problems[J].Numer Math,2013,125(4):639-677.
[10]Kupradze V A.Dynamical Problems in Elasticity[M].Amersterdam:North-Holland Publishing Co,1963.
[2]Bramble J H,Pasciak J E,Prieto A.A Note on the Existence and Uniqueness of Solutions of the Frequency Domain Elastic Wave Problems:A Priori Estimates in H1[J].J Math Anal Appl,2009,345(1):396-404.
[3]张德悦,马富明,方明.求解洞穴散射问题的带有吸收边界层的有限元方法[J].计算物理,2008,25(3):301-308.(ZHANG Deyue,MA Fuming,FANG Ming.Finite Element Method with Perfectly Matched Absorbing Layers for Wave Scattering from aCavity[J].Chinese Journal of Computational Physics,2008,25(3):301-308.)
[4]杨孝英,马富明,杜新伟.计算开洞穴电磁散射问题的一种优化PML方法[J].吉林大学学报(理学版),2009,47(2):185-190.(YANG Xiaoying,MA Fuming,DU Xinwei.An Optimal PML Method for Electromagnetic Scattering from Cavities[J].Journal of Jilin University(Science Edition),2009,47(2):185-190.)
[5]CHEN Zhiming,WU Xinming.Long-Time Stability and Convergence of the Uniaxial Perfectly Matched Layer Method for Time-Domain Acoustic Scattering Problems[J].SIAM J Numer Anal,2012,50(5):2632-2655.
[6]杨孝英,王旖旎.解声波散射问题的一种优化PML方法[J].吉林大学学报(理学版),2013,51(1):89-91.(YANG Xiaoying,WANG Yini.An Optimal Perfectly Matched Layer Technique for Uniaxial Scattering Problems[J].Journal of Jilin University(Science Edition),2013,51(1):89-91.)
[7]杨孝英.双层介质散射问题单轴优化PML方法的收敛性[J].吉林大学学报(理学版),2016,54(5):983-988.(YANG Xiaoying.Convergence of Single Axis Optimal Perfectly Matched Layer Method for Scattering Problems in Two-Layered Media[J].Journal of Jilin University(Science Edition),2016,54(5):983-988.)
[8]杨孝英.散射问题优化PML方法的收敛性[J].吉林大学学报(理学版),2016,54(4):709-714.(YANG Xiaoying.Convergence of Optimal Perfectly Matched Layer Method for Scattering Problems[J].Journal of Jilin University(Science Edition),2016,54(4):709-714.)
[9]CHEN Zhiming,CUI Tao,ZHANG Linbo.An Adaptive Anisotropic Perfectly Matched Layer Method for 3-D Time Harmonic Electromagnetic Scattering Problems[J].Numer Math,2013,125(4):639-677.
[10]Kupradze V A.Dynamical Problems in Elasticity[M].Amersterdam:North-Holland Publishing Co,1963.