基于混合有限元格式的完美匹配层与多次透射公式人工边界比较研究
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摘要
介绍了完美匹配层(PML)人工边界可以吸收不同频率和任意角度入射波的原理以及PML人工边界的构造方法.在此基础上,将PML人工边界应用于地震波动数值模拟的速度-应力混合有限元格式中,探讨了PML应用的可行性,并通过数值试验研究了PML人工边界的反射率,比较了PML人工边界与多次透射公式(MTF)人工边界应用于体波和面波模拟中数值反射的差异,对两种边界的透射效果进行了分析.结果表明,尽管数值离散后PML人工边界不再保持完美匹配特性,但PML人工边界在近场波动数值模拟中可获得比MTF人工边界更为理想的吸收效果,在角点透射、大角度掠射情形下尤为明显;PML人工边界在混合有限元格式的数值算法中,未见失稳等不良反应,比MTF人工边界有更好的稳定性;在合理选择参数的情况下,PML人工边界的运算量可接受.
The theory of perfectly matched layer (PML) artificial boundary condition (ABC), which is characterized by absorption any wave motions with arbitrary frequency and arbitrarily incident angle, is introduced. The construction process of PML boundary based on elastodynamic partial differential equation (PDE) system is developed. Combining with velocity-stress hybrid finite element formulation, the applicability of PML boundary is investigated and the numerical reflection of PML boundary is estimated. The reflectivity of PML and multi-transmitting formula (MTF) boundary is then compared based on body wave and surface wave simulations. The results show that although PML boundary yields some reflection, its absorption performance is superior to MTF boundary in the numerical simulations of near-fault wave propagation, especially in corner and large angle grazing incidence situations. The PML boundary does not arise any unstable phenomenon and the stability of PML boundary is better than MTF boundary in hybrid finite element method. For a specified problem and analysis tolerance, the computational efficiency of PML boundary is only a little lower than MTF boundary.
The theory of perfectly matched layer (PML) artificial boundary condition (ABC), which is characterized by absorption any wave motions with arbitrary frequency and arbitrarily incident angle, is introduced. The construction process of PML boundary based on elastodynamic partial differential equation (PDE) system is developed. Combining with velocity-stress hybrid finite element formulation, the applicability of PML boundary is investigated and the numerical reflection of PML boundary is estimated. The reflectivity of PML and multi-transmitting formula (MTF) boundary is then compared based on body wave and surface wave simulations. The results show that although PML boundary yields some reflection, its absorption performance is superior to MTF boundary in the numerical simulations of near-fault wave propagation, especially in corner and large angle grazing incidence situations. The PML boundary does not arise any unstable phenomenon and the stability of PML boundary is better than MTF boundary in hybrid finite element method. For a specified problem and analysis tolerance, the computational efficiency of PML boundary is only a little lower than MTF boundary.
引文
李宁.2006.完美匹配层理论及其在地震波模拟中的应用[D]:[学位论文].哈尔滨:中国地震局工程力学研究所:51-56.
廖振鹏.2002.工程波动理论导论[M].北京:科学出版社:253-263.
张晓志.2005.近断层强地面运动数值模拟研究[D].[学位论文].中国地震局工程力学研究所:47-51.
赵海波,王秀明,王东,等.2007.完全匹配层吸收边界在孔隙介质弹性波模拟中的应用[J].地球物理学报,50(2):581-591.
周正华,廖振鹏.2001.多次透射公式的一种稳定实现措施[J].地震工程与工程振动,21(1):10-14.
Basu U,Chopra A K.2003.Perfectly matchedlayers for ti me-harmonic electrodynamics of unbounded domains:Theoryand finite-element i mplementation[J].Compu Methods Appl Mech Engrg,192(11-12):1337-1375.
Basu U,Chopra A K.2004.Perfectly matchedlayers for transient elastodynamics of unbounded domains[J].Int J Num-er Meth Eng,59(8):1039-1074.
Brenger J P.1994.A perfectly matched layer for the absorption of electromagnetic waves[J].J Comput Phys,114:185-200.
B啨renger J P.1996.Three-di mensional perfectly matchedlayer for the absorption of electromagnetic waves[J].JComputPhys,127:363-379.
Chew WC,Weedon W H.1994.A3-Dperfectly matched mediumfrom modified Maxwell′s equations with stretched co-ordinates[J].MicrowOpt Techn Let,13(4):599-604.
Chew WC,Liu Q H.1996.Perfectly matched layers for elastodynamics:A new absorbing boundary condition[J].Comput Acoust,4(4):341-359.
Collino F,Monk P.1998.The perfectly matchedlayerin curvilinear coordinates[J].SIAMJ Sci Comput,19(6):2061-2090.
Collino F,Tsogka C.2001.Application of the PML absorbinglayer model to the linear elastodynamic problemin aniso-tropic heterogeneous media[J].Geophysics,66(1):294-307.
Hagstrom T,Hariharan S I.1998.Aformulation of asymptotic and exact boundary conditions usinglocal operators[J].Appl Num Math,27:403-416.
Hastings F D,Schneider J B,Broschat S L.1996.Application of the perfectly matchedlayer(PML)absorbing boundarycondition to elastic wave propagation[J].J Acoust Soc Amer,100(5):3061-3069.
Hesthaven J S.1998.On the analysis and construction of perfectly matchedlayers for the linearized Euler equations[J].J Comput Phys,142:129-147.
Komatitsch D,Tromp J.2003.Aperfectly matchedlayer absorbing boundary conditionfor the second-order seismic waveequation[J].Geophys J Int,154:146-153.
Liao Z P,Wong H L.1984.Atransmitting boundary for the numerical si mulation of elastic wave propagation[J].Soil Dyn Earthq Eng,3(4):174-183.
Liu Q H,Tao J.1997.The perfectly matched layer(PML)for acoustic waves in absorptive media[J].J Acoust SocAmer,102(4):2072-2082.
Qi Q,Geers T L.1998.Evaluation of the perfectly matchedlayer for computational acoustics[J].J Comput Phys,139166-183.
Virieux J.1986.P-SV wave propagationin heterogeneous media:Velocity-stress finite difference method[J].Geophys-ics,51:889-901.
Zeng Y Q,He QE,Liu Q H.2001.The application of the perfectly matchedlayerin a numerical modeling of wave prop-agationin poroelastic media[J].Geophysics,66(4):1258-1266.
廖振鹏.2002.工程波动理论导论[M].北京:科学出版社:253-263.
张晓志.2005.近断层强地面运动数值模拟研究[D].[学位论文].中国地震局工程力学研究所:47-51.
赵海波,王秀明,王东,等.2007.完全匹配层吸收边界在孔隙介质弹性波模拟中的应用[J].地球物理学报,50(2):581-591.
周正华,廖振鹏.2001.多次透射公式的一种稳定实现措施[J].地震工程与工程振动,21(1):10-14.
Basu U,Chopra A K.2003.Perfectly matchedlayers for ti me-harmonic electrodynamics of unbounded domains:Theoryand finite-element i mplementation[J].Compu Methods Appl Mech Engrg,192(11-12):1337-1375.
Basu U,Chopra A K.2004.Perfectly matchedlayers for transient elastodynamics of unbounded domains[J].Int J Num-er Meth Eng,59(8):1039-1074.
Brenger J P.1994.A perfectly matched layer for the absorption of electromagnetic waves[J].J Comput Phys,114:185-200.
B啨renger J P.1996.Three-di mensional perfectly matchedlayer for the absorption of electromagnetic waves[J].JComputPhys,127:363-379.
Chew WC,Weedon W H.1994.A3-Dperfectly matched mediumfrom modified Maxwell′s equations with stretched co-ordinates[J].MicrowOpt Techn Let,13(4):599-604.
Chew WC,Liu Q H.1996.Perfectly matched layers for elastodynamics:A new absorbing boundary condition[J].Comput Acoust,4(4):341-359.
Collino F,Monk P.1998.The perfectly matchedlayerin curvilinear coordinates[J].SIAMJ Sci Comput,19(6):2061-2090.
Collino F,Tsogka C.2001.Application of the PML absorbinglayer model to the linear elastodynamic problemin aniso-tropic heterogeneous media[J].Geophysics,66(1):294-307.
Hagstrom T,Hariharan S I.1998.Aformulation of asymptotic and exact boundary conditions usinglocal operators[J].Appl Num Math,27:403-416.
Hastings F D,Schneider J B,Broschat S L.1996.Application of the perfectly matchedlayer(PML)absorbing boundarycondition to elastic wave propagation[J].J Acoust Soc Amer,100(5):3061-3069.
Hesthaven J S.1998.On the analysis and construction of perfectly matchedlayers for the linearized Euler equations[J].J Comput Phys,142:129-147.
Komatitsch D,Tromp J.2003.Aperfectly matchedlayer absorbing boundary conditionfor the second-order seismic waveequation[J].Geophys J Int,154:146-153.
Liao Z P,Wong H L.1984.Atransmitting boundary for the numerical si mulation of elastic wave propagation[J].Soil Dyn Earthq Eng,3(4):174-183.
Liu Q H,Tao J.1997.The perfectly matched layer(PML)for acoustic waves in absorptive media[J].J Acoust SocAmer,102(4):2072-2082.
Qi Q,Geers T L.1998.Evaluation of the perfectly matchedlayer for computational acoustics[J].J Comput Phys,139166-183.
Virieux J.1986.P-SV wave propagationin heterogeneous media:Velocity-stress finite difference method[J].Geophys-ics,51:889-901.
Zeng Y Q,He QE,Liu Q H.2001.The application of the perfectly matchedlayerin a numerical modeling of wave prop-agationin poroelastic media[J].Geophysics,66(4):1258-1266.
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