基于紧凑存储的任意四边形网格有限元法地震波场数值模拟
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摘要
采用有限元法精确模拟复杂介质条件下的地震波场,一般采用三角单元,但在节点数相同的情况下,三角单元的计算精度不如矩形单元高,采用三角单元模拟复杂界面时,编制确定结构刚度矩阵非零元素位置的程序也较为麻烦。采用矩形单元离散含有倾斜或起伏界面的地质模型时,无法避免绕射噪声,若加密网格又会增加计算量。为此,本文基于任意四边形单元模拟声波的传播,在倾斜或起伏界面条件下,可以有效避免因离散引起的"阶梯状"界面,在不增加计算量以及内存占用的前提下,有效地消除离散绕射噪声。采用对角的集中质量矩阵代替一致质量矩阵,避免矩阵的求逆运算,从而提高显式有限元法的计算效率;对结构刚度矩阵采用紧凑存储格式,每一行需要存储的元素最多为5个,同时零元素不参与运算,既减少内存的占用,又极大地提高计算效率。
In general, we apply finite element method with triangular meshes to accurately modeling wave field under complex structures; however, in the condition of the same number of nodes, computational accuracy of triangular element is lower than that of rectangular element. It is difficult to implement the computer program which locates the non-zero elements of global stiffness matrix when we adopt triangular elements to fit complex geometrical interface. Using rectangular elements to divide geological model containing dipping or rugged interface will result in diffraction noise, and infilling grid will increase computational amount. For these reasons, this paper applies finite element method to solve 2-D acoustic wave equation, arbitrary quadrilateral meshes is adopted to fit dipping or rugged interface, which can avoid introducing "ladder" interface resulted from discrete rectangular grid, and discrete diffraction noise is eliminated effectively in the condition of no increasing computational amount and memory occupation. Diagonal lumped mass matrix is used to replace consistent massmatrix to avoid matrix inversion and improve the computational efficiency of explicit finite element method. In addition, we employ compact storage format to store global stiffness matrix, and the elements need to be stored in each row is no more than 5, the zero elements are not involved in computing at the same time, by this method, not only reducing the memory occupation, but also improving the computational efficiency.
In general, we apply finite element method with triangular meshes to accurately modeling wave field under complex structures; however, in the condition of the same number of nodes, computational accuracy of triangular element is lower than that of rectangular element. It is difficult to implement the computer program which locates the non-zero elements of global stiffness matrix when we adopt triangular elements to fit complex geometrical interface. Using rectangular elements to divide geological model containing dipping or rugged interface will result in diffraction noise, and infilling grid will increase computational amount. For these reasons, this paper applies finite element method to solve 2-D acoustic wave equation, arbitrary quadrilateral meshes is adopted to fit dipping or rugged interface, which can avoid introducing "ladder" interface resulted from discrete rectangular grid, and discrete diffraction noise is eliminated effectively in the condition of no increasing computational amount and memory occupation. Diagonal lumped mass matrix is used to replace consistent massmatrix to avoid matrix inversion and improve the computational efficiency of explicit finite element method. In addition, we employ compact storage format to store global stiffness matrix, and the elements need to be stored in each row is no more than 5, the zero elements are not involved in computing at the same time, by this method, not only reducing the memory occupation, but also improving the computational efficiency.
引文
[1]吴国忱,王华忠.波场模拟中的数值频散分析与校正策略[J].地球物理学进展,2005,20(1):58-65.Wu GC,Wang HZ.Analysis of numerical dispersion in wave-field simulation[J].Progress in Geophysics,2005,20(1):58-65.(in Chinese).
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[6]薛东川,王尚旭.波动方程有限元叠前逆时偏移[J].石油地球物理勘探,2008,43(1):17-21.Xue DC,Wang SX.Wave-equation finite element prestack reverse-time migration[J].Oil Geophysical Prospecting,2008,43(1):17-21.(in Chinese).
[7]薛东川,王尚旭.利用组合质量矩阵压制数值频散[J].石油地球物理勘探,2008,43(3):318-320.Xue DC,Wang SX.Using combined mass matrix to suppress numerical dispersion[J].Oil Geophysical Prospecting,2008,43(3):318-320.(in Chinese).
[8]刘有山,滕吉文,刘少林,等.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[J].地球物理学报,2013,56(9):3085-3099.Liu YS,Teng JW,Liu SL,et al.Explicit finite element method with triangle meshes stored by sparse format and its perfectly matched layers absorbing boundary condition[J].Chinese Journal of Geophysics,2013,56(9):3085-3099.(in Chinese).
[9]Ke BX,Zhao B,Cai JM,et al.2-D finite element acoustic wave modeling including rugged topography[C]//SEG Int’l Exposition and Annual Meeting.San Antonio,Texas,2001:131-139.
[10]Du X,Bancroft JC,Dong Y.A new migration method using a finite element and finite difference approach[C]//Canada Society of Explorations Geophysicists(CSEG)National Convention.2004.
[11]Yang JH,Liu T,Tang GY,et al.Modeling seismic wave propagation within complex structures[J].Applied Geophysics,2009,6(1):30-41.
[12]Liu T,Hu T,Sen MK,et al.A hybrid scheme for seismic modelling based on Galerkin method[J].Geophysical Journal International,2011,186(3):1165-1178.
[13]郭建.一种有限元快速算法[J].石油物探,1991,30(2):36-43.Guo J.A kind of fast finite element algorithm[J].Geophysical Prospecting for Petroleum,1991,30(2):36-43.(in Chinese).
[14]Sarma GS,Mallick K,Gadhinglajkar VR.Nonreflecting boundary condition in finite-element formulation for an elastic wave equation[J].Geophysics,1998,63(3):1006-1016.
[15]Celep Z,Ba?ant ZP.Spurious reflection of elastic waves due to gradually changing finite element size[J].International Journal for Numerical Methods in Engineering,1983,19(5):631-646.
[16]孙林洁,印兴耀.基于PML边界条件的高倍可变网格有限差分数值模拟方法[J].地球物理学报,2011,54(6):1614-1623.Sun LJ,Yin XY.A finite-difference scheme based on PML boundary condition with high power grid step variation[J].Chinese Journal of Geophysics,2011,54(6):1614-1623.(in Chinese).
[17]王勖成.有限单元法[M].北京:清华大学出版社,2003:472-475.Wang MC.Finite element method[M].Beijing:Tsinghua University Press,2003:472-475.(in Chinese).
[18]张剑锋.有限元波动模拟中傍轴近似透射边界[J].大连理工大学学报,1995,35(5):741-744.Zhang JF.Transmission boundary of paraxial approximations for numerical wave simulation by finite element method[J].Journal of Dalian University of Technology,1995,35(5):741-744.(in Chinese).
[19]冯德山,陈承申,王洪华.基于混合边界条件的有限单元法GPR正演模拟[J].地球物理学报,2012,55(11):3774-3785.Feng DS,Chen CS,Wang HH.Finite element method GPR forward simulation based on mixed boundary condition[J].Chinese Journal of Geophysics,2012,55(11):3774-3785.(in Chinese).
[20]Zhao JG,Shi RQ.Perfectly matched layer-absorbing boundary condition for finite-element time-domain modeling of elastic wave equations[J].Applied Geophysics,2013,10(3):323-336.
[21]王洪华,戴前伟.基于UPML吸收边界条件的GPR有限元数值模拟[J].中国有色金属学报,2013,23(7):2003-2011.Wang HH,Dai QW.Finite element numerical simulation for GPR based on UPML boundary condition[J].The Chinese Journal of Nonferrous Metals,2013,23(7):2003-2011.(in Chinese).
[22]王月英,宋建国.波场正演模拟中Sarma边界条件的改进[J].石油物探,2007,46(4):359-362.Wang YY,Song JG.Improvement of Sarma boundary condition in wavefield forward modeling[J].Geophysical Prospecting for Petroleum,2007,46(4):359-362.(in Chinese).
[23]徐世浙.地球物理中的有限元法[M].北京:科技出版社,1994:112-115.Xu SZ.Finite element method for geophysics[M].Beijing:Science Press,1994:112-115.(in Chinese).
[2]殷文,印兴耀,吴国忱,等.高精度频率域弹性波方程有限差分方法及波场模拟[J].地球物理学报,2006,49(2):561-568.Yin W,Yin XY,Wu GC,et al.The method of finite difference of high precision elastic wave equations in the frequency domain and wave-field simulation[J].Chinese Journal of Geophysics,2006,49(2):561-568.(in Chinese).
[3]张永刚.地震波场数值模拟方法[J].石油物探,2003,42(2):143-148.Zhang YG.On numerical simulations of seismic wave-field[J].Geophysical Prospecting for Petroleum,2003,42(2):143-148.(in Chinese).
[4]张美根,王妙月,李小凡,等.各向异性弹性波场的有限元数值模拟[J].地球物理学进展,2002,17(3):384-389.Zhang MG,Wang MY,Li XF,et al.Finite element forward modeling of anisotropic elastic waves[J].Progress in Geophysics,2002,17(3):384-389.(in Chinese).
[5]薛东川,王尚旭,焦淑静.起伏地表复杂介质波动方程有限元数值模拟方法[J].地球物理学进展,2007,22(2):522-529.Xue DC,Wang SX,Jiao SJ.Wave equation finite-element modeling including rugged topography and complicated medium[J].Progress in Geophysics,2007,22(2):522-529.(in Chinese).
[6]薛东川,王尚旭.波动方程有限元叠前逆时偏移[J].石油地球物理勘探,2008,43(1):17-21.Xue DC,Wang SX.Wave-equation finite element prestack reverse-time migration[J].Oil Geophysical Prospecting,2008,43(1):17-21.(in Chinese).
[7]薛东川,王尚旭.利用组合质量矩阵压制数值频散[J].石油地球物理勘探,2008,43(3):318-320.Xue DC,Wang SX.Using combined mass matrix to suppress numerical dispersion[J].Oil Geophysical Prospecting,2008,43(3):318-320.(in Chinese).
[8]刘有山,滕吉文,刘少林,等.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[J].地球物理学报,2013,56(9):3085-3099.Liu YS,Teng JW,Liu SL,et al.Explicit finite element method with triangle meshes stored by sparse format and its perfectly matched layers absorbing boundary condition[J].Chinese Journal of Geophysics,2013,56(9):3085-3099.(in Chinese).
[9]Ke BX,Zhao B,Cai JM,et al.2-D finite element acoustic wave modeling including rugged topography[C]//SEG Int’l Exposition and Annual Meeting.San Antonio,Texas,2001:131-139.
[10]Du X,Bancroft JC,Dong Y.A new migration method using a finite element and finite difference approach[C]//Canada Society of Explorations Geophysicists(CSEG)National Convention.2004.
[11]Yang JH,Liu T,Tang GY,et al.Modeling seismic wave propagation within complex structures[J].Applied Geophysics,2009,6(1):30-41.
[12]Liu T,Hu T,Sen MK,et al.A hybrid scheme for seismic modelling based on Galerkin method[J].Geophysical Journal International,2011,186(3):1165-1178.
[13]郭建.一种有限元快速算法[J].石油物探,1991,30(2):36-43.Guo J.A kind of fast finite element algorithm[J].Geophysical Prospecting for Petroleum,1991,30(2):36-43.(in Chinese).
[14]Sarma GS,Mallick K,Gadhinglajkar VR.Nonreflecting boundary condition in finite-element formulation for an elastic wave equation[J].Geophysics,1998,63(3):1006-1016.
[15]Celep Z,Ba?ant ZP.Spurious reflection of elastic waves due to gradually changing finite element size[J].International Journal for Numerical Methods in Engineering,1983,19(5):631-646.
[16]孙林洁,印兴耀.基于PML边界条件的高倍可变网格有限差分数值模拟方法[J].地球物理学报,2011,54(6):1614-1623.Sun LJ,Yin XY.A finite-difference scheme based on PML boundary condition with high power grid step variation[J].Chinese Journal of Geophysics,2011,54(6):1614-1623.(in Chinese).
[17]王勖成.有限单元法[M].北京:清华大学出版社,2003:472-475.Wang MC.Finite element method[M].Beijing:Tsinghua University Press,2003:472-475.(in Chinese).
[18]张剑锋.有限元波动模拟中傍轴近似透射边界[J].大连理工大学学报,1995,35(5):741-744.Zhang JF.Transmission boundary of paraxial approximations for numerical wave simulation by finite element method[J].Journal of Dalian University of Technology,1995,35(5):741-744.(in Chinese).
[19]冯德山,陈承申,王洪华.基于混合边界条件的有限单元法GPR正演模拟[J].地球物理学报,2012,55(11):3774-3785.Feng DS,Chen CS,Wang HH.Finite element method GPR forward simulation based on mixed boundary condition[J].Chinese Journal of Geophysics,2012,55(11):3774-3785.(in Chinese).
[20]Zhao JG,Shi RQ.Perfectly matched layer-absorbing boundary condition for finite-element time-domain modeling of elastic wave equations[J].Applied Geophysics,2013,10(3):323-336.
[21]王洪华,戴前伟.基于UPML吸收边界条件的GPR有限元数值模拟[J].中国有色金属学报,2013,23(7):2003-2011.Wang HH,Dai QW.Finite element numerical simulation for GPR based on UPML boundary condition[J].The Chinese Journal of Nonferrous Metals,2013,23(7):2003-2011.(in Chinese).
[22]王月英,宋建国.波场正演模拟中Sarma边界条件的改进[J].石油物探,2007,46(4):359-362.Wang YY,Song JG.Improvement of Sarma boundary condition in wavefield forward modeling[J].Geophysical Prospecting for Petroleum,2007,46(4):359-362.(in Chinese).
[23]徐世浙.地球物理中的有限元法[M].北京:科技出版社,1994:112-115.Xu SZ.Finite element method for geophysics[M].Beijing:Science Press,1994:112-115.(in Chinese).