全局弱式无网格法在大地电磁二维正演中的应用研究
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摘要
无网格法作为有限元法的一种重要补充和发展具有重要的研究意义和实用价值,其避免了网格剖分,具有前期处理简单,精度高,自适应分析方便,适用于复杂模型和复杂边界条件等优点.本文重点研究基于全局弱式无网格法的大地电磁二维正演问题.本文首先论述了无网格理论,并推导出基于全局弱式无网格法的大地电磁二维问题对应的最终总体矩阵表达式,通过标准模型数值模拟实验研究了相关重要参数对计算精度及计算效率的影响,并确定其最优值,选取高斯点数量为2～3、无量纲支持域尺寸为1.1～1.3.后续分别用有限元法和全局弱式无网格法计算典型一维模型和二维模型,通过对比计算结果及计算精度,验证本文中提出方法在处理大地电磁二维问题上的有效性及在处理复杂模型上的优越性.
The mesh-free method is a kind of important supplement and development to finite element method, it has important research significance and practical value. It avoids the mesh generation with the advantages of simple pre-processing, high precision, simple adaptive analysis and suitable for complex model and complex boundary conditions. This thesis focuses on the magnetotelluric forward modeling based on global weak-form mesh-free methods. In this paper, the mesh-free theory is discussed and the final global matrix expression corresponding that based on the global weak-form mesh-free methods of the two dimensional magnetotelluric problem are derived. Last we give the best relevant parameters by the numerical simulation of several one-dimensional models, for example, the number of Gauss points is 2～3, and the dimensionless support domain size is 1.1～1.3. Subsequently, several typical one-dimensional model and two-dimensional model are calculated by finite element method and Element-free Galerkin method. The effectiveness and superiority in dealing with complex models of the proposed method is verified by comparing the calculated results.
The mesh-free method is a kind of important supplement and development to finite element method, it has important research significance and practical value. It avoids the mesh generation with the advantages of simple pre-processing, high precision, simple adaptive analysis and suitable for complex model and complex boundary conditions. This thesis focuses on the magnetotelluric forward modeling based on global weak-form mesh-free methods. In this paper, the mesh-free theory is discussed and the final global matrix expression corresponding that based on the global weak-form mesh-free methods of the two dimensional magnetotelluric problem are derived. Last we give the best relevant parameters by the numerical simulation of several one-dimensional models, for example, the number of Gauss points is 2～3, and the dimensionless support domain size is 1.1～1.3. Subsequently, several typical one-dimensional model and two-dimensional model are calculated by finite element method and Element-free Galerkin method. The effectiveness and superiority in dealing with complex models of the proposed method is verified by comparing the calculated results.
引文
Belytschko T, Lu YY, Gu L. 1994. Element-free Galerkin Methods [J]. International Journal for Numerical Method in Engineering, 37(2): 229-256.
FENG DeShan, WANG HongHua, DAI QianWei.2013. Forward simulation of Ground Penetrating Radar based on the element-free Galerkin method [J].Chinese Journal of Geophysics (in Chinese), 56(1): 298-308, doi: 10.6038/cjg20130131.
GU Yuantong, DING Hua. 2005. Recent developments of meshless method [J]. Advances in Mechanics (in Chinese), 35(3): 323-337.
Jia Xiao-feng, Hu Tian-Yue, Wang Run-qiu. 2006a. A mesh-free algorithm of seismic wave simulation in complex medium [J]. Oil Geophysical Prospecting (in Chinese), 41(1): 43- 48.
Jia X F, Hu T Y, Wang R Q. 2006b. Wave equation modeling and imaging by using element-free method [J]. Progress in Geophysics (in Chinese), 21(1): 11-17, doi: 10.3969/j.issn.1004-2903.2006.01.003.
Li J J. 2014. Magnetotelluric two-dimensional forward by weak-form meshfree methods[Master’s thesis].Changsha:Central South University.
Li J J, Yan J B. 2014. Developments of meshless method and applications in geophysics [J]. Progress in Geophysics (in Chinese), 29(1): 452- 461, doi: 10.6038/pg20140166.
Liu G R, Gu Y T. 2007. An Introduction to Meshfree Methods and Their Programming [M]. Shandong: Shandong University Press.
LU J, Li Y G. 2017. Two-dimensional magnetotelluric modeling using the Meshfree Local Petrov-Galerkin method [J]. Chinese J. Geophys. (in Chinese), 60(3): 1189-1200, doi: 10.6038/cjg20170329.
Su Z, Hu W B, Zhu Y. 2012. Meshfree method used in two-dimensional magnetotelluric forwarding [J]. Journal of Oil and Gas Technology (in Chinese), 34(5): 87-90.
Tan H D, Yu Q F, Booker J, et al. 2003. Magnetotelluric three-dimensional modeling using the staggered-grid finite difference method [J]. Chinese Journal of Geophysics (in Chinese), 46(5): 705-711, doi: 10.3321/j.issn:0001-5733.2003.05.019.
Wang J G, Liu G R. 2002. A point interpolation meshless method based on radial basis functions [J]. International Journal for Numerical Methods in Engineering, 54(11): 1623-1648.
WANG Yueying. 2007. Study of element-free galerkin method in the seismic forward modeling [J]. Progress in Geophysics (in Chinese), 22(5): 1539-1544, doi: 10.3969/j.issn.1004-2903.2007.05.029.
XU Shizhe. 1994. Finite element method for geophysics [M]. Beijing: Science Press.
ZHANG Xiong, LIU Yan, et al. 2004. Meshless method [M]. Beijing: Tsinghua University Press, 1-15.
Belytschko T, Lu YY, Gu L, 等. 1994. 无单元Galerkin法[J]. 国际工程数值方法, 37(2): 229-256.
冯德山, 王洪华, 戴前伟.2013. 基于无单元Galerkin法探地雷达正演模拟[J]. 地球物理学报, 56(1): 298-308, doi: 10.6038/cjg20130131.
顾元通, 丁桦. 2005. 无网格法及其最新进展[J]. 力学进展, 35 (3): 323-337.
贾晓峰, 胡天跃, 王润秋. 2006a. 复杂介质中地震波模拟的无单元法[J]. 石油地球物理勘探, 41(1): 43- 48.
贾晓峰, 胡天跃, 王润秋. 2006b. 无单元法用于地震波波动方程模拟与成像[J]. 地球物理学进展, 21(1): 11-17, doi: 10.3969/j.issn.1004-2903.2006.01.003.
Liu G R, Gu YU. 2007. 无网格法理论及程序设计[M]. 山东: 山东大学出版社.
李俊杰, 严家斌. 2014. 无网格法进展及在地球物理学中的应用[J]. 地球物理学进展, 29(1): 452- 461, doi: 10.6038/pg20140166.
李俊杰. 2014. 基于弱式无网格法的大地电磁二维正演[硕士论文]. 长沙:中南大学.
卢杰, 李予国. 2017. 无网格局部Petrov-Galerkin法大地电磁场二维正演模拟[J]. 地球物理学报, 60(3): 1189-1200, doi: 10.6038/cjg20170329.
苏洲, 胡文宝, 朱毅. 2012. 二维大地电磁正演中无网格算法研究[J]. 石油天然气学报, 34(5): 87-90.
谭捍东, 余钦范, Booker J, 等. 2003. 大地电磁法三维交错采样有限差分数值模拟[J].地球物理学报, 46(5): 705-711, doi: 10.3321/j.issn:0001-5733.2003.05.019.
Wang J G, Liu G R. 2002. 一种基于径向基函数法的无网格点插法[J]. 国际工程数值方法, 54(11): 1623-1648.
王月英. 2007. 地震波正演模拟中无单元Galerkin法初探[[J]. 地球物理学进展, 22(5): 1539-1544, doi: 10.3969/j.issn.1004-2903.2007.05.029.
徐世浙. 1994. 地球物理中的有限单元法[M]. 北京:科学出版社.
张雄, 刘岩. 2004. 无网格法[M]. 北京:清华大学出版社,1-15.
FENG DeShan, WANG HongHua, DAI QianWei.2013. Forward simulation of Ground Penetrating Radar based on the element-free Galerkin method [J].Chinese Journal of Geophysics (in Chinese), 56(1): 298-308, doi: 10.6038/cjg20130131.
GU Yuantong, DING Hua. 2005. Recent developments of meshless method [J]. Advances in Mechanics (in Chinese), 35(3): 323-337.
Jia Xiao-feng, Hu Tian-Yue, Wang Run-qiu. 2006a. A mesh-free algorithm of seismic wave simulation in complex medium [J]. Oil Geophysical Prospecting (in Chinese), 41(1): 43- 48.
Jia X F, Hu T Y, Wang R Q. 2006b. Wave equation modeling and imaging by using element-free method [J]. Progress in Geophysics (in Chinese), 21(1): 11-17, doi: 10.3969/j.issn.1004-2903.2006.01.003.
Li J J. 2014. Magnetotelluric two-dimensional forward by weak-form meshfree methods[Master’s thesis].Changsha:Central South University.
Li J J, Yan J B. 2014. Developments of meshless method and applications in geophysics [J]. Progress in Geophysics (in Chinese), 29(1): 452- 461, doi: 10.6038/pg20140166.
Liu G R, Gu Y T. 2007. An Introduction to Meshfree Methods and Their Programming [M]. Shandong: Shandong University Press.
LU J, Li Y G. 2017. Two-dimensional magnetotelluric modeling using the Meshfree Local Petrov-Galerkin method [J]. Chinese J. Geophys. (in Chinese), 60(3): 1189-1200, doi: 10.6038/cjg20170329.
Su Z, Hu W B, Zhu Y. 2012. Meshfree method used in two-dimensional magnetotelluric forwarding [J]. Journal of Oil and Gas Technology (in Chinese), 34(5): 87-90.
Tan H D, Yu Q F, Booker J, et al. 2003. Magnetotelluric three-dimensional modeling using the staggered-grid finite difference method [J]. Chinese Journal of Geophysics (in Chinese), 46(5): 705-711, doi: 10.3321/j.issn:0001-5733.2003.05.019.
Wang J G, Liu G R. 2002. A point interpolation meshless method based on radial basis functions [J]. International Journal for Numerical Methods in Engineering, 54(11): 1623-1648.
WANG Yueying. 2007. Study of element-free galerkin method in the seismic forward modeling [J]. Progress in Geophysics (in Chinese), 22(5): 1539-1544, doi: 10.3969/j.issn.1004-2903.2007.05.029.
XU Shizhe. 1994. Finite element method for geophysics [M]. Beijing: Science Press.
ZHANG Xiong, LIU Yan, et al. 2004. Meshless method [M]. Beijing: Tsinghua University Press, 1-15.
Belytschko T, Lu YY, Gu L, 等. 1994. 无单元Galerkin法[J]. 国际工程数值方法, 37(2): 229-256.
冯德山, 王洪华, 戴前伟.2013. 基于无单元Galerkin法探地雷达正演模拟[J]. 地球物理学报, 56(1): 298-308, doi: 10.6038/cjg20130131.
顾元通, 丁桦. 2005. 无网格法及其最新进展[J]. 力学进展, 35 (3): 323-337.
贾晓峰, 胡天跃, 王润秋. 2006a. 复杂介质中地震波模拟的无单元法[J]. 石油地球物理勘探, 41(1): 43- 48.
贾晓峰, 胡天跃, 王润秋. 2006b. 无单元法用于地震波波动方程模拟与成像[J]. 地球物理学进展, 21(1): 11-17, doi: 10.3969/j.issn.1004-2903.2006.01.003.
Liu G R, Gu YU. 2007. 无网格法理论及程序设计[M]. 山东: 山东大学出版社.
李俊杰, 严家斌. 2014. 无网格法进展及在地球物理学中的应用[J]. 地球物理学进展, 29(1): 452- 461, doi: 10.6038/pg20140166.
李俊杰. 2014. 基于弱式无网格法的大地电磁二维正演[硕士论文]. 长沙:中南大学.
卢杰, 李予国. 2017. 无网格局部Petrov-Galerkin法大地电磁场二维正演模拟[J]. 地球物理学报, 60(3): 1189-1200, doi: 10.6038/cjg20170329.
苏洲, 胡文宝, 朱毅. 2012. 二维大地电磁正演中无网格算法研究[J]. 石油天然气学报, 34(5): 87-90.
谭捍东, 余钦范, Booker J, 等. 2003. 大地电磁法三维交错采样有限差分数值模拟[J].地球物理学报, 46(5): 705-711, doi: 10.3321/j.issn:0001-5733.2003.05.019.
Wang J G, Liu G R. 2002. 一种基于径向基函数法的无网格点插法[J]. 国际工程数值方法, 54(11): 1623-1648.
王月英. 2007. 地震波正演模拟中无单元Galerkin法初探[[J]. 地球物理学进展, 22(5): 1539-1544, doi: 10.3969/j.issn.1004-2903.2007.05.029.
徐世浙. 1994. 地球物理中的有限单元法[M]. 北京:科学出版社.
张雄, 刘岩. 2004. 无网格法[M]. 北京:清华大学出版社,1-15.
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