Precision of meshfree methods and application to forward modeling of two-dimensional electromagnetic sources
Li Jun-Jie1, Yan Jia-Bin2, and Huang Xiang-Yu2
1. Zhejiang Design Institute of Water Conservancy and Hydroelectric Power, Hangzhou 310002, China.
2. Key Laboratory of Non-ferrous Resources and Geological Hazard Detection, School of Geosciences and Info-Physics, Central South University, Changsha 410083, China.
Abstract Meshfree method offers high accuracy and computational capability and constructs the shape function without relying on predefined elements. We comparatively analyze the global weak form meshfree methods, such as element-free Galerkin method (EFGM), the point interpolation method (PIM), and the radial point interpolation method (RPIM). Taking two dimensional Poisson equation as an example, we discuss the support-domain dimensionless size, the field nodes, and background element settings with respect to their effect on calculation accuracy of the meshfree method. RPIM and EFGM are applied to controlled-source two-dimensional electromagnetic modeling with fixed shape parameters. The accuracy of boundary conditions imposed directly and by a penalty function are discussed in the case of forward modeling of two-dimensional magnetotellurics in a homogeneous medium model. The coupling algorithm of EFG–PIM and EFG–RPIM are generated by integrating the PIM or RPIM and EFGM. The results of the numerical modeling suggest the following. First, the proposed meshfree method and corresponding coupled methods are well-suited for electromagnetic numerical modeling. The accuracy of the algorithm is the highest when the support-domain dimensionless size is 1.0 and the distribution of field nodes is consistent with the nodes of background elements. Second, the accuracy of PIM and RPIM are lower than that of EFGM for the Poisson equation but higher than EFGM for the homogeneous medium MT response. Third, RPIM overcomes the matrix inversion problem of PIM and has a wider selection of support-domain dimensionless sizes as compared to RPIM.
This research was supported by the National Nature Science Foundation of China (Grant No. 40874055) and the Natural Science Foundation of the Hunan Province, China (Grant No. 14JJ2012).
Cite this article:
Li Jun-Jie,Yan Jia-Bin,Huang Xiang-Yu. Precision of meshfree methods and application to forward modeling of two-dimensional electromagnetic sources[J]. APPLIED GEOPHYSICS, 2015, 12(4): 503-515.
[1]
Belytschko, T., Lu, Y. Y., and Gu, L., 1994a, Fracture and crack growth by element-free Galerkin methods: Modelling and Simulation in Materials Science and Engineering, 2(3), 519−534.
[2]
Belytschko, T., Lu, Y. Y., and Gu, L., 1994b, Element-free Galerkin methods: International Journal for Numerical Methods in Engineering, 37(2), 229−256.
[3]
Dong, H., Wei, W. B., Ye, G. F., Jin, S., and Jing, J. E., 2014, Study of three-dimensional magnetotelluric inversion including surface topography based on finite-difference method: Chinese Journal of Geophysics, 57(3), 939−952.
[4]
Dai, Q. W., and Wang, H. H., 2013, Element free method forward modeling of GPR based on random medium model: The Chinese Journal of Nonferrous Metals, 23(9), 1−9.
[5]
Feng, C., Li, S. H., and Liu, X.Y., 2014, A 2D particle contact-based meshfree method and its application to hypervelocity impact simulation: Explosion and Shock Waves, 34(3), 292−299.
[6]
Feng, D. S., Wang, H. H., and Dai, Q. W., 2013, Forward simulation of Ground Penetrating Radar based on the element-free Galerkin method: Chinese Journal of Geophysics, 56(1), 298−308.
[7]
Franke, C., and Schaback, R., 1998, Solving partial differential equations by collocation using radial basis functions: Applied Mathematics and Computation, 93(1), 73−82.
[8]
Hu, Y. C., Li, T. L., Fan, C. S., Wang, D. Y., and Li, J. P., 2015, Three-dimensional tensor controlled-source electromagnetic modeling based on the vector finite-element method: Applied Geophysics, 12(1), 35−46.
[9]
Jia, X. F., Hu, T. Y., and Wang, R. Q., 2005, A meshless method for acoustic and elastic modeling: Applied Geophysics, 2(1), 1−6.
[10]
Jia, X. F., Hu, T. Y., and Wang, R. Q., 2006, A mesh-free algorithm of seismic wave simulation in complex medium: Oil Geophysical Prospecting, 41(1), 43−48
[11]
Li, J., 2010, Frequency domain controlled-source two-dimensional forward numerical simulation using finite element method: PhM Thesis, Central South University of Geosciences and Info-Physics, Changsha.
[12]
Li, J. J., and Yan, J. B., 2014a, Developments of meshlessmethod and applications in geophysics: Progress in Geophysics, 29(1), 452−461.
[13]
Li, J. J., and Yan J. B., 2014b, Magnetotelluric two-dimensional forward numerical modeling by meshfree point interpolation method: Geophysical Prospecting for Petroleum, 53(5), 617−626.
[14]
Li, J. J., and Yan, J. B., 2015, Magnetotelluric twodimensional forward by finite element-radial point interpolation method: The Chinese Journal of Nonferrous Metals, 25(5), 1314−1324.
[15]
Liu, G. R., and Gu, Y. T., 2001, A point interpolation method for two-dimensional solids: International Journal for Numerical Methods in Engineering, 50(4), 937−951.
[16]
Liu, G. R., and Gu, Y. T., 2005, An Introduction to Meshfree Methods and Their Programming: Springer, Berlin.
[17]
Liu, L. C., Dong, X. H., and Li, C. X., 2009, Adaptive finite element-element-free Galerkin coupling method for bulk metal forming processes: Journal of Zhejiang University-Science A, 10(3), 353−360.
[18]
Liu, Y., Wang, X. B., and Wang, Y., 2013, Numerical modeling of the 2D time-domain transient electromagnetic secondary field of the line source of the current excitation: Applied Geophysics, 10(2), 134−144.
[19]
Lu, P., Zhao, G. Q., Guan Y. J., and Wu, X., 2008, Bulk metal forming process simulation based on rigid-plastic/visco-plastic element free Galerkin method: Materials Science and Engineering A, 479(1), 197−212.
[20]
Ma, S., Zhang, X., and Qiu, X. M., 2009, Comparison study of MPM and SPH in modeling hypervelocity impact problems: International Journal of Impact Engineering, 36(2), 272−282.
[21]
Ren, Z. Y., and Tang, J. T., 2010, 3D direct current resistivity modeling with unstructured mesh by adaptive finite-element method: Geophysics, 75(1), 7−17.
[22]
Wang, J. G., and Liu, G. R., 2002, A point interpolation meshless method based on radial basis functions: International Journal for Numerical Methods in Engineering, 54(11), 1623−1648.
[23]
Wang, R., Wang, M. Y., and Di, Q. Y., 2006, Electromagnetic modeling due to line source in frequency domain using finite element method: Chinese Journal of Geophysics, 49(6), 1858−1866.
Xu, J. R., 2010, Research and applications of 2-D MT forward modeling and inversion with topography: PhD Thesis, Central South University of Geosciences and Info-Physics, Changsha.
[26]
Xu, S. Z., 1994, Finite element method for geophysics: Science Press, Beijing, 229−235.
[27]
Yan, J. B., and Li, J. J., 2014, Magnetotelluric forward calculation by meshless method: Journal of Central South University (Science and Technology), 45(10), 3513−3520.
[28]
Zhang, J. F., Tang, J. T., Yu, Y., and Liu, C. S., 2009, Finite element numerical simulation on line controlled source based on quadratic interpolation: Journal of Jilin University (Earth Science Edition), 39(5), 929−935.
[29]
Zhdanov, M. S., Varentsov, I. M., Weaver, J. T., Golubev, N. G., and Krylov V. A., 1997, Methods for modelling electromagnetic fields results from COMMEMI-the international project on the comparison of modelling methods for electromagnetic induction: Journal of Applied Geophysics, 37(3), 133−271.
[30]
Zhang, Y. Y., and Chen, L., 2008, A simplified meshless method for dynamic crack growth: Cmes-computer Modeling in Engineering and Sciences, 31(3), 189−200.
2015, Vol. 12 
