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Second-generation wavelet finite element based on the lifting scheme for GPR simulation

  • Ground penetrating radar
  • Published:
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Abstract

Ground-penetrating radar (GPR) is a highly efficient, fast and non-destructive exploration method for shallow surfaces. High-precision numerical simulation method is employed to improve the interpretation precision of detection. Second-generation wavelet finite element is introduced into the forward modeling of the GPR. As the finite element basis function, the second-generation wavelet scaling function constructed by the scheme is characterized as having multiple scales and resolutions. The function can change the analytical scale arbitrarily according to actual needs. We can adopt a small analysis scale at a large gradient to improve the precision of analysis while adopting a large analytical scale at a small gradient to improve the efficiency of analysis. This approach is beneficial to capture the local mutation characteristics of the solution and improve the resolution without changing mesh subdivision to realize the efficient solution of the forward GPR problem. The algorithm is applied to the numerical simulation of line current radiation source and tunnel non-dense lining model with analytical solutions. Result show that the solution results of the second-generation wavelet finite element are in agreement with the analytical solutions and the conventional finite element solutions, thereby verifying the accuracy of the second-generation wavelet finite element algorithm. Furthermore, the second-generation wavelet finite element algorithm can change the analysis scale arbitrarily according to the actual problem without subdividing grids again. The adaptive algorithm is superior to traditional scheme in grid refinement and basis function order increase, which makes this algorithm suitable for solving complex GPR forward-modeling problems with large gradient and singularity.

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References

  • Amaratunga, K., and Williams, J. R., 1993, Wavelet based Green’s function approach to 2D PDEs: Engineering Computations, 10(4), 349–367.

    Article  Google Scholar 

  • Amaratunga, K., Williams, J. R., Qian, S., and Weiss, J., 1994, Wavelet-Galerkin solutions for one-dimensional partial differential equations: International Journal for Numerical Methods in Engineering, 37(16), 2703–2716.

    Article  Google Scholar 

  • Berenger, J. P., 1994, A Perfectly Matched Layer for the Absorption of Electromagnetic Waves: Journal of Computational Physics, 114(2), 185–200.

    Article  Google Scholar 

  • Chen, H. B., and Li, T. L., 2019, 3-D marine controlled-source electromagnetic modelling in an anisotropic medium using a Wavelet-Galerkin method with a secondary potential formulation: Geophysical Journal International, 219(1), 373–393.

  • Dai, Q. W., Wang, H. H., Feng, D. S., and Chen, D. P., 2012, Finite element method forward simulation for complex geoelectricity GPR model based on triangle mesh dissection: Journal of Central South University (Science and Technology), 43(7), 2668–2673.

    Google Scholar 

  • D’Heedene, S., Amaratunga, K., and Castrillón-Candás, J., 2005, Generalized hierarchical bases: a Wavelet-Ritz-Galerkin framework for Lagrangian FEM: Engineering Computations, 22(1), 15–37.

    Article  Google Scholar 

  • Fang, H. Y., and Lin, G., 2013, Simulation of GPR wave propagation in complicated geoelectric model using symplectic method: Chinese Journal of Geophysics, 56(2), 653–659.

    Google Scholar 

  • Feng, D. S., and Dai, Q. W., 2011, GPR numerical simulation of full wave field based on UPML boundary condition of ADI-FDTD: NDT & E International, 44(6), 495–504.

    Article  Google Scholar 

  • Feng, D. S., and Wang, X., 2017, Convolution perfectly matched layer for the Finite-Element Time-Domain method modeling of ground penetrating radar: Chinese Journal of Geophysics, 60(1), 413–423.

    Google Scholar 

  • Feng, D. S., and Wang, X., 2018, Adaptive multi-scale second-generation wavelet collocation method numerical simulation of Ground Penetrating Radar: Chinese Journal Geophysics, 61(9), 3851–3864.

    Google Scholar 

  • Feng, D. S., Wang, X., and Zhang, B., 2018, Specific evaluation of tunnel lining multi-defects by all-refined GPR simulation method using hybrid algorithm of FETD and FDTD: Construction and Building Materials, 185, 220–229.

    Article  Google Scholar 

  • Ge, D. B., and Wei, B., 2014, Time domain computational method for electromagnetic wave (Volume II) (in Chinese): Xidian University Press, Xi’an, 188–191.

    Google Scholar 

  • Gedney, S. D., and Navsariwala, U., 1995, An unconditionally stable finite element time-domain solution of the vector wave equation: IEEE Microwave and Guided Wave Letters, 5(10), 332–334.

    Article  Google Scholar 

  • Giannopoulos, A., 2005, Modelling ground penetrating radar by GprMax: Construction and Building Materials, 19(10), 755–762.

    Article  Google Scholar 

  • He, W. Y., Zhu, S., and Chen, Z. W., 2018, A multi-scale wavelet finite element model for damage detection of beams under a moving load: International Journal of Structural Stability and Dynamics, 18(6), 1850078.

    Article  Google Scholar 

  • He, Y. M., Chen, X. F., and Xiang, J. W., 2007, Adaptive multiresolution finite element method based on second generation wavelets: Finite Elements in Analysis and Design, 43(6–7), 566–579.

    Article  Google Scholar 

  • He, Z. J., Chen, X. F., Li, B., and Xian, J. W., 2006, Theory of the wavelet based Finite Element Methods and the application in Engineering (in Chinese): Science Press, Beijing, 229–246

    Google Scholar 

  • Huang, Q. H., Li, Z. H., and Wang, Y. B., 2010, A parallel 3-D staggered grid pseudospectral time domain method for ground-penetrating radar wave simulation: Journal of Geophysical Research: Solid Earth, 115(B12).

    Google Scholar 

  • Irving, J., and Knight, R., 2006, Numerical modeling of ground-penetrating radar in 2-D using MATLAB: Computers & Geosciences, 32(9), 1247–1258.

    Article  Google Scholar 

  • Kevin, A., John, R. W., Sam, Q., and John, W., 2010, Wavelet-Galerkin solutions for one-dimensional partial differential equations: International Journal for Numerical Methods in Engineering, 37(16), 2703–2716.

    Google Scholar 

  • Li, J., Zeng, Z. F., Huang, L., and Liu, F. S., 2012, GPR simulation based on complex frequency shifted recursive integration PML boundary of 3D high order FDTD: Computers & Geosciences, 49, 121–130.

    Article  Google Scholar 

  • Liu, Q. H., and Fan, G. X., 1999, Simulations of GPR in dispersive media using a frequency-dependent PSTD algorithm: IEEE Transactions on Geoscience and Remote Sensing, 37(5), 2317–2324.

    Article  Google Scholar 

  • Liu, X. J., Wang, J. Z., and Zhou, Y. H., 2016, A space-time fully decoupled wavelet Galerkin method for solving two-dimensional Burgers’ equations: Computers & Mathematics with Applications, 72(12), 2908–2919

    Article  Google Scholar 

  • Lu, T., Cai, W., and Zhang, P. W., 2005, Discontinuous galerkin time-domain method for GPR simulation in dispersive media: IEEE Transactions on Geoscience and Remote Sensing, 43(1), 72–80.

    Article  Google Scholar 

  • Mishra, V. S., 2011, Wavelet Galerkin solutions of ordinary differential equations: International Journal of Mathematical Analysis, 5(9), 407–424.

    Google Scholar 

  • Nazabat, H., Mohd, N. K., Varun, J., et al. 2016, Use of wavelets in marine controlled source electromagnetic method for geophysical modeling: International Journal of Applied Electromagnetics and Mechanics, 51(4), 431–443.

    Article  Google Scholar 

  • Quraishi, S. M., and Sandeep, K., 2011, A second generation wavelet based finite elements on triangulations: Computational Mechanics, 48(2), 163–174.

    Article  Google Scholar 

  • Quraishi, S. M., and Sandeep, K., 2013, Multiscale modeling of beam and plates using customized second-generation wavelets: Journal of Engineering Mathematics, 83(1), 185–202.

    Article  Google Scholar 

  • Ren, Q., Tobon, L. E., and Liu, Q. H., 2013, A new 2D non-spurious discontinuous Galerkin finite element time domain (DG-FETD) method for Maxwell’s equations: Progress in Electromagnetics Research, 143, 385–404

    Article  Google Scholar 

  • Restrepo, J. M., and Leaf, G. K., 1997, Inner product computations using periodized Daubechies wavelets: International Journal for Numerical Methods in Engineering, 40(19), 3557–3578.

    Article  Google Scholar 

  • Sarkar, T. K., Adve, R. S., García-Castillo, L. E., and Salazar-Palma, M., 1994, Utilization of wavelet concepts in finite elements for an efficient solution of Maxwell’s equations: Radio Science, 29(4), 965–977.

    Article  Google Scholar 

  • Shi, L. K., Shen, X. Q., Yan, W. L., et al. 2001. A wavelet interpolation Galerkin method for the solution of boundary value problems in 2D electrostatic field: Journal of Hebei University of Technology (in Chinese), 30(1), 62–66.

    Google Scholar 

  • Somchai, S. I., and Eckart, S., 2013, Wavelet-Galerkin solution of a partial differential equation with nonlinear viscosity: Applied Mathematical Sciences, 38(7), 1849–1880.

    Google Scholar 

  • Sweldens, W., 1994, The construction and application of wavelets in numerical analysis: PhD Thesis, Department of Computer Science, Katholieke Universiteit Leuven, Belgicko.

    Google Scholar 

  • Sweldens, W., 1998, The lifting scheme: a construction of second generation wavelet: SIAM Journal on Mathematical Analysis, 29(2), 511–546.

    Article  Google Scholar 

  • Sweldens, W., and Schröder, P., 2000, Wavelets in the Geosciences: Springer, Berlin, 72–107.

    Book  Google Scholar 

  • Wang, Y. M., Chen, X. F., and He, Z. J., 2012, A second-generation wavelet-based finite element method for the solution of partial differential equations: Applied Mathematics Letters, 25(11), 1608–1613

    Article  Google Scholar 

  • Yang, S. Y., Ni, G. Z., 1999, Wavelet-Galerkin method for the numerical calculation of electromagnetic fields: Proceedings of the CSEE (in Chinese), 19(1), 56–61.

    Google Scholar 

  • Ye, J. J., He, Y. M., Chen, X. F., et al., 2010, Pipe crack identification based on finite element method of second generation wavelets: Mechanical Systems and Signal Processing, 24(2), 379–393.

    Article  Google Scholar 

  • Yee, K. S., 1996, Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media: IEEE Transactions on Antennas and Propagation, 14(3), 302–307.

    Google Scholar 

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Authors and Affiliations

  1. School of Geosciences and Info-Physics, Central South University, Changsha, 410083, China

    De-Shan Feng, Hua Zhang & Xun Wang

  2. Key Laboratory of Metallogenic Prediction of Nonferrous Metals, Ministry of Education, Changsha, 410083, China

    De-Shan Feng, Hua Zhang & Xun Wang

Authors
  1. De-Shan Feng
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  2. Hua Zhang
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  3. Xun Wang
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Corresponding author

Correspondence to Xun Wang.

Additional information

This work was supported by the National Natural Science Foundation of China (Nos. 41574116 and 41774132) and Hunan Provincial Innovation Foundation for Postgraduate (Grant Nos. CX2017B052) and the Fundamental Research Funds for the Central Universities of Central South University (Nos. 2018zzts693)

Feng De-shan, professor, doctoral supervisor. He received Ph.D. (2006) in Geophysical Prospecting and Information Technology from Central South University. Visiting scholar at Rice University, USA, 2013–2014. He is interested in the theory and application of GPR, forward modeling and inversion, and wavelet analysis. E-mail: fengdeshan@126.com

Wang Xun, Corresponding author, received his M.S. (2016) in Geological Engineering from Central South University. He is presently a Ph.D. candidate in Geophysical Prospecting and Information Technology at Central South University. His main interests is numerical simulation of electromagnetic method. E-mail: wangxun0727@csu.edu.cn.

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Feng, DS., Zhang, H. & Wang, X. Second-generation wavelet finite element based on the lifting scheme for GPR simulation. Appl. Geophys. 17, 143–153 (2020). https://doi.org/10.1007/s11770-020-0801-2

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  • DOI: https://doi.org/10.1007/s11770-020-0801-2

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