Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-25T12:42:23.383Z Has data issue: false hasContentIssue false
  • English
  • Français

Directly Simulation of Second Order Hyperbolic Systems in Second Order Form via the Regularization Concept

Part of: Partial differential equations, boundary value problems Hyperbolic equations and systems

Published online by Cambridge University Press:  22 June 2016

Hassan Yousefi ,
Seyed Shahram Ghorashi  and
Timon Rabczuk
Hassan Yousefi*
Affiliation:
Institute of Structural Mechanics, Bauhaus-Universität Weimar, 99423 Weimar, Germany
Seyed Shahram Ghorashi*
Affiliation:
Research Training Group 1462, Bauhaus-Universität Weimar, 99423 Weimar, Germany
Timon Rabczuk*
Affiliation:
Division of Computational Mechanics, Ton Duc Thang University, Ho Chi Minh City, Viet Nam Faculty of Civil Engineering, Ton Duc Thang University, Ho Chi Minh City, Viet Nam
*
*Corresponding author. Email addresses:[email protected]; [email protected] (H. Yousefi), [email protected] (S. Sh. Ghorashi), [email protected] (T. Rabczuk)
*Corresponding author. Email addresses:[email protected]; [email protected] (H. Yousefi), [email protected] (S. Sh. Ghorashi), [email protected] (T. Rabczuk)
*Corresponding author. Email addresses:[email protected]; [email protected] (H. Yousefi), [email protected] (S. Sh. Ghorashi), [email protected] (T. Rabczuk)
Get access

Abstract

We present an efficient and robust method for stress wave propagation problems (second order hyperbolic systems) having discontinuities directly in their second order form. Due to the numerical dispersion around discontinuities and lack of the inherent dissipation in hyperbolic systems, proper simulation of such problems are challenging. The proposed idea is to denoise spurious oscillations by a post-processing stage from solutions obtained from higher-order grid-based methods (e.g., high-order collocation or finite-difference schemes). The denoising is done so that the solutions remain higher-order (here, second order) around discontinuities and are still free from spurious oscillations. For this purpose, improved Tikhonov regularization approach is advised. This means to let data themselves select proper denoised solutions (since there is no pre-assumptions about regularized results). The improved approach can directly be done on uniform or non-uniform sampled data in a way that the regularized results maintenance continuous derivatives up to some desired order. It is shown how to improve the smoothing method so that it remains conservative and has local estimating feature. To confirm effectiveness of the proposed approach, finally, some one and two dimensional examples will be provided. It will be shown how both the numerical (artificial) dispersion and dissipation can be controlled around discontinuous solutions and stochastic-like results.

Keywords

Second order hyperbolic systemsTikhonov regularizationNumerical (artificial) dispersion

MSC classification

Secondary: 35L51: Second-order hyperbolic systems 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
Type
Research Article
Information
Communications in Computational Physics , Volume 20 , Issue 1 , July 2016 , pp. 86 - 135
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Appelö, D., Banks, J.W., Henshaw, W. D., and Schwendeman, D.W., 2012: Numericalmethods for solid mechanics on overlapping grids: Linear elasticity, J. Comput. Phys., 231(18), 6012–6050.Google Scholar
[2] Arcangeli, R., and Ycart, B., 1993: Almost sure convergence of smoothing Dm -splines for noisy data, Numer. Math., 66(3), 281–294.CrossRefGoogle Scholar
[3] Banks, J.W., and Henshaw, W. D. 1, 2012: Upwind schemes for the wave equation in second-order form, J. Comput. Phys., 231(17), 5854–5889.Google Scholar
[4] Barakat, V., Guilpart, B., Goutte, R., and Prost, R., 1997: Model-based Tikhonov-Miller image restoration, IEEExplore, Proceedings International conference on Image processing (ICIP’97), pp. 310-31, ISBN: 0-8186-8183-7, Washington, DC, USA, October 26-29.Google Scholar
[5] Boulsina, F., Berrabah, M., and Dupuy, J. C., 2008: Deconvolution of SIMS depth profiles: Towards simple and faster techniques, Appl. Surf. Sci., 255(5), 1946–1958.Google Scholar
[6] Boyd, J. P., 1992: Defeating the Runge phenomenon for equispaced polynomial interpolation via Tikhonov regularization, Appl. Math. Lett., 5(6), 57–59.CrossRefGoogle Scholar
[7] Boyd, J. P., and Ong, J. R., 2009: Exponentially-convergent strategies for defeating the Runge phenomenon for the approximation of non-periodic functions, Part I: Single-interval schemes, Commun. Comput. Phys., 5(2-4), 484–497.Google Scholar
[8] Calhoun-Lopez, M., and Gunzburger, M. D., 2005: A finite element, multiresolution viscosity method for hyperbolic conservation laws, SIAM J. Num. Anal., 43, 1988–2011.CrossRefGoogle Scholar
[9] Calhoun-Lopez, M., and Gunzburger, M. D., 2007: The efficient implementation of a finite element, multi-resolution viscosity method for hyperbolic conservation laws, J. Comput. Phys., 225, 1288–1313.CrossRefGoogle Scholar
[10] Chartrand, R. and Staneva, V., 2008: Total variation regularization of images corrupted by non-Gaussian noise using a quasi-Newton method, IEEE T. Image Processing, 2(6), 295–303.CrossRefGoogle Scholar
[11] Cullum, J., 1971: Numerical differentiation and regularization, SIAM J. Num. Anal., 8(2), 254–265.CrossRefGoogle Scholar
[12] Day, S. M., and Ely, G. P., 2002: Effect of a shallow weak zone on fault rupture: Numerical simulation of scale-model experiments, Bull. Seismol. Soc. Am., 92(8), 3022–3041.CrossRefGoogle Scholar
[13] Day, S. M., Dalguer, L. A., Lapusta, N., and Liu, Y., 2005: Comparison of finite difference and boundary integral solutions to three-dimensional spontaneous rupture, J. Geophys. Res., 110.Google Scholar
[14] Fornberg, B., 1998: Calculation of weights in finite difference formulas, SIAM Rev. 40(3), 685–691.CrossRefGoogle Scholar
[15] Gautier, B., Prudon, G., and Dupuy, J. C., 1998: Toward a better reliability in the deconvolution of SIMS depth profiles, Surf. Interface Anal., 26(13), 974–983.Google Scholar
[16] Gottlieb, D., and Hesthaven, J. S., 2001: Spectral methods for hyperbolic problems, J. Comput. Appl. Math., 128(1-2), 83–131.CrossRefGoogle Scholar
[17] Groetsch, C.W., 1998: Lanczo's generalized derivative, Amer. Math. Monthly, 105, 320–326.Google Scholar
[18] Gu, Y., and Wei, G.W., 2003: Conjugate filter approach for shock capturing, Commun. Numer. Meth. Engng, 19, 99–110.CrossRefGoogle Scholar
[19] Hansen, P. C., 1998: Rank-Deficient and Discrete Ill-Posed Problems, Philadelphia: SIAM.Google Scholar
[20] Hilber, H. M., Hughes, T. J. R., and Taylor, R. L., 1977: Improved numerical dissipation for time integration algorithms in structural dynamics, Earthq. Eng. Struct. Dyn., 5(3), 283–292.Google Scholar
[21] Hoff, C., and Pahl, P. J., 1988: Development of an implicit method with numerical dissipation from a generalized single step algorithm for structural dynamics, Comput. Methods Appl. Mech. Eng., 67(3), 367–385.CrossRefGoogle Scholar
[22] Hoff, C., and Pahl, P. J., 1988: Practical Performance of the θ1- method and comparison with other dissipative algorithms in structural dynamics, Comput. Methods Appl. Mech. Eng., 67, 87–110.Google Scholar
[23] Hoover, W. G., 2006: Smooth Particle Applied Mechanics, The State of the Art, Word Scientific.CrossRefGoogle Scholar
[24] Horn, B. K. P., and Schunck, B. G., 1981: Determining optical flow, Artificial Intelligence, 17(1-3), 185–203.Google Scholar
[25] Hughes, T. J. R., 1987: The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Englewood Cliffs: Prentice Hall.Google Scholar
[26] Hulbert, G.H., and Chung, J., 1996: Explicit time integration algorithms for structural dynamics with optimal numerical dissipation, Comput. Methods Appl. Mech. Eng., 137, 175–188.Google Scholar
[27] Hutchinson, M. F., and de Hoog, F. R., 1985: Smoothing noisy data with spline functions, Numer. Math., 47(1), 99–106.CrossRefGoogle Scholar
[28] Jauberteaua, F., and Jauberteaub, J. L., 2009: Numerical differentiation with noisy signal, Appl. Math. Comput., 215(6), 2283–2297.Google Scholar
[29] Kawahara, J., and Yamashita, T., 1992: Scattering of elastic waves by a fracture zone containing randomly distributed cracks, Pure Appl. Geophys., 139, 121–144.Google Scholar
[30] Kersey, S. N., 2006: Mixed interpolating-smoothing splines and the ν-spline, J. Math. Anal. Appl., 322(1), 28–40.Google Scholar
[31] Khan, I. R., and Ohba, R., 2000: New finite difference formulas for numerical differentiation, J. Comput. Appl. Math., 126, 269–276.Google Scholar
[32] Kurganov, A., and Pollack, M., 2011: Semi-discrete central-upwind schemes for elasticity in heterogeneous media, (submitted for publication).Google Scholar
[33] Lanczos, C., 1956: Applied Analysis, Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
[34] Le, T., Chartrand, R., and Asaki, Th. J., 2007: A variational approach to reconstructing images corrupted by Poisson noise, J. Math. Imaging Vision, 27(3), 257–263.Google Scholar
[35] Lele, S. K., 1992: Compact finite difference schemes with spectral-like resolution, J. Comput. Phys., 103(1), 16–42.Google Scholar
[36] Lee, T. C. M., 2003: Smoothing parameter selection for smoothing splines: A simulation study. Comput. Stat. Data Anal., 42(1-2), 139–148.CrossRefGoogle Scholar
[37] LeVeque, R. J., 1992: Numerical Methods for Conservation Laws, second edition, Berlin, Birkhäuser.Google Scholar
[38] Li, X. D., and Wiberg, N. E., 1996: Structural dynamic analysis by a time-discontinuous Galerkin finite element method, Int. J. Numer. Methods Eng., 39, 2131–2152.Google Scholar
[39] Loader, C., 2004: Smoothing: Local regression techniques, in Gentle, J. E., Härdle, W., and Mori, Y., Hand-Book of Computational Statistics Concepts and Methods, Berlin: Springer.Google Scholar
[40] Nakamura, G., Wang, S., and Wang, Y., 2008: Numerical differentiation for the second order derivatives of functions of two variables, J. Comput. Appl. Math., 212, 341–358.Google Scholar
[41] Petrov, Y. P., and Sizikov, V. S., 2005: Well-Posed, Ill-Posed, and Intermediate Problems with Applications, Berlin, Boston: De Gruyter.Google Scholar
[42] Pruess, S., 1976: Properties of splines in tension, J. Approx. Theory, 17(1), 86–96.CrossRefGoogle Scholar
[43] Pruess, S., 1978: An algorithm for computing smoothing splines in tension, Computing, 19(4), 365–373.Google Scholar
[44] Ragozin, D. L., 1983: Error bounds for derivative Estimates based on spline smoothing of exact or noisy data, J. Approx. Theory, 37, 335–355.Google Scholar
[45] Ramm, A. G., and Smirnova, A. B., 2001: On stable numerical differentiation, Math. Comp., 70(235), 1131–1153.CrossRefGoogle Scholar
[46] Ramm, A. G., and Smirnova, A., 2003: Stable numerical differentiation: When is it possible?, Jour. Korean SIAM, 7(1), 47–61.Google Scholar
[47] Rangarajana, S. K., and Purushothaman, S. P., 2005: Lanczo's generalized derivative for higher orders, J. Comput. Appl. Math., 177, 461–465.Google Scholar
[48] Rentrop, P., 1980: An algorithm for the computation of the exponential spline, Numer.Math., 35(1), 81–93.Google Scholar
[49] Rentrop, P., and Wever, U., 1987: Computational strategies for the tension parameters of the exponential spline, in Thoma, M., Allgöwer, F., and Morari, M., Lecture Notes in Control and Information Sciences, Springer: ISSN: 0170-8643, pp 122–134.Google Scholar
[50] Rodríguez, P., 2013: Total variation regularization algorithms for images corrupted with different noise models: A review, J. Electr. Comput. Engrg, 2013, 1–18.Google Scholar
[51] Rohlfing, T., Maurer, C. R. J. r., Bluemke, D. A., and Jacobs, M. A., 2003: Volume-preserving nonrigid registration of MR breast images using free-formdeformation with an incompressibility constraint, IEEE T. Med Imaging, 22(6), 730-41.Google Scholar
[52] Ruan, D., Fessler, J. A., Roberson, M., Balter, J., and Kessler, M., 2006: Nonrigid registration using regularization that accomodates local tissue rigidity, Proc. SPIE 6144, Medical Imaging 2006: Image Processing, 614412; doi:10.1117/12.653870.Google Scholar
[53] Rudin, L. I., Osher, S., and Fatemi, E., 1992: Nonlinear total variation based noise removal algorithms, Physica D (Nonlinear Phenomena), 60(1-4), 259–268.Google Scholar
[54] de Silanes, M. C. L., 1997: Convergence and error estimates for (m,l,s)-splines, J. Comput. Appl. Math., 87(2), 373–384.Google Scholar
[55] Sochacki, J., Kubichek, R., George, J., Fletcher, W. R., and Smithson, S., 1987: Absorbing boundary conditions and surface waves, Geophysics, 52(1), 60–71.Google Scholar
[56] Tadmor, E., 1989: Convergence of spectral methods for nonlinear conservation laws, SIAM J. Num. Anal., 26(1), 30–44.Google Scholar
[57] Terzopoulos, D., 1986: Regularization of inverse visual problems involving discontinuities, IEEE T. Pattern Anal. Mach. Intell., 8(4), 413–424.Google Scholar
[58] Trangenstein, J. A., 2007: Numerical Solution of Hyperbolic Partial Differential Equations, Cambridge University Press, New York.Google Scholar
[59] Walden, J., 1999: Filter bank methods for hyperbolic PDEs, SIAM J. Num. Anal., 36(4), 1183–1233.Google Scholar
[60] Wang, Y., Jia, X., Cheng, J., 2002: A numerical differentiation method and its application to reconstruction of discontinuity, Inverse Probl., 18, 1461–1476.CrossRefGoogle Scholar
[61] Wang, Z., and Wen, R., 2010: Numerical differentiation for high orders by an integration method, J. Comput. Appl. Math., 234(3), 941–948.Google Scholar
[62] Wei, G. W., and Gu, Y., 2002: Conjugate filter approach for solving Burgers equation, J. Comput. Appl. Math., 149, 439–456.Google Scholar
[63] Wei, T., Hon, Y. C., Wang, Y., 2005: Reconstruction of numerical derivatives from scattered noisy data, Inverse Probl., 21, 657–672.CrossRefGoogle Scholar
[64] Yomogida, K., and Benites, R., 2002: Scattering of seismic waves by cracks with the boundary integral method, Pure Appl. Geophys., 159, 1771–1789.Google Scholar
[65] Youn, S. K., and Park, S. H., 1995: A new direct higher-order Taylor-Galerkin finite element method, Comput. & Structures, 56(4), 651–656.CrossRefGoogle Scholar
[66] Yousefi, H., Noorzad, A., and Farjoodi, J., 2010: Simulating 2D waves propagation in elastic solid media using wavelet based adaptive method, J. Sci. Comput., 42(3), 404–425.Google Scholar
[67] Yousefi, H., Noorzad, A., and Farjoodi, J., 2013: Multiresolution based adaptive schemes for second order hyperbolic PDEs in elastodynamic problems, Appl. Math. Modell., 37(12-13), 7095–7127.CrossRefGoogle Scholar
[68] Zienkiewicz, O. C., and Taylor, R. L., 2000: The Finite Element Method, Volume 1, The Basis, John Wiley & Sons Canada, Ltd., 5th edition.Google Scholar
[69] Zingg, D. W., Lomax, H., and Jurgens, H., 1996: High-accuracy finite-difference schemes for linear wave propagation, SIAM J. Sci. Comput., 17(2), 328–346.Google Scholar
[70] Zingg, D. W., 2000; Comparison of high-accuracy finite-difference methods for linear wave propagation, SIAM J. Sci. Comput., 22(2), 476–502.Google Scholar