×

zbMATH — the first resource for mathematics

A coupled finite and boundary spectral element method for linear water-wave propagation problems. (English) Zbl 07163389
Summary: A coupled boundary spectral element method (BSEM) and spectral element method (SEM) formulation for the propagation of small-amplitude water waves over variable bathymetries is presented in this work. The wave model is based on the mild-slope equation (MSE), which provides a good approximation of the propagation of water waves over irregular bottom surfaces with slopes up to \(1:3\). In unbounded domains or infinite regions, space can be divided into two different areas: a central region of interest, where an irregular bathymetry is included, and an exterior infinite region with straight and parallel bathymetric lines. The SEM allows us to model the central region, where any variation of the bathymetry can be considered, while the exterior infinite region is modelled by the BSEM which, combined with the fundamental solution presented by A. Cerrato et al. [Eng. Anal. Bound. Elem. 62, 22–34 (2016; Zbl 1403.76063)] can include bathymetries with straight and parallel contour lines. This coupled model combines important advantages of both methods; it benefits from the flexibility of the SEM for the interior region and, at the same time, includes the fulfilment of the Sommerfeld’s radiation condition for the exterior problem, that is provided by the BSEM. The solution approximation inside the elements is constructed by high order Legendre polynomials associated with Legendre-Gauss-Lobatto quadrature points, providing a spectral convergence for both methods. The proposed formulation has been validated in three different benchmark cases with different shapes of the bottom surface. The solutions exhibit the typical \(p\)-convergence of spectral methods.

MSC:
76 Fluid mechanics
65 Numerical analysis
Software:
RIDE
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Patera, A. T., A spectral element method for fluid dynamics: laminar flow in a channel expansion, J. Comput. Phys., 54, 468-488 (1984) · Zbl 0535.76035
[2] Eskilsson, C.; Sherwin, S. J., Discontinuous Galerkin spectral/hp element modelling of dispersive shallow water systems, J. Sci. Comput., 22, 1, 269-288 (2005) · Zbl 1067.76057
[3] Eskilsson, C.; Sherwin, S. J., Spectral/HP discontinuous Galerkin methods for modelling 2D Boussinesq equations, J. Comput. Phys., 212, 2, 566-589 (2006) · Zbl 1084.76058
[4] Blaise, S.; St-Cyr, A., A dynamic hp-adaptive discontinuous Galerkin method for shallow-water flows on the sphere with application to a global tsunami simulation, Mon. Weather Rev., 140, 3, 978-996 (2012)
[5] Engsig-Karup, A.; Eskilsson, C.; Bigoni, D., A stabilised nodal spectral element method for fully nonlinear water waves, J. Comput. Phys., 318, 1-21 (2016) · Zbl 1349.76570
[6] Berkhoff, W., Computation of combined refraction-diffraction, Proceedings of 13th International Conference on Coastal Engineering (1972), ASCE
[7] Berkhoff, W., Mathematical Models for Simple Harmonic Linear Water Waves. Wave Diffraction and Refraction (1976), Delft Hydraulics Laboratory, Ph.D. thesis
[8] Tsay, T.-K.; Liu, P. L.-F., A finite element model for wave refraction and diffraction, Appl. Ocean Res., 5, 1, 30-37 (1983)
[9] Booij, N., A note on the accuracy of the mild-slope equation, Coastal Eng., 7, 3, 191-203 (1983)
[10] Porter, D.; Staziker, D., Extensions of the mild-slope equation, J. Fluid Mech., 300, 367-382 (1995) · Zbl 0848.76010
[11] Massel, S. R., Extended refraction-diffraction equation for surface waves, Coastal Eng., 19, 97-126 (1993)
[12] Maa, J.-Y.; Hsu, T.-W.; Lee, D.-Y., The RIDE model: an enhanced computer program for wave transformation, Ocean Eng., 29, 11, 1441-1458 (2002)
[13] Chamberlain, P.; Porter, D., The modified mild-slope equation, J. Fluid Mech., 291, 393-407 (1995) · Zbl 0843.76006
[14] Suh, K. D.; Lee, C.; Park, W. S., Time-dependent equations for wave propagation on rapidly varying topography, Coastal Eng., 32, 91-117 (1997)
[15] Chandrasekera, C. N.; Cheung, K. F., Extended linear refraction-diffraction model, J. Waterway Port Coastal Ocean Eng., 123, 5, 280-286 (1997)
[16] Lee, C.; Park, W. S.; Cho, Y.-S.; Suh, K. D., Hyperbolic mild-slope equations extended to account for rapidly varying topography, Coastal Eng., 34, 243-257 (1998)
[17] Li, B., An evolution equation for water waves, Coastal Eng., 23, 3, 227-242 (1994)
[18] Hsu, T.-W.; Wen, C.-C., A study of using parabolic model to describe wave breaking and wide-angle wave incidence, J. Chin. Inst. Eng., 23, 4, 515-527 (2000)
[19] Hsu, T.-W.; Wen, C.-C., A parabolic equation extended to account for rapidly varying topography, Ocean Eng., 28, 11, 1479-1498 (2001)
[20] Panchang, V. G.; Pearce, B. R.; Wei, G.; Cushman-Roisin, B., Solution of the mild-slope wave problem by iteration, Appl. Ocean Res., 13, 4, 187-199 (1991)
[21] Li, B.; Anastasiou, K., Efficient elliptic solvers for the mild-slope equation using the multigrid technique, Coastal Eng., 16, 3, 245-266 (1992)
[22] Mehdizadeh, O. Z.; Paraschivoiu, M., Investigation of a two-dimensional spectral element method for Helmholtz’s equation, J. Comput. Phys., 189, 111-129 (2003) · Zbl 1024.65112
[23] Bayliss, A.; Goldstein, C.; Turkel, E., On accuracy conditions for the numerical computation of waves, J. Comput. Phys., 59, 3, 396-404 (1985) · Zbl 0647.65072
[24] Ihlenburg, F.; Babus̆ka, I., Finite element solution of the Helmholtz equation with high wave number part I: the h-version of the FEM, Comput. Math. Appl., 30, 9, 9-37 (1995) · Zbl 0838.65108
[25] Ihlenburg, F.; Babus̆ka, I., Finite element solution of the Helmholtz equation with high wave number part II: the h-p version of the FEM, SIAM J. Numer. Anal., 34, 1, 315-358 (1997) · Zbl 0884.65104
[26] Deraemaeker, A.; Babus̆ka, I.; Bouillard, P., Dispersion and pollution of the FEM solution for the Helmholtz equation in one, two and three dimensions, Int. J. Numer. Methods Eng., 46, 4, 471-499 (1999) · Zbl 0957.65098
[27] Thompson, L. L.; Pinsky, P. M., A Galerkin least-squares finite element method for the two-dimensional Helmholtz equation, Int. J. Numer. Methods Eng., 38, 3, 371-397 (1995) · Zbl 0844.76060
[28] Liu, G. R.; Dai, K. Y.; Nguyen, T. T., A smoothed finite element method for mechanics problems, Comput. Mech., 39, 6, 859-877 (2006) · Zbl 1169.74047
[29] Bettess, P.; Zienkiewicz, O. C., Diffraction and refraction of surface waves using finite and infinite elements, Int. J. Numer. Methods Eng., 11, 8, 1271-1290 (1977) · Zbl 0367.76014
[30] Lau, S.; Ji, Z., Efficient 3-D infinite element for water wave diffraction problems, Int. J. Numer. Methods Eng., 28, 6, 1371-1387 (1989) · Zbl 0687.76009
[31] Givoli, D.; Keller, J. B., Non-reflecting boundary conditions for elastic waves, Wave Motion, 12, 3, 261-279 (1990) · Zbl 0708.73012
[32] Keller, J. B.; Givoli, D., Exact non-reflecting boundary conditions, J. Comput. Phys., 82, 1, 172-192 (1989) · Zbl 0671.65094
[33] Givoli, D., Non-reflecting boundary conditions, J. Comput. Phys., 94, 1, 1-29 (1991) · Zbl 0731.65109
[34] Bonet, R. P., Refraction and diffraction of water waves using finite elements with a DNL boundary condition, Ocean Eng., 63, 77-89 (2013)
[35] Giorgiani, G.; Fernández-Méndez, S.; Huerta, A., Hybridizable discontinuous Galerkin p-adaptivity for wave propagation problems, Int. J. Numer. Methods Fluids, 73, 10, 883-903 (2013)
[36] Modesto, D.; Zlotnik, S.; Huerta, A., Proper generalized decomposition for parameterized Helmholtz problems in heterogeneous and unbounded domains: application to harbor agitation, Comput. Methods Appl. Mech. Eng., 295, 127-149 (2015) · Zbl 1423.76261
[37] Beltrami, G.; Bellotti, G.; Girolamo, P.; Sammarco, P., Treatment of wave breaking and total absorption in a mild-slope equation FEM model, J. Waterway Port Coastal Ocean Eng., 127, 5, 263-271 (2001)
[38] Steward, D.; Panchang, V., Improved coastal boundary condition for surface water waves, Ocean Eng., 28, 1, 139-157 (2001)
[39] Chen, W., Finite Element Modeling of Wave Transformation in Harbors and Coastal Regions with Complex Bathymetry and Ambient Currents (August 2002), Department of Civil Engineering, University of Maine, Ph.D. thesis
[40] Liu, S.-X.; Sun, B.; Sun, Z.-B.; Li, J.-X., Self-adaptive FEM numerical modeling of the mild-slope equation, Appl. Math. Model., 32, 12, 2775-2791 (2008) · Zbl 1167.86300
[41] Hauguel, A., A combined FE-BIE method for water waves, 16th International Conference on Coastal Engineering, 715-721 (1978)
[42] Shaw, R.; Falby, W., FEBIE—a combined finite element-boundary integral equation method, Comput. Fluids, 6, 3, 153-160 (1978) · Zbl 0385.76027
[43] Hamanaka, K.i., Open, partial reflection and incident-absorbing boundary conditions in wave analysis with a boundary integral method, Coastal Eng., 30, 3-4, 281-298 (1997)
[44] Isaacson, M.; Qu, S., Waves in a harbour with partially reflecting boundaries, Coastal Eng., 14, 3, 193-214 (1990)
[45] Lee, H. S.; Williams, A., Boundary element modeling of multidirectional random waves in a harbor with partially reflecting boundaries, Ocean Eng., 29, 1, 39-58 (2002)
[46] Lee, H. S.; Kim, S. D.; Wang, K.-H.; Eom, S., Boundary element modeling of multidirectional random waves in a harbor with a rectangular navigation channel, Ocean Eng., 36, 17-18, 1287-1294 (2009)
[47] Zhu, S., A new DRBEM model for wave refraction and diffraction, Eng. Anal. Boundary Elem., 12, 4, 261-274 (1993)
[48] Liu, H., A modified GDRBEM model for wave scattering, International Conference on Estuaries and Coasts, 749-755 (2003)
[49] Zhu, S.-P.; Liu, H.-W.; Chen, K., A general DRBEM model for wave refraction and diffraction, Eng. Anal. Boundary Elem., 24, 5, 377-390 (2000) · Zbl 0980.76056
[50] Zhu, S.-P.; Liu, H.-W.; Marchant, T. R., A perturbation DRBEM model for weakly nonlinear wave run-ups around islands, Eng. Anal. Boundary Elem., 33, 1, 63-76 (2009) · Zbl 1188.76233
[51] Hsiao, S.-S.; Chang, C.-M.; Wen, C.-C., Solution for wave propagation through a circular cylinder mounted on different topography ripple-bed profile shoals using DRBEM, Eng. Anal. Boundary Elem., 33, 11, 1246-1257 (2009) · Zbl 1244.76045
[52] Naserizadeh, R.; Bingham, H. B.; Noorzad, A., A coupled boundary element-finite difference solution of the elliptic modified mild slope equation, Eng. Anal. Boundary Elem., 35, 1, 25-33 (2011) · Zbl 1259.76031
[53] Cerrato, A.; González, J. A.; Rodríguez-Tembleque, L., Boundary element formulation of the mild-slope equation for harmonic water waves propagating over unidirectional variable bathymetries, Eng. Anal. Boundary Elem., 62, 22-34 (2016) · Zbl 1403.76063
[54] Belibassakis, K., The Green’s function of the mild-slope equation: the case of a monotonic bed profile, Wave Motion, 32, 339-361 (2000) · Zbl 1074.76598
[55] Harwood, A. R.; Dupère, I. D., Calculation of acoustic Green’s functions using BEM and Dirichlet-to-Neumann-type boundary conditions, Appl. Math. Model., 39, 14, 4134-4150 (2015) · Zbl 1443.76201
[56] He, Y.; Min, M.; Nicholls, D. P., A spectral element method with transparent boundary condition for periodic layered media scattering, J. Sci. Comput., 68, 2, 772-802 (2016) · Zbl 1373.78442
[57] Kumar, P.; Zhang, H.; Kim, K. I.; Shi, Y.; Yuen, D. A., Wave spectral modeling of multidirectional random waves in a harbor through combination of boundary integral of Helmholtz equation with Chebyshev point discretization, Comput. Fluids, 108, 13-24 (2015) · Zbl 1390.76654
[58] Vos, P. E.; Sherwin, S. J.; Kirby, R. M., From h to p efficiently: implementing finite and spectral/HP element methods to achieve optimal performance for low- and high-order discretisations, J. Comput. Phys., 229, 13, 5161-5181 (2010) · Zbl 1194.65138
[59] Bergmann, P. G., The equation in a medium with a variable index of refraction, J. Acoust. Soc. Am., 17, 4, 329-333 (1946)
[60] Radder, A. C., On the parabolic equation method for water-wave propagation, J. Fluid Mech., 95, 179-186 (1979) · Zbl 0415.76012
[61] Wu, T., Boundary Element Acoustics, Fundamentals and Computer Codes, vol. 3 (2000), WIT Press: WIT Press Boston · Zbl 0987.76500
[62] Aliabadi, M.; Hall, W.; Phemister, T., Taylor expansions for singular kernels in the boundary element method, Int. J. Numer. Meth. Eng., 21, 2221-2236 (1985) · Zbl 0599.65011
[63] Aliabadi, M.; Hall, W., The regularising transformation integration method for boundary element kernels. Comparison with series expansion and weighted Gaussian integration methods, Eng. Anal. Bound. Elem., 6, 66-71 (1989)
[64] Guiggiani, M.; Krishnasamy, G.; Rudolphi, T.; Rizzo, E., General algorithm for the numerical solution of hypersingular boundary integral equations, J. Appl. Mech. Trans. ASME, 59, 604-614 (1992) · Zbl 0765.73072
[65] Maday, Y.; Ronquist, E. M., Optimal error analysis of spectral methods with emphasis on non-constant coefficients and deformed geometries, Comput. Methods Appl. Mech. Eng., 80, 1, 91-115 (1990) · Zbl 0728.65078
[66] Ito, Y.; Tanimoto, K., A method of numerical analysis of wave propagation-application to wave diffraction and refraction, Proceedings of 13th International Conference on Coastal Engineering, ASCE (1972)
[67] Berkhoff, J.; Booy, N.; Radder, A., Verification of numerical wave propagation models for simple harmonic linear water waves, Coastal Eng., 6, 3, 255-279 (1982)
[68] Belibassakis, K.; Athanassoulis, G.; Gerostathis, T., A coupled-mode model for the refraction-diffraction of linear waves over steep three-dimensional bathymetry, Appl. Ocean Res., 23, 6, 319-336 (2001)
[69] Li, B.; Fleming, C. A., A three dimensional multigrid model for fully nonlinear water waves, Coastal Eng., 30, 3, 235-258 (1997)
[70] Rodríguez-Tembleque, L.; González, J. A.; Cerrato, A., Partitioned solution strategies for coupled BEM-FEM acoustic fluid-structure interaction problems, Comput. Struct., 152, 45-58 (2015)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.