Panagiotopoulos, Dionysios; Deckers, Elke; Desmet, Wim Krylov subspaces recycling based model order reduction for acoustic BEM systems and an error estimator. (English) Zbl 1441.76083 Comput. Methods Appl. Mech. Eng. 359, Article ID 112755, 22 p. (2020). Summary: Boundary Element Method frequency sweep analyses in acoustics are usually accompanied by a vast numerical cost of assembling and solving numerous linear systems. In that context, this work proposes a model order reduction technique to mitigate the resulting computational cost of such analyses. First, a series expansion of the Green’s function BEM kernel is leveraged to construct a series of frequency independent matrices. Next, in a model order reduction way, the arising matrices are projected on a reduced basis utilizing a Galerkin projection. By this off-line matrix projection, both the assembly and the solution of the BEM full-size linear systems degenerate into assembling and solving a reduced system for all frequencies. Significant speed-up factors can, thus, be achieved for both operations. The projection basis employed in this model reduction scheme is developed through an Arnoldi algorithm for the BEM systems on a grid of master frequencies. The method is based on Krylov subspaces recycling, as the subspaces produced at master frequencies are recycled to approximate the surface distribution of the acoustic variables on the whole frequency range of interest. Utilizing Krylov subspaces facilitates as well the definition of a robust error estimator that indicates the quality of the reduced system. The performance of the proposed method is assessed for both an exterior and an interior problem for a simple and more complicated geometry, respectively. Cited in 8 Documents MSC: 76M15 Boundary element methods applied to problems in fluid mechanics 65N38 Boundary element methods for boundary value problems involving PDEs 76Q05 Hydro- and aero-acoustics Keywords:model order reduction; BEM; Krylov subspaces recycling; Arnoldi; error estimator Software:Gmsh; OpenBEM PDFBibTeX XMLCite \textit{D. Panagiotopoulos} et al., Comput. Methods Appl. Mech. Eng. 359, Article ID 112755, 22 p. (2020; Zbl 1441.76083) Full Text: DOI Link References: [1] Brebbia, C. A., The Boundary Element Method for Engineers (1980), Pentech press [2] Banerjee, P. K.; Butterfield, R., Boundary Element Methods in Engineering Science, Vol. 17 (1981), McGraw-Hill London · Zbl 0499.73070 [3] Costabel, M., Principles of Boundary Element Methods (1986), Techn. Hochsch., Fachbereich Mathematik [4] Brebbia, C.; Butterfield, R., Formal equivalence of direct and indirect boundary element methods, Appl. Math. Model., 2, 2, 132-134 (1978), URL http://www.sciencedirect.com/science/article/pii/0307904X78900525 · Zbl 0385.35050 [5] Rieckh, G.; Kreuzer, W.; Waubke, H.; Balazs, P., A 2.5 d-fourier-bem model for vibrations in a tunnel running through layered anisotropic soil, Eng. Anal. Bound. Elem., 36, 6, 960-967 (2012) · Zbl 1351.74131 [6] Bonnet, M.; Maier, G.; Polizzotto, C., Symmetric galerkin boundary element methods, Appl. Mech. Rev., 51, 11, 669-704 (1998) [7] Marburg, S.; Nolte, B., Computational Acoustics of Noise Propagation in Fluids: Finite and Boundary Element Methods, Vol. 578 (2008), Springer [8] Bettess, P., Infinite elements, Int. J. Numer. Methods Eng., 11, 1, 53-64 (1977) · Zbl 0362.65093 [9] Qi, Q.; Geers, T. L., Evaluation of the perfectly matched layer for computational acoustics, J. Comput. Phys., 139, 1, 166-183 (1998) · Zbl 0903.76073 [10] Liu, Q.-H.; Tao, J., The perfectly matched layer for acoustic waves in absorptive media, J. Acoust. Soc. Am., 102, 4, 2072-2082 (1997) [11] Harari, I.; Hughes, T. J., A cost comparison of boundary element and finite element methods for problems of time-harmonic acoustics, Comput. Methods Appl. Mech. Engrg., 97, 1, 77-102 (1992) · Zbl 0775.76095 [12] Marburg, S., Six boundary elements per wavelength: is that enough?, J. Comput. Acoust., 10, 01, 25-51 (2002) · Zbl 1360.76168 [13] Fischer, M.; Gauger, U.; Gaul, L., A multipole galerkin boundary element method for acoustics, Eng. Anal. Bound. Elem., 28, 2, 155-162 (2004) · Zbl 1049.76042 [14] Liu, Y.; Nishimura, N., The fast multipole boundary element method for potential problems: a tutorial, Eng. Anal. Bound. Elem., 30, 5, 371-381 (2006) · Zbl 1187.65134 [15] Hackbusch, W.; Khoromskij, B. N., A sparse h-matrix arithmetic, Computing, 64, 1, 21-47 (2000) · Zbl 0962.65029 [16] Kirkup, S. M.; Henwood, D., Methods for speeding up the boundary element solution of acoustic radiation problems, J. Vib. Acoust., 114, 3, 374-380 (1992) [17] Schenck, H.; Benthien, G., The application of a coupled finite-element boundary-element technique to large-scale structural acoustic problems, Adv. Bound. Elem., 2, 309-317 (1989) [18] Benthien, G. W.; Schenck, H., Structural-acoustic coupling, (Boundary Element Methods in Acoustics (1991), Computational Mechanics Publications Southampton: Computational Mechanics Publications Southampton UK), 109-129 [19] Wu, T.; Li, W.; Seybert, A., An efficient boundary element algorithm for multi-frequency acoustical analysis, J. Acoust. Soc. Am., 94, 1, 447-452 (1993) [20] Koopmann, G. H.; Benner, H., Method for computing the sound power of machines based on the helmholtz integral, J. Acoust. Soc. Am., 71, 1, 78-89 (1982) [21] Wang, Z.; Zhao, Z.; Liu, Z.; Huang, Q., A method for multi-frequency calculation of boundary integral equation in acoustics based on series expansion, Appl. Acoust., 70, 3, 459-468 (2009) [22] Zhang, Q.; Mao, Y.; Qi, D.; Gu, Y., An improved series expansion method to accelerate the multi-frequency acoustic radiation prediction, J. Comput. Acoust., 23, 01, 1450015 (2015) · Zbl 1360.76179 [23] Oh, J.-S.; Youn, S.-K., An efficient technique for the indirect bem for multi-frequency acoustic analysis using green’s function approximation, J. Mech. Sci. Technol., 32, 5, 1965-1973 (2018) [24] Lefteriu, S.; Lenzi, M. S.; Beriot, H.; Tournour, M.; Desmet, W., Fast frequency sweep method for indirect boundary element models arising in acoustics, Eng. Anal. Bound. Elem., 69, 32-45 (2016) · Zbl 1403.76087 [25] Slone, R. D.; Lee, R.; Lee, J.-F., Broadband model order reduction of polynomial matrix equations using single-point well-conditioned asymptotic waveform evaluation: derivations and theory, Internat. J. Numer. Methods Engrg., 58, 15, 2325-2342 (2003) · Zbl 1036.78019 [26] Slone, R. D.; Lee, R.; Lee, J.-F., Well-conditioned asymptotic waveform evaluation for finite elements, IEEE Trans. Antennas and Propagation, 51, 9, 2442-2447 (2003) · Zbl 1368.78146 [27] Bai, Z., Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems, Appl. Numer. Math., 43, 1-2, 9-44 (2002) · Zbl 1012.65136 [28] Grimme, E. J., Krylov projection methods for model reduction (1992), URL https://perso.uclouvain.be/paul.vandooren/ThesisGrimme.pdf [29] Antoulas, A. C., Approximation of Large-Scale Dynamical Systems, Vol. 6 (2005), Siam [30] Wolf, J., Modal synthesis for combined structural-acoustic systems, AIAA J., 14, 1, 33-38 (1976) [31] Sung, S. H.; Nefske, D. J., Component mode synthesis of a vehicle structural-acoustic system model, AIAA J., 24, 6, 1021-1026 (1986) [32] Kirkup, S. M.; Amini, S., Solution of the helmholtz eigenvalue problem via the boundary element method, Internat. J. Numer. Methods Engrg., 36, 2, 321-330 (1993) · Zbl 0825.76470 [33] Chatterjee, A., An introduction to the proper orthogonal decomposition, Curr. Sci., 808-817 (2000) [34] Liang, Y.; Lee, H.; Lim, S.; Lin, W.; Lee, K.; Wu, C., Proper orthogonal decomposition and its applications—part i: theory, J. Sound Vib., 252, 3, 527-544 (2002) · Zbl 1237.65040 [35] Liang, Y.; Lin, W.; Lee, H.; Lim, S.; Lee, K.; Sun, H., Proper orthogonal decomposition and its applications-part ii: Model reduction for mems dynamical analysis, J. Sound Vib., 256, 3, 515-532 (2002) [36] Chinesta, F.; Ladevèze, P., Separated representations and pgd-based model reduction, (Fundamentals and Applications, vol. 554 (2014), International Centre for Mechanical Siences, Courses and Lectures, Springer) · Zbl 1300.93001 [37] Saad, Y.; Schultz, M. H., Gmres: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 7, 3, 856-869 (1986) · Zbl 0599.65018 [38] Marburg, S.; Schneider, S., Performance of iterative solvers for acoustic problems. part i. solvers and effect of diagonal preconditioning, Eng. Anal. Bound. Elem., 27, 7, 727-750 (2003) · Zbl 1060.76606 [39] Schneider, S.; Marburg, S., Performance of iterative solvers for acoustic problems. part ii. acceleration by ilu-type preconditioner, Eng. Anal. Bound. Elem., 27, 7, 751-757 (2003) · Zbl 1060.76610 [40] Ryckelynck, D.; Hermanns, L.; Chinesta, F.; Alarcón, E., An efficient ‘a priori’model reduction for boundary element models, Eng. Anal. Bound. Elem., 29, 8, 796-801 (2005) · Zbl 1182.76913 [41] Liang, T.; Wang, J.; Xiao, J.; Wen, L., Coupled be-fe based vibroacoustic modal analysis and frequency sweep using a generalized resolvent sampling method, Comput. Methods Appl. Mech. Engrg., 345, 518-538 (2019) · Zbl 1440.65217 [42] Xiao, J.; Meng, S.; Zhang, C.; Zheng, C., Resolvent sampling based rayleigh-ritz method for large-scale nonlinear eigenvalue problems, Comput. Methods Appl. Mech. Engrg., 310, 33-57 (2016) · Zbl 1439.65141 [43] Arnoldi, W. E., The principle of minimized iterations in the solution of the matrix eigenvalue problem, Q. Appl. Math., 9, 1, 17-29 (1951) · Zbl 0042.12801 [44] Golub, G. H.; Van Loan, C. F., Matrix Computations, Vol. 3 (2012), JHU press [45] Parks, M. L.; De Sturler, E.; Mackey, G.; Johnson, D. D.; Maiti, S., Recycling krylov subspaces for sequences of linear systems, SIAM J. Sci. Comput., 28, 5, 1651-1674 (2006) · Zbl 1123.65022 [46] Keuchel, S.; Biermann, J.; von Estorff, O., A combination of the fast multipole boundary element method and krylov subspace recycling solvers, Eng. Anal. Bound. Elem., 65, 136-146 (2016) · Zbl 1403.65202 [47] Carlberg, K.; Farhat, C., An adaptive pod-krylov reduced-order model for structural optimization, (8th World Congress on Structural and Multidisciplinary Optimization, Lisbon, Portugal, Vol. 1 (2009)), 1-11 [48] Carlberg, K.; Forstall, V.; Tuminaro, R., Krylov-subspace recycling via the pod-augmented conjugate-gradient method, SIAM J. Matrix Anal. Appl., 37, 3, 1304-1336 (2016) · Zbl 1348.15005 [49] Ryckelynck, D., A priori hyperreduction method: an adaptive approach, J. Comput. Phys., 202, 1, 346-366 (2005), URL http://www.sciencedirect.com/science/article/pii/S002199910400289X · Zbl 1288.65178 [50] S. Li, J. Trevelyan, W. Zhang, A model reduction method for boundary element method, in: Proceedings of the 24th UK Conference of the Association for Computational Mechanics in Engineering, 2016. [51] Li, S.; Trevelyan, J.; Wu, Z.; Lian, H.; Wang, D.; Zhang, W., An adaptive svd-krylov reduced order model for surrogate based structural shape optimization through isogeometric boundary element method, Comput. Methods Appl. Mech. Engrg., 349, 312-338 (2019) · Zbl 1441.74187 [52] Binion, D.; Chen, X., A krylov enhanced proper orthogonal decomposition method for efficient nonlinear model reduction, Finite Elem. Anal. Des., 47, 7, 728-738 (2011) [53] Xie, X.; Zheng, H.; Jonckheere, S.; van de Walle, A.; Pluymers, B.; Desmet, W., Adaptive model reduction technique for large-scale dynamical systems with frequency-dependent damping, Comput. Methods Appl. Mech. Engrg., 332, 363-381 (2018) · Zbl 1440.74452 [54] van de Walle, A., The power of model order reduction in vibroacoustics and its applications in model-based sensing (2018), KU Leuven, (Ph.D. thesis) [55] Burton, A.; Miller, G., The application of integral equation methods to the numerical solution of some exterior boundary-value problems, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 323, 1553, 201-210 (1971) · Zbl 0235.65080 [56] Schenck, H. A., Improved integral formulation for acoustic radiation problems, J. Acoust. Soc. Amer., 44, 1, 41-58 (1968) · Zbl 0187.50302 [57] Kirkup, S. M., The Boundary Element Method in Acoustics (2007), Integrated sound software [58] Mason, J. C.; Handscomb, D. C., Chebyshev Polynomials (2002), Chapman and Hall/CRC · Zbl 1015.33001 [59] Bartle, R. G.; Sherbert, D. R., Introduction to real analysis (2011), Wiley: Wiley Hoboken, NJ · Zbl 0810.26001 [60] Ciskowski, R. D.; Brebbia, C. A., Boundary Element Methods in Acoustics (1991), Springer · Zbl 0758.76036 [61] Björck, Å., Numerics of gram-schmidt orthogonalization, Linear Algebra Appl., 197, 297-316 (1994) · Zbl 0801.65039 [62] Henriquez, V. C.; Juhl, P. M., Openbem-an open source boundary element method software in acoustics, (Internoise 2010, vol. 7 (2010), Sociedade Portuguesa de Acustica (SPA) Lisbon, Portugal), 5796-5805 [63] Geuzaine, C.; Remacle, J.-F., Gmsh: A 3-d finite element mesh generator with built-in pre-and post-processing facilities, Internat. J. Numer. Methods Engrg., 79, 11, 1309-1331 (2009) · Zbl 1176.74181 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.