×

A comparison of FE-BE coupling schemes for large-scale problems with fluid-structure interaction. (English) Zbl 1156.74374

Summary: To predict the sound radiation of structures, both a structural problem and an acoustic problem have to be solved. In case of thin structures and dense fluids, a strong coupling scheme between the two problems is essential, since the feedback of the acoustic pressure onto the structure is not negligible. In this paper, the structural part is modeled with the finite element (FE) method. An interface to a commercial FE package is set up to import the structural matrices. The exterior acoustic problem is efficiently modeled with the Galerkin boundary element (BE) method. To overcome the well-known drawback of fully populated system matrices, the fast multipole method is applied. Different coupling formulations are investigated. They are either based on the Burton-Miller approach or use a mortar coupling scheme. For all cases, iterative solvers with different preconditioners are used. The efficiency with respect to their memory consumption and computation time is compared for a simple model problem. At the end of the paper, a more complex structure is simulated.

MSC:

74S05 Finite element methods applied to problems in solid mechanics
74S15 Boundary element methods applied to problems in solid mechanics
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)

Software:

ANSYS
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Fahy, Sound and Structural Vibration: Radiation, Transmission and Response (2007)
[2] Ohayon, Structural Acoustics and Vibration (1998)
[3] Conca, Fluids and Periodic Structures (1995)
[4] Junge M, Fischer M, Maess M, Gaul L. Acoustic simulation of an idealized exhaust system by coupled FEM and fast multipole BEM. Proceedings of the Twelfth International Congress on Sound and Vibration (ICSV12), Lisboa, 2005.
[5] Junger, Acoustic fluid-elastic structure interaction: basic concepts, Computers and Structures 65 pp 287– (1997) · Zbl 0936.74506
[6] Junger, Sound, Structures, and Their Interaction (1986)
[7] Pretlove, Forced vibration of a rectangular panel backed by a closed rectangular cavity, Journal of Sound and Vibration 3 pp 252– (1966)
[8] Zienkiewicz, The Finite Element Method: Its Basis and Fundamentals (2000)
[9] Wang, Displacement/pressure based mixed finite element formulations for acoustic fluid-structure interaction problems, International Journal for Numerical Methods in Engineering 40 pp 2001– (1997) · Zbl 0886.73073
[10] Bathe, A mixed displacement-based finite element formulation for acoustic fluid-structure interaction, Computers and Structures 56 pp 225– (1995) · Zbl 1002.76536
[11] Dreyer, Effectiveness and robustness of improved infinite elements for exterior acoustics, Computer Methods in Applied Mechanics and Engineering 195 pp 3591– (2006) · Zbl 1123.76030
[12] Ihlenburg, Finite Element Analysis of Acoustic Scattering (1998) · Zbl 0908.65091 · doi:10.1007/b98828
[13] Gerdes, A review of infinite element methods for exterior Helmholtz problems, Journal of Computational Acoustics 8 pp 43– (2000) · Zbl 1360.65248
[14] WuTW (ed.). Boundary Element Acoustics: Fundamentals and Computer Codes. WIT Press: Southampton, U.K., 2000.
[15] EstorffO von (ed.). Boundary Elements in Acoustics: Advances and Applications. WIT Press: Southampton, U.K., 2000. · Zbl 0987.76515
[16] Gaul, Boundary Element Methods for Engineers and Scientists (2003) · Zbl 1071.65162 · doi:10.1007/978-3-662-05136-8
[17] Fischer M. The fast multipole boundary element method and its application to structure-acoustic field interaction. Ph.D. Thesis, University of Stuttgart, 2004.
[18] Greengard, A fast algorithm for particle simulations, Journal of Computational Physics 73 pp 325– (1987) · Zbl 0629.65005
[19] Rokhlin, Diagonal forms of translation operators for the Helmholtz equation in three dimensions, Applied and Computational Harmonic Analysis 1 pp 82– (1993) · Zbl 0795.35021
[20] Giebermann K. Schnelle Summationsverfahren zur numerischen Lösung von Integralgleichungen für Streuprobleme im \(\mathbb{R}\)3. Ph.D. Thesis, Universität Karlsruhe, 1997. · Zbl 0930.65130
[21] Nishimura, Fast multipole accelerated boundary integral equation methods, Applied Mechanics Reviews 55 pp 299– (2002)
[22] Schneider S. Efficient usage of the boundary element method for solving the time harmonic Helmholtz equation in three dimensions. Ph.D. Thesis, Technische Universität Dresden, 2003.
[23] Brunner D, Junge M, Gaul L. Strong coupling of the fast multilevel multipole boundary element method with the finite element method for vibro-acoustic problems. Proceedings of the 14th International Congress on Sound and Vibration (ICSV14), Cairns, 2007.
[24] Shen, An adaptive fast multipole boundary element method for three-dimensional acoustic wave problems based on the Burton-Miller formulation, Computational Mechanics 40 pp 461– (2007) · Zbl 1176.76083
[25] Rjasanow, The Fast Solution of Boundary Integral Equations (2007) · Zbl 1119.65119
[26] Bebendorf M. Hierarchical matrices: a means to efficiently solve elliptic boundary value problems. Habilitation Thesis, University of Leipzig, 2007. · Zbl 1151.65090
[27] Saad, Iterative Methods for Sparse Linear Systems (1996) · Zbl 1031.65047
[28] Ochmann, An iterative GMRES-based boundary element solver for acoustic scattering, Engineering Analysis with Boundary Elements 27 pp 714– (2003) · Zbl 1060.76608
[29] Marburg, Performance of iterative solvers for acoustic problems. Part I. Solvers and effect of diagonal preconditioning, Engineering Analysis with Boundary Elements 27 pp 727– (2003) · Zbl 1060.76606
[30] Amini, Coupled Boundary and Finite Element Methods for the Solution of the Dynamic Fluid-Structure Interaction Problem (1992) · doi:10.1007/978-3-642-51727-3
[31] Márquez, A new BEM-FEM coupling strategy for two-dimensional fluid-solid interaction problems, Journal of Computational Physics 199 pp 205– (2004) · Zbl 1127.74328
[32] Chen, A Galerkin-type BE-FE formulation for elasto-acoustic coupling, Computer Methods in Applied Mechanics and Engineering 152 pp 147– (1998)
[33] Moosrainer M. Fluid-Struktur-Kopplung. Ph.D. Thesis, Universität der Bundeswehr München, 2000.
[34] Jeans, Solution of fluid-structure interaction problems using a coupled finite element and variational boundary element technique, Journal of the Acoustical Society of America 88 pp 2459– (1990)
[35] Everstine, Coupled finite element/boundary element approach for fluid-structure interaction, Journal of Sound and Vibration 87 pp 1938– (1990)
[36] Seybert, Application of the boundary element method to acoustic cavity response and muffler analysis, Journal of Vibration and Acoustics 109 pp 15– (1987) · doi:10.1115/1.3269388
[37] Hughes, An efficient preconditioned iterative solver for solving a coupled fluid structure interaction problem, International Journal of Computer Mathematics 81 pp 583– (2004) · Zbl 1093.76051
[38] Langer, Analyses of sound transmission through windows by a coupled finite and boundary element method, Acta Acoustica 89 pp 78– (2003)
[39] Bernardi, Nonlinear Partial Differential Equations and their Applications pp 13– (1994)
[40] Belgacem, The mortar finite element method with Lagrange multipliers, Numerische Mathematik 84 pp 173– (1999) · Zbl 0944.65114
[41] Flemisch, Elasto-acoustic and acoustic-acoustic coupling on nonmatching grids, International Journal for Numerical Methods in Engineering 67 pp 1791– (2006) · Zbl 1127.74042
[42] Gaul, Large-scale simulations of acoustic-structure interaction using the fast multipole BEM, Zeitschrift für Angewandte Mathematik und Mechanik 86 pp 4– (2006) · Zbl 1298.76153
[43] Wagner, The hybrid boundary element method in structural acoustics, Zeitschrift für Angewandte Mathematik und Mechanik 84 (12) pp 780– (2002)
[44] Gaul, A coupled symmetric BE-FE-method for acoustic fluid-structure interaction, Engineering Analysis with Boundary Elements 26 pp 629– (2002) · Zbl 1037.74054
[45] Howe, Acoustics of Fluid-Structure Interactions. Cambridge Monographs on Mechanics (2004) · Zbl 0921.76002
[46] Païdoussis, Acoustics of Fluid-Structure Interactions: Slender Structures and Axial Flow 1 (1998)
[47] Morand, Fluid Structure Interaction: Applied Numerical Methods (1995)
[48] Kergourlay G, Balmès E, Clouteau D. Model reduction for efficient FEM/BEM coupling. Proceedings of ISMA 25, International Conference on Noise and Vibration Engineering, Leuven, Belgium, 2000; 1167-1174.
[49] Maess M. Methods for efficient acoustic-structure simulation of piping systems. Ph.D. Thesis, University of Stuttgart, 2006.
[50] Petyt, Introduction to Finite Element Vibration Analysis (1990) · Zbl 0789.73002 · doi:10.1017/CBO9780511624292
[51] Zienkiewicz, The Finite Element Method for Solid and Structural Mechanics (2005) · Zbl 1084.74001
[52] ANSYS Inc. Documentation Release 11.0, 2007.
[53] Kinsler, Fundamentals of Acoustics (1999)
[54] Morse, Vibration and Sound (1948)
[55] Nedéléc, Integral equations with nonintegrable kernels, Integral Equations Operator Theory 5 pp 562– (1982)
[56] van der Vorst, Iterative Krylov Methods for Large Linear Systems (2003) · doi:10.1017/CBO9780511615115
[57] Coifman, The fast multipole method for the wave equation: a pedestrian description, IEEE Antennas and Propagation Magazine 35 pp 7– (1993)
[58] Gyuru, A prescription for the multilevel Helmholtz FMM, IEEE Computational Science and Engineering 5 pp 39– (1998)
[59] Brakhage, Über das Dirichlet’sche Außenraumproblem für die Helmholtz’sche Schwingungsgleichung, Archiv der Mathematik 16 pp 325– (1965) · Zbl 0132.33601
[60] Schenk, Improved integral formulation for acoustic radiation problems, Journal of the Acoustical Society of America 44 pp 41– (1968) · Zbl 0187.50302
[61] Burton, The application of the integral equation method to the numerical solution of some exterior boundary-value problems, Proceedings of the Royal Society of London A 323 pp 201– (1971) · Zbl 0235.65080
[62] Amini, On the choice of the coupling parameter in boundary integral formulations of the exterior acoustic problem, Applicable Analysis 35 pp 75– (1990) · Zbl 0663.35013
[63] Wilton, Acoustic radiation and scattering from elastic structures, International Journal for Numerical Methods in Engineering 13 pp 123– (1978) · Zbl 0384.76059
[64] Benzi, Preconditioning techniques for large linear systems: a survey, Journal of Computational Physics 182 pp 418– (2002) · Zbl 1015.65018
[65] Satish B, Buschelman K, Gropp WD, Kaushik B, Knepley MG, McInnes LC, Smith BF, Zhang H. PETSc Webpage. http://www.mcs.anl.gov/petsc, 2007.
[66] Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method, ACM Transactions on Mathematical Software 30 pp 165– (2004) · Zbl 1072.65036
[67] Chen, Efficient preconditioners for iterative solution of the boundary element equations for the three-dimensional Helmholtz equation, Applied Numerical Mathematics 36 pp 475– (2001) · Zbl 0979.65107
[68] Chow, Approximate inverse techniques for block-partitioned matrices, SIAM Journal on Scientific Computing 18 pp 1657– (1997) · Zbl 0888.65035
[69] Gaunaurd, RST analysis of monostatic and bistatic acoustic echoes from an elastic sphere, Journal of the Acoustical Society of America 73 pp 1– (1982) · Zbl 0551.73027
[70] Gaunaurd, Nearfield effects in acoustic scattering by submerged rigid bodies and elastic shells, Journal of the Acoustical Society of America 85 pp 2465– (1989)
[71] Jeans, Elastoacoustic analysis of submerged fluid-filled thin shells, International Journal for Numerical Methods in Engineering 37 pp 2911– (1994) · Zbl 0825.73892
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.