Application and analysis of an adaptive wave-based technique based on a boundary error indicator for the sound radiation simulation of a combustion engine model

• Autorzy: Mócsai T. , Diwoky F. , Hepberger A. , Priebsch H.-H. , Augusztinovicz F.
• Języki publikacji: EN

Abstrakty

EN

In recent years, Trefftz methods have received increasing attention, as being alternatives of the already well-established element-based simulation methods (e.g., finite element and boundary element methods). The wave-based technique is based on the indirect Trefftz approach for the solution of steady-state, timeharmonic acoustic problems. The dynamic field variables are expanded in terms of wave functions, which satisfy the governing partial differential equation, but do not necessarily satisfy the imposed boundary conditions. Therefore, the approximation error of the method is exclusively caused by the error on the boundary, since there is no additional error present in the domain. The authors investigate the potentials of a novel boundary error indicator-controlled adaptive local refinement strategy. Practical, industrial-oriented application of the method is presented on the 3D free-field sound radiation model of a simplified combustion engine. Results and efficiency of the approach are compared to a priori, frequency-dependent global refinement strategies.

Słowa kluczowe

EN

Trefftz method; adaptivity; error indicator; boundary error; Wave Based Technique; engine sound; numerical acoustics

PL

metoda Trefftza; adaptacyjność; wskaźnik błędów; błąd graniczny; dźwięk silnika; akustyka numeryczna

• Wydawca: Instytut Podstawowych Problemów Techniki PAN
• Czasopismo: Computer Assisted Methods in Engineering and Science
• Rocznik: 2015
• Tom: Vol. 22, no. 1
• Strony: 3--30
• Opis fizyczny: Bibliogr. 79 poz., rys., tab., wykr.

Twórcy

autor

Mócsai T.

• Budapest University of Technology and Economics Laboratory of Vibroacoustics Department of Telecommunications H-1117, Magyar Tudosok korutja 3., Budapest, Hungary

autor

Diwoky F.

• AVL List GmbH Hans-List Platz 1, A-8020, Graz, Austria

autor

Hepberger A.

• AVL List GmbH Hans-List Platz 1, A-8020, Graz, Austria

autor

Priebsch H.-H.

• Virtual Vehicle Competence Center Inffeldgasse 21, A-8010, Graz, Austria

autor

Augusztinovicz F.

• Budapest University of Technology and Economics Laboratory of Vibroacoustics Department of Telecommunications H-1117, Magyar Tudosok korutja 3., Budapest, Hungary

Bibliografia

• [1] O.C. Zienkiewicz. The finite element method. McGraw-Hill, London, New York, 3rd expanded and revised edition, 1977.
• [2] F. Ihlenburg. Finite element analysis of acoustic scattering. Springer, Berlin, 1998.
• [3] S. Kirkup. The boundary element method (BEM) in acoustics. Integrated Sound Software, 1998.
• [4] T.W. Wu. Boundary element acoustics: fundamentals and computer codes. WIT Press, 2000.
• [5] I. Harari, T.J.R. Hughes. Finite element methods for the Helmholtz equation in an exterior domain: model problems. Computer Methods in Applied Mechanics and Engineering, 87(1): 59–96, 1991.
• [6] F. Ihlenburg, I. Babuska. Finite element solution of the Helmholtz equation with high wave number, part I: the h-version of the FEM. Computers and Mathematics with Applications, 30(9): 9–37, 1995.
• [7] I.G. Graham, M. Löhndorf, J.M. Melenk, E.A. Spence. When is the error in the h-BEM for solving the Helmholtz equation bounded independently of k?. BIT Numerical Mathematics, 55(1): 171–214, 2015.
• [8] S. Marburg, B. Nolte. Computational acoustics of noise propagation in fluids – finite and boundary element methods. Springer-Verlag, Berlin, Heidelberg, 2008.
• [9] I.M. Babuska, S.A. Sauter. Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers? SIAM Review, 42(3): 451–484, 2000.
• [10] F. Ihlenburg. The medium-frequency range in computational acoustics: practical and numerical aspects. Journal of Computational Acoustics, 11(2): 175–193, 2003.
• [11] J.T. Oden, S. Prudhomme, L. Demkowicz. A posteriori error estimation for acoustic wave propagation problems. Archives of Computational Methods in Engineering, 12(4): 343–389, 2005.
• [12] M. Ainsworth, J.T. Oden. A posteriori error estimation in finite element analysis. Pure and Applied Mathematics Series. John Wiley, 2000.
• [13] J.R. Stewart, T.J.R. Hughes. An a posteriori error estimator and hp-adaptive strategy for finite element discretizations of the Helmholtz equation in exterior domains. Finite Elements in Analysis and Design, Adaptive Meshing, Part 1, 25(1–2): 1–26, 1997.
• [14] J.R. Stewart, T.J.R. Hughes. A posteriori error estimation and adaptive finite element computation of the Helmholtz equation in exterior domains. Finite Elements in Analysis and Design, Special Issue: Robert J. Melosh Medal Competition, 22(1): 15–24, 1996.
• [15] J.R. Stewart, T.J.R. Hughes. Explicit residual-based a posteriori error estimation for finite element discretizations of the Helmholtz equation: computation of the constant and new measures of error estimator quality. Computer Methods in Applied Mechanics and Engineering, 131(3–4): 335–363, 1996.
• [16] S. Irimie, Ph. Bouillard. A residual a posteriori error estimator for the finite element solution of the Helmholtz equation. Computer Methods in Applied Mechanics and Engineering, 190(31): 4027–4042, 2001.
• [17] F. Ihlenburg, T. Strouboulis, S.K. Gangaraj, I. Babuska. A posteriori error estimation for finite element solutions of Helmholtz’ equation, part I: the quality of local indicators and estimators. International Journal for Numerical Methods in Engineering, 40(18): 3443–3462, 1997.
• [18] T. Strouboulis, I. Babuska, S.K. Gangaraj. Guaranteed computable bounds for the exact error in the finite element solution, part I: one-dimensional model problem. Computer Methods in Applied Mechanics and Engineering, 176(1–4): 51–79, 1999.
• [19] T. Strouboulis, I. Babuska, S.K. Gangaraj. Guaranteed computable bounds for the exact error in the finite element solution, part II: bounds for the energy norm of the error in two dimensions. International Journal for Numerical Methods in Engineering, 47(1–3): 427–475, 2000.
• [20] F. Ihlenburg, Ph. Bouillard. Error estimation and adaptivity for the finite element method in acoustics: 2D and 3D applications. Computer Methods in Applied Mechanics and Engineering, 176(1–4): 147–163, 1999.
• [21] F. Ihlenburg, T. Strouboulis, S.K. Gangaraj, I. Babuska. A posteriori error estimation for finite element solutions of Helmholtz equation – part II: estimation of the pollution error. International Journal for Numerical Methods in Engineering, 40(21): 3883–3900, 1997.
• [22] S. Prudhomme, J.T. Oden. On goal-oriented error estimation for elliptic problems: application to the control of pointwise errors. Computer Methods in Applied Mechanics and Engineering, 176(1–4): 313–331, 1999.
• [23] J. Sarrate, J. Peraire, A. Patera. A posteriori finite element error bounds for non-linear outputs of the Helmholtz equation. International Journal for Numerical Methods in Fluids, 31(1): 17–36, 1999.
• [24] L. Banjai, S. Sauter. A refined Galerkin error and stability analysis for highly indefinite variational problems. SIAM J. Numer. Anal., 45(1): 37–53, January 2007.
• [25] A. Karafiat. Adaptive boundary element method for acoustic scattering problems. Journal of Theoretical and Applied Mechanics, 36(2), 423–436, 1998.
• [26] L. Demkowicz, J.T. Oden. Recent progress on application of hp-adaptive BE/FE methods to elastic scattering. International Journal for Numerical Methods in Engineering, 37(17): 2893–2910, 1994.
• [27] E. Kita, N. Kamiya. Error estimation and adaptive mesh refinement in boundary element method. an overview. Engineering Analysis with Boundary Elements, 25(7): 479–495, 2001.
• [28] J.M. Melenk, I. Babuška. The partition of unity finite element method: basic theory and applications. TICAM report. Texas Institute for Computational and Applied Mathematics, University of Texas at Austin, 1996.
• [29] P. Massimi, R. Tezaur, Ch. Farhat. A discontinuous enrichment method for three-dimensional multiscale harmonic wave propagation problems in multi-fluid and fluid-solid media. International Journal for Numerical Methods in Engineering, 76(3): 400–425, 2008.
• [30] I. Harari. A survey of finite element methods for time-harmonic acoustics. Computer Methods in Applied Mechanics and Engineering, A Tribute to Thomas J.R. Hughes on the Occasion of his 60th Birthday, 195(13–16): 1594–1607, 2006.
• [31] M. Fischer, U. Gauger, L. Gaul. A multipole Galerkin boundary element method for acoustics. Engineering Analysis with Boundary Elements, 28(2): 155–162, 2004.
• [32] D.W. Herrin, F. Martinus, T.W. Wu, A.F. Seybert. An assessment of the high frequency boundary element and Rayleigh integral approximations. Applied Acoustics, 67(8): 819–833, 2006.
• [33] N.A Gumerov, R. Duraiswami. Fast multipole methods for the Helmholtz equation in three dimensions. Elsevier Science, 2005.
• [34] E. Trefftz. Ein Gegenstück zum ritzschen Verfahren, A counterpart to Ritz’s method. Proceedings of the 2nd International Congress on Applied Mechanics, pp. 131–137, 1926.
• [35] E. Kita, N. Kamiya. Trefftz method: an overview. Advances in Engineering Software, 24(1–3): 3–12, 1995.
• [36] B. Pluymers, B. van Hal, D. Vandepitte, W. Desmet. Trefftz-based methods for time-harmonic acoustics. Archives of Computational Methods in Engineering, 14: 343–381, 2007.
• [37] E. Deckers, O. Atak, L. Coox, R. D’Amico, H. Devriendt, S. Jonckheere, K. Koo, B. Pluymers, D. Vandepitte, W. Desmet. The wave based method: an overview of 15 years of research. Wave Motion, 51(4): 550–565, 2014.
• [38] W. Desmet. A wave based prediction technique for coupled vibro-acoustic analysis. PhD thesis, Katholieke Universiteit Leuven, Leuven, 1998.
• [39] C. Vanmaele, D. Vandepitte, W. Desmet. An efficient wave based prediction technique for plate bending vibrations. Computer Methods in Applied Mechanics and Engineering, 196(33–34): 3178–3189, 2007.
• [40] C. Vanmaele, D. Vandepitte, W. Desmet. An efficient wave based prediction technique for dynamic plate bending problems with corner stress singularities. Computer Methods in Applied Mechanics and Engineering, 198(30–32): 2227–2245, 2009.
• [41] B. Pluymers, W. Desmet, D. Vandepitte, P. Sas. Application of an efficient wave-based prediction technique for the analysis of vibro-acoustic radiation problems. Journal of Computational and Applied Mathematics, Selected Papers from the Second International Conference on Advanced Computational Methods in Engineering (ACOMEN 2002), 168(1–2): 353–364, 2004.
• [42] B. Van Genechten, K. Vergote, D. Vandepitte, W. Desmet. A multi-level wave based numerical modelling framework for the steady-state dynamic analysis of bounded Helmholtz problems with multiple inclusions. Computer Methods in Applied Mechanics and Engineering, 199(29–32): 1881–1905, 2010.
• [43] B. Van Genechten, B. Bergen, D. Vandepitte, W. Desmet. A Trefftz-based numerical modelling framework for Helmholtz problems with complex multiple-scatterer configurations. Journal of Computational Physics, 229(18): 6623–6643, 2010.
• [44] W. Desmet, B. van Hal, P. Sas, D. Vandepitte. A computationally efficient prediction technique for the steadystate dynamic analysis of coupled vibro-acoustic systems. Advances in Engineering Software, 33(7–10): 527–540, 2002.
• [45] B. van Hal, W. Desmet, D. Vandepitte, P. Sas. Coupled finite element-wave-based approach in steady-state structural acoustics. Computational Fluid and Solid Mechanics 2003, K.J. Bathe [Ed.], pp. 1552–1555, Elsevier Science Ltd, Oxford, 2003.
• [46] B. van Hal, W. Desmet, D. Vandepitte. Hybrid finite element – wave-based method for steady-state interior structural-acoustic problems. Computers and Structures, Advances in Analysis of Fluid Structure Interaction, 83(2–3): 167–180, 2005.
• [47] B. Pluymers, A. Hepberger, W. Desmet, H.H. Prebsch, D. Vandepitte, P. Sas. Experimental validation of the wave-based prediction technique for the analysis of the coupled vibro-acoustic behavior of a 3D cavity. Computational Fluid and Solid Mechanics 2003, K.J. Bathe [Ed.], pp. 1483–1487, Elsevier Science Ltd, Oxford, 2003.
• [48] B. Van Genechten, D. Vandepitte, W. Desmet. A direct hybrid finite element – wave-based modelling technique for efficient coupled vibro-acoustic analysis. Computer Methods in Applied Mechanics and Engineering, 200(5–8): 742–764, 2011.
• [49] F. Diwoky, A. Hepberger, T. Mocsai, H. Pramberger, H.-H. Priebsch. Application of the wave based technique to 3D radiation problems. Proc. of the LSAME 2008 Trefftz Symposium, Leuven, Belgium, 2008.
• [50] T. Mocsai, A. Hepberger, F. Diwoky, H.-H. Priebsch. Investigations on potential improvements of the wave based technique for the application to radiation problems under anechoic conditions. Proc. of ISMA 2008 – International Conference on Noise and Vibration Engineering, Leuven, Belgium, 2008.
• [51] T. Mocsai, A. Hepberger, F. Diwoky, H.-H. Priebsch. Engine radiation simulation up to 3 kHz using the wave based technique. Proc. of the ICSV 2009 – The 16th International Congress on Sound and Vibration, Krakow, Poland, 2009.
• [52] T. Mocsai, F. Diwoky, H.-H. Priebsch. The application of the wave based technique for 3D free-field radiation calculation using a residual error controlled adaptive strategy. In Proc. of ISMA 2010 – International Conference on Noise and Vibration Engineering, Leuven, Belgium, 2010.
• [53] F. Augusztinovicz, T. Mócsai. Application and analysis of a boundary error indicator based adaptive wave based technique. AIA-DAGA 2013 Conference on Acoustics, 2013.
• [54] J. Jegorovs. On the convergence of the WBM solution in certain non-convex domains. Proc. of the International Conference on Noise and Vibration Engineering ISMA 2006, 2006.
• [55] B. Pluymers. Wave based modelling methods for steady-state vibro-acoustics. PhD thesis, Katholieke Universiteit Leuven, Leuven, 2006.
• [56] R. Courant, D. Hilbert. Methods of mathematical physics, Vol. 1. Wiley, 2008.
• [57] P. Gamallo, R.J. Astley. A comparison of two Trefftz-type methods: the ultraweak variational formulation and the least-squares method, for solving shortwave 2-D Helmholtz problems. International Journal for Numerical Methods in Engineering, 71(4): 406–432, 2007.
• [58] M.H. Protter, H.F. Weinberger. Maximum principles in differential equations. Englewood Cliffs, N.J., PrenticeHall, 1967.
• [59] R. Schaback. A posteriori error bounds for meshless methods. Proc. of ECCOMAS 2007 (European Congress on Computational Methods in Applied Sciences and Engineering), 2007.
• [60] V.G. Sigillito. Explicit a priori inequalities with applications to boundary value problems. Pitman Publishing, 1977.
• [61] G. Still. Computable bounds for eigenvalues and eigenfunctions of elliptic differential operators. Numerische Mathematik, 54: 201–223, 1989.
• [62] J.R. Kuttler, V.G. Sigillito. Bounding eigenvalues of elliptic operators. SIAM Journal on Mathematical Analysis, 9(4): 768–773, 1978.
• [63] A.H. Barnett, T. Betcke. Stability and convergence of the method of fundamental solutions for Helmholtz problems on analytic domains. Journal of Computational Physics, 227(14): 7003–7026, 2008.
• [64] A.H. Barnett. Perturbative analysis of the method of particular solutions for improved inclusion of high-lying Dirichlet eigenvalues. SIAM Journal on Numerical Analysis, 47(3): 1952–1970, 2009.
• [65] A.H. Barnett, A. Hassell. Boundary quasi-orthogonality and sharp inclusion bounds for large Dirichlet eigenvalues, 2010. arXiv:1006.3592 [math.AP]
• [66] Z.C. Lu. Trefftz and collocation methods. WIT Press, 2008.
• [67] Z.C. Li. The Trefftz method for the Helmholtz equation with degeneracy. Applied Numerical Mathematics, 58: 131–159, February 2008.
• [68] J.B. Fahnline. The generalized inverse source method for the computation of acoustic fields. PhD thesis, Pennsylvania State University, 1993.
• [69] G.H. Koopmann, L. Song, J.B. Fahnline. A method for computing acoustic fields based on the principle of wave superposition. J. Acoust. Soc. Am., 86(6): 2433–2438, 1989.
• [70] L. Song, G.H. Koopmann, J.B. Fahnline. Numerical errors associated with the method of superposition for computing acoustic fields. J. Acoust. Soc. Am., 89(6): 2625–2633, 1991.
• [71] J.A. Kolodziej, R. Starosta. Self-adaptive Trefftz procedure for harmonic problems. First International Workshop on the Trefftz Method, Computer Assisted Mechanics and Engineering Sciences, 4: 491–500, 1997.
• [72] B. Van Hal. Automation and performance optimization of the wave based method for interior structural-acoustic problems. PhD thesis, Katholieke Universiteit Leuven, Leuven, 2004.
• [73] L.F. Shampine. Vectorized adaptive quadrature in MATLAB. Journal of Computational and Applied Mathematics, 211(2): 131–140, 2008.
• [74] G. Offner. Modelling of condensed flexible bodies considering non-linear inertia effects resulting from gross motions. Journal of Multi-body Dynamics. Proceedings of the Institution of Mechanical Engineers, Part K, 2009.
• [75] H.H. Priebsch, G. Offner, H.M. Herbst. A generic simulation model for cylinder kit vibro-acoustics, part II: piston slap and engine structure interaction. Proceedings of ASME ICE Spring Technical Conference, 2003.
• [76] B. Loibnegger, H.H. Priebsch, I. McLuckie, M.T. Ma, G. Offner. A fast approach to model hydrodynamic behaviour of journal bearings for analysis of crankshaft and engine dynamics. 30th Leeds-Lyon Symposium on Tribology, 2003.
• [77] D. Colton, R. Kress. Inverse acoustic and electromagnetic scattering theory (applied mathematical sciences). Springer, 2nd edition, January 1998.
• [78] R. Coifman, V. Rokhlin, S. Wandzura. The fast multipole method for the wave equation: a pedestrian prescription. IEEE Antennas and Propagation Magazine, 35(3): 7–12, 1993.
• [79] E. Darve. The fast multipole method, part I: error analysis and asymptotic complexity. SIAM Journal on Numerical Analysis, 38(1): 98–128, 2001.