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

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, time-harmonic 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.

[1]  S. Kirkup,et al.  The Boundary Element Method in Acoustics: A Development in Fortran , 1998 .

[2]  Wim Desmet,et al.  Coupled finite element-wave-based approach in steady-state structural acoustics , 2003 .

[3]  Wim Desmet,et al.  An efficient wave based prediction technique for plate bending vibrations , 2007 .

[4]  Andrew Hassell,et al.  Boundary Quasi-Orthogonality and Sharp Inclusion Bounds for Large Dirichlet Eigenvalues , 2010, SIAM J. Numer. Anal..

[5]  Bert Pluymers,et al.  Experimental validation of the wave based prediction technique for the analysis of the coupled vibro-acoustic behavior of a 3D cavity , 2003 .

[6]  Martin Ochmann,et al.  Boundary Element Acoustics Fundamentals and Computer Codes , 2002 .

[7]  Lehel Banjai,et al.  A Refined Galerkin Error and Stability Analysis for Highly Indefinite Variational Problems , 2007, SIAM J. Numer. Anal..

[8]  A. Cheng,et al.  Trefftz and Collocation Methods , 2008 .

[9]  Ivo Babuška,et al.  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† , 2000 .

[10]  V. G. Sigillito,et al.  Explicit A Priori Inequalities with Applications to Boundary Value Problems. , 1978 .

[11]  Josep Sarrate,et al.  A POSTERIORI FINITE ELEMENT ERROR BOUNDS FOR NON-LINEAR OUTPUTS OF THE HELMHOLTZ EQUATION , 1999 .

[12]  Zi-Cai Li,et al.  The Trefftz method for the Helmholtz equation with degeneracy , 2008 .

[13]  Wim Desmet,et al.  A multi-level wave based numerical modelling framework for the steady-state dynamic analysis of bounded Helmholtz problems with multiple inclusions , 2010 .

[14]  Isaac Harari,et al.  A survey of finite element methods for time-harmonic acoustics , 2006 .

[15]  G. Still,et al.  Computable bounds for eigenvalues and eigenfunctions of elliptic differential operators , 1988 .

[16]  Bert Pluymers.  Wave Based Modelling Methods for Steady-State Vibro-Acoustics (Golfgebaseerde modelleringsmethoden voor stationaire vibro-akoestiek) , 2006 .

[17]  Leszek Demkowicz,et al.  Recent progress on application of hp-adaptive BE/FE methods to elastic scattering , 1994 .

[19]  Miss A.O. Penney.  (b) , 1974, The New Yale Book of Quotations.

[20]  V. G. Sigillito,et al.  Bounding Eigenvalues of Elliptic Operators , 1978 .

[21]  A. F. Seybert,et al.  An assessment of the high frequency boundary element and Rayleigh integral approximations , 2006 .

[22]  Bas Van Hal.  Automation and performance optimization of the wave based method for interior structural-acoustic problems , 2004 .

[23]  Wim Desmet,et al.  Application of an efficient wave-based prediction technique for the analysis of vibro-acoustic radiation problems , 2004 .

[24]  Ivo Babuška,et al.  A posteriori error estimation for finite element solutions of Helmholtz' equation—Part II: estimation of the pollution error , 1997 .

[25]  M.-T. Ma,et al.  A fast approach to model hydrodynamic behaviour of journal bearings for analysis of crankshaft and engine dynamics , 2003 .

[26]  Wim Desmet,et al.  A direct hybrid finite element – Wave based modelling technique for efficient coupled vibro-acoustic analysis , 2011 .

[27]  Ivo Babuška,et al.  Guaranteed computable bounds for the exact error in the finite element solution. Part I : One-dimensional model problem , 1999 .

[28]  李幼升,et al.  Ph , 1989 .

[29]  E. Kita,et al.  Trefftz method: an overview , 1995 .

[30]  Thomas J. R. Hughes,et al.  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 , 1996 .

[31]  R. Courant,et al.  Methods of Mathematical Physics , 1962 .

[32]  E. Kita,et al.  Error estimation and adaptive mesh refinement in boundary element method, an overview , 2001 .

[33]  Philippe Bouillard,et al.  Error estimation and adaptivity for the finite element method in acoustics , 1998 .

[34]  Wim Desmet,et al.  The wave based method: An overview of 15 years of research , 2014 .

[35]  Thomas J. R. Hughes,et al.  A posteriori error estimation and adaptive finite element computation of the Helmholtz equation in exterior domains , 1996 .

[36]  F. Ihlenburg.  Finite Element Analysis of Acoustic Scattering , 1998 .

[37]  P. Bouillard,et al.  Efficiency of A Residual a posteriori Error Estimator for the Finite Element Solution of the Helmholtz Equation , 2001 .

[38]  Wim Desmet,et al.  An efficient wave based prediction technique for dynamic plate bending problems with corner stress singularities , 2009 .

[39]  Stefan A. Sauter,et al.  Is the Pollution Effect of the FEM Avoidable for the Helmholtz Equation Considering High Wave Numbers? , 1997, SIAM Rev..

[40]  J. B. Fahnline,et al.  A method for computing acoustic fields based on the principle of wave superposition , 1989 .

[41]  H. Weinberger,et al.  Maximum principles in differential equations , 1967 .

[42]  R. J. Astley,et al.  A comparison of two Trefftz‐type methods: the ultraweak variational formulation and the least‐squares method, for solving shortwave 2‐D Helmholtz problems , 2007 .

[43]  T. Hughes,et al.  Finite element methods for the Helmholtz equation in an exterior domain: model problems , 1991 .

[44]  C. Farhat,et al.  A discontinuous enrichment method for three‐dimensional multiscale harmonic wave propagation problems in multi‐fluid and fluid–solid media , 2008 .

[45]  A Hepberger,et al.  ENGINE RADIATION SIMULATION UP TO 3 KHZ USING THE WAVE BASED TECHNIQUE , 2009 .

[46]  R. Kress,et al.  Inverse Acoustic and Electromagnetic Scattering Theory , 1992 .

[47]  J. B. Fahnline,et al.  Numerical errors associated with the method of superposition for computing acoustic fields , 1991 .

[48]  J. Z. Zhu,et al.  The finite element method , 1977 .

[49]  Alex H. Barnett,et al.  Perturbative Analysis of the Method of Particular Solutions for Improved Inclusion of High-Lying Dirichlet Eigenvalues , 2009, SIAM J. Numer. Anal..

[50]  G Offner.  Modelling of condensed flexible bodies considering non-linear inertia effects resulting from gross motions , 2011 .

[51]  Eric Darve,et al.  The Fast Multipole Method , 2000 .

[52]  P. Bouillard,et al.  Error estimation and adaptivity for the finite element method in acoustics: 2D and 3D applications , 1999 .

[53]  Bert Pluymers,et al.  Trefftz-Based Methods for Time-Harmonic Acoustics , 2007 .

[54]  Timo Betcke,et al.  Stability and convergence of the method of fundamental solutions for Helmholtz problems on analytic domains , 2007, J. Comput. Phys..

[55]  Eric Darve,et al.  The Fast Multipole Method I: Error Analysis and Asymptotic Complexity , 2000, SIAM J. Numer. Anal..

[56]  A Hepberger,et al.  Investigations on potential improvements of the Wave Based Technique for the application to radiation problems under anechoic conditions , 2008 .

[57]  Serge Prudhomme,et al.  On goal-oriented error estimation for elliptic problems: application to the control of pointwise errors , 1999 .

[58]  Ivo Babuška,et al.  A posteriori error estimation for finite element solutions of Helmholtz’ equation. part I: the quality of local indicators and estimators , 1997 .

[59]  Thomas J. R. Hughes,et al.  An a posteriori error estimator and hp -adaptive strategy for finite element discretizations of the Helmholtz equation in exterior domains , 1997 .

[60]  S. Marburg,et al.  Computational acoustics of noise propagation in fluids : finite and boudary element methods , 2008 .

[61]  Wim Desmet,et al.  Hybrid finite element-wave-based method for steady-state interior structural-acoustic problems , 2005 .

[62]  R. Coifman,et al.  The fast multipole method for the wave equation: a pedestrian prescription , 1993, IEEE Antennas and Propagation Magazine.

[63]  J. Oden,et al.  A Posteriori Error Estimation in Finite Element Analysis , 2000 .

[64]  J. Tinsley Oden,et al.  A posteriori error estimation for acoustic wave propagation problems , 2005 .

[65]  W. Desmet.  A wave based prediction technique for coupled vibro-acoustic analysis , 1998 .

[66]  Wim Desmet,et al.  A computationally efficient prediction technique for the steady-state dynamic analysis of coupled vibro-acoustic systems , 2000 .

[67]  Frank Ihlenburg,et al.  The Medium-Frequency Range in Computational Acoustics: Practical and Numerical Aspects , 2003 .

[68]  Jens Markus Melenk,et al.  When is the error in the $$h$$h-BEM for solving the Helmholtz equation bounded independently of $$k$$k? , 2015 .

[69]  I. Babuska,et al.  Finite element solution of the Helmholtz equation with high wave number Part I: The h-version of the FEM☆ , 1995 .

[70]  Wim Desmet,et al.  A Trefftz-based numerical modelling framework for Helmholtz problems with complex multiple-scatterer configurations , 2010, J. Comput. Phys..

[71]  Hans H. Priebsch,et al.  A Generic Simulation Model for Cylinder Kit Vibro-Acoustics: Part II — Piston Slap and Engine Structure Interaction , 2003 .

[72]  L. Shampine.  Vectorized adaptive quadrature in MATLAB , 2008 .

[73]  John B. Shoven,et al.  I , Edinburgh Medical and Surgical Journal.

[74]  Lothar Gaul,et al.  A multipole Galerkin boundary element method for acoustics , 2004 .

[75]  R. Duraiswami,et al.  Fast Multipole Methods for the Helmholtz Equation in Three Dimensions , 2005 .