Accurate acoustic computations using a meshless method

• Autorzy: Suleau S. , Bouillard P
• Konferencja: International Workshop on the Trefftz Method (2 ; 1999 ; Lisbon, Portugal)
• Języki publikacji: EN

Abstrakty

EN

It is well known today that the standard finite element method (FEM) is unreliable to compute approximate solutions of the Helmholtz equation for high wavenumbers due to the pollution effect, consisting mainly of the dispersion, i.e. the numerical wavelength is longer than the exact one. Unless highly refined meshes are used, FEM solutions lead to unacceptable solutions in terms of precision. The paper presents an application of the Element-Free Galerkin Method (EFGM) leading to extremely accurate results in comparison with the FEM. Moreover, the present meshless formulation is not restricted to regular distribution of nodes as some stabilisation methods and a simple but real-life problem is investigated in order to show the improvement in the accuracy of the numerical results, as compared with FEM results.

Słowa kluczowe

EN

Trefftz method

PL

metoda Trefftza

• Wydawca: Instytut Podstawowych Problemów Techniki PAN
• Czasopismo: Computer Assisted Mechanics and Engineering Sciences
• Rocznik: 2001
• Tom: Vol. 8, No. 2-3
• Strony: 455--468
• Opis fizyczny: Bibliogr. 20 poz., rys., wykr.

Twórcy

autor

Suleau S.

• Department of Continuum Mechanics, Université Libra de Bruxelles, CP 194/5, Av. F. D. Roosevelt 50, 1050 Brussels, Belgium

autor

Bouillard P

• Department of Continuum Mechanics, Université Libra de Bruxelles, CP 194/5, Av. F. D. Roosevelt 50, 1050 Brussels, Belgium

Bibliografia

• [1] I. Babuška, F. Ihlenburg, E. Paik, S. Sauter. A Generalized Finite Element Method for solving the Helmholtz equation in two dimensions with minimal pollution. Comput. Methods Appl. Mech. Eng. 128:325-359, 1995.
• [2] I. Babuška, F. Ihlenburg, T. Strouboulis, S. K. Gangaraj. A posteriori Error Estimation for Finite Element Solutions of Helmholtz' Equation. Part I: the Quality of Local Indicators and Estimators. Int. j. numer. methods eng. 40:3443-3462, 1997.
• [3] I. Babuška, J. M. Melenk. The partition of unity method. Int. j. numer. methods eng. 40:727-758, 1997.
• [4] T. Belytschko, Y. Krongauz, D. Organ, M. Fleming, P. Krysl. Meshless methods: An overview and recent developments. Comput. Methods Appl. Mech. Eng. 139:3-47, 1996.
• [5] T. Belytschko, Y. Y. Lu, L. Gu, Element-Free Galerkin Methods. Int. j. numer. methods eng. 37:229-256, 1994.
• [6] Ph. Bouillard, F. Ihlenburg. Error estimation and adaptivity for the ¯nite element solution in acoustics. In P. Ladevµeze & J. T. Oden eds., Advances in Adaptive Computational Methods in Mechanics, Elsevier, 1998.
• [7] Ph. Bouillard, S. Suleau. Element-free Galerkin solutions for Helmholtz problems: formulation and numerical assessment of the pollution effect. Comput. Methods Appl. Mech. Eng. 162:317-335, 1998.
• [8] A. Deraemaeker, I. Babuška, Ph. Bouillard. Dispersion and pollution of the FEM solution for the Helmholtz equation in one, two and three dimensions. Int. j. numer. methods eng. 46:471-500, 1999.
• [9] W. Desmet. A wave based prediction technique for coupled vibro-acoustic analysis, PhD dissertation, KU Leuven, Belgium, 1998.
• [10] L. Franca, C. Farhat, A. Macedo, M. Lessoine. Residual-Free Bubbles for the Helmholtz Equation. Int. j. numer. methods eng. 40:4003-4009, 1997.
• [11] I. Harari, K. Grosh, T. J. R. Hughes, M. Malhotra, P. M. Pinsky, J. R. Stewart, L. L. Thompson. Recent Developments in Finite Element Methods for Structural Acoustics. Arch. of Comp. Meth. Eng. 3:131-311, 1996.
• [12] I. Herrera. Trefftz method. Topics in Boundary Element Research, Vol. 1: Basic Principles and Applications, Brebbia ed., 225-253, Springer-Verlag, 1985.
• [13] F. Ihlenburg, I. Babuška. Finite Element Solution of the Helmholtz Equation with High Wave Number. Part 1: The h-Version of the FEM. Computers Math. Applic.. 38(9):9-37, 1995.
• [14] F. Ihlenburg, I. Babuška. Dispersion Analysis and Error Estimation of Galerkin Finite Element Methods for the Helmholtz Equation. Int. j. numer. methods eng. 38:3745-3774, 1995.
• [15] I. Kaljevic, S. Saigal. An Improved Element Free Galerkin Formulation, Int. j. numer. methods eng., 40:2953-2974, 1997.
• [16] Y. Krongauz, T. Belytschko. Enforcement of essential boundary conditions in meshless approximations using finite elements. Comput. Methods Appl. Mech. Eng. 131:133-145, 1996.
• [17] P. Lancaster, K. Salkausas. Surfaces generated by moving least squares methods. Math. Comput. 37:141- 158, 1981.
• [18] D. J. Nefske. Sound in small enclosures. In L. Beranek and I. Vér, eds., Noise and Vibration Control Engineering. Principles and Applications, 1st edition, J. Wiley & Sons, ISBN 0-471-61751-2, London, 1992.
• [19] S. Suleau, Ph. Bouillard. 1D Dispersion analysis for the element-free Galerkin method for the Helmholtz equation. Int. j. numer. methods eng., 47(6):1169-1188, 1999.
• [20] S. Suleau, Ph. Bouillard. Dispersion and pollution of meshless solutions for the Helmholtz equation. Comput. Methods Appl. Mech. Eng., in print, 2000.