An efficient numerical method, using integral equations, is developed to calculate precisely the acoustic eigenfrequencies and their associated eigenvectors, located in a given high frequency interval. It is currently known that the real symmetric matrices are well adapted to numerical treatment. However, we show that this is not the case when using integral representations to determine with high accuracy the spectrum of elliptic, and other related operators. Functions are evaluated only in the boundary of the domain, so very fine discretizations may be chosen to obtain high eigenfrequencies. We discuss the stability and convergence of the proposed method. Finally we show some examples.
Issue Section:
Research Papers
1.
Babuška
, I.
, and Osborn
, J. E.
, 2000, “Eigenvalue Problems
,” Handbook of Numerical Analysis: Finite Element Methods (Part 1)
, Vol. 2
, P. G.
Ciarlet
and J. L.
Lions
, eds., North-Holland
, Amsterdam
.2.
Zienkiewicz
, O. C.
, 1997, ”Origins, Milestones and Directions of the Finite Element Method—A Personal View
,” Handbook of Numerical Analysis: Techniques of Scientific Computing (Part 2)
, Vol. 5
, P. G.
Ciarlet
and J. L.
Lions
, eds., North-Holland
, Amsterdam
.3.
Zienkiewicz
, O. C.
, 2000, The Finite Element Method
, 5th ed., McGraw-Hill
, Oxford
.4.
Strang
, G.
, and Fix
, G. J.
, 1973, An Analysis of the Finite Element Method
, Prentice-Hall
, Englewood Cliffs, NJ
.5.
Oden
, J. T.
, and Reddy
, J. N.
, 1976, An Introduction to the Mathematical Theory of Finite Elements
, Wiley Interscience
, New York
.6.
Wait
, R.
, and Mitchell
, A. R.
, 1985, Finite Element Analysis and Applications
, Wiley
, Chichester
.7.
Mercier
, B.
, Osborn
, J. E.
, Rappaz
, J.
, and Raviart
, P. A.
, 1981, “Eigenvalue Approximation by Mixed and Hybrid Methods
,” Math. Comput.
0025-5718, 36
, pp. 427
–453
.8.
Raviart
, P. A.
, and Thomas
, J. M.
, 1983, Introduction à l’Analyse Numérique des Équations aux Dérivées Partielles
, Masson
, Paris
.9.
Kolata
, W. G.
, 1978, “Approximation of Variationally Posed Eigenvalue Problems
,” Numer. Math.
0029-599X, 29
, pp. 159
–171
.10.
Brezzi
, F.
, and Fortin
, M.
, 1991, Mixed and Hybrid Finite Elements Methods
, Springer-Verlag
, Berlin
.11.
Rannacher
, R.
, 1979, “Nonconforming Finite Element Methods for Eigenvalue Problems in Linear Plate Theory
,” Numer. Math.
0029-599X, 33
, pp. 23
–42
.12.
Grégoire
, J. P.
, Nedelec
, J. -C.
, and Planchard
, J.
, 1975, “A Method for Computing Eigenfrequencies of an Acoustic Resonator
,” Applications of Methods of Functional Analysis to Problems of Mechanics
(Lecture Notes in Mathematics
), Vol. 503
, pp. 343
–353
.13.
Grégoire
, J. P.
, Nedelec
, J. -C.
, and Planchard
, J.
, 1976, “A Method of Finding the Eigenvalues and Eigenfunctions of Self-Adjoint Operators
,” Comput. Methods Appl. Mech. Eng.
, 8
, pp. 201
–214
. 0045-782514.
Bernardi
, C.
, and Maday
, Y.
, 2000, “Spectral Methods
,” Handbook of Numerical Analysis: Techniques of Scientific Computing (Part 2)
, Vol. 5
, P. G.
Ciarlet
and J. L.
Lions
, eds., North-Holland
, Amsterdam
.15.
Gottlieb
, D.
, and Orszag
, S. A.
, 1977, Numerical Analysis of Spectral Methods, Theory and Applications
, SIAM
, Philadelphia, PA
.16.
Vandeven
, H.
, 1990, “On the Eigenvalues of Second-Order Spectral Differentiation Operators
,” Comput. Methods Appl. Mech. Eng.
, 80
, pp. 313
–318
. 0045-782517.
Forsythe
, G. E.
and Wasow
, W. R.
, 1960, Finite Difference Methods for Partial Differential Equations
, Wiley
, New York
.18.
Banerjee
, P. K.
, Ahmad
, S.
, and Wang
, H. C.
, 1988, “A New BEM Formulation for the Acoustic Eigenfrequencies Analysis
,” Int. J. Numer. Methods Eng.
0029-5981, 26
, pp. 1299
–1309
.19.
Coyette
, J. P.
, and Fyfe
, K. R.
, 1990, “An Improved Formulation for Acoustic Eigenmode Extraction From Boundary Element Models
,” ASME J. Vibr. Acoust.
0739-3717, 112
, pp. 392
–398
.20.
Ali
, A.
, Rajakumar
, C.
, and Yunus
, S. M.
, 1991, “On the Formulation of the Acoustic Boundary Element Eigenvalue Problems
,” Int. J. Numer. Methods Eng.
, 31
, pp. 1271
–1282
. 0029-598121.
Kirkup
, S. M.
, and Amini
, S.
, 1993, “Solution of the Helmholtz Eigenvalue Problem Via the Boundary Element Method
,” Int. J. Numer. Methods Eng.
0029-5981, 36
, pp. 321
–330
.22.
Chen
, J. T.
, Liu
, L. W.
, and Hong
, H. -K.
, 2003, “Spurious and True Eigensolutions of Helmholtz BIEs and BEMs for a Multiply Connected Problem
,” Proc. R. Soc. London, Ser. A
0950-1207, 459
, pp. 1891
–1924
.23.
Kuo
, S. R.
, Chen
, J. T.
, and Huang
, C. X.
, 2000, “Analytical Study and Numerical Experiments for True and Spurious Eigensolutions of a Circular Cavity Using the Real Part Dual BEM
,” Int. J. Numer. Methods Eng.
0029-5981, 48
(9
), pp. 1401
–1422
.24.
Alves
, C. J.
, and Antunes
, P. R.
, 2005, “The method of Fundamental Solutions Applied to the Calculation of Eigenfrequencies and Eigenmodes of 2D Simply Connected Shapes
,” Comput., Mater., Continua
1546-2218, 2
(4
), pp. 251
–265
.25.
Pierce
, A. D.
, 1989, Acoustics: An Introduction to Its Physical Principles and Applications
, Acoustical Society of America
, New York
.26.
Morse
, P. M.
, 1981, Vibration and Sound
, Acoustical Society of America
, New York
.27.
Beranek
, L. L.
, 1996, Acoustics
, Acoustical Society of America
, New York
.28.
Nédélec
, J. -C.
, 2001, Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems
, Springer-Verlag
, Berlin
.29.
Kellog
, O. D.
, 1929, Foundations of Potential Theory
, Springer
, Berlin
.30.
Mikhlin
, S. G.
, 1957, Integral Equations
, Pergamon
, Oxford
.31.
Abramowitz
, M.
, and Stegun
, I. A.
, 1970, Handbook of Mathematical Functions
, Dover
, New York
.32.
Ha Duong
, T.
, 1987, “Équations Intégrales pour la Résolution Num´erique de Problèmes de Diffraction d’Ondes Acoustiques dans R3
,” Ph.D. thesis, Universit´e Pierre et Marie Curie, Paris VI, France.33.
Hamdi
, M. A.
, 1982, “Formulation Variationelle par Équations Intégrales pour le Calcul de Champs Acoustiques Lin´eaires Proches et Lointains
,” Ph.D. thesis, Université de Technologie de Compiègne, France.34.
Ciarlet
, P. G.
, 1978, The Finite Element Method for Elliptic Problems
, North-Holland
, Amsterdam
.35.
Conca
, C.
, Duran
, M.
, and Rappaz
, J.
, 1998, “Rate of Convergence Estimates for the Spectral Approximation of a Generalized Eigenvalue Problem
,” Numer. Math.
0029-599X, 79
, pp. 349
–370
.36.
Hamdi
, M. A.
, 1984, “Rayonnement de structure à géométrie axisymétrique
,” 9ème Colloque d’Acoustique Aéronautique
, Compiègne
, France
.37.
Durán
, M.
, Miguez
, M.
, and Nédélec
, J. -C.
, 2001, “Numerical Stability in the Calculation of Eigenfrequencies Using Integral Equations
,” J. Comput. Appl. Math.
0377-0427, 130
, pp. 323
–336
.38.
Bendali
, A.
, and Devys
, C.
, 1986, “Calcul numérique du rayonnement de cornets électromagnétiques dont l’ouverture est partiellement remplie par un diélectrique
,” L’Onde électrique
, 66
, pp. 77
–81
.39.
Christiansen
, S. H.
, and Nédélec
, J. -C.
, 2000, ““Des préconditionneurs pour la résolution numérique des équations intégrales de frontière de l’acoustique
,” C. R. Acad. Sci., Ser. I: Math.
0764-4442, 330
, pp. 617
–622
.Copyright © 2009
by American Society of Mechanical Engineers
You do not currently have access to this content.
