an eigenvalue search algorithm for the modal

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Journal of Computational Acoustics, Vol. 19, No. 1 (2011) 95–109 c IMACS DOI: 10.1142/S0218396X11004304

AN EIGENVALUE SEARCH ALGORITHM FOR THE MODAL ANALYSIS OF A RESONATOR IN FREE SPACE STEFANIE FUß∗ , STUART C. HAWKINS† and STEFFEN MARBURG∗,‡

J. Comp. Acous. 2011.19:95-109. Downloaded from www.worldscientific.com by Professor Steffen Marburg on 09/03/12. For personal use only.

∗ LRT4 — Institute of Mechanics Universit¨ at der Bundeswehr M¨ unchen D-85579 Neubiberg, Germany †

Department of Mathematics, Macquarie University Sydney, NSW 2109, Australia ‡ steff[email protected] Received 15 November 2010 Revised 20 December 2010

In this article we present an algorithm for the three-dimensional numerical simulation of the sound spectrum and the propagation of acoustic radiation inside and around long slender hollow objects. The fluid inside and close to the object is meshed by Lagrangian tetrahedral finite elements. To obtain results in the far field of the object, complex conjugated Astley-Leis infinite elements are used. To apply these infinite elements the finite element domain is meshed either in a spherical or an ellipsoidal shape. Advantages and disadvantages of both shapes regarding the form of the object are discussed in this article. The formulation leads to a quadratic eigenvalue problem with real, large and nonsymmetric matrices. An eigenvalue search algorithm is implemented to concentrate on the computation of the interior eigenmodes. This algorithm is based on a linearization of the quadratic problem in a state space formulation. The search algorithm uses a complex shift to efficiently extract the relevant eigenvalues only. Keywords: Infinite elements; eigenvalue problem; Arnoldi method; recorder.

1. Introduction In a static fluid, the sound propagation is described by the scalar wave equation. For acoustic problems it is often assumed that the wave equation is harmonic in time. Therefore, the wave equation can be reduced to the harmonic wave equation, also known as Helmholtz equation. In this article, we discuss the simulation of the sound radiation in an unbounded domain. Only for simple examples is it possible to do such a simulation analytically. Therefore, it is advisable to use numerical methods, e.g. the finite element method (FEM) or the boundary element method (BEM). BEM is not an alternative, because a modal analysis of the fluid is not compatible with BEM. At least, the authors are not aware of any formulation of the algebraic eigenvalue problem using BEM for open domains. When using FEM a suitable formulation is required to present the radiation to infinity.28 In this article only the finite element method will be considered. 95

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There are three main concepts to present the radiation condition in a finite element model. They are the formulation of absorbing boundary conditions,23,25,30 the formulation of infinite elements,5,8,15 and the formulation of a perfectly matching layer.12,13 Generally, the sound propagation is simulated numerically with finite elements in the near field of the radiating body. A surrounding (artificial) surface is created at a sufficient distance from the radiating body to apply one of the above mentioned treatments to verify the radiation condition. Good overviews to this topic are found in the book by Kollmann32 and in articles by Thompson43 and Givoli.24 In this article, as example of a long slender hollow object we choose to study the radiation of a recorder fluid. The advantage of this example is that the exact frequencies of different recorders are known. The finite element model consists of the fluid inside and in the near field of a soprano recorder. The radiation to infinity is taken into consideration by adding infinite elements. The aim of this work is the examination of the interior eigenmodes for different notes and the comparison of these eigenmodes to the exact frequencies of such a recorder. This is a first step towards a sensitivity analysis and the investigation of the influence of slight modifications of the recorder model (e.g. the tone holes). A quadratic eigenvalue problem is solved iteratively to concentrate on the computation of the interior eigenmodes. The spectrum of an unbounded acoustic domain is continuous. However, beside the continuous spectrum there are so-called trapped modes, i.e. additional discrete eigenvalues.29,47,48 These trapped modes may correspond to resonance phenomena even in the external domain,33,35 Some recent theoretical work about eigenvalues, eigenvectors and resonances of open systems has been published by Koch and co-workers.26,27,31 There are several scientific works, experimental as well as numerical, that deal with the sound evolution and sound propagation in recorders or similar music instruments. As early as 1960, Benade11 published his work on the influence of recorder tone holes. The main focus of this article was on the interaction between the inner tube and the tone hole. Barjau et al.10 developed a CTIM algorithm that allows the exact simulation of an acoustic wave with arbitrarily placed discontinuities in the time domain. Linear acoustic behavior and few viscothermic losses are required for a successful use of this algorithm. Additionally, a comparison at different discrete places at different times verified the CTIM algorithm. In this conjunction, the work of Dubos et al.20 also has to be mentioned. Nederveen et al.38 analyzed the partition of the flow at tone holes of woodwind instruments. Furthermore, there are several publications by Wolfe and co-authors in the field of experimental investigations on woodwind instruments.21,22,49,50 More works in experimental investigations can be found in Refs. 1 and 14 and d’Andréa-Novel et al.18 present a comparison between numerical and experimental data for the pressure of a slide flute. At the Technische Universit¨ at Dresden, an experimental and numerical modal analysis of a recorder and the recorder fluid was performed to examine the influence of the structure on the acoustic behavior.37 It was found that the recorder structure has no significant influence on the acoustic field. The comparison of experimental and numeric data showed that experiments are not precise

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enough for a sensitivity analysis as is planned in the course of this work. Therefore, the implementation of numeric methods is essential. The outline of this article is as follows. Section 2 discusses the theory of the finite and infinite element formulation. The weak formulation of the Helmholtz equation is deduced for finite and infinite element domain and its discretization is explained. In Sec. 3 the eigenvalue problem is derived and the eigenvalue search algorithm, which is used to concentrate on the computation of specific eigenvalues, is depicted. In Sec. 4 numerical examples are presented and conclusions are given in Sec. 5.


2. Theory A numerical method was developed to compute eigenvalues and eigenvectors of a fluid inside and around a three-dimensional recorder model. Infinite elements, in particular complex conjugated Astley-Leis elements, are used to represent far field effects.4,7–9,34 2.1. Subdivision of the domain Infinite elements perform best when they are applied to a convex surface which can be described in separable coordinates. Therefore, we start with a definition of a convex surface ΓC , which is defined at the distance a(x) from the point of origin. Following Astley,3 the inner domain ΩC which lies between the radiating body B with the boundary Γ and ΓC is discretized by finite elements. The outer domain ΩR lies between ΓC and ΓR , with the radius R defining the distance of ΓR to the point of origin of the coordinate system of B. By taking R → ∞ the unbounded domain ΩR can be entirely taken into consideration. This domain is discretized by infinite elements. An example of such an exterior problem is shown in Fig. 1.

ΓR n

ΓC a(x)

Γ n

B

O

ΩC R ΩR

Fig. 1. Division in FE domain ΩC and IE domain ΩR for radiation problems.

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Fig. 2. Tetrahedra element of first order (left) and second order (right).

Fig. 3. P1 global infinite element (left) and reference element (right) with three interpolation points.

Lagrangian tetrahedral elements of first (P1 ) or second (P2 ) order are used as finite elements, see Fig. 2. In Fig. 3 the global infinite element and its reference element for P1 can be seen. The P2 case is analogous. Due to the triangular surface of the tetrahedral elements on ΓC the reference element for the infinite elements is a pentahedral element. A beam on which the interpolation points are situated runs in the radial direction through each node of the global element. The number of interpolation points, which is identical on each beam, determines the radial order of the infinite element and can be chosen arbitrarily. The z coordinate in the reference element runs from −1 to 1 while x and y coordinates run from 0 to 1. z = −1 corresponds to the node of the global element which is on ΓC and z = 0 to the point with a distance a(x) from ΓC (which has distance 2a(x) from the origin), while z = 1 corresponds to infinity in the global coordinate system. 2.2. Helmholtz equation and boundary conditions The Helmholtz equation describes the fluid pressure inside and around the recorder in the frequency domain. Assuming that the pressure is harmonic in time, p(x, t) = Re{p(x)e−iωt }. Then the Helmholtz equation can be written as2,28 ∆p(x) + k2 p(x) = 0 for x ∈ Ω,

(1)

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with k = ω/c being the wave number. For the problem discussed here, the Neumann boundary condition is used for rigid boundaries on the entire recorder ∂p(x) = 0 for x ∈ Γ, ∂n

(2)

and the Sommerfeld radiation condition is necessary to ensure that only outwardly propagating components exist at large distance from the radiating surface ∂p(x) − ikp(x) → 0 as r → ∞. (3) r ∂r


2.3. Weak formulation The weak formulation of the Helmholtz equation is obtained by multiplying Eq. (1) with a test function q and integrating over the domain Ω. Under consideration of Neumann boundary conditions, the weak formulation of the eigenvalue problem in the finite element domain results in 2 ∇p · ∇qdΩC − k pqdΩC = 0. (4) ΩC

ΩC

For the infinite element domain, with consideration of the Sommerfeld radiation condition, this yields ∇p · ∇qdΩR − ik pqdΓR − k2 pqdΩR = 0. (5) lim R→∞

ΩR

ΓR

ΩR

2.4. Discretization The pressure p is approximated in the entire domain as a continuous function. Since global continuity is imposed, the values ξi are defined uniquely in the nodes of the mesh, belonging to more than one tetrahedron and our approximation is p(x) ∼ =

N

ξi wiFE (x).

i=1

Here N is the number of nodes in the finite element domain and wiFE are the linear or quadratic interpolation functions which take the value 1 at node i and 0 at the other nodes. The values ξi represent the values of the pressure in each node. The same interpolation functions are used for the test function q. The discretized weak formulation of the Helmholtz equation for the computation with finite elements is then N FE FE 2 FE FE ξi ∇w i · ∇w j dΩ − k w i w j dΩ = 0, i=1

Ω

Ω

j = 1, 2, . . . , N.

(6)

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The first integral yields the stiffness matrix and the second yields the mass matrix wiFE wjFE dΩCk , Mij =


Kij =

(7)

ΩCk

ΩCk

∇wiFE · ∇wjFE dΩCk .

(8)

For infinite elements, the computation of discretized interpolation and test functions is more complicated and is explained in the following. We describe these functions in spherical polar coordinates (r, ϑ, ϕ). Since the interpolation functions of the finite elements and the infinite elements have to be compatible on ΓC , the interpolation functions of the infinite elements have to be reduced to the two-dimensional interpolation functions on the boundary. The interpolation functions are ikµ(r,ϑ,ϕ) N IE i (r, ϑ, ϕ) = Pi (r, ϑ, ϕ)e

(9)

and the conjugated complex test functions are W iIE (r, ϑ, ϕ) = w(r, ϑ, ϕ)Pi (r, ϑ, ϕ)e−ikµ(r,ϑ,ϕ) .

(10)

Here µ is the phase function, w is the weighting for polar coordinates and Pi represents the real interpolation function, µ = r − a(x) = a(x) w=

1−r 2

2

1+z , 1−z

,

Pi (r, ϑ, ϕ) = λwr (ϑ, ϕ)(1 − r)Lm s (r). Herein, λ = 1/(1 − r) is a scale factor depending on the type of element and dimension. wr is the two-dimensional shape function and Lm s is a Lagrangian polynomial of order (m − 1) Lm s =

m r − rq rs − rq

for s = q,

q=1

m determines the number of interpolation points, with s = 1, 2, . . . , m. There are various ways to compute the positions of the interpolation points, for example with Lagrange, Legendre3,6 or Jacobi polynomials.19,46 The index i of the interpolation function derives from i = m(rad − 1) + s, with rad being the number of the radial beam considered at that moment, and corresponds to the node numbers of the element. For Lagrangian elements of order one rad runs from 1 to 3 and for Lagrangian elements of order two from 1 to 6. Using the interpolation functions (9) and the test functions (10), the discretized form of the Helmholtz equation can be obtained.8 Discretized mass, stiffness and damping

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Mij =

ΩRk

wPi Pj (1 − ∇µ · ∇µ)dΩRk ,

(11)

(Pi ∇w + w∇Pi ) · ∇Pj dΩRk ,

(12)

(wPi ∇µ · ∇Pj − Pi Pj ∇w · ∇µ − wPj ∇Pi · ∇µ)dΩRk .

(13)

Kij =

ΩRk


Dij =

ΩRk

The global matrices obtained with complex conjugated Astley-Leis elements are real and nonsymmetric. To describe the complete problem, finite and infinite element matrices are combined and the resulting system of equations is written as matrix formulation (K − ikD − k2 M)ϕ = 0.

(14)

Only the infinite elements have nonzero entries in the damping matrix D. 3. Eigenvalue Problem 3.1. State space formulation The finite element domain shape has to be either spherical or ellispoidal,15,16 due to the infinite element method discussed in this article. These two possible domain shapes can be seen in Fig. 4. The advantage of a spherical domain shape is that, due to the symmetry, the entries of the mass matrix associated with the infinite elements are zero. On the other hand, when thinking of the recorder, an ellipsoid can be placed very close around the structure to keep the entire fluid model as small as possible. Consequently, the number of degrees of freedom of the model is significantly smaller than with a spherical domain shape. For exterior problems the complex conjugated eigenvalues of (14) are situated on rays in the complex plane. The number depends on the radial order of the infinite elements.33 The interior eigenmodes, also known as trapped modes, are distinguished by their imaginary

Fig. 4. Spherical domain shape (left) and ellipsoidal domain shape (right).

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part being significantly bigger than their real part, and therefore they are positioned very close to the imaginary axis. Only the interior eigenmodes are of interest in the following computations. They account for a small part of the entire spectrum of eigenmodes only. To efficiently compute these eigenvalues numerically it is necessary to employ a scheme that focused only on the eigenmodes. To solve the large, sparse and unsymmetric eigenvalue problem efficiently, we require an iterative solver. Our method is based on the Arnoldi method, after converting the quardratic eigenvalue problem to a linear eigenvalue problem. In a first step Eq. (14) is linearized. This is done by using a state space formulation. The most common state space formulation of a general eigenvalue problem is17,36,40

M 0 0 M ϕ 0 −λ = , 0 −K M D ψ 0 with ϕ = λψ as eigenvectors and λ = −ik. This equation can also be written as a generalized eigenvalue problem (A − λB)Φ = 0.

(15)

To obtain a standard eigenvalue problem either matrix A or B has to be inverted. The mass matrix can be singular in certain circumstances, for example when using a spherical domain shape. Thus it is not possible to use the common state space formulation, because an inversion of the mass matrix would be difficult to handle. Therefore the state space equation was modified to meet these requirements.39 The new matrices A and B are

I 0 0 I A= , B= . (16) 0 −K M D The standard formulation of Eq. (15), using the matrices from (16), is 1 Φ = A−1 BΦ. λ

(17)

For the action of A−1 in this transformation we use a preconditioned GMRes algorithm.42 The system is preconditioned by an incomplete LU preconditioner.41 3.2. Eigenvalue search algorithm When using the iterative Arnoldi algorithm for (17), the first eigenvalues that are computed are the ones closest to zero or, when implementing a shift, the ones closest to the shift. As stated earlier, only the interior eigenmodes, the ones close to the imaginary axis, are of interest. Therefore shifts are applied along the imaginary axis and for each shift we terminate the Arnoldi method after only few eigenvalues are computed. The quadratic eigenvalue problem is (K + λD + λ2 M)ϕ = 0,

(18)

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¯ + s. With this substitution, with λ = −ik. When using a shift s, λ is substituted by λ = λ Eq. (18) becomes ¯ ¯ 2 M)ϕ = 0 (K + sD + s2 M + λ(D + 2sM) + λ

(19)

and can also be written as


¯ +λ ¯D ¯ +λ ¯ 2 M)ϕ = 0. (K

(20)

¯ and damping matrix D ¯ are complex and a complex Arnoldi Now, the new stiffness matrix K eigenvalue solver has to be used. The application of the eigenvalue search algorithm is explained in more detail by an example. In this example we consider the domain inside and around a box, that is open on one side. For this problem, with a small number of degrees of freedom, all eigenvalues are computed using Matlabs eig function. Then the search algorithm is applied to that model to compute all eigenvalues along the imaginary axis between 0 i and 100 i. In the search algorithm it is determined that four eigenvalues are to be computed at each step. The criteria for an eigenvalue to be computed is that the residual norm for this eigenvalue is less than a prescribed tolerance. This is necessary, because lots of eigenvalues are computed by the Arnoldi algorithm, but most of them are poor approximations to eigenvalues and those have to be discarded. In the first step the first four eigenvalues closest to the shift s = 0 have been computed. The distance between zero and the furthest eigenvalue from zero determines the range in which all eigenvalues have been computed. In Fig. 5(a) the first step of the search algorithm can be seen. The blue crosses represent all eigenvalues of the model. The red circles are the eigenvalues obtained with the Arnoldi eigenvalue solver by using the search algorithm. The green circle represents the disk in which we know all eigenvalues are computed. The center of the green circle corresponds to the position of the shift. In the next step a shift is implemented at 100 i and the eigenvalues are computed around that shift, Fig. 5(b). Then the next shifts are chosen at the midpoints between two shifts until the green circles overlap each other, see Fig. 6. This is a good way to make sure, that all eigenvalues are obtained within the range of 0 i and 100 i. 4. Example For all computations a soprano recorder with German fingering is used. It is tuned to 442 Hz. A three-dimensional finite element model is build in Ansys 11.0, meshing everything inside and around the recorder within an ellipsoidal domain. Figure 7 shows a section of the fluid model, meshed with tetrahedral elements. Herein second order Lagrangian elements are used. In this figure the mesh inside the recorder can be seen, as well as the not-meshed part of the recorder itself and part of the outer finite element mesh. This mesh is read into a noncommercial code, written in Fortran 90, that was developed by the authors, and there the infinite elements are added before starting with the computations.

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80

80

60

60

40

40

20

20

0

0

−30

−20

−10

0 Re

10

20

−30

30

−20

(a) step 1

−10

0 Re

10

20

30

10

20

30

(b) step 2

Fig. 5. First steps of search algorithm.

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Im


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0

0

−30

−20

−10

0 Re

(a) step 4

10

20

30

−30

−20

−10

0 Re

(b) step 8

Fig. 6. Different steps of search algorithm.

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(a) Note c with all tone holes closed.

(b) Note f with the lowest three tone holes open.

Fig. 8. First eigenfrequency (left) and first harmonic (right).

Figure 8 shows the obtained eigenvectors for the notes c and f . For note c all tone holes are closed and half of the wavelength equals the length of the inner recorder hole, while for note f the three lowest tone holes are open and it can be seen that most of the sound propagation is through the labium and open tone holes and not at the end of the 527

705.5

526.8

705

526.6

frequency

704.5

frequency


Fig. 7. Section of FE model of the fluid.

526.4

526.2

704

703.5 526 703

525.8

525.6

0

0.5

1

1.5

2

dof

2.5

3

3.5 5 x 10

702.5

0

0.5

1

1.5

2

2.5

dof

Fig. 9. Convergence of c = 525.630 Hz (left) and f = 701.630 Hz (right).

3

3.5

4 x 10

5

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recoder. In the left graphics, the first eigenfrequency can be seen and in the right ones the first harmonic. In Fig. 9 the convergence behavior of notes c and f is shown. Displayed is the frequency against the number of the degrees of freedom. According to the “Musical Instrument Digital Interface” table (MIDI-table) and subsequent conversion of the frequencies to the tuning of the recorder being considered in this work, the frequency of c should be at 525.630 Hz and for f at 701.630 Hz. The slight difference between the required and the obtained frequencies was expected, since the frequencies according to the MIDI table are obtained when a musician is actually playing the recorder, which means, that a volume flow is present. This volume flow has not been taken into consideration in these computations, but is planned for further studies. 5. Conclusion In this paper, a three-dimensional numerical simulation of the sound spectrum and the propagation of acoustic radiation inside and around a recorder has been presented. The fluid inside and close to the recorder has been meshed by Lagrangian finite elements. To verify the radiation condition to infinity, a layer of complex conjugated Astley-Leis infinite elements has been attached on the finite element mesh. Both the finite and infinite element method have been described in detail. The differences of spherical and ellipsoidal domain shapes have been discussed and it has been shown, that when using an ellipsoidal domain shape, the degree of freedom of the model can be reduced significantly. Furthermore, the formulation of the eigenvalue problem has been presented. In this work, only the interior eigenmodes have been of interest, because these are the frequencies that one hears when a musician is playing a recorder. To save computation time, an eigenvalue search algorithm has been developed and implemented in the source code. With this algorithm it has been possible to focus on the computation of the interior eigenmodes. For two different notes the results of the modal analysis have been depicted. Comparing numerically computed and expected eigenvalues showed only slight differences between these values. This difference is reasonable, since the volume flow, which is present when playing a recorder, has not been taken into consideration in the computations. An important future prospect is the inclusion of the volume flow in the computations. When analyzing the sound propagation of a nonstatic fluid, the harmonic wave equation, as discussed in this article, is not suitable for the computations. One possibility to include a displacement field is given by the Galbrun equation.44,45 This will provide the ability to examine the influence of the flow on the eigenfrequencies. Acknowledgment These investigations were extracted from the research project Numerical Simulation of the Sound Spectrum and Sound Propagation Inside and Close to a Recorder, GR 1388/18–1, which has been funded by Deutsche Forschungsgemeinschaft (DFG). We further acknowledge the support by the German Ministry of Education and Research (BMBF) in the context

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of the WTZ project AUS 06/001. The computation was run on the PC-Farm Deimos at the Zentrum f¨ ur Hochleistungsrechnen of the Technische Universit¨ at Dresden.


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18. B. d’ Andréa-Novel, B. Fabre and J.-M. Coron, An acoustic model for automatic control of a slide flute, Acta Acustica United with Acustica 96 (2010) 713–721. 19. D. Dreyer and O. von Estorff, Improved conditioning of infinite elements for exterior acoustics, International Journal for Numerical Methods in Engineering 58 (2003) 933–953. 20. V. Dubos, J. Kergomard, A. Khettabi, J. Dalmont, D. Keefe and C. Nederveen, Theory of sound propagation an a duct with a branched tube using modal decompostion, Acta Acustica United with Acustica 85(2) (1999) 153–169. 21. N. H. Fletcher, J. Smith, A. Z. Tarnopolsky and J. Wolfe, Acoustic impedance measurements– correction for probe geometry mismatch, Journal of the Acoustical Society of America 117(5) (2005) 2889–2895. 22. C. Fritz and J. Wolfe, How do clarinet players adjust the resonances of their vocal tracts for different playing effects? Journal of the Acoustical Society of America 118(5) (2005) 3306–3315. 23. D. Givoli, Recent advances in the DtN FE–method, Archives of Computational Methods in Engineering 6(2) (1999) 71–116. 24. D. Givoli, Computational absorbing boundaries, in Computational Acoustics of Noise Propagation in Fluids — Finite and Boundary Element Methods, eds. S. Marburg and B. Nolte (Springer-Verlag, Berlin, Heidelberg, 2008) 145–166. 25. I. Harari, K. Grosh, T. R. Hughes, M. Malhotra, P. M. Pinsky, J. R. Stewart and L. L. Thompson, Recent development in finite element methods for structural acoustics, Archives of Computational Methods in Engineering 3(2–3) (1996) 131–309. 26. S. Hein, T. Hohage and W. Koch, On resonances in open systems, Journal of Fluid Mechanics 506 (2004) 255–284. 27. S. Hein, T. Hohage, W. Koch and J. Sch¨ obel, Acoustic resonances in a high-lift configuration, Journal of Fluid Mechanics 582 (2007) 179–202. 28. F. Ihlenburg, Finite element analysis of acoustic scattering, Applied Mathematical Sciences, Vol. 132 (Springer-Verlag, New York, Berlin, Heidelberg, 1998). 29. D. S. Jones, The eigenvalues of ∇2 u + λu = 0 when the boundary conditions are given on semi-Infinite domains, Proceedings of Cambridge Philosophers Society 49 (1953) 668–684. 30. J. B. Keller and D. Givoli, Exact nonreflecting boundary conditions, Journal of Computational Physics 82 (1989) 172–192. 31. W. Koch, Acoustic resonances and trapped modes in annular plate cascades, Journal of Fluid Mechanics 628 (2009) 155–180. 32. F. G. Kollmann, Maschinenakustik: Grundlagen, Messtechnik, Berechnung, Beeinflussung, 2nd edn. (Springer-Verlag, Berlin, Heidelberg, New York, 2000). 33. S. Marburg, F. Dienerowitz, T. Horst and S. Schneider, Normal modes in external acoustics, part II, Acta Acustica United with Acustica 92 (2006) 97–111. 34. S. Marburg and B. Nolte, Computational Acoustics of Noise Propagation in Fluids — Finite and Boundary Element Methods (Springer-Verlag, Berlin, Heidelberg, 2008) 1–34. 35. S. Marburg, Normal modes in external acoustics. Part III: Sound power evaluation based on superposition frequency–independent of modes, Acta Acustica United with Acustica 92(2) (2006) 296–311. 36. K. Meerbergen, Locking and restarting quadratic eigenvalue solvers, SIAM Journal on Scientific Computing 22(5) (2001) 1814–1839. 37. F. M¨ obius, Experimentelle und rechnerische Modalanalyse an einer Blockfl¨ ote, Technische Universit¨ at Dresden, Diplomarbeit (2008). 38. C. Nederveen, J. Jansen and R. van Hassel, Correction for woodwind tone–hole calculation, Acta Acustica United with Acustica 84 (1998) 957–966. 39. P. Ruge, Eigenvalues of damped strucutres: Vectoriteration in the original space of dof, Computational Mechanics 22 (1998) 167–173.

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109

40. Y. Saad, Numerical Methods for large Eigenvalue Problems, Algorithms and Architectures for Advanced Scientific Computing (Manchester University Press, Manchester, 1992). 41. Y. Saad, Interative Methods for Sparse Linear Systems, SIAM, 2003. 42. Y. Saad and M. H. Schultz, Gmres: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM Journal on Scientific and Statistical Computing 7(3) (1986) 856–869. 43. L. Thompson, A review of finite-element methods for time–harmonic acoustics, Journal of the Acoustical Society of America 119(3) (2006) 1315–1330. 44. F. Treyssède, G. Gabard and M. B. Tahar, A mixed finite element method for acoustic wave propagation in moving fluids based on an Eulerian-Lagrangian description, Journal of the Acoustical Society of America 113(2) (2003) 705–716. 45. F. Treyssède and M. B. Tahar, Comparison of a finite element model with a multiple-scales solution for sound propagation in varying ducts with swirling flows, Journal of the Acoustical Society of America 115(6) (2004) 2716–2730. 46. O. von Estorff, S. Petersen and D. Dreyer, Efficient infinite elements based on Jacobi polynomials, in Computational Acoustics of Noise Propagation in Fluids — Finite and Boundary Element Methods, eds. S. Marburg and B. Nolte (Springer-Verlag, Berlin, Heidelberg, 2008) 231–250. 47. P. Werner, Ein Resonanzph¨ anomen in der Theorie akustischer und elektromagnetischer Wellen, Mathematical Methods in the Applied Sciences 6 (1984) 104–128. 48. P. Werner, Zur Asymptotik der Wellengleichung und der W¨ armeleitungsgleichung in zweidimensionalen Außenr¨ aumen, Mathematical Methods in the Applied Sciences 7 (1985) 170–201. 49. J. Wolfe and J. Smith, Cutoff frequencies and cross fingerings in baroque, classical, and modern flutes, Journal of the Acoustical Society of America 114(4) (2003) 2263–2272. 50. J. Wolfe, J. Smith, J. Tann and N. H. Fletcher, Acoustic impedance spectra of classical and modern flutes, Journal of Sound and Vibration 243(1) (2001) 127–144.

an eigenvalue search algorithm for the modal

an eigenvalue search algorithm for the modal

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