Aeroelastic Vibrations and Stability of Plates and Shells 9783110338379, 9783110338362

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Sergey D. Algazin, Igor A. Kijko Aeroelastic Vibrations and Stability of Plates and Shells

De Gruyter Studies in Mathematical Physics

| Edited by Michael Efroimsky, Bethesda, Maryland, USA Leonard Gamberg, Reading, Pennsylvania, USA Dmitry Gitman, São Paulo, Brazil Alexander Lazarian, Madison, Wisconsin, USA Boris Smirnov, Moscow, Russia

Volume 25

Sergey D. Algazin, Igor A. Kijko

Aeroelastic Vibrations and Stability of Plates and Shells |

Physics and Astronomy Classification Scheme 2010 302.60.Cb, 02.60.Lj, 02.70.Hm Authors Dr. Sergey D. Algazin Russian Academy of Sciences Institute for Problems in Mechanics Prospect Vernadskogo 101 Moscow, 119526 Russia [email protected] Prof. Dr. Igor A. Kijko Matveevsky str. 10-2-321 Moscow, 119517 Russia [email protected]

ISBN 978-3-11-033836-2 e-ISBN (PDF) 978-3-11-033837-9 e-ISBN (EPUB) 978-3-11-038945-6 Set-ISBN 978-3-11-040491-3 ISSN 2194-3532 Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2015 Walter de Gruyter GmbH, Berlin/Munich/Boston Typesetting: PTP-Berlin, Protago-TEX-Production GmbH, Berlin Printing and binding: CPI books GmbH, Leck ♾ Printed on acid-free paper Printed in Germany www.degruyter.com

| This book is dedicated to the memory of a prominent scientist in mechanics and our teacher, A. A. Ilyushin.

Preface Vibrations of engineering structures, aircraft elements (wings, fins), and thin-walled structural elements occurring upon their interaction with gas flow (as a rule, air flow) are referred to as “flutter”. One has to distinguish three main types of such vibrations: the classical flutter, exemplified by vibrations of aircraft wings and fins; stall flutter, exemplified by vibrations of suspension bridges, tall stacks; and panel flutter, to which vibrations of thin-walled elements (plates, shallow shells) of aircraft or rockets at (for the most part) supersonic speeds belong. The growth of scientific interest in these phenomena was especially pronounced in the 1930s because of developments in aviation. We quote Russian test pilot M. L. Gallay: “With new high-speed aircraft becoming available, a wave of mysterious and unexplained air accidents rolled over almost all the developed countries. Casual eyewitnesses who spotted these accidents from the ground in all the cases described nearly the same picture: the aircraft was flying absolutely normally with nothing alarming noticed, and then suddenly some unknown force, as if by explosion, destroyed the aircraft – and the next moment twisted debris, wings, fins, body, are falling to the ground . . . All the eyewitnesses independently described what they saw as an explosion . . . However, investigation did not confirm this version: no traces of explosives, soot, or any burnt material were found on the debris . . . This new dangerous phenomenon was named “flutter”, but, if I remember correctly, it was Molière who said that a sick person does not get well sooner only because he knows what his illness is called in Latin”¹. This is a description of classical flutter. A dramatic example of stall flutter is the Tacoma Narrows Bridge catastrophe in the USA, in which a suspension bridge (span 854 m, width 11.9 m) collapsed in 1940; see description of this accident in the above-quoted book. A classical example of panel flutter is plate vibration in supersonic gas flow. The solution of many particular problems of this class became possible after A. A. Ilyushin discovered in 1947 the law of plane sections in high supersonic speed aerodynamics, after which the problem of panel flutter for plates (and, later, shallow shells) was formulated in a closed (by that time) form, leading to the development of effective analytical research methods. This (and other) questions are discussed in this book. When writing this book, we did not aim to encompass or generalize the extensive bibliography on panel flutter available today (more than 700 works have been published on the subject since the 1930s). The main purpose was different: within the framework of mathematical models of the phenomenon which have been developed up till now, to present analytical and efficient numerical methods by which dif-

1 Quoted from: Ya. G. Panovko and I. I. Gubanova. Stability and Vibrations of Elastic Systems. Moscow, Fizmatlit, 1964, pp. 251–252.

viii | Preface ferent classes of panel flutter problems can be solved for plates and shallow shells. For this reason, only a few particular examples are considered in the book; we give preference to new problem formulations, mathematical substantiation of the research methods developed, and clarification of new mechanical effects. Some aspects of the approach, especially mathematical, have not yet been well-developed; we have noted some such aspects within the text, while others can be noticed by the thoughtful reader. We would greatly appreciate any comments with respect either to its content, or to possible further developments. We hope that this book will be of interest to everyone involved in the analysis of dynamic stability of thin-walled structures. We wish to acknowledge the financial support by the Russian Foundation for Basic Research (Grants No. 95-01-00407, 97-01-00923, and 05-01-00250).

Contents Preface | vii Introduction | 1 Part I

Flutter of plates

1

Statement of the problem | 5

2

Determination of aerodynamic pressure | 6

3

Mathematical statement of problems | 11

4

Reduction to a problem on a disk | 14

5

Test problems | 20

6 6.1 6.2 6.3 6.4 6.5 6.6

Rectangular plate | 36 Problem statement and analytical solution | 36 Numerical–analytical solution | 38 Results | 41 Bubnov–Galerkin (B–G) method | 42 Dependence of critical flutter velocity on plate thickness | 46 Dependence of critical flutter velocity on altitude | 46

7 7.1 7.2

Flutter of a rectangular plate of variable stiffness or thickness | 48 Strip with variable cross section | 48 Rectangular plates | 52

8

Viscoelastic plates | 57

Part II Flutter of shallow shells 9

General formulation | 63

10

Determination of aerodynamic pressure | 66

11

The shallow shell as part of an airfoil | 71

12

The shallow shell of revolution | 74

x | Contents 13

The conical shell: external flow | 78

14 14.1 14.2

The conical shell: internal flow | 82 Statement of the problem | 82 Determination of dynamic pressure | 87

15

Example calculations | 91

Part III Numerical methods for non-self-adjoint eigenvalue problems 16 16.1 16.2 16.3 16.4 16.5 16.6 16.7 17

Discretization of the Laplace operator | 99 The Sturm–Liouville problem | 99 Interpolation formula for a function of two variables on a disk, and its properties | 104 Discretization of the Laplace operator | 108 Theorem of h-matrices | 109 Construction of h-matrix cells by discretization of Bessel equations | 112 Fast multiplication of h-matrices by vectors using the fast Fourier transform | 114 Symmetrization of the h-matrix | 116

17.1 17.2

Discretization of linear equations in mathematical physics with separable variables | 118 General form of equations with separable variables | 118 Further generalization | 119

18 18.1 18.2 18.3 18.4

Eigenvalues and eigenfunctions of the Laplace operator | 122 The Dirichlet problem | 123 Mixed problem | 135 The Neumann problem | 136 Numerical experiments | 140

19 19.1 19.2 19.3

Eigenvalues and eigenfunctions of a biharmonic operator | 142 Boundary-value problem of the first kind | 145 Boundary-value problem of the second kind | 145 Numerical experiments | 148

20

Eigenvalues and eigenfunctions of the Laplace operator on an arbitrary domain | 151 20.1 Eigenvalues and eigenvectors of the Laplace operator | 151 20.1.1 The Dirichlet problem | 158

Contents

20.1.2 20.1.3 20.1.4 20.2 20.3

| xi

Mixed problem | 158 The Neumann problem | 159 Description of the program LAP123C | 159 Program for conformal mapping | 164 Numerical Experiments | 166

21

Eigenvalues and eigenfunctions of a biharmonic operator on an arbitrary domain | 168 21.1 Eigenvalues and eigenfunctions of a biharmonic operator | 168 21.1.1 Boundary-value problem of the first kind | 173 21.1.2 Boundary-value problem of the second kind | 173 21.1.3 Description of the program BIG12AG | 173 21.2 Program for conformal mapping | 177 21.3 Numerical experiments | 179 22 22.1 22.2 22.3 22.4

Error estimates for eigenvalue problems | 180 Localization theorems | 180 A priori error estimate in eigenvalue problems | 183 A posteriori error estimate for eigenvalue problems | 185 Generalization for operator pencil | 185

Conclusion | 187 Bibliography | 189

Introduction In this book, vibrations of plates and shallow shells interacting with air flow are considered. As a rule, the main problem to be solved is to find out the parameter domain in which the vibrations are stable. The geometry and mechanical properties of the vibrating structural element are usually known; therefore, the question is the determination of flow velocity beyond which vibrations become unstable. The phenomenon of possible vibration instability is referred to as panel flutter, and the corresponding velocity is known as the critical flutter velocity. The panel flutter problem became of particular interest during the post-war years of the 20th century, due to rapid development of aerospace engineering. Theoretical progress was promoted by the discovery of the law of plane sections in the aerodynamics of high supersonic velocities, which enabled, generally speaking, the coupled aeroelasticity problem to be resolved by a simple formula of the “piston theory”. The first studies relying on the piston theory were performed in the 1950s by A. A. Movchan et al. They considered the flutter problem for a rectangular plate in the simplest case where the flow velocity vector lies in the plane of the plate and is parallel to one of its edges. If asymptotic stability is of interest (which has been the case in the majority of past and current flutter studies), the problem is reduced to the analysis of the dependence of spectrum of a non-self-adjoint operator of the fourth order (its main part is the biharmonic operator) on the flow velocity. Evidently, even in this simplest formulation the flutter problem is far from trivial. Nevertheless, A. A. Movchan et al. obtained results which highlighted the essential points of the problem and, therefore, for a long time were considered as reference solutions. Further development of approaches to flutter problem which followed these fundamental results did not touch upon the basic points of the theory: the forces of aerodynamic interaction of flow and vibrating element were described by the piston theory formula even in the cases where its applicability is questionable (a dramatic example is the flutter of a conical shell subjected to internal gas flow at high supersonic speed). At the same time, no attempts were made to formulate the flutter problem for a plate or shallow shell of arbitrary plan view shape; and such mathematical aspects as existence of the solution, general properties, structure of the spectrum, etc., were not touched upon. The large number of papers published in that period is attributed to the consideration of a variety of boundary condition combinations, physical effects of different nature (temperature, electromagnetic field), mechanical properties (viscoelastic, multilayered, anisotropic plates and shells), etc. The situation changed in the mid1990s. On the one hand, new statements of flutter problems for plates and shells as parts of the aircraft cover at high supersonic velocities were formulated. On the other hand, a numerical-analytical nonsaturating method was developed which enabled an efficient solution of eigenvalue problems for non-self-adjoint operators (or systems of such operators) arising in the flutter studies. Taken together, these achievements

2 | Introduction allowed the class of flutter problems to be significantly extended and new mechanical effects to be obtained. These results, belonging, for the most part, to the authors and their colleagues, comprise the content of this book. When presenting the material, as a rule we do not make reference to the original works and their authors. However, each part of the book begins with a brief introduction which indicates which works form the basis of the respective material.

| Part I: Flutter of plates

4 | Part I Flutter of plates The flutter problem for a rectangular plate was first formulated and studied to large extent by A. A. Movchan in the 1950s [495–498]. These results, today regarded as classical, became possible after A. A. Ilyushin discovered in 1947 the law of plane sections in high-speed aerodynamics [310]. One of its consequences is the local formula of the piston theory for additional pressure exerted by gas onto the vibrating plate, by which the flutter problem is reduced to an eigenvalue problem for a nonself-adjoint operator. A. A. Movchan et al. considered the problem in a rather limited particular case, where the flow velocity vector is parallel to one of the plate sides; numerous further works by other authors were devoted to a rather straightforward extension of the theory onto multilayered plates, account for the effects of different physical fields, etc. The situation changed in mid-1990s when new formulations of panel flutter problems were presented by A. A. Ilyushin and I. A. Kijko [311]. Based on these formulations, the general spectral properties of the flutter operator were obtained, a numerical-analytical method was developed which enabled the spectrum to be studied, new classes of problems were solved, and new mechanical effects were discovered (I. A. Kijko and S. A. Algazin) [33–39, 41]. These results form the basis of the material presented in this section. Also presented are some particular results on the flutter of plates of variable thickness or stiffness, as well as one special case of the optimization problem (V. I. Isaev and A. K. Kadyrov) [316, 330]. A new solution for the flutter of viscoelastic plate is given which resolves the long-existing critical flutter velocity paradox [350, 356].

1 Statement of the problem Consider a plate occupying a domain S on the x, y plane, bounded by the contour Γ (hereafter, Γ is supposed to be piecewise-smooth). One side of the plate is subjected to gas flow with the velocity vector v = {vx , vy } = {v cos θ , v sin θ } , v = |v|. If, in addition to the unperturbed state w0 ≡ 0, we consider a perturbation w = w(x, y, t), then the aerodynamic pressure Δp caused by interaction with the perturbed flow appears. It will be shown in the following analysis that Δp is a linear operator of w, which will allow us to present the solution in the form w = φ (x, y) exp(ω t), Δp = Δp0 (x, y) exp(ω t) in all cases except the flutter problem for a viscoelastic plate. The equation for vibrations of a constant-thickness plate takes the form DΔ2 w + ρ h

𝜕2 w = Δp, 𝜕t2

(1.1)

where D = Eh3 /(12(1 − 𝜈2 )) is the cylindrical rigidity, E, 𝜈, and ρ are Young’s modulus, Poisson’s ratio, and density of plate material, and h is its thickness. From the above considerations, we have Δp0 = L1 (φ ) + L2 (φ , ω ), and therefore, (1.1) can be rewritten in the form DΔ2 φ + L1 (φ ) + ρ hw2 φ + L2 (φ ; ω ) = 0. (1.2) On the contour Γ, the deflection amplitude φ (x, y) satisfies the boundary conditions x, y ∈ Γ, M1 (φ ) = 0, M2 (φ ) = 0, (1.3) where the boundary operators M1 and M2 are problem-specific and will be given in each particular case. We assume hereafter that the plate is not subjected to any loads in its median plane. The system of equations (1.2) and (1.3) represents a complicated eigenvalue problem with a non-self-adjoint operator; its eigenvalues are denoted by ω . By definition, we take that the perturbed motion of the plate is stable if Re ω < 0, and unstable if Re ω > 0; the critical parameters of the system (plate, flow) are determined by the condition Re ω = 0. In the further analysis, we consider the following main questions: determination of Δp, formulation of new problems; development of an efficient analytical approach, and identification of new mechanical effects.

2 Determination of aerodynamic pressure Numerous studies on the vibrations and stability of plates in supersonic high-speed flows are carried out on the basis of the piston theory for the aerodynamic pressure Δp caused by the interaction of the flow and vibrating plate. This formulation has become so common that it was applied even in the cases where its validity is questionable. Here, we derive Δp in the cases of “moderate” supersonic (M ∼ 1.5–2) and low subsonic velocities. Consider an elastic strip occupying domain S : {0 ≤ x ≤ l, y = 0, |z| < ∞}. On the side y ≥ 0, the strip is placed in a gas flow with unperturbed parameters (planar problem) v = {u0 , 0}, p0 , ρ0 , a0 = (𝛾p0 /ρ0 )1/2 , so that the unperturbed flow potential is φ0 = u0 x. Small vibrations of the strip w(x, t) (with w/ℓ ≪ 1) cause flow perturbation; the perturbed flow potential is denoted by φ1 = φ0 + φ . Then we proceed in the usual way: from the Cauchy–Lagrange integral, equations of motion, mass conservation, and equation of state we obtain an equation for φ1 and linearize it with respect to the perturbation φ to obtain 𝜕2 φ 1 𝜕2 φ M 𝜕2 φ 𝜕2 φ 2 (M + − 1) + 2 = 0, − a0 𝜕x𝜕t 𝜕y2 𝜕x2 a20 𝜕t2

(2.1)

where M = u0 /a0 . The potential φ must vanish at infinity and satisfy the impermeability condition on the line y = 0: 𝜕φ 𝜕w 𝜕w = + u0 𝜕y 𝜕t 𝜕x 𝜕φ = 0. x ≥ l, 𝜕y

y = 0,

0 ≤ x ≤ l,

(2.2)

y = 0,

x ≤ 0,

(2.3)

The overpressure in the flow is obtained from p = −ρ0 (

𝜕φ 𝜕φ ). + u0 𝜕t 𝜕x

(2.4)

We search for the solution in the class of functions φ (x, y, t) = f (x, y) exp(ω t), w(x, t) = W(x) exp(ω t), and p(x, y, t) = q(x, y) exp(ω t). Introduce now the nondimensional coordinates x/l and y/l, retaining hereafter the previous notation. Also, introduce the nondimensional frequency lω /a0 = Ω. The system of equations (2.1)– (2.4) is transformed to 𝜕φ 𝜕2 f 𝜕2 f + 2MΩ + Ω2 f − 2 = 0 2 𝜕x 𝜕x 𝜕y 𝜕W 𝜕f ) = a0 (ΩW + M y = 0, 0 ≤ x ≤ 1, 𝜕y 𝜕x 𝜕φ =0 y = 0, x ≤ 0, x ≥ 1, 𝜕y ρ a 𝜕f q = − 0 0 (Ωf + M ) . l 𝜕x (M 2 − 1)

(2.5) (2.6) (2.7) (2.8)

2 Determination of aerodynamic pressure

| 7

In what follows, it is necessary to distinguish the cases of M < 1 and M > 1; we consider them one by one. For M > 1, perturbations are absent to the left of the point x = 0; therefore, it is possible to apply the Laplace transform along the x coordinate; condition (2.7) is not relevant, and the function W(x) can be prolonged to x ≥ 1 arbitrarily (as long as applicability conditions for the Laplace transform are satisfied), and this will not affect the overpressure q(x, 0) acting on the strip. From (2.5) we obtain for the Laplace ̃ y) transform f (s, β 2f ̃ −

𝜕2 f ̃ = 0, 𝜕y2

A solution bounded at infinity is

β 2 = (M 2 − 1) s2 + 2MΩs + Ω2 . f ̃ = c1 e−β y .

(2.9)

From the boundary condition (2.8) for Laplace transform 󵄨 𝜕f ̃ 󵄨󵄨󵄨 󵄨󵄨 = −β c1 = a0 (Ω + Ms)W̃ 𝜕y 󵄨󵄨󵄨y=0 it is possible to determine the parameter c1 , and therefore it follows from (2.9) that f ̃ = −a0

Ω + Ms ̃ −β y We . β

(2.10)

The overpressure (in terms of Laplace transforms) is now obtained from equation (2.8): ̃ 0) = Δp(s) ̃ = q(s,

ρ0 a20 (Ω + Ms)2 ̃ W(s). l β

(2.11)

The inverse Laplace transform is found from tables and convolution theorem. We first write β = √M 2 − 1√(s + s1 )(s + s2 ) ≡ √M 2 − 1β0 ; s1 = Ω/(M −1), s2 = Ω/(M + 1); (s1 + s2 )/2 = MΩ/ (M 2 − 1) ≡ α1 ; (s1 − s2 )/2 = Ω/ (M 2 − 1) ≡ α2 . We now have 1 L(−1) ( ) = I0 (α2 x)e−α1 x ≡ H(x), β0 where I0 (z) is the modified Bessel function; therefore x

(−1)

L

W̃ ( ) = ∫ H(x − τ )W(τ ) dτ β0 0 x

L(−1) (

𝜕W sW̃ ) = ∫ H(x − τ ) dτ β0 𝜕τ 0

x

L(−1) (

𝜕 𝜕W s2 W̃ )= ∫ H(x − τ ) dτ . β0 𝜕x 𝜕τ 0

8 | 2 Determination of aerodynamic pressure We perform the necessary calculations and substitute the results in equation (2.11) to obtain finally Δp(x) =

ρ0 a20 M l (M 2 − 1) +

−

+

1/2

2 [ M − 2 ΩW + M 𝜕W 𝜕x M2 − 1 [ x

(M 2 + 2) Ω2 2M (M 2 − 1)

(2.12)

−α1 (x−τ )

(M 2 − 1)

2 (M 2

0

x

2Ω2

MΩ

2

∫ e−α1 (x−τ ) I0 (α2 (x − τ )) W(τ ) dτ

2

∫e

I1 (α2 (x − τ )) W(τ ) dτ

0 x

2

∫ e−α1 (x−τ ) I2 (α2 (x − τ )) W(τ ) dτ ] , − 1) 0 ] 2

where I𝜈 (z), 𝜈 = 1, 2, are the modified Bessel functions. There are important implications of equation (2.12): 1. The formula of the piston theory is obtained in the limit M ≫ 1, and it is valid only 󵄨 󵄨 for the calculation of a few first eigenvalues Ωn such that |Ωn |/M 2 ∼ 1 (or 󵄨󵄨󵄨α2 󵄨󵄨󵄨 ∼ 1), because I𝜈 (z) grow exponentially with increasing argument. This important point has not been taken into account so far. 2. If |z| < 1, then I𝜈 (z) ∼ (z/2)𝜈 , therefore for “moderately” supersonic velocities M 2 > 2 the first few eigenvalues Ωn can be calculated with the last two integral terms in equation (2.12) omitted and, also, with Δp(x) taken in the form Δp(x) ≅

ρ0 a20 M [ M 2 − 2 𝜕W ΩW + M 2 −1 2 𝜕x √ M ℓ M −1 [ +

(M 2 + 2) Ω2 2M(M 2 − 1)2

x

(2.13)

∫ e−α1 (x−τ ) W(τ ) dτ ] 0 ]

Assume now that the plate occupies a domain S on plane x, y with a boundary Γ, and it is subjected to a gas flow with velocity v = v0 n0 = {v0 cos θ , v0 sin θ }. We assume that the overpressure Δp(x, y) can be expressed by a formula which generalizes equation (2.13) (and, accordingly, (2.12)): Δp(x, y) =

2 ρ0 a20 M [ M − 2 ΩW + Mn0 grad W 2 l√M 2 − 1 M − 1 [

+

(M 2 + 2) Ω2 2M (M 2 − 1)

2

−α1 (s−τ )

∫e AB

W (x(τ ), y(τ )) dτ ] , ]

(2.14)

2 Determination of aerodynamic pressure

| 9

y Γ

S

n0

B(x, y)

A(x0, y0) 0

x

Fig. 2.1. On the calculation of integral for Δp in equation (2.14)

where s = [(x −x0 )2 + (y−y0 )2 ]1/2 , parameter τ varies from zero to s along the line AB, l is the characteristic size of domain S. Integration along AB is performed only for x, y ∈ S; any straight line AB intersects Γ only at two points (see Figure 2). We now nondimensionalize Δp from equation (2.14) and substitute it into (1.1) to arrive at an equation in the form (1.2) (W ↔ φ ), in which the operator L2 denotes the integral term. This problem has not been studied thus far in this complex formulation. The following results will be obtained for flow velocities such that (M 2 − 2)/ (M 2 − 1) ≅ 1, and the integral term can be neglected. For subsonic flows (M < 1) it is convenient to solve the system (2.5)–(2.7) by exponential Fourier transform; we have for f ̃(s, y) f ̃(s, y) = ce−β1 (s)y ,

β12 = (1 − M 2 ) s2 + 2isMΩ + Ω2 .

(2.15)

The parameter c is determined from equation (2.15) taking into account equation (2.7): c1 = −

a0 ̃ Φ(s), β1 (s)

1

𝜕W(τ ) ̃ ) e−isτ dτ . Φ(s) = ∫ (ΩW(τ ) + M 𝜕t

(2.16)

0

The function f (x, y) is restored by ∞

̃ a Φ(s) f (x, y) = − 0 ∫ e−β1 (s)y+ixs ds, 2π β1 (s)

(2.17)

−∞

and the overpressure on the plate is found from (2.8): ∞

Δp =

̃ ρ0 a20 Φ(s) 𝜕 (Ω + M ) ∫ eixs ds. 2π ℓ 𝜕x β1 (s)

(2.18)

−∞

Evidently, the main difficulty consists in calculating the integral in equation (2.18); this cannot be done in the general case. Therefore, in order to obtain some estimates

10 | 2 Determination of aerodynamic pressure we consider only the critical state Ω = iΩ0 for low subsonic velocities M 2 ≪ 1 . In this case, β12 ≅ s2 − Ω20 , and for the integral in (2.18) we obtain ∞

∞

̃ Φ(s)

J(x) = ∫

√s2

−∞

−

Ω20

eixs ds = ∫ −∞

eixs ds √s2

−

1

∫ (iΩ0 W + M

Ω20 0

𝜕W ) e−isτ dτ . 𝜕τ

We now change the integration order: ∞

1

J(x) = ∫ (iΩ0 W + M 0

𝜕W eis(x−τ ) ) dτ ∫ ds. 𝜕τ 2 − Ω2 √s −∞ 0

The internal integral is straightforward to calculate; therefore, we obtain finally x

J(x) =π ∫ (iΩ0 W + M 0 0

𝜕W ) [−N0 (Ω0 (x − τ )) + H0 (Ω1 (x − τ ))] dτ 𝜕τ (2.19)

𝜕W ) [N0 (Ω0 (τ − x)) + H0 (Ω0 (τ − x))] dτ , − π ∫ (iΩ0 W + M 𝜕τ x

where N0 (z) is the cylindrical function of the second kind and H0 (z) is the Struve function. Further transformations (integration by parts, differentiation of the integrals with respect to variable limit or parameter) in equatioins (2.19) and (2.18) after substituting J(x) into them are impractical, because N0 (z) has logarithmic singularity at zero point. The obtained result cannot be formally generalized to the flutter of a plate of arbitrary planform; the only problem which, in our opinion, is worth of studying approximately is the flutter of a rectangular plane elongated across the flow. Note in conclusion that there are no solutions of flutter problems based on equations (2.18) and (2.19) for aerodynamic pressure known to the authors.

3 Mathematical statement of problems It was shown in Chapter 2 that for relatively high supersonic velocities the aerodynamic pressure Δp is described by the piston theory formula. Therefore, the plate vibration equation takes the form DΔ2 w +

𝛾p0 0 𝛾p 𝜕w 𝜕2 w vn grad w + 0 + gh 2 = 0. a0 a0 𝜕t 𝜕t

If we take w = ψ (x, y) exp(ω t), then DΔ2 ψ +

𝛾p0 0 𝛾p vn grad ψ + ( 0 ω + ghω 2 ) ψ = 0. a0 a0

Instead of frequency ω , it is convenient to introduce the eigenvalue λ defined by (𝛾p0 /a0 )ω + ghω 2 + λ = 0; the vibration equation then takes the form 𝛾p0 0 vn grad ψ = λψ . a0

DΔ2 ψ +

(3.1)

Together with the condition on the contour x , y ∈ Γ,

M1 (φ ) = 0,

M2 (φ ) = 0,

(3.2)

the system (3.1) and (3.2) represents an eigenvalue problem. Since the domain S is arbitrary, it is convenient to rotate the coordinate system, directing the Ox axis along the velocity vector; we then obtain from (3.1) DΔ2 φ + 𝛾p0

vx 𝜕φ = Δφ . a0 𝜕x

(3.3)

Denote by l the characteristic size of domain S and introduce nondimensional (primed) coordinates and parameters: x = x󸀠 l,

y = y󸀠 l,

ρ = ρ 󸀠 p0 /a20 ,

h = h󸀠 l,

vx = vx a0 ,

E = E󸀠 p0 , φ = φ 󸀠 l,

ω = ω 󸀠 a0 /l λ = λ 󸀠 p0 /l.

In the new variables (we now drop the primes), equation (3.3) takes the form DΔ2 φ + 𝛾 vx

𝜕φ = λφ , 𝜕x

ρ hω 2 + 𝛾ω + λ = 0.

(3.4)

Hereafter, we consider two types of boundary conditions: rigidly fixed (clamped): x, y ∈ Γ,

φ = 0,

𝜕φ = 0, 𝜕n

(3.5)

1 − 𝜈 𝜕φ = 0. R0 𝜕n

(3.6)

and simply supported: x, y ∈ Γ,

φ =0

Δφ −

12 | 3 Mathematical statement of problems Here, R0 is the curvature radius of the contour, and n is the outward pointing normal to it. Before formulating the flutter problems, we prove some statements on the properties of the problem just obtained. First of all, we show that in each of the problems (3.4), (3.5) or (3.4), (3.6) a condition Re λ > 0 holds true. Let φ = φ1 + iφ2 be the solution; multiply both sides of the first equation from (3.4) by φ̄ = φ1 − iφ2 and integrate over the domain S to obtain D ∬ φ̄ Δ2 φ ds + 𝛾0 ∬ S

S

𝜕φ φ̄ ds = λ ∬ φ φ̄ ds, 𝜕x

𝛾0 = 𝛾vx .

(3.7)

S

Apply now Green’s formula: ∬ (φ̄ Δ2 φ − Δφ ⋅ Δφ̄ ) ds = ∫ ( Γ

S

𝜕φ̄ 𝜕Δφ ) d𝛾, φ̄ − Δφ 𝜕n 𝜕n

(3.8)

and consider separately the boundary conditions (3.5) and (3.6). For the conditions (3.5), the right-hand side of equation (3.8) vanishes, and relation (3.7) takes the form 𝜕φ 󵄨 󵄨2 󵄨 󵄨2 λ ∬ 󵄨󵄨󵄨φ 󵄨󵄨󵄨 ds = D ∬ 󵄨󵄨󵄨Δφ 󵄨󵄨󵄨 ds + 𝛾0 ∬ φ̄ ds. 𝜕x S

S

(3.9)

S

We then have

𝜕φ 𝜕φ 𝜕φ 1 𝜕 󵄨󵄨 󵄨󵄨2 φ̄ = 󵄨󵄨φ 󵄨󵄨 + i (φ1 2 − φ2 1 ) . 𝜕x 2 𝜕x 𝜕x 𝜕x Substituting this relation into equation (3.7), we obtain 𝜕φ 𝜕φ 𝜕 󵄨󵄨 󵄨󵄨2 1 󵄨 󵄨2 󵄨 󵄨2 λ ∬ 󵄨󵄨󵄨φ 󵄨󵄨󵄨 ds = D ∬ 󵄨󵄨󵄨Δφ 󵄨󵄨󵄨 ds + 𝛾0 ∬ 󵄨φ 󵄨 ds + i𝛾0 ∬ (φ1 2 − φ2 1 ) ds 2 𝜕x 󵄨 󵄨 𝜕x 𝜕x S

S

S

S

Due to the boundary condition (φ = 0 on Γ), the second integral on the right-hand side vanishes, and therefore 𝜕φ 𝜕φ 󵄨 󵄨2 󵄨 󵄨2 λ ∬ 󵄨󵄨󵄨φ 󵄨󵄨󵄨 ds = D ∬ 󵄨󵄨󵄨Δφ 󵄨󵄨󵄨 ds + i𝛾0 ∬ (φ1 2 − φ2 1 ) ds, 𝜕x 𝜕x S

S

S

from which follows that Re λ > 0. Note that in the particular case where S is a rectangle, and the velocity vector is parallel to one of its sides, this property of eigenvalues was obtained previously by A. A. Movchan. Consider now the case of simply supported boundary conditions. Note first of all that if the contour is a polygon, then on each of its sides R−1 0 = 0, and from equation (3.6) follows Δφ = 0 on Γ. In this case the right-hand side of Green’s formula (3.8) turns to zero, and the proof is similar to the one given above. In the general case, we have from Green’s formula (3.8) under conditions (3.6) that 1 󵄨󵄨󵄨󵄨 𝜕φ 󵄨󵄨󵄨󵄨2 󵄨 󵄨2 ∬ φ̄ Δ2 φ ds = ∬ 󵄨󵄨󵄨Δφ 󵄨󵄨󵄨 ds − (1 − 𝜈) ∫ 󵄨 d𝛾. 󵄨 R0 󵄨󵄨󵄨 𝜕n 󵄨󵄨󵄨 S

S

Γ

3 Mathematical statement of problems

|

13

After the well-known transformation of the right-hand side of this relation, we obtain ∬ φ̄ Δ2 φ ds = L(φ1 ) + L(φ2 ) S 2

L(φ ) = ∬ [( S

2

𝜕2 φ 𝜕2 φ 𝜕2 φ 𝜕2 φ ) + 2𝜈 + 2(1 − 𝜈) ( ) ] ds. 𝜕x𝜕y 𝜕x2 𝜕x2 𝜕y2

Substituting the result into equation (3.7), we obtain 𝜕φ 𝜕φ 󵄨 󵄨2 λ ∬ 󵄨󵄨󵄨φ 󵄨󵄨󵄨 ds = D(L(φ1 ) + L(φ2 )) + i𝛾0 ∬ (φ1 2 − φ2 1 ) ds, 𝜕x 𝜕x S

S

from which follows

󵄨 󵄨2 Re λ ∬ 󵄨󵄨󵄨φ 󵄨󵄨󵄨 ds = D (L(φ1 ) + L(φ2 )) .

(3.10)

S

Now use the inequality 2

L(u) ≥ (1 − 𝜈) ∬ ( S

2

𝜕2 u 𝜕2 u 𝜕2 u 𝜕2 u ) ds ≡ L1 (u) > 0, ) ( + 2 + 𝜕x2 𝜕x2 𝜕2 y 𝜕2 y

from which, on the basis of (3.10), follows that 󵄨 󵄨2 Re λ ∬ 󵄨󵄨󵄨φ 󵄨󵄨󵄨 ds ≥ D (L1 (φ1 ) + L1 (φ2 )) > 0, S

which completes the proof. The second statement concerns the behavior of eigenvalues with increasing flow velocity; we stress that it is of a purely mathematical nature and is not related to the conditions under which equation (3.1) was derived. For zero flow velocity, the eigenvalues of the formulated problem are the squares of the eigen frequencies for plate vibration (discrete spectrum λk ). As the flow velocity is increasing, λk are converging and then merge (some of λk ), forming a complex conjugate pair afterwards. The critical system parameters are determined by the condition Re ω = 0; let λk = Re λk ± i Im λk : then we obtain from the second equation (3.4) (with ω = iωk ): −hωk2 + i𝛾ωk + Re λk ± i Im λk = 0, and, therefore, Re λk = h(Im λk )2 /𝛾2 . On the complex plane, λ forms a wellknown curve, the stability parabola. If, therefore, the system parameters are such that all λk lie within the stability parabola, then the plate vibrations are stable; if at least one λk moves beyond the parabola, vibrations are unstable. From this discussion, two principal problems of panel flutter of plates follow: 1. All system parameters, except the flow velocity, are given; it is required to determine the critical flow velocity. The overwhelming majority of flutter studies have been carried out in this formulation. 2. The shape and material of a plate are known, together with the flow parameters; it is required to determine the relative plate thickness h/l, which provides its strength according to the vibration stability criterion.

4 Reduction to a problem on a disk We now rewrite the problem formulation; for the sake of convenience, vector n0 is pointing in the opposite direction: DΔ2 φ − vn0 grad φ = λφ

(4.1)

For a clamped contour, we have x, y ∈ Γ,

𝜕φ = 0, 𝜕n

(4.2)

𝜕2 φ 𝜈 𝜕φ + = 0. R0 𝜕n 𝜕n2

(4.3)

φ = 0,

while for a simply supported edge x, y ∈ Γ, φ = 0,

Here, R0 is the contour curvature, and n is the outward pointing normal to the contour. Substitute now the Cartesian coordinates x, y by the curvilinear coordinates r, θ defined by x = U(r, θ ), y = V(r, θ ). If the Cauchy–Riemann conditions 𝜕U 1 𝜕V = , 𝜕r r 𝜕θ

𝜕V 1 𝜕U =− 𝜕r r 𝜕θ

are satisfied, the coordinate system r, θ is orthogonal. We now choose the functions U(r, θ ) and V(r, θ ) in such a way that the function ψ (ζ ) = U(r, θ ) + iV(r, θ ),

ζ = r exp(iθ )

󵄨 󵄨 defines the conformal mapping of the disk 󵄨󵄨󵄨ζ 󵄨󵄨󵄨 = r ≤ 1 onto the domain S. Then, in the coordinates r, θ , equation (4.1) takes the form 𝜕φ 𝜕φ 1 󵄨 󵄨−2 DΔ (󵄨󵄨󵄨󵄨ψ 󸀠 (ζ )󵄨󵄨󵄨󵄨 Δφ ) − 𝛾 ((vx Ur + vy Vr ) + (vy Ur − Vr vx ) ) = λ |ψ 󸀠 (ζ )|2 φ 𝜕r r 𝜕θ ψ 󸀠 (ζ )ζ ψ 󸀠 (ζ )ζ ) , Vr = Im ( ). Ur = Re ( r r

(4.4)

The boundary condition (4.2)–(4.3) takes the form φ |r=1 = 0

𝜕φ 󵄨󵄨󵄨󵄨 󵄨 =0 𝜕r 󵄨󵄨󵄨r=1 󵄨 𝜕2 φ 𝜕φ ψ 󸀠󸀠 (ζ ) 𝜕φ 󵄨󵄨󵄨 󵄨󵄨 = 0. + 𝜈 + (𝜈 − 1) Re (ζ ) 𝜕r ψ 󸀠 (ζ ) 𝜕r 󵄨󵄨󵄨r=1 𝜕r2 Relations (4.4)–(4.7) form the statement of eigenvalue problem. Denote now f (r, θ ) = 𝛾 ((vx Ur + vy Vr )

𝜕φ 𝜕φ 1 ) + λ |ψ 󸀠 (ζ )|2 φ , + (vy Ur − Vr vx ) 𝜕r r 𝜕θ

(4.5) (4.6) (4.7)

4 Reduction to a problem on a disk

|

15

and consider, instead of the differential equation (4.4), an integro-differential equation 2π

󵄨 󵄨2 󵄨 󵄨2 DΔφ = 󵄨󵄨󵄨󵄨ψ 󸀠 (ξ )󵄨󵄨󵄨󵄨 ∫ K(ξ , ζ )f (ζ ) dζ + 󵄨󵄨󵄨󵄨ψ 󸀠 (ξ )󵄨󵄨󵄨󵄨 ∫ K0 (ξ , θ )w(eiθ ) dθ . 0 |ζ |≤1 Here, K(ξ , ζ ) is Green’s function of the Laplace operator on a disk with boundary condition (4.5): K0 (ξ , θ ) =

1 − r2 1 , 2 2π 1 + r − 2r cos(θ − ε )

ξ = reiε

w(eiθ ) = |ψ 󸀠 (ζ )|−2 Δφ (ζ )|ζ =eiθ .

Denote R(ξ ) = |ψ 󸀠 (ξ )|2 ∫ K(ξ , ζ )f (ζ ) dζ |ζ |≤1 2π 󸀠

2

S(ξ ) = |ψ (ξ )| ∫ K0 (ξ , θ )w(eiθ ) dθ , 0

then DΔφ = R(ξ ) + S(ξ ) Inverting once again the Laplace operator, we obtain 2π

1 1 φ (ξ ) = ∫ K(ξ , q)[R(q) + S(q)] dq + ∫ K0 (ξ , θ )w(eiθ ) dθ . D D |q|≤1

(4.8)

0

Note that the rightmost integral vanishes due to the boundary condition (4.5). We now have to determine in relation (4.8) an unknown function w(eiθ ) from one of the boundary conditions (4.6) or (4.7). We apply the trigonometric interpolation to function w(eiθ ): w(eiθ ) =

2 2n ∑ D (θ − θj )wj + ρn (θ ; w), N j=0 n

n

Dn (θ ) = 0.5 + ∑ cos kθ , k=1

where ρn (θ ; w) is the interpolation error. The functions S(q) and R(q) are interpolated by the formula (see Part III, Chapter 16, Section 16.2): ∫ K(ξ , q)S(q) dq = − ∑ H𝜈l (ξ )S𝜈l + RM (ξ ; S), 𝜈,l

|q|≤1

where H𝜈l (ξ ) are defined in Part III, Chapter 16, Section 16.3, RM (ξ ; S) = ∫ K(ξ , q)ρM (q, S) dq, |q|≤1

16 | 4 Reduction to a problem on a disk where ρM (q, S) is the the interpolation error: 2π

S𝜈l =z𝜈l ∫ K0 (ζ𝜈l , θ )w(eiθ ) dθ 0 2π

=

2z𝜈l 2n ∑ ( ∫ K0 (ζ𝜈l , θ )Dn (θ − θj ) dθ ) wj + Rn (ζ𝜈l ; w) N j=0 0

2π

󵄨 󵄨2 z𝜈l = 󵄨󵄨󵄨󵄨ψ 󸀠 (ζ𝜈l )󵄨󵄨󵄨󵄨 .

Rn (ζ𝜈l ; w) = ∫ K0 (ζ𝜈l , θ )ρn (θ ; w) dθ , 0

Denoting additionally 2π

Hj0 (ζ𝜈l )

2 = ∫ K0 (ζ𝜈l , θ )Dn (θ − θj ) dθ . N 0

We write down the integrals in (4.8): 2n

∫ K(ξ , q)S(q) dq = − ∑ H𝜈l (ξ )z𝜈l ∑ Hj01 (ζ𝜈l )wj1 𝜈,l

|q|≤1

j1 =0

(4.9)

− ∑ H𝜈l (ξ )Rn (ζ𝜈l ; w) + RM (ξ , S) 𝜈,l

∫ K(ξ , q)R(q) dq = − ∑ H𝜈l (ξ )R𝜈l + ∫ K(ξ , q)RM (q; R) dq 𝜈,l

|q|≤1

|q|≤1

(4.10)

R𝜈l = z𝜈l ∫ K(ξ𝜈l , ζ )f (ζ ) dζ . |ζ |≤1

Now apply to function f (ζ ) the same interpolating formula and substitute it into equation (4.10): ∫ K(ξ , ζ )f (ζ ) dζ = − ∑ Hj (ζ )fj + ∫ K(ξ , ζ )RM (ζ ; f ) dζ . j

|q|≤1

|q|≤1

We then obtain Ri = −zi ∑ Hj (ζi )fj + zi ∫ K(ξi , ζ )RM (ζ ; f ) dζ . j

(4.11)

|ζ |≤1

Here, a single index i is used to number the interpolation nodes instead of two indices 𝜈, l: the interpolation nodes are numbered counterclockwise, starting from the first circle; Hij = Hj (ζi ) is the Dirichlet problem matrix for the Laplace operator on a disk (see Part III, Chapter 16, Section 16.3).

4 Reduction to a problem on a disk

|

17

Substitute now equation (4.11) into (4.10) to obtain ∫ K(ξ , q)R(q) dq = ∑ Hi (ξ )zi ∑ Hij fj i

|q|≤1

j

(4.12) − ∑ Hi (ξ )zi ∫ K(ξi , ζ )RM (ζ ; f ) dζ + ∫ K(ξ , q)RM (q; R) dq. i

|q|≤1

|ζ |≤1

From (4.8), with reference to equations (4.12) and (4.9), we obtain φ (ξ ) =

1 ∑ H (ξ )zi ∑ Hij fj D i i j −

2n 1 ∑ Hi (ξ )zi ∑ Hj01 (ζi )wj1 + Rn,M (ξ ; f , R, S) D i j =0 1

Rn,M (ξ ; f , R, S) =

1 D

{ { − ∑ H (ξ )zi ∫ K(ξi , ζ )RM (ζ ; f )dζ { { j i |ζ |≤1 {

(4.13)

} } + ∫ K(ξ , q)RM (q, R)dq} } |q|≤1 } 1 − {∑ Hi (ξ )Rn (ξi ; w) + RM (ξ ; S)} . D i We have to define w = (w0 , w1 , . . . , w2n ) in equation (4.13) in such a way that the boundary condition (4.6) or (4.7) are satisfied. Denote by L the differential operator on the left-hand side of the boundary condition. Then, applying this operator to (4.13), we obtain 2n

∑ L(Hi (ξ ))zi Hij fj − ∑ L (Hi (ξ )) zi ∑ Hj01 (ζi )wj1 + LRn,M (ξ ; f , R, S) = Lφ (ξ ). i,j

i

j1 =0

Introduce now a new notation: 󵄨 L (Hi (ξ ))󵄨󵄨󵄨ξ =eiθj2 = H̄ i,j2 ,

j2 = 0, 1, . . . , 2n,

∑ H̄ i,j2 zi Hij fj = Rj2 i,j

∑ H̄ i,j2 zi Hj01 (ζi ) = B̄ j2 ,j1 i

󵄨 δj2 = LRn,M (ξ ; f , R, S)󵄨󵄨󵄨ξ =eiθj2 . For the vector w = (w0 , w1 , . . . , w2n )T we then have a system of linear equations: 2n

∑ B̄ j2 ,j1 wj1 = Rj2 + δj2 . j1 =0

18 | 4 Reduction to a problem on a disk Therefore, 2n

wj1 = ∑ Cj1 ,j2 (Rj2 + δj2 ),

C = B̄ −1 .

j2 =0

Substituting into equation (4.13), we obtain φ (ξ ) =

2n 1 1 ∑ Hi (ξ )zi Hij fj − ∑ Hi (ξ )zi ∑ Hj01 (ζi ) D i,j D i j =0 1

2n

(4.14)

× ∑ Cj1 ,j2 (Rj2 + δj2 ) + Rn,M (ξ ; f , R, S). j2 =0

Here, fj = λ zj φj + Φj Φj = 𝛾 ((vx Ur + vy Vr )

𝜕φ 𝜕φ 󵄨󵄨󵄨󵄨 1 + (vy Ur − Vr vx ) )󵄨 , 𝜕r r 𝜕θ 󵄨󵄨󵄨ζ =ζj

j = 1, . . . , M.

Let ξ run through all interpolation nodes ξi , i = 1, 2, . . . , M, then φi =

λ 1 ∑ (B2ij − ∑ Bil Elj ) φj + ∑ (∑ Bil Hlj − ∑ Bil Elj∗ ) Φj + R̄ i D j D j l l l 2n

2n

j1 =0

j2 =0

2n

2n

Elj = ∑ Hj01 (ζl ) ∑ Cj1 ,j2 ∑ H̄ i,j2 zi Bij Elj∗

= ∑

Hj01 (ζl )

j1 =0

i

(4.15)

∑ Cj1 ,j2 ∑ H̄ i,j2 zi Hij j2 =0

i

2n 2n 1 λ R̄ i = Rn,M (ξi ; f , R, S) + ( − ) ∑ Bil ∑ Hj01 (ξl ) ∑ Cj1 ,j2 δj2 . D D l j =0 j =0 1

2

Denote G = B2 − BE; then equation (4.15) takes the form φi −

1 λ ∑ G z−1 Φj = ∑ Gij φj + R̄ i , D j ij j D j

i = 1, 2, . . . , M.

(4.16)

Denote by D(r) and D(θ ) the matrices of differentiation with respect to r and θ obtained by the differentiation of the interpolation formula (r) ) (θ ) ) + bj (∑ D(θ ), Φj = aj (∑ D(r) jl φl + δj jl φl + δj l

l

󵄨 aj = k(vx Ur + vy Vr )󵄨󵄨󵄨󵄨ζ =ζ , j

bj =

k (v U − vx Vr )|ζ =ζj , r y r

while a and b denote the corresponding diagonal matrices.

j = 1, 2, . . . , M

4 Reduction to a problem on a disk

|

19

Represent equation (4.16) in the matrix form λ 1 −1 GZ (aD(r) + bD(θ ) ) φ = Gφ + R̄ + δ D D 1 δ = GZ −1 (aδ (r) + bδ (θ ) ) . D φ−

Denote A=I−

(4.17)

1 −1 GZ (aD(r) + bD(θ ) ) . D

Inverting in equation (4.17) the matrix A, we finally obtain φ=

λ −1 A Gφ + R∗ , D

R∗ = A−1 (R̄ + δ ).

(4.18)

Neglecting the discretization error R∗ , we obtain an approximate eigenvalue problem, which will be studied below.

5 Test problems In this section, we first consider relatively simple flutter problems for an infinite plate and a strip; these results are mainly of methodological value, though the new mechanical effects obtained are also observed in the general case. After that, flutter of circular and nearly-circular plates, as well as of an elliptic plate is studied. 1. Infinite plate. The equation for plate vibrations is DΔ2 φ + 𝛾v (

𝜕φ 𝜕φ cos θ + sin θ ) = λφ . 𝜕x 𝜕y

(5.1)

The boundary conditions mean that the solution is bounded at infinity. The perturbed motion limited everywhere at the initial instant is chosen in the form φ = A exp(iax + iβ y), where α and β are real-value parameters. Substituting these relations into equation (5.1), we obtain D(α 2 + β 2 )2 + i𝛾v(α cos θ + β sin θ ) = λ = λ1 + iλ2 , from which the stability parabola equation D(α 2 + β 2 )2 = hv2 (α cos θ + β sin θ )2 follows. Therefore, D(α 2 + β 2 )2 v2 = ≡ v02 . (5.2) h(α cos θ + β sin θ )2 Since α and β are arbitrary numbers, we conclude from (5.2) that, whatever the nonzero flow velocity is, one can find out such values of α and β that the inequality v > v0 is satisfied, i.e. that the corresponding eigenvalue is located beyond the stability parabola. Therefore, the perturbed motion is unstable for any velocity v ≠ 0, which means that the critical flutter velocity is equal to zero. 2. Infinitely long strip. Consider an elastic strip occupying on anxy-plane a domain Γ : {|x| < ∞, 0 ≤ y ≤ l}; consider also the case of longitudinal flow V = {vx , 0}; the strip sides are assumed to be simply supported. The system of equations takes the (nondimensional) form

y = 0,

𝜕φ = λφ , ghω 2 + 𝛾ω + λ = 0 𝜕x 𝜕2 φ 𝜕2 φ = 0; y = 1, φ = 0, = 0. 2 𝜕y 𝜕y2

DΔ2 φ + 𝛾vx

(5.3)

φ = 0,

(5.4)

We take the following form for for the solution bounded at infinity: φ = A0 exp(ky− iα x) with real-valued α ; substitute it into equation (5.3) to obtain the characteristic equation D(k2 − α 2 )2 = λ + i𝛾α vx

5 Test problems

| 21

with the roots k12 = α 2 + √(λ + i𝛾α vx )/D;

k22 = α 2 − √(λ + i𝛾α vx )/D.

A general solution to equation (5.3) takes the form φ = (C1 sh k1 y + C2 ch k1 y + C3 sh k2 y + C4 ch k2 y) exp(−iα x). From the first two conditions (5.4) we obtain C2 + C4 = 0 and k12 C2 + k22 C4 = 0, and therefore C2 = C4 = 0, because the alternative case, k12 − k22 = √(λ + i𝛾α vx )/D = 0 and λ = −i𝛾α vx , is impossible, due to the property Re λ > 0 established above. The remaining conditions (5.4) result in the system C1 sh k1 + C3 sh k2 = 0,

C1 k12 sh k1 + C3 k22 sh k2 = 0

with the determinant δ = (k12 − k22 ) sh k1 sh k2 . The nontriviality condition for δ = 0 entails k1 = inπ (or k2 = inπ ), and we finally obtain 2

λn = (α 2 + n2 π 2 ) D − i𝛾α vx

(5.5)

φn = sin(nπ y) exp(−iα x). Since α is arbitrary, the spectrum (5.5) is not discrete, although to each α = α0 there corresponds a certain sequence 2

λn (α0 ) = D (α02 + n2 π 2 ) − i𝛾α0 vx . The critical flutter velocity is obtained in the following way. For stable vibrations of the strip, all λ are contained within the stability parabola Re λ ⋅ 𝛾2 = ρ h(Im λ )2 ; therefore, from (5.5) we obtain the inequality vx < vx(n) (α ) = (

D 1/2 α 2 + n2 π 2 ) . ρh α

(5.6)

For each n, the curves vx(n) (α ) have a minimum vx(n)min = 2nπ (D/ρ h)1/2 at α = nπ ; the lowest of all values is reached for n = 1, and we take the corresponding velocity vx(1)min as the critical velocity vx,cr = 2π (

C0 h π D 1/2 ) = , ρh √3(1 − 𝜈2 ) a0 l

where C0 = √E/ρ is the speed of sound for longitudinal waves in long, thin rods of the strip material. Therefore, the most “dangerous” perturbations, from the possible vibration instability point of view, will be perturbations of the form φ = sin π y exp(−iπ x). In the case V = {vx , vy }, an approximate solution of the problem can be obtained by different methods; for example, by reduction to the Volterra integral equation, or

22 | 5 Test problems by the Bubnov–Galerkin method. We make use of the latter, because the qualitative results (which are of primary interest to us at the moment) will be the same; the accuracy of the Bubnov–Galerkin method will be discussed later for the rectangular plate problem. For simplicity, divide (5.3) by D and denote 𝛾1 = 𝛾/D, gh/D = a1 ; for the eigenvalue λ1 = λ /D we retain the previous notation. As a result, we arrive at the eigenvalue problem for the equation Δ2 φ + 𝛾1 vx

𝜕φ 𝜕φ + 𝛾1 vy = λ, 𝜕x 𝜕y

a1 ω 2 + 𝛾1 ω + λ = 0,

(5.7)

with boundary conditions (5.4). We now set φ = (C1 sin π y + C2 sin 2π y) exp(−iα x) and, after applying the wellknown transformation, obtain a homogeneous system: C1 (β1 − iα 𝛾1 vx − λ ) −

8 𝛾 v C =0 3 1 y 2

(5.8)

8 𝛾 v C + (β2 − iα 𝛾1 vx − λ ) C2 = 0. 3 1 y 1

The condition that its determinant is equal to zero gives the characteristic equation (λ − λ1 )(λ − λ2 ) = 0: λ1,2 =

β2 + β1 1 − iα 𝛾1 vx ± √δ , 2 2 2

β1 = (π 2 + α 2 ) ,

2

δ = (β2 − β1 ) − 4 (8𝛾1 vy /3) 2

2

(5.9)

β2 = (4π 2 + α 2 ) .

First of all, let us consider the case of “small” transverse velocities: 4(

8𝛾1 vy 3

) ≪ (β2 − β1 )2

and limit ourselves by √δ represented in the form √δ ≅ (β2 − β1 ) (1 − 2

(8𝛾1 vy )2 9(β2 − β1 )2

).

In (5.9) we choose the eigenvalue λ1 with the smallest real part and substitute it into the stability parabola equation 𝛾12 Re λ1 = a1 (Im λ1 )2 ; as a result, we obtain an expression for vx = vx (α , vy ) with the same accuracy with respect to the small parameter: (8𝛾1 vy )2 1 β vx (α , vy ) ≅ √ 1 (1 + ) α a1 18β1 (β2 − β1 ) (5.10) (8𝛾1 vy )2 1 (1) = vx (α ) + . 18√a1 α (π 2 + α 2 )(β2 − β1 )

5 Test problems

| 23

∗ = Here, vx(1) (α ) is obtained from (5.6) with n = 1. It is straightforward to show that vx,cr min vx (α , vy ) > vx,cr . Indeed, rewrite equation (5.10) in the form

vx (α , vy ) =

1 π2 + α2 ε ( + ). α ψ (α ) √a1

With the derivative of the right-hand side equal to zero, we obtain 1−

π2 − εφ (α ) = 0, α2

φ (α ) = ψ 󸀠 (α )/ψ 2 (α ).

Its approximate (with respect to parameter ε ) solution takes the form α ∗ ≅ π (1 +

ε φ (π )) . 2

Substituting this result into equation (5.9), we obtain finally ∗ ≅ vx,cr + vx,cr

ε . √a1 ψ (π )

We refer to this result as the effect of stabilization of strip vibrations with respect to fluctuations of velocity vector in the case of longitudinal flow. From the second equation (5.8) it follows that in this case Re(C1 /C2 ) ≅ 3(β2 − β1 )/(8𝛾1 vy ) ≫ 1, which means that the vibration modes are practically the same as in the case vy = 0. With the increase in transverse velocity, the discriminant δ decreases to vanish at vy = vy(0) (α ): 3 β2 − β1 . (5.11) vy(0) (α ) = 16 𝛾1 From equation (5.9) we then find that λ1 = λ2 = (β2 + β1 )/2 − iα 𝛾1 vx . Substituting this into the stability parabola equation, we obtain vx(1) (α ) = ((β2 + β1 )/(2α 2 a1 )) therefore, 1/2 π 4, 64π (1) = min vx(1) (α ) = (2√34 + 10) ≅ . vx,cr α √a1 √a1

1/2

, and, (5.12)

and the critical waviness parameter is equal αcr(1) = π √4 8.5. From equation (5.11) we obtain 9(5 + √17) 4 (1) = vy(1) (αcr(1) ) = π . (5.13) vy,cr 16𝛾1 For vy > vy(1) (α ), as follows from (5.9), the eigenvalues become λ1,2 =

β2 + β1 − i (α 𝛾1 vx ± √−δ ) . 2

Evidently, the first to reach the stability parabola will be λ1 with a positive sign before the root; after simple transformations we obtain (

16𝛾1 vy 3

2

2

2(β2 + β1 ) ) = (β2 − β1 ) + (𝛾1 √ − 2α 𝛾1 vx ) . a1 2

(5.14)

24 | 5 Test problems

vx

(1) vx,cr

vx,cr I 0

II v

(2) vy,cr

(1) y,cr

vy

Fig. 5.1. Dependence of vx,cr vs vy,cr in the flutter problem for a strip.

It follows from the latter formula with vx = 0 (transverse flow) that vy (α ) =

3 (β2 − β1 )2 2(β2 + β1 ) + ] [ 16 a1 𝛾12

1/2

.

Evidently, min vy (α ) is reached for α = 0 (cylindrical bending); this velocity has to be α taken as the critical flutter velocity: 1/2

vy,cr = min vy (α ) = α

3π 2 15 2 34 ] ⋅ [( ) + 16 𝛾1 a1

.

The domain of stable vibrations on the (vx , vy ) plane is sketched in Figure 5.1; at (1) (1) ) the left curve has a vertical tangent. In subthe point with coordinates (vx,cr , vy,cr domain (I), we have dvx /dvy > 0, while in subdomain (II) dvx /dvy < 0; for purely transverse flow, velocity vector fluctuations destabilize the vibrations. 3. Round plate. In this section we consider the flutter problem for a round plate; this problem can be considered as methodological, but it is also of interest in its own right. 󵄨 󵄨 Since the domain is the unity disk, we have ψ (ζ ) = ζ and 󵄨󵄨󵄨ζ 󵄨󵄨󵄨 ≤ 1, and therefore 󸀠 󸀠 Ur = Re(ψ ζ /r) = cos θ and Vr = Im(ψ ζ /r) = sin θ . The spectral problem is formulated as 𝜕φ 𝜕φ ] = λφ DΔ2 φ − 𝛾 [(vx cos θ + vy sin θ ) + (vx sin θ − vy cos θ ) 𝜕r r𝜕θ 󵄨 φ 󵄨󵄨󵄨r=1 = 0 𝜕φ 󵄨󵄨󵄨󵄨 󵄨 =0 𝜕r 󵄨󵄨󵄨r=1 󵄨 𝜕2 φ 𝜕φ 󵄨󵄨󵄨 ( 2 +𝜈 )󵄨󵄨󵄨 = 0. 𝜕r 󵄨󵄨r=1 𝜕r Calculations were performed for the following parameters: p0 = 1.0126 ⋅ 105 Pa, ρ0 = 1.2928 kg/m3 , 𝜈 = 0.33, 𝛾 = 1.4, E = 6.86 ⋅ 1010 Pa, ρ = 2.7 ⋅ 103 kg/m3 . The nondimensional plate thickness, unless otherwise stated, is taken to be h = 3⋅10−3 . In this

5 Test problems

| 25

case, the nondimensional parameter 𝛾2 ρ0 /ρ h = 0.2234, and the stability parabola equation is y2 = 0.2234x, i.e. it is tangent to the real axis. Below, the values of Re λ and Im λ are given as functions of velocity v = vx . V=0.00 0.178514E+00 0.0000000E+00 0.773161E+00 0.0000000E+00 0.773161E+00 0.0000000E+00 0.208068E+01 0.0000000E+00 0.208068E+01 0.0000000E+00

V=0.01 0.1795529E+00 0.0000000E+00 0.7731649E+00 0.0000000E+00 0.7737315E+00 0.0000000E+00 0.2080810E+01 0.0000000E+00 0.2080810E+01 0.0000000E+00

V=0.05 0.204827E+00 0.0000000E+00 0.772916E+00 0.0000000E+00 0.787438E+00 0.0000000E+00 0.208370E+01 0.0000000E+00 0.208374E+01 0.0000000E+00

V=0.1 0.2893258E+00 0.0000000E+00 0.7671301E+00 0.0000000E+00 0.8307226E+00 0.0000000E+00 0.2091960E+01 0.0000000E+00 0.2092567E+01 0.0000000E+00

V=0.15 0.467860E+00 0.0000000E+00 0.721530E+00 0.0000000E+00 0.904503E+00 0.0000000E+00 0.210292E+01 0.0000000E+00 0.210600E+01 0.0000000E+00

V=0.16 0.5367171E+00 0.0000000E+00 0.6859869E+00 0.0000000E+00 0.9231474E+00 0.0000000E+00 0.2105080E+01 0.0000000E+00 0.2109058E+01 0.0000000E+00

V=0.161 0.546201E+00 0.0000000E+00 0.679959E+00 0.0000000E+00 0.925086E+00 0.0000000E+00 0.210529E+01 0.0000000E+00 0.210936E+01 0.0000000E+00

V=0.162 0.5567974E+00 0.0000000E+00 0.6728446E+00 0.0000000E+00 0.9270390E+00 0.0000000E+00 0.2105500E+01 0.0000000E+00 0.2109678E+01 0.0000000E+00

V=0.163 0.569116E+00 0.0000000E+00 0.664029E+00 0.0000000E+00 0.929005E+00 0.0000000E+00 0.210570E+01 0.0000000E+00 0.210998E+01 0.0000000E+00

V=0.164 0.5847359E+00 0.0000000E+00 0.6519364E+00 0.0000000E+00 0.9309856E+00 0.0000000E+00 0.2105914E+01 0.0000000E+00 0.2110301E+01 0.0000000E+00

V=0.1645 0.595469E+00 0.0000000E+00 0.642974E+00 0.0000000E+00 0.931980E+00 0.0000000E+00 0.210601E+01 0.0000000E+00 0.211045E+01 0.0000000E+00

V=0.1648 0.6047802E+00 0.0000000E+00 0.6347301E+00 0.0000000E+00 0.9325796E+00 0.0000000E+00 0.2106079E+01 0.0000000E+00 0.2110550E+01 0.0000000E+00

V=0.1649 0.609408E+00 0.0000000E+00 0.630457E+00 0.0000000E+00

V=0.16495 0.6126738E+00 0.0000000E+00 0.6273703E+00 0.0000000E+00

26 | 5 Test problems

0.932779E+00 0.0000000E+00 0.210609E+01 0.0000000E+00 0.211058E+01 0.0000000E+00

0.9328794E+00 0.0000000E+00 0.2106110E+01 0.0000000E+00 0.2110597E+01 0.0000000E+00

V=0.16497 0.614464E+00 0.0000000E+00 0.625650E+00 0.0000000E+00 0.932919E+00 0.0000000E+00 0.210611E+01 0.0000000E+00 0.211060E+01 0.0000000E+00

V=0.16499 0.6171668E+00 0.0000000E+00 0.6230197E+00 0.0000000E+00 0.9329595E+00 0.0000000E+00 0.2106118E+01 0.0000000E+00 0.2110610E+01 0.0000000E+00

V=0.164995 0.618404E+00 0.0000000E+00 0.621800E+00 0.0000000E+00 0.932969E+00 0.0000000E+00 0.210611E+01 0.0000000E+00 0.211061E+01 0.0000000E+00

V=0.164997 0.6193236E+00 0.0000000E+00 0.6208878E+00 0.0000000E+00 0.9329735E+00 0.0000000E+00 0.2106119E+01 0.0000000E+00 0.2110612E+01 0.0000000E+00

V=0.164998 0.615611E+00 0.0000000E+00 0.624539E+00 0.0000000E+00 0.932939E+00 0.0000000E+00 0.210611E+01 0.0000000E+00 0.211060E+01 0.0000000E+00

V=0.164999 0.6201093E+00 -0.1288604E-02 0.9329775E+00 0.0000000E+00 0.2106120E+01 0.0000000E+00 0.2110613E+01 0.0000000E+00 0.2743295E+01 0.0000000E+00

V=0.165 0.620111E+00-0.1672301E-02 0.932979E+00 0.0000000E+00 0.210612E+01 0.0000000E+00 0.211061E+01 0.0000000E+00 0.274329E+01 0.0000000E+00

V=0.17 0.6291597E+00 -0.7610678E-01 0.9431576E+00 0.0000000E+00 0.2107125E+01 0.0000000E+00 0.2112181E+01 0.0000000E+00 0.2745945E+01 0.0000000E+00

V=0.20 0.689641E+00-0.2130506E+00 0.101179E+01 0.0000000E+00 0.211207E+01 0.0000000E+00 0.212159E+01 0.0000000E+00 0.276452E+01 0.0000000E+00

V=0.2798 0.9049170E+00 -0.4511531E+00 0.1269552E+01 0.0000000E+00 0.2106709E+01 0.0000000E+00 0.2136002E+01 0.0000000E+00 0.2842563E+01 0.0000000E+00

V=0.3 0.972456E+00-0.5110125E+00 0.135808E+01 0.0000000E+00 0.209906E+01 0.0000000E+00 0.213166E+01 0.0000000E+00 0.287062E+01 0.0000000E+00

V=0.4 0.1364564E+01 -0.8722558E+00 0.2001214E+01 -0.3055128E+00 0.2059245E+01 0.0000000E+00 0.3072732E+01 0.0000000E+00 0.4516915E+01 0.0000000E+00

5 Test problems

| 27

y 1,0 0,5

x −1,0

−0,5

0,0

0,5

1,0

−0,5 −1,0

Fig. 5.2. Plan-form of the plate: epitrochoid with n = 4 and ε = 0.1.

The first calculation was carried out for the velocity v = 0. As expected, the first eigenvalue is simple, while the other two are multiple. As the velocity is increased, the multiple eigenvalues are split (shifting to the right), but remain real-valued. Then, the first and second eigenvalues begin to approach each other (probably, at some instant they will completely merge to form a multiple eigenvalue, but such studies are impossible to perform, because the corresponding eigenvalue problem has a Jordan cell). For v = 0.164999, a complex-valued pair with small imaginary part appears (in the table, only the eigenvalue with a negative imaginary part is presented). The real part of this complex-valued pair is close to the real eigenvalues obtained for the previous velocity v = 0.164998. As the velocity increases further, the absolute value of the complex pair also increases (note that this complex pair remains the only one available) and at v = 0.2798 it reaches the stability parabola. Thus, this velocity, is the critical flutter velocity. For v = 0.4, a second complex-valued pair appears, but it is located within the stability parabola. Thus, the mechanism of flutter instability is revealed. For the round plate, the study of flutter instability conditions was performed by the first eigenvalue. The condition of flutter onset detected by the appearance of a complex-valued pair for the spectral problem in question gives an underestimated value. Also, the functional form of the spectrum perturbation with the increase in the velocity is clarified for the spectral problem being considered. Consider now the results of the numerical computation of the critical flutter velocity for a round plate and a plate obtained from a disc by the conformal mapping 󵄨 󵄨 z = ζ (1 + εζ n ), 󵄨󵄨󵄨ζ 󵄨󵄨󵄨 ≤ 1 (the curve obtained by this mapping is called epitrochoid). For ε = 1/n, this curve has n angular points; therefore all calculations were carried out for ε < 1/n. In particular, two domains were considered: n = 4, ε = 0.1, 0.2, 0.24, and n = 12, ε = 0.0625, as well as two boundary-value problems: rigid clamping, and simple support of the edge. The former of these domains is presented in Figure 5.2; the second domain is not shown here.

28 | 5 Test problems

1,0

0,5

x

−1,0

−0,5

0,0

0,5

1,0

Fig. 5.3. Functions Re φ (x, 0) and Re φ (0, y) at v = 0.2798 for epitrochoid with n = 4 and ε = 0.1.

First, we consider a round plate with a clamped edge. Calculations were carried out on 9 × 15 and 15 × 31 grids (the first number corresponds to the number of circular grid lines, the second one denotes the number of grid nodes on each circle). On both grids, the same critical velocity v = 0.2798 was obtained. The second calculation was carried out for a four-petal epitrochoid with ε = 0.1. For θ = 0 (where θ is the angle between the flow velocity vector and axis 0x) the same value of critical velocity was obtained. The first eigenvalue to reach the stability parabola is the one with the minimum absolute value, λ = (0.935906, 0.457245). In Figure 5.3 the functions Re φ (x, 0) and Re φ (0, y) are plotted. The curve which has no intersections with the axis 0x corresponds to the function Re φ (0, y), The other curve is for Re φ (x, 0). Thus, after intersecting the axis 0x, the function Re φ (x, 0) gradually tends to zero. For θ = π /4, the critical flutter velocity is v = 0.2789. Thus, for the domain being considered, the critical flutter velocity depends weakly upon the direction of flow. The second calculation was carried out for the same domain, but with ε = 0.2. The critical flutter velocity obtained on the 9 × 15 grid is v = 0.2771. A close value v = 0.2796 was obtained on the fine grid. The graphs of the real part of eigenfunctions are presented in Figure 5.4. The first eigenvalue to reach the stability parabola is the one which has the minimum absolute value, λ = (0.996053, 0.471697). Thus, the first eigenvalue and the critical flutter velocity changed by a small fraction in comparison with the previous calculation. The shape of the eigenfunction (compare Figures 5.3 and 5.4) has changed, but the oscillations of Re φ (x, 0) near the right boundary are most probably attributed to the computational error (note that the boundary of this domain has very large curvature at four points). For θ = π /4, on both grids we obtained the same value of critical flutter velocity v = 0.2826. The eigenvalue with the smallest absolute value λ = (0.940322, 0.458382) is the first to reach the stability parabola. The last calculation carried out on this domain with the clamping boundary condition was for ε = 0.24. The critical velocities obtained for θ = 0 are v = 0.2724 on the 9 × 15 grid, and v = 0.2751 on the 15 × 31 grid. Stability was determined by the first eigenvalue λ = (0.987082, 0.469646). For θ = π /4, we obtained v = 0.2821 on the 9 × 15 grid, while a close value v = 0.2809 was obtained on the 15 × 31 grid. Stability was determined by the first eigenvalue λ = (0.940836, 0.458527).

5 Test problems

| 29

1,0

0,5

x

−1,0

−0,5

0,0

0,5

1,0

−0,5

Fig. 5.4. Real part of eigenfunction for θ = π /4, v = 0.2796, n = 4 and ε = 0.2.

−1,0

1,0

0,5

−1,0

−0,5

0,0

0,5

1,0

−0,5

−1,0

Fig. 5.5. Functions Re φ , v = 0.2581, n = 4 and ε = 0.2.

Then the boundary value problem of the second kind was considered with ε = 0.2, n = 4. This problem is computationally more challenging than the first one, and therefore it was solved on 13 × 25 and 15 × 31 grids. For θ = 0, we obtained v = 0.2653 on the first grid, and v = 0.2581 on the second. It is interesting to note that in this problem the first eigenvalue is real-valued λ1 = 0.63323, while the stability is determined by the second eigenvalue λ2 = (0.680571, 0.390052). The plot of the real part of the eigenvalue is presented in Figure 5.5. Thus, for a nonround plate, the stability is not necessarily determined by the first eigenvalue. However, for θ = π /4, the stability is determined by the first eigenvalue. It was obtained on a 13 × 25 grid with v = 0.2611, while on the 15 × 31 grid we obtained v = 0.2613, with λ1 = (0.610680, 0.369400) being the eigenvalue which determined the stability. It is interesting to note that the plot of real part of eigenfunction (see Figure 5.5) is qualitatively different: Re φ (0, y) has zeroes, because in this case the stability is determined by the second eigenvalue. We then considered a domain bounded by a 12-petal epitrochoid (n = 12, ε = 0.0625) with two boundary condition types: clamping and simple support. The air flow velocity vector is directed at an angle θ = 0, π /12, and π /4 with respect to the axis 0x. Due to symmetry, the critical velocities for the latter two angles must coincide, and checking this was one of the purposes of our calculations. The boundary value

30 | 5 Test problems 1,0

0,5

x

−1,0

−0,5

0,0

0,5

1,0

Fig. 5.6. Functions Re φ (boundary-value problem of the first kind), v = 0.2848, n = 12, and ε = 0.0625.

1,0

0,5

−1,0

−0,5

0,0

0,5

1,0

−0,5

−1,0

Fig. 5.7. Functions Re φ (boundary-value problem of the second kind), v = 0.2291, n = 12, and ε = 0.0625.

problem of the first kind was considered first. For θ = 0, we obtained on the 9 × 15 grid the critical flutter velocity v = 0.2805, while on the 15 × 31 grid it was found that v = 0.2848. Stability is determined by the first eigenvalue λ = (0.940611, 0.458456). the real parts of the eigenfunction are plotted in Figure 5.6. The second calculation was carried out for θ = π /12. On a 9 × 15 grid we obtained the critical flutter velocity v = 0.2803, while on a 15 × 31 grid it was found that v = 0.2849. Stability is determined by the first eigenvalue λ = (0.940768, 0.458494). For θ = π /4, we obtained on a 9 × 15 grid that v = 0.2796, whereas on the 15 × 31 grid it was found that v = 0.2851. Thus, the last two calculations gave practically coinciding results, which confirms the reliability of the proposed approach. Then the boundary value problem of the second kind was considered on the same domain. For θ = 0, we obtained on the 9 × 15 grid that v = 0.2152, while on the 15×31 grid it was found that v = 0.2291. Stability is determined by the first eigenvalue λ = (0.618777, 0.371835). The real parts of the eigenfunction are plotted in Figure 5.7. The next calculation on this domain was carried out for simply supported edges with θ = π /12. On the 13 × 25 grid, it was obtained that v = 0.2351, while on the 15 × 31 grid we obtained v = 0.2305. Stability is determined by the first eigenvalue λ = (0.634463, 0.373240). For θ = π /4, the value obtained on the15 × 31 grid is v = 0.2385. As can be seen, the results of the latter two calculations agree with high accuracy.

5 Test problems

| 31

1.5

1 0.75 0.5

Eigenfunction

1.25

0.25 0 –1

–1 –0.5

–0.5 0 Y

0 0.5

0.5 1

1

X

Fig. 5.8. Clamped elliptic plate (a = 1) with eccentricity e = 0.7: shape of Re φ (φ is the amplitude) for θ = 0.

4.1. Clamped elliptic plate. Consider an elliptic plate (a = 1) with eccentricity e = 0.7. The angle between the flow velocity vector and x-axis takes the values θ = 0, π /8, π /4, 3π /8, and π /2. Calculations were carried out on 9 × 15 and 15 × 31 grids using standard parameters (see problem 3). On both grids, close results were obtained. Here we consider the values obtained on the fine grid: 0.3622, 0.3742, 0.4076, 0.4441, and 0.4505. Thus, in comparison with the unity disk (0.2798, see problem 3) higher critical velocities are obtained. The critical velocity increases with the increase in the flow vector angle from 0 to π /2. The dependence of Re φ (where φ is the amplitude) for θ = 0 is presented in Figure 5.8. This surface is featured by a distinct “hump” elongated in the flow direction. When the flow velocity angle varies, this hump is rotating along with the flow velocity vector, but the qualitative shape of the eigen mode remains the same. 4.1. Simply supported elliptic plate. For a simply supported elliptic plate with the same parameters and flow velocity vector angles the following critical velocities were obtained: 0.2783, 0.2833, 0.2946, 0.3006, and 0.2987(2). The shape of the eigen mode for θ = 0 is shown in Figure 5.9. Clearly, it changed dramatically. Qualitatively the same shape of the eigen mode is obtained for θ = π /8, however at θ = π /4 the shape of the eigen mode changes abruptly and becomes qualitatively similar to that presented in Figure 5.8. Thus, on the interval from θ = π /8 to θ = π /4 sharp changes of the natural vibration mode occurs. For θ = 3π /8 and θ = π /2, the natural vibration mode is qualitatively similar to that shown in Figure 5.8. Note that for θ = π /2 the critical velocity was determined by the second eigenvalue. It is somewhat lower than that for θ = 3π /8, i.e. for simply supported boundary condition there is no monotonic increase in the critical velocity when the flow velocity vector orientation angle is varied between θ = 0 and θ = π /2. In comparison with the unity disk (0.2241), the critical

32 | 5 Test problems

Eigenfunction

0.5

0

–0.5 –1 –0.5 –0.5

Fig. 5.9. Simply supported elliptic plate (a = 1) with eccentricity e = 0.7: shape of Re φ (φ is the amplitude) for θ = 0.

0 0

Y

X

0.5 0.5

1

1,00

0

8 0, 0,00

0

4 0,

0 0,6 0 0,4

0

0 0,

0 0,2 00 –0,

0

,4 –0

20

–0,

40 –0,

60

–0,

0 ,8 –0

Fig. 5.10. Contour lines of the eigen mode surface plotted from value 0.0 in positive and negative directions with step 0.1 (clamped boundary).

velocity is higher. The shape of the eigen mode for the disk is similar to that shown in Figure 5.8, i.e. the shape of the eigen mode for the ellipse is qualitatively different for angles θ = 0 to π /8. Analysis of the calculation results shows that the qualitative behavior of flutter eigen modes for an elliptic plate is different for clamped and simply supported boundary conditions. The critical flutter velocity is higher in the case of a clamped plate, as compared to a simply supported plate. Let us now discuss the results obtained. In Figures 5.10 and 5.11, the contour map of the eigen mode surface are plotted, starting with value 0.0 in positive and negative directions with step 0.1. It can be seen that in Figure 5.10 (clamped boundary condition) there are no contours on the right side of the plate. Here, the eigenfunction takes small negative values ∼ −0.01. Then a straight line is visible which divides the plate into two nearly equal parts. This line is the 0.0-level contour (note that the plate edge

5 Test problems

|

33

0,00

0 0,8

0 0,6

0 0,4

0 0,4

0 0,2 00 –0, 20 –0,

0 0,0 ,40 –0

40

–0,

60

–0,

,80 –0

Fig. 5.11. Contour lines of the eigen mode surface plotted from value 0.0 in positive and negative directions with step 0.1 (simply supported boundary).

is also a 0.0-level contour). Calculations show that this line is somewhat shifted from the center of the ellipse towards the flow and is practically straight. A similar picture is observed in the case of simply supported boundary (see Figure 5.10). However, in the latter case the eigen mode takes rather large negative values on the right side of the plate, whereas the 0.0-level contour is similarly located. To check the generality of these conclusions, a set of calculations was performed: (i) ellipse, boundary-value problem of the first kind, θ = π /8, Vcr = 0.3742; (ii) epitrochoid, boundary-value problem of the first kind, ε = 0.1, n = 4, θ = π /6, Vcr = 0.2795; (iii) epitrochoid, boundary-value problem of the first kind, ε = 0.1, n = 4, θ = π /8, Vcr = 0.2799; (iv) epitrochoid, boundary-value problem of the first kind, ε = 0.1, n = 4, θ = π /4, Vcr = 0.2789; (v) epitrochoid, boundary-value problem of the first kind, ε = 0.1, n = 4, θ = 0, Vcr = 0.2798. In all the cases there is a straight nodal line (0.0-level contour) which is perpendicular to the freestream velocity vector (rotating together with it) and somewhat shifted from the center of symmetry towards the free stream. These are the features of the eigen mode of the the spectral problem being considered. 5. Dependence of the critical flutter velocity on plate thickness. In this section we consider the computational experiments in which the dependence of the critical flutter velocity on the plate thickness is studied. The following approach is taken. For a round plate and a plate bounded by an epitrochoid (ε = 0.1, n = 4) we perform calculations in which the plate thickness is varied from h = 0.001 to h = 0.01 with the step 0.001. Then the critical velocities obtained are fitted by an analytical formula v = v(h). The calculations were carried out for the same parameters of the plate as were used in the previous section. For a round plate clamped at its edge, the following critical flutter velocities are obtained: 0.1404(3), 0.1544(2), 0.2791(1), 0.4801(1), 0.8361(1), 1.3806(1), 2.1482(1), 3.1745(1), 4.4955(1), and 6.1476(1). Here the number in parentheses denotes the number of the eigenvalue by which stability was determined. Interestingly, for a thin plate the stability is determined by the eigenvalue other than the first one (compare with calculations for problem 4). It turned out that the dependence of the

34 | 5 Test problems 7 6 5 4 3

h

Fig. 5.12. Dependence of the critical flutter velocity on plate thickness (clamped round plate); v = a + bh3 , a = 0.091048718; b = 6124609.2.

h

Fig. 5.13. Dependence of the critical flutter velocity on plate thickness (simply supported round plate); v = a + bh3 , a = 0.10004173; b = 6028457.

2 1 0 0,001

0,003

0,005

0,007

0,009

7 6 5 4 3 2 1 0 0,001

0,003

0,005

0,007

0,009

critical flutter velocity on the plate thickness takes the form v = a + bh3 . The values of constants a and b are given in Figure 5.12. It is evident from this graph that the above analytical formula approximates the data of numerical experiments with high accuracy. The second calculation was carried out for a round simply supported plate. The following critical flutter velocities were obtained: 0.1283(3), 0.1306(3), 0.2241(1), 0.3236(1), 0.5237(1), 0.8386(1), 1.2862(1), 1.8870(1), 2.6618(1), and 3.6317(1). The dependence v = v(h) has the same form as in the first calculation, but with different constants a and b (Figure 5.13). The last calculation was carried out for a plate with the outer contour formed by an epitrochoid (ε = 0.1, n = 4) and clamped edge (the air velocity vector was directed along the axis 0x); see Figure 5.14. The following critical flutter velocities were obtained: 0.08913(3), 0.1531(2), 0.2797(1), 0.4878(1), 0.8487(1), 1.4001(1), 2.1776(1), 3.2171(1), 4.5553(1), and 6.2289(1). The dependence v = v(h) has the same form as in the first calculation, albeit with different constants a and b (see Figure 5.14). All the above-mentioned calculations were carried out on a 9 × 15 grid. To check the accuracy, calculations were also performed for h = 0.01 on a 15 × 31 grid. For the round plate, the critical flutter velocities coincided within all the decimal places presented. For the epitrochoid, a close value of 6.2310(1) was obtained. Looking ahead, we note that for a rectangular plate the dependence v = v(h) has the same form as in the calculations just considered, althgouth with different constants a and b. The result

5 Test problems

4

|

35

υ

3,5 3 2,5 2 1,5 1 0,5 0 0,001

0,003

0,005

0,007

0,009

h

Fig. 5.14. Dependence of the critical flutter velocity on plate thickness (plate with an epitrochoid outer contour (e = 0.1, n = 4) and clamped edge, air velocity vector directed along 0x axis); v = a + bh3 , a = 0.10124565; b = 3511783.

obtained has important practical implications. It is sufficient to calculate the critical flutter velocity for two values of the thickness, and then determine a and b, after which the formula v = a + bh3 can be applied.

6 Rectangular plate 6.1 Problem statement and analytical solution Consider a rectangular plate occupying on the xy-plane a domain S1 : {0 ≤ x ≤ a, 0 ≤ y ≤ b}. Take the size a as the linear scale and introduce nondimensional parameters in the same way as before, and denote a/b = δ . As a result, we obtain an eigenvalue problem on the domain S : {0 ≤ x ≤ 1, 0 ≤ y ≤ 1/δ }: DΔ2 φ + 𝛾vn0 grad φ = λφ , x, y ∈ Γ,

φ = 0,

ρ hω 2 + 𝛾ω + λ = 0

(6.1)

M(φ ) = 0.

(6.2)

For further convenience, divide both sides of equation (6.1) by D, denote 𝛾v/D = v0 , while for all other parameters retain the previous notation. Assume that n0 = {1, 0}, and the plate edges y = 0 and y = 1/δ are simply supported; in this case only by substitution φ = X(x) sin(nδπ y) the problem is reduced to the eigenvalue problem for a fourth-order ordinary differential equation: X IV − 2k2 X 󸀠󸀠 + k4 X + v0 X 󸀠 = λ X, x = 0,

X = 0,

M1 (X) = 0;

x = 1,

k = nπδ X = 0,

M2 (X) = 0.

(6.3) (6.4)

It is known from general theory that this problem has a discrete spectrum with know asymptotic behavior; however, in practical applications it is required to know the first few eigenvalues, i.e. to have explicit expressions for the roots of characteristic equation of fourth order. Substituting X = exp(sx) into equation (6.3), we obtain 2

(s2 − k2 ) + v0 s − λ = 0.

(6.5)

If s𝜈 = s𝜈 (k, v0 λ ) are the roots of this equation, then its general solution takes the form X = C𝜈 exp(s𝜈 x). Taking the boundary conditions (6.4) into account, we obtain the characteristic determinant Δ(k, v0 , λ ), which, being an integer function, has a countable set of (generally speaking, complex-valued) isolated roots. A trick by which the determinant Δ(k, v0 , λ ) can be reduced to a form convenient for the analysis was proposed by A. A. Movchan. Let s1 and s2 be two roots of equation (6.5); if we set s1 = α + β i and s2 = α − β i, it is easy to show that the other two roots of equation (6.5), as well as the parameters v0 and λ are expressed in terms of α and β by s3,4 = −α ± √β 2 − 4α 2 v0 = −4α (β 2 − α 2 − k2 ) 4

2

2

2

(6.6) 2

2

λ = k + (α + β )(β − 3α + 2k ).

6.1 Problem statement and analytical solution

| 37

λ

λ04 A2 λ03 λ02

A1 λ01 v01

0

v02

v0

Fig. 6.1. Eigenvalue dependence on flow velocity.

The determinant Δ(k, v0 , λ ) is also reduced to a function of α and β ; for example, for simply supported edges x = 0 and x = 1 we obtain Δ(k, α , β ) = α 2 (ch 2α − ch √β 2 − 2α 2 + 2k2 cos β ) +

(β 2 − α 2 − k2 )2 + 2α 2 (α − k2 ) sin β sh √β 2 − 2α 2 + 2k2 = 0, 2β 2 2 2 √β − 2α + 2k

(6.7)

while for clamped edges x = 0 and x = 1 we obtain Δ(k, α , β ) = ch 2α − ch √β 2 − 2α 2 + 2k2 cos β +

k2 − 3α 2

sin β sh √β 2 − 2α 2 + 2k2 = 0. √β 2 − 2α 2 + 2k2 β

(6.8)

Further calculations are carried out in the following way. For v0 = 0, we obtain from equation (6.5) that s1−4 = ±(k2 ± √λ )1/2 , and the characteristic determinant is reduced to the well-known form (in fact, the problem corresponds to free plate vibrations); its roots λi(0) are real-valued (see Figure 6.1). Then, v0 is increased by some step, and new roots α and β of equation Δ = 0 satisfying the second relation from (6.6) are obtained; the corresponding curves (λ10 A1 , λ20 A1 , etc.) are shown in Figure 6.1. At some velocity v0 = v0(1) the two roots merge, and at v0 > v0(1) they become complex conjugate. At v0 = v0(2) the same occurs with the other pair of roots, etc. Thus, it is necessary to determine which of the roots is the first to reach the stability parabola; the corresponding flow velocity v0 cr will be the critical flutter velocity.

38 | 6 Rectangular plate Detailed analysis of the equation Δ(k, α , β ) = 0 was carried out in the original works by A. A. Movchan et al. where numerous qualitative and quantitative results were obtained; these results concern a particular problem and are not repeated here.

6.2 Numerical–analytical solution Consider a rectangular plate occupying a domain S1 : {|x| ≤ a, |y| ≤ b1 } on the xyplane. Take the size a as the linear scale, denote b = b1 /a, and introduce nondimensional parameters in the same way as before. As a result, we obtain an eigenvalue problem on the domain S : {|x| ≤ 1 |y| ≤ b}: DΔ2 φ − 𝛾vn0 grad φ = λφ , φ = φ (x, y) 󵄨 φ 󵄨󵄨󵄨Γ = 0 𝜕φ 󵄨󵄨󵄨󵄨 𝜕φ 󵄨󵄨󵄨󵄨 = 0, =0 󵄨󵄨 󵄨 𝜕x 󵄨󵄨|x|=1 𝜕y 󵄨󵄨󵄨|y|=b 󵄨 󵄨 𝜕2 φ 󵄨󵄨󵄨 𝜕2 φ 󵄨󵄨󵄨 󵄨 󵄨󵄨 = 0, = 0, 󵄨 𝜕x2 󵄨󵄨󵄨|x|=1 𝜕y2 󵄨󵄨󵄨|y|=b

(6.9) (6.10) (6.11) (6.12)

where the boundaries conditions (6.10) and (6.11) correspond to clamped edges, while (6.10) and (6.12) describe simply supported edges. On the complex plane λ , the domains of stable and unstable vibrations are divided by the stability parabola 𝛾2 Re λ − ρ h(Im λ )2 = 0; since V = {v cos θ , v sin θ } and λ = λ (v, θ ), equation 𝛾2 Re λ − ρ h(Im λ )2 = f (v, θ ) = 0 defines, on the parameter plane v, θ , the neutral curve bounding the domain of subcritical parameters, i. e., stable vibrations. In the previous sections, general properties of eigenvalues were established: Re λ > 0; vibrations corresponding to real-valued λ are stable; for a fixed θ with increase in v, the eigenvalues sequentially enter the complex domain; when v is fixed, the number of complex λ is finite. Thus, the following analysis scheme was adopted: (i) a discrete analogue for the problem (6.9)–(6.12) is developed; (ii) for fixed θ , the critical velocity is determined by the first eigenvalue; (iii) for this velocity, stability analysis is performed with respect to other complex eigenvalues; (iv) if a complex λ is found beyond the stability parabola, a critical velocity corresponding to this eigenvalue is calculated; (v) of all “critical” velocities found in this way, the minimum one is determined. We first construct the discrete Laplacian H with boundary condition (6.10). Introduce on the xy-plane a grid consisting of nodes (2𝜈 − 1)π ) , 𝜈 = 1, 2, . . . , n 2n (2μ − 1)π ) , μ = 1, 2, . . . , m. yμ = b cos ( 2m x𝜈 = cos (

(6.13) (6.14)

6.2 Numerical–analytical solution

|

39

Let A be the matrix of discrete operator corresponding to the differential operator 𝜕2 φ /𝜕x2 with boundary condition φ (−1) = φ (1) = 0 on the grid (6.13); B be the matrix of discrete operator corresponding to the differential operator 𝜕2 φ /𝜕y2 with boundary condition φ (−b) = φ (b) = 0 on the grid (6.14). The discrete Laplacian takes the form (see Part III, Chapter 17, Section 17.2): H = Im ⊗ A + B ⊗ I n ,

(6.15)

where In and Im are unity matrices of dimensions n × n and m × m; by the symbol ⊗, the Kronecker matrix product is denoted. The eigenvector of matrix H is u = w ⊗ v, where v is the eigenvector of matrix A and w is the eigenvector of matrix B. The grid points are enumerated first in x, then in y (left to right, bottom to top). One can say that the matrix (6.15) inherits the variable separability property of the differential Laplace operator. Discretization of the operator 𝜕2 φ /𝜕x2 with boundary condition φ (a) = φ (b) = 0 is carried out in the following way: (i) on the grid (6.13) (a = −1, b = 1) or (6.14) (a = −b, b = b) the Lagrange interpolating formula satisfying the boundary conditions is written; (ii) second-order derivatives at the grid nodes are obtained by differentiation of the interpolation formula. As a result, we obtain Dij = (

k−1 2 2 ∑ cos(qψj ) [(2 + q2 ) cos qψi ) b − a k sin2 ψj q=0

+ 3q cos ψi + 3q cos ψi ψj =

(2j − 1)π , 2k

sin qψi ] sin ψi

(6.16)

i, j = 1, 2, . . . , k.

Here, k = n, a = −1, b = 1 for matrix A; k = m, a = −b, b = b for matrix B. Discretization of variables 𝜕φ /𝜕x and 𝜕φ /𝜕y is performed similarly. On the grid (6.13) (a = −1, b = 1) or (6.14) (a = −b, b = b) the Lagrange interpolating formula satisfying the boundary conditions is written, derivatives at the grid nodes are obtained by differentiation of this interpolation formula. As a result, we obtain the differentiation matrix k−1 q cos qψm sin qψ𝜈 4 ∑ D𝜈μ = k(b − a) q=0 sin ψ𝜈 (6.17) (2𝜈 − 1)π ψ𝜈 = , 𝜈, μ = 1, 2, . . . , k. 2k For k = n, a = −1, and b = 1 we obtain matrix Dx of differentiation with respect to x; for k = m, a = −b, b = b we obtain matrix Dy of differentiation with respect to y. To obtain the derivatives of the function φ at the grid nodes, it is necessary to multiply matrix D by the vector of the function φ values at the grid nodes. From the boundary conditions (6.10) and (6.12) follows the condition 󵄨 Δφ 󵄨󵄨󵄨𝜕Γ = 0,

(6.18)

40 | 6 Rectangular plate and in this case the matrix of the biharmonic operator with boundary condition (6.10) and (6.18) is H 2 . This is straightforward, because matrix H 2 has the same eigenvectors as matrix H, and its corresponding eigenvalues are λi2 , i = 1, 2, . . . , N, where λi are eigenvalues of matrix H of dimensions N × N (N = mn). Consider now discretization of (6.9) with boundary conditions (6.10) and (6.11), i.e. a clamped plate. Apply to the function φ = φ (x, y) on a rectangle the interpolation formula n

m

φ (x, y) = ∑ ∑ Mi0 (z)Lj0 (x)φ (xj , yi ) j=1 i=1

y = bz,

z ∈ [−1, 1],

l(x) Lj0 (x) = 󸀠 , l (xj )(x − xj )

x ∈ [−1, 1] l(x) = (x2 − 1)2 Tn (x) (6.19)

Tn (x) = cos n arccos x xj = cos θj , Mi0 (z) =

θj = (2j − 1)π /2n,

M(z) , M 󸀠 (zi )(z − zi )

zi = cos θi ,

j = 1, 2, . . . , n

M(z) = (z2 − 1)2 Tm (z)

θi = (2i − 1)π /2m,

i = 1, 2, . . . , m.

This clearly satisfies the clamped boundary conditions. To obtain the matrix of the discrete biharmonic operator H, it is necessary to apply the biharmonic operator to the interpolation formula (6.19), i.e. to differentiate equation (6.19) four times with respect to x and y. As a result, we obtain a nonsymmetric matrix H of dimension N ×N, N = mn. Enumerate the grid points (xj , yi ) in the rectangle first in y, then in x, i.e. top to bottom, right to left. As a result we obtain that Δ2 φ us approximated by the Hφ , where φ is the vector of function φ = φ (x, y) values at the grid points. Note that matrix H is nonsymmetric, although the biharmonic operator is self-adjoint. Therefore, matrix H can possess complex eigenvalues. When solving stability problems, it is highly undesirable that the discrete biharmonic operator has complex eigenvalues arising from a discretization error. Modification of the approach was required: instead of matrix H, we consider a matrix (H + H 󸀠 )/2. The rationale behind this trick is the following. The original problem is self-adjoint (biharmonic operator with clamped boundary condition). However, due to the discretization error we obtain a nonsymmetric matrix H. Represent H in the form H = (H + H 󸀠 )/2 + (H − H 󸀠 )/2, and include the nonsymmetric part into the discretization error. A relevant perturbation introduced into the eigenvalue of matrix H depends on how close the resolvents of matrices H and (H + H 󸀠 )/2 are in the part of the complex plane which is of interest for the flow stability analysis. This perturbation can be theoretically evaluated by the approach presented in Chapter 22 of Part III; here, numerical evaluation was undertaken.

6.3 Results

| 41

Matrix H (for b = 1) of dimension 361 × 361 (361 = 19 × 19) has the first eigenvalue √λ1 /π 2 = 2.4902, it was compared with the well-known result √λ1∗ /π 2 = 2.489; matrix (H + H 󸀠 )/2 has the eigenvalue √λ1󸀠 /π 2 = 2.3961. Thus, perturbation introduced into eigenvalues by symmetrization of matrix H is regarded acceptable. Discretization of qrad φ in the boundary-value problem (4.1)–(4.3) was performed similarly.

6.3 Results Consider the results obtained for a simply supported plate. The mechanical parameters were taken the same as before, while the (relative) size b, thickness h, flow velocity v, and angle θ were variable. Test calculations were first performed for a rectangular plate (b = 1, h = 0.003). The following results were obtained: vcr (0) = vcr (π /2) = 0.2103, vcr (π /4) = 0.2001; in all cases vcr was determined by the first eigenvalue (hereafter, λk are enumerated in the increasing order of their magnitudes). The plots Re φ (x, 0) and Re φ (0, y) are identical. Results obtained for a plate with b = 0.5 and h = 0.003 are presented in Table 6.1; the eigenvalue number is given in the parentheses next to each eigenvalue. We need to mention a point which can be important in the numerical analysis of 󵄨 󵄨 󵄨 󵄨 flutter: it turned out that 󵄨󵄨󵄨λ1 󵄨󵄨󵄨 = 1.5658 and 󵄨󵄨󵄨λ2 󵄨󵄨󵄨 = 1.56660 are very close, while λ1 > 0 is real-valued and does not give rise to an unstable vibration mode. In the third row of ∗ the table, the values of vcr obtained by the Bubnov–Galerkin method in eight-term approximation are given; evidently, this method underestimates the results significantly (capabilities of the Bubnov–Galerkin method in application to this problem will be discussed below). Two other points are worth mentioning: (i) a relatively sharp increase in the critical flutter velocity in the angle range θ ∈ (π /4, 3π /8) and its gradual variation for other angles; (ii) the maximum of critical flutter velocity is located near the point θ = 15π /32: this is the so-called effect of plate vibration stabilization with respect to fluctuations of velocity vector direction near θ = π /2. Note that this effect was previously obtained in the flutter problem for a strip. In Figures 6.2–6.4, the real parts of eigenfunctions are shown for different angles θ (π /4, 5π /16, 3π /8) at v = vcr . Table 6.1. Simply supported rectangular plate, b = 0.5, h = 0.003. θ

0

π /8

π /4

5π /16

3π /8

15π /32

π /2

vcr ∗ vcr

0.35434(1) 0.3042

0.3737(1) 0.3307

0.4346(1) —

0.4801(1) 0.4207

0.5235(1) —

0.5275(2) 0.4022

0.5257(2) 0.4121

42 | 6 Rectangular plate Table 6.2. Simply supported rectangular plate, b = 0.25, h = 0.0015. θ

0

π /8

π /4

5π /16

vcr

0.2655(3)

0.2832(3)

0.3453(1)

0.4014(1)

θ

3π /8

7π /16

15π /32

π /2

vcr

0.4803(1)

0.4912(2)

0.4867(3)

0.4851(4)

Calculations were also carried out for an elongated plate of the size b = 0.25, h = 0.0015 (with the same ratio of thickness to length of the short side ratio as in the previous simulation); results are summarized in Table 6.2. The behavior of vcr = vcr (θ ) is qualitatively the same as in the previous simulation. In Figures 6.5–6.9, the real parts of eigenfunctions are shown for different angles θ (0, π /4, 5π /16, 3π /8, 7π /16) at v = vcr . It is clear that in the parameter range where the critical flutter velocity increases sharply, the plate vibration mode changes qualitatively. One can conclude that in this parameter range the plate is most sensitive to the variation of magnitude and direction of flow velocity vector.

6.4 Bubnov–Galerkin (B–G) method It is generally accepted that for rectangular plate flutter problems (in the traditional formulation V = {vx , 0}) the Bubnov–Galerkin method gives a reasonable value of the critical velocity even in the two-term approximation. However, it has long been noted that in the case of a plate elongated in the streamwise direction the effectiveness of the method deteriorates abruptly, and to reach some reasonable accuracy it is necessary to retain a considerable (generally speaking, unknown beforehand) number of terms in the approximating sum. The applicability of the Bubnov–Galerkin method to plate flutter problems in the general formulation has not yet been sufficiently studied. The results given below fill in this gap to some extent. It follows from the figures presented in this section that the characteristic dimension of perturbation is of the order of half the smaller side of the plate; therefore, an approximate solution was sought in the form φ = cmn sin(mπ y) sin(βπ x), m = 1, 2; n = 1, . . . , 4 (the plate occupies the region K = {0 ≤ x ≤ 1/β , 0 ≤ y ≤ 1}). The standard Bubnov–Galerkin procedure for solving equation (6.9) is reduced to the examination of the roots of the characteristic determinant of the eighth order, which is not written here because of its cumbersome form. The main purpose of the procedure is to determine the dependence λ = λ (v, θ ). The calculated results are presented in the third columns of Tables 6.3 and 6.4 for a plate with dimensions b/a = 0.5, h/a = 0.003. Eigenvalues are presented as pairs of numbers, Re λ (the first one) and Im λ (the second one); the asterisk (∗ ) denotes the

6.4 Bubnov–Galerkin (B–G) method

|

43

Fig. 6.2. Rectangular plate (b = 0.5, h = 0.003), real part of eigenfunction for θ = π /4 and v = vcr ; v = 0.4346.

Fig. 6.3. Rectangular plate (b = 0.5, h = 0.003), real part of eigenfunction for θ = 5π /16 and v = vcr ; v = 0.4801.

Fig. 6.4. Rectangular plate (b = 0.5, h = 0.003), real part of eigenfunction for θ = 3π /8 and v = vcr ; v = 0.5235.

Fig. 6.5. Rectangular plate (b = 0.25, h = 0.0015), real part of eigenfunction for θ = 0 and v = vcr ; v = 0.2665.

44 | 6 Rectangular plate

Fig. 6.6. Rectangular plate (b = 0.25, h = 0.0015), real part of eigenfunction for θ = π /4 and v = vcr ; v = 0.3541.

Fig. 6.7. Rectangular plate (b = 0.25, h = 0.0015), real part of eigenfunction for θ = 5π /16 and v = vcr ; v = 0.4014.

Fig. 6.8. Rectangular plate (b = 0.25, h = 0.0015), real part of eigenfunction for θ = 3π /8 and v = vcr ; v = 0.4803.

Fig. 6.9. Rectangular plate (b = 0.25, h = 0.0015), real part of eigenfunction for θ = 7π /16 and v = vcr ; v = 0.4912.

6.4 Bubnov–Galerkin (B–G) method

|

45

∗ Table 6.3. Rectangular plate, b = 0.5, h = 0.003, θ = 0, vcr = 0.3546, vcr = 0.3041. ∗ v/vcr

0.2 0.5 0.8 1.0 2.0

Current method

B–G

0.3054; 0.6758; 0.6381; 1.7361; 0.9426; 1.6468; 1.2452; 1.5355; 3.1764; 2.6887; 4.9558; 6.8751;

0.2933; 0.6723; 0.5159; 0.6522; 0.7525; 1.9168; 0.8601; 2.5660; 1.0475; 2.3787; 4.2831; 7.8906;

0 0 0.0666 0 0.3118 0 0.5275 (*) 0 0.7483 2.0591 (*) 0.1886 0

0 0 0 0 0.2822 0 0.4384 (*) 0.4498 1.1202 (*) 3.0369 (*) 0.5942 1.7927

∗ Table 6.4. Rectangular plate, b = 0.5, h = 0.003, θ = π /4, vcr = 0.4346, vcr = 0.4121 ∗ v/vcr

0.5 1.0 2.0

Current method

B–G

0.7066; 1.8095; 1.5252; 1.8760; 3.4437; 4.0025; 4.8677; 7.1122;

0.6499; 0.7027; 1.3085; 2.7396; 2.8552; 2.3907; 3.3375; 7.0163;

0.02435 0 0.5838 (*) 0 2.5618 (*) 1.6035 (*) 0.3876 0

0 0 0.5407 (*) 0.3776 0.1886 2.3365 (*) 3.2750 (*) 1.1392

∗ values of λ on the stability parabola or beyond it. The relative flow velocity v/vcr is presented in the first columns of the tables. The second column of the tables contains the eigenvalues obtained by the method described here for the respective values of relative velocity v/vcr . Analysis of the results allows the following conclusions to be drawn: ∗ 1. The Bubnov–Galerkin method gives satisfactory estimates for the values of vcr if the number of terms in the formula for φ is not less than N ∼ 4a/b (two “halfwaves” along the smaller side and 2a/b “half-waves” along the larger side). 2. Upon the determination of the dependence λ = λ (v, θ ) and, consequently, the vibration modes, the Bubnov–Galerkin method gives an error that increases with the increase in the flow velocity and leads to the deterioration of qualitative results.

These conclusions are not final; investigation for plates of different geometries and different combinations of boundary conditions is required.

46 | 6 Rectangular plate Table 6.5. Clamped plate. θ

a

b

0 3π /8 π /2

0.18086698 0.28562681 0.088122509

88565587 16948368 29134275

Table 6.6. Simply supported plate. θ

a

b

0 3π /8 π /2

0.15705135 0.23725284 0.075931991

6172557.5 10966926 17242086

6.5 Dependence of critical flutter velocity on plate thickness Calculations were carried out for a rectangular plate with aspect ratio (1 : 2) for the same parameters as in the other examples and for three velocity vector angles θ = 0, 3π /8, and π /2. In all the cases, the dependence vcr = a + bh3 was confirmed. The values of a and b are given in Tables 6.5 and 6.6.

6.6 Dependence of critical flutter velocity on altitude As an example, consider a simply supported rectangular plate with aspect ratio 1 : 2 (b = 0.5) and relative thickness 0.005. Let the flow velocity vector be directed along the x-axis. Known data on air pressure and density, gravity acceleration and sound speed as functions of altitude above sea level were used, all other parameters were the same as in the above examples. Critical flutter velocity was calculated for altitudes from 0 to 11 km above sea level, with a step of 1 km. As before, calculations were carried out on two grids, 9 × 9 and 19 × 19. The results obtained coincided within the accuracy of calculation of the root of transcendent stability equation. In all the cases, stability was determined by the first eigenvalue. The following critical flutter velocities were obtained: 0.9322, 1.0181, 1.1203, 1.2416, 1.3853, 1.5556, 1.7577, 1.9997, 2.2836, 2.6249, 3.0340, and 3.5261. Thus, the critical flutter velocity increases with altitude above sea level. The results obtained are fitted quite well by the analytical formula v = (a + cx + ex2 + g3 + i4 + kx5 )/(1 + bx + dx2 + fx3 + hx4 + jx5 ), where v is the flutter velocity, x is the altitude above sea level. Here, a = 0.93220144, b = −0.03922232, c = 0.041901991, d = −0.0019833293, e = 0.002180686, f = 0.0002874881, g = 0.00012557829, h = −2.2255284e − 5, i = 2.7092604e − 6, j = 6.2616321e − 7, k = −9.2719828e − 7. This analytical formula is quite com-

6.6 Dependence of critical flutter velocity on altitude |

47

plex and does not allow a clear view of the increase in the critical flutter velocity. Therefore, results were also fitted by a simpler approximating formula v = a + bxc with a = 1.0064589, b = 0.021827843, and c = 1.9714133. On the altitudes below one kilometer, the error of this approximation is within a few percent, and for larger heights the quality of approximation is even better. Thus, a nearly quadratic dependence of the critical velocity on height is obtained.

7 Flutter of a rectangular plate of variable stiffness or thickness 7.1 Strip with variable cross section Vibrations of the strip are described by the equation 𝜕2 𝜕2 w 𝜕2 w 𝜕2 w 𝜕2 [h3 ( 2 + 𝜈 2 )] + 2(1 − 𝜈) (h3 ) 2 𝜕x𝜕y 𝜕x𝜕y 𝜕x 𝜕x 𝜕y +

𝜕2 w 𝜕2 𝜕2 w [h3 ( 2 + 𝜈 2 )] 2 𝜕y 𝜕y 𝜕x

+

𝛾p0 𝜕w ρ H 𝜕2 w ( = 0. + vn0 grad w) + a0 D0 𝜕t D0 𝜕t2

(7.1)

Here, p0 and a0 are freestream pressure and speed of sound, 𝛾 is the polytropic index, H = h0 h(x, y) is the plate thickness, D0 = Eh30 / (12(1 − 𝜈2 )), ρ , E, and 𝜈 are the density, Young’s modulus and Poisson’s ratio of the strip material. We seek the solution to equation (7.1) in the class of functions w = φ (y) exp(ω t − iax) subject to condition h = h(y); α is a positive real-valued parameter; coordinates x and y are nondimensionalized by l (strip width), with previous notation retained. As a result, we obtain from equation (7.1) 󸀠

h3 (α 4 φ − 𝜈α 2 φ 󸀠󸀠 ) − 2α 2 (1 − 𝜈) (h3 φ 󸀠 ) + [h3 (φ 󸀠󸀠 − 𝜈α 2 φ )]

󸀠󸀠

+ A1 M (−iαφ cos θ + φ 󸀠 sin θ ) + (A1 Ω + A2 hΩ2 ) φ = 0 𝛾p l3 A1 = 12(1 − 𝜈 ) 0 3 ; E h0 2

a2 l2 A2 = 12(1 − 𝜈 ) 20 2 ; c0 h0 2

c20

E = ; ρ

(7.2)

lω . Ω= a0

On the edges y = 0 and y = 1, boundary conditions of simple support φ = 0, φ 󸀠󸀠 = 0 are posed. The problem (7.2) consists of finding the eigenvalues Ω for a given function h(y), this can be solved numerically; however, in order to solve the optimization problem as well as reveal new mechanical effects, we obtain an approximate solution by the twoterm Bubnov–Galerkin method: φ = c1 sin π y + c2 sin 2π y. Following the standard procedure, we obtain from (7.2) a system of equations: iα 1 A1 M cos θ + A1 Ω + A2 c20 Ω2 ) 2 2 4 2 2 + c2 (β2 a11 + 4π α (1 − 𝜈)b11 − A1 M sin θ + A2 c11 Ω2 ) = 0 3 (7.3) 4 2 2 c1 (μ1 a11 + 4π α (1 − 𝜈)b11 + A1 M sin θ + A2 c11 Ω2 ) 3 iα 1 2 2 + c2 (μ2 a02 + 8π α (1 − 𝜈)b02 − A1 M cos θ + A1 Ω + A2 c02 Ω2 ) = 0. 2 2

c1 (β1 a20 + 2π 2 α 2 (1 − 𝜈)b20 −

7.1 Strip with variable cross section

|

49

Here, the following notation is introduced: β1 = α 4 + 2𝜈π 2 α 2 + π 4 ;

β2 = α 4 + 5𝜈π 2 α 2 + 4π 4

μ2 = α 4 + 8𝜈π 2 α 2 + 16π 4

μ1 = β2 ; 1

1 3

p

q

bpq = ∫ h3 cosp π y cosq 2π y dy

apq = ∫ h sin π y sin 2π y dy; 0

(7.4)

0

1 p

q

cpq = ∫ h sin π y sin 2π y dy;

p, q = 1, 2.

0

The condition of a characteristic determinant of the system (7.3) equal to zero relates M and Ω, and the main problem is to separate the subdomains of stable (Re Ω < 0) and unstable (Re Ω > 0) vibrations. The boundary between these subdomains corresponds to Ω = iΩ1 ; in this case we obtain from equation (7.3) (with straightforward notation) c1 (Au − iA0 ) + c2 A12 = 0;

c1 A21 + c2 (A22 − iA0 ) = 0.

The characteristic equation splits into two: A0 (A11 + A22 ) = 0;

A11 A22 − A12 A21 − A20 = 0.

(7.5)

It is easy to obtain that for small θ (θ 2 ≪ 1) in equation (7.5) one can take A0 = α A1 M cos θ − A1 Ω1 = 0, A11 A22 − A12 A21 = 0, then obtain from the former relation Ω1 = α M cos θ and substitute it into the latter one. The minimum with respect to α of the smallest positive root M0 of the obtained equation determines the critical velocity. On the other hand, for θ = π /2, A0 = −A1 Ω ≠ 0, therefore A11 + A22 = 0, A211 + A20 + A12 A21 = 0, and the subsequent procedure is similar to the one described above. (Note that for θ = π /2, due to symmetry, we have α = 0). From the continuity of the solution with respect to θ it follows that there exists a value θ = θ0 such that three conditions are satisfied: A0 = 0, A11 + A22 = 0, and A211 + A12 A21 = 0; analysis of this system is quite straightforward. Consider in detail a practically important case of h = 1 + ε f (y), (ε f )2 ≪ 1. We obtain for the parameters (7.4) 1

a20 =

1 1 + 3ε ∫ f sin2 π y dy = + 3ε a󸀠20 2 2 0

1

a02

1 1 = + 3ε ∫ f sin2 2π y dy = + 3ε a󸀠02 2 2 0

b20 c20

1 1 = − 3ε a󸀠20 ; b02 = − 3ε a󸀠02 2 2 1 1 = + ε a󸀠20 ; c02 = + ε a󸀠02 2 2

50 | 7 Flutter of a rectangular plate of variable stiffness or thickness 1

a11 = 3ε ∫ f sin π y sin 2π y dy = 3ε a󸀠11 0 1

b11 = 3ε ∫ f cos π y cos 2π y dy = 3ε b󸀠11 0 1

c11 = ε a󸀠11 ; it is assumed that ∫ f dy = 1. 0

System (7.3) is transformed as c1 [β11 + 3ε A󸀠11 a󸀠20 − A2 Ω21 (1 + ε a󸀠20 ) − iA0 ] + c2 (3εβ12 − c1 (3εβ21 +

8 A M sin θ ) = 0 3 1

8 A M sin θ ) + c2 [β22 + 3ε A󸀠22 a󸀠02 − A2 Ω21 (1 + ε a󸀠02 ) − iA0 ] = 0. 3 1

(7.6)

Here, β11 = (π 2 + α 2 )2 , β12 = β21 = (α 4 + 5𝜈π 2 α 2 + 4π 4 )a󸀠11 + 4π 2 α 2 b󸀠11 −A2 Ω21 a󸀠11 /3, β22 = (4π 2 + α 2 )2 , A󸀠11 = (π 2 − α 2 )2 + 4𝜈π 2 α 2 , and A󸀠22 = (4π 2 − α 2 )2 + 16𝜈π 2 α 2 ; it is assumed that ε 󸀠 = 2ε , hereafter the prime is omitted. According to the general scheme, for 0 ≤ θ < θ0 we have A0 = 0 and therefore Ω1 = α M cos θ ; denote A2 Ω21 = z to obtain (8A1 sin θ /3)2 M 2 = (8A1 sin θ /3)2 (Ω21 /α 2 cos2 θ ) = (8A1 tan θ /3α )2 (z/A2 ) ≡ C1 z. Now the characteristic equation (7.6) can be represented, within the accuracy up to ε 2 , in the form (the notation is straightforward): (b1 − a1 z)(b2 − a2 z) + C1 z = 0.

(7.7)

For the angles θ for which the estimate C1 ∼ ε holds true, an approximate solution to equation (7.7) takes the form z≅

β + 3ε A󸀠11 a󸀠20 b1 b1 β11 + C1 ≅ 11 + C1 . a1 b2 − b1 β22 − β11 1 + ε a󸀠20

(7.8)

According to the notation adopted, M 2 = (1/A2 cos2 θ )(z/α 2 ); by definition we take Mcr = minα (M), which is equivalent to minα (z/α 2 ); substituting (7.8) we obtain af1/4

2

ter simple transformations: αcr = π (1 + 36C0 /7π 4 ) , C0 = (8A1 tan θ /(3√3A2 π )) . Evidently, with high accuracy αcr = π . Substituting this result into (7.8), we obtain 2 = Mcr

=

1 z ( 2) 2 A2 cos θ α cr 8A1 tan θ 2 4π 2 4 󸀠 (1 ) . ( − ε a (1 − 3𝜈)) + 20 21 π 2 A2 cos θ A2 cos2 θ

(7.9)

The solution depends on the linear functional a󸀠20 ; we formulate the optimization problem in the following way. To find a function f (y) which maximizes Mcr (7.9) subject

7.1 Strip with variable cross section

| 51

to additional restrictions 1

1 2

∫ f (y) dy = 0;

∫ (f 󸀠 (y)) dy < c∗ ,

0

0

(7.10)

∗

where c is some constant. We write a functional 1 2

I(f ) = ∫ [f (y) sin2 π y + λ1 f (y) + λ2 (f 󸀠 (y)) ] dy. 0

From its stationary condition, after normalization, we obtain the solution f (y) = cos2 π y + k1 + k2 y + k3 y2 .

(7.11)

The parameters ks are obtained from the first equation (7.10) and the boundary conditions which we take in the form y = 0, f 󸀠 = −δ0 ; y = 1, f 󸀠 = δ0 . After that we obtain from equation (7.11) f (y) = cos 2π y − δ0 y(1 − y) + δ0 /6. The functional a󸀠20 turns out to be a󸀠20 = −0.25 (1 + δ0 /π 2 ). Substituting this into equation (7.9), we obtain 2 Mcr =

δ0 8A1 tan θ 2 ε 4 4π 2 (1 (1 ) . )) ( − 3𝜈) + + + (1 4 21 π 2 A2 cos θ A2 cos2 θ π2

Evidently, the optimization effect substantially depends upon 𝜈: if 3𝜈 < 1, it is necessary to choose ε > 0, and then h = 1 + ε f (y); if 3𝜈 > 1, it is necessary to choose h = 1 − ε f (y). Consider the limiting case θ = π /2; the solution, with high accuracy, will be valid in the domain θ0 ≤ θ ≤ π /2. From the system (7.6) we obtain the critical velocity (in his case αcr = 0): 2 2 A2 β + β11 β − β11 8 2 ( A1 ) Mcr = ( 22 + ε B0 ) + 1 ( 22 + ε Ω0 ) . 3 2 A2 2

(7.12)

Here, 2B0 = 3a󸀠02 A󸀠22 − 3a󸀠20 A󸀠11 − (β22 + β11 )a󸀠02 , 2Ω0 = 3a󸀠02 A󸀠22 − 3a󸀠20 A󸀠11 − (β22 + β11 )(a󸀠02 + a󸀠20 ). Denote additionally D = (β22 − β11 )2 /4 + A21 (β22 + β11 )/(2A2 ) and linearize equation (7.12) with respect to ε to obtain 2 A2 β − β11 8 2 ( A1 ) Mcr = D [1 + ε ( 22 B0 + 1 Ω0 )] . 3 D A2 D

52 | 7 Flutter of a rectangular plate of variable stiffness or thickness In the case being considered, α = 0, therefore β11 = A󸀠11 = π 4 , β22 = A󸀠22 = 16π 4 , B0 = (π 4 /2)(31a󸀠02 − 3a󸀠20 ), Ω0 = (π 4 /2)(31a󸀠02 − 14a󸀠20 ); we then write A21 β22 − β11 B0 + Ω ≡ g1 B0 + g2 Ω0 D0 A2 D0 0 π4 󸀠 [a02 (31g1 + 31g2 ) − a󸀠20 (3g1 + 14g2 )] 2 π 4 λ1 󸀠 λ π4 ≡ λ1 (a󸀠02 − 2 a󸀠20 ) ≡ (a02 − λ a󸀠20 ). 2 λ1 2 =

The optimization problem consists of finding the maximum value of the linear functional J1 = a󸀠02 − λ a󸀠20 subject to additional restrictions (7.10). The Lagrange function for this problem is L(f ) = f (y)(sin2 2π y − λ sin2 π y) + λ3 [f 󸀠 (y)]2 + λ4 f , and the stationarity condition leads to the equation 2λ3 f 󸀠󸀠 − sin2 2π y + λ sin2 π y − λ4 = 0. The normalized solution satisfying the first restriction (7.10) and boundary conditions f 󸀠 = 0 at y = 0 and y = 1 is f (y) =

1 cos 4π y − λ cos 2π y. 4

Accordingly, we obtain a󸀠02 = −1/16, a󸀠20 = λ /4, J1 = a󸀠02 − λ a󸀠20 = −(1/16 + λ 2 /4), g1 B0 + g2 Ω0 = −(π 4 /8)(1/4 + λ 2 ). Remember that during the derivation we substituted 2ε = ε 󸀠 (with the prime omitted); we revert to the former meaning of the parameter and change its sign, so that the effect of Mcr was positive; finally we obtain 1 h = 1 − ε ( cos 4π y − λ cos 2π y) 4 2 λ π4 1 8 2 ( A1 ) Mcr = D [1 + ε 1 ( + λ 2 )] . 3 4 4

The estimates show that for ordinary values of parameters the principal term in the expression for D is the first one: D ∼ (β22 − β11 )2 /4 = (15 ⋅ π 4 /2)2 , and therefore 2ε

β22 − β11 15π 4 ⋅ 4 ⋅ π 4 ⋅ 31 3 󸀠 (−a󸀠02 + B0 ∼ 2ε a ) ≅ ε /2, D 31 20 (15π 4 )2 ⋅ 2

2 and we obtain the ultimate estimate: (8A1 /3)2 Mcr ≅ D(1 + 0.5ε ) .

7.2 Rectangular plates Consider a rectangular plate occupying a domain S : {0 ≤ x ≤ x1 , 0 ≤ y ≤ l} on the x, y-plane. We take the flexural rigidity in the form D = D0 (h(y) + f2 (x)), D0 =

7.2 Rectangular plates |

53

E0 h2 (12(1 − 𝜈2 )), nondimensionalize the coordinates by the scale l, and assume, as usual, w = φ (x, y) exp(ω t); plate vibrations are then governed by the equation f1 (y)

𝜕4 φ 𝜕4 φ 𝜕4 φ + f (x) + 𝜈 (f (y) + f (x)) 2 1 2 𝜕x4 𝜕y2 𝜕x2 𝜕y2 +

𝜕2 φ 𝜕2 φ 𝜕2 𝜕2 (f ) (f ) + (x) (y) 2 1 𝜕x2 𝜕x2 𝜕y2 𝜕y2

+𝜈

𝜕2 φ 𝜕2 φ 𝜕2 𝜕2 (f (x) ) + 𝜈 (f (y) ) 𝜕x2 2 𝜕y2 𝜕y2 1 𝜕x2

+ 2(1 − 𝜈)

+ A1 Mn0 grad φ = λφ ; A1 = 12(1 − 𝜈2 )

(7.13)

𝜕2 φ 𝜕2 ((f1 (y) + f2 (x)) ) 𝜕x𝜕y 𝜕x𝜕y

𝛾p0 l3 ; E h3

A1 Ω + A2 Ω2 + λ = 0,

A2 = 12(1 − 𝜈2 )

a20 l2 , c20 h2

c20 =

Ω=

lω a0

E0 . ρ

In what follows, two types of boundary conditions are considered. For simple support, we have 𝜕2 φ x = 0, x = x1 : φ = 0, =0 𝜕x2 (7.14) 2 𝜕 φ y = 0, y = 1 : φ = 0, = 0. 𝜕y2 In the second case we consider the edge x = x1 as free, with corresponding conditions x = x1 :

𝜕2 φ 𝜕2 φ + 𝜈 =0 𝜕x2 𝜕y2 𝜕2 φ 𝜕2 φ 𝜕 [(f1 (y) + f2 (x)) ( 2 + 𝜈 2 )] 𝜕x 𝜕x 𝜕y + 2(1 − 𝜈)

(7.15)

𝜕2 φ 𝜕 [(f1 (y) + f2 (x) ] = 0, 𝜕y 𝜕x𝜕y

while on all other edged, x = 0, y = 0, andy = 1, boundary conditions (7.14) are posed. Problem 1. It was shown in Section 7.1 that for an infinitely long strip with simply supported boundaries placed in the flow parallel to its edges the optimal profile is h(y) = 1 − ε cos 2π y, ε 2 ≪ 1. Therefore, for a plate elongated in the flow direction (x1 ∼ 2 ÷ 3) we take f1 = 1 − ε f11 (y) and f2 (x) = 0, keeping in mind that the solution will be valid for θ 2 ≪ 1.

54 | 7 Flutter of a rectangular plate of variable stiffness or thickness Table 7.1. Critical flutter velocity for a simply supported plate. M(1) 0

M(1) 0.2

M(1) 0.4

M(2) 0

M(2) 0.2

M(2) 0.4

0.95 0.8

0.9525 0.7875

0.9335 0.775

0.3482 0.280

0.3482 0.277

0.351 0.275

Table 7.2. Critical flutter velocity for a plate with one free edge. M(1) 0

M(1) 0.2

M(1) 0.4

M(2) 0

M(2) 0.2

M(2) 0.4

0.883 0.770

0.8875 0.750

0.8625 0.7375

0.3194 0.2749

0.3168 0.2696

0.3141 0.2771

We take φ = u(x) sin π y, substitute it into equation (7.13) and apply the Bubnov– Galerkin procedure to obtain (1 − 2ε b1 ) uIV − 2π 2 [1 + 2ε (1 − 2𝜈)b1 − 2ε (1 − 𝜈)b2 ] uII + π 4 (1 − 2ε b1 )u + A1 M cos θ uI = λ u 1

1 2

b1 = ∫ f11 (y) sin π y dy; 0

(7.16)

b2 = ∫ f11 (y) dy. 0

This equation is solved with the boundary conditions (7.14) on the edges x = 0 and x = x1 . Calculations were performed for the following parameters: 𝛾 = 1.4; p0 /E = 1.43 ⋅ 10−6 ; a20 /c20 = 43.56; the relative thickness h/l, Poisson’s ratio and parameter of thickness variation ε were variable. It was taken that f11 = cos 2π y; therefore, b2 = 0 and b1 = 0.25. The results obtained are summarized in Table 7.1 (𝜈 = 0.3). Problem 2. Consider the boundary condition (7.15) on the plate edge x = x1 ; substitute into it f2 (x) = 0, f1 (y) = 1 − ε cos 2π y, multiply by sin π y and integrate with respect to y between zero and one to obtain x = x1 : uII − 𝜈π 2 u = 0 ε 3 uIII (1 + ) − π 2 [2 − 𝜈 − ε (1 − 𝜈)] u1 = 0. 2 2

(7.17)

Thus, we obtain an eigenvalue problem for the system (7.16) and (7.17). Results of calculations performed for the same parameter values are summarized in Table 7.2. In the tables, the critical M is presented; the subscript indicates the value of ε , superscript “1” corresponds to parameters A1 = 172.8 and A2 = 1881.8, superscript “2” corresponds to A1 = 825 and A2 = 5560. The first row is for x0 = 2, the second one is for x0 = 3.

7.2 Rectangular plates |

55

Table 7.3. Dependence of the critical flutter velocity on Poisson’s ratio.

𝜈 0.4 0.45

Problem 1 M(1) 0.2

M(1) 0.4

Problem 2 M(1) 0.2

M(1) 0.4

0.9625 0.800 0.9625 0.8125

0.9625 0.800 0.975 0.8125

0.900 0.7625 0.9125 0.775

0.900 0.7625 0.9125 0.775

Table 7.4. Dependence of the critical flutter velocity the sign of cross-section shape parameter.

ε

Bnd. cond. (7.14) x1 = 2 x1 = 3

Bnd. cond. (7.15) x1 = 2 x1 = 3

0 0.2 0.4 −0.2 −0.4

0.3482 0.3482 0.3508 0.3508 0.3456

0.3194 0.3168 0.3142 0.3142 0.3247

0.2801 0.2775 0.2749 0.2749 0.2880

0.2749 0.2697 0.2671 0.2671

It was obtained in Section 7.1 that for a strip in a flow parallel to its edges, the optimum profile depends substantially upon Poisson’s ratio of the strip material. Therefore, additional simulations were carried out; the results are summarized in Table 7.3. For each value of Poisson’s ratio, the upper row corresponds to x1 = 2, the lower one to x1 = 3. Analysis of the results confirms (quantitatively) the evident conclusions that the increase in relative length, the decrease in thickness, and the loosening of the right edge of the plate all diminish the critical flutter velocity; however, the interaction of parameters 𝜈 and ε turns out quite complex and irregular. Additional calculations were carried out for negative values of ε (with 𝜈 = 0.3, A1 = 825, A2 = 5560); the results are presented in Table 7.4. In the second and third columns, the results for the boundary conditions (7.14) are presented, while in the last two columns the corresponding values for the boundary conditions (7.15) are given, with f2 = 0. Clearly, in some cases the change in sign of ε leads to the increase in Mcr , while in other cases the critical velocity becomes lower. Additional studies are therefore necessary. Problem 3. In equation (7.13) we set f1 = 1 − ε cos 2π y, f2 = −0.2 cos βπ x, β x1 = 1, φ = u(x) sin π y and apply the Bubnov–Galerkin procedure to obtain φ uIV + 2φ1I uIII + (φ1II − 2π 2 φ2 ) uII + (A1 M cos θ − 2π 2 φ2I )uI + π 4 φ1 u = λ u ε φ1 (x) = 1 + − 0.2 cos2 βπ x (7.18) 2 ε φ2 (x) = 1 − (1 − 2𝜈) − 0.2 cos2 βπ x. 2 Equation (7.18) with the boundary conditions (7.14) constitutes an eigenvalue problem.

56 | 7 Flutter of a rectangular plate of variable stiffness or thickness Table 7.5. Dependence of the critical flutter velocity on Poisson’s ratio. 𝜈

M(1) 0.2

M(1) 0.4

M(2) 0.2

M(2) 0.4

M(3) 0.2

M(3 0.4

0.3

0.975 0.825 0.9875 0.8375 1.000 0.850

0.975 0.8125 1.000 0.8375 1.0125 0.850

0.4519 0.3556 0.4556 0.3593 0.4593 0.3630

0.4481 0.3519 0.4593 0.3593 0.4630 0.3630

8.250 5.000 8.375 5.125 8.375 5.125

8.125 4.750 8.375 5.000 8.500 5.125

0.4 0.45

The results of calculations are summarized in Table 7.5. The superscript on the critical value M has the following meaning: “1” means A1 = 172.8 and A2 = 1881.8; “2” means A1 = 583.2 and A2 = 4234.032; “3” means A1 = 17.28 and A2 = 1881.8; for each 𝜈 the upper row corresponds to x1 = 2, the lower one to x1 = 3. Comparative analysis of the data in Tables 7.3, 7.4, and 7.5 shows unambiguously the positive effect of stiffness variation in the direction of flow. One can expect, therefore, that the optimal stiffness function D(x, y) will be dependent on the flow velocity vector direction (in the above study, it was assumed that θ = 0). On the other hand, the data presented in Table 7.5 confirm the conclusion on the significant dependence of Mcr on Poisson’s ratio. The new mechanical effects revealed need to be studied in a wider range of problem parameters and for different types of function D(x, y).

8 Viscoelastic plates The flutter of viscoelastic plates was first studied by G. S. Larionov and V. I. Matesh. In the case of small viscosity, it was shown by application of the averaging method and the Bubnov–Galerkin method that the critical flow velocity is approximately two times lower than that of the corresponding elastic plate with instantaneous modulus, and this ratio is independent on the viscous properties of the material. Since only asymptotic stability was studied, this result raised some concerns. It was almost evident that the sufficient stability condition and its corresponding critical flow velocity can be found by solving the elastic problem with the instantaneous modulus substituted by its limit value. This assumption was substantiated by I. A. Kijko for lowviscosity material just on the basis of the Bubnov–Galerkin method. In what follows, we present a flutter problem solution for an infinitely long strip (exact and obtained from Galerkin approximation) with no assumption of low viscosity. A new principal result is obtained: the critical velocity is equal to the instantaneous-modulus velocity, while the viscous properties of the material affect the strip motion only in the subcritical domain. In a Cartesian system of coordinates, the strip occupies a domain 0 ≤ y ≤ l, |x| < ∞. One side of the strip is subjected to gas flow with the velocity vector v = vn0 , n0 = {cos θ , sin θ } and unperturbed parameters p0 , ρ0 , a0 (pressure, density, and speed of sound, respectively). The material of the strip is linearly viscoelastic, with stress related to strain by t

̂ (t). σ = E0 (ε (t) − ε0 ∫ Γ(t − τ )ε (τ )dτ ) ≡ E0 (1 − ε0 Γ)ε 0

where E0 is the instantaneous modulus, and ε0 is the viscosity parameter. Vibrations of the strip are described by the equation D0 (1 − ε0 Γ)̂ Δ2 w + ρ h

𝜕2 w 𝛾p0 𝜕w + ( + vn0 grad w) = 0, a0 𝜕t 𝜕t2

(8.1)

where D0 = E0 h3 / (12(1 − 𝜈2 )), h is the strip thickness, 𝛾 is the polytropic index, and ρ and 𝜈 are the density and constant Poisson’s ratio of its material. Equation (8.1) is solved for the simply supported boundary conditions y = 0,

w = 0,

𝜕2 w = 0; 𝜕y2

y = l,

w = 0,

𝜕2 w = 0, 𝜕y2

(8.2)

and the initial conditions are determined by the type of perturbation. The problem consists of finding the minimum flow velocity vcr , such that for v < vcr the perturbed motion is stable, while for v > vcr it is unstable. Assume that Γ(t) = exp(−β t), and introduce nondimensional coordinates x/l and y/l, time β t, and velocity M = v/a0 , retaining the previous notation. Equation (8.1)

58 | 8 Viscoelastic plates takes the form 2 ̂ 2 w + a3 Mn0 grad w + a2 𝜕w + a1 𝜕 w = 0 (1 − λ Γ)Δ 𝜕t 𝜕t2

(8.3)

with the following notation: a1 = 12(1 − 𝜈2 )

β 2 l4 ρ ; h2 E0

a3 = 12(1 − 𝜈2 )

𝛾p0 l 3 ( ) ; E0 h

a2 = 12(1 − 𝜈2 ) λ =

ε0 ; β

β l4 𝛾p0 h3 a0 E0

Γ = e−t .

For θ = 0, the perturbation which is limited at infinity and satisfies the conditions (8.2) is chosen in the form t = 0, w = c1 exp(−iα x) sin π y, 𝜕w/𝜕t = c2 exp(−iα x) sin π y, α ∈ R. Accordingly, we take w = A(t) exp(−iα x) sin π y, substitute it into equation (8.3), and perform Laplace transform with parameter s: ̃ = p2 (s) , A(s) p3 (s)

p2 (s) = (s + 1)(a1 c1 s + a1 c2 + a2 c1 )

(8.4)

p3 (s) = a1 s3 + (a1 + a2 )s2 + (M + a2 − iα a3 M)s + (1 − λ )M − iα a3 M. The original of (8.4) is restored in a straightforward way: A(t) = A1 es1 t + A2 es2 t + A3 es3 t a

Am =

(1 + sm ) (c2 + c1 ( a2 + sm )) 1

(sm − sj )(sm − sk )

(8.5) ,

m ≠ k ≠ j.

In the latter formulas, sm are roots of equation p3 (s) = 0, while μ = (α 2 + π 2 )2 . The solution of (8.5) decays exponentially (i.e. asymptotically stable) if ∀m, and we have Re sm < 0; if, however, for some root Re s∗m > 0, the solution is asymptotically unstable. On the boundary between the stable and unstable regions we have Re s∗m = 0, provided that the other two roots belong to the left half-plane. For M = 0 (free vibrations) we have Re sm < 0 for ∀m; with an increase in M, as will be demonstrated by calculations, one of the roots (let it be s1 ) is the first to approach the imaginary axis, and at some M ∗ it becomes purely imaginary. The values s∗1 = iy and M ∗ are easily found from equation p3 (iy) = 0; as a result, we obtain M ∗ = (μ + a2 − a1 y2 ) y/(α a3 ), where y must be substituted by the positive root of biquadratic equation y4 − (

μ μ − 1) y2 − (1 − λ ) = 0. a1 a1

The velocity M ∗ depends on the waviness parameter α ; by definition we take Mcr = M (αcr ), where αcr is obtained from the condition minα M ∗ . ∗

8 Viscoelastic plates |

59

Table 8.1. Comparison of critical velocities for elastic and viscoelastic strips.

β = 0.1 β = 0.01

λ =0

λ = 0.1

λ = 0.2

λ = 0.3

0.96029 0.96029

0.96065 0.96033

0.96102 0.96037

0.96138 0.96040

Calculations were performed for particular values of the parameters: p0 /E0 = 5 ⋅ 10−5 , ρ = 8 ⋅ 103 kg/m3 , 𝛾 = 1.4, 𝜈 = 0.3, a0 = 330 m/s , l/h = 3 ⋅ 102 . The results are summarized in Table 8.1; it turned out in all cases that αcr = π with an accuracy of four decimal digits. The value λ = 0 corresponds to the instantaneous-modulus velocity obtained from Melcr = 2π a2 /(a3 √a1 ). An important conclusion can be derived from the data of Table 8.1: Mcr and Melcr differ only in the fourth decimal digit, and this difference is reduced with an increase in relaxation time. In the domain M < Mcr , the variation of root s1 with the increase in M was studied. Up to M ∼ 0.1Mcr , we have Re s1 < Re sel 1 (the difference is within a few percent); for el M > 0.1Mcr the roots s1 and s1 coincide with the accuracy up to the fourth decimal digit; this is also a new obtained mechanical effect. For arbitrary angles θ , we represent the solution of equation (8.3) in the form of a three-term Bubnov–Galerkin approximation: w = (ck (t) sin kπ y) exp(−iα x), k = 1, 3. After application of this well-known procedure to the Laplace transforms, we arrive at a system of linear equations with respect to ck (s) : 8 a M sin θ ⋅ c2 (s) = Q1 (s) 3 3 24 8 a3 M sin θ ⋅ c1 (s) + (λ1 μ2 + B2 )c2 (s) − a M sin θ ⋅ c3 (s) = Q2 (s) 3 5 3 24 a M sin θ ⋅ c2 (s) + (λ1 μ3 + B2 ) ⋅ c3 (s) = Q3 , 5 3 (λ1 μ1 + B2 ) ⋅ c1 (s) −

(8.6)

where λ1 = 1 − λ /(s + 1), B2 = s(a1 s + a2 ) − iα Ma3 cos θ , μk = (α 2 + k2 π 2 )2 ; polynomials Qk (s) are determined by the initial conditions. The behavior of solutions ck (t) of the system (8.6) is determined by the roots of its determinant (polynomial of the ninth order); as before, a root s1 which is the fastest to approach the imaginary axis with an increase in M, was found, and the critical flutter velocity was obtained: Mcr = minα M ∗ (α ), provided that P9 (iy) = 0, iy = s∗1 . The results of the calculations are summarized in Table 8.2 (the parameters have the same values; β = 0.1, λ = 0.1).

60 | 8 Viscoelastic plates Table 8.2. Dependence of critical velocity on flow direction.

Mcr Mel cr

0

3π /8

58π /128

59π /128

62π /128

π /2

0.096066 0.096030

0.253636 0.253543

0.714807 0.714590

1.737786 1.737783

1.727327 1.727324

1.725347 1.725344

The same principal conclusion holds true: the critical flutter velocity Mcr practically el coincides with Mcr . The second important conclusion is that, similar to the elastic problem, in the vicinity of θ = π /2 there exists an angle θ0 near which (to the left) Mcr increases sharply, i.e. the plot Mcr (θ ) has a vertical tangent line; as θ passes θ0 , the vibrations of the strip change abruptly.

| Part II: Flutter of shallow shells

62 | Part II Flutter of shallow shells Vibration and stability problems for thin-walled structural elements of aircraft has attracted much attention in the rapidly developing realm of aerospace engineering. Some arising problems (flutter of conical shells, cylindrical panels, etc.) were considered by many authors, however, most of these works are not satisfactory from their statement point of view because the aerodynamic pressure Δp is described by the piston theory formula (sometimes, with some inessential corrections). An attempt at a more strict consideration of the “aerodynamic” part of the problem was recently undertaken (I. A. Kijko and M. A. Najafov) [355]. It was found that for an external flow past a slender axisymmetric body or airfoil the formula for Δp consists of two parts: quasistatic and a dynamic; the latter, in addition to the terms “of the kind of” those present in piston theory, involves a term with a mechanical meaning of membrane stresses in the mid-plane of the shell. Similar terms were obtained in the flutter problem formulation for a conical shell with internal supersonic gas flow [352].

9 General formulation Consider a shallow shell which forms a part of aircraft cover; the shell contour is formed by relatively stiff structural (load-bearing) elements of the aircraft structural frame. The shell is subjected to external loads of two types: vibration passed through the load-bearing elements, and aerodynamic forces exerted by air flow past the aircraft. Generally speaking, some other external forces of various physical nature (i.e. thermal, radiation, etc.) can act upon the shell, but we will not be focusing on them for now. As a rule, vibrations passed on the shell on its contour do not excite deflections dangerous to shell performance (fatigue effects are also not considered here). On the contrary: small-amplitude shell oscillations occurring randomly, including those due to the vibrations, at some particular (critical) flow velocities can become unstable under the influence of aerodynamic forces. This well-known effect is referred to as panel flutter. The mathematical model for the physical phenomenon just described includes equations for shell vibrations for the perturbed flow. Evidently, these equations are coupled: on the vibrating shell surface, a nonpermeability condition is posed for the flow, while shell vibrations are affected by the aerodynamic pressure. In such a general formulation, the problem is practically beyond the reach of (analytical, in the first instance) predictive methods, and some well-grounded simplifications are therefore inevitable. These simplifications relate to both the “shell” and “aerodynamic” sides of the model. Hereafter, we accept the following: 1. The simplest version of the nonlinear theory of shallow shells and plates is adopted; particular flutter problems are considered with their appropriate linearization. 2. Aerodynamic pressure is determined within the law of plane sections. 3. Shell motion is described by a class of functions containing the time factor in the form exp(ω t), and regimes determined for which the condition Re ω < 0 holds true, i.e. asymptotic stability is studied. The stress-strain state of a shallow shell is described by the normal deflection w(x, y, t) and stress function Φ(x, y, t); orthogonal coordinates x, y of the support plane coincide with high accuracy with the projections onto this plane of the principal curvature lines, and kx , ky are the principal (initial) curvatures. The membrane stresses in the mid-plane are related to Φ by the well-known formulas σx =

𝜕2 Φ , 𝜕y2

σy =

𝜕2 Φ , 𝜕x2

τ =−

𝜕2 Φ . 𝜕x𝜕y

64 | 9 General formulation The system of equations is taken in the form DΔ2 w = hL (w, Φ) + h (kx

𝜕2 w 𝜕2 Φ 𝜕2 Φ ) + q(x, y, t) − ρ h + k y 𝜕y2 𝜕x2 𝜕t2

𝜕2 w 𝜕2 w E Δ2 Φ = − L (w, w) − E (kx 2 + ky 2 ) 2 𝜕y 𝜕x L(u, v) =

(9.1)

𝜕2 u 𝜕2 v 𝜕2 u 𝜕2 v 𝜕2 u 𝜕2 v + − 2 . 𝜕x𝜕y 𝜕x𝜕y 𝜕x2 𝜕y2 𝜕y2 𝜕x2

Here, D = Eh3 / (12(1 − 𝜈2 )); also, E, 𝜈, and ρ are Young’s modulus, Poisson’s ratio, and density of plate material, respectively, and h is its thickness. The aerodynamic pressure q(x, y, t) will be shown below to consist of two parts, quasi-static q0 (x, y) and dynamic q1 (x, y, t); accordingly, we consider the basic state of the shell w0 (x, y) and Φ0 (x, y), and its perturbed state w1 (x, y, t), Φ1 (x, y, t): w = w0 (x, y) + w1 (x, y, t),

Φ = Φ0 (x, y) + Φ1 (x, y, t).

Substitute these relations into equation (9.1) and linearize it with respect to perturbations to obtain for the basic state DΔ2 w0 = hL(w0 , Φ0 ) + h (kx

𝜕2 Φ0 𝜕2 Φ0 + k ) + q0 (x, y, t) y 𝜕y2 𝜕x2

𝜕2 w 𝜕2 w E Δ Φ0 = − L(w0 , w0 ) − E (kx 20 + ky 20 ) . 2 𝜕y 𝜕x

(I)

2

This system is supplemented by the boundary conditions x, y ∈ Γ,

w0 = 0,

M1 (w0 ) = 0,

M2 (Φ0 ) = 0,

M3 (Φ0 ) = 0.

(I󸀠 )

The differential operators Ms are problem-specific and will be given in each particular case. The system of equations for perturbations takes the form DΔ2 w1 = hL (w1 , Φ0 ) + hL (w0 , Φ1 ) + h (kx Δ2 Φ1 = −

𝜕2 w 𝜕2 Φ1 𝜕2 Φ1 ) + q1 (x, y, t) − ρ h 21 + ky 2 2 𝜕y 𝜕x 𝜕t 2

(II)

2

𝜕 w 𝜕 w E L(w0 , w1 ) − E (kx 21 + ky 21 ) . 2 𝜕y 𝜕x

Perturbations w1 and Φ1 are subject to initial conditions and the same homogeneous boundary conditions x, y ∈ Γ,

w1 = 0,

M1 (w1 ) = 0,

M2 (Φ1 ) = 0,

M3 (Φ1 ) = 0.

(II󸀠 )

In what follows we retain the previous notation for the basic state, and denote perturbations by the same symbols with subscripts omitted. Possible simplifications of equation systems (I) and (II) are considered individually in the cases of the shell and plate.

9 General formulation |

65

Shell. If the quasistatic variations of curvatures 𝜕2 w0 /𝜕x2 𝜕2 w0 /𝜕y2 can be neglected in comparison with the initial curvatures kx and ky , then the terms L(w0 , Φ0 ) and L(w0 , w0 ) in equation (I) can be omitted, and the system (I) becomes linear. Accordingly, in equations (II) the terms L(w0 , Φ) and L(w0 , w) are omitted. Plate. In this case, kx = ky = 0 and equations (I) are, in fact, the well-known equation system by T. von Karman. In equations (II), the principal role will be played by membrane stresses described by the term L(w, Φ0 ); therefore the term L(w0 , Φ) can be omitted, and the second equation can be totally dropped from consideration. As a result, we obtain the governing equation for a plate: DΔ2 w = hL(w, Φ0 ) + q1 (x, y, t) − ρ h

𝜕2 w , 𝜕t2

(III)

supplemented by conditions on the contour x, y ∈ Γ,

w = 0,

M(w) = 0

(III󸀠 )

and corresponding initial conditions. One can see that the problem can be closed in the mathematical sense, provided that the aerodynamic pressure is known in the basic and perturbed states, q0 (x, y) and q1 (x, y, t); this problem is tackled in the next chapter.

10 Determination of aerodynamic pressure Consider a slender axisymmetric body or an airfoil in a gas flow with high supersonic velocity and zero angle of attack; the flow velocity vector is directed along the body axis (or orthogonally to the airfoil edge), the x-axis is aligned with the velocity vector, the y-axis is directed along the airfoil edge, while the z-axis forms a right-handed coordinate system (for the body, the direction of the y-axis is arbitrary). Suppose that the deformable part of a body or airfoil surface occupies the domain [x1 , x2 ] and consider the axisymmetric deflections of the shell of the revolution, or the cylindrical bending of a shallow cylindrical shell. Let the underformed genera󵄨 󵄨 trix be described by the equation z1 = kx + φ (x), with 󵄨󵄨󵄨φ (x)/(kx)󵄨󵄨󵄨 ≪ 1, i.e. the body of revolution has only small deviations from a cone, and the airfoil from a wedge. Then ̃ t), w(x, ̃ t) = w cos(n, z), where n on the deformable part we have z = kx + φ (x) − w(x, is the outer normal to the surface z1 . Within the same accuracy as that of the law of plane sections, we have w̃ ≅ w, and therefore z = kx + φ (x) − w(x, t).

(10.1)

According to the law of plane sections, the gas state between the body and shock wave (SW) is determined from the solution of a problem on time-dependent planar flow in the plane x = vx t, where vx = const is the flow velocity caused by expansion by a piston according to the law z(t) = kvx t + φ (vx t) − w(vx t, t).

(10.2)

We construct the solution by expansion with respect to a small parameter, the ratio of gas densities in front of SW and behind it: 2a20 𝛾−1 ρ0 ] ≡ ε a(D), [1 + = ∗ ρ 𝛾+1 (𝛾 − 1)D2

(10.3)

where a0 is the freestream speed of sound, D is the speed of SW. If the expansion performed with respect to parameter ε , then the requirement that (ρ 0/ρ ∗ )2 ≪ 1 entails the condition a(D) ∼ 1, which will be used below. Now we estimate the flow velocity vx which ensures that the above inequality holds true. From the condition (ρ 0 /ρ ∗ )2 < 1 follows that a20 /D2 ≪ 1; due to the assumption |(φ − w)/(kx)| ≪ 1 an estimate D ∼ δ kvx is valid for the speed of SW, with δ ∼ 1; therefore we finally obtain that (δ kM)2 ≫ 1. Introduce the Lagrangian variables time t and coordinate z, so that dz = ρ 0 rμ −1 dr, where r is the distance from the particle to the axis or plane of symmetry at the initial time instant, ρ 0 is the initial density, μ = 2 for cylindrical waves, and μ = 1 for planar ones. The unknown functions to be found are the distance from the particle to the axis or plane of symmetry ζ = ζ (t, z), pressure p = p(t, z), and density ρ = ρ (t, z).

10 Determination of aerodynamic pressure

|

67

The momentum, mass, and energy equations are 𝜕2 ζ 𝜕p = −ζ μ −1 ; 𝜕z 𝜕t2

𝜕ζ 1 ; = 𝜕z ρζ μ −1

𝜕 p ( ) = 0. 𝜕t ρ 𝛾

(10.4)

The solution must satisfy the conditions on the shock wave z = z∗ : p∗ =

2 ρ 0 D2 − ε p0 , 𝛾+1

ρ∗ =

−1 1 0 ρ (1 + 2a20 /(𝛾 − 1)D2 ) ε

(10.5)

and the boundary condition on the piston ζ (t, 0) = kvx t + φ (vx t) − w(vx t, t),

z = 0,

(10.6)

where p0 is the freestream pressure. We represent the solution by expansion with respect to parameter ε : ζ = ζ0 + εζ1 + ⋅ ⋅ ⋅ ;

ρ = ε −1 ρ0 + ρ1 + ⋅ ⋅ ⋅

p = p0 + ε p1 + ⋅ ⋅ ⋅ ;

and substitute it into equation (10.4); in this way we obtain equation systems for the zero-th and first-order approximations having solutions representable either in the explicit form or in quadratures. The zero-th-order approximation is ζ0 = ζ0 (t),

1−μ

p0 = P(t) − zζ0

𝜕2 ζ0 , 𝜕t

1/𝛾

ρ0 =

p0 ϑ0 (z)

(10.7)

and the first-order approximation is ζ1 =

1

z −1/𝛾

∫ ϑ0 (z)p0

μ −1 ζ0 z ∗

dz + ζ1∗ (t)

z

z

𝜕2 ζ 1 𝜕2 ζ 1 p1 = (μ − 1) 20 μ ∫ ζ1 dz − μ −1 ∫ 21 dz + p∗1 (t) 𝜕t ζ0 ∗ ζ0 z∗ 𝜕t z

(10.8)

p1 ρ − 𝛾 1 = ϑ1 (z) p0 ρ0 In the latter relations, ζ0 (t), P(t), ϑ0 (z), ζ1∗ (t), p∗1 (t), and ϑ1 (z) are other arbitrary functions; the lower integration limit z∗ is to be determined below. Assume that ζ0 (t) is the shock wave propagation law: then on the shock wave we μ have z∗ = ρ 0 ζ0 (t)/μ . Substituting the above expansions into the conditions (10.5) on μ SW, we obtain that at z = z∗ = ρ 0 ζ0 (t)/μ ζ0 = ζ0 (t),

p0 =

ζ1 = 0,

2 ρ 0 ζ0̇ 2 , 𝛾+1 0

p1 = −p ,

where the dot denotes the time derivative.

ρ0 =

ρ1 = 0,

ρ0 a (ζ ̇ )

(10.9)

68 | 10 Determination of aerodynamic pressure In the following, it is convenient to turn from the Lagrangian variable z to variμ able τ defined by z = ρ 0 ζ0 (τ )/μ ; evidently, τ (z) is the time instant at which the shock μ wave passes the particle with the Lagrangian coordinate z; therefore, z∗ = ρ 0 ζ0 (t)/μ . Substitute equations (10.7) and (10.8) into equation (10.9), and determine the arbitrary functions involved in the solution; the required expressions for p0 , p1 , and ζ1 take the form p0 =

2 1 1−μ ρ 0 ζ0̇ 2 + ρ 0 ζ0 ζ0̈ − ζ0̈ ζ0 z 𝛾+1 μ t

p1 = − (μ − 1)

+

ζ1 = −

t

ρ0 μ −1

ζ0

∫ τ

𝜕2 ζ1 μ −1 ζ (ξ )ζ0̇ (ξ ) dξ − p0 𝜕t2 0

(10.10)

t

1 μ −1

ζ0

ρ 0 ζ0̈ μ −1 (ξ )ζ0̇ (ξ ) dξ μ ∫ ζ1 (t, ξ )ζ0 ζ0 τ

1+2/𝛾 ∫ a (ζ0̇ (ξ )) ψ (t, ξ )ζ0̇ (ξ ) dξ ≡ ζ1 (t, τ ) τ −1/𝛾

μ

ζ (ξ ) 1 ψ (t, ξ ) = [ζ0̇ 2 (t) + ζ0 (t)ζ0̈ (t) (1 − 0μ )] μ ζ0 (t)

.

Thus, the solution is expressed in terms of a yet unknown law of SW propagation ζ0 (t). An equation from which this law can be found is obtained from the condition on the piston: for τ = 0 (z = 0) the following equality must be satisfied (if we limit ourselves to the solution terms linear in ε ): ζ (t) = ζ0 (t) + εζ1 (t, 0) = z(t), which can be rewritten, using the notation ξ =t

ζ1 (t, 0) = −F {t, ξ ; ζ0 (t), ζ0 (ξ )}ξ =0 , in the form ξ =t

ζ0 (t) = ε F {t, ξ ; ζ0 (t), ζ0 (ξ )}ξ =0 + z(t),

(10.11)

with the function z(t) given by (10.2). The functional F is essentially nonlinear, which makes an analytical solution of equation (10.11) practically impossible. However, the presence of a small parameter indicates that it is possible to solve this equation by the method of sequential approximations: ξ =t ζ0(0) (t) = z(t), ζ0(n+1) (t) = ε F {t, ξ ; ζ0n (t), ζ0n (ξ )}ξ =0 + z(t). (10.12) Leaving aside the general analysis, we point out one consideration in favor of the possible convergence of the sequence (10.12). In the case φ = w = 0, equation (10.11) has an exact solution ζ0̃ (t) = Dt, where D is the root of quadratic equation D = ε Da(D)/μ + u,

u = kvx .

10 Determination of aerodynamic pressure

|

69

In this case, the sequence (10.12) leads to an iterative process for the determination of D: D(0) = u, D(n+1) = u + ε D(n) a(D(n) )/μ , which converges for a0 /u < [(𝛾 + 1)(μ + ε )/2]1/2 . In particular, for μ = 1 it follows that k𝛾M > 1, which agrees with the estimate obtained above. In the linear theory of thin shallow shells it is assumed that (φ /ut)2 ≪ 1, (w/φ ) ≪ 1, and therefore, one can expect that the sequence (10.12) will be convergent under some conditions which are, in some sense, very close to that presented above. We no perform some preliminary estimates. Let l be the characteristic shell size in the flow direction: then the characteristic flow time is t1 = l/vx . The characteristic shell vibration time is t2 = l2 /(ch), where A = (E/ρ̃ )1/2 , with E and ρ̃ being Young’s modulus and shell material density, while h is its thickness; therefore t1 /t2 = ch/(vx l) ≤ 1 and φ̇ /u ∼ φ /(t1 u) ≪ 1,

ζ0 φ̈ /ζ0̇ 2 ∼ φ /(kl) ≪ 1.

Since (w/φ )2 ≪ 1, the following estimates hold true a fortiori: ̇ ≪ 1, w/u

̈ ζ0̇ 2 ≪ 1. ζ0 w/

These estimates will be relied upon in the further derivations. Calculate the first-order approximation in equation (10.12): ξ =t

ζ0(1) (t) = ε F{t, ξ ; Z(t), Z(ξ )}ξ =0 + Z(t).

(10.13)

All expressions of the type of 1 + (φ − w)/(ut) and similar to them, involved in ζ1 (t, τ ) and F in different powers, are expanded in series, with necessary further transformations performed. Equation (10.13) is presented in the final form containing only the terms linear in φ − w = W and respective derivatives ζ0(1) (t) = ut + ε uta(u) + (1 +

ε a(u)) W(t) μ

ε 2ε 2 ̇ ̈ a(u)W(t)t − a(u)W(t)t μ𝛾 2μ 2 𝛾

−

(10.14)

t

+

2𝛾 a20 1−μ 2ε ̇ ) dτ . )t ∫ τ μ −1 W(τ (a(u) − 𝛾 𝛾 − 1 u2 0

In this and subsequent relationships it is assumed that φ (τ ) ≡ φ (vx τ ), w(τ ) ≡ w(vx τ , t). Calculate now the second-order approximation in equation (10.12), retaining only the terms linear in ε , of the kind of ε W, ε Ẇ etc. It is easy to see that ζ0(2) is obtainable from ζ0(1) by the following substitutions in equation (10.14): u1 t = (u + ε ua(u)/μ )t → u2 t = (u + ε u1 a(u1 )/μ )t,

a(u) → a(u1 ).

70 | 10 Determination of aerodynamic pressure Evidently, this solution structure will hold true in approximations of any order (within the above accuracy in ε ). The iterative process for un , as was shown above, converges to D, and therefore we finally obtain ζ0 (t) = Dt + (1 + ε a(D)/μ ) W(t) −

2ε ε 2 ̇ ̈ a(D)W(t)t − a(D)W(t)t μ𝛾 2μ 2 𝛾 (10.15)

t

2ε ̇ ) dτ . + ((1 − 𝛾)a(D) + 𝛾) t1−μ ∫ τ μ −1 W(τ 𝛾 0

From equation (10.10) we now find out ζ1 (t, τ ): ζ1 (t, τ ) = −

a(D) 2a(D) ̇ τm τm Dt (1 − m ) + W(t)t (1 − m ) μ t μ𝛾 t t

−

2 ̇ ) dτ ((1 − 𝛾)a(D) + 𝛾)t1−μ ∫ τ μ −1 W(τ 𝛾

(10.16)

τ

τm + (μ − 1)a(D)W(t) (1 − m ) t a(D) ̈ τm τ μ −1 W(t)t2 (1 − m ) . )+ μ −1 2 t t 2μ 𝛾 2

− a(D) (W(t) − W(τ )

In fact, the problem was solved because p1 is expressed in terms of ζ0 and ζ1 by simple quadratures.

11 The shallow shell as part of an airfoil In this case (planar waves) μ = 1 and we obtain from equations (10.15) and (10.16) that 2ε a(D)) W(t) 𝛾 ε 2ε 2 ̇ ̈ a(D)W(t)t − a(D)W(t)t − 𝛾 2𝛾 2a(D) ζ1 (t, τ ) = − Da(D)(t − τ ) − (2 − 3a(D) + ) (W(t) − W(τ )) 𝛾 a(D) ̈ 2a(D) ̇ W(t)(t − τ ) + W(t)(t − τ )2 . + 𝛾 2𝛾 ζ0 (t) = Dt + (1 + 2ε +

Substitute these relations into the first two relations from equation (10.10) to obtain the pressure on the piston (z = 0, i. e., τ = 0): p|τ =0 ≅ (p0 + ε p1 )τ =0 =

2ρ 0 D2 − ε p0 𝛾+1

̈ 4ρ 0 DW(t) 12ε a(D) ̈ + [1 − (1 + 2ε − ε a(D)) + ρ 0 DW(t)t ] + ⋅⋅⋅ . 𝛾+1 𝛾(𝛾 + 1)

(11.1)

The dots in equation (11.1) denote the terms of higher order in comparison with the previous ones; these terms are dropped hereafter. We now consider the flow past an airfoil in the Eulerian coordinate system fixed with the body. Take into account that t = x/vx ,

Ẇ = 𝜕W/𝜕t + vx 𝜕W/𝜕x.

Substitute these relations into equation (11.1) to obtain the pressure drop on the airfoil side surface: Δp = p − p0 =

2 (ρ 0 D2 − 𝛾p0 ) 𝛾+1

−

4ρ 0 D 𝜕w 𝜕w ) (1 + 2ε − ε a(D)) ( + vx 𝛾+1 𝜕t 𝜕x

−

ρ 0 Dx 12a(D) 𝜕2 w 𝜕2 w 𝜕2 w (1 − ε ) ( 2 + 2vx + vx2 2 ) vx 𝛾(𝛾 + 1) 𝜕t𝜕x 𝜕t 𝜕x

+

4ρ 0 Dvx 𝜕φ 𝜕2 φ 12a(D) )x 2 . (1 + 2ε − ε a(D)) + ρ 0 Dvx (1 − ε 𝛾+1 𝜕x 𝛾(𝛾 + 1) 𝜕x

In accordance with the above assumption, Δp = q0 (x, y) + q1 (x, y, t).

(11.2)

72 | 11 The shallow shell as part of an airfoil The quasi-static loading part q0 corresponds to the basic state w0 (x, y), Φ0 (x, y). From equation (11.2) we obtain q0 (x, y) =

2 (ρ 0 D2 − 𝛾p0 ) 𝛾+1 −

4ρ 0 Dvx 𝜕φ 𝜕w0 (1 + 2ε − ε a(D)) ( ) − 𝛾+1 𝜕x 𝜕x

+ ρ 0 Dvx x (1 − ε q1 (x, y, t) = −

(11.3)

𝜕2 φ 𝜕2 w0 12a(D) ) )( 2 − 𝛾(𝛾 + 1) 𝜕x 𝜕x2

4ρ 0 D 𝜕w 𝜕w (1 + 2ε − ε a(D)) ( + vx ) 𝛾+1 𝜕t 𝜕x

ρ 0 Dx 𝜕2 w 𝜕2 w 12a(D) 𝜕2 w (1 − ε ) ( 2 + 2vx + vx2 2 ) . − vx 𝛾(𝛾 + 1) 𝜕t𝜕x 𝜕t 𝜕x

(11.4)

Here, the function of dynamic (perturbed) deflection was written with subscript “1” omitted. Estimate now the order of magnitude of different terms in the formulas obtained. 󵄨 󵄨 In the linear theory of shallow shells it is adopted that 󵄨󵄨󵄨w0 /φ 󵄨󵄨󵄨 ≪ 1; therefore in equation (11.3) the terms containing w0 can be neglected; it is easy to prove that the remaining terms are of the same order of magnitude. In equation (11.4) compare the “added mass” (ρ 0 Dx/vx ) (1 − 12ε a(D)/(𝛾(𝛾 + 1))) with the specific mass of the shell ρ̃ h; since the second multiplier is of the order of unity, we obtain ρ 0 Dx/(vx ρ̃ h) ∼ ρ 0 kl/(ρ̃ h). For the usual ranges of these parameters the ratio will be of the order from 10−1 to 10−3 , i.e. in the first approximation the term with the “added mass” can be neglected. The characteristic shell oscillation time is t2 = l2 /(ch), and therefore, the ratio (2vx 𝜕2 w/𝜕t𝜕x)/(vx2 𝜕2 w/𝜕x2 ) ∼ ch/(vx l) will be of the order of 10−1 to 10−2 . The ultimate conclusion is therefore that the main input into the dynamic component of pressure acting on the shell is provided by the terms 4ρ 0 D 𝜕w 𝜕w ) q1 (x, y, t) = − (1 + 2ε − ε a(D)) ( + vx 𝛾+1 𝜕t 𝜕x (11.5) 12a(D) 𝜕2 w 0 ) . − ρ Dvx x (1 − ε 𝛾(𝛾 + 1) 𝜕x2 The first of them is the traditional term of the piston theory, however, with a coefficient which depends in rather complicated way on the flow velocity; the second term represents the compressive normal load in the mid-surface of the shall which, evidently, can affect substantially the vibration mode and critical flutter velocity.

11 The shallow shell as part of an airfoil |

73

In the problem of the vibration of a plate occupying some domain on a wedge face, φ = 0 and, generally speaking, there are no a priori reasons to neglect any terms in the quasistatic loading q0 from (11.3). The expression for the dynamic loading (11.5) also remains intact.

12 The shallow shell of revolution In this case (cylindrical waves) μ = 2, and we obtain from equations (10.15) and (10.16) that ε ε ̇ ζ0 (t) = Dt + (1 + 2ε + a(D)) W(t) − a(D)W(t)t 2 𝛾 t

1 ε 2 ̈ − 2ε ∫ W(ξ ) dξ − a(D)W(t)t 8𝛾 t 0

2

ζ1 (t, τ ) =

ξ2 Da(D) ξ a(D) ̇ ( ) − t) + W(t) (t − 2 t 𝛾 t t

ξ2 2 ∫ W(s) ds + W(t) (a(D) 2 − 2 (1 + a(D))) + 𝛾t t τ

ξ4 a(D) ̈ + W(t) (t2 − 2ξ 2 + 2 ) . 8𝛾 t We substitute these relations into equation (10.10), and perform estimates similar to those of the previous section; as a result, we obtain the pressure drop on the shell surface: Δp = (p + ε p1 − p0 )τ =0 = +

2ρ 0 D2 a(D) 𝛾p0 (𝛾 + 1) ] [1 + ε − 𝛾+1 4 2ρ 0 D2

4ρ 0 D 3ε 11a(D) ̇ (1 + −ε ) W(t) 𝛾+1 4 8𝛾

+ 2ε

ρ 0 DW(t) 9a(D) 5𝛾 + 1 (1 + ) t 8 𝛾(𝛾 + 1)

(12.1) t

+

ρ 0D ̈ ρ 0D 3a(D) 4 ) + 2ε ( W(t)t (1 − ε − 1) 2 ∫ W(ξ ) dξ 2 2𝛾(𝛾 + 1) 𝛾+1 t 0

0

+ 2ε

t

1 + 2𝛾 ρ D ∫ ξ 2 W(ξ ) dξ . 𝛾 t4 0

̇ ̈ The coefficients of W(t), W(t)t, and W(t)/t were regrouped in such a way that the terms of the order of ε 2 were singled out and then dropped. In comparison with equation (11.1), equation (12.1) contains new terms, the ones of the Winkler foundation type, and the integral ones. Let us estimate their order of magnitude. After transformation to Eulerian coordinates vx t = x, vx t1 = x1 , we obtain

12 The shallow shell of revolution

|

75

sequentially ερ 0 D

φ −w W(t) φ (x) − w(x, t) 0 = ερ Dvx = 2kρ 0 Dvx t x kx

ερ 0 D

1 1 ∫ W(ξ ) dξ − ερ 0 D 2 (φ (t)t − w(t,̃ t)(t − t1 )) t2 t

t

0

= ε kρ 0 Dvx

w(x0󸀠 , t)(x − x1 ) φ (x0 ) − kερ 0 Dvx kx kx

t

ερ 0 D

1 1 ∫ ξ 2 W(ξ ) dξ = ερ 0 D 4 (φ (t0̃ )t3 − w(x0󸀠󸀠 , t)(t3 − t13 )) t4 3t 0

=

x3 φ (x̃0 ) ε k w(x̃1 , t) εk ρ0 Dvx − ρ0 Dvx (1 − 13 ) , 3 kx 3 kx x

where the latter two relations were obtained from the mean value theorem, with tilde and prime denoting some mean values of corresponding quantities. Since 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨φ (x)/(kx)󵄨󵄨󵄨 < 1, a fortiori 󵄨󵄨󵄨w(x, t)/(kx)󵄨󵄨󵄨 < 1, k ∼ ε , and, therefore, ε k ≪ 1. We conclude that the terms written above are of the order of ε 2 and can be dropped in the following considerations. Estimates for the “added mass” and second-order mixed derivative 𝜕2 w/𝜕t𝜕x are identical to those obtained in the previous section; therefore we finally obtain the dynamic pressure acting on the flow side of shell surface: q1 (x, t) = − −

4ρ 0 D 𝜕w 3ε 11a(D) 𝜕w (1 + )( ) −ε + vx 𝛾+1 4 8𝛾 𝜕t 𝜕x ρ 0 Dρ 0 vx x 3a(D) 𝜕2 w (1 − ε ) 2. 2 2𝛾(𝛾 + 1) 𝜕x

(12.2)

The quasi-static pressure part will take the form q0 (x) =

𝛾p0 2ρ 0 D2 a(D) ) (1 + ε − 𝛾+1 4 2ρ 0 D2 +

4ρ 0 Dvx 𝜕φ 𝜕w0 3ε 11a(D) (1 + −ε )( − ) 𝛾+1 4 8𝛾 𝜕x 𝜕x

+

𝜕 2 φ 𝜕 2 w0 ρ 0D 3a(D) ). vx x (1 − ε )( 2 − 2 2𝛾(1 + 𝛾) 𝜕x 𝜕x2

(12.3)

Evidently, the principal result of thischapter is that the dynamic pressure part at a given point of a vibrating shell is expressed in terms of the shell deflection and its derivatives at this point; of course, this is a consequence of the law of plane sections. In this case, the system of equations governing the shell motion becomes linear and homogeneous with respect to w, and the boundary conditions are also homogeneous. Therefore, looking for a solution in the form w = Ψ(x, y) exp(ω t), Φ = F(x, y) exp(ω t), we arrive at a spectral problem with eigenvalues λ generated by the equation α1 ω 2 + α2 ω + λ = 0.

(12.4)

76 | 12 The shallow shell of revolution The non-self-adjoint operator of flutter problems possesses, generally speaking, complex-valued roots, depending on the flow velocity. It follows from equation (12.4) that the domains of stable (Re ω < 0) and unstable (Re ω > 0) vibrations are separated on the complex plane λ by the stability parabola α22 Re λ = α1 (Im λ )2 . For vx = 0, all λ are real-valued (these are natural vibration frequencies); with the increase in flow velocity some eigenvalues enter the complex domain. Therefore, the problem consists in finding out the eigenvalue which is the first to reach the stability parabola; the flow velocity corresponding to this number will be the critical flutter velocity. The eigenvalue problem for a system of non-self-adjoint operators can be studied analytically only in some simple cases (as a rule, approximate solutions are obtained by the Bubnov–Galerkin method), and therefore, modern computational methods must be applied. A method of this type, a numerical-analytical nonsaturating method, was developed by the authors and is presented below. In the end of this chapter we formulate the spectral problems, for which detailed studies will be carried out below. A plate in a flow at zero angle of attack. In this case, q0 = 0, and therefore Φ0 = 0; the influence of a reflected wave can be neglected, and for q1 (x, y, t) we obtain a 𝛾p generalized formula of the piston theory q1 = − a 0 ( 𝜕w + vn0 grad w) 𝜕t 0

By setting w = Ψ(x, y) exp(ω t), we obtain from (III) and (III󸀠 ) 𝛾ρ 0 0 vn grad Ψ = λ Ψ a0 (x, y) ∈ Γ, Ψ = 0, M(Ψ) = 0, D∗ Δ2 Ψ +

where D∗ is the cylindrical stiffness. A plate as a part of a wedge face. Assume V = {vx , 0}, then we obtain from equation (11.5) 𝜕2 w 𝜕w 𝜕w ) − a2 (D)x 2 . q1 = −a1 (vx , D) ( + vx 𝜕t 𝜕x 𝜕x As before, for the amplitude function Ψ(x, y) we obtain an equation D∗ Δ2 Ψ + a2 (D)xvx

𝜕2 Ψ 𝜕Ψ + a01 (D)vx + hL(Ψ, Φ0 ) = λ Ψ, 𝜕x 𝜕y2

supplemented by the boundary conditions; the stress function Φ0 (x, y) of the base state is obtained from the system (I), while q0 is obtained from (11.3) with φ = 0.

12 The shallow shell of revolution

| 77

The shallow cylindrical panel; velocity vector belongs to the base plane. In this problem q0 = 0, Φ0 = 0, kx = 1/R, ky = 0, where R is the shell radius. The linear formulation of the problem follows from (II) and (II󸀠 ), while q1 is described by a formula of the generalized piston theory: D∗ Δ2 Ψ +

𝛾ρ 0 0 h 𝜕2 F vn grad Ψ − = λΨ 0 R 𝜕y2 a Δ2 F +

x, y ∈ Γ,

Ψ = 0,

E 𝜕2 Ψ =0 R 𝜕y2

M1 (Ψ) = 0,

M2 (F) = 0,

M3 (F) = 0.

The conical shell as part of a slender cone. The position of a point on the cone surface can be described conveniently by coordinates s and θ , where s is the distance from the cone tip (x = s cos α ), θ = χ sin α , χ is the polar angle reckoned from some fixed generatrix. The system of governing equations differs from (II), therefore we present it here: 𝜕2 w DΔ2 w − hΔk Φ − hL(Φ, w) = q0 + q1 − ρ h 2 𝜕t (12.5) 1 2 Δ Φ + EΔk w + EL(w, w) = 0. 2 Here, 1 𝜕w 1 𝜕Φ 1 𝜕2 w 𝜕2 Φ 1 𝜕2 Φ 𝜕2 w ) + 2 2) 2 + ( + 2 s 𝜕s s 𝜕s s 𝜕θ 𝜕s s 𝜕θ 2 𝜕s2 𝜕 1 𝜕Φ 𝜕 1 𝜕w )( ) − 2( 𝜕s s 𝜕θ 𝜕s s 𝜕θ 1 𝜕2 (. . . ) 1 𝜕(. . . ) 𝜕2 (. . . ) 1 𝜕2 (. . . ) ; Δ(. . . ) = + . + 2 Δk (. . . ) = cot α 2 2 s s 𝜕s 𝜕s 𝜕s s 𝜕θ 2

L(Φ, w) = (

the boundary conditions are described by (II󸀠 ). As before, we set w = w0 + w1 , Φ = Φ0 + Φ1 , w1 = Ψ(s, θ ) exp(ω t), Φ1 = F(s, θ ) exp(ω t), substitute into equation (12.5), and perform linearization; as a result, we obtain equations for the basic and perturbed states. The basic state is described by equation (12.5), on the right-hand side of which only q0 from (12.3) must be retained with φ = 0. For the amplitude of perturbed motion, we have a simplified formulation: DΔ2 Ψ − hΔk F − hL(Φ0 , Ψ) + b1 (D)

vx x 𝜕2 Ψ v 𝜕Ψ + b2 (D) x = λΨ cos α 𝜕s2 cos α 𝜕s

Δ2 F + EΔk Ψ = 0, with the boundary conditions following in a straightforward way from equation (II󸀠 ).

13 The conical shell: external flow Consider a cone occupying in the spherical coordinates (r, θ , φ ) a domain V : {0 ≤ r ≤ ∞; θ = α ; 0 ≤ φ ≤ 2π }. Part of the cone r1 ≤ r ≤ r2 is occupied by an elastic shell, and the rest of the cone is rigid. The cone is subjected to a symmetric gas flow at high supersonic velocity, so that v0 /a0 = M ≫ 1. Here, v0 is the freestream velocity, 1/2 a0 ≃ (𝛾p0 /ρ0 ) is the freestream speed of sound, p0 and ρ0 are the unperturbed pressure and gas density, and 𝛾 is the polytropic index. According to the results obtained in the previous chapter, aerodynamic pressure is described by q(r, t) =

𝛾p0 2ρ0 D2 a(D) ) (1 + ε − 𝛾+1 4 2ρ0 D2 4ρ D 𝜕w 3ε 11a(D) 𝜕w )( ) − 0 (1 + −ε + v0 𝛾+1 4 8𝛾 𝜕t 𝜕r −

(13.1)

ρ0 Drv0 3a(D) 𝜕2 w (1 − ε ) 2. 2 2𝛾(𝛾 + 1) 𝜕r

Here, ε = (𝛾 − 1)/(𝛾 + 1); a(D) = 1 + 2a20 /[(𝛾 − 1)D2 ], D = v0 tan β , and β is the cone angle of the bow shock in the unperturbed flow determined from the quadratic equation ε Da(D) + 2v0 tan α = 2D. Introducing the notation M tan β = z, M tan α = z0 , we obtain this equation in the form (3 + 𝛾)z2 − 2(𝛾 + 1)z0 z − 2 = 0.

(13.2)

The stress-strain state of the shell is described by the equations of engineering shell theory in the mixed form: D0 Δ2 w − Δk F − L(w, F) = q(r, t) − ρ h

𝜕2 w 𝜕t2

Δ2 F + EhΔk w − 0.5L(w, w) = 0.

(13.3) (13.4)

Here, F is the stress function, D0 = Eh3 /(12(1 − 𝜈2 )) is the cylindrical stiffness, h is the shell thickness, ρ is its material density, and Δ is the Laplace operator in spherical coordinates, Δk = cot α (𝜕2 /𝜕r2 )/r. The operator L(u, v) has the form L(u, v) =

𝜕2 u 1 𝜕2 v 1 𝜕v 1 𝜕u 𝜕2 v 1 𝜕2 u ( + ( + ) + ) r 𝜕r r 𝜕r 𝜕r2 r2 𝜕ψ 2 𝜕r2 r2 𝜕ψ 2 − 2(

1 𝜕v 1 𝜕u 1 𝜕2 v 1 𝜕2 u )( ), − 2 − r 𝜕r𝜕ψ r 𝜕ψ r 𝜕r𝜕ψ r2 𝜕ψ

where ψ = φ sin α . It can be seen from equation (13.1) that the pressure q(r, t) has a purely “static” component (first term); therefore, we assume that the solution to the systems (13.3) and (13.4) can be represented by a sum of the basic (quasistatic) and

13 The conical shell: external flow |

79

perturbed (dynamic) states: w = w0 (r) + w1 (r, t) and F = F0 (r) + F1 (r, t). Substitute these relations into equation (13.3) and (13.4), linearize them with respect to the perturbations and perform necessary simplifications to obtain the following: Basic (quasistatic) state: D0 Δ2 w0 −

cot α 𝜕2 F0 = q0 (r) r 𝜕r2

(13.5)

cot α 𝜕2 w0 =0 r 𝜕r2

(13.6)

Δ2 F0 + Eh

q0 (r) =

2ρ0 D2 a(D) 1 (1 + ε − 2) 𝛾+1 4 2z 4ρ Dv 3ε 11a(D) 𝜕w0 − 0 0 (1 + −ε ) 𝛾+1 4 8𝛾 𝜕r −

ρ0 Drv0 𝜕 2 w0 3a(D) (1 − ε ) . 2 2𝛾(1 + 𝛾) 𝜕r2

Perturbed (dynamic) state: D0 Δ2 w1 −

𝜕 2 w1 cot α 𝜕2 F1 − L(w , F ) = q (r, t) − ρ h 1 0 1 r 𝜕r2 𝜕t2

(13.7)

cot α 𝜕2 w1 =0 r 𝜕r2

(13.8)

Δ2 F1 + Eh

q1 (r, t) = − −

4ρ0 D 𝜕w 𝜕w 3ε 11a(D) (1 + ) ( 1 + v0 1 ) −ε 𝛾+1 4 8𝛾 𝜕t 𝜕r ρ0 Drv0 𝜕2 w1 3a(D) . (1 − ε ) 2 2𝛾(𝛾 + 1) 𝜕r2

Consider the basic state. Introduce the nondimensional coordinate s = r/r2 , as well as the nondimensional deflection w0 /h and stress function F0 /(Eh2 r2 ), retaining for them the previous notation. Substituting into equation (13.5) and (13.6), we obtain 1 𝜕 2 F0 tan α h2 2 Δ w − = q∗0 0 s 𝜕s2 12(1 − 𝜈2 ) r22 tan α Δ2 F0 + q∗0 = B1 (1 +

(13.9)

1 𝜕 2 w0 =0 s 𝜕s2

(13.10)

𝜕w ε ∗ 3 1 11ε ∗ a (z) − 2 ) − B2 (1 + ε − a (z)) 0 4 4 8𝛾 𝜕s 2z

− B3 (1 −

𝜕2 w 3ε a∗ (z)) s 20 ; 2𝛾(1 + λ ) 𝜕s

a∗ (z) = 1 +

2 (𝛾 − 1)z2

(13.11)

80 | 13 The conical shell: external flow

B1 =

2𝛾 p0 r22 2 z tan α ; 𝛾 + 1 E h2

B2 =

4𝛾 p0 r2 zz ; 𝛾+1 E h 0

B3 =

𝛾 p0 r2 zz . 2 E h 0

Supplement the system (13.9) and (13.10) by the boundary conditions of simple support; in terms of w0 and F0 these conditions are written in the form s = s1 ,

s=1:

w0 = 0,

𝜕2 w0 v 𝜕w0 + =0 s 𝜕s 𝜕s2

𝜕2 F0 = 0. 𝜕s2

𝜕F0 = 0, 𝜕s

(13.12)

Finally, the basic state problem is reduced to the solution of the system (13.9) and (13.10) with boundary conditions (13.12). The order of magnitude of the terms on the right-hand side of equation (13.11) for q∗0 is determined, evidently, by the coefficients Bi . We have B2 2h = ; B1 r2 tan β

B3 𝛾 + 1 h = . B1 4 r2 tan β

Realistically, h/ (r2 tan β ) ≪ 1, and therefore we can adopt with high accuracy that q∗0 (s) ≅ B1 (1 +

ε ∗ 1 a (z) − 2 ) . 4 2z

(13.13)

Consider now the dynamic state. Introduce the nondimensional deflection w = w1 /h and stress function F = F1 / (Eh2 r2 ), and substitute them into equations (13.7) and (13.8) to obtain 1 𝜕2 w =0 s 𝜕s2 tan α h2 2 1 𝜕2 F h 𝜕F0 1 𝜕2 w Δ w− − tan α 2 2 2 s 𝜕s r2 𝜕s s 𝜕s2 12(1 − 𝜈 ) r2 tan α Δ2 F +

− tan α

2 h 𝜕2 F0 1 𝜕w tan α r2 1 𝜕2 w ( ) = q + 1 r2 𝜕s2 s 𝜕s E h2 s2 𝜕ψ 2

We search for the solution of the above system of equations in the class of functions: w = W(s) cos nφ exp(ω t), F = Φ(s) cos nφ exp(ω t); for the amplitudes W(s) and Φ(s) we obtain the following system of equations: tan α Δ2n Φ +

1 󸀠󸀠 W =0 s

tan α h2 2 1 h 󸀠1 Δn W − Φ󸀠󸀠 − tan α F0 W 󸀠󸀠 2 2 s r2 s 12(1 − 𝜈 ) r2 h 1 n2 − tan α F0󸀠󸀠 ( W 󸀠 − W) + A3 sW 󸀠󸀠 + A2 W 󸀠 = λ W, 2 2 r2 s s sin α

(13.14)

(13.15)

where primes denote derivatives with respect to s, and also operator Δn ≡ 𝜕2 /𝜕s2 + (𝜕/𝜕s)/s − n2 /(sin2 α ) is introduced; the spectral parameter λ is generated by A4 Ω2 + A1 Ω + λ = 0, Ω = r2 ω /c2 .

13 The conical shell: external flow |

81

The following notation is used: 4𝛾 p0 r2 c0 3ε 11ε ∗ z tan α (1 + − a (z)) 𝛾 + 1 E h a0 4 8𝛾 4𝛾 p0 r2 3ε 11ε ∗ zz (1 + (z) − a (z)) ; c20 = E/ρ A2 = 𝛾+1 E h 0 4 8𝛾 𝛾 p0 r2 3ε zz (1 − a∗ (z)) ; A4 = tan α . A3 = 2 E h 0 2𝛾(𝛾 + 1)

A1 =

The boundary conditions of simple support are expressed in terms of functions W and Φ as v s = s1 , s = 1 : W = 0, W 󸀠󸀠 + W 󸀠 = 0 s (13.16) 2 n 󸀠󸀠 Φ = 0; Φ = 0. Φ󸀠 − s sin2 α Thus, we finally obtained an eigenvalue problem (13.14)–(13.16); since the problem is non-self-adjoint, it has a countable set of complex-valued eigenvalues λk corresponding to complex-valued vibration frequencies Ωk . By definition, the perturbed motion is stable if ∀k we have Re(Ωk ) < 0, and unstable if for at least one k0 we have Re(Ωk0 ) > 0; the domains of stable and unstable vibrations are separated on the complex plane λ by the stability parabola A4 (Im λ )2 = A21 Re λ , its equation is used for determination of critical values for the system parameters. Traditionally, the panel flutter problem is formulated in the following way. For a given system of characteristics (geometry, mechanical properties, etc.) the critical flutter velocity Mcr is determined, such that for M ≤ Mcr the perturbed state is stable. Two essential points must be emphasized: (i) for M ≤ Mcr the basic state w0 , F0 must be statically stable; (ii) the system (13.14)–(13.16) involves the waviness parameter n, and therefore Mcr = Mcr (n); the true critical flutter velocity is Mcr (ncr ) = min Mcr (n). n

14 The conical shell: internal flow 14.1 Statement of the problem Consider a circular conical shell occupying in the spherical coordinates a domain V0 : {s1 ≤ s ≤ s2 ; 0 ≤ θ ≤ α ; 0 ≤ φ ≤ 2π }. The shell is elastic, with material density ρ1 , Young’s modulus E, and Poisson’s ratio 𝜈; its thickness is h. Inside the shell there is a supersonic gas flow with the freestream parameters: pressure q0 (s), density ρ0 (s), velocity u0 (s), local speed of sound a0 (s). The stress-strain state of the shell is described by the equations of engineering shell theory in the mixed form: DΔ2 w − Δk F − L(w, F) − q = 0

(14.1)

Δ2 F + EhΔk w + 0.5L(w, w) = 0.

(14.2)

Here, w is the normal deflection, F is the stress function, D = Eh3 /(12(1 − 𝜈2 )) is the cylindrical stiffness, Δ is the Laplace operator, Δk = cot α (𝜕2 /𝜕s2 )/s; L(U, V) =

𝜕2 U 1 𝜕2 V 1 𝜕V 1 𝜕U 𝜕2 V 1 𝜕2 U ( + ( + ) + ) s 𝜕s s 𝜕s 𝜕s2 s2 𝜕ψ 2 𝜕s2 s2 𝜕ψ 2 − 2(

1 𝜕2 V 1 𝜕V 1 𝜕2 U 1 𝜕U − 2 )( − 2 ), s 𝜕s𝜕ψ s 𝜕ψ s 𝜕s𝜕ψ s 𝜕ψ

with ψ = φ sin α ; in equation (14.1), q is the transverse load to be specified later. Let us now consider the variational formulation of the shell vibration problem. Denote by w0 , F0 the basic solution corresponding to unperturbed gas flow parameters. Assume that, in addition to the basic solution, there exists a perturbed solution w0 + w, F0 + F corresponding to the perturbed pressure q0 + q. Substitute it into equations (14.1) and (14.2) and linearize them with respect to perturbations to obtain: Basic state:

𝜕2 F 1 cot α 20 − L(w0 , F0 ) − q0 = 0 s 𝜕s 2 w0 𝜕 1 1 𝜕w0 𝜕2 w0 + Eh = 0. Δ2 F0 + Eh cot α s s 𝜕s 𝜕s2 𝜕s2 DΔ2 w0 −

(14.3) (14.4)

Perturbed state: DΔ2 w −

1 𝜕2 F cot 𝛾 2 − L(w0 , F) − L(w, F0 ) − q = 0 s 𝜕s

1 1 𝜕2 w0 𝜕2 w 𝜕2 w + Δ2 F + Eh cot α 2 + Eh ( 2 s 𝜕s s 𝜕s2 𝜕ψ 2 1 𝜕2 w0 𝜕w 1 𝜕w0 𝜕2 w ) = 0. + + s 𝜕s2 𝜕s s 𝜕s 𝜕s2

(14.5)

(14.6)

14.1 Statement of the problem

| 83

We introduce nondimensional parameters retaining the previous notation: s ⇒ s/s2 , w0 ⇒ w0 /h, F0 ⇒ F0 /(Eh2 s2 ), w ⇒ w/h, and F ⇒ F/(Eh2 s2 ). Simplify the equation set (14.3)–(14.6): in equations (14.6)–(14.4) drop the quadratic terms, do the same in equation (14.6), while in equation (14.6) neglect the term L(w0 , F). As a result, we obtain the following: Basic state: tan α Δ2 F0 +

1 𝜕 2 w0 =0 s 𝜕s2

2 1 𝜕2 F0 q0 s2 tan α h2 2 Δ w − = tan α . 0 s 𝜕s2 E h2 12(1 − 𝜈2 ) s22

(14.7) (14.8)

Perturbed state: (it is assumed that the basic state possesses axial symmetry: w0 = w0 (s), F0 = F0 (s)) 1 𝜕2 w tan α Δ2 F + =0 (14.9) s 𝜕s2 1 𝜕2 F h 𝜕F0 1 𝜕2 w tan α h2 2 Δ w− − tan 𝛾 2 2 2 s 𝜕s s2 𝜕s s 𝜕s2 12 (1 − 𝜈 ) s2 2 tan α s2 h 𝜕2 F0 1 𝜕w 1 𝜕2 w ( ) = q − tan α . + s2 𝜕s2 s 𝜕s E h2 s2 𝜕ψ 2

(14.10)

Consider first the basic state. Equation set (14.7) and (14.9) must be supplemented by boundary conditions; assume that the left edge s = s1 is clamped, the right one s = s2 is free; the boundary conditions are taken in the form 𝜕w0 = 0, 𝜕s 𝜕2 w0 1 𝜕w0 + = 0, 𝜕s2 s 𝜕s

s = s1 : w0 = 0, s = s2 :

𝜕F0 = 0, 𝜕s

𝜕2 F0 𝜈 𝜕F0 − =0 s 𝜕s 𝜕s2 𝜕3 w0 1 𝜕2 w0 1 𝜕w0 + − 2 =0 s 𝜕s2 𝜕s3 s 𝜕s

F0 = 0,

(14.11)

𝜕3 F0 1 𝜕2 F0 1 𝜕F + − 2 0 = 0. s 𝜕s2 𝜕s3 s 𝜕s

To solve the problem (14.7), (14.8), and (14.11) for the basic state, it is necessary to specify the gas pressure q0 = q0 (s) on the shell walls. We adopt the model of onedimensional flow (this is justified for small enough cone angles α ); in this case, of primary importance is the relationship between the cross-section area S(s) and Mach number M(s) = u0 (s)/a0 (s): k

S(s) [2 + (𝛾 − 1)M] 1 , = Scr (𝛾 + 1)k1 M

(14.12)

where Scr = S(scr ) is the area of critical cross-section, 𝛾 is the gas polytropic index, and k1 = (𝛾 + 1)/(2(𝛾 − 1)). By definition, Mcr = 1; in the current case S1 = S(s1 ), and

84 | 14 The conical shell: internal flow therefore, assuming that M1 = M(s1 ), we obtain from equation (14.12) that Scr =

S(s1 )(𝛾 + 1)k1 k1

[2 + (𝛾 − 1)M12 ]

.

(14.13)

Denote by p∗ the stagnation pressure; then from relations p∗ = pcr (

𝛾 + 1 k/(k+1) ) , 2

𝛾 − 1 2 𝛾/(𝛾−1) p∗ = (1 + M ) p 2

we obtain an expression for p(s): p(s) = pcr

(k + 1)k2 [2 + (k − 1)M 2 ]

k2

,

(14.14)

where k2 = 𝛾/(𝛾 − 1). If pcr is given, and then p(s) is determined from equation (14.14); on the other hand, it is possible that the inlet pressure is given as p1 = p(s1 ), then pcr is calculated from equation (14.6) and (14.13). Finally, q0 (s) = p(s), and the boundaryvalue problem for the basic state (14.7), (14.8), and (14.11) is written in the closed form. We now turn to equations (14.9) and (14.10) for the perturbed state; we search for their solution in the class of functions w = W(s) cos nφ exp(ω t); F = Φ(s) cos nφ exp(ω t). After substitution into equation (14.9) and (14.10), we obtain tan α Δ2n Φ + [

1 󸀠󸀠 W =0 s

1 h 1 h2 2 tan α Δ W − Φ󸀠󸀠 − tan α F0󸀠 W 󸀠󸀠 − 2 s s2 s 12 (1 − 𝜈 ) s22 n tan 𝛾 s22 n2 h 1 ωt W)] e cos nφ = q , − tan α F0󸀠󸀠 ( W 󸀠 − s2 s E h2 s2 sin2 α

where an operator Δn =

d2 ds2

+

1 d s ds

−

n2 s2 sin2 α

(14.15)

(14.16)

is introduced.

The dynamic loading, q, consists of the sum of inertial and aerodynamic loading forces: q = q1 + q2 , with q1 = −ρ1 h𝜕2 w/𝜕t2 , where ρ1 in the density of shell material; for q2 we take a simplified formula which will be derived later on: h ω s2 1 [ W + MW 󸀠 + W s2 a0 (s) 2s sin α 𝜕u 𝛾 − 1 1 −M 0 ( M + ) W] exp(ω t) cos nφ . 𝜕s 2 M

q2 (s) = 𝛾q0 (s)

14.1 Statement of the problem

| 85

Substituting this relation into equation (14.6), we obtain 1 h 1 h2 2 tan α Δ W − Φ󸀠󸀠 − 2 tan α F0󸀠󸀠 W 󸀠󸀠 2 s s s 12 (1 − 𝜈 ) s22 n h n2 󸀠󸀠 1 󸀠 ( W) tan α F − W 0 s s2 s2 sin2 α ω s2 1 tan α s2 =− 𝛾q (s) [ W + MW 󸀠 + W E h 0 a0 (s) 2s sin α −

−M

(14.17)

s2 ω 2 tan 𝛾 𝜕u0 𝛾 − 1 1 W, ( M + ) W] − 2 2 𝜕s 2 M c1

where a0 (s) is the local speed of sound, c21 = E/ρ1 , while the basic state pressure q0 (s) is determined by equation (14.14). Expressions for a0 (s) and u0 (s) are obtained by considering a one-dimensional flow p κ3 1 a0 (s) = a0 √𝛾 + 1 ( cr ) (14.18) p0 √2 + (𝛾 − 1)M 2 u0 (s) = Ma0 (s) = a0 √𝛾 + 1 (

pcr κ3 M(s) ) , p0 √2 + (𝛾 − 1)M 2

κ3 =

𝛾−1 , 2𝛾

(14.19)

where pcr is the pressure in the critical cross-section, p0 and a0 are pressure and speed of sound in stagnant gas. Introduce the following notation: s2 p0 κ3 pcr ( ) h pcr E p s A2 = 𝛾 tan α (𝛾 + 1)κ2 2 cr h E ω s2 Ω= a0 A1 = 𝛾 tan α (𝛾 + 1)κ1

f1 (s) = [2 + (𝛾 − 1)M 2 ] f2 (s) = [2 + (𝛾 − 1)M 2 ]

−κ1 −κ2

,

and denote the operator in the right-hand side of (14.17) by Ln (W, Φ, F0 ); as a result, equation (14.17) takes the form Ln (W, Φ, F0 ) + A2 f2 (s)MW 󸀠 𝜕u 𝛾 − 1 1 1 −M 0 ( M + )] W 2s sin α 𝜕s 2 M a2 + (A1 f1 (s)Ω + tan α 20 Ω2 ) W = 0. c1 + A2 f2 (s) [

From equation (14.19) we have p κ3 𝜕u0 M󸀠 = 2a0 √𝛾 + 1 ( cr ) 3/2 𝜕s p0 [2 + (𝛾 − 1)M 2 ]

(14.20)

86 | 14 The conical shell: internal flow Thus, we finally obtained the system of equations (14.15) and (14.20) for the determination of W and Φ; it has to be supplemented by proper boundary conditions. Consider the boundary conditions which are of interest in applications: the left edge of the shell (s = s1 /s2 ) will be assumed to be clamped, the right edge (s = 1) is assumed to be free. In terms of deflection W and loading function Φ, these conditions are written as s = s1 /s2 = s0 :

s=1:

W = 0;

W 󸀠 = 0;

Φ = 0;

Φ󸀠󸀠 −

𝜈 󸀠 Φ =0 s

(14.21)

1 𝜈n2 { { W 󸀠󸀠 + W 󸀠 − W=0 { { s s2 sin2 α { 2 { (1 − 𝜈)n2 { { W 󸀠󸀠󸀠 + 1 W 󸀠󸀠 − 𝜈n W󸀠 − W=0 2 s { s2 sin α s3 sin2 α

(14.22)

n2 󸀠 { Φ Φ=0 − { { { s sin2 α 2 { { 󸀠 1 1 1 { {Φ − Φ + h W = 0. 2 2 ) tan α s2 s (1 12 − 𝜈 s { 2

(14.23)

For small cone angles we have 12 tan α ∼ 1, and therefore the last term in the second equation (14.23) can be dropped; then we obtain instead of (14.23) s=1:

Φ = 0,

Φ󸀠 = 0.

(14.24)

The validity of this assumption i still to be confirmed in practical simulations. The equation set (14.15), (14.20) with boundary conditions (14.21), (14.22), and (14.23) (or (14.21), (14.22), and (14.24)) constitute an eigenvalue problem for Ω. Since the problem is non-self-adjoint, eigenvalues Ωk , generally speaking, are complex-valued. If for at least one Ωk we have Re Ωk > 0, the basic state is unstable with respect to perturbation of the given class of functions; if, on the contrary, ∀k we have Re Ωk < 0, the basic state is considered stable. Accordingly, the following statements of panel flutter problems are possible for a conical shell: 1. Shell parameters are given, the variable parameters are the inlet pressure σ = q(s1 ), and the number of perturbation half-waves n. Suppose that ∀n and for some σ all Ωk (σ , n) belong to the left half-plane. By increasing σ and simultaneously varying n, we can determine the values min σ = σ0 (n0 ) and n0 , at which n one of Ωk reaches the imaginary axis for the first time. By definition, σ0 and n0 determine the critical regime. Evidently, any state for which σ < σ0 ∀n will be stable. 2. In the subcritical regime, σ = σ ∗ < σ0 , of interest is to find out the perturbation mode (n∗ ) with the slowest damping; in fact, this means to find out the eigenvalue Ω∗k (n∗ ) w closest to the imaginary axis. 3. The shell thickness h is the value of interest, and all other shell and flow parameters are given. Suppose that ∀n and some h all Ωk (h, n) belong to the left half-plane.

14.2 Determination of dynamic pressure

| 87

We gradually decrease h, simultaneously varying n, and determine the values max h = h0 (n0 ) and n0 at which one of Ωk reaches the imaginary axis for the first n time. The value h0 will give the critical thickness, since for h < h0 ∀n, the state will be unstable.

14.2 Determination of dynamic pressure Consider (in the spherical coordinates r, θ , ψ ) a conical surface θ = α , r ∈ [r0 , ∞); its part [r1 > r0 , r2 ] is occupied by a thin elastic shell, and the rest of the cone is rigid. The cone angle α is assumed to be small, so that α 2 ≪ 1. Inside this structure, there is a gas flow in the positive direction of axis r. In the unperturbed state (as if the shall is not moving) the gas flow is assumed to be radial, and its parameters ρ0 , p0 , u0 , and a0 are the density, pressure, radial velocity, and speed of sound, respectively; these are some known functions of the polar radius r. By assumptions, M 2 = (u0 /a0 )2 ≫ 1. Small-amplitude shell vibrations cause perturbations in the basic flow generating additional aerodynamic pressure Δp. With good accuracy, the perturbed flow can be considered as potential. Denote by u the flow velocity vector: u = {ur , uθ , uψ } = {u0 +

𝜕φ 𝜕φ 1 𝜕φ , , }, 𝜕r r𝜕θ r sin θ 𝜕ψ

(14.25)

and determine the perturbation of the speed of sound from the Cauchy–Lagrange integral 𝛾−1 𝜕φ 𝜕φ a=− (u0 + ). (14.26) 2a0 𝜕r 𝜕t For the gas flow, we make use of a well-known equation: a2 ∇u =

𝜕2 φ 𝜕u + 2u + u [(u∇) u ] 𝜕t 𝜕t2

Substitute into it equations (14.26) and (14.26), linearize with respect to small perturbations, introduce the nondimensional variable r󸀠 = r/l, l = r2 − r1 , and set, as usual, φ = φ1 exp (ω t) cos mψ , retaining the previous notation. As a result, we obtain (M 2 − 1)

𝜕2 φ lω M 2 𝜕u0 𝛾 − 1 1 2 𝜕φ [2M ( + + 2 M + )− ] a0 a0 𝜕r 2 M r 𝜕r 𝜕r2 M 2 lω 𝜕u0 l2 ω 2 m2 ]φ (𝛾 + − 1) + a0 a0 𝜕r a20 r2 sin2 θ 𝜕φ 1 𝜕 (sin θ ) = 0. − r sin θ 𝜕θ r𝜕θ +[

(14.27)

Denote by W0 (r, ψ , t) the normal shell deflections and set W0 = W exp (ω t) cos mψ ; the impermeability condition on the shell surface is then written in the following form

88 | 14 The conical shell: internal flow (after linearization): 𝜕φ 󵄨󵄨󵄨󵄨 𝜕W lω 1 = u0 ( W − W) . + 󵄨 r𝜕θ 󵄨󵄨󵄨θ =α 𝜕r u0 r

(14.28)

On the shell axis, the velocity u0 vanishes: 1 𝜕φ 󵄨󵄨󵄨󵄨 󵄨 = 0. uθ 󵄨󵄨󵄨θ =0 = 󵄨 l r𝜕θ 󵄨󵄨󵄨θ =0

(14.29)

For the additional pressure we take Δp = Δq exp (ω t) cos mψ , in which case Δq is determined from a known linearized formula: Δq = −

ρ0 u0 𝜕φ ωl φ) . ( + l 𝜕r u0 θ =α

(14.30)

Simplification of the problem (14.27)–(14.29) is based on the assumption of the smallness of cone angle α 2 ≪ 1, which entails sin θ = θ , cos θ ≈ 1 − θ 2 /2 ≈ 1. Over the shell, the nondimensional variable r changes insignificantly (r ≫ 1), and therefore introduce a new variable ζ = r sin θ , so that dζ = rdθ . Substitute this into equation (14.27), and take into account that M 2 ≫ 1 and 2/r ≪ 1 to obtain 𝜕φ 𝜕2 φ 𝜕2 φ 1 𝜕φ m2 + − 2 φ − M 2 2 − A(r) − B(r)φ = 0. 2 ζ 𝜕ζ 𝜕r 𝜕ζ ζ 𝜕r

(14.31)

The last term on the right-hand side of equation (14.28) is small in comparison with the first one; therefore 𝜕φ 󵄨󵄨󵄨󵄨 𝜕W lω = u0 ( W) , + 󵄨 𝜕ζ 󵄨󵄨󵄨ζ =ζ0 𝜕r u0

ζ0 = r sin α ,

(14.32)

and we obtain, instead of equation (14.30), Δq = −

ρ0 u0 𝜕φ ωl φ) . ( + l 𝜕r u0 ζ =ζ0

(14.33)

In equation (14.31) the following notation is used: A(r) = 2M B(r) =

M2 𝛾 − 1 1 𝜕u lω ( +2 M+ ) 0 a0 a0 2 M 𝜕r

l2 ω 2 M 2 lω 𝜕u0 + (𝛾 − 1) . 2 ao a0 𝜕r a0

Approximate solution of the problem (14.31) and (14.32) is based on the fact that for small cone angles M(r), A(r), and B(r) are slowly and smoothly varying functions. Introduce instead of r a variable z reckoned from the left shell edge, retaining the previous notation for all three functions. In supersonic flow, u0 = 0 and Δp = 0 for z ≤ 0, therefore φ = 0, 𝜕φ /𝜕z = 0 for z ≤ 0. Apply the Laplace transform to

14.2 Determination of dynamic pressure

|

89

equations (14.31) and (14.32), assuming M(z), A(z), and B(z) to be “nearly constant” parameters; we then obtain (s is the transformation parameter) 𝜕2 φ ∗ 1 𝜕φ ∗ m2 + − (β 2 + 2 ) φ ∗ = 0, 2 ζ 𝜕ζ 𝜕ζ ζ 𝜕φ ∗ 󵄨󵄨󵄨󵄨 lω = u0 (s + ) W∗ 󵄨 𝜕ζ 󵄨󵄨󵄨ζ =ζ0 u0

(14.34)

(14.35)

β 2 = M 2 s2 + As + B. Condition (14.29) is written in the form 1 𝜕φ ∗ 󵄨󵄨󵄨󵄨 󵄨 u∗θ 󵄨󵄨󵄨θ =0 = = 0. 󵄨 l 𝜕ζ 󵄨󵄨󵄨ζ =0

(14.36)

The solution to equation (14.34), subject to the condition (14.36), is written in terms of the modified Bessel function: φ ∗ = C Im (βζ ); determining C from (14.36), we finally obtain I (βζ ) lω φ ∗ = u0 (s + W ∗. ) m u0 β I󸀠m (βζ0 ) The Laplace transform for aerodynamic pressure Δq∗ is found from equation (14.30): ρ u2 ω l 2 Im (βζ0 ) Δq∗ = − 0 0 (s + W ∗, ) (14.37) l u0 β I󸀠m (βζ0 ) where prime denotes the derivative of Im by its argument. The inverse Laplace transform of equation (14.37) gives a result which is difficult to analyze analytically, so we will therefore perform some preliminary estimates. The natural vibration frequencies of a conical shell can be obtained by the formula ωj =

π C0 h 1/2 ( ) ζj . l R

(14.38)

Here, C0 is the speed of sound in long, thin rods of the shell material, h is its thickness, R = (r1 + r2 ) sin α /2, and the parameter ζj depends on the boundary conditions, vibration mode and other problem parameters: it varies from few unities to values of the order of ten. From equation (14.38) we obtain 1/2 |ω | l ωj l π C h ∼ = ( ) ζj . a0 a0 a0 R

For practically encountered shells an estimate follows |ω | l/a0 ∼ (0.1−0.7)⋅10, taking 󵄨 󵄨 into account which one can obtain for M 2 ≫ 1 that 󵄨󵄨󵄨βζ0 󵄨󵄨󵄨 ≫ 1 and use asymptotic expansion Im (βζ0 ). According to the Laplace transform property, Re β > 0, and we obtain after some straightforward manipulations Im (βζ0 ) 1 ∼1+ . 󸀠 2βζ Im (βζ0 ) 0

(14.39)

90 | 14 The conical shell: internal flow We now set β 2 = M 2 (s + s1 )(s + s2 ), 2M 2 s1,2 = A ± √A2 − 4M 2 B and denote lω /u0 = Ω; substituting it together with (14.39) into (14.37), we obtain Δq∗ = −

ρ0 u20 ρ0 u20 (s + Ω)2 W ∗ (s) (s + Ω)2 W ∗ (s) − . Ml (s + s1 )1/2 (s + s2 )1/2 2ζ0 M 2 l (s + s1 )(s + s2 )

The inverse transform Δq(z) is restored by lookup tables and the convolution theorem, and integral terms are evaluated on the basis of straightforward inequality |(s2 − s1 )/(s2 + s1 )| ≪ 1. After relevant transformations, we obtain 𝛾ρ0 a0 𝜕W [ω W + u0 + W a0 l𝜕z 2lζ0 (z) 𝜕u 𝛾 − 1 1 −M 0 ( M + )W l𝜕z 2 M

Δq(z) = −

(14.40)

z

−

1 𝜕u0 𝛾 − 1 1 ( M + ) ∫ e−Ω(z−τ ) W(τ )dτ ], ζ0 (z) l𝜕z 2 M 0

where in writing the integral term it was taken approximately that (s1 + s2 )/2 = A(r)/ (2M 2 ) ≅ lω /u0 = Ω. The first two terms on the right-hand side of equation (14.40) correspond to the piston theory; the other two terms have the physical meaning of the pressure arising upon interaction of bended shell with Winkler foundation having the stiffness K∗ =

𝛾p0 a0 𝜕u 𝛾 − 1 1 [ −M 0 ( M + )] . a0 2lζ0 l𝜕z 2 M

The last term in equation (14.40), unlike the previous ones, is nonlocal: the pressure at a point z depends (with damped memory) on the deflections on the whole preceding interval.

15 Example calculations In Chapter 9, the formulation of the shallow shell flutter problem is given under the assumption that the pressure drop on the shell surface is determined from the law of plane sections for supersonic aerodynamics. If the piston theory formula is used for the pressure drop, then the shallow shell flutter problem, from the mathematical point of view, is reduced to an eigenvalue problem for a system of two equations with biharmonic higher-order operators written in terms of the amplitude deflection function φ and stress function F. For some types of boundary conditions, the stress function F can be (numerically) eliminated; the remaining equation for φ will contain two nondimensional constants of the order of 10−3 and 102 (for characteristic parameter values) as coefficients of the high-order derivatives, which renders the problem ill-posed. On the other hand, the presence of a boundary layer requires refinement of the grid near the edge contour. To overcome these computational difficulties, a nonsaturating numerical method is applied which was successfully used to study the flutter of a plate with an arbitrary plan-view shape. In this chapter, the method is extended to the panel flutter problem for shallow shells of a rectangular plan view shape; particular calculations were carried out for cylindrical and spherical shells. New mechanical effects concerning the vibration modes and dependence of the critical flutter velocity on the flow velocity vector direction were obtained. In the nondimensional form, the initial system of differential equations is written as (the simplest form of linear theory) DΔ2 φ − hL(F) − κ (v, grad φ ) = λφ 2

Δ F + EL(φ ) = 0 2

λ = −ρ hω − κω 2

L(f ) = ky

(15.1) (15.2) (15.3)

2

𝜕 f 𝜕 f + kx 2 , 𝜕x2 𝜕y

(15.4)

where D = Eh3 /12(1 − 𝜈2 ) is the flexural rigidity, E is Young’s modulus, 𝜈 is Poisson’s ratio, κ is the polytropic index, v = (v cos θ , v sin θ ) is the air flow velocity, h is the shell thickness, ρ is the density of shell material, and ω is the complex frequency of vibrations; kx and ky are the principal curvatures (the lines of principal curvatures coincide with the coordinate lines); φ = φ (x, y) is the shell deflection, and F = F(x, y) is the stress function. All the above-mentioned quantities are dimensionless. Nondimensionalization was performed as described above; the stress function is nondimensionalized by p0 a2 . The above equations are considered on the rectangular domain S = −1 ≤ x ≤ 1, −b ≤ y ≤ b.

92 | 15 Example calculations

1.

2.

We solve equations (15.1)–(15.3) for two types of boundary conditions: hinged (simply supported) edges: x = 1, −1 :

φ = 0,

𝜕2 φ = 0, 𝜕x2

𝜕2 F = 0, 𝜕y2

𝜕2 F =0 𝜕x𝜕y

(15.5)

y = b, −b :

φ = 0,

𝜕2 φ = 0, 𝜕y2

𝜕2 F = 0, 𝜕x2

𝜕2 F = 0; 𝜕x𝜕y

(15.6)

clamped edges with slip: x = 1, −1 :

φ = 0,

𝜕φ = 0, 𝜕x

𝜕2 F = 0, 𝜕y2

𝜕2 F =0 𝜕x𝜕y

(15.7)

y = b, −b :

φ = 0,

𝜕φ = 0, 𝜕y

𝜕2 F = 0, 𝜕x2

𝜕2 F = 0. 𝜕x𝜕y

(15.8)

It is easily shown that the boundary conditions imposed on the stress function F can be replaced, without loss of generality, by the equivalent conditions (x, y) ∈ 𝜕G,

F = 0,

𝜕F = 0, 𝜕n

(15.9)

where n is the vector of outward pointing normal to the contour of the shell. The problem is solved by the same method as before: for v = 0 all eigenvalues are real-valued, with the increase in flow velocity some eigenvalues enter the complex plane; the problem is, therefore, to find out (for a given θ ) the complex eigenvalue which is the first to reach the stability parabola. Thus this eigenvalue determines the critical velocity and vibration mode corresponding to it (eigenfunction). It follow that, in order to solve the problem correctly, quite a long initial spectral interval must be calculated. Thus, on each iteration step one has to solve the complete eigenvalue problem for a N ×N nonsymmetric matrix, where N is the number of grid points. Arising difficulties are overcome by the use of a nonsaturating method which provides high accuracy for smooth solutions even on relatively coarse grids. To discretize the boundary-value problems described above, it is necessary to discretize the biharmonic operators Δ2 φ and Δ2 F with the boundary conditions of simple support or clamping, respectively. Also, the operator L(f ) and the terms containing first-order derivatives (v, grad φ ) have to be discretized. We assume that kx and ky are constants. We have kx = 0 and ky = 1/R for a cylindrical shell, and kx = ky = 1/R for a spherical shell (R is the shell radius). Therefore, discretization of the operator L(f ) under the homogeneous Dirichlet boundary condition is needed. This discretization is performed according to the approach described in Part III. The terms with first-order derivatives (v, grad φ ) are discretized in a similar way. It should be noted that since L(f ) is a second-order operator, it suffices to satisfy only one boundary condition, namely, f = 0 on 𝜕G, in order to discretize this operator. Since there are no nodes on the boundary, an interpolation formula that does not

15 Example calculations

|

93

satisfy (compulsorily) the boundary condition φ = 0 on 𝜕G was used for discretization of the terms with first-order derivatives. In the discretization of biharmonic operators, both boundary conditions were satisfied. Calculations (see below) show that the solution obtained for φ satisfies the boundary conditions. Discretization of the biharmonic operator subject to boundary conditions of simple support or clamping is described in Part III. For a function F = F(x, y) on a rectangle we use the interpolation formula n

m

F(x, y) = ∑ ∑ Mi0 (z)Lj0 (x)F(xj , yi ) j=1 i=1

y = bz, Lj0 (x) = xj = cos ϑj , Mi0 (z) =

z ∈ [−1, 1],

l(x) , l󸀠 (xj )(x − xj )

x ∈ [−1, 1]

l(x) = (x2 − 1)2 Tn (x)

ϑj = (2j − 1)π /2/n,

M(z) , M 󸀠 (zi )(z − zi )

(15.10)

j = 1, 2, . . . , n

M(z) = (z2 − 1)2 Tm (z)

zi = cos ϑi, ϑi = (2i − 1)π /2m,

i = 1, 2, . . . , m.

This formula satisfies the boundary conditions (15.9). We enumerate the nodes on the rectangle (xj , yi ) first along y, and then along x, i.e. from top to bottom and from right to left, and substitute (15.8) into (15.1) and (15.2). As a result, we obtain Hφ − hLh F = λφ

(15.11)

H3 F + ELh φ = 0.

(15.12)

Here, H is a N × N matrix of the discrete problem for the plate, where N = mn. Its construction for the case of simple support is described in Part III. Resolving equation (15.12) with respect to F and substituting it into relation (15.11), we obtain (H + hELh H3−1 Lh )φ = λφ .

(15.13)

Here, φ is a vector containing the approximate values of shell deflection at the grid nodes, λ is an approximate eigenvalue, and Lh and H3 are N × N matrices resulting from the discretization of operators L and Δ2 F. Further analysis was carried out using the finite-dimensional eigenvalue problem (15.13). As was mentioned before, this problem contains a large parameter hE (of the order of 102 for the data used in calculations). This parameter is multiplied by a matrix which is nonsymmetric and can therefore have complex eigenvalues for the flow velocity v = 0, as obtained in practical calculations. Therefore, the approach was updated: the matrix H3 in (15.13) was replaced by the matrix H3 = 0.5(H3 + H3󸀠 ), where the prime denotes a transposed matrix. Similar symmetrization was applied to the matrices Lh and H0 (here, H0 is the matrix of the discrete biharmonic operator for the deflection φ ).

94 | 15 Example calculations As a result, the matrix Lh H3−1 Lh became symmetric with an accuracy of 10−6 . This, however, was insufficient, since the discrete problem still had complex eigenvalues for v = 0. After repeated symmetrization of matrix Lh H3−1 Lh , the eigenvalues of the discrete problem for v = 0 became real-valued and positive. In the calculations of critical velocity, convergence was achieved. Calculations were performed with the same values of parameters as before: the relative thickness and dimensionless radius of the shell were taken to be 3 ⋅ 10−3 and 2.5, respectively. Preliminary calculations on the 9 × 9, 13 × 13, and 19 × 19 grids showed that the results obtained on the 13 × 13 and 19 × 19 grids are close enough; below, we give the critical velocities obtained for the 19 × 19 grid. Calculations for a spherical shallow shell of square planform can be regarded as test calculation. For the angles θ = 0, π /8, π /4, 3π /8, and π /2 the following critical velocities were obtained: 1.4263(20), 1.4924(18), 1.5813(18), 1.4924(18), and 1.4263(20). The numbers in parentheses indicate the first eigenvalue that reaches the stability parabola. As was expected from the symmetry of the problem, the critical velocities are symmetric about the straight line θ = π /4. This supports the correctness of the method and its program implementation. Furthermore, to control the calculations, the following two diagrams were plotted: the deflection function Re φ (x, 0) and Re φ (0, y), as well as the two-dimensional function Re φ (x, y). For θ = π /4, the curves Re φ (x, 0) and Re φ (0, y) coincide, which confirms the validity of the calculations. Eigenfunctions Re φ (x, y) are also identical at the angles θ = 0 and θ = π /2. This all shows that the calculations are correct. For a clamped spherical shallow shell, for the same directions of flow velocity vector the following critical velocities were obtained: 1.6424(20), 1.7038(16), 1.6876(17), 1.7038(16), and 1.62384(20). Generally speaking, the results are similar to those obtained in the previous case. Calculations for a spherical shallow shell of rectangular planform (b = 0.5) were performed. For the case of simple support, the following critical velocities were obtained for the same values of angle θ : 1.7752(9), 1.8787(9), 1.8414(5), 1.8558(4), and 1.7469(4). For the boundary conditions of clamping, we obtained, respectively, 1.6138(9), 1.6902(9), 1.8935(5), 1.7335(5), and 1.6602(5). Further calculations were performed for a cylindrical, simply supported shell of square planform. For the same values of angle θ we obtained the following critical velocities: 2.7654(7), 0.5606(1), 0.3004(1), 0.2205(1), and 0.2120(1). The principal difference is seen between these and the previous results: the velocity decreases abruptly for θ close to π /2. We note that, for a square plate, the critical velocity is equal to 0.2103 at θ = 0 and π /2. Thus, the critical flutter velocity for the flow directed along the generatrix of the cylindrical shell is an order of magnitude higher than that across the generatrix. This effect can be easily explained: the bending rigidity of a cylindrical shell along the generatrix is much higher than that across the generatrix. The evolution of the eigen modes is shown in Figures 15.1–15.5

15 Example calculations

|

95

Fig. 15.1. Square planform cylindrical simply supported shell, θ = π /2, v = 0.2120.

Fig. 15.2. Square planform cylindrical simply supported shell, θ = 3π /8, v = 0.2295.

Fig. 15.3. Square planform cylindrical simply supported shell, θ = π /4, v = 0.3004.

96 | 15 Example calculations

Fig. 15.4. Square planform cylindrical simply supported shell, θ = π /8, v = 0.5606.

Fig. 15.5. Square planform cylindrical simply supported shell, θ = 0, v = 2.7654.

In addition, calculations were performed for cylindrical shells with the radii of 10 and 40 with the same directions of the flow velocity vector. The critical velocity values obtained for the radius equal to 10 are 0.8216(14), 0.4629(1), 0.2287(1), 0.1727(1), and 0.1591(1); those obtained for the radius 40 are 0.3378(6), 0.3439(1), 0.2433(1), 0.1673(1), and 0.1514(1). Thus, as R → ∞, the critical flutter velocity decreases for the flows directed both along and across the generatrix. This conclusion is very important, since a small initial convexity of the shell in a transverse flow (for a radius equal to 40, the shell rise is just 0.0125) leads to the decrease in the critical flutter velocity.

| Part III: Numerical methods for non-self-adjoint eigenvalue problems

98 | Part III Numerical methods for non-self-adjoint eigenvalue problems At the present time, the most popular solution technique in the mechanics of solids is the finite element method. Its deficiencies are well known: a piecewise-linear approximation of displacements results in stress discontinuity. Most problems in mechanics of solids, however, are governed by equations of the elliptic type having smooth solutions. Therefore, of great importance is the development of algorithms which could take advantage of this smoothness. The idea of such algorithms was K. I. Babenko’s [65, 66]. Since then, their efficiency has been proven in many applications to elliptic eigenvalue problems solved by the authors of this book. In particular, problems of this type include flutter of plates and shallow shells described in the new framework by A. A. Ilyushin and I. A. Kijko [311]. A principal advantage of this problem statement is that a complex aeroelasticity problem is reduced to an elliptic eigenvalue problem, for which effective nonsaturating algorithms can be developed. With this approach, it became possible to solve new problems which are beyond the reach of the classical methods. The main result of the current work is that expensive and difficult real experiments can be substituted by computational experiments. Examples of such studies were presented above.

16 Discretization of the Laplace operator In the previous parts of this book it was shown that panel flutter problems are reduced to the eigenvalue analysis for a non-self-adjoint operator (or a system of such operators). For problems of this type, exact analytical methods have only limited success, as will be demonstrated below, which stimulates the development of approximate and numerical methods. Of the approximate methods, the Bubnov–Galerkin method is worth mentioning; however, difficulties with the choice of a proper system of coordinate functions for arbitrary domains and boundary conditions limit significantly its capabilities. The finite element method (FEM), being universal, finds wide applications in flutter studies; however, it possesses some serious drawbacks which will be considered in the following discussions. This and the subsequent chapters are devoted to the development of an alternative to the FEM numerical-analytical method, which is nonsaturating (according to K. I. Babenko’s terminology) and, in our opinion, performs much better in eigenvalue problems. It is convenient to demonstrate the approach by considering the Sturm– Liouville problem.

16.1 The Sturm–Liouville problem Consider the classical problem y󸀠󸀠 (x) − q(x)y(x) = λρ (x)y(x)

(16.1)

y(−1) = y(+ 1) = 0.

(16.2)

Here, q(x) and ρ (x) are given functions, and λ is a spectral parameter. This problem is traditionally solved by the finite-difference method; if h is the grid step, we introduce on the interval (−1, + 1) a total of n grid points: xi = −1 + hi, h = 2/(n + 1), i = 1, 2, . . . , n; x0 = −1, xn+1 = 1. Thus, on the interval [−1, 1] we choose (n + 2) nodes. Let y(x) ∈ C3 [−1, 1], then y(x + h) = y(x) +

y󸀠 (x) y󸀠󸀠 (x) 2 y󸀠󸀠󸀠 (x) 3 h+ h + h + O(h4 ) 1! 2! 3!

(16.3)

y(x − h) = y(x) −

y󸀠󸀠 (x) 2 y󸀠󸀠󸀠 (x) 3 y󸀠 (x) h+ h − h + O(h4 ). 1! 2! 3!

(16.4)

Summing up (16.3) and (16.3), we obtain y(x + h) + y(x − h) = 2y(x) + y󸀠󸀠 (x)h2 + O(h4 ) ⇒ y󸀠󸀠 (x) =

y(x + h) − 2y(x) + y(x − h) + O(h2 ) h2

(16.5)

100 | 16 Discretization of the Laplace operator Denote y(xi ) = yi ;

y󸀠󸀠 (xi ) = yi󸀠󸀠 ;

then it follows from (16.5) that yi󸀠󸀠 =

yi+1 − 2yi + yi−1 + O(h2 ), h2

i = 1, 2, . . . , n.

(16.6)

The first term on the right-hand side of equation (16.6) is the second-order finitedifference approximation for the second derivative yi󸀠󸀠 with the truncation error O(h2 ). Substitute equations (16.6) into (16.3) to obtain at each grid point yi+1 − 2yi + yi−1 − qi yi = λρi yi + O(h2 ), h2 y0 = yn+1 = 0.

i = 1, 2, . . . , n

(16.7) (16.8)

Neglecting the truncation error O(h2 ), we obtain an approximate finite-dimensional problem for a tridiagonal symmetric matrix. It will be shown in Chapter 22 that the perturbation introduced into the eigenvalues by dropping O(h2 ) is of the order of the discretization error multiplied by a factor depending on the distance between the given eigenvalue and the remaining part of the spectrum. Thus, independently of the smoothness of the solution to the Sturm–Liouville problem (16.1)–(16.2), the eigenvalues are obtained with an error of the order of O(h2 ). According to terminology introduced by K. I. Babenko, the finite-difference method for the Sturm–Liouville problem possesses saturation. A similar drawback is also characteristic of the finiteelement method. We shall now present an alternative method for the solution of problem (16.1)–(16.2), which is free of the just mentioned deficiency. We reduce the boundary-value problem (16.1)–(16.2) to an integral equation. Let G(x, ξ ) be Green’s function for the operator d2 /dx2 with boundary conditions (16.2). We then have +1

y(x) = ∫ G(x, ξ ) [q(ξ ) + λρ (ξ )y(ξ )] dξ .

(16.9)

−1

Discretization of the integral equation (16.9) is performed by using, for the functions qy and ρ y, an interpolation formula n

(Pn f )(x) = ∑ f (xk )lk (x) + Rn (x; f ), k=1

where the fundamental interpolation functions are given by lk (x) =

Tn (x) , (x − xk )Tn󸀠 (xk )

Tn (x) = cos(n arccos x),

k = 1, 2, . . . , n

xk = cos[(2k − 1)π /2n],

16.1 The Sturm–Liouville problem

|

101

with Rn (x; f ) being the interpolation error. As a result, we obtain n

y(x)q(x) = ∑ yk qk lk (x) + Rn (x; yq) k=1 n

y(x)ρ (x) = ∑ yk ρk lk + Rn (x; yρ ) k=1

yk = y(xk ),

ρk = ρ (xk ),

qk = q(xk ),

k = 1, 2, . . . , n.

Substitute these relations into equation (16.9) to obtain n

n

yj = ∑ Djk qk yk + λ ∑ Djk ρk yk + rn (xj ; yq) + λ rn (xj ; yρ ). k=1

k=1

Here, +1

Djk = ∫ G(xj , ξ )lk (ξ ) dξ ,

j, k = 1, 2, . . . , n

(16.10)

−1 +1

rn (xj ; yq) = ∫ G(xj , ξ )Rn (ξ , yq) dξ ,

j = 1, 2, . . . , n

(16.11)

j = 1, 2, . . . , n.

(16.12)

−1 +1

rn (xj ; yρ ) = ∫ G(xj , ξ )Rn (ξ , yρ ) dξ , −1

As a result, we obtain the following algebraic eigenvalue problem: (An − λ Bn )y = ra + λ rb .

(16.13)

Here, An = I − DQ and Bn = DP are n × n matrices; Q = diag(q1 , . . . , qn ), P = diag(ρ1 , . . . , ρn ) are diagonal matrices.Elements of matrix D are defined by equation (16.10), and the elements of error vectors ra and rb are determined by equations (16.11) and (16.12), respectively. Note that λ in equation (16.13) is the exact eigenvalue, and y is an n-vector with elements equal to the values of the corresponding eigenfunction at the grid points. By dropping the discretization errors ra and rb in equation (16.13), we obtain an approximate eigenvalue problem: (An − λ ̃ Bn )ỹ = 0,

(16.14)

where λ ̃ is an approximate eigenvalue, while ỹ is an n-vector with elements equal to the approximate values of the corresponding eigenfunction at the grid points. Some theorems which can be used to evaluate the error introduced into the eigenvalues by dropping the discretization error will be considered in Chapter 22; here, we give only the final result.

102 | 16 Discretization of the Laplace operator For example, let λ be a simple eigenvalue of the Sturm–Liouville problem (16.1)– (16.2), while Γλ be a convex rectifiable contour containing inside only one eigenvalue λ . Then, if r0−1 = sup Spr[R(ζ )(An − ζ Bn ) − I] < 1, ζ ∈Γλ

−1

where R(ζ ) = (A − ζ B) is the resolvent of the matrix pencil A, B, with the matrices A and B defined by relations A = An − (y, y)−1 ra y∗ ,

B = Bn − (y, y)−1 rb y∗ ,

where y∗ is a row vector complex conjugate to the column vector y, then inside Γλ there exists one eigenvalue λ ̃ of the problem (16.14), and the following inequality holds: r−1 󵄨󵄨 󵄨 󵄨󵄨λ − λ ̃ 󵄨󵄨󵄨 ≤ ρ 0 −1 , 󵄨 󵄨 1−r 0

󵄨 󵄨 ρ = max 󵄨󵄨󵄨λ − ζ 󵄨󵄨󵄨 . ζ ∈Γ

(16.15)

λ

Calculations give r0−1 = sup Spr[R(ζ )(ra − ζ rb )y∗ (y, y)−1 ]. ζ ∈Γλ

Substitute in this relation the spectral radius by the Euclidean norm, which makes inequality (16.15) even stronger. Thus, 󵄩 󵄩 󵄩 󵄩 r0−1 ≤ sup Spr 󵄩󵄩󵄩R(ζ )󵄩󵄩󵄩2 󵄩󵄩󵄩ra − ζ rb 󵄩󵄩󵄩2 . ζ ∈Γλ

The rate at which the discretization errors ra and rb are reduced depends on the smoothness of functions qy and ρ y. Namely, an inequality holds: 󵄨 󵄨 max 󵄨󵄨󵄨Rn (x; y)󵄨󵄨󵄨 ≤ c(1 + ωn )En−1 (y), |x|≤1

(16.16)

where c is an absolute constant, ωn = O(ln(n)) is the Lebesgue interpolation constant, while En−1 (y) is the best approximation of function y by a polynomial of degree less than (n − 1) in the norm C. According to the Weierstrass theorem, for any continuous function we have lim En (y) = 0,

n→∞

and the rate at which En (y) decreases as n → ∞ depends on the smoothness of y. Thus, let y(x) be a continuous function defined on the interval [−1, + 1], and Pn (x) be a polynomial of degree n which has the smallest deviation from y(x) on the interval being considered, En (y) is the best approximation to y(x) by a polynomial of degree n, so that 󵄨 󵄨 En (y) = max 󵄨󵄨󵄨y(x) − Pn (x)󵄨󵄨󵄨 . |x|≤1

Then the following theorem holds true:

16.1 The Sturm–Liouville problem

|

103

Theorem 1 (D. Jackson). If a function y(x) has on the interval −1 ≤ x ≤+ 1 a continuous derivative y(p) (x) satisfying the Lipschitz condition 󵄨 󸀠 󵄨 󵄨 󸀠󸀠 󵄨 󵄨 󸀠 󵄨󵄨󵄨 (p) 󸀠 (p) 󸀠󸀠 󵄨󵄨 󸀠󸀠 󵄨 󸀠 󸀠󸀠 󵄨󵄨y (x ) − y (x )󵄨󵄨󵄨 < K 󵄨󵄨󵄨󵄨x − x 󵄨󵄨󵄨󵄨 , (x ≠ x ; 󵄨󵄨󵄨󵄨x 󵄨󵄨󵄨󵄨 , 󵄨󵄨󵄨󵄨x 󵄨󵄨󵄨󵄨 ≤ 1), 󵄨 󵄨 then for its best approximation by ordinary polynomials the following inequality holds: En (y)
0, and assume that q ≡ 0 (the case q ≠ 0 is tackled in a similar way). Left-multiply (16.35) by matrix C and make substitution u = (ZPC)−1/2 w. Then we obtain an eigenvalue problem for matrix A = (ZPC)1/2 B(ZPC)1/2 , where B = CH, while matrix (ZPC)1/2 is a diagonal one, with elements √zi pi /ci on the diagonal. It is evident that matrix A, as well as B, is asymptotically symmetric.

17 Discretization of linear equations in mathematical physics with separable variables The approach to the discretization of Laplace operator presented in Chapter 16 is based on the fact that the discretized problem inherits properties of the differential problem. In particular, the separability of variables is inherited. An important question is how this discretization approach can be extended to other equations of mathematical physics with separable variables. In particular, this is essential for the construction of discrete biharmonic operator on a rectangular domain with simply supported boundary conditions.

17.1 General form of equations with separable variables Consider a linear operator S in the Banach function space B, having an eigenfunction uk = vk (⋅) exp(ikφ ), k = 0, 1, . . . , where vk (⋅) denotes a function of one or more arguments, φ ∈ [0, 2π ]. Separability of variables means that Suk = (sk vk ) exp(ikφ ),

(17.1)

where sk is some linear operator. Suppose also that the linear operators S and sk have real-valued coefficients, then S(Re uk ) = (sk vk ) Re(exp(ikφ )).

(17.2)

From the linearity of operators S and sk it follows that the properties (17.1) and (17.2) also hold for exponential functions of the form exp[ik(φ − φp )], where φp is some number. Let u ∈ B. We apply the following interpolation in φ: u≈

2 2n ∑ u D (φ − φp ), N p=0 p n

N = 2n + 1, φp = 2π p/N.

Here, up = u(⋅, φp ); the dot denotes one or more variables: n

󸀠

Dn (φ − φp ) = ∑ cos [k (φ − φp )] , k=0

where the prime on the summation sign means that the term for k = 0 is taken with coefficient 12 .

17.2 Further generalization

| 119

We then have u≈ Su =

2 n 2n ∑ { ∑ u cos [k(φ − φp )]} N k=0 p=0 p 2 n 2n ∑ { ∑ (s u ) cos [k(φ − φp )]} . N k=0 p=0 k p

Discretize now the operator sk . We apply to function up interpolation of the form m

up ≈ ∑ lq (⋅)uqp , q=1

where lq , q = 1, 2, . . . , m are the fundamental interpolation functions, m is the number of grid nodes, uqp is the value if function up at q-th grid node. Denote aqk (⋅) = sk lq (⋅) Hqp (⋅, φ ) =

2 n ∑ a (⋅) cos [k (φ − φp )] . N k=0 qk

Then

(17.3)

m 2n

Su ≈ ∑ ∑ Hqp (⋅, φ )uqp . q=1 p=0

If (⋅) and φ run through all the grid nodes, then we obtain from equation (17.3) that Su ≈ Hu, where H is an h-matrix, and u is a column vector containing the values of corresponding function at grid points (the grid points are numbered first in φ , then in the other spatial variables). To construct the blocks of h-matrix Λk , k = 0, 1, . . . , n, it is necessary to discretize the operators sk . The discretization method for equations with separable variables presented in this section was applied for a fast solution of Poisson’s equation in a torus and in the exterior of a body of revolution [26]. The right-hand side of the Poisson’s equation was arbitrary, i.e. the problems considered were three-dimensional; an approximate solution of the Poisson’s equation was based on the properties of the h-matrix.

17.2 Further generalization The next logical step to generalization of the discretization method presented in Section 17.1 is to consider the case where the eigenfunction of the linear operator can be presented in the form of a product of a function of many variables and a function of a single variable (in Section 17.1 the function of a single variable was exp(ikφ )). As an example, consider a Poisson’s equation on a rectangle G = {[−1, 1] × [−b, b]}. It is required to find a matrix which inherits the property of separability of variables for

120 | 17 Discretization of linear equations in mathematical physics with separable variables the eigenfunction of Laplace operator on a rectangle; such a matrix has the following form: C = In ⊗ A + B ⊗ Im . (17.4) Here, n is the number of grid points along the height of the rectangle, m is the number of grid points along the width of the rectangle, In is a n × n unity matrix, A is a m × m matrix (one-dimensional discrete Laplacian on the interval [−1, 1]), B is an n × n matrix (one-dimensional discrete Laplacian on the interval [−b, b]);, and Im is an m × m unity matrix. To construct the matrices A and B, it is necessary to discretize the one-dimensional spectral problem u󸀠󸀠 = λ u with boundary conditions u(−1) = u(1) = 0 and u(−b) = u(b) = 0, respectively. This discretization is performed by the method described in Section 16.1. The eigenvalue of matrix C is a sum of eigenvalues of matrices A and B, while the corresponding eigenvector is given by the Kronecker product of eigenvectors of these matrices. This property demonstrates that the discrete Laplacian inherits properties of the differential Laplace operator. Representation of the eigenfunction of the differential Laplace operator in the form of the product of two functions of a single variable is matched by the Kronecker product of the eigenvectors of matrices A and B. The matrix property (17.4) shows that, instead of the Laplace operator, some other linear operator of mathematical physics can be considered, whereas instead of the rectangle G we can consider any domain on which the eigenfunction of the operator is represented as a product of two functions (for example, the product of a function of two variables and a function of single variable). The next question to be answered is to what extent the properties of the class of h-matrices are relevant to the matrices (17.4): namely, how to invert matrix C analytically, and if an algorithm is possible for fast multiplication of matrix C−1 by a vector. Consider these questions in detail. Let n

B = ∑ λk bk ,

b2k = bk ,

bk bp = 0,

k ≠ p

k=1

be the spectral decomposition of matrix B. This decomposition can always be constructed, provided that B has simple structure, i.e. it possesses a complete eigenvector system; this particular case will be considered in our further discussion. Here, bk , k = 1, 2, . . . , n are eigen projectors onto the one-dimensional invariant subspace, and λk is the corresponding eigenvalue. In the practical calculations, the size of matrix B is quite moderate (n ≤ 19), and the spectral decomposition can be constructed by solving the complete eigenvalue problem for matrices B and B󸀠 .

17.2 Further generalization

| 121

Note that ∑nk=1 bk = In , because matrix ∑nk=1 bk coincides with its inverse, and transforms equation (17.4) in the following way: n

n

C = ( ∑ bk ) ⊗ A + ( ∑ λk bk ) ⊗ Im k=1 n

k=1 n

= ∑ (bk ⊗ A + λk bk ⊗ Im ) = ∑ bk ⊗ (A + λk Im ) . k=1

k=1

Then the inverse to matrix C is n

C−1 = ∑ bk ⊗ (A + λk Im )−1 ,

(17.5)

k=1

which is verified by immediate multiplication. Equation (17.5) is a generalization of equation (16.44) (see Chapter 16). The case where fast multiplication of matrix C−1 by a vector (circular cylinder) is possible is considered in detail in [26]. The discretization of the Laplace operator presented in this section is used for construction of a discrete biharmonic operator with simply supported boundary conditions.

18 Eigenvalues and eigenfunctions of the Laplace operator In Chapter 17, the discretization of the Laplace operator was considered which relied on an interpolation formula for a function of two variables on a disk with equal number of nodes on each circular grid line. Here, we extend these results to a more generic interpolation formula (16.28a) with an unequal number of grid points on the circular grid lines. We consider three boundary-value problems: Δu + (Q + λ P)u = 0,

z∈G

(18.1)

u|𝜕G = 0

(18.2)

𝜕u 󵄨󵄨󵄨󵄨 󵄨 =0 𝜕n 󵄨󵄨󵄨𝜕G 𝜕u 󵄨󵄨󵄨󵄨 Au + 󵄨 = 0, 𝜕n 󵄨󵄨󵄨𝜕G

(18.3) (18.4)

where Q, P, and A are some functions defined on domain G, n is the outward-pointing normal to 𝜕G. We assume that Q, P, A, and 𝜕G ∈ C∞ . 󵄨 󵄨 Let z = φ (ζ ), 󵄨󵄨󵄨ζ 󵄨󵄨󵄨 ≤ 1 be the conformal mapping of the unity disc onto the domain Γ; then , from equations (18.1) and (18.4) we obtain that 󵄨 󵄨2 Δu + 󵄨󵄨󵄨󵄨φ 󸀠 (ζ )󵄨󵄨󵄨󵄨 (q + λ p)u = 0

(18.5)

u|r=1 = 0 𝜕u 󵄨󵄨󵄨󵄨 󵄨 =0 𝜕r 󵄨󵄨󵄨r=1 𝜕u 󵄨󵄨󵄨󵄨 αu + 󵄨 = 0, 𝜕r 󵄨󵄨󵄨r=1

(18.6) (18.7) (18.8)

󵄨 󵄨 where q(ζ ) = Q (z(ζ )), p(ζ ) = P (z(ζ )), and α (ζ ) = A (z(ζ )) 󵄨󵄨󵄨󵄨φ 󸀠 (ζ )󵄨󵄨󵄨󵄨. Example 1. Let φ (ζ ) = ζ (1 + εζ n ), 0 < ε < 1/(n + 1); then 󵄨󵄨 󸀠 󵄨󵄨2 󵄨󵄨φ (ζ )󵄨󵄨 = 1 + ε 2 (n + 1)2 r2n + 2ε rn (n + 1) cos nθ , 󵄨 󵄨 󵄨2 󵄨 The calculation of 󵄨󵄨󵄨󵄨φ 󸀠 (ζ )󵄨󵄨󵄨󵄨 is performed by subroutine MOD2.

ζ = reiθ .

(18.9)

18.1 The Dirichlet problem

| 123

Listing of the subroutine MOD2

1

SUBROUTINE MOD2 (Z,M,NL,EPS,N) IMPLICIT REAL*8 (A-H,O-Z) DIMENSION Z(1),NL(M) PI=3.14159265359D0 I=0 DO 1 NU=1,M R= COS((2.*NU-1.)*PI/4./M) R1=R**N*(N+1) R2= R1*R1 N1=NL(NU) DO 1 L=1,N1 I=I+1 T=2.*PI*(L-1.)/N1 Z(I)=1.+2.*R1*EPS*COS(N*T)+EPS*EPS*R2 RETURN END

Parameters. Z is an array containing the calculated values of |φ 󸀠 (ζ )|2 at the interpolation nodes inside the disk, the length of this array is equal to the number of nodes inside the disk, M is the number of circles in the disk, NL is a one-dimensional array of length M, its i-th element contains the number of points (odd) on i-th circular grid line, and EPS and N are ε and n, respectively. 1 Let K(ζ , ξ ) = − 2π ln |(1 − ζ ξ ̄ )/(ζ − ξ )| be Green’s function for the Laplace operator with Dirichlet boundary condition. With this function, we reduce our problem to the integral equation u(ζ ) = − ∫ K(ζ , ξ )|φ 󸀠 (ξ )|2 [q(ξ ) + λ p(ξ )] u(ξ ) dξ |ξ |≤1

+ ∫ |ξ |=1

𝜕K(ζ , ξ ) ψ (θ ) dθ , ζ = ρ eiφ , 𝜕r

ξ = reiθ .

(18.10)

Here, ψ (θ ) ≡ u(eiθ ) is the value of u on the boundary. For the Dirichlet problem, we have ψ (θ ) ≡ 0, whereas for the Neumann and mixed problems it must be found from the boundary condition.

18.1 The Dirichlet problem In this case the integral equation (18.10) is simplified to 󵄨󵄨 dz 󵄨󵄨2 󵄨 󵄨 u(ζ ) = − ∫ r ∫ K(ζ , reiθ ) 󵄨󵄨󵄨 󵄨󵄨󵄨 (q + λ p) u dθ dr. 󵄨󵄨 dξ 󵄨󵄨 1

2π

0

0

(18.11)

124 | 18 Eigenvalues and eigenfunctions of the Laplace operator dz 2 Interpolate | dξ | (q + λ p) u by formula (16.28a) to obtain

u(ζ ) = λ H𝜈l (ζ )(q𝜈l + λ p𝜈l )u𝜈l .

(18.12)

Note. Hereafter, we make use of tensor notation. Summation is implied over repeating Greek indices in the range from 1 to m/2, and over Latin indices in the range from 0 to (2n𝜈 + 1). Implicit formulas for the integrals H𝜈l (ζ ) can be written as H𝜈l (ρ eiφ ) =

n𝜈 { L (ρ ) 1 2(−1)𝜈−1 sin ψ𝜈 ∑ mk cos k(φ − θ𝜈l ) { (2n𝜈 + 1)m k k=1(2) { n𝜈 L (ρ ) + 2r𝜈 ∑ 0k cos k(φ − θ𝜈l ) k k=1(2) n𝜈

m−1

+ 2r𝜈 ∑ cos sψ𝜈 ∑ s=1

k=1

1 − (−1)s+k Lsk (ρ ) cos k(φ − θ𝜈l ) k

} + 4r𝜈 ∑ cos sψ𝜈 Ls0 (ρ )} , s=1(2) } m−1

where ψ𝜈 = (2𝜈−1)π , m/2 is the number of circles, (2n𝜈 + 1) is the number of points on 2m n𝜈 𝜈-th circle, and ∑k=1(2) denotes summation from k = 1 to k = n𝜈 with step 2 etc.: ρ

Lsk (ρ ) = ρ

−k

1

1

∫ Ts (r)rk dr + ρ k ∫ Ts (r)r−k dr − ρ k ∫ Ts (r)rk dr ρ

0

0

Ts (r) = cos(s arccos r) L0k (ρ ) = (ρ k − ρ ) ρ

2k , 1 − k2

L01 (ρ ) = −ρ ln ρ

1

Lk0 (ρ ) = − ∫ Tk (ρ ) ln ρ dr − ∫ Tk (r) ln r dr, 0

ρ

L10 (ρ ) =

1 − ρ2 . 4

Let the point ζ run through all interpolation nodes, we then obtain an algebraic eigenvalue problem u = H(Q + λ P)u, (18.13) where u = (u1 , u2 , . . . ,n )󸀠 is the approximate value of the eigenfunction at the interpolation nodes, Q = diag(q1 z1 , . . . , qn zn ) and P = diag(p1 z1 , . . . , pn zn ) are diagonal matrices with pi and qi being the values of functions q = Q (z(ζ )) and p = P (z(ζ )) at 󵄨 󵄨2 the interpolation nodes, while zi are the values of 󵄨󵄨󵄨󵄨φ 󸀠 (ζ )󵄨󵄨󵄨󵄨 at the interpolation node, n = ∑m/2 𝜈=1 (2n𝜈 + 1) is the number of interpolation nodes, Hji = Hi (ζj ) is a n × n matrix. The nodes are enumerated counterclockwise, starting with the first circle. This problem is solved by program LAP123G.

18.1 The Dirichlet problem

Listing of the program LAP123G PROGRAM LAP123G C 29.09.94 IMPLICIT REAL*8 (A-H,O-Z) PARAMETER(M=9,M1=18,M2=17,M3=19,NM=13,NMAX=27,NT=135, *NT2=18225,NG=31,NTG=4185,NMG=837) DIMENSION A(NT2),AB(NT2),X(NT),Y(NT),IANA(NT),HN(NTG), *NL(M),B(M,M),C(M,NMAX),D(M,NMAX),BC(M1,NM),E(M3,M) DIMENSION B1(M),C1(NMG),B2(NG,NG),E1(M2,M),H(NG), *U(NT),Z(NT),Y2(11) DIMENSION E2(NG,NG) CHARACTER*1 CH1,CH2 EQUIVALENCE (AB(1),HN(1)) COMMON//EPS1,NP COMMON /DM/ DM EXTERNAL QMOD2,PMOD2 DATA NL/27, 25, 23, 21, 17, 9, 7, 3, 3/ DM=17.75D0 DM=1.D0 NP=4 EPS1=1.D0/6.D0 WRITE(*,*) ’ NP = ? , EPS1 = ? ’ READ(*,*) NP,EPS1 NONL=9 NONL=8 WRITE (*,*) ’Calculate by program LAPLAS or read from disk 1 (Y/N)’ READ (*,14) CH1 IF (CH1.EQ.’Y’) CALL LAPLAS (A,M,M1,M3,NL,NM,B,X,Y,C,D,BC,E) IF (CH1.EQ.’N’) THEN REWIND NONL READ (NONL ) (A(I),I=1,NT2) ENDIF WRITE (*,*) ’Write result of LAPLAS to disk (Y/N)’ READ (*,14) CH2 14 FORMAT (A) IF (CH2.EQ.’Y’) THEN REWIND NONL WRITE (NONL) A END FILE NONL ENDIF WRITE(*,*) ’Enter the boundary-value problem type ? ’ READ(*,*) IP IF(IP.EQ.1.OR.IP.EQ.2) GO TO 100 IF(IP.EQ.3) CALL LAP3 (A,NG,M,NL,NM,HN,B1,X,C1,B2,BC,E1,H) 100 CALL TRANSP (A,NT) IF (IP.EQ.2) GO TO 200 C

| 125

126 | 18 Eigenvalues and eigenfunctions of the Laplace operator

CALL MOD2(Y,M,NL,EPS1,NP) I1=0 DO 5 I=1,NT DO 5 J=1,NT I1=I1+1 5 A(I1)=A(I1)*Y(I) IF (IP.EQ.1.OR.IP.EQ.3) GO TO 400 200 CONTINUE C Neumann problem CALL BIJ(A,NT,NG,M,NL,NM,HN,B1,X,C1,BC,E,H,E2) I1=0 DO 55 I=1,NT DO 55 J=1,NT I1=I1+1 55 AB(I1)=A(I1) CALL LDUDN (A,AB,NT,Y,C,NL,M,QMOD2,PMOD2) CALL DMINV (A,NT,DD1,X,Y) CALL DIVAB (NT,A,AB,Y) 400 CONTINUE CALL ELMHES (NT,NT,1,NT,A,IANA) WRITE(*,*) ’ELMHES’ CALL ELTRAN (NT,NT,1,NT,A,IANA,AB) WRITE(*,*) ’ELTRAN’ CALL HQR2 (NT,NT,1,NT,A,X,Y,AB,IERR) WRITE(*,*) ’HQR2’ NOUT=4 OPEN (NOUT,FILE=’NOUT’) WRITE (NOUT,*) ’ IERR = ’, IERR 13 FORMAT (13I5) 12 FORMAT (4E18.11) WRITE(*,12) (X(I),I=1,NT) WRITE (NOUT,*) ’X’ WRITE(NOUT,12) (X(I),I=1,NT) WRITE (NOUT,*) ’Y’ WRITE(NOUT,12) (Y(I),I=1,NT) IF (IP.EQ.1.OR.IP.EQ.3) THEN DO 210 I=1,NT 210 Y(I)=1.D0/SQRT(ABS(X(I))) ELSE Y(1)=0.D0 DO 1 I=2,NT 1 Y(I)=SQRT(1.D0+1.D0/X(I)) ENDIF WRITE (NOUT,*) ’Eigenvalues’ WRITE(NOUT,12) (Y(I),I=1,NT) WRITE (*,*) ’Eigenvalues’ WRITE(*,12) (Y(I),I=1,NT) PAUSE C

18.1 The Dirichlet problem

22

4

20

21 120

NT1=NT/3 NT1=6 M11=2*M IF(IP.EQ.2) KN=2 IF(IP.NE.2) KN=1 DO 21 K=KN,NT1 I2=NT*(K-1) DO 22 I=1,NT I3=I2+I U(I)=AB(I3) CALL URT (0.D0,M,NL,U,Y) CALL URT (3.14159265359D0,M,NL,U,Z) DO 4 I=1,M I1=M11-I+1 Y(I1)=Z(I) DO 20 LL=1,11 IF (IP.EQ.1.OR.IP.EQ.3) THEN X2=0.1*(LL-1)*SQRT(X(K)) ELSE X2=0.1*(LL-1)/SQRT(1.D0+1.D0/X(K)) X2=0.1*(LL-1) ENDIF Y2(LL)=EIGEN (X2,Y,Z,M11,-1.D0,+1.D0) CALL NORM1(Y2,11) WRITE (NOUT,12) Y2 PRINT 12,Y2 PAUSE FORMAT(A) END FUNCTION ALFA (X) IMPLICIT REAL*8 (A-H,O-Z) COMMON//EPS1,NP COMMON /DM/ DM ALFA=DM*SQRT(1.+2.*EPS1*(NP+1.)*COS(NP*X)+EPS1*EPS1*(NP+1)**2) RETURN END

1

2

SUBROUTINE NORM1(Y,N) IMPLICIT REAL*8 (A-H,O-Z) DIMENSION Y(1) P=0.D0 DO 1 I=1,N IF (ABS(Y(I)).GT.P) IP=I IF (ABS(Y(I)).GT.P) P=ABS(Y(I)) CONTINUE P=Y(IP) DO 2 I=1,N Y(I)=Y(I)/P

| 127

128 | 18 Eigenvalues and eigenfunctions of the Laplace operator

RETURN END

1

1

SUBROUTINE PMOD2 (Z,M,NL) IMPLICIT REAL*8 (A-H,O-Z) DIMENSION Z(1),NL(1) COMMON // EPS,NP I=0 DO 1 NU=1,M N=NL(NU) DO 1 L=1,N I=I+1 Z(I)=1.D0 CALL MOD2(Z,M,NL,EPS,NP) RETURN END SUBROUTINE QMOD2 (Z,M,NL) IMPLICIT REAL*8 (A-H,O-Z) DIMENSION Z(1),NL(1) COMMON // EPS,NP I=0 DO 1 NU=1,M N=NL(NU) DO 1 L=1,N I=I+1 Z(I)=1.D0 CALL MOD2(Z,M,NL,EPS,NP) RETURN END

Parameters of the program are set up in the operator PARAMETER(M=9,M1=18, M2=17,M3=19,NM=13,NMAX=27,NT=135,NT2=18225,NG=31,NTG=4185,NMG=837). Here, M=9 is the number of circles on the disk, M1=2*M, M2=2*M-1, M3=2*M+1, NM= max n𝜈 , 2n𝜈 + 1 is the number of points on 𝜈-th circle, NMAX=27 is the maximum 𝜈 number of points on the circles, NT=135 is the total number of grid points, NT2=NT*NT, NG=31 is the number of points on the disk boundary, NTG=NG*NT, NMG=(2*NM+1)*NG. The number of points on each circle is set in the operator DATA NL/27, 25, 23, 21, 17, 9, 7, 3, 3/, the parameter A in the boundary condition is set in the operator DM=1; if α is defined as a function, it has to be done in FUNCTION ALFA(X). Required subroutines. LAPLAS, LSK, LS0, LAP3, TRANSP, MOD2, BIJ, LDUDN, DMINV, DIVAB, ELMHES, ELTRAN, HQR2, URT, EIGEN, NORM1, PMOD2, QMOD2. Subroutines LAPLAS, LSK, and LS0 are described below; NORM1(Y,N) is the normalization subroutine which calculates the Chebyshev norm of vector Y of length N. All other subroutines are described in [26]. The matrix H is calculated by the subroutine LAPLAS.

18.1 The Dirichlet problem

Listing of the program LAPLAS

1

11

3 2

6

9

SUBROUTINE LAPLAS (A,M,M1,M3,NL,NM,B,X,Y,C,D,BC,E) IMPLICIT REAL*8 (A-H,O-Z) DIMENSION A(1),B(M,M),X(M),Y(M3),NL(M),C(M,1),D(M,1),BC(M1,NM), 1 E(M3,M) DOUBLE PRECISION BC,X,Y,PD ,PID,EPS,E EPS=1.D-19 PID=3.14159265358979323846264D+0 PI=PID M2=M1-1 DO 1 NU=1,M PD=PID*FLOAT(2*NU-1)/FLOAT (2*M1) X(NU)=DCOS(PD) E(1,NU)=X(NU) DO 1 KS=2,M3 E(KS,NU)=DCOS(FLOAT(KS)*PD) DO 2 MU=1,M PD=X(MU) DO 11 I=1,M2 Y(I)=E(I,MU) CALL LS0 (PD,Y,M2) DO 2 NU=1,M P1=0. DO 3 KS=1,M2,2 PD=X(NU)*E(KS,NU)*Y(KS) IF (DABS(PD).LE.EPS) PD=0.D+0 P1=P1+4.*PD B(NU,MU)=P1 DO 6 NU=1,M N1=NL(NU) DO 6 L=1,N1 L1=L-1 P=2.*PI*L1/N1 C(NU,L)=COS(P) D(NU,L)=SIN(P) I1=0 DO 7 MU=1,M DO 9 I=1,M3 Y(I)=E(I,MU) PD=X(MU) P=PD CALL LSK (BC,Y,M1,NM,PD) N1= NL(MU) DO 7 KA=1,N1 DO 7 NU=1,M N2= NL(NU) N3=(N2-1)/2 MNU=M-NU+1

| 129

130 | 18 Eigenvalues and eigenfunctions of the Laplace operator

13

8

20 10 7

DO 7 L=1,N2 I1=I1+1 A(I1)=B(NU,MU) DO 13 K=1,N3 LK=MOD(L*K,N2) +1 KAK=MOD(KA*K,N1)+1 Y(K)=C(NU,LK)*C(MU,KAK)+D(NU,LK)*D(MU,KAK) DO 8 K=1,N3,2 K1=K+1 IF (K.NE.1) P1=(P**K-P)*2.*K/(1-K*K) IF (K.EQ.1) P1=-DLOG(P)*P PD=Y(K)*X(NU) PID=Y(K)*X(MNU)*BC(M1,K) IF (DABS(PID).LE.EPS ) PID=0. D+0 IF ( DABS(PD).LE.EPS ) PD=0. D+0 A(I1)=A(I1)+(2.*PD *P1+2.*(-1)**(NU-1)*PID)/FLOAT (K) DO 10 KS=1,M2 I=(1-(-1)**KS)/2+1 IF(I.GT.N3) GO TO 10 DO 20 K=I,N3,2 PD=Y(K)*X(NU)*E(KS,NU)*BC(KS,K) IF ( DABS(PD).LE.EPS ) PD=0. D+0 A(I1)=A(I1)+4.*PD /FLOAT(K) CONTINUE A(I1)=A(I1)/FLOAT(N2*M1) RETURN END

Parameters. A is an array which on output contains the matrix A ordered rowwise. The array length is equal to he square of the number of interpolation nodes; M is the number of circles, M1=2*M; M3=2*M+1; NL is a one-dimensional array of length M, its 𝜈th element contains the (odd) number of points on the 𝜈-th circle; NM= max n𝜈 , 2n𝜈 + 1 𝜈 is the number of points on the 𝜈-th circle; B, X, Y, C, D, BC, E are work arrays of dimensions B(M,M), X(M), Y(M3), C,D=(M,(2*NM+1)), BC(M1,NM), E(M3,M). All arrays must be defined with double precision. Required subroutines. LS0, LSK. Listing of the program LS0 SUBROUTINE LS0 (R0,Y,M) DIMENSION Y(M) DOUBLE PRECISION R0,Y,P1,P2 DOUBLE PRECISION R1,R2,R3 DO 2 K=3,M,2 K1=M-K+3 P=K1+1

18.1 The Dirichlet problem

2

3

| 131

Y(K1)=R0*Y(K1)/P-FLOAT(K1)*Y(K1-1)/P/FLOAT(K1-1) P1=1.-R0*R0 Y(1)=0.25*P1 DO 3 K=3,M,2 R1=K+1 R2=K-1 R3=K P2=-1./R1/R1/R2-Y(K)/R1-R3*P1/R1/R2 P1=-2./R1/R2-2.*Y(K)-P1 Y(K)=P2 RETURN END

This subroutine calculates the integral Lk0 by the recurrent formulas J (ρ ) 1 k − − k P (ρ ) (k + 1)(k2 − 1) k + 1 k2 − 1 k−1 2 Pk+1 (ρ ) = − 2 − 2Jk (ρ ) − Pk−1 (ρ ) k −1 ρ Tk (ρ ) kTk−1 (ρ ) P2 (ρ ) = 1 − ρ 2 , Jk (ρ ) = − 2 k+1 k −1 2 1−ρ L10 (ρ ) = , Tk (ρ ) = cos(k arccos ρ ), 4

Lk0 (ρ ) = −

where k ≠ 1 is an odd integer. Parameters. R0 is the value of ρ (double precision); Y is an array of length M2 which contains, as odd elements, the calculated integral (double precision). On entry, this array is initialized by the table y𝜈 = T𝜈 (ρ ), 𝜈 = 1, 2, . . . ,M2; M2 is the dimension of array Y. Listing of the program LSK

1

SUBROUTINE LSK (A,X,M1,N,R0) DIMENSION A(M1,N),X(M1) DOUBLE PRECISION A,X,R0,P DOUBLE PRECISION R1,R2,R3 M2=M1+1 DO 1 K=2,M2 K1=M2-K+2 R1=K1 R2=K1+1 R3=K1-1 X(K1)=R0*X(K1)/R2-R1*X(K1-1)/R2/R3 CALL IKJ (A,X,M1,N,R0) DO 2 K=1,M1 I=(1-(-1)**K)/2+1 I1=K-(-1)**K

132 | 18 Eigenvalues and eigenfunctions of the Laplace operator

6 2

7 3

4

8 5

IF (I.GT.N) GO TO 2 DO 6 J=I,N,2 P=R0**J A(I1,J)=A(K,J)/P A(K,J)=0. CONTINUE CALL IKJ1(A,X,M1,N,R0) DO 3 K=1,M1 I=(1-(-1)**K)/2+1 I1=K-(-1)**K IF (I.GT.N) GO TO 3 DO 7 J=I,N,2 P=R0**J A( I1,J)=A( I1,J)+A(K,J)*P A(K,J)=0. CONTINUE DO 4 K=2,M2 R1=K*K-1 X(K)=-1./R1 CALL IKJ (A,X,M1,N,1.0D+0) DO 5 K=1,M1 I=(1-(-1)**K)/2+1 I1=K-(-1)**K IF (I.GT.N) GO TO 5 DO 8 J=I,N,2 P=R0**J A(K,J)=-P*A(K,J)+A(I1,J) CONTINUE RETURN END

This subroutine calculates the integrals Lsk with double precision for odd (s + k). Parameters. A is the output array of size M1×N which contains in appropriate elements the calculated integrals (double precision); X is an array of length M3 which contains on entry the values x𝜈 = T𝜈 (ρ ), 𝜈 = 1, 2, . . . ,M3; M1 and N are dimensions of array A; R0 is the value of ρ (double precision). Required subroutines. IKJ, IKJ1. Listing of the program IKJ SUBROUTINE IKJ (A,X,M1,N,BE) DIMENSION A(M1,N),X( 1) DOUBLE PRECISION A,X,BE,P,P1,P4 P=BE*BE A(2,1)=0.5*P*(P-1.) DO 3 K=4,M1,2

18.1 The Dirichlet problem

3

2

5 4

|

133

P1=K*K-4 A(K,1)=0.5*(X(K+1)+X(K-1))-FLOAT ((-1)**(K/2))/P1 IF (N.EQ.1) RETURN DO 2 J=2,N,2 K1=J+2 P1=K1 A(1,J)=BE**K1/P1 M=M1-1 N1= N-1 DO 4 K=1,M I=(1-(-1)**K)/2 +1 P=X(K+1) IF (I.GT.N1) GO TO 4 DO 5 J=I,N1,2 P4=K+J+3 P1=(K+1)*(J+1) A(K+1,J+1)=A(K,J)*P1/P4/FLOAT(K)+P*FLOAT(K+2)/P4*BE**(J+1) CONTINUE RETURN END

β This subroutine calculates the integral Ikj̃ = ∫0 Tk (r)rj dr, 0 < β < 1 with double precision by the formula

̃ Ik+1,j+1 =

k+2 (k + 1)(j + 1) ̃ I + Jk+1 (β )β j+1 , k(k + j + 3) kj k+j+3

k ≥ 2, j ≥ 1,

where (k + j) is an odd integer. The first column is calculated from ̃ = Ik,1

1 (−1)k/2 [Jk+1 (β ) + Jk−1 (β )] − 2 , 2 k −4

̃ = 0.5β 2 (β 2 − 1). The first row is calculated from where k is an even integer, k ≠ 2, I2,1 ̃ = I1,j

β j+2 . 2+j

Remark. Here, Jk (ρ ) has the same meaning as in the description of subroutine LS0. Parameters. A is the output array of size M1×N, which contains in appropriate elements the calculated integrals (double precision); X is an array of length M1+1 which nT (β ) β Tn (β ) contains on entry the table Jk (β ), k = 2, . . . , M1 + 1; Jn (β ) = n+1 − nn−1 (double 2 −1 precision); M1 and N are dimensions of array A; BE is the value of β (double precision).

134 | 18 Eigenvalues and eigenfunctions of the Laplace operator Listing of the subroutine IKJ1

2

5 9

7 10

8 6

SUBROUTINE IKJ1 (B,Y,M1,N, AL) DIMENSION B(M1,N),Y(M1) DOUBLE PRECISION B,Y,P,AL DOUBLE PRECISION R1 P=-DLOG(AL) M=M1-1 B(2,1)=1.-AL*AL-P DO 2 K=3,M,2 R1=K*K-1 B(K+1,1)=-2./R1-2.*Y(K)-B(K-1,1) IF (N.EQ.1) RETURN B(1,2)=P IF (N.EQ.2) GO TO 10 IF (N.LT.4) GO TO 9 DO 5 J=4,N,2 J1=2-J R1=J1 B(1,J)=1./R1-AL**J1/R1 B(2,3)=2.*B(1,2)+0.5*(1.-1./AL/AL) IF (N.LE.4) GO TO 10 DO 7 J=5,N,2 J1=J-3 J2=J-1 R1=J1 R2=J2 B(2,J)=-2.*(1.-1./AL**J1)/R1+(1.-1./AL**J2)/R2 CONTINUE DO 6 K=2,M I=((1-(-1)**K))/2+2 IF (I.GT.N) GO TO 6 DO 8 J=I,N,2 B(K+1,J)=2.*B(K,J-1)-B(K-1,J) CONTINUE RETURN END 1

This subroutine calculates the integral Ikj = ∫α Tk (r)r−j dr, α > 0, (k + j) is odd, 1

with double precision by the recurrent formula. Denote Pk (α ) = ∫α Tk (r)r−1 dr; then Pk+1 (α ) = − k22−1 − 2Jk (α ) − Pk−1 (α ), k ≥ 2, P2 = 1 − α 2 + ln α is the formula for the first 3−j

1−j

−2

column; I2j = 2 1−α − 1−α , j ≠ 1, 3; I21 = P2 , I31 = −2 ln α + 1−α2 are relations for 3−j 1−j the second row. The remaining integrals are calculated from Ik+1,j = 2Ik,j−1 − Ik−1,j with double precision. Parameters. B is the output array of size M1×N, which contains in appropriate elements the calculated integrals (double precision); Y is a double precision array of

18.2 Mixed problem

|

135

dimension M1+1 which contains on entry the same values as array X in subroutine IKJ; M1 and N are dimensions of array B; AL is the value of α (double precision). Thus, by using the subroutines LAPLAS, LS0, LSK, IKJ, and IKJ1 we reduce the eigenvalue and eigenfunction problem to an algebraic eigenvalue problem solution to which is obtained by standard methods [26].

18.2 Mixed problem As was shown above, this problem is reduced to the integral equation (18.11), where the function ψ (θ ) must be determined from the boundary condition (18.9). Denote 𝜕K(ζ , ξ ) 󵄨󵄨󵄨󵄨 󵄨 ≡ K0 (ζ , θ ), 𝜕r 󵄨󵄨󵄨r=1 K0 (ζ , θ ) =

ζ = ρ eiφ ,

ξ = reiθ

1 − ρ2 . 2π (1 + ρ 2 − 2ρ cos(θ − φ ))

dz 2 Apply to function | dξ | (q + λ p)u the same interpolation as was used for the Dirichlet problem, while to ψ (θ ) apply the trigonometric interpolation:

ψ (θ ) =

2 2n ∑ D (θ − θj )ψj + ρn (θ ; ψ ) N j=0 n

ψj = ψ (θj ),

θj =

Dn (θ − θj ) =

2π j , N

j = 0, 1, . . . , 2n

n 1 + ∑ cos k(θ − θj ). 2 k=1

Denote 2π

Hj0 (ζ ) =

2 ∫ K0 (ζ , θ )Dn (θ − θj ) dθ 2n + 1 0

(18.14)

n 2 1 { + ∑ ρ L cos L(φ − θj )} , = 2n + 1 2 L=1

ζ = ρ eiφ ;

then we have u(ζ ) = H𝜈l (ζ )z𝜈l (q𝜈l + λ p𝜈l )u𝜈l + Hj0 (ζ )ψj ,

󵄨 󵄨2 z𝜈l = 󵄨󵄨󵄨󵄨φ 󸀠 (ζ𝜈l )󵄨󵄨󵄨󵄨 .

(18.15)

Here, summation over j is performed from 0 to 2n, while summation over 𝜈 and l is performed as stated above. The integrals Hj0 (ζ ) are calculated by subroutine HJ0. Parameters and listing of this subroutine are given in [26]. Now choose the vector ψ = (ψ0 , ψ1 , . . . , ψ2n )󸀠 in such a way that the boundary conditions (18.8) are satisfied. To achieve this, we differentiate relation (18.15) with

136 | 18 Eigenvalues and eigenfunctions of the Laplace operator respect to ρ , then set ρ = 1 and substitute the result into the boundary condition to obtain the following system of linear equations: 󸀠 Bij ψj + H𝜈l (θi )f𝜈l = 0,

(18.16)

where Bij =

n 2 ∑ l cos l(θi − θj ), 2n + 1 l=1

n(n + 1) 2n + 1 󵄨 𝜕H𝜈l (ρ eiθj ) 󵄨󵄨󵄨󵄨 󸀠 H𝜈l (θj ) = 󵄨󵄨󵄨 , 𝜕ρ 󵄨󵄨ρ =1

f𝜈l = z𝜈l (q𝜈l + λ p𝜈l )u𝜈l ;

i ≠ j;

Bii = αi +

As a result, we obtain 󸀠 (θi ) = H𝜈l

n𝜈 { L󸀠 (1) 1 2(−1)𝜈−1 sin ψ𝜈 ∑ mk cos k(θi − θ𝜈l ) { (2n𝜈 + 1)m k k=1(2) { n𝜈 L󸀠 (1) + 2r𝜈 ∑ 0k cos k(θi − θ𝜈l ) k k=1(2) n𝜈

m−1

+ 2r𝜈 ∑ cos sψ𝜈 ∑ s=1

k=1

1 − (−1)s+k 󸀠 Lsk (1) cos k(θi − θ𝜈l ) k

} + 4r𝜈 ∑ cos sψ𝜈 L󸀠s0 (1)} , s=1(2) } m−1

where L󸀠0k (1) = − L󸀠s0 (1) =

2k , 1+k −1

1 + (−1)

k = 1, 3, 5, . . . s−1 2

s

,

s = 1, 3, 5, . . . is odd

1

L󸀠sk (1)

= −2k ∫ Ts (r)rk dr. 0

󸀠 Derivatives H𝜈l (θi ) are calculated by subroutine HNLI; its listing and description of parameters are given in [26].

18.3 The Neumann problem In contrast to Section 18.2, we have to determine the function in relation (18.10) from the boundary condition (18.7). Similar to the above approach, we start with relation (18.15) to obtain for the vector ψ = (ψ0 , ψ1 , . . . , ψ2n )T a system of linear equa-

18.3 The Neumann problem

| 137

tions (18.15) for which, in the current case, matrix B takes the form Bij =

n 2 ∑ l cos l(θi − θj ), 2n + 1 l=1

θj =

2π (j − 1) , 2n + 1

j = 1, 2, . . . , 2n + 1 ≡ N.

(18.17)

It is easy to see that matrix B is degenerate. This occurs due to the fact that the boundary-value problem Δu + f = 0, 𝜕u/𝜕r|r=1 = 0 is not always resolvable. The condition for its resolvability on L2 is ∫ f (ζ ) dζ = 0.

(18.18)

|ζ |≤1

󵄨2 󵄨 In our case, f (ζ ) = 󵄨󵄨󵄨󵄨φ 󸀠 (ζ )󵄨󵄨󵄨󵄨 (q(ζ ) + λ p(ζ )). Substituting f (ζ ) by the interpolation formula (16.28a) and taking the integrals, we obtain the following approximate resolvability condition: m/2 2n𝜈

∑ ∑ c𝜈 f𝜈l = 0,

𝜈=1 l=0

where m−1 } { cos ψ𝜈 8π r𝜈 ∑ ts cos sψ𝜈 } + m(2n𝜈 + 1) { 2 s=3(2) (18.19) } { s−1 (2𝜈 − 1)π , s ≥ 1 is odd. ts = 1/(1 + (−1) 2 s), ψ𝜈 = 2m Consider now the solution of the degenerate equation set Bψ = d with matrix B 󸀠 defined by (18.17) and di = − ∑𝜈,l H𝜈l (θi )f𝜈l . This solution must depend on one arbitrary n 2 constant. Denote by aj = 2n+1 ∑l=1 l cos lφj the first row of matrix B. Matrix B has a symmetric circulant and, therefore, its eigenvalues are

c𝜈 =

λk = a1 + a2 θk + ⋅ ⋅ ⋅ + aN θkN−1 ,

θk = e

2π k i N

= eiφk+1 ,

k = 1, 2, . . . , N,

while the corresponding eigenvectors are xk = (1, θk , θk2 , . . . , θkN−1 ). Since θN = 1, we have λN = a1 + a2 + ⋅ ⋅ ⋅ + aN = 0, while the corresponding eigenvector is (1, 1, . . . , 1)󸀠 . All other eigenvalues are real-valued and multiple. However, in the calculations it is convenient to consider them in the complex-valued form: B = ΩΛΩ−1 ,

138 | 18 Eigenvalues and eigenfunctions of the Laplace operator where λ1 0 Λ=( ⋅⋅⋅ 0

⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅

0 λ2 ⋅⋅⋅ 0

0 0 ) ⋅⋅⋅ λN

1 θ1 Ω = ( θ12 ⋅⋅⋅ θ1N−1 θ1−1 θ2−1 ⋅⋅⋅ θN−1

1 1 1 = ( ⋅⋅⋅ N 1

Ω−1

⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅

1 θ2 θ22 ⋅⋅⋅ θ2N−1

⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅

1 θN θN2 ) ⋅⋅⋅ θNN−1

θ1N−1 θ2N−1 ). ⋅⋅⋅ θNN−1

Therefore, we obtain ΩΛΩ−1 ψ = d. Denote Ω−1 ψ = ξ and Ω−1 d = η ; then Λξ = η . Since λN = 0, we must have ηN = 0 (the system is consistent), and therefore, ξN is an arbitrary real-valued number: ξj =

1 η , j = 1, 2, . . . , N − 1. λj j

Further, we have ψ = Ωξ and, therefore, N−1

ψi = ΩiN ξN + ∑ Ωij ξj . j=1

However, ξj =

ηj λj

N

,

ηj = ∑ i=1

θj1−i N

di .

Ultimately, we arrive at p−l N 1 N N−1 θj ) dl = ξN + ∑ Epl dl , ψp = ξN + ∑ ( ∑ N l=1 j=1 λj l=1

(18.20)

where Epq =

2π j 2 n 1 2 n cos(p − q) N ∑ Re θjp−q = ∑ N j=1 λj N j=1 λj

λj =

n n(n + 1) + 2 ∑ ak+1 cos kφj+1 . 2n + 1 k=1

Denote μj = N 2 λj ; then Epq == μj =

2π j 2 n cos(p − q) N ∑ N j=1 μj

n n(n + 1) 2π k 2π k + ∑ l [cos (l − j) + cos (l + j)] . 2 N N k,l=1

(18.21) (18.22)

18.3 The Neumann problem

|

139

Substitute (18.21) into (18.16) to obtain 2n

N

p=0

q=1

󸀠 (θq )) f𝜈l } . u(ζ ) = H𝜈l (ζ )f𝜈l + ∑ {Hp0 (ζ )(ξN − ∑ Epq H𝜈l

(18.23)

Let the point ζ run through all interpolation nodes ζi . Then (I − (A − B)Q)u − ξN e0 = λ (A − B)Pu,

(18.24)

where A is the matrix of Dirichlet problem, 2n

N

p=0

q=1

Bij = ∑ Hp0 (ζi ) ∑ Epq Hj󸀠 (φq ). Here, one index j was used instead of 𝜈 and l; Q and P are diagonal matrices, the same as appeared in the Dirichlet problem. The matrix A − B is performed by subroutine BIJ. Its listing and description can be found in [26] together with the subroutines HNLI and HJ0 called from it. Left-multiply equation (18.24) by the matrix 1 −1 E =( ⋅⋅⋅ −1 ∗

0 1 ⋅⋅⋅ 0

⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅

0 0 ). ⋅⋅⋅ 1

Then the unknown parameter ξN will be eliminated from all equations except the first one because 1 0 ∗ E e0 = ( ) . ⋅⋅⋅ 0 Substitute the first row containing the unknown parameter by the relation ∑ cj zj qj uj = −λ ∑ cj zj pj uj j

j

to obtain an algebraic eigenvalue problem (A0 − λ B0 )u = 0, where zj is the value of |φ 󸀠 (ζ )|2 at the j-th interpolation node. The coefficients cj (constant on each circle) are calculated in subroutine CNU by relations (18.20); see [26] for the listing and description of this subroutine. Calculation of the final matrix of Neumann problem is performed by the subroutine LDUDN; its listing and description of parameters are given in [26].

140 | 18 Eigenvalues and eigenfunctions of the Laplace operator Table 18.1. Epitrochoid, np = 4, ε = 1/6. √λi i

135

9 × 15 = 135

1 2 3 4 5 6

2.389 3.74 3.74 4.68 5.22 5.37

2.3844462 3.73479 3.73479 4.6028 5.2112 5.409

18.4 Numerical experiments The first calculation was carried out for the epitrochoid with np = 4, ε = 1/6. On the disc, a grid with 135 nodes was introduced, the number of nodes for each circle was NL = 27, 25, 23, 21, 17, 9, 7, 3, 3 (starting from the first circle). In the calculations, many complex-valued pairs with the imaginary parts of the order of 10−4 were obtained. The second calculation was carried out on a uniform grid with 9 × 15 = 135 points; all the eigenvalues obtained were real-valued. The results of calculations are presented in Table 18.1. The digits which coincide with those obtained on a fine grid 30 × 41 = 1230 (see Table 3.2 in [26]) are presented. The results in the third column of Table 18.1 coincide with those from [26]. The accuracy on the nonuniform grid is much worse (the second column in Table 18.1). The reasons for this are analyzed in detail in [26]. In short, one can say that in the second calculation the discrete problem inherits the properties of infinite-dimensional problem. The second problem considered is the eigenvalue problem on the disc: 2

Δu + λ e−ar u = 0 u|r=1 = 0. The calculation were performed on a 15 × 31 grid using a modified version (LAP123P) of the subroutine LAP123 from [26], and the second eigenvalue was evaluated. The results are presented in Table 18.2, the second column contains the results obtained by S. V. Nesterov. In the third calculation, higher-order eigenvalues were evaluated on the domain considered in the previous example. The following boundary-value problem was considered: −Δu(x) = λ u(x),

x∈G

u|𝜕G = 0. The eigenvalues are described by the asymptotic formula λn =

4π n + O(n1/2 ln n), mes G

18.4 Numerical experiments

| 141

Table 18.2. Eigenvalue problem on the disk domain. a 0.2 0.3 0.4 0.5 0.6 0.8

λ1 Authors

S. V. Nesterov

6.0355037914 6.1616424673 6.2878601882 6.4139790729 6.5400232874 6.7917939883

6.036051 6.1617386 6.2875 6.413976 6.540042 6.791812

where mes G is the area of domain G. For epitrochoid, mes G = π (1 + ε 2 (np + 1)). The asymptotic formula was sought for in the form λn = a + bn + c√n ln n. By knowing λn for two values of n, one can obtain a and c. In the calculation on a 15×31 grid with n = 100 and n = 200, we obtained a = −257.8878054, b = 3.51219512, c = 7.3121964. Then, the results obtained from this formula were compared with those calculated directly for 100 < n < 200. The difference was between 0.5%–4.7%. However, a better agreement could not be reached. Remark. It is not recommended to perform calculations on grids finer than 15 × 31 due to a possible accumulation of roundoff errors in the formula (18.13) (calculation of integrals Lsk involves multiplication by a big parameter ρ −k ).

19 Eigenvalues and eigenfunctions of a biharmonic operator We consider two boundary value problems: (19.1)–(19.3) and (19.1), (19.2), (19.4): Δ2 u(z) = (Q + λ P)u(z),

z∈G

u|𝜕G = 0 𝜕u 󵄨󵄨󵄨󵄨 󵄨 =0 𝜕n 󵄨󵄨󵄨𝜕G

(19.1) (19.2) (19.3)

󵄨 𝜕2 u 𝜕2 u 1 𝜕u 󵄨󵄨󵄨 )󵄨󵄨󵄨 = 0 + 𝜈( 2 + 2 ρ 𝜕n 󵄨󵄨𝜕G 𝜕n 𝜕s

(19.4)

Here, G is a domain in the complex plane with a sufficiently smooth boundary 𝜕G, n is the unit vector of outer normal to 𝜕G, 𝜕/𝜕s denotes differentiation with respect to the arc length (the length is measured counterclockwise), 1/ρ is the curvature of 𝜕G, and 𝜈 is a constant (Poisson’s ratio), −1 < 𝜈 < 1. Note that for Q = 0 and P = 1 the problem reduces to the problem of free plate vibrations; boundary conditions (19.2) and (19.3) correspond to plate clamping along its edge, while boundary conditions (19.2) and (19.4) correspond to a simply supported plate. Let z = φ (ζ ), ζ ≤ 1 be a conformal transformation mapping the unit disk onto the domain G. In the plane ζ we obtain, instead of equations (19.1)–(19.4), the following relations: 󵄨 󵄨−2 󵄨 󵄨2 Δ (󵄨󵄨󵄨󵄨φ 󸀠 (ζ )󵄨󵄨󵄨󵄨 Δu) = 󵄨󵄨󵄨󵄨φ 󸀠 (ζ )󵄨󵄨󵄨󵄨 (q + λ p)u(ζ ),

ζ = reiφ , r < 1

u|r=1 = 0 𝜕u 󵄨󵄨󵄨󵄨 󵄨 =0 𝜕r 󵄨󵄨󵄨r=1

󵄨 φ 󸀠󸀠 (ζ ) 𝜕2 u 𝜕u 󵄨󵄨󵄨 󵄨󵄨 = 0, + {𝜈 + (𝜈 − 1) Re (ζ )} φ 󸀠 (ζ ) 𝜕r 󵄨󵄨󵄨r=1 𝜕r2 where q(ζ ) = Q (z(ζ )), p(ζ ) = P (z(ζ )). Indeed, for the curvature we have

1 = ρ 1 𝜕u 𝜕u = , 𝜕n |φ 󸀠 (ζ )| 𝜕r

φ 󸀠󸀠 (ζ ) ) φ 󸀠 (ζ ) 󵄨󵄨 󸀠 󵄨󵄨 󵄨󵄨ζφ (ζ )󵄨󵄨

1 + Re (ζ

φ 󸀠󸀠 (ζ ) 𝜕u 1 1 𝜕2 u 𝜕2 u (ζ ) = − Re φ 󸀠 (ζ ) 𝜕r 𝜕n2 |φ 󸀠 (ζ )|2 𝜕r2 r|φ 󸀠 (ζ )|2

(19.5) (19.6) (19.7) (19.8)

19 Eigenvalues and eigenfunctions of a biharmonic operator

|

143

because 𝜕 󸀠 |φ (ζ )| 𝜕 𝜕 𝜕r = ln |φ 󸀠 (ζ )| = Re ln φ 󸀠 (ζ ) |φ 󸀠 (ζ )| 𝜕r 𝜕r = Re [

dζ φ 󸀠󸀠 (ζ ) ζφ 󸀠󸀠 (ζ ) d 1 ). ln φ 󸀠 (ζ ) ] = Re [eiθ 󸀠 ] = Re ( 󸀠 dζ dr φ (ζ ) r φ (ζ )

Note that the boundary condition (19.6) is taken into account in the boundary condition (19.8), i.e. we set 𝜕2 u/𝜕s2 = 0. Example 2. Let φ (ζ ) = ζ (1 + εζ n ), 0 < ε < 1/(n + 1), then Re (

cos nθ + ε1 ζφ 󸀠󸀠 (ζ ) ) = ε1 n , 󸀠 φ (ζ ) 1 + 2ε1 cos nθ + ε12

ε1 = ε (n + 1),

ε1 < 1.

Let us consider instead of differential equation (19.6) an integral equation. Denote 󵄨󵄨 dz 󵄨󵄨2 󵄨󵄨 (q + λ p)u. Then, inverting in (19.5) the first Laplace operator, we obtain f (ζ ) = 󵄨󵄨󵄨 dξ 󵄨 󵄨󵄨 2π

Δu = |z󸀠 (ξ )|2 ∫ K(ξ , ζ )f (ζ )dζ + |z󸀠 (ξ )|2 ∫ K0 (ξ , θ )v(eiθ ) dθ . |ζ |≤1

0

Let 2π 󸀠

R(ξ ) = |z (ξ )|

2

∫ K(ξ , ζ )f (ζ ) dζ ,

󸀠

2

S(ξ ) = |z (ξ )| ∫ K0 (ξ , θ )v(eiθ ) dθ ;

|ζ |≤1

0

󵄨 then we obtain Δu = R(ξ ) + S(ξ ), where v(e ) = |z (ζ )| Δu(ζ )󵄨󵄨󵄨󵄨ζ =eiθ . Inverting once again the Laplace operator, we obtain 󸀠

iθ

−2

2π

u(ξ ) = ∫ K(ξ , y)[R(y) + S(y)]dy + ∫ K0 (ξ , θ )u(eiθ ) dθ . |y|≤1

(19.9)

0

In equation (19.9), we have yet to define the unknown function v(eiθ ), relying on the second boundary condition (19.7) or (19.8). We apply to v(eiθ ) the trigonometric 2 interpolation: v(eiθ ) = 2n+1 ∑2n j=0 Dn (θ − θj )vj , while for functions S(y) and R(y) we apply the interpolation formula (16.28a) (see Chapter 16) to obtain the following approximate equation: ∫ K(ξ , y)S(y)dy = −H𝜈l (ξ )S𝜈l , |y|≤1

where 2π

S𝜈l = z𝜈l ∫ K0 (ξ𝜈l , θ )v(eiθ )dθ 0 2π

=

2n 2z𝜈l 2n ∑ ( ∫ K0 (ξ𝜈l , θ )Dn (θ − θj )dθ ) vj = z𝜈l ∑ Hj0 (ξ𝜈l )vj . 2n + 1 j=0 j=0 0

144 | 19 Eigenvalues and eigenfunctions of a biharmonic operator Here, z𝜈l = |φ 󸀠 (ζ𝜈l )|2 , while the values Hj0 (ξ ) are calculated by subroutine HJ0 presented in [26]. Thus, 2n 0 (ξ𝜈l )vj1 ∫ K(ξ , y)S(y)dy = −H𝜈l (ξ )z𝜈l ∑ Hj1 j1=0

|y|≤1

∫ K(ξ , y)R(y)dy = −H𝜈l R𝜈l ,

R𝜈l = z𝜈l ∫ K(ξ𝜈l , ζ )f (ζ ) dζ .

|y|≤1

|ζ |≤1

Applying now to f (ζ ) the same interpolation formula, and substituting it then to equation (19.9), we obtain 2n 0 u(ξ ) = ∑ Hi (ξ )zi Aij zj (pj λ + qj )uj − ∑ Hi (ξ )zi ∑ Hj1 (ζi )vj1 . i,j

i

(19.10)

j1=0

Here, instead of two indices 𝜈 and l, a single one is introduced, i.e. the points are enumerated counterclockwise, starting from the first circular grid line; Aij is the Dirichlet problem matrix for the Laplace equation. In relation (19.10), we have yet to define the unknown vector v = (v0 , . . . , v2n ), using the boundary condition (19.7) or (19.8). Denote by M the differential operator on the left-hand side of the boundary condition; then, applying this operator to (19.10) we obtain 2n 0 ∑ M (Hi (ξ )) zi Aij zj (pj λ + qj )uj − ∑ M (Hi (ξ )) zi ∑ Hj1 (ζi )vj1 = 0. i,j

i

(19.11)

j1=0

Introduce the notation 󵄨 M (Hi (ξ ))󵄨󵄨󵄨ξ =eiθj2 = H̄ i,j2 ,

j2 = 0, . . . , 2n

∑ H̄ i,j2 zi Aij zj (pj λ + qj )uj = Rj2 i,j 0 ∑ H̄ i,j2 zi Hj1 (ζi )vj1 = B̄ j2,j1 . i

Thus, to determine the vector v = (v0 , . . . , v2n ) we have a system of linear equa2n ̄ ̄ −1 tions ∑2n j1=0 Bj2,j1 vj1 = Rj2 , from which we obtain vj1 = ∑j2=0 Cj1,j2 Rj2 , where C = B . Substituting vj1 into (19.11), we obtain 2n

2n

0 (ζi ) ∑ Cj1,j2 Rj2 . u(ξ ) = ∑ Hi (ξ )zi Aij zj (pj λ + qj )uj − ∑ Hi (ξ )zi ∑ Hj1 i,j

i

j1=0

(19.12)

j2=0

Let ξ run through all points; then ui = ∑(B2ij − ∑ Bil Elj )(pj λ + qj )uj , j

l

(19.13)

19.1 Boundary-value problem of the first kind

|

145

where B = AZ, and Z = diag(z1 , . . . , zNT ) is a diagonal matrix, zi = |φ 󸀠 (ζi )|2 , i = 1, 2, . . . , NT, A is the Dirichlet problem matrix for the Laplace equation, NT is the number of interpolation points on the disk, i.e. B is the Dirichlet problem matrix for our domain with P = 1, Q = 0. For the matrix E we have 2n

2n

j1=0

j2=0

0 Elj = ∑ Hj1 (ζl ) ∑ Cj1,j2 ∑ H̄ i,j2 zi Bij . i

2

Denote D = B − BE, then we obtain the following algebraic eigenvalue problem: u = D(Q + λ P)u,

(19.14)

where Q = diag(q1 , . . . , qNT ) and P = diag(p1 , . . . , pNT ) are diagonal matrices. Matrix C is calculated by subroutine CN, while the values Hj0 (ξ ) are calculated by subroutine HJ0M [26].

19.1 Boundary-value problem of the first kind We refer to the problem (19.5)–(19.7) as the boundary-value problem of the first kind. Note that the solution method described above is applicable for the boundary condition (19.6), while the second boundary condition can be arbitrary. The arising boundary-value problems differ only in the calculation of array H̄ i,j2 . For the boundary condition (19.3), this array is calculated by subroutine HNLI presented in [26]. Matrix D is calculated in subroutines EBIGM and BIGM [26].

19.2 Boundary-value problem of the second kind We refer to the problem (19.5), (19.6), and (19.8) as the boundary-value problem of the second kind. In this case, in order to calculate the array H̄ i,j2 , we have to evaluate the derivatives

𝜕2 H (ρ eiφ )|ρ =1 . 𝜕ρ 2 𝜈l

Calculations give

n𝜈 󵄨󵄨 { −L󸀠mk (1) − 2k 𝜕2 1 iφ 󵄨󵄨 𝜈−1 󵄨 ∑ H (ρ e ) = sin ψ 2(−1) cos k(φ − θ𝜈l ) 󵄨󵄨 𝜈 k 𝜕ρ 2 𝜈l 󵄨󵄨ρ =1 (2n𝜈 + 1)m { k=1(2) { n𝜈 kL󸀠0k (1) + 2r𝜈 ∑ cos k(φ − θ𝜈l ) k k=1(2) m−1

n𝜈

+ 2r𝜈 ∑ cos sψ𝜈 ∑ s=1

k=1

1 − (−1)s+k [−L󸀠sk (1) − 2k] cos k(φ − θ𝜈l ) k

} + 4r𝜈 ∑ cos sψ𝜈 (−L󸀠s0 (1) − 1)} , s=1(2) } m−1

146 | 19 Eigenvalues and eigenfunctions of a biharmonic operator where 2k , k = 1, 3, 5, . . . 1+k −1 , s = 1, 3, 5, . . . is odd L󸀠s0 (1) = s−1 1 + (−1) 2 s L󸀠0k (1) = −

1

L󸀠sk (1) = −2k ∫ Ts (r)rk dr. 0

Practically, the array H̄ i,j2 for the boundary condition (19.8) is calculated in subroutine HNLI2M [26]. Calculation of matrix D is performed by subroutine BIG12G. Listing of the program BIG12G

C C C C

C C C

PROGRAM BIG12G IMPLICIT REAL*8 (A-H,O-Z) PARAMETER (M=9,M1=18,M2=17,M3=19,NT=135,NT2=18225,NG=31, *NTG=4185,NMAX=27,NM=13,NMG=837) PARAMETER (M=9,M1=18,M2=17,M3=19,NT=135,NT2=18225,NG=15, *NTG=4185,NMAX=27,NM=13,NMG=837) PARAMETER (M=15,M1=30,M2=29,M3=31,NM=15,NMAX=31,NT=465, *NT2=216225,NG=31,NTG=14415,NMG=961) DIMENSION A(NT2),AB(NT2),X(NT),Y(NT),IANA(NT),HN(NTG), *NL(M),B(NG,NG),C(NTG),BA(NTG) DIMENSION BC(M1,NM),E(M2,M),H(NG), *U(NT),Z(NT),Y2(11) EQUIVALENCE (A(1),BA(1)),(A(7204),HN(1)) DATA NL /27,25,23,21,17,9,7,3,3/ DATA NL /9*15/ DATA NL /15*31/

C PUAS=0.25D0 WRITE(*,*) ’ NP = ? , EPS1 = ? ’ READ(*,*) NP,EPS1 NONL=8 REWIND (NONL) READ (NONL) (A(I),I=1,NT2) CALL TRANSP (A,NT) C NZAP = 9 WRITE(NZAP) (A(I),I=1,NT2) END FILE (NZAP) C WRITE(*,*) ’Enter the type of boundary problem ? ’ READ(*,*) IP C IF(IP.EQ.2) CALL PSI (Y,NG,PUAS,EPS1,NP,C) IF(IP.EQ.1) CALL HNLI (HN,M,M1,M2,NG,NL,NM,B,X,C,BC,E)

19.2 Boundary-value problem of the second kind

30

100 C

C

13 12

3

22

4

IF(IP.EQ.2) CALL HNLI2M (HN,M,M1,M2,NG,NL,NM,B,X,C,BC,E,NT,Y) CALL MOD2 (Y,M,NL,EPS1,NP) DO 30 I=1,NT I2=(I-1)*NG DO 30 J1=1,NG I3=I2+J1 HN(I3)=HN(I3)*Y(I) CALL CN (B,NG,HN,NL,X,M,H,NT) CALL EBIGM (AB,BA,NT,NG,NL,B,HN,H,C,X,M,NZAP,Y) CALL BIGM (A,AB,NT,NZAP,Y) CONTINUE CALL ELMHES (NT,NT,1,NT,A,IANA) WRITE(*,*) ’ELMHES’ CALL ELTRAN (NT,NT,1,NT,A,IANA,AB) WRITE(*,*) ’ELTRAN’ CALL HQR2 (NT,NT,1,NT,A,X,Y,AB,IERR) CALL HQR2M(NT,NT,1,NT,A,X,Y,AB,IERR,100) WRITE(*,*) ’HQR2’ NOUT = 4 OPEN(UNIT=4,FILE=’NOUT’) WRITE (NOUT,*) ’ IERR = ’, IERR FORMAT (13I5) FORMAT (4E18.11) WRITE(NOUT,12) (X(I),I=1,NT) WRITE(NOUT,12) (Y(I),I=1,NT) DO 3 I=1,NT Y(I)=1.D0/X(I) WRITE (NOUT,*) ’Eigenvalues’ WRITE(NOUT,12) (Y(I),I=1,NT) WRITE (*,*) ’Eigenvalues’ PRINT 12,(Y(I),I=1,NT) PAUSE NT1=NT/4 NT1=5 DO 21 K=1,NT1 I2=NT*(K-1) DO 22 I=1,NT I3=I2+I U(I)=AB(I3) CALL URT (0.D0,M,NL,U,Y) WRITE (*,*) ’URT’ CALL URT (3.141592653589D0,M,NL,U,Z) WRITE (*,*) ’URT’ DO 4 I=1,M I1=M1-I+1 Y(I1)=Z(I) DO 20 LL=1,11 X2=0.1*(LL-1)

|

147

148 | 19 Eigenvalues and eigenfunctions of a biharmonic operator

20

21

1

2

Y2(LL)=EIGEN(X2,Y,Z,M1,-1.D0,+1.D0) WRITE (*,*) ’EIGEN’ CALL NORM1(Y2,11) WRITE (*,*) ’NORM1’ PRINT 12,Y2 PAUSE STOP END SUBROUTINE NORM1(Y,N) IMPLICIT REAL*8 (A-H,O-Z) DIMENSION Y(1) P=0.D0 DO 1 I=1,N IF (ABS(Y(I)).GT.P) IP=I IF (ABS(Y(I)).GT.P) P=ABS(Y(I)) CONTINUE P=Y(IP) DO 2 I=1,N Y(I)=Y(I)/P RETURN END

The program BIG12G is similar to LAP123G. The program reads in the matrix A calculated by program LAP123G.

19.3 Numerical experiments In this section, an attempt is undertaken to repeat the calculations from [26] (see Tables 4.3 and 4.4 therein). This attempt was unsuccessful. Grids with 1230 = 30 × 41 and 820 = 20 × 41 nodes are not reachable on modern computers, because in the process of evaluating integrals Lsk accumulation of round-off errors occurs. When calculations were performed on BESM-6 mainframe computers, this could be overcome by using double precision (the length of mantissa on that machine was 48 bits); on modern computers calculations with similar accuracy are not possible. The first calculation was performed for epitrochoid (n = 4, ε = 1/6) where the boundary-value problem of the first kind was solved. On the disc, a grid with 9 circular lines was introduced, with the number of nodes on the circles equal to NL = 27, 25, 23, 21, 17, 9, 7, 3, 3; the total number of nodes was NT = 135, and the number of nodes on the outer boundary of the disc was NG = 31. The results obtained are presented in Table 19.1 (first column), in the second column the results obtained on the 15 × 31 grid are shown, while in the third column the results obtained on the 30 × 41 grid (see Table 4.3 in [26]) are presented.

19.3 Numerical experiments

| 149

Table 19.1. Epitrochoid, np = 4, ε = 1/6. i 1 2 3 4 5 6

λi 117. 391. 391. 845. 1304. 1305.

122.3 461.2 461.2 827.1 1329.1 1698.

122.6037 461.8864 461.9196 827.2753 1329.6937 1701.4344

From Table 19.1 it is evident that accuracy of the calculations is rather low, which is explained by the challenging nature of this problem. The second calculation was carried out for the same domain; boundary-value problem of the second kind was solved with Poisson’s ratio taken to be 0.25. In Table 19.2, the first column contains the results obtained on 15 × 31 grid, while in the second column the results obtained on 30 × 41 grid are given (see Table 4.4 in [26]). Table 19.2. Epitrochoid, np = 4, ε = 1/6 (boundary-value problem of second kind). i 1 2 3 4 5

λi 60. 227. 235. 389.2 717.

68.2813 242.6973 245.1974 389.3203 726.9001

One can see that the accuracy is even worse, which is explained by the more computationally challenging nature of the boundary-value problems of the second kind. Remark. During the calculation of eigenvalues of matrix D, subroutine HQR2 from EISPACK package returned an error due to reaching the maximum number of iterations equal to 31. This subroutine was modified, and the maximum number of iterations was increased to 100. The modified subroutine is named HQR2M. The final calculation to be considered in the boundary-value problem of the first kind on a disc: 2 𝜕u 󵄨󵄨󵄨󵄨 Δ2 u − λ e−ar u = 0, u|r=1 = 󵄨 =0 𝜕r 󵄨󵄨󵄨r=1 The first eigenvalue was calculated on 13 × 25 grid. The results obtained are presented in Table 4.9, in the second column. In the third column, results obtained by S. V. Nesterov are presented. Calculations were performed by the program BIG12P [26]. The data presented in the table can be approximated by a simple linear function λ1 = 104.26316 + a ⋅ 16.626537 (S. V. Nesterov, private communication).

150 | 19 Eigenvalues and eigenfunctions of a biharmonic operator Table 19.3. Boundary-value problem of the first kind on a disk. a 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0

λi 105.99635034 107.63336375 109.27398562 110.91806178 112.56544328 114.21598673 115.86955396 117.52601185 119.18523224 120.84709172 124.17825731 127.51861130 130.86732337 134.22362586 137.58680951

105.996279 107.6332949 109.2739274 110.91801 112.56540 114.222951 115.869517 117.5259747 119.1851870 120.8686813 124.178184 127.518526 130.866855 134.223645 137.58700

20 Eigenvalues and eigenfunctions of the Laplace operator on an arbitrary domain In this chapter, a program for calculation of eigenvalues and eigenfunctions of the Laplace operator on an arbitrary domain is presented. The method assumes that parametric equations describing the domain boundary are available.

20.1 Eigenvalues and eigenvectors of the Laplace operator We consider three boundary-value problems: Δu(z) + (Q + λ P)u = 0,

z∈G

u|𝜕G = 0 𝜕u 󵄨󵄨󵄨󵄨 󵄨 =0 𝜕n 󵄨󵄨󵄨𝜕G 𝜕u 󵄨󵄨󵄨󵄨 Au + 󵄨 = 0, 𝜕n 󵄨󵄨󵄨𝜕G

(20.1) (20.2) (20.3) (20.4)

where Q, P, and A are some functions defined on domain G, and n is the outward pointing normal to 𝜕G. We assume that Q, P, A, and 𝜕G ∈ C∞ . 󵄨 󵄨 Let z = φ (ζ ), 󵄨󵄨󵄨ζ 󵄨󵄨󵄨 ≤ 1 be the conformal mapping of the unity disc onto the domain Γ. Then on ζ -plane we formally obtain the same equations (20.1)–(20.4) in which, however, instead of u(z) and (Q + λ P)u one should write u(ζ ) = u (z(ζ )) and 󵄨󵄨 󸀠 󵄨󵄨2 󵄨󵄨φ (ζ )󵄨󵄨 (q + λ p)u, q(ζ ) = Q(z(ζ )), p(ζ ) = P(z(ζ )), while instead of A one should 󵄨 󵄨 󵄨 󵄨 write α (θ ) = A(z(eiθ )) 󵄨󵄨󵄨󵄨φ 󸀠 (eiθ )󵄨󵄨󵄨󵄨. The boundary condition is now posed at r = 1. Also, the derivatives along the normal in relations (20.3) and (20.4) are now substituted by respective derivatives along the radius. Eigenvalues and eigenfunctions of the boundary-value problems (20.1)–(20.4) are calculated by the subroutine LAP123C. Listing of the program LAP123C PROGRAM LAP123C IMPLICIT REAL*8 (A-H,O-Z) DIMENSION A(378225),AB(378225),X(615),Y(615),IANA(615), *HN(25215),NL(15),B(49,49),C(50625) DIMENSION B1(15),BC(30,24),E(29,15),H(49), *U(615),Z(656),Y2(11) DIMENSION E2(49,49) DIMENSION C0(225),C1(225) EQUIVALENCE (AB(1),HN(1)) COMMON//EPS1,NP

152 | 20 Eigenvalues and eigenfunctions of the Laplace operator on an arbitrary domain

COMMON /DM/ DM COMMON /CZ/ ZY(615) COMMON /CZG/ ZG(615) CHARACTER*1 IC EXTERNAL QMOD2,PMOD2 DM=1.D0 15 WRITE (*,*) ’Analytical conformal mapping or calculation (Y/N)’ READ (*,14) IC 14 FORMAT(A) IF (IC.NE.’Y’.AND.IC.NE.’N’) GO TO 15 IF (IC.EQ.’Y’) THEN WRITE(*,*) ’ NP = ? , EPS1 = ? ’ READ(*,*) NP,EPS1 ENDIF WRITE(*,*) ’M = ? ’ READ(*,*) M1 WRITE(*,*) ’N = ? ’ READ (*,*) N C C

C

UNIT FOR DATA INPUT NREAD = 3 OPEN(UNIT=3,FILE=’DATA’) UNIT FOR INTERMEDIATE DATA OUTPUT NOUT = 4 OPEN(UNIT=4,FILE=’NOUT’)

C 300

IM = 0 IM = IM + 1 IF (IM.GT.10) STOP READ(NREAD,*) M

C NT=M*N NM=(N-1)/2 M2=M*M C

10

READ (NREAD,*) (C0(I),I=1,M2) READ (NREAD,*) (C1(I),I=1,M2) IF (M1.NE.M) GO TO 300 DO 10 I = 1,M NL(I) = N WRITE (NOUT,*) ’ M = ’, M WRITE(NOUT,*) ’LAMDA0’ WRITE(NOUT,*) (C0(I),I=1,M2) WRITE(NOUT,*) ’LAMDA1’ WRITE(NOUT,*) (C1(I),I=1,M2)

C IF (IC.EQ.’N’) THEN OPEN (2,FILE=’FILEZ’)

20.1 Eigenvalues and eigenvectors of the Laplace operator

16 17

100

6

7

READ (2,*) (Z(I),I=1,NT+N) DO 16 I=1,N ZG(I)=SQRT(Z(I)) DO 17 I=1,NT ZY(I)=Z(N+I) ENDIF CALL HMATR1 (A,M,N,C0,C1,C) CALL RASPAK(A,M,NM) WRITE(*,*) ’Enter the boundary-value problem type ? ’ READ(*,*) IP IF(IP.EQ.1.OR.IP.EQ.2) GO TO 100 IF (IC.EQ.’Y’) CALL MOD2G (ZG,N) IF(IP.EQ.3) CALL LAP3 (A,N,M,NL,NM,HN,B1,X,C,B,BC,E,H) CALL TRANSP (A,NT) IF (IC.EQ.’Y’) THEN CALL MOD2(Y,M,N) DO 6 I=1,NT ZY(I)=Y(I) ENDIF IF (IC.EQ.’N’) THEN DO 7 I=1,NT Y(I)=ZY(I) ENDIF IF (IP.EQ.2) GO TO 200

C

5 200 C

55

400

I1=0 DO 5 I=1,NT DO 5 J=1,NT I1=I1+1 A(I1)=A(I1)*Y(I) IF (IP.EQ.1.OR.IP.EQ.3) GO TO 400 CONTINUE Neumann problem CALL BIJ(A ,NT,N,M,NL,NM,HN,B1,X,C,BC,E,H,E2) I1=0 DO 55 I=1,NT DO 55 J=1,NT I1=I1+1 AB(I1)=A(I1) CALL LDUDN (A,AB,NT,Y,C,NL,M,QMOD2,PMOD2) CALL DMINV (A,NT,DD1,X,Y) CALL DIVAB (NT,A,AB,Y) CONTINUE NT2=NT*NT CALL ELMHES (NT,NT,1,NT,A,IANA) WRITE(*,*) ’ELMHES’ CALL ELTRAN (NT,NT,1,NT,A,IANA,AB) WRITE(*,*) ’ELTRAN’ CALL HQR2 (NT,NT,1,NT,A,X,Y,AB,IERR)

|

153

154 | 20 Eigenvalues and eigenfunctions of the Laplace operator on an arbitrary domain

13 12 C C

WRITE(*,*) ’HQR2’ WRITE (NOUT,*) ’ IERR = ’, IERR FORMAT (13I5) FORMAT (4E18.11) WRITE(*,12) (X(I),I=1,NT) WRITE WRITE WRITE WRITE

(NOUT,*) ’X’ (NOUT,12) (X(I),I=1,NT) (NOUT,*) ’Y’ (NOUT,12) (Y(I),I=1,NT)

C RMAX=0.D0 IJ=1 I1=1 IF (IP.EQ.2) THEN Y(1)=X(1) IANA(1)=1 IJ=2 I1=2 ENDIF 110 DO 60 I=I1,NT IF (X(I).GT.RMAX) THEN RMAX=X(I) IANA(IJ)=I Y(IJ)=X(I) ENDIF 60 CONTINUE X(IANA(IJ))=0.D0 RMAX=0.D0 IJ=IJ+1 IF(IJ.LE.NT) GO TO 110 C WRITE(NOUT,12) (X(I),I=1,NT) WRITE(NOUT,12) (Y(I),I=1,NT) IF (IP.EQ.1.OR.IP.EQ.3) THEN DO 210 I=1,NT C 210 Y(I)=1.D0/SQRT(ABS(Y(I))) 210 Y(I)=1.D0/Y(I) ELSE DO 1 I=1,NT 1 Y(I)=SQRT(ABS(1.D0+1.D0/Y(I))) C1 Y(I)=1.D0+1.D0/Y(I) C1 Y(I)=SQRT(1.D0+1.D0/Y(I)) Y(1)=0.D0 ENDIF WRITE (NOUT,*) ’Eigenvalue’ WRITE(NOUT,12) (Y(I),I=1,NT) C M11=2*M

20.1 Eigenvalues and eigenvectors of the Laplace operator

22

4

C 20

21 120

C

5

IF(IP.EQ.2) KN=2 IF(IP.NE.2) KN=1 DO 21 K=KN,10 WRITE (*,*) ’Enter the eigenvalue number ?’ READ (*,*) IJ WRITE (*,*) IJ, Y(IJ) I2=NT*(IANA(IJ)-1) DO 22 I=1,NT I3=I2+I U(I)=AB(I3) CALL URT (0.D0,M,NL,U,X) CALL URT (3.14159265359D0,M,NL,U,Z) DO 4 I=1,M I1=M11-I+1 X(I1)=Z(I) DO 20 LL=1,11 IF (IP.EQ.1.OR.IP.EQ.3) THEN X2=0.1*(LL-1)/Y(IJ) ELSE X2=0.1*(LL-1) X2=0.1*(LL-1)/Y(IJ) ENDIF Y2(LL)=EIGEN (X2,X,Z,M11,-1.D0,+1.D0) CALL NORM1(Y2,11) WRITE (NOUT,12) Y2 PRINT 12,Y2 PAUSE FORMAT(A) END FUNCTION ALFA (X) IMPLICIT REAL*8 (A-H,O-Z) COMMON//EPS1,NP COMMON /DM/ DM ALFA=DM RETURN END SUBROUTINE MOD2G (Z,N) IMPLICIT REAL*8 (A-H,O-Z) DIMENSION Z(1) COMMON // EPS,L PI=3.141592653589D0 I0=0 DO 5 K=1,N T=2.*PI*(K-1)/N I0=I0+1 Z(I0)=SQRT(1.+2.*EPS*(L+1)*COS(L*T)+EPS**2*(L+1)**2) RETURN

|

155

156 | 20 Eigenvalues and eigenfunctions of the Laplace operator on an arbitrary domain

END SUBROUTINE MOD2 (Z,M,N) IMPLICIT REAL*8 (A-H,O-Z) DIMENSION Z(1) COMMON // EPS,L PI=3.141592653589D0 I0=0 DO 5 NU=1,M R=COS((2.*NU-1.)*PI/4./M) DO 5 K=1,N T=2.*PI*(K-1)/N I0=I0+1 Z(I0)=1.+2.*EPS*(L+1)*R**L*COS(L*T)+EPS**2*(L+1)**2*(R*R)**L RETURN END

5

1

2

1

SUBROUTINE NORM1(Y,N) IMPLICIT REAL*8 (A-H,O-Z) DIMENSION Y(1) P=0.D0 DO 1 I=1,N IF (ABS(Y(I)).GT.P) IP=I IF (ABS(Y(I)).GT.P) P=ABS(Y(I)) CONTINUE P=Y(IP) DO 2 I=1,N Y(I)=Y(I)/P RETURN END SUBROUTINE PMOD2 (Z,M,NL) IMPLICIT REAL*8 (A-H,O-Z) DIMENSION Z(1),Y(615),NL(1) COMMON /CZ/ ZY(615) N=NL(1) I=0 DO 1 NU=1,M DO 1 L=1,N I=I+1 Z(I)=ZY(I) RETURN END SUBROUTINE QMOD2 (Z,M,NL) IMPLICIT REAL*8 (A-H,O-Z) DIMENSION Z(1),Y(615),NL(1) COMMON /CZ/ ZY(615) N=NL(1) I=0

20.1 Eigenvalues and eigenvectors of the Laplace operator

1

2 1 3

|

157

DO 1 NU=1,M DO 1 L=1,N I=I+1 Z(I)=ZY(I) RETURN END SUBROUTINE BN(B,N,C) IMPLICIT REAL*8 (A-H,O-Z) DIMENSION B(N,N) ,C(1) DIMENSION LL(49),MM(49) COMMON /CZG/ ZG(615) PI=3.141592653590D0 N1=(N-1)/2 N2=N-1 DO 1 I=1,N2 J1=I+1 DO 1 J=J1,N B(I,J)=0.D0 DO 2 L=1,N1 B(I,J)=B(I,J)+2.D0*L*COS(L*2.D0*PI*(I-J)/N)/N B(J,I)=B(I,J) DO 3 I=1,N B(I,I)=ZG(I)*ALFA(2.D0*PI*(I-1)/N)+0.25D0*(N-1.D0/N) CALL DMINV (B,N,D,LL,MM) IF (ABS(D).LT.1.E-3) WRITE(*,*) ’IN BN D = ’,D RETURN END

The subroutine MOD2(Z,M,N) calculates |φ 󸀠 (ζ )|2 for the epitrochoid. Parameters. Z is an array containing on output the values of |φ 󸀠 (ζ )|2 at the interpolation points inside the disk; the length of this array is equal to the number of interpolation points; M is the number of circular grid lines on the disk; N is the (odd) number of points on i-th circle; EPS and N are ε and N, respectively. Remark. Parameters of the domain are denoted by EPS1 and NP and are passed to the subroutine MOD2 via an unnamed COMMON block. Subroutine MOD2G(Z,N) calculates |φ 󸀠 (ζ )|2 at the interpolation nodes on the disk boundary. Its parameters are similar to those described above.

158 | 20 Eigenvalues and eigenfunctions of the Laplace operator on an arbitrary domain 20.1.1 The Dirichlet problem This case is reduced to an algebraic eigenvalue problem u = HZf + R.

(20.5)

Here, u is a column vector containing the eigenvalues at the grid nodes; H is an M × M matrix obtained from equation (18.8) when ζ runs through all grid points; Z is a diagonal matrix with elements z𝜈l , 𝜈 = 1, 2, . . . , m; l = 0, 1, . . . , 2n on the diagonal (see Chapter 18); f is either a given column vector containing values of the corresponding function at the grid nodes, or f = (Q + λ P)u, where Q and P are diagonal matrices with diagonal elements equal to the values of the corresponding function at the grid nodes (in the latter case, an eigenvalue problem is considered); R is the residual vector containing the values of function RM (ζ ; F) at the grid nodes. Discarding the discretization error R in equation (20.5), we arrive at a finite-dimensional approximate problem. Calculation of the matrix H is performed by the program HMATR1 [26]. Initial data for this program must be contained in a file DATA. For m = 3 and 5, the initial data are given in [26]. In practical calculations, we used the values m = 3, 5, 7, 9, 11, 13, and 15. the corresponding data can be obtained by writing to the Institute for Problems in Mechanics RAS, or by e-mail ([email protected]). The calculated h-matrix H is output in the compressed form, i.e. only noncoinciding elements are written [26]. Unpacking of the compressed data and the rowwise output of array H is performed by the program RASPAK [26].

20.1.2 Mixed problem This problem is reduced to an algebraic eigenvalue problem by the method presented in Chapter 18, the difference being in the calculation of the matrix BN. The subroutine BN calculates an inverse of matrix B (see Chapter 18): Listing of the subroutine BN SUBROUTINE BN(B,N,C) IMPLICIT REAL*8 (A-H,O-Z) DIMENSION B(N,N),C(1) DIMENSION LL(49),MM(49) COMMON /CZG/ ZG(615) PI=3.141592653590D0 N1=(N-1)/2 N2=N-1 DO 1 I=1,N2 J1=I+1 DO 1 J=J1,N B(I,J)=0.D0

20.1 Eigenvalues and eigenvectors of the Laplace operator

2 1 3

|

159

DO 2 L=1,N1 B(I,J)=B(I,J)+2.D0*L*COS(L*2.D0*PI*(I-J)/N)/N B(J,I)=B(I,J) DO 3 I=1,N B(I,I)=ZG(I)*ALFA(2.D0*PI*(I-1)/N)+0.25D0*(N-1.D0/N) CALL DMINV (B,N,D,LL,MM) IF (ABS(D).LT.1.E-3) WRITE(*,*) ’IN BN D = ’,D RETURN END

Parameters. B is a N ×N array, on exit contains matrix B; N is the dimension of array B; C is a work array of length 1 (not used in this version of subroutine). 󵄨 󵄨2 Remark. Calculated values of 󵄨󵄨󵄨󵄨φ 󸀠 (ζ )󵄨󵄨󵄨󵄨 are passed via COMMON /CZG/ ZG(615). Required functions and subroutines. ALFA, DMINV. FUNCTION ALFA(T) calculates the values of function α on the boundary of the disk. DMINV is a double precision version of subroutine MINV [1]. The subroutine LAP3 calculates matrix H − E (see Chapter 18) of the mixed problem. After calling LAP3, the matrix is transposed by the subroutine TRANSP [26].

20.1.3 The Neumann problem Matrix (H − E)Z (see Chapter 18) is calculated by the subroutine BIJ. Subtract the first row of equation (18.24) from all the other rows, and substitute it by the discrete resolvability condition. Then we obtain the final form of the algebraic eigenvalue problem: A1 u = λ A2 u + δ2 , where the matrices A1 and A2 are obtained from matrices R(I − (H − E)ZQ) and R(H − E)ZP by changing the first row to rows c1 q1 z1 . . . cM qM zM and −c1 p1 z1 ⋅ ⋅ ⋅ − c1 p1 zM , respectively. Matrices A1 and A2 are calculated by the subroutine LDUDN [26].

20.1.4 Description of the program LAP123C 1. 2.

The program performs calculations for α ≡ DM (see the beginning of the program). Parameters of the domain NP and EPS1 are entered in dialog. If the conformal mapping is defined analytically, calculations are performed for epitrochoid, i.e. 󵄨 󵄨 for a domain obtained from a disk by conformal mapping z = ζ (1 + εζ n ), 󵄨󵄨󵄨ζ 󵄨󵄨󵄨 ≤ 1, ε < (n + 1). For other domains, the user has to provide specific subroutines MOD2 󵄨 󵄨2 󵄨 󵄨 and MOD2G, which calculate 󵄨󵄨󵄨󵄨z󸀠 󵄨󵄨󵄨󵄨 at the grid points inside the domain and 󵄨󵄨󵄨󵄨z󸀠 󵄨󵄨󵄨󵄨 and on its boundary, respectively.

160 | 20 Eigenvalues and eigenfunctions of the Laplace operator on an arbitrary domain Grid parameters M and N are entered in dialog. Here, M is the number of circular grid lines on the disk, while N is the number of points on each circle. 4. Program data must be contained in the file DATA. 5. The type of boundary-value problem is entered in dialog. 6. Results of calculation are output on the screen and written into the file NOUT. 7. The calculated eigenfunction is output at 11 points on the real axis (see the program). 8. The program either performs conformal mapping analytically, or reads in the precalculated data from the file FILEZ. The program which performs these calculations is presented below. 9. To solve the algebraic eigenvalue problem, the program calls subroutines ELMHES, ELTRAN, and HQR2 from the EISPACK package available at http://www.netlib.org/ eispack. 3.

In what follows, we present three files, NOUT1, NOUT2, and NOUT3 with the results obtained by solving the boundary-value problems of the first, second, and third kinds (Dirichlet, Neumann, and mixed) for the domain presented by epitrochoid EPS1 = 1/6, NP = 4 on a grid with M=3 and N=7 (the first five eigenvalues are given). These results are output by the program into the file NOUT. NOUT1 M = 3 LAMDA0 411.941825936000 -45.1418752798000 35.1897397936000 -61.1418752812000 29.3333333334000 -24.1914580529000 7.47692687292000 -8.19145805268000 14.7248407322000 LAMDA1 457.798232344000 -58.2837505577000 26.6666666663000 -80.1401570151000 37.3333333328000 -21.4734903555000 26.6666666637000 -15.6170838946000 32.8684342717000 IERR = 0 X 0.17652925285E+00 0.73617203152E-01 0.44307523376E-01 0.72789702502E-01 0.37933028422E-01 0.25209236536E-01 0.18579192222E-01 0.31389198480E-01 0.27305791331E-01 0.13342756833E-01 0.19932879110E-01 0.12722448607E-01 0.69245000609E-02 0.61940716769E-02 0.53605109241E-02 0.48574010821E-03 0.26756420900E-02 0.49854482653E-03 0.26029335063E-02 0.49981975563E-02 0.66914956087E-02 Y 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00

20.1 Eigenvalues and eigenvectors of the Laplace operator

| 161

0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.17652925285E+00 0.73617203152E-01 0.72789702502E-01 0.44307523376E-01 0.37933028422E-01 0.31389198480E-01 0.27305791331E-01 0.25209236536E-01 0.19932879110E-01 0.18579192222E-01 0.13342756833E-01 0.12722448607E-01 0.69245000609E-02 0.66914956087E-02 0.61940716769E-02 0.53605109241E-02 0.49981975563E-02 0.26756420900E-02 0.26029335063E-02 0.49854482653E-03 0.48574010821E-03 Eigenvalue 0.23800805673E+01 0.36856181936E+01 0.37065087338E+01 0.47507399945E+01 0.51344182305E+01 0.56442973753E+01 0.60516334741E+01 0.62982537422E+01 0.70829631703E+01 0.73364605404E+01 0.86571952877E+01 0.88657329883E+01 0.12017269154E+02 0.12224705383E+02 0.12706088814E+02 0.13658308167E+02 0.14144685353E+02 0.19332409706E+02 0.19600559253E+02 0.44786579280E+02 0.45373054559E+02 0.10000000000E+01 0.99802215301E+00 0.99125814730E+00 0.97981342549E+00 0.96381673166E+00 0.94341753548E+00 0.91878345637E+00 0.89009768761E+00 0.85755642049E+00 0.82136626845E+00 0.78174169119E+00 -0.39235917486E-01 0.68412781674E-01 0.17713877991E+00 0.28623724160E+00 0.39499409353E+00 0.50268916235E+00 0.60859931202E+00 0.71200158122E+00 0.81217632080E+00 0.90841033122E+00 0.10000000000E+01 -0.17226619411E+00 -0.48064622992E-01 0.76536074970E-01 0.20080391700E+00 0.32400436971E+00 0.44540298138E+00 0.56426801424E+00 0.67987307681E+00 0.79149975616E+00 0.89844025020E+00 0.10000000000E+01 0.10000000000E+01 0.80542115797E+00 0.62594796099E+00 0.46261829022E+00 0.31637743587E+00 0.18807240851E+00 0.78446250259E-01-0.11867653888E-01 -0.82351264858E-01-0.13260757823E+00-0.16236631297E+00 0.95908825529E+00 0.97546924726E+00 0.98787130129E+00 0.99610555065E+00 0.10000000000E+01 0.99940085118E+00 0.99417382894E+00 0.98420550670E+00 0.96940463235E+00 0.94970345395E+00 0.92505904551E+00

NOUT2 M = 3 LAMDA0 411.941825936000 -61.1418752812000 7.47692687292000 LAMDA1 457.798232344000 -80.1401570151000 26.6666666637000 IERR = 0

-45.1418752798000 29.3333333334000 -8.19145805268000

35.1897397936000 -24.1914580529000 14.7248407322000

-58.2837505577000 37.3333333328000 -15.6170838946000

26.6666666663000 -21.4734903555000 32.8684342717000

162 | 20 Eigenvalues and eigenfunctions of the Laplace operator on an arbitrary domain

X -0.10000000000E+01 0.62953408990E+00 0.61213257805E+00 0.18765111000E+00 0.98460527735E-01 0.88027329915E-01 0.49947964838E-01 0.65645629867E-01 0.36180878204E-01 0.26131114490E-01 0.22968741051E-01 0.38113729920E-01 0.34822108734E-02 0.68032921003E-02 0.14249506809E-01 0.12365957609E-01 0.19425215436E-01 0.37394698619E-02 0.15067823070E-01 0.64320635145E-02 0.12067584339E-01 Y 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 -0.10000000000E+01 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 -0.10000000000E+01 0.62953408990E+00 0.61213257805E+00 0.18765111000E+00 0.98460527735E-01 0.88027329915E-01 0.65645629867E-01 0.49947964838E-01 0.38113729920E-01 0.36180878204E-01 0.26131114490E-01 0.22968741051E-01 0.19425215436E-01 0.15067823070E-01 0.14249506809E-01 0.12365957609E-01 0.12067584339E-01 0.68032921003E-02 0.64320635145E-02 0.37394698619E-02 0.34822108734E-02 Eigenvalue 0.00000000000E+00 0.16088742427E+01 0.16228471995E+01 0.25157580477E+01 0.33401129108E+01 0.35156945667E+01 0.40290577680E+01 0.45848484979E+01 0.52189332396E+01 0.53515333311E+01 0.62664627426E+01 0.66736370841E+01 0.72442722574E+01 0.82077151279E+01 0.84366976624E+01 0.90480478727E+01 0.91578723596E+01 0.12165017963E+02 0.12508840937E+02 0.16383454614E+02 0.16975684901E+02 0.10000000000E+01 0.10000000000E+01 0.10000000000E+01 0.10000000000E+01 0.10000000000E+01 0.10000000000E+01 0.10000000000E+01 0.10000000000E+01 0.10000000000E+01 0.10000000000E+01 0.10000000000E+01 -0.21364488345E-01 0.91108227687E-01 0.20911512544E+00 0.33097101221E+00 0.45441286502E+00 0.57648541245E+00 0.69342671659E+00 0.80055375491E+00 0.89214800215E+00 0.96134101219E+00 0.10000000000E+01 -0.33765506507E-01 0.83223567787E-01 0.20428791557E+00 0.32807841757E+00 0.45268199452E+00 0.57546610469E+00 0.69292324134E+00 0.80051543015E+00 0.89251872672E+00 0.96186771406E+00 0.10000000000E+01 0.15493359455E+00 0.35714586529E-02-0.67247686799E-01-0.59479877622E-01 0.20188724090E-01 0.16019224271E+00 0.34393550182E+00 0.54964443693E+00 0.75021650823E+00 0.91307111334E+00 0.10000000000E+01 0.97313245166E+00 0.10000000000E+01 0.96436432935E+00 0.86250722528E+00 0.69600528881E+00 0.47270044705E+00 0.20767046372E+00-0.75800550228E-01 -0.34525162601E+00-0.55807442647E+00-0.66054273544E+00

20.1 Eigenvalues and eigenvectors of the Laplace operator

|

NOUT3 M = 3 LAMDA0 411.941825936000 -45.1418752798000 35.1897397936000 -61.1418752812000 29.3333333334000 -24.1914580529000 7.47692687292000 -8.19145805268000 14.7248407322000 LAMDA1 457.798232344000 -58.2837505577000 26.6666666663000 -80.1401570151000 37.3333333328000 -21.4734903555000 26.6666666637000 -15.6170838946000 32.8684342717000 IERR = 0 X 0.62323422018E+00 0.19532366583E+00 0.10514928125E+00 0.19395409756E+00 0.74952042045E-01 0.64157947750E-01 0.43100531422E-01 0.52419378772E-01 0.32556701459E-01 0.23798498313E-01 0.34325033708E-01 0.21029739648E-01 0.32646339842E-02 0.67157774395E-02 0.13507873256E-01 0.11767577470E-01 0.18044140786E-01 0.34898681298E-02 0.14279479382E-01 0.63594589437E-02 0.11471884947E-01 Y 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.00000000000E+00 0.62323422018E+00 0.19532366583E+00 0.19395409756E+00 0.10514928125E+00 0.74952042045E-01 0.64157947750E-01 0.52419378772E-01 0.43100531422E-01 0.34325033708E-01 0.32556701459E-01 0.23798498313E-01 0.21029739648E-01 0.18044140786E-01 0.14279479382E-01 0.13507873256E-01 0.11767577470E-01 0.11471884947E-01 0.67157774395E-02 0.63594589437E-02 0.34898681298E-02 0.32646339842E-02 Eigenvalue 0.12667017028E+01 0.22626770232E+01 0.22706516919E+01 0.30838755645E+01 0.36526517282E+01 0.39479784030E+01 0.43677127157E+01 0.48168008131E+01 0.53975231513E+01 0.55421694676E+01 0.64822417408E+01 0.68957745136E+01 0.74444376438E+01 0.83684266362E+01 0.86041210444E+01 0.92184194873E+01 0.93364679051E+01 0.12202585306E+02 0.12539779966E+02 0.16927604036E+02 0.17501801399E+02 0.99684794599E+00 0.10000000000E+01 0.99823344715E+00 0.99157303817E+00 0.97999348128E+00 0.96338410170E+00 0.94151350121E+00 0.91399421771E+00 0.88024738476E+00 0.83946739114E+00 0.79058654041E+00 -0.80998473338E-01 0.24661769048E-01 0.13282644682E+00 0.24285655239E+00

163

164 | 20 Eigenvalues and eigenfunctions of the Laplace operator on an arbitrary domain

0.35406102873E+00 0.46569402235E+00 0.57695213630E+00 0.68697168306E+00 0.79482593762E+00 0.89952239034E+00 0.10000000000E+01 -0.15949195899E+00 -0.42895395428E-01 0.75424745859E-01 0.19485271187E+00 0.31473067766E+00 0.43435355107E+00 0.55296377751E+00 0.66974614461E+00 0.78382258704E+00 0.89424699121E+00 0.10000000000E+01 0.10000000000E+01 0.68023971736E+00 0.40637213303E+00 0.17964833843E+00 0.94047458905E-03-0.12926890261E+00-0.21089745717E+00-0.24427370772E+00 -0.23014766227E+00-0.16970145296E+00-0.64559970779E-01 0.97430805300E+00 0.98792723621E+00 0.99655743709E+00 0.10000000000E+01 0.99808493329E+00 0.99067295367E+00 0.97765753053E+00 0.95896693031E+00 0.93456626084E+00 0.90445951571E+00 0.86869161858E+00

20.2 Program for conformal mapping The program implementing numerical conformal mapping was developed by E. P. Kazandzhan [345]. Here, we present an example of how this program is invoked. PROGRAM KAZAN PARAMETER(N=201,N1=202,M=210) C 2.06.94 IMPLICIT REAL*8 (A-H,O-Z) DIMENSION AL(N,N),PR(N),PSI(N),SL(N),FK(N) DIMENSION TJ(N1),TJ1(N1),TJ2(N1),MD(N1) COMPLEX*16 T(N),T1(N),T2(N),ZJ(N),SUM(N) COMPLEX*16 CW(M),CZ(M),CZ1(M),CZ2(M),CC DIMENSION Z(M),ZR(M),ZG(M) COMMON /D/ LL(301),MM(301) C This COMMON block must also be added in subroutine KONT COMMON /S/ S COMMON // EPS,NP WRITE(*,*) ’ NP = ? , EPS = ? ’ READ(*,*) NP,EPS S=2.D0*3.141592653589D0 CC=(0.,0.) WRITE (*,*) ’M = ?’ READ (*,*) MG WRITE (*,*) ’N =?’ READ (*,*) NG IJ=0 DO 1 I=1,NG IJ=IJ+1 TT=S*(I-1)/NG 1 CW(IJ)=(COS(TT))+(0.,1.)*(SIN(TT)) DO 2 NU=1,MG R=COS((2.*NU-1.)*S/8./MG) DO 2 L=1,NG TT=S*(L-1)/NG

20.2 Program for conformal mapping

IJ=IJ+1 CW(IJ)=(R*COS(TT))+(0.,1.)*(R*SIN(TT)) CALL CONFOR *(CC,AL,PR,PSI,T,T1,T2,SL,FK,ZJ,SUM,N,TJ,TJ1,TJ2, *MD,N1,CW,CZ,CZ1,CZ2,M) DO 3 I=1,M 3 Z(I)=CZ1(I)*CONJG(CZ1(I)) CALL MOD2G (ZG,NG) ENORM=0.D0 DO 4 I=1,NG IF (ABS(SQRT(Z(I))-ZG(I)).GT.0.D0) THEN ENORM=ABS(SQRT(Z(I))-ZG(I)) ENDIF 4 CONTINUE WRITE (*,*) ’ENORM G’,ENORM CALL MOD2 (ZR,MG,NG) ENORM=0.D0 DO 5 I=NG+1,M IF (ABS(Z(I)-ZR(I-NG)).GT.0.D0) THEN ENORM=ABS(Z(I)-ZR(I-NG)) ENDIF 5 CONTINUE WRITE (*,*) ’ENORM R’,ENORM PAUSE WRITE (*,*) ’Writing on disk’ OPEN (4,FILE=’FILEZ’) WRITE (4,*) (Z(I),I=1,M) STOP END 2

C

SUBROUTINE KONT(C) IMPLICIT REAL*8 (A-H,O-Z) Epitrochoid COMPLEX*16 CT,CT1,CT2 COMMON /TRI/ CT,CT1,CT2 COMMON // EPS,NP A=EPS CC=COS(C) SS=SIN(C) C4=COS((NP+1)*C) S4=SIN((NP+1)*C) CT=(CC+A*C4)+(0.,1.)*(SS+A*S4) RETURN ENTRY UR(C) CT1=(-SS-(NP+1)*A*S4)+(0.,1.)*(CC+(NP+1)*A*C4) CT2=(-CC-(NP+1)**2*A*C4)+(0.,1.)*(-SS-(NP+1)**2*A*S4) RETURN END

|

165

166 | 20 Eigenvalues and eigenfunctions of the Laplace operator on an arbitrary domain The user has to provide the subroutine KONT(C) which calculates τ (θ ), τ 󸀠 (θ ), and τ 󸀠󸀠 (θ ), θ ∈ [0, 2π ] for the given contour [26]. The results are assigned to variables CT, CT1, and CT2.

20.3 Numerical Experiments Calculations by the presented method were carried out for an elliptic domain on a 15 × 31 grid, the results were compared with those obtained by L. D. Akulenko and S. V. Nesterov [9, 11]. The length of larger semi-axis of the ellipse was taken equal to 1, while the eccentricity varied from e = 0.1 to 0.9 with step 0.1. Two boundary-value problems were considered: Dirichlet [9] and Neumann [11]. The results coincided for all e except 0.9. The number of coinciding decimal digits was four to one, depending on the eigenvalue number and eccentricity. Calculated results for e = 0.9 are presented in Table 20.1 for the Dirichlet problem, and in Table 20.2 for the Neumann problem. Table 20.1. Dirichlet problem for an ellipse.

i 1 2 3 4

λi Authors’ Results 15 × 31

13 × 25

[9]

17.7577 29.1706 44.2093 60.2676

17.7431 28.9868 43.6043 60.2555

18.1101 30.3330 — 61.6536

Table 20.2. Neumann problem for an ellipse. √λi i 1 2 3 4 5 6 7 8

Authors’ Results 15 × 31

13 × 25

[11]

1.8309 3.3798 4.0155 5.0292 5.1105 6.3250 6.8103 7.6841

1.8064 3.3920 4.0207 5.1307 5.1506 6.4210 7.0534 7.6912

1.8766 3.4355 4.0149 — 5.1115 — — 7.3689

20.3 Numerical Experiments

|

167

In Table 20.1, the results obtained by the authors are shown in the first and second columns, while the third column contains the results by L. D. Akulenko and S. V. Nesterov. Evidently, it follows from Table 20.1 that what was obtained in [9] are not the three lowest vibration frequencies, but the first, second, and fourth ones. In Table 20.2, the results obtained by the authors are shown in the first and second columns, while the third column contains the results by L. D. Akulenko and S. V. Nesterov. It follows from Table 20.2 that obtained in [11] are not the five lowest vibration frequencies, but the first, second, third, fifth, and eighth ones. Also, calculations were carried out for an epitrochoid; the results obtained coincided with those presented in Chapter 18. Remark. The program which calculates the conformal mapping of disk onto ellipse is named CONFE [345]. It has to be run first, which asks for the excentricity and grid parameters to be entered, and the results are written on the disk. After that, the program LAP123C has to be run, which finds out the eigenvalues and eigenfunctions.

21 Eigenvalues and eigenfunctions of a biharmonic operator on an arbitrary domain In this chapter, a program for calculation of eigenvalues and eigenfunctions of the biharmonic operator on an arbitrary domain is presented. The method assumes that parametric equations describing the domain boundary are available.

21.1 Eigenvalues and eigenfunctions of a biharmonic operator Algorithms are considered for the numerical solution of the boundary value problems (21.1)–(21.3) and (21.1), (21.2), (21.4): Δ2 u(z) = F(z),

z∈G

(21.1)

u|𝜕G = 0 𝜕u 󵄨󵄨󵄨󵄨 󵄨 =0 𝜕n 󵄨󵄨󵄨𝜕G 󵄨 𝜕2 u 𝜕2 u 1 𝜕u 󵄨󵄨󵄨 + 𝜈( 2 + )󵄨󵄨󵄨 = 0. 2 ρ 𝜕n 󵄨󵄨𝜕G 𝜕n 𝜕s

(21.2) (21.3) (21.4)

Here G is a domain in the complex z-plane with a sufficiently smooth boundary 𝜕G; n is the unit vector of the external normal to 𝜕G; 𝜕/𝜕s denotes differentiation with respect to the arc length (the length is measured counterclockwise); 1/ρ is the curvature of 𝜕G; and 𝜈 is a constant (Poisson’s ratio). The function F(z) is either given or has the form F(z) = (Q(z) + λ P(z))u(z), where Q and P are certain functions, and we have an eigenvalue problem for the biharmonic equation in this case. In particular, for Q = 0 and P = 1 we obtain the free-vibration problem for a plate, where the natural frequency ω is related to the spectral parameter λ by √λ = ω √ρ0 /D, where ρ0 is the density, while D is the cylindrical stiffness. Boundary conditions (21.2) and (21.3) mean that the plate is clamped along the edge, while boundary conditions (21.2) and 󵄨 󵄨 (21.4) correspond to the case of edge support. Let z = φ (ζ ), 󵄨󵄨󵄨ζ 󵄨󵄨󵄨 ≤ 1 be a conformal transformation mapping the unit disk onto the domain G. Then, in place of (21.1)– (21.4) we obtain the following relations in the ζ -plane: 󵄨 󵄨−2 󵄨 󵄨2 Δ (󵄨󵄨󵄨󵄨φ 󸀠 (ζ )󵄨󵄨󵄨󵄨 Δu) = 󵄨󵄨󵄨󵄨φ 󸀠 (ζ )󵄨󵄨󵄨󵄨 f (ζ ),

ζ = reiφ ,

u|r=1 = 0 𝜕u 󵄨󵄨󵄨󵄨 󵄨 =0 𝜕r 󵄨󵄨󵄨r=1 󵄨 φ 󸀠󸀠 (ζ ) 𝜕2 u 𝜕u 󵄨󵄨󵄨 󵄨󵄨 = 0. + {𝜈 + (𝜈 − 1) Re (ζ )} φ 󸀠 (ζ ) 𝜕r 󵄨󵄨󵄨r=1 𝜕r2

r 1.

(22.2)

With |κ | < r0 , we obtain n

(T(κ ) − ζ I)−1 = R(ζ ) + ∑ κ k Rk (ζ ) k=1

(22.3)

Rk (ζ ) = (−1)k (R(ζ )T (1) )k R(ζ ), i.e. ζ ∈ ρ (t(κ )) and (T(κ ) − ζ I)−1 can be represented uniformly with respect to ζ ∈ Γ by the convergent series (22.3). The coefficients of this series are also bounded operators, and, hence, (T(κ )−ζ I)−1 is also a bounded operator defined in the whole of B. Integrating (22.3) term by term, we obtain that the natural projector P(κ ) of the operator T(κ ) is defined by a series, convergent for |κ | 1, which proves the theorem. Remarks. 1. Instead of condition (22.1), we can introduce a cruder condition 󵄩 󵄩 sup 󵄩󵄩󵄩R(ζ )(Tn − ζ I) − I 󵄩󵄩󵄩 < 1.

(22.4)

ζ ∈Γ

If the perator T is bounded, relations (22.1) and (22.4) hold true, provided that 󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩R(ζ )󵄩󵄩󵄩 󵄩󵄩󵄩Tn − T 󵄩󵄩󵄩 < 1, 2. 3.

∀ζ ∈ Γ,

󵄩 󵄩 where 󵄩󵄩󵄩R(ζ )󵄩󵄩󵄩 is a continuous function on a compact subset of the complex plane. If ζ ∈ ρ (T) and Spr(R(ζ )(Tn − ζ I) − I) < 1, then ζ ∈ ρ (Tn ). Condition (22.1) implies that the resolvents of operators T and Tn are closely similar.

If we additionally assume in Theorem 6 that operator T is bounded, then T and Tn can be interchanged in the condition (22.1). For instance, if the interior of Γλ ̃ , where λ ̃ is a simple eigenvalue of operator Tn (we can find λ ̃ and the corresponding isolating

182 | 22 Error estimates for eigenvalue problems neighborhood by numerical calculations), contains no other eigenvalues of Tn , and we have the condition sup Spr(Rn (ζ )(T − ζ I) − I) < 1, ζ ∈Γλ ̃

Rn (ζ ) = (Tn − ζ I)−1 .

Then the interior of Γλ ̃ contains a unique eigenvalue of operator T. In other words, by using the results of the calculations, we can prove a strict theorem on the localization of the eigenvalues of a bounded operator T. A task of this kind is also encountered in Gauss’s problem [67]. Let us exemplify the application of Theorem 6 in the finite-dimensional case. Let A be an n × n matrix with complex-valued elements aij . Denote A1 = diag(a11 , . . . , ann ), A2 = A − A1 , A3 (ζ ) = (A1 − ζ I)−1 A2 , i.e. 0 a21

a12 a11 −ζ

A3 (ζ ) = ( a22 −ζ ⋅⋅⋅ an1 ann −ζ

a1n a11 −ζ a2n a22 −ζ

⋅⋅⋅

0

⋅⋅⋅

⋅⋅⋅ ⋅⋅⋅

⋅⋅⋅ ⋅⋅⋅

⋅⋅⋅ 0

),

(22.5)

where ζ ∈ Γ is a boundary point of the domain formed by the union of circles with centers aii and radii ri (Γ can consist of several closed nonintersecting contours). Let Pi = ∑i=j̸ |aij |, i = 1, . . . , n; then it was shown by Gerschgorin that all the eigenvalues of matrix A lie within the domain formed by the union of circles with centers aii and radii Pi . This widely known result can easily be obtained as a corollary to Theorem 6. Indeed, let |A|∞ = max ∑j |aij | be the Chebyshev norm of the matrix; then i

|A3 (ζ )|∞ = max ∑ i

i=j̸

|aij | |aii − ζ |

≤ max i

Pi . ri

(22.6)

P

If maxi r i < 1, then all eigenvalues of matrix A lie within Γ. From this, Geri schgorin’s result follows. Let ri = Pi + ε , ε > 0; then the right-hand side of (22.6) is less than unity, but ε > 0 is arbitrary and, hence, ri = Pi , all the eigenvalues of matrix A lie within, or on the boundary of the corresponding domain. In fact, this is Gerschgorin’s theorem. Notice that condition (22.4) has been utilized in these considerations; if the finer condition (22.1) is used, we arrive at the following theorem. Theorem 7. Let A be an n×n matrix with complex elements aij , and let Γ be a rectifiable contour (or a finite set of such contours that are pairwise nonintersecting), which contains inside it the diagonal elements aii , i = 1, 2, . . . , n of matrix A; then, if the condition sup Spr A3 (ζ ) < 1 ζ ∈Γ

is satisfied, then all eigenvalues of matrix A lie within Γ.

22.2 A priori error estimate in eigenvalue problems

| 183

The result of Theorem 7 is thus a generalization of Gerschgorin’s result. By using Theorem 6, other results of a similar type can easily be obtained. Notice that the eigenvalue localization theorems are of great importance for practical evaluation of eigenvalues.

22.2 A priori error estimate in eigenvalue problems Theorem 8. Assume that all assumptions of Theorem 6 hold true, except that Γ is a convex contour Γλ which contains in its interior an eigenvalue λ of operator T of algebraic multiplicity m, and contains no other spectral points of this operator. Denote ρ = max |λ −ζ |, while λ ̂ = 1 (λ + ⋅ ⋅ ⋅ + λ ) is the arithmetic mean of the eigenvalues ζ ∈Γλ

m

1

m

of operator Tn , lying inside Γλ ; then we have |λ − λ ̂ | ≤ ρ

r0−1 , 1 − r0−1

where r0 is defined in (22.2). Proof. Function λ ̂ (κ ) = phic for |κ | < r0 , i.e.

1 m

(λ1 (κ ) + ⋅ ⋅ ⋅ + λm (κ )) (see proof of Theorem 6) is holomorλ ̂ (κ ) = λ + κ λ1̂ + κ 2 λ2̂ + ⋅ ⋅ ⋅ ,

(22.7)

while, since Γλ is a convex contour, λ ̂ (κ ) lies within Γλ , and for the coefficients of series (22.7) we have the Cauchy formulas: 󵄨󵄨 ̂ 󵄨󵄨 󵄨󵄨λk 󵄨󵄨 ≤ ρ r0−k , 󵄨 󵄨

k = 1, 2, . . . .

But r0 > 1, and hence the series (22.7) is majorized by a convergent geometric progression with the ratio q = r0−1 . Hence follows the theorem. Corollary. Let T be a bounded operator; then the operator T (1) = Tn −T is also bounded. Assume that 󵄩󵄩 󵄩󵄩 󵄩󵄩󵄩 (1) 󵄩󵄩󵄩 󵄩󵄩R(ζ )󵄩󵄩 󵄩󵄩󵄩T 󵄩󵄩󵄩 < 1, ζ ∈ Γλ . Then the inequality 󵄩󵄩 󵄩󵄩 |λ − λ ̂ | ≤ Cn 󵄩󵄩󵄩T (1) 󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩 holds true, where Cn = ρ 󵄩󵄩󵄩R(ζ0 )󵄩󵄩󵄩 (1 − 󵄩󵄩󵄩R(ζ0 )󵄩󵄩󵄩 󵄩󵄩󵄩󵄩T (1) 󵄩󵄩󵄩󵄩)−1 , while ζ0 ∈ Γλ is the point at 󵄩 󵄩 which 󵄩󵄩󵄩R(ζ )󵄩󵄩󵄩 reaches its maximum for ζ ∈ Γλ . It is probably fair to say that existing methods for calculating the eigenvalues of operator (differential, integral, etc.) equations are reduced, for the most part, to a finite-dimensional problem of the type Av = μ v obtained from the relation Au = λ u + r,

(22.8)

where A is a n × n matrix, while u and r are n-dimensional vectors. Notice that λ is an exact eigenvalue of the corresponding operator T. Further, u = (u1 . . . un )T , where ui

184 | 22 Error estimates for eigenvalue problems are the exact values at the interpolation nodes (grid points, coefficients of series expansion, etc.) of the eigenfunction of the initial operator, corresponding to eigenvalue λ ; r = r1 . . . rn )T is the discretization error. Here, r = r(u, λ ), i.e. the discretization error has some individual value for each eigenfunction. Let λ be a simple eigenvalue of operator T, and Pn the projector onto the finitedimensional subspace Ln ⊂ B. We call the operator Tn = Pn TPn the discretization of 󵄨 operator T, and denote by A the matrix of the finite-dimensional operator Pn TPn 󵄨󵄨󵄨Ln in the basis l1 , . . . , ln ∈ Ln . Let the condition (22.1) be satisfied, where Γ is a contour Γλ satisfying the conditions of Theorem 6. Thus, there is a single eigenvalue of operator Tn inside the contour Γλ . Hence, there is one eigenvalue of matrix A inside Γλ . The exact eigenvalue of the initial operator T satisfies a relation of the type (22.8). We introduce the matrix B = A − (u, u)−1 ru∗ , where u∗ = (ū 1 , . . . , ū n ) is the matrix adjoint to column matrix u, while (u, v) = (u1 v̄1 + ⋅ ⋅ ⋅ + un v̄n ) is the scalar product in Cn . It is easy to see that Bu = λ u, i. e., λ and u are an eigenvalue and an eigenvector of matrix B . Denote by ‖ . ‖2 the matrix norm subordinate to the vector norm in Cn ; then ‖A − B‖2 ≤ ‖r‖2 . We recall that there is exactly one eigenvalue of matrix A inside the contour Γλ . If we have the condition sup Spr(A − ζ I)(B − ζ I) − I) < 1,

(22.9)

ζ ∈Γλ

then there are no eigenvalues of matrix B apart from λ within Γλ . Notice that condition (22.9) holds provided that 󵄩󵄩 󵄩 󵄩󵄩R(ζ , A)󵄩󵄩󵄩2 ‖r‖2 < 1,

∀ζ ∈ Γλ .

(22.10)

Hence, if the discretization error is sufficiently small; then there are no “parasitic” eigenvalues of matrix B within Γλ , i.e. eigenvalues other than λ . It now remains to apply the corollary to Theorem 7 in order to obtain the error estimate 󵄨󵄨󵄨λ − λ ̃ 󵄨󵄨󵄨 ≤ C ‖r‖ 󵄨󵄨 󵄨󵄨 n 2 (22.11) 󵄩 󵄩 󵄩 󵄩 Cn = ρ 󵄩󵄩󵄩R(ζ0 , A)󵄩󵄩󵄩2 (1 − 󵄩󵄩󵄩R(ζ0 , A)󵄩󵄩󵄩2 ‖r‖2 )−1 , 󵄩 󵄩 where ζ0 ∈ Γλ is the point at which the maximum of 󵄩󵄩󵄩R(ζ , A)󵄩󵄩󵄩2 is reached for ζ ∈ Γλ . Now let λ be a semisimple eigenvalue of the closed operator T of multiplicity m, while M is the corresponding m-dimensional geometric eigensubspace, and dim Pn M = m for sufficiently large n. As a result of the discretization of the eigenvalue problem for operator T, we generally obtain m finite-dimensional problems of type (22.8): Aui = λ ui + ri , i = 1, 2, . . . , m, where (ui , uj ) = δij . If the contour Γλ satisfies the conditions of Theorem 7, and the condition (22.1) holds true, then there are m eigenvalues λ1 , . . . , λm of operator Tn (of matrix A) within Γλ , counting each eigenvalue as many times as its multiplicity. Consider the matrix B = A − r1 u∗1 − ⋅ ⋅ ⋅ − rm u∗m .

22.3 A posteriori error estimate for eigenvalue problems

| 185

It is easy to see that λ is the m-tuple eigenvalue of matrix B, while u1 , . . . um are the corresponding eigenvectors. If the condition (22.9) holds true, then there are no other eigenvalues of matrix B within Γλ . Denote R = A − B; then ‖R‖2 ≤ m maxi ‖r‖2 . Relation (22.9) is satisfied provided that 󵄩󵄩 󵄩 󵄩󵄩R(ζ , A)󵄩󵄩󵄩2 ‖R‖2 < 1, ∀ζ ∈ Γλ . Now, in the same way as for a simple eigenvalue, we obtain the estimate 󵄨󵄨 󵄨 󵄨󵄨λ − λ ̂ 󵄨󵄨󵄨 ≤ Cn ‖R‖2 󵄨 󵄨 󵄩 󵄩 󵄩 󵄩 Cn = ρ 󵄩󵄩󵄩R(ζ0 , A)󵄩󵄩󵄩2 (1 − 󵄩󵄩󵄩R(ζ0 , A)󵄩󵄩󵄩2 ‖R‖2 )−1 . 󵄩 󵄩 Here, λ ̂ = m1 (λ1 + ⋅ ⋅ ⋅ + λm ), ζ0 ∈ Γλ is the point at which the maximum of 󵄩󵄩󵄩R(ζ , A)󵄩󵄩󵄩2 is reached for ζ ∈ Γλ .

22.3 A posteriori error estimate for eigenvalue problems The theorems proved in the previous sections can also be used for obtaining an a posteriori error estimate for the eigenvalue problem for a bounded operator T. Indeed, let Tn be a sequence of bounded operators (discretization of operator T), for which the eigenvalues can be calculated directly. For example, if Tn = Pn TPn , then calculation of eigenvalues of operator Tn is equivalent to calculation of eigenvalues of an n×n matrix A, a problem for which reliable algorithms are available. Let λ ̃ be a simple eigenvalue of operator Tn , and let Γλ ̃ be a closed contour containing within the point λ ̃ and not containing any other spectral points of operator Tn . To evaluate the accuracy to which λ ̃ is an eigenvalue of operator T, we have to calculate the quantity r0−1 = sup Spr(Rn (ζ )(T − ζ I) − I) < 1, ζ ∈Γλ ̃

Rn (ζ ) = (Tn − ζ I)−1

If r0−1 < 1, then a single eigenvalue λ of operator T lies within Γλ ̃ , and we have an inequality r−1 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨λ − λ ̃ 󵄨󵄨󵄨 ≤ ρ 0 −1 , ρ = max 󵄨󵄨󵄨λ ̃ − ζ 󵄨󵄨󵄨 󵄨 󵄨 󵄨 ζ ∈λ ̃ 󵄨 1 − r0 It is important to note that the eigenvalue of maximum modulus of the bounded operator Rn (ζ )(T − ζ I) − I can be calculated roughly. We only have to make sure that Spr(Rn (ζ )(T − ζ I) − I) < 1 and to note the order of this quantity.

22.4 Generalization for operator pencil Let B be a Banach space, and let A and B be a pair of bounded operators. We denote by P(A, B) the resolvent set, i.e., the set of complex numbers ζ ∈ C, for which a bounded

186 | 22 Error estimates for eigenvalue problems operator (A − ζ B)−1 exists. The complement Σ(A, B) = C − P(A, B) is called the spectrum of the operator pair A, B. If for some number λ ∈ C there is a solution u ≠ 0 of equation Au = λ Bu, then λ is called the eigenvalue corresponding to the eigenvector u for the operator pair A, B. The eigenvalues of the operator pair A, B lie within the spectrum Σ(A, B). The notation r(ζ ) = A − ζ B)−1 is used hereafter. Let λ be an eigenvalue of the operator pair A, B, and let Γ ⊆ P(A, B) be a rectifiable contour enclosing only this eigenvalue. Denote 1 ∫ R(ζ ) dζ E=− 2π i Γ

Operator P = EB is a projector. If the space P(B) is finite-dimensional, then dim P = dim P(B) is referred to as algebraic multiplicity of the eigenvalue λ (see [651]). Theorem 9. Let A, B be a pair of bounded operators in the Banach space B, and An , Bn be another pair of bounded operators. Let Γ be a rectifiable closed contour (or finite set of such contours, which are pairwise non-intersecting), which encloses m eigenvalues of the operator pair A, B counted with their algebraic multiplicity, and let r0−1 = sup, ζ ∈Γ

Spr(R(ζ )(An − ζ Bn ) − I) < 1.

(22.12)

Then, there are exactly m eigenvalues of the operator pair An , Bn within Γ, each eigenvalue being counted as many times as its algebraic multiplicity. The proof is similar to that of Theorem 6, except that now, unlike the classical theory of resolvents, the projection operator on the algebraic characteristic subspace is the operator P = EB (see above). Theorem 10. Under the conditions of Theorem 9 but taking the contour Γ as a convex contour Γλ which encloses the eigenvalue λ of the operator pair A, B of algebraic multiplicity m and does not enclose any other points of the spectrum of that operator, let 󵄨 󵄨 ρ = maxζ ∈Γλ 󵄨󵄨󵄨λ − ζ 󵄨󵄨󵄨, and λ ̂ = m1 (λ1 + ⋅ ⋅ ⋅ + λm ) be the arithmetic mean of the eigenvalues of the operator pair An , Bn which lie within Γλ . Then r−1 󵄨󵄨 󵄨 󵄨󵄨λ − λ ̂ 󵄨󵄨󵄨 ≤ ρ 0 −1 , 󵄨 󵄨 1−r 0

where r0 is defined by (22.12). The proof is a word-for-word repetition of Theorem 9. The implication of Theorem 10 is that the resolvents of the operator pairs A, B and An , Bn are close enough. Error estimates for the Sturm–Liouville problem relying on the theorems proven above was given in Chapter 16. Error estimates in the eigenvalue problems for the Laplace operator are performed along exactly the same lines.

Conclusion In the book, new statements of panel flutter problems are considered. In the new formulation by A. A. Ilyushin and I. A. Kijko, panel flutter problems for plates and shallow shells are reduced to a non-self-adjoint eigenvalue problem. At the present time, the most popular solution method in mechanics of deformable solids is the finite element method. Its deficiencies are well known: piecewiselinear approximation of displacements results in stress discontinuity. Most problems in the mechanics of solids, however, are governed by equations of the elliptic type with smooth solutions. Therefore, of great importance is the development of algorithms which could take advantage of this smoothness. The idea of such algorithms originated with K. I. Babenko [65, 66] in the 1970s. Since then, their efficiency has been proven in many applications to elliptic eigenvalue problems solved by the first author of this book. For example, eigenvalue problem was considered for the zero-th-order Bessel equation; on a grid with 20 points, the first eigenvalue of this problem was obtained with 22 decimal digits. Unlike the classical finite-different methods and finite element method, possessing power-law convergence rate with respect to the number of grid points, the method developed is featured by exponential error decrease. To the above-mentioned class of eigenvalue problems belong the plate and shallow shell flutter problem in the new formulation by A. A. Ilyushin and I. A. Kijko [311]. A principal advantage of this problem statement is that a complex aeroelasticity problem is reduced to an elliptic eigenvalue problem for which effective nonsaturating algorithms can be developed. With this approach, it became possible to solve new problems which are beyond the reach of classical methods. The main result of the current work is that expensive and difficult real experiments can be substituted by computational experiments. Full versions of the programs presented in the book can be obtained by e-mail [email protected] or by writing to the Institute for Problems in Mechanics RAS, Ave. Vernadskogo 101 Bldg 1, Moscow 119526, Russia.

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