LOCATING REAL EIGENVALUES OF A SPECTRAL PROBLEM IN FLUID-SOLID TYPE STRUCTURES

Exploiting minmax characterizations for nonlinear and nonoverdamped eigenvalue problems, we prove the existence of a countable set of eigenvalues converging to ∞ and inclusion theorems for a rational spectral problem governing mechanical vibrations of a tube bundle immersed in an incompressible viscous fluid. The paper demonstrates that the variational characterization of eigenvalues is a powerful tool for studying nonoverdamped eigenproblems, and that the appropriate enumeration of the eigenvalues is of predominant importance, whereas the natural ordering of the eigenvalues may yield false conclusions.


Introduction
Characterizations of eigenvalues as minmax or maxmin values of the Rayleigh quotient are known to be very powerful tools when studying selfadjoint linear operators on a Hilbert space H.To name just a few applications, bounds for eigenvalues, comparison theorems, interlacing results, and monotonicity of eigenvalues can be proved easily.Generalizations to families of operators depending nonlinearly on an eigenparameter were derived for overdamped problems, that is, if a generalized Rayleigh quotient called Rayleigh functional is defined on the entire space H [6,7,8,9,10,15], and counting the eigenvalue in an appropriate way it was shown in [11,13] that variational characterizations of eigenvalues hold in the nonoverdamped case, too.In this paper, we apply this variational characterization to a model governing small vibrations of a tube bundle immersed in an incompressible viscous fluid.
The governing nonclassical eigenvalue problem involving the Stokes system of equations with nonlocal and rational boundary conditions was studied by Conca et al. [3] (see also the recent monograph [5]).Transforming this problem to one of determining the characteristic values of a compact (nonselfadjoint) operator they proved that there exists a countable set of eigenvalues which converge to infinity.Moreover, it was shown that the number of eigenvalues with nonvanishing imaginary part is finite, and that they are all lying in a semicircle about the origin in the left half-plane.In [2] an upper bound of the number of nonreal eigenvalues was provided, and upper and lower bounds of the real eigenvalues were derived.The proof of these bounds however is false.
Taking advantage of variational characterizations of real eigenvalues for selfadjoint nonlinear eigenproblems, we obtain the existence of countably many real eigenvalues in a more transparent and less technical way than in [3], and comparing the Rayleigh functional of the rational problem to the Rayleigh quotients of suitable linear eigenvalue problems, we prove upper and lower a priori bounds for the real eigenvalues outside the semicircle mentioned above.
A crucial point when applying minmax or maxmin characterizations of eigenvalues for nonoverdamped problems is to enumerate the eigenvalues correctly.The natural ordering which was used in [2] is inappropriate, and a finite element discretization of the rational eigenproblem (to which the techniques from [2,3] and the ones used here apply as well), demonstrates that the bounds developed in [2] actually do not hold.
The aim of this paper is twofold.Firstly, it demonstrates by a fluid-structure interaction problem the direct study of which requires a mathematical analysis which is far from being trivial (cf.[2,3,5]), that the variational characterization of eigenvalues is a powerful tool for studying nonoverdamped nonlinear eigenvalue problems.Secondly, enumerating the eigenvalues appropriately is of predominant importance, whereas the naive enumeration of the eigenvalues may yield false conclusions.
Our paper is organized as follows.Section 2 summarizes the minmax characterization of eigenvalues of nonoverdamped eigenproblems where the eigenparameter appears nonlinearly.Section 3 outlines the rational eigenvalue problem governing small vibrations of a tube bundle immersed in an incompressible viscous fluid and collects the results in [2,3] on the number and location of the eigenvalues.In Section 4, we derive the existence of a countable set of real eigenvalues, converging to infinity in a transparent way, from the variational characterizations in Section 2, and we obtain lower and upper bounds of these real eigenvalues.The paper closes with a numerical example revealing that the bounds given in [2] do not hold.

Variational characterization of eigenvalues of nonlinear eigenproblems
We consider the nonlinear eigenvalue problem where T(λ) for every λ in an open real interval J is a selfadjoint and bounded operator on a Hilbert space H.As in the linear case, λ ∈ J is called an eigenvalue of problem (2.1) if (2.1) has a nontrivial solution x = 0.Such an x is called an eigenelement or eigenvector corresponding to λ.We assume that f : Heinrich Voss 39 is continuously differentiable, and that for every fixed x ∈ H 0 , H 0 := H \ {0}, the real equation has at most one solution in J. Then (2.3) implicitly defines a functional p on some subset D of H 0 which we call the Rayleigh functional.We assume that Then it follows from the implicit function theorem that D is an open set and that p is continuously differentiable on D.
For the linear eigenvalue value problem T(λ) := λI − A where A : H → H is selfadjoint and continuous the assumptions above are fulfilled, and p is the Rayleigh quotient and D = H 0 .If A additionally is completely continuous, then A has a countable set of eigenvalues which can be characterized as minmax and maxmin values of the Rayleigh quotient by the principles of Poincaré and of Courant, Fischer, and Weyl (cf.[14]).
The nonlinear eigenproblems variational properties using the Rayleigh functional were proved for overdamped systems (i.e., if the Rayleigh functional is defined on H 0 ) by Duffin [6] and Rogers [9] for the finite-dimensional case and by Hadeler [7,8], Rogers [10], and Werner [15] for the infinite-dimensional case.For nonoverdamped systems Werner and the author [13] proved a minmax characterization of Poincaré type; a maxmin characterization generalizing the principle of Courant, Fischer, and Weyl is contained in [11].
In this section, we assemble the results in [11,13] for the nonlinear nonoverdamped eigenvalue problem (2.1).
We denote by H j the set of all j-dimensional subspaces of H and by V 1 := {v ∈ V : v = 1} the unit sphere of the subspace V of H.
We already stressed the fact that the eigenvalues of problem (2.1) have to be enumerated appropriately to derive variational characterizations for nonoverdamped problems.To this end we assume that for every fixed λ ∈ J there exists ν(λ) > 0 such that the linear operator T(λ) + ν(λ)I is completely continuous.Then the essential spectrum of T(λ) contains only the point −ν(λ), and every eigenvalue µ > −ν(λ) of T(λ) can be characterized as maxmin value of the Rayleigh quotient of T(λ).In particular, if λ is an eigenvalue of the nonlinear problem (2.1), then µ = 0 is an eigenvalue of the linear problem T(λ)y = µy, and therefore there exists n ∈ N such that (2.5) In this case, we call λ an nth eigenvalue of the nonlinear eigenvalue problem (2.1).With this enumeration the following minmax characterization of the eigenvalues of problem (2.1) holds which was proved in [13].
Theorem 2.1.Under the conditions given above the following assertions hold.
(i) For every n ∈ N there is at most one nth eigenvalue of problem (2.1) which can be characterized by (2.6) The minimum is attained by the invariant subspace W of T(λ n ) corresponding to the n largest eigenvalues of T(λ n ), and sup v∈W∩D p(v) is attained by all eigenvectors of (2.1) corresponding to λ n .The set of eigenvalues of (2.1) is at most countable.
(ii) If, conversely, for some n ∈ N, then λ n is the nth eigenvalue of (2.1) and the characterization (2.6) holds.
The characterization of the eigenvalues in Theorem 2.1 is a generalization of the minmax principle of Poincaré for linear eigenvalue problems.In a similar way as in [13], the maxmin characterization of Courant, Fischer, and Weyl can be generalized to the nonlinear case (cf.[11]). (2.8)

A rational eigenvalue problem in fluid-structure interaction
This section is devoted to the presentation of the mathematical model which describes the problem governing free vibrations of a tube bundle immersed in an incompressible viscous fluid whose velocity field and pressure satisfy the steady Stokes equations.The tubes are assumed to be rigid, assembled in parallel inside the fluid, and elastically mounted in such a way that they can vibrate transversally, but they cannot move in the direction perpendicular to their sections.The fluid is assumed to be contained in a cavity which is infinitely long, and each tube is supported by an independent system of springs (which simulates the specific elasticity of each tube).Due to these assumptions, threedimensional effects are neglected, and so the problem can be studied in any transversal section of the cavity.Considering small vibrations of the fluid (and the tubes) around the state of rest, and assuming that the fluid is viscous and incompressible, this is a nonclassical eigenvalue problem involving the Stokes system of equations with nonlinear conditions on the boundaries of the tubes, which model the fluid-solid interaction.On the boundary of the cavity we assume the standard nonslip conditions.
Mathematically, the problem can be described in the following way as shown in Figure 3.1 (cf.[3,5]): let Ω 0 ⊂ R 2 (the section of the cavity) be an open bounded set with locally Lipschitz continuous boundary Γ 0 .We assume that there exists a family Ω j = ∅, j = 1,...,K, (the sections of the tubes) of simply connected open sets such that Ωj ⊂ Ω 0 for every j, Ωj ∩ Ωi = ∅ for j = i, and each Ω j has a locally Lipschitz continuous boundary Γ j .With these notations we set Ω := Ω 0 \ K j=1 Ωj .Then the boundary Γ of Ω consists of K + 1 connected components which are Γ 0 and Γ j , j = 1,...,K.
If u(x)e −ωt is the velocity field of the fluid, p(x)e −ωt denotes its pressure, and ν its kinematic viscosity, then the eigenvalue problem governing the free vibrations of the fluidsolid structure which was derived by Conca et al. [3] obtains the following form: Here m i is the mass per unit length of the ith tube, and k i represents the stiffness constant of the spring system supporting the ith tube.e(u) is the linear strain tensor of the fluid defined by and σ(u, p) denotes its stress tensor satisfying the Stokes law To rewrite problem (3.1)-(3.4) in variational form let be the standard Sobolev space equipped with the usual scalar product.Then clearly is a closed subspace of H 1 (Ω) 2 .e i j (u)e i j (v)dx (3.9) defines a norm on H which is equivalent to the standard Sobolev norm.Hence, H equipped with this scalar product is a Hilbert space.Multiplying (3.1) by v ∈ H and integrating by parts, one gets (cf.[3]) the variational form of problem (3.1)- (3.4).
Find ω ∈ C and u ∈ H, u = 0 such that for every v ∈ H, Conca et al. [3] multiplied the rational eigenproblem (3.10) by ω obtaining a quadratic problem.They proved that the eigenvalues of this problem are the characteristic values of a compact operator acting on a Hilbert space.Hence, they obtained that the set of eigenvalues of problem (3.10) is countable, and its only cluster point is ∞.Moreover, they proved the following location result.Theorem 3.1.Let (ω,u) be a solution of the rational eigenvalue problem (3.10).Then the following assertions hold: where µ denotes the smallest eigenvalue of the linear eigenproblem.Find µ ∈ C and v ∈ H, v = 0 such that for every w ∈ H, From Theorem 3.1(ii), it follows at once that problem (3.10) has only a finite number of nonreal eigenvalues.In [2] Conca et al. proved that the maximum number of nonreal eigenvalues is 4K, and [1] contains a numerical example that demonstrates that this bound is attained, which is approved by our numerical example in Section 5 as well.

Comparison results
In this section, we prove inclusion results for the real eigenvalues ω j > √ k/m taking advantage of the minmax characterization for these eigenvalues and comparing the Rayleigh functional with Rayleigh quotients R 1 of the linear eigenvalue problem (3.12) and R 2 of the linear problem.
Find ω ∈ C and v ∈ H such that for every w ∈ H, Problem (3.10) fulfills the conditions of the minmax theory for the interval J := ( we have Hence, all eigenvalues ω j ∈ J of problem (3.10) can be characterized by where the Rayleigh functional p is defined by F(ω,u) = 0, and F is given in (4.2).By D we denote the domain of definition of p.
Lemma 4.1.Let R 1 be the Rayleigh quotient of the linear eigenproblem (3.12).Then it holds that Proof.For every u ∈ H, u = 0 it holds that Thus, u ∈ D, and p(u) ≥ R 2 (u).
We are now in the position to prove an inclusion theorem for real eigenvalues of problem (3.10).
Theorem 4.3.(i) Assume that the jth eigenvalue of problem (4.1) is contained in J. Then the nonlinear eigenproblem (3.10) has a jth eigenvalue ω j ∈ J, and µ j is a lower bound of ω j : Since the linear problem (4.1) has a countable set of eigenvalues the only accumulation point of which is ∞, the nonlinear problem (3.10) must have countably many real eigenvalues as well which also converge to ∞.
(ii) If (3.10) has a jth eigenvalue ω j ∈ J, then and Lemma 4.2 yields In particular V ∩ D = ∅ for every V ∈ H j .
We already pointed out in [12] that the natural enumeration whereby we call the smallest eigenvalue the first one, the second smallest the second one, and so forth is not appropriate for the quadratic eigenvalue problem (4.16), and therefore the proof of these bounds is not correct.The numerical example in the next section demonstrates that the bounds (4.17) and (4.18) actually do not hold.
For those eigenvalues ω j contained in J, the bounds (4.17) and (4.18) can be adjusted if we replace the natural ordering of the eigenvalues ωj by the enumeration introduced in Section 2. The upper bound ω j ≤ η j is already contained in Theorem 4.3(ii).
The lower bound is obtained from the maxmin characterization in Theorem 2.1.Let W = span{u 1 ,...,u n−2K−1 } denote the subspace of H spanned by the eigenelements of problem (3.12) corresponding to the n − 2K − 1 smallest eigenvalues, and let Then obviously p(u) = R 1 (u) for every u ∈ D ∩ Z, and we obtain from Rayleigh's principle and the maxmin characterization in Theorem 2.1 (4.20)

Numerical experiments
While the variational form (3.10) was convenient for the theoretical study of problem (3.1)-(3.4),its numerical treatment requires to deal with the incompressibility condition div u = 0 implicitly, and to use a mixed variational formulation, which reads as follows (cf.[1,4]).Find (u, p,ω) Ω q div udx = 0. ( Here H denotes the space as follows: which again is a closed subspace of H 1 (Ω) 2 .We discretized this problem by finite elements using piecewise quadratic ansatz functions on a regular triangulation of Ω for the velocity field, and piecewise linear functions on the same triangulation for the pressure yielding a rational matrix eigenvalue problem which can be reduced to a general matrix eigenvalue problem and solved using standard numerical software.The convergence properties of this approach are studied in [4].
For k ≥ 30.82 the discrete version of (5.1) has nonreal eigenvalues, and for k ≥ 106.03 there exist 16 nonreal eigenvalues demonstrating that the bound 4K on the number of nonreal eigenvalues is attained.
For k = 400 the smallest real eigenvalue is ω1 = 13.478,whereas the smallest eigenvalue of (3.12) is η 1 = 9.671 demonstrating that (4.17) does not hold.Finally, k = 1 contradicts the lower bound in (4.18), since ω9 = 9.605, whereas η 1 = 9.672.5.1 contains the smallest eigenvalues µ j and η j of the linear problems (4.1) and (3.12), respectively, which for m = 1 and identical k j are bounds for eigenvalues greater than √ k.In columns 4 and 5 we added the smallest real eigenvalues ω j of the rational eigenproblem for k = 1 and k = 400 satisfying ω j > √ k where these eigenvalues are enumerated in the way introduced in Section 2.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.
However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning 42 A spectral problem in fluid-solid type structures It is well known from Korn's inequality that the scalar product u,v := Ω e(u) : e(v)dx :=

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation