Structured Eigenvalue Problems Research Articles

Checking dissipativity of linear behavior systems given in kernel representation

The behavior approach and the problem of dissipativity have both been introduced and studied extensively by Willems and others. Current methods to check dissipativity will either check the solvability of a linear matrix inequality or rely on symbolic computations with image representations via computer algebra packages. We will discuss a new characterization for linear behavior systems in kernel representation that allows to check dissipativity via the solution of structured eigenvalue problems. The complexity and efficiency of our new method will be compared to the existing methods.

On eigenvalue problem of bar structures with stochastic spatial stiffness variations

This paper presents an analysis of stochastic eigenvalue problem of plane bar structures. Particular attention is paid to the effect of spatial variations of the flexural properties of the structure on the first four eigenvalues. The problem of spatial variations of the structure properties and their effect on the first four eigenvalues is analyzed in detail. The stochastic eigenvalue problem was solved independently by stochastic finite element method (stochastic FEM) and Monte Carlo techniques. It was revealed that the spatial variations of the structural parameters along the structure may substantially affect the eigenvalues with quite wide gap between the two extreme cases of zero- and full-correlation. This is particularly evident for the multi-segment structures for which technology may dictate natural bounds of zero- and full-correlation cases.

Transonic Aeroelastic Instability Searches Using Sampling and Aerodynamic Model Hierarchy

DOI: 10.2514/1.J050509 A hierarchy of flow models is exploited for transonic aeroelastic stability analysis using the kriging interpolation technique applied within the Schur complement eigenvalue framework. In the Schur framework, a modified structural eigenvalue problem describes the coupled aeroelastic system with a precomputed interaction term depending on the response frequency. The interaction term, representing the influence of the high-dimensional computational fluid dynamics system, is approximated by reconstruction based on samples that can be computed using a frequency or time domain solver. The computationally cheap approximation model is developed and discussed in this paper for two-degree-of-freedom aerofoil cases. The approximation model is used for both the parametric blind search of aeroelastic instability and for updating predictions based on aerodynamic models of different fidelities.

Modified Finite Element Transfer Matrix Method for Eigenvalue Problem of Flexible Structures

The speedy computation of eigenvalue problems is the key point in structure dynamics. In this paper, by combining transfer matrix method and finite element method, the modified finite element-transfer matrix method and its algorithm for eigenvalue problems are presented. By using this method, the speedy computation of eigenvalue problem of flexible structures can be realized, and the repeated eignvalue problem can be solved simply and conveniently. This method has the low order of system matrix, high computational efficiency, and stability. Formulations of this method, as well as some numerical examples, are given to validate the method.

An adaptive inverse iteration algorithm using interpolating multiwavelets for structural eigenvalue problems

Structural eigenvalues have been broadly applied in modal analysis, damage detection, vibration control, etc. In this paper, the interpolating multiwavelets are custom designed based on stable completion method to solve structural eigenvalue problems. The operator-orthogonality of interpolating multiwavelets gives rise to highly sparse multilevel stiffness and mass matrices of structural eigenvalue problems and permits the incremental computation of the eigenvalue solution in an efficient manner. An adaptive inverse iteration algorithm using the interpolating multiwavelets is presented to solve structural eigenvalue problems. Numerical examples validate the accuracy and efficiency of the proposed algorithm.

An efficient method for evaluating the natural frequencies of structures with uncertain-but-bounded parameters

Using interval theory and the second-order Taylor series, the eigenvalue problems of structures with multi-parameter can be transformed into those with single parameter. The epsilon-algorithm is used to accelerate the convergence of the Neumann series to obtain the bounds of eigenvalues of structures with single interval parameter, thus increasing the computing accuracy and reducing the computational effort. Finally, the effect of uncertain parameters on natural frequencies is evaluated. Two engineering examples show that the proposed method can give better results than those obtained by the first-order approximation, even if the uncertainties of parameters are fairly large.

Improved group-theoretical method for eigenvalue problems of special symmetric structures, using graph theory

Group-theoretical methods for decomposition of eigenvalue problems of skeletal structures with symmetry employ the symmetry group of the structures and block-diagonalize their matrices. In some special cases, such decompositions can further be continued. This particularly happens when submatrices resulted from the decomposition process, correspond to substructures with new symmetrical properties which are not among the properties of the original structure. Thus, a group-theoretical method is not able to recognize such additional symmetry from the original problem. In this paper, an algorithm is presented based upon a combination of group-theoretical ideas and graph-methods. This algorithm identifies the cases where the structure has the potential of being further decomposed, and also finds the symmetry group, and subsequently the transformation which can further decompose the system. It is also possible to find out when the block-diagonalization is complete and no further decomposition is possible. This is particularly useful for large eigenvalue problems such as calculation of the buckling load or natural frequencies of vibrating systems with special symmetries.

Structured Backward Errors and Pseudospectra of Structured Matrix Pencils

Structured backward perturbation analysis plays an important role in the accuracy assessment of computed eigenelements of structured eigenvalue problems. We undertake a detailed structured backward perturbation analysis of approximate eigenelements of linearly structured matrix pencils. The structures we consider include, for example, symmetric, skew-symmetric, Hermitian, skew-Hermitian, even, odd, palindromic, and Hamiltonian matrix pencils. We also analyze structured backward errors of approximate eigenvalues and structured pseudospectra of structured matrix pencils.

Smooth orthogonal decomposition for modal analysis of randomly excited systems

Modal parameter estimation in terms of natural frequencies and mode shapes is studied using smooth orthogonal decomposition for randomly excited vibration systems. This work shows that under certain conditions, the smooth orthogonal decomposition eigenvalue problem formulated from white-noise induced response data can be tied to the unforced structural eigenvalue problem, and thus can be used for modal parameter estimation. Using output response ensembles only, the generalized eigenvalue problem is formed to estimate eigenfrequencies and modal vectors for an eight-degree-of-freedom lightly damped vibratory system. The estimated frequencies are compared against system frequencies obtained from structural eigenvalue problem and estimated modal vectors are checked using the modal assurance criterion. For lightly damped simulations, satisfactory results are obtained for estimating both system frequencies and modal vectors.

Studies of refinement and continuity in isogeometric structural analysis

We investigate the effects of smoothness of basis functions on solution accuracy within the isogeometric analysis framework. We consider two simple one-dimensional structural eigenvalue problems and two static shell boundary value problems modeled with trivariate NURBS solids. We also develop a local refinement strategy that we utilize in one of the shell analyses. We find that increased smoothness, that is, the “ k-method,” leads to a significant increase in accuracy for the problems of structural vibrations over the classical C 0 -continuous “ p-method,” whereas a judicious insertion of C 0 -continuous surfaces about singularities in a mesh otherwise generated by the k-method, usually outperforms a mesh in which all basis functions attain their maximum level of smoothness. We conclude that the potential for the k-method is high, but smoothness is an issue that is not well understood due to the historical dominance of C 0 -continuous finite elements and therefore further studies are warranted.

Exact bounds for the sensitivity analysis of structures with uncertain-but-bounded parameters

Based on interval mathematical theory, the interval analysis method for the sensitivity analysis of the structure is advanced in this paper. The interval analysis method deals with the upper and lower bounds on eigenvalues of structures with uncertain-but-bounded (or interval) parameters. The stiffness matrix and the mass matrix of the structure, whose elements have the initial errors, are unknown except for the fact that they belong to given bounded matrix sets. The set of possible matrices can be described by the interval matrix. In terms of structural parameters, the stiffness matrix and the mass matrix take the non-negative decomposition. By means of interval extension, the generalized interval eigenvalue problem of structures with uncertain-but-bounded parameters can be divided into two generalized eigenvalue problems of a pair of real symmetric matrix pair by the real analysis method. Unlike normal sensitivity analysis method, the interval analysis method obtains informations on the response of structures with structural parameters (or design variables) changing and without any partial differential operation. Low computational effort and wide application rang are the characteristic of the proposed method. Two illustrative numerical examples illustrate the efficiency of the interval analysis.

Parallel computation of arbitrarily shaped waveguide modes using BI‐RME and Lanczos methods

AbstractThis paper is devoted to the parallelization of a new method for solving large, structured eigenvalue problems, which appear in the electromagnetic modal analysis of arbitrarily shaped waveguides, typically present in many modern passive devices. This new method, based on the boundary integral‐resonant mode expansion (BI‐RME) technique and in the Lanczos method (for solution of the eigenvalue problem), was recently proposed by the authors, showing important advantages in terms of CPU time and memory over previously used solutions. As it will be fully described in this paper, the parallel version of such a new method allows further important savings in the overall CPU computation time. Comparative benchmarks and scalability issues related to the implemented parallel algorithm are discussed. Copyright © 2006 John Wiley & Sons, Ltd.

Eigenvalues of structures with uncertain elastic boundary restraints

The eigenvalue problems of structures with random elastic boundary supports are studied in this paper. Using the perturbation method, the differential equations including stochastic distributed parameters and random boundary conditions that govern the eigenproblems are transformed to a series of deterministic differential equations and boundary conditions. Then the differential equations and boundary conditions are discretized utilizing the finite element method (FEM). This process is easy to be implemented since the resulting perturbation equations with different orders own the same FEM meshes. The first-order and second-order sensitivities of eigenvalues are derived through solving the deterministic algebraic equations. Upon determining these sensitivities of eigenvalues, the approximate statistic expressions of random eigenvalues considering uncertain elastic boundary supports are established. At the end, several numerical examples are given to illustrate the application and effectiveness of the present method. Comparison of the present numerical results with those from the Monte-Carlo simulation method verifies the accuracy of the developed method.

Efficient and accurate waveguide mode computation using BI-RME and Lanczos methods

In this paper, a special purpose algorithm for solving large eigenvalue problems based on the Lanczos method is successfully applied to an engineering problem: the electromagnetic analysis and design of passive waveguide devices. For dealing with such complex problems, the boundary integral-resonant mode expansion (BI-RME) technique has been recently proposed. This technique solves integral equations (IEs) through the well-known method of moments (MoM), thus leading to structured eigenvalue problems. These problems frequently become very large when solving complex arbitrary geometries with high accuracy. In such cases, the eigenvalue problem cannot be efficiently solved with standard methods by means of personal computers, essentially due to CPU time and memory allocation requirements. In this paper, we propose an alternative technique, based on the Lanczos method, for the fast and accurate solution of large BI-RME generalized eigenvalue problems. The novel theoretical aspects of this approach, as well as the impacton the original BI-RME formulation, are described. Comparative benchmarks are also successfully presented for the full-wave analysis and design of real passive microwave devices. Copyright © 2005 John Wiley Sons, Ltd.

Structured Eigenvalue Condition Numbers

This paper investigates the effect of structure‐preserving perturbations on the eigenvalues of linearly and nonlinearly structured eigenvalue problems. Particular attention is paid to structures that form Jordan algebras, Lie algebras, and automorphism groups of a scalar product. Bounds and computable expressions for structured eigenvalue condition numbers are derived for these classes of matrices, which include complex symmetric, pseudo‐symmetric, persymmetric, skew‐symmetric, Hamiltonian, symplectic, and orthogonal matrices. In particular we show that under reasonable assumptions on the scalar product, the structured and unstructured eigenvalue condition numbers are equal for structures in Jordan algebras. For Lie algebras, the effect on the condition number of incorporating structure varies greatly with the structure. We identify Lie algebras for which structure does not affect the eigenvalue condition number.

A Nonlinear Eigenvalue Hybrid FEM Formulation for Two Dimensional Open Waveguiding Structures

A non linear hybrid finite element formulation, for the two dimensional eigenvalue analysis of open waveguides is proposed. The infinite solution domain of this kind of problems is divided into two region using a fictitious cylindrical surface-C. Inside the surface-C the finite element method is employed. Outside the surface-C the infinite domain is modeled through an infinite sum of cylindrical harmonics. The two solutions are coupled considering the continuity of the tangential field components along the surface-C. The overall procedure ends up to a nonlinear eigenvalue problem of the form A(β) � x = 0 where β is the propagation constant along the axis of the waveguide. For the solution of the nonlinear eigenvalue problem the Regula Falsi method is considered. The solution procedure is based on the initial values provided from a linear approximation of the problem. Finally, the validity of the method is verified by comparison with measurements presented in the bibliography. Introduction During the last years a particular research effort is directed towards the solution of the eigenvalue problem of arbitrary cross-section waveguiding structures in a unified and general way. All of the numerical techniques developed towards this direction try to convert the open-radiating problem to an equivalent closed one and in that way truncating the solution domain. This can be achieved either by making use of an artificial boundary transparent to the solution or by combining the Finite Element Method (FEM) with methods, such as the method of moments, capable of modeling the unbounded region. When the artificial boundary is considered one method to truncate the solution domain is to impose on it either the Absorbing Boundary Conditions (ABCs) or employ the Perfect Matching Layer (PML). An alternative method is to express the field in the unbounded region as an expansion of solution satisfying both the Maxwell equations and the radiation condition. However, for the two dimensional (2D) open waveguides, the performance of the already proposed techniques in the solution of the corresponding eigenvalue problem, is very poor. In particular, while PML is quite efficient in the estimation of the field distribution generated by a specific source, when this is used in the solution of eigenvalue problem leads to spurious (or corrupted) solutions. In the present work a hybrid finite element method capable of handling problems involving open arbitrary shaped waveguides is described. The problem at a first stage is approximated by means of a linear eigenvalue formulation. The formulation is derived by combining the finite element method and an approximate expansion in cylindrical harmonics (1). Namely, the radial wavenumber in the unbounded media is considered approxi- mately equal to that of free space. This is a reasonable approximation for the spectral region only around cut-off. The eigenvalues calculated using this approach were in good agreement with experimental results. However, aiming at a more generally valid method, the present effort considers the accurate radial wavenumber, which unfortunately (as it is already expected) yields a non-linear eigenvalue problem. The final nonlinear algebraic system is formulated employing a full electric field FEM formulation discretized by mixed edge/node triangular elements. The final nonlinear system is solved using the Regula Falsi technique (2) and employing the solution of the first linear approach as an initial guess.

Solution theorems for the standard eigenvalue problem of structures with uncertain-but-bounded parameters

Generalized eigenvalue problems from the modal analysis are often converted to the standard eigenvalue problems. In this paper, it evaluates the upper and lower bounds on the eigenvalues of the standard eigenvalue problem of structures subject to severely deficient information about the structural parameters. Here, we focus on non-probabilistic interval analysis models of uncertainty, which are adapted to the case of severe lack of information on uncertainty. Non-probabilistic, interval analysis method in which uncertainties are defined by interval numbers appears as an alternative to the classical probabilistic models. For the standard eigenvalue problem of structures with uncertain-but-bounded parameters, the vertex solution theorem, the positive semi-definite solution theorem and the parameter decomposition solution theorem for the standard eigenvalue problem are presented, and compared with Deif's solution theorem in numerical examples. It is shown that, for the upper and lower bounds on the eigenvalues of the standard eigenvalue problem with uncertain-but-bounded parameters, the presented vertex solution theorem is unconditional, and the positive semi-definite solution theorem and the parameter decomposition solution theorem have less limitary conditions compared with Deif's solution theorem. The effectiveness of the vertex solution theorem, the positive semi-definite solution theorem and the parameter decomposition solution theorem are illustrated by numerical examples

A Chart of Backward Errors for Singly and Doubly Structured Eigenvalue Problems

We present a chart of structured backward errors for approximate eigenpairs of singly and doubly structured eigenvalue problems. We aim to give, wherever possible, formulae that are inexpensive to compute so that they can be used routinely in practice. We identify a number of problems for which the structured backward error is within a factor $\sqrt{2}$ of the unstructured backward error. This paper collects, unifies, and extends existing work on this subject.

Estimating Natural Frequencies and Mode Shapes from Forced Response Calculations

A method is proposed for condensing a structural eigenvalue problem by using more than one forced response vector in a condensation. The method approximates an eigenvector by a linear combination of forced response vectors, resulting in a significantly smaller eigenvalue problem that retains the eigenpairs that were most excited by the force vectors. The condensation is extremely economical in cases where the forced response vectors have been computed for other purposes. In cases where some eigenpairs are approximately known, the method produces accurate eigenvalue estimates by choosing the force vectors to be parallel to the approximate eigenvectors. This idea forms the basis of an iterative procedure that finds all of the eigenvalues in a specified frequency range. Theoretical error bounds on the estimated eigenvalues are also presented. Numerical examples illustrate unique features of the condensation in its noniterative and iterative forms.

Interval eigenvalue analysis for structures with interval parameters

In this paper, a new method to solve the uncertain eigenvalue problem of structures with interval parameters is presented. The expressions of the interval stiffness matrix and interval mass matrix are developed directly from the interval parameters. With matrix perturbation and interval extension theory, the algorithm for interval eigenvalue analysis is discussed. The computations are based on the element level thus simplifying the calculations greatly, while the sharp bounds are guaranteed by the interval operations. Two numerical examples, a box cantilever beam and an automobile frame both with interval parameters, are given to show the validity of the presented method.