Comparison of parametric and non-parametric methods for bladed disks with random geometric mistuning

This paper presents different strategies to quantify the influence of random geometric mistuning on the dynamic behavior of bladed disk. These strategies are compared for different correlation lengths of the random field describing the blade geometry. The results allow to determine the kind of random geometric mistuning for which these strategies are appropriate. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Production Matrices and Riordan Arrays

We translate the concept of succession rule and the ECO method into matrix notation, introducing the concept of production matrix. This allows us to combine our method with other enumeration techniques using matrices, such as the method of Riordan matrices. Finally we treat the case of rational production matrices, i.e., those leading to rational generating functions. L. Ferrari and S. Rinaldi have been partially supported by MIUR project: Linguaggi formali e automi: metodi, modelli e applicazioni.     

The cocyclic Hadamard matrices of order less than 40

In this paper all cocyclic Hadamard matrices of order less than 40 are classified. That is, all such Hadamard matrices are explicitly constructed, up to Hadamard equivalence. This represents a significant extension and completion of work by de Launey and Ito. The theory of cocyclic development is discussed, and an algorithm for determining whether a given Hadamard matrix is cocyclic is described. Since all Hadamard matrices of order at most 28 have been classified, this algorithm suffices to classify cocyclic Hadamard matrices of order at most 28. Not even the total numbers of Hadamard matrices of orders 32 and 36 are known. Thus we use a different method to construct all cocyclic Hadamard matrices at these orders. A result of de Launey, Flannery and Horadam on the relationship between cocyclic Hadamard matrices and relative difference sets is used in the classification of cocyclic Hadamard matrices of orders 32 and 36. This is achieved through a complete enumeration and construction of (4t, 2, 4t, 2t)-relative difference sets in the groups of orders 64 and 72.     

Displacement structure approach to q-adic polynomial-Vandermonde and related matrices

In the present paper a new class of the so-called q-adic polynomial-Vandermonde-like matrices over an arbitrary non-algebraically closed field is introduced. This class generalizes both the simple and the confluent polynomial-Vandermonde-like matrices over the complex field, and the q-adic Vandermonde and the q-adic Chebyshev-Vandermonde-like matrices studied earlier by different authors. Three kinds of displacement structures and two kinds of fast inversion formulas are obtained for this class of matrices by using displacement structure matrix method, which generalize the corresponding results of the polynomial-Vandermonde-like and the q-adic Vandermonde-like matrices.     

Estimation of parameter matrices based on measured data

Finite element structural updating based on measured data may inherent significant errors due to uncertainties in the updated physical parameter matrices. This study presents analytical equations to estimate the change in the physical parameter matrices based on the measured modal data of dynamic systems and the measured displacement data of static systems. The equations for the parameter estimation are derived by minimizing cost functions in the satisfaction of the eigenvalue equation, the mode shape orthogonality requirements for the dynamic system, and the satisfaction of the measured displacement data for the static systems. The proposed method utilizes the Moore–Penrose inverse for the inverse of the rectangular matrices without using Lagrange multipliers. Comparing the analytical results with Berman & Nagy’s method and Yang & Chen’s method, this study demonstrates that the derived equations take simpler forms and produce more accurate results. The proposed method can be widely utilized in predicting static or dynamic parameter matrices for the design and analysis of any structure.     

Hessenberg matrix for sums of Hermitian positive definite matrices and weighted shifts

In this work, we introduce an algebraic operation between bounded Hessenberg matrices and we analyze some of its properties. We call this operation m-sum and we obtain an expression for it that involves the Cholesky factorization of the corresponding Hermitian positive definite matrices associated with the Hessenberg components.This work extends a method to obtain the Hessenberg matrix of the sum of measures from the Hessenberg matrices of the individual measures, introduced recently by the authors for subnormal matrices, to matrices which are not necessarily subnormal.Moreover, we give some examples and we obtain the explicit formula for the m-sum of a weighted shift. In particular, we construct an interesting example: a subnormal Hessenberg matrix obtained as the m-sum of two not subnormal Hessenberg matrices.     

Mechanical model of a mistuned 2DOF-structure coupled by viscous damping

In this paper the vibration amplitude reduction of a mechanical system is investigated when applying a damper between different DOF of that structure. Thereby, an elementary mechanical model consisting of two mass-spring elements with slightly differing eigenfrequencies and which are connected via a viscous dashpot damper is regarded. To investigate the systems behavior, an analysis of the transfer function and its poles and zeros in the L APLACE domain is performed when varying the damping constant. This reveals that the achievable amplitude reduction depends on the mistuning of the system and that even ideal viscous damping leads to a coupling which results in a shift of eigenfrequencies. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A robust AINV‐type method for constructing sparse approximate inverse preconditioners in factored form

This paper suggests a new method, called AINV‐A, for constructing sparse approximate inverse preconditioners for positive‐definite matrices, which can be regarded as a modification of the AINV method proposed by Benzi and Túma. Numerical results on SPD test matrices coming from different applications demonstrate the robustness of the AINV‐A method and its superiority to the original AINV approach. Copyright © 2001 John Wiley & Sons, Ltd.     

阶梯矩阵及其一般化在迭代法中的应用

Lu Hao首先给出了阶梯矩阵及其一般性的定义和性质．这类矩阵为迭代法提供了新矩阵分裂的基础．基于此新矩阵类的迭代方法的显著特征是它对于并行计算很容易被实现．应用这一新的分解方法,给出了一般的加速松弛方法（GAOR）,而关于AOR方法的一些性质可以被延伸到该新方法中,并针对Hermite正定矩阵进行了新方法收敛性的分析．最后,给出了一些例子来表明新方法的优越性．     

Kron's method for symmetric eigenvalue problems

Kron's method has been used successfully by engineers for over 30 years to find eigenvalues of large symmetric matrices. These matrices arise from domain decomposition of nonoverlapping domains of self-adjoint partial differential operators. This paper discusses some theoretical aspects of Kron's method. Specifically, the poles and zeroes of eigenvalues of the Kron matrix and their relationships with the eigenvalues of the original matrix are given.     

Energy harvesting by coupled oscillators forced by excitations with harmonic and random components

We examine an energy harvesting system of two magnetopiezoelastic oscillators coupled by electric circuit and driven by a combination of harmonic and stochastic ambient disturbances. The effects of synchronization and escape from a single potential well are discussed. In the system with relative mistuning in the stiffness of the harvesting oscillators, we show the dependence of the voltage output for different noise levels and excitation frequency. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Schur complement‐based domain decomposition preconditioners with low‐rank corrections

This paper introduces a robust preconditioner for general sparse matrices based on low‐rank approximations of the Schur complement in a Domain Decomposition framework. In this ‘Schur Low Rank’ preconditioning approach, the coefficient matrix is first decoupled by a graph partitioner, and then a low‐rank correction is exploited to compute an approximate inverse of the Schur complement associated with the interface unknowns. The method avoids explicit formation of the Schur complement. We show the feasibility of this strategy for a model problem and conduct a detailed spectral analysis for the relation between the low‐rank correction and the quality of the preconditioner. We first introduce the SLR preconditioner for symmetric positive definite matrices and symmetric indefinite matrices if the interface matrices are symmetric positive definite. Extensions to general symmetric indefinite matrices as well as to nonsymmetric matrices are also discussed. Numerical experiments on general matrices illustrate the robustness and efficiency of the proposed approach. Copyright © 2016 John Wiley & Sons, Ltd.     

A novel approach to solve linear systems arising from BEM using fast wavelet transforms

This paper uses Daubechies orthogonal wavelets to change dense and fully populated matrices of boundary element method (BEM) systems into sparse and semi‐banded matrices. Then a novel algorithm based on hierarchical nature of multiresolution analysis is introduced to solving resultant sparse linear systems. This algorithm decomposes NS‐form of transformed parent matrix into descendant systems with reduced sizes and solves them iteratively using GMRES algorithm. Both parts, changing dense matrices to sparse systems and the novel solver, can be added as a black box to the existing BEM codes. Transforming matrices into wavelet space needs less time than saved by solving sparse large systems. Numerical results with a precise study on sensitivity of solution for physical variables to the thresholding parameter, and savings in computer time and memory are presented. Also, the suitable value for thresholding parameter is recommended for elasticity problems. The results indicate that the proposed method is efficient for large problems. Copyright © 2009 John Wiley & Sons, Ltd.     

Displacement structure approach to q-adic Chebyshev–Vandermonde-like matrices

In this paper, a new class of so-called q-adic Chebyshev–Vandermonde-like matrices over an arbitrary non-algebraically closed field is introduced. This class generalizes both the ordinary Chebyshev–Vandermonde-like matrices over the complex field studied earlier by Kailath and Olshevsky [T. Kailath, V. Olshevsky, Displacement structure approach to Chebyshev–Vandermonde and related matrices, Integral Equations Operator Theory 22 (1995) 65–92], and the classical q-adic Vandermonde-like matrices with respect to power basis by Yang and Hu [Z.H. Yang, Y.J. Hu, Displacement structure and fast inversion formulas for q-adic Vandermonde-like matrices, J. Comput. Appl. Math. 176 (2005) 1–14]. Three kinds of displacement structures and consequently, three kinds of fast inversion formulas are presented for this class of matrices by using displacement structure theory method, which generalize the corresponding results for Chebyshev–Vandermonde-like and q-adic Vandermonde-like matrices.     

A new data envelopment analysis method for priority determination and group decision making in the analytic hierarchy process

The DEAHP method for weight deviation and aggregation in the analytic hierarchy process (AHP) has been found flawed and sometimes produces counterintuitive priority vectors for inconsistent pairwise comparison matrices, which makes its application very restrictive. This paper proposes a new data envelopment analysis (DEA) method for priority determination in the AHP and extends it to the group AHP situation. In this new DEA methodology, two specially constructed DEA models that differ from the DEAHP model are used to derive the best local priorities from a pairwise comparison matrix or a group of pairwise comparison matrices no matter whether they are perfectly consistent or inconsistent. The new DEA method produces true weights for perfectly consistent pairwise comparison matrices and the best local priorities that are logical and consistent with decision makers (DMs)’ subjective judgments for inconsistent pairwise comparison matrices. In hierarchical structures, the new DEA method utilizes the simple additive weighting (SAW) method for aggregation of the best local priorities without the need of normalization. Numerical examples are examined throughout the paper to show the advantages of the new DEA methodology and its potential applications in both the AHP and group decision making.     

Accurate and fast computations with positive extended Schoenmakers–Coffey matrices

Schoenmakers–Coffey matrices are correlation matrices with important financial applications. Several characterizations of positive extended Schoenmakers–Coffey matrices are presented. This paper provides an accurate and fast method to obtain the bidiagonal decomposition of the conversion of these matrices, which in turn can be used to compute with high relative accuracy the eigenvalues and inverses of positive extended Schoenmakers–Coffey matrices. Numerical examples are included.     

Random orthogonal matrix simulation

This paper introduces a method for simulating multivariate samples that have exact means, covariances, skewness and kurtosis. We introduce a new class of rectangular orthogonal matrix which is fundamental to the methodology and we call these matrices L matrices. They may be deterministic, parametric or data specific in nature. The target moments determine the L matrix then infinitely many random samples with the same exact moments may be generated by multiplying the L matrix by arbitrary random orthogonal matrices. This methodology is thus termed “ROM simulation”. Considering certain elementary types of random orthogonal matrices we demonstrate that they generate samples with different characteristics. ROM simulation has applications to many problems that are resolved using standard Monte Carlo methods. But no parametric assumptions are required (unless parametric L matrices are used) so there is no sampling error caused by the discrete approximation of a continuous distribution, which is a major source of error in standard Monte Carlo simulations. For illustration, we apply ROM simulation to determine the value-at-risk of a stock portfolio.     

Constructing weighing matrices by the method of weaving

We describe a way of obtaining new weighing matrices from old, which we call weaving because it involves distributing rows and columns of given matrices along the columns and rows of an array to form a partitioned matrix whose blocks have rank one. The method generalizes both the direct sum and direct product of matrices. We use the method to construct some new weighing matrices of relatively small order. © 1995 John Wiley & Sons, Inc.     

Curve veering and mode localization in a buckling problem

Summary It is shown on a simple example that small disorder, or mistuning, may alter drastically the eigensolution of buckling problems in nearly periodic structures with weak internal coupling. Specifically, the phenomena of eigenvalue loci veering and mode shape localization, which are known to occur for free vibration problems, are evidenced in the case of structural buckling as well.