lanczos algorithm for the quadratic eigenvalue problem in engineering applications abstract a new approach of employing the lanczos two sided recursion to solve the quadratic eigenvalue problem is presented the methodology employed retains the n order quadratic problems as posed without the need to use the method of matrix augmentation traditionally used to cast the problem as a linear eigenvalue problem of order 2n pursuing the concept of finding a characteristic matrix to yield an eigenvalue square matrix the lanczos recursion has been devised to solve the quadratic eigenvalue problem appropriate proofs showing the lanczos biorthogonal transformation are included the example problems presented validate the algorithm and show the effectiveness of the method for large quadratic eigenproblems

measurement selection for vibration based structural damage identification many researchers have intensively investigated the methods of using vibration properties of a structure to identify its damage in the past two decades it has been realized that the modal data included in damage identification analysis have a great influence on the accuracy of identification results in this paper a technique is presented to select a subset of measurement points and corresponding mode to be used in model updating for damage identification a new concept of damage measurability is introduced in terms of two factors namely the sensitivity of a residual vector to the structural damage and the sensitivity of the damage to the measurement noise based on the two factors the structural damage measurability is estimated quantitatively it yields more accurate and reliable damage identification results if the mode and measurement points corresponding to the largest damage measurability value are chosen in field test and numerical analysis the advantage of the technique is that it is based on the undamaged state of structures and thus independent of the damage configuration therefore it is applicable in practice to determine the measurement selection prior to field testing and damage identification analysis a frame structure is used to demonstrate the validity and versatility of the method it demonstrates a significant improvement on the accuracy and reliability of damage identification results when the measurement points and mode are determined according to the present technique

two level scheme for selection of primary degrees of freedom and semi analytic sensitivity based on the reduced system a number of approximate techniques have been developed to calculate the eigenvalues in a reduced manner these schemes approximate the lower eigenvalues that represent the global behavior of the structures the present study proposes a two level condensation scheme tlcs this scheme consists of two steps in the first step the candidate elements are selected for constructing the reduced system by energy estimation in the second step primary degrees of freedom are selected by sequential elimination from the candidate degrees of freedom linked to the selected elements through the first step the proposed method saves computational cost effectively and recovers eigenvalues of the full system with high accuracy from the lowest to the truncated range of frequency moreover besides the accuracies of the eigenvalue well constructed reduced system assures the accuracy of sensitivity of the eigenvalue compared to full system in this paper the semi analytic design sensitivity of eigenvalues is obtained by the rigid body separation technique in reduced system for the sensitivity calculation the eigenvectors obtained from the reduced system are transformed into global system field numerical examples demonstrate the reliability of the eigenvalue and its sensitivity obtained by the reduced system through the comparison with conventional sensitivity schemes

brief paper output feedback stabilization for stochastic nonlinear systems whose linearizations are not stabilizable this brief paper investigates the problem of output feedback stabilization for a class of high order stochastic nonlinear systems which are neither necessarily feedback linearizable nor affine in the control input based on the ideas of the homogeneous systems theory and the adding a power integrator technique an output feedback controller is constructed to ensure that the equilibrium at the origin of the closed loop system is globally asymptotically stable gas in probability the efficiency of the output feedback controller is demonstrated by a simulation example

symmetric positive definite flux continuous full tensor finite volume schemes on unstructured cell centered triangular grids novel cell centered full tensor finite volume methods are presented for general unstructured grids in two spatial dimensions the numerical schemes are flux continuous and based on computing the transmissibilities in a local subcell transform space ensuring that local flux matrices are symmetric as a result the global discretization matrix is shown to be symmetric positive definite for any grid type a symmetric physical space method is also introduced and the symmetric methods are shown to be closely related discrete ellipticity conditions are derived for positive definiteness of the physical space and subcell space schemes computational examples are presented for unstructured triangular grids demonstrating good performance of the scheme the schemes are compared with the so called multipoint flux approximation mpfa o method i aavatsmark t barkve o boe and t mannseth siam j sci comput 19 1998 pp 1700 1716 good agreement between the methods is obtained but the new scheme shows improved behavior in challenging cases

dynamic condensation method for structural eigenvalue analysis an improved dynamic condensation approach is presented for accurate calculation of structural eigenproperties the approach is iterative the kept and reduced degrees of freedom in the approach are related through a condensation matrix that is used to form a condensed eigenvalue problem an initial condensation matrix can be defined in terms of the system submatrices the eigenproperties calculated in an iterative step are utilized to update the condensation matrix matrix inversions or gaussian eliminations are avoided in the updating process by employing the orthogonality of the eigenvectors the process of updating the condensation matrix and the eigenvalue problem is repeated until a desired convergence in the eigenvalues is achieved usually a few iterations are quite adequate iterations for both the lower as well as the higher modes can be performed as the condensation matrices for the two sets of modes are very simply related numerical examples are presented to show the applicability of the proposed approach several methods that have been used in practice to select the kept degrees of freedom for condensing are also evaluated numerically with respect to their effectiveness in providing accurate estimates of the eigenproperties with a minimum number of iterations

delaunay s mesh of a convex polyhedron in dimension d application to arbitrary polyhedra this paper presents a method for creating a delaunay triangulation connected to a set of specified points the theoretical aspect is recalled for an arbitrary dimension and the method is discussed in order to derive a practical approach valid for dimensions 2 and 3 which is simple robust and well adapted to computation convex polyhedral and arbitrary polyhedral situations are introduced

a discrete duality finite volume approach to hodge decomposition and div curl problems on almost arbitrary two dimensional meshes we define discrete differential operators such as gradient divergence and curl on general two dimensional nonorthogonal meshes these discrete operators verify discrete analogues of usual continuous theorems discrete green formulae discrete hodge decomposition of vector fields and vector curls have a vanishing divergence and gradients have a vanishing curl we apply these ideas to discretize div curl systems we give error estimates based on the reformulation of these systems into equivalent equations for the potentials numerical results illustrate the use of the method on several types of meshes some of which are degenerating triangular meshes and nonconforming locally refined meshes

a finite volume method for approximating 3d diffusion operators on general meshes a finite volume method is presented for discretizing 3d diffusion operators with variable full tensor coefficients this method handles anisotropic non symmetric or discontinuous variable tensor coefficients while distorted non matching or non convex n faced polyhedron meshes can be used for meshes of polyhedra whose faces have not more than four edges the associated matrix is positive definite and symmetric if the diffusion tensor is symmetric a second order resp first order accuracy is numerically observed for the solution resp gradient of the solution

iterative method for dynamic condensation combined with substructuring scheme an iterated improved reduced system iirs procedure combined with substructuring scheme for both undamped and nonclassically damped structures is presented iterated iirs method is an efficient reduction technique because the highly accurate eigenproperties from the repeatedly updated condensed matrices can be obtained without consuming expensive computational cost however single domain direct approach of this method to large structures requires much computational resources and even makes analysis intractable in the case only limited computer storage is available these problems can be overcome by combining the substructuring scheme with iirs procedure the newly developed iirs method combined with a substructuring scheme can provide an efficient methodology for large scale eigenvalue problems the validation of the present method and the evaluation of computational efficiency are demonstrated through the numerical examples

a new iterative order reduction ior method for eigensolutions of large structures order reduction is a computationally efficient method to estimate some lowest eigenvalues and the corresponding eigenvectors of large structural systems by reducing the order of the original model to a smaller one but its accuracy is limited to a small range of frequencies that depends on the selection of the retained degrees of freedom this paper proposes a new iterative order reduction ior technique to obtain accurately the eigensolutions of large structural systems the technique retains all the inertia terms associated with the removed degrees of freedom this hence leads to the reduced mass matrix being in an iterated form and the reduced stiffness matrix constant from these mass and stiffness matrices the eigensolutions of the reduced system can be obtained iteratively on convergence the reduced system reproduces the eigensolutions of the original structure a proof of the convergence property is also presented applications of the method to a practical garteur structure as well as a plate have demonstrated that the proposed method is comparable to the commonly used subspace iteration method in terms of numerical accuracy moreover it has been found that the proposed method is computationally more efficient than the subspace iteration method copyright 2003 john wiley sons ltd

scaling sensitivity and stability in the numerical solution of quadratic eigenvalue problems the most common way of solving the quadratic eigenvalue problem qep l 2 m l d k x 0 is to convert it into a linear problem l x y z 0 of twice the dimension and solve the linear problem by the qz algorithm or a krylov method in doing so it is important to understand the influence of the linearization process on the accuracy and stability of the computed solution we discuss these issues for three particular linearizations the standard companion linearization and two linearizations that preserve symmetry in the problem for illustration we employ a model qep describing the motion of a beam simply supported at both ends and damped at the midpoint we show that the above linearizations lead to poor numerical results for the beam problem but that a two parameter scaling proposed by fan lin and van dooren cures the instabilities we also show that half of the eigenvalues of the beam qep are pure imaginary and are eigenvalues of the undamped problem our analysis makes use of recently developed theory explaining the sensitivity and stability of linearizations the main conclusions of which are summarized as well as arguing that scaling should routinely be used we give guidance on how to choose a linearization and illustrate the practical value of condition numbers and backward errors

improvement of reduction method combined with sub domain scheme in large scale problem for a few decades various approximate techniques have been developed to calculate the eigenvalues in a reduced manner in order to construct reliable reduced systems it is essential to select the proper primary degrees of freedom pdofs unless the pdofs are selected properly the selection of pdofs might be localized and the eigenvalue prediction might emphasize excessively the lower modes or lose the important modes moreover sometimes it takes considerable amount of computing time to construct a reduced system in large scale problem these troubles in constructing reduced system can be avoided by applying reduction scheme in sub domain level after dividing global system into a number of subdomains reduced system which has only the pdofs is constructed in each sub domain this paper presents new algorithms to construct efficient reduction system through three different schemes they are version 1 version 2 and version 3 systems the version 3 system is constructed by combining the advantages of the version 1 and the version 2 systems numerical examples demonstrate that the proposed version 3 method saves computational cost effectively and provides a reduced system which can predict accurate eigenvalues of global system

dynamic condensation in a damped system through rational selection of primary degrees of freedom abstract various system reduction methods have been proposed to calculate eigenvalues and eigenvectors with computational efficiency in the case of system condensation the method of selecting proper primary degrees of freedom pdofs is crucial because they are closely related to the accuracy and reliability of eigensystem prediction in a reduced system although reasonable selection methods for pdofs have been proposed for undamped systems these methods cannot be directly extended to damped systems we propose a pdof selection method for a damped system the proposed method is based on the ratio of dof wise energy distributions in order to estimate the energy distribution of the structure ritz vectors are obtained using a two sided lanczos algorithm energy distribution matrices are calculated from the obtained ritz vectors then degrees of freedom dofs corresponding to the lowest rayleigh quotients are selected as the pdofs the efficiency and reliability of the proposed method are demonstrated using numerical examples

discretisation and multigrid solution of elliptic equations with mixed derivative terms and strongly discontinuous coefficients this paper introduces a new conservative cell centred finite volume scheme for the accurate solution of elliptic diffusion equations with strongly varying coefficients the discretisation is designed to model a wide range of heterogeneities and anisotropies and in particular the case where the diffusivity is represented by a nondiagonal matrix which may occur if the medium is anisotropic in a general direction such problems arise for example in oil reservoir simulation when renormalisation techniques are used to model the reservoir geology it is in this context that the method is described although it is applicable to a far wider class of problems in heat conduction electrostatics and potential theory the paper also describes the application of a mulitigrid scheme for efficient numerical solution of the algebraic equations derived using the new discretisation

model reduction using dynamic and iterated irs techniques static or guyan reduction is widely used to reduce the number of degrees of freedom in a finite element model but it is exact only at zero frequency the improved reduced system irs method makes some allowance for the inertia terms and produces a reduced model which more accurately estimates the modal model of the full system in this paper the irs method is extended by obtaining the equivalent transformation based on dynamic rather than static reduction an iterative algorithm based on the irs method is also described on convergence this algorithm provides a reduced model which reproduces a subset of the modal model of the full system the iterative version of the irs method based on dynamic reduction is also described

the lanczos algorithm applied to unsymmetric generalized eigenvalue problem application of the two sided lanczos recursion to the unsymmetric generalized eigenvalue problem is presented the system matrices are real and unsymmetric therefore the recursions are performed in real arithmetic and complex arithmetic is employed in the qr algorithm used to extract the eigenvalues of the transformed tridiagonal matrix the biorthonormal transformation of the unsymmetric generalized eigenvalue problem is considered in detail with appropriate proofs presented in appendices issues relating to the computer implementation of the unsymmetric generalized eigenvalue problem are discussed the example problems solved demonstrate the working of the algorithm in extracting the complex and or real eignevalues of an unsymmetric system of matrices also the algorithm is applied to extract a few of the eigenvalues of a large fluid structure interaction problem and the results are compared with the eigenfrequencies extracted by an unsymmetric subspace iteration procedure presented in the literature

the convergence of the iterated irs method abstract static or guyan reduction is widely used to reduce the number of degrees of freedom in a finite element model but it is exact only at zero frequency the improved reduced system irs method makes some allowance for the inertia terms and produces a reduced model which more accurately estimates the modal model of the full system the irs method may be extended to produce an iterative algorithm of the reduction transformation it has already been shown that this reduced model reproduces a subset of the modal model of the full system if the algorithm converges in this paper it is proved that the iterated irs method converges it is also shown that the lower modes converge more quickly than the higher modes and that the master co ordinates should be chosen to give an accurate static reduction

element based node selection method for reduction of eigenvalue problems dynamic analysis of structural systems requires a considerable amount of computing time because the dynamic behavior of a structure is dominated by lower modes all of the eigenvalues of the system need not be calculated the previously proposed methods approximate the eigenvalues of a global system but they take great amount of computing time to construct a reduced system an element level energy estimation technique is proposed to construct a reduced finite element model through several examples it is demonstrated that the proposed method effectively saves computational time and accurately predicts the eigenvalues of a global system

energy method for selection of degrees of freedom in condensation an analytical method is presented for the selection of degrees of freedom in condensation of eigenproblems the method isbasedon theenergiesassociatedwiththedegrees offreedom in theeigenmodesofstructuralsystems for theenergyestimationritzvectorsarecalculated usingthestiffnessandmassmatrices theenergiesaddedthrough themodesortheweighted rowsum oftheenergy distribution matrix can be used asan effective guideline on which degrees of freedom should be retained in the analysis another approach of sequential selection can be employed in which a e nite number of new degrees of freedom with the largest energy are taken in each mode and the e nal union becomes the analysis set the energy of the selected degrees of freedom or the column sum of the matrix can beused to predict thesolution accuracy in each mode the erroranalysisshowsthattheperturbation in eigenvalue is related with the energy of the degrees of freedom the row and column sums indicate the completeness of the selected set and give a clue to how many degrees of freedom to be included the energy criterion has shown that the conventional practice of choosing translational degrees of freedom is appropriate only for the lowest modes numerical investigations were performed to test the convergence criterion the energy method proved to work well for typical example problems