taylor approximation of second degree for complex mode in asymmetrical damped system algorithms for the first and second order sensitivities of the complex mode for an asymmetrical damped linear discrete dynamic system with respect to design parameters are presented the first an

eigenderivative analysis of asymmetric non conservative systems in general the damping matrix of a dynamic system or structure is such that it can not be simultaneously diagonalized with the mass and stiffness matrices by any linear transformation for this reason the eigenvalues and eigenvectors and consequently their derivatives become complex expressions for the first and second order derivatives of the eigenvalues and eigenvectors of these linear non conservative systems are given traditional restrictions of symmetry and positive definiteness have not been imposed on the mass damping and stiffness matrices the results are derived in terms of the eigenvalues and left and right eigenvectors of the second order system so that the undesirable use of the first order representation of the equations of motion can be avoided the usefulness of the derived expressions is demonstrated by considering a non proportionally damped two degree of freedom symmetric system and a damped rigid rotor on flexible supports copyright 2001 john wiley sons ltd

an iterative method for simultaneously computing the derivatives of simple and multiple eigenpairs of a quadratic eigenvalue problem abstract a new iterative method is proposed to compute the first order and high order derivatives of several eigenpairs of a quadratic eigenvalue problem qep the derivatives of simple and semi simple eigenpairs are simultaneously computed the proposed method avoids to solve many different linear systems with the size of the qep only the eigenpairs to be differentiated are required in our method convergence properties of the proposed iterative method are established in order to accelerate the proposed iterative method vector epsilon algorithm is applied and then an acceleration algorithm for our iterative method is presented and analyzed numerical experiments show that the proposed method is efficient for the computation of the first order and high order derivatives of eigenpairs of a qep especially for several eigenpair derivatives in moderate and large qep

qz based algorithm for system pole transmission zero and residue derivatives the qz algorithm gives a robust way of computing solutions to the generalized eigenvalue problem the generalized eigenvalue problem is used in linear control theory to find solutions to ricatti equations as well as to determine system transmission zeros in state space linear system analysis the system poles and transmission zeros are particularly important for determining system time and frequency response here we embed calculation of the eigenvalue derivatives in the qz algorithm such that the derivatives of system poles and transmission zeros are computed simultaneously with the poles and zeros themselves the resulting method is further exercised in finding generalized eigenvalues and their sensitivities required for finding the derivatives of system residues this technique should open the door to solutions of problems of interest by unconstrained gradient based methods typical numerical results are presented

second order sensitivity derivatives in structural analysis haug s 1981 expressions for the calculation of static constraint function second derivatives have resulted in an order of magnitude saving over conventional methods for the calculation of such derivatives a similar but simplified procedure is presented which results in a reduction of computational cost by an additional factor of two the results are presented in a notation that is different from that of haug but more common in other work on structural synthesis and analysis

rounding errors of partial derivatives of simple eigenvalues of the quadratic eigenvalue problem in this paper we derive rounding errors of partial derivatives of a simple eigenvalue of the quadratic eigenvalue problem dependent on several parameters we prove a series of lemmas and finally get theorems of rounding errors of both nonsymmetric and symmetric qeps examples are given to show the validity of our theorems and numerical results show that our rounding error is a very good upper bound estimation of the relative error of the eigenvalue

second order design sensitivity analysis of mechanical system dynamics dependence of dynamic response of nonlinear mechanical systems on design variables is analysed an adjoint variable method is used to derive first and second order derivatives of measures of dynamic response with respect to design variables a computational algorithm is presented for numerical calculation of first and second design derivatives a simple oscillator example is solved analytically and by the adjoint variable method with identical results a burst fire automatic weapon mechanism with linear and nonlinear damping is treated numerically it is shown that quadratic appriximations of dynamic response using results of second order design sensitivity analysis can be substantially better than conventional linear approximations

sensitivity analysis of non conservative eigensystems an expression for the derivatives of eigenvalues and eigenvectors of non conservative systems is presented contrary to previous methods that use state space form 2n space to consider damping proposed method solves the eigenpair derivatives of damped system explicitly the computation size of n order is maintained and the eigenpair derivatives are obtained simultaneously from one equation so that it is efficient in cpu time and storage capacity moreover this method can be extended to asymmetric non conservative damped systems although additional problems are generated contrary to the eigenpair sensitivity methods of symmetric systems in asymmetric case an algebraic method for the eigenpair derivatives can be obtained through similar procedure the proposed expression is derived by combining the differentiations of the eigenvalue problem and normalization condition into one linear algebraic equation the numerical stability is proved by showing non singularity of the proposed equation and the efficiency of the derived expression is illustrated by considering a cantilever beam with lumped dampers and a whirling beam

asymptotic green s function of a surface magnetic current element on a perfect electric conductor plane covered by a lossy dielectric substrate published analyses of radiation modeling for slot structures on dielectric substrate are empirical or numerical this paper proposes exact analytical asymptotic expressions of the far field green s functions of a surface magnetic current element on a perfect electric conductor plane covered by a lossy dielectric substrate of finite thickness from these expressions the radiation pattern of both the space wave and surface wave far away from an arbitrary shaped slot antenna structure can be calculated provided the source distribution across the slot is known the potentials used in the analysis are defined and their boundary conditions are expressed the helmholtz equation is solved in the laplace domain and the solutions are transformed into the space domain using the inverse hankel transform and steepest descent method the influences of the substrate thickness and dielectric constant are analyzed using the calculated expressions the model is validated by comparison with surface wave and space wave measurements and with numerical results obtained from a commercial electromagnetic simulator

the lieb yau conjecture for ground states of pseudo relativistic boson stars abstract it is known that ground states of the pseudo relativistic boson stars exist if and only if the stellar mass n 0 satisfies n n where the finite constant n is called the critical stellar mass lieb and yau conjecture in comm math phys 1987 that ground states of the pseudo relativistic boson stars are unique for each n n in this paper we prove that the above uniqueness conjecture holds for the particular case where n 0 is small enough

optimum design sensitivity based on approximation concepts and dual methods an optimum design sensitivity analysis capability is reported which exploits the approximation concepts dual method formulation of the minimum weight structural sizing problem an efficient iterative solution technique is used to facilitate determination of sensitivity derivatives for both primal and dual variables estimates on the useful range of parameter perturbations over which the optimum design sensitivity projections can be expected to yield satisfactory revised optimum designs are also obtained numerical results for several example problems will be presented to illustrate the effectiveness of the capability reported

asymptotic extraction approach for antennas in a multilayered spherical media an efficient algorithm is introduced to enhance the convergence of dyadic green s functions dgf in a layered spherical media where asymptotic expressions have been developed the formulated expressions involve an infinite series of spherical eigenmodes that can be reduced to the simple homogenous media green s function using the addition theorem of spherical hankel functions substantial improvements in the convergence speed have been attained by subtracting the asymptotic series representation from the original dgf the subtracted components are then added to the solution using the homogenous media green s function format

derivatives of complex eigenvectors using nelson s method 6shyy w an adaptive grid method for navier stokes flow computation applied mathematics and computation vol 21 no 3 1987 pp 201 219 7jeng y n andliou s c modi edmultipleone dimensional adaptive grid method numerical heat transfer part b vol 15 no 2 1989 pp 241 247 8jeng y n and lee z s revisit to the modi ed muliple onedimensional adaptive grid method numerical heat transfer part b vol 29 no 3 1996 pp 305 323 9bockelie m j and smith r e an adaptive grid method for computing the high speed 3d viscous flow about a re entry vehicle aiaa paper 92 2685 june 1992 10yee h c and harten a implicit tvd schemes for hyperbolic conservation laws in curvilinear coordinates aiaa journal vol 25 no 2 1987 pp 266 274 11jeng y n and liou y c adaptive grid generation by elliptic equations with grid control at all of the boundaries numerical heat transfer part b vol 23 no 2 1993 pp 135 151

hankel transform domain analysis of open circular microstrip radiating structures a full wave analysis of a circular disk conductor printed on a dielectric substrate backed by a ground plane is presented the method is based on the galerkin s method applied in the hankel transform domain and is numerically quite efficient several numerical examples are compared with available experimental data the agreement is found to be quite good and the validity of the present method is confirmed

nonlinear scattering with nonlocal interaction we consider the scattering problem for the hartree type equation in n withn 2 i frac partial u partial t frac 1 2 delta u v u 2 u where v x sum limits j 1 2 lambda j x gamma j lambda 1 lambda 2 ne 0 0 lambda j in mathbb r gamma j 0 and denotes the convolution in n we prove the existence of wave operators inh0 k l2 n x k l2 n for any positive integerk under the assumption 1 1 2 2 this is an optimal result in the sense that the existence of wave operators breaks down if min 1 2 1 the case where 1 1 2 2 is also treated according to the sign of 2

variational approach by means of adjoint systems to structural optimization and sensitivity analysis ii structure shape variation abstract for a linear elastic structure the first variation of an arbitrary stress strain and displacement functionals corresponding to variation of shape of external boundaries or interfaces is derived by using the solutions for primary and adjoint systems the application to optimal design is next presented and the relevant optimally conditions are derived from general expressions the path independent integrals used in fracture mechanics are rederived as a particular case of general expressions

natural frequency and mode shape sensitivities of damped systems part ii multiple natural frequencies abstract an efficient algorithm with proven numerical stability is derived for computation of eigenvalue and eigenvector derivatives of damped vibratory systems with multiple eigenvalues in the proposed method adjacent eigenvectors and orthonormal condition are used to compose an algebraic equation whose order is n m n m where n is the number of co ordinates and m the number of multiplicity of a multiple natural frequency the mode shape derivatives of the damped systems can be obtained by solving the algebraic equation the method can be consistently applied to both structural systems with structural design parameters and mechanical systems with lumped design parameters as an example of a structural system to demonstrate the theory of the proposed methods and its possibilities in the case of multiple eigenvalues the finite element model of the cantilever beam is considered and also a 5 dof mechanical system in the case of a non proportionally damped system the design parameter of the cantilever beam is its height and that of the 5 dof mechanical system is a spring

an annular slot antenna on a dielectric half space the radiation properties of both singly and multiply fed annular slot antennas on a dielectric air interface have been analyzed using galerkin s method applied in the hankel transform domain the validity of our method has been confirmed through comparison with both theoretical and experimental results the magic slot configuration has also been studied the polarization diversity of this radiator makes it suitable for implementing a quasioptical balanced mixer

nonlinear schrodinger equations and sharp interpolation estimates a sharp sufficient condition for global existence is obtained for the nonlinear schrodinger equation begin array 20 c nls 2i phi t delta phi left phi right 2 sigma phi 0 x in mathbb r n t in mathbb r end array in the case 2 n this condition is in terms of an exact stationary solution nonlinear ground state of nls it is derived by solving a variational problem to obtain the best constant for classical interpolation estimates of nirenberg and gagliardo

vector hankel transform analysis of annular ring microstrip antenna a rigorous analysis of the resonant frequency characteristics of an annular ring conductor printed on a dielectric substrate backed by a ground plane is presented the problem is cast into a set of vector dual integral equations by using vector hankel transform vht galerkin s method is then applied to compute the resonant frequencies of both transverse magnetic modes tm 11 and tm 12 of the microstrip antenna the real part of the resonant frequency shift of the tm 11 mode increases with increasing dielectric thickness in contrast to the tm 12 mode while the tm 11 mode has a high q it is a poor radiating mode the tm 12 mode is more suitable for antenna applications for a probe excitation the input impedance is obtained using single mode approximation for both the tm 11 and tm 12 modes it is shown that the tm 12 mode of the annular ring has a greater bandwidth than the tm 11 mode

rectangular and zonal slots on a sphere with a backing shell theory and experiment a rectangular slot antenna on a spherical cavity with a conducting backing shell is investigated rigorously the green s function approach is used to formulate an integral equation for the magnetic slot current which is solved using the moment method two previously developed methods to evaluate the admittance integrals are used and compared with each other both methods use recurrence formulas and analytical integration to speed the computation the slot and cavity resonances of the structure are studied in particular two simple formulas for the first two natural resonances of the cavity structure are given the effects of the shell size slot length and cavity dielectric constant on the input impedance are discussed measurements were carried out to verify the theory and very good agreement between theory and experiment is obtained in addition the zonal slot antenna on a spherical cavity with a conducting backing shell is studied the effect of the shell radius on the input impedance is investigated and again very good agreement between theory and experiment is found

approximate method for eigensensitivity analysis of a defective matrix based on the exact modal expansion method an arbitrary high order approximate method is developed for calculating the second order eigenvalue derivatives and the first order eigenvector derivatives of a defective matrix the numerical example shows the validity of the method if the different eigenvalues m 1 m q of the matrix are arranged so that m 1 m q and satisfy the condition that m q 1 m q 1 1 for some q 1

statistical model based damage detection and localization subspace based residuals and damage to noise sensitivity ratios the vibration based structural health monitoring problem is addressed as the double task of detecting damages modeled as changes in the eigenstructure of a linear dynamic system and localizing the detected damages within a fem of the monitored structure the proposed damage detection algorithm is based on a residual generated from a stochastic subspace based covariance driven identification method and on the statistical local approach to the design of detection algorithms this algorithm basically computes a global test which performs a sensitivity analysis of the residuals to the damages relative to uncertainties and noises how this residual relates to some residuals for damage localization and model updating is discussed damage localization is stated as a detection problem this problem is addressed by plugging aggregated sensitivities of the modes and mode shapes w r t fem structural parameters in the above setting this results in directional tests which perform the same type of damage to noise sensitivity analysis of the residual as for damage detection how the sensitivity aggregation mechanism relates to sub structuring is outlined numerical results obtained on one example are reported

eigenvalue and eigenvector derivatives of second order systems using structure preserving equivalences abstract a structure preserving equivalence in the sense intended here is a mapping between the stiffness damping and mass matrices describing some initial second order system and the corresponding three matrices of another second order system having identical spectrum most second order systems can be diagonalised through a mapping of this sort the mapping provides a new approach to the evaluation and the understanding of eigenvalue and eigenvector derivatives in place of pairs of eigenvalues we think of real scalar stiffness damping and mass quantities representing decoupled single degree of freedom systems in place of pairs of eigenvectors we think of individual columns of the matrices involved in the mapping this approach resolves the completely artificial phenomenon that the eigenvalue and eigenvector derivatives become undefined at instants when modification of say a damping parameter causes a pair of complex eigenvalues to turn into a pair of real eigenvalues or vice versa it also has the advantage of being applicable to cases where any one or more of the system matrices are singular

novel modal method for efficient calculation of complex eigenvector derivatives a novel modal superposition method is presented for calculating eigenvector derivatives in self adjoint or non self adjoint systems the method is especially appropriate for determination of derivatives of many eigenvectors and requires only one eigenvalue and its corresponding right and left eigenvectors two parameters for the eigenvalue under consideration are introduced to obtain a generic derivative formula which contains most of the available modal superposition methods as a branch matrices of the modified system are decomposed with available lower modes to avoid repeatedly solving equations for different eigenvalues the contribution of truncated higher modes is expanded in a continued product of a few matrices polynomials and one generalized power series whereby rapid convergence can be achieved easily even in the vicinity of the convergent boundary without additional computation cost all other modal superposition coefficients can be neglected intrinsically with the determination of introduced parameters but a simple condition for coefficients associated with lower order eigenvalues should be satisfied in a special case numerical examples show that the method is efficient and can give results comparable to the exact solutions the method is applicable to various damped systems and closed mode cases

a direct algebraic method for eigensolution sensitivity computation of damped asymmetric systems in general the derivative of an eigenvector of a vibrating symmetric system is the solution of a singular problem further complications are encountered in dealing with asymmetric damped systems for which the left and right eigenvectors and their derivatives become distinct and complex several approaches have been proposed to overcome this singularity such as nelson s method and the modal method in the present work a new approach is presented for calculating simultaneously the derivatives of the eigenvalues and their associated derivatives of the left and right eigenvectors for asymmetric damped systems with the proposed method the exact eigenderivatives can be obtained by solving a first order linear algebraic system of equations the method is applied on a 104 dof ventilator rotor system which is used as an example of an asymmetric damped system with distinct eigenvalues the diameter of the shaft has been chosen as the design parameter the comparison of the computational time shows that the proposed method is more efficient than both nelson s approach and the modal method copyright 2006 john wiley sons ltd

classification of positive solitary solutions of the nonlinear choquard equation in this paper we settle the longstanding open problem concerning the classification of all positive solutions to the nonlinear stationary choquard equation delta u u 2u left frac 1 x u 2 right 0 quad u in h 1 mathbb r 3 which can be considered as a certain approximation of the hartree fock theory for a one component plasma as explained in lieb and lieb simon s papers starting from 1970s we first prove that all the positive solutions of this equation must be radially symmetric and monotone decreasing about some fixed point interestingly to use the new method of moving planes introduced by chen li ou we deduce the problem into an elliptic system as a key step we transform this differential system into a system of integral equations with the help of riesz and bessel potentials and then use the method of a moving plane in an integral form next using radial symmetry we deduce the uniqueness result from lieb s work our argument can be adapted well to study the radial symmetry of positive solutions of the equation in the generalized form u k 1 f left u k 2 u right

iterative computation of eigenvector derivatives for middle eigenvalues the computation of eigenvector derivatives for middle eigenvalues in the real symmetric eigensystems is discussed based on finite element modeling the coefficient matrices of the governing equations for eigenvector derivatives are singular this work concentrates on a new method for such computations the method is based on an iteration technique and its convergence is proved the number of iterations is adaptively determined based on some error estimation highly accurate approximations to eigenvector derivatives can be achieved with low computational cost the proposed method can deal with both cases of simple and repeated eigenvalues in a unified manner it is especially suitable for the large sparse matrices from finite element models finally three examples including an aircraft wing structure are used to demonstrate the effectiveness of the proposed method

minimal blow up solutions of mass critical inhomogeneous hartree equation in this paper we are concerned with the cauchy problem of the inhomogeneous hartree equation iut u k x rnk y x y 2 u t y 2dyu t x x rn n 3 first we establish the mass concentration property of the blow up solutions second we show that the blow up solutions with minimal mass should concentrate at a critical point of k finally under certain assumptions on global maximum points of k we establish nonexistence of such solutions

sensitivity analysis for problems of dynamic stability abstract in mechanics as well as in physics the most general and important thing is to study the dependence of the characteristics of a physical process on problem parameters problems of dynamic stability for non conservative systems involve determination of eigenvalues and eigenvectors for these problems it is shown in general how the different sensitivity analyses can be performed without any new eigenvalue analyses the main question relates to the change in flutter load as a function of change in stiffness mass boundary conditions or load distribution discretized as well as non discretized examples are presented in details

rates of change of eigenvalues and eigenvectors in damped dynamic system rates of change of eigenvalues and eigenvectors of a damped linear discrete dynamic system with respect to the system parameters are presented a nonproportional viscous damping model is assumed because of the nonproportional nature of the damping the mode shapes and natural frequencies become complex and as a consequence the sensitivities of eigenvalues and elgenvectors are also complex the results are presented in terms of the complex modes and frequencies of the second order system and the use of rather undesirable state space representation is avoided the usefulness of the derived expressions is demonstrated by considering an example of a nonproportionally damped two degree of freedom system

techniques for generating analytic covariance expressions for eigenvalues and eigenvectors one of the key problems in data fusion is the estimation of a parameter vector from a set of noisy measurements in many cases the optimal estimate of such a parameter vector can be solved through the least squares problem there are however a number of important problems whose solution instead takes the form of an eigenvalue eigenvector problem furthermore knowledge of the uncertainty of the optimal estimate is as important as the estimate itself for many practical applications and this uncertainty is typically captured in the form of a covariance matrix while the covariance matrix for a least squares estimate has been well studied there has been substantially less research on the covariance of an optimal estimate originating from an eigenvalue eigenvector problem in this paper we develop general expressions that determine the uncertainty in a vector estimate obtained from an eigenvalue eigenvector problem given the uncertainty of the matrix this includes developing expressions for the analytic derivatives of the eigenvalues and eigenvectors with respect to the matrix from which they come finally the techniques developed are numerically validated with forward finite differencing and a monte carlo analysis and then used to determine covariance expressions for an ellipse fitting technique and the estimation of attitude quaternions

eigensensitivity analysis of a defective matrix with zero first order eigenvalue derivatives a direct method is developed for calculating the first to third order eigenvalue derivatives and first to second order eigenvector derivatives of a defective matrix with a zero first order eigenvalue derivative associated with jordan blocks of order higher than the lowest it is found under some conditions that the corresponding eigen solution of the perturbed problem has a distinct structure that in the fractional power series of the perturbation parameter the denominator of the power can not only he the orders of jordan blocks as in the normal cases but also can be the sum of two successive orders of jordan blocks the solution method is composed of three parts one of them is simular to known work and the other two are very different numerical example show the correctness of the method

calculation of eigensolution derivatives for nonviscously damped systems using nelson s method a method to calculate the derivatives of the eigenvalues and eigenvectors of multiple degree of freedom damped linear dynamic systems with respect to arbitrary design parameters is presented in contrast to the traditional viscous damping model a more general nonviscous damping model is considered the nonviscous damping model is such that the damping forces depend on the past history of velocities via convolution integrals over given kernel functions because of the general nature of the damping eigensolutions are generally complex valued and eigenvectors do not satisfy the classical orthogonality relationship the proposed method to calculate the eigenvector derivative depends only on the eigenvector concerned numerical examples are provided to illustrate the derived results nomenclature c j constant associated with the derivative of u j d s dynamic stiffness matrix d j constant associated with the derivative of v j g s laplace transform of g t g t nonviscous damping matrix k stiffness matrix l laplace transform of m mass matrix m total number of eigenvectors n number of degrees of freedom of the system n number of relaxation parameters p design parameter rank rank of a matrix s laplace domain parameter u j jth eigenvector t displacement vector v j jth adjoint left eigenvector x j vector associated with the derivative of u j yj vector associated with the derivative of v j j normalization constant for the jth eigenvector j jth eigenvalue k relaxation parameters subscripts t matrix transpose derivative with respect to s

modal expansion method for eigensensitivity with repeated roots extensions are made to the modal expansion eigensensitivity procedure for self adjoint systems originally formulated by fox and kapoor these allow the computation of sensitivity information for mode shapes associated with repeated roots and repeated root derivatives the extensions follow a similar line to additions to nelson s direct method proposed first by ojalvo and later generalized independently by mills curran and dailey the formulation based on modal expansion is simpler and more computationally efficient than the direct method although some accuracy is sacrificed by modal truncation the method generally yields acceptable results for the primary application in optimization

sensitivity theory for nonlinear systems i nonlinear functional analysis approach concepts of nonlinear functional analysis are employed to investigate the mathematical foundations underlying sensitivity theory this makes it possible not only to ascertain the limitations inherent in existing analytical approaches to sensitivity analysis but also to rigorously formulate a considerably more general sensitivity theory for physical problems characterized by systems of nonlinear equations and by nonlinear functionals as responses two alternative formalisms labeled the forward sensitivity formalism and the adjoint sensitivity formalism are developed in order to evaluate the sensitivity of the response to variations in the system parameters the forward sensitivity formalism is formulated in normed linear spaces and the existence of the g teaux differentials of the operators appearing in the problem is shown to be both necessary and sufficient for its validity this formalism is conceptually straightforward and can be advantageously used to assess the effects of relatively few parameter alterations on many responses on the other hand for problems involving many parameter alterations or a large data base and comparatively few functional type responses the alternative adjoint sensitivity formalism is computationally more economical however it is shown that this formalism can be developed only under conditions that are more restrictive than those underlying the validity of the forward sensitivity formalism in particular the requirement that operators acting on the state vector and on the system parameters must admit densely defined g teaux derivatives is shown to be of fundamental importance for the validity of this formalism the present analysis significantly extends the scope of sensitivity theory and provides a basis for still further generalizations

efficient solution of integral equation based electromagnetic problems with the use of macromodels the idea of so called macromodels is applied to problems of computational electromagnetics involving integral equation method of moments formulations it is shown that with the use of equivalence principles it is possible to exclude a part of the problem calculate its wideband response without an a priori knowledge of the excitations and then efficiently combine the result into the original situation the idea is validated using examples involving slot excited dielectric resonator antennas

computational methods for complex eigenproblems in finite element analysis of structural systems with viscoelastic damping treatments in this paper efficient numerical methods to approximate the complex eigenvalues and eigenvectors in non proportional and non viscous systems are presented these methods are specially conceived for practical engineering applications making use of the finite element analysis to determinate the effect that potential damping treatments have on the natural frequencies and mode shapes of structural systems considering the solution of the undamped problem the complex eigenpair is estimated by finite increments using the eigenvector derivatives for non proportional viscous systems with low and medium damping a simple single step technique is presented whose rapidity and accuracy is verified by means of numerical applications for higher damped systems an incremental approach is proposed that keeps the accuracy without significantly increasing the computational time for non viscously damped systems a fast iterative modality is suggested which allows to approximate in an efficient and simple way the complex eigenpair as numerical applications the study of a metallic beam with free layer damping treatment is completed using finite element procedures where the damping material is modelized by an exponential model whose parameters are obtained from curve fitting to experimental data

partial derivatives of the eigen triplet of the quadratic eigenvalue problem depending on several parameters this paper concerns the partial derivatives of the eigen triplet of the quadratic matrix polynomial q p l email protected 2m p email protected p k p where m p c p k p c n x n are complex analytic matrix valued functions email protected c m is a complex parameter vector first the analyticity theorem for simple eigenvalues and the corresponding eigenvectors is given second a new method is proposed to compute partial derivatives of the eigen triplet the derivatives of the eigen triplet can be obtained by solving algebraic linear equations of order n 1 where it only requires the information of the eigen triplet whose partial derivatives are to be computed and what is more important the condition numbers of the coefficient matrices are better than those of the nonsingular coefficient matrices arisen in the bordered matrix method 1 and nelson s method 5 numerical tests show the feasibility and efficiency of the new method the results are better or at least comparable with current methods

eigenvector derivatives with repeated eigenvalues in this paper an algorithm is derived for computing the derivatives of eigenvalues and eigenvectors for real symmetric matrices in the case of repeated eigenvalues where the matrices are functions of real parameters such as mass density or moment of inertia the algorithm is an extension of recent work by i u ojalvo the key step in this extended derivation is to differentiate the eigenvalue equation twice the algorithm preserves the symmetry and band structure of the matrices allowing efficient computer storage and solution techniques applications include sensitivity analysis and optimization of the normal modes of finite element modeled structures such as large space structures a cantilever beam finite element example is included

normalized solutions and asymptotical behavior of minimizer for the coupled hartree equations abstract we prove the existence and nonexistence of l 2 r n normalized solutions of coupled hartree equations which arisen from the studies of the nonlinear optics and multi component bose einstein condensates under certain type trapping potentials by proving some delicate energy estimates we give a precise description on the concentration behavior of minimizer solutions of the system furthermore an optimal blowing up rate for the minimizer solutions of the system is also proved

computation of eigensolution derivatives for nonviscously damped systems using the algebraic method t he computation of the eigensolution derivatives plays a significant role in dynamic model updating structural design optimization structural dynamic modification damage detection andmany other applications themethods to calculate eigensolution derivatives are well established for undamped and viscous damped systems these common methods can be divided into the modal method nelson s method and the algebraic method fox and kapoor 1 first proposed the modal method for symmetric undamped systems by approximating the derivative of each eigenvector as a linear combination of all undamped eigenvectors later adhikari and friswell 2 and adhikari 3 extended the modal method to the more general asymmetric systems with viscous and nonviscous damping respectively to simplify the computation of eigensolution derivatives nelson 4 proposed a method which requires only the eigenvector of interest by expressing the derivative of each eigenvector as a particular solution and a homogeneous solution for symmetric undamped systems later friswell and adhikari 5 extended nelson s method to symmetric and asymmetric systemswith viscous damping recently adhikari and friswell 6 extended nelson s method to symmetric and asymmetric nonviscously damped systems however nelson s method is lengthy and clumsy for programming lee et al 7 derived an efficient algebraic method which has a compact form to compute the eigensolution derivatives by solving a nonsingular linear system of algebraic equations for symmetric systems with viscous damping later guedria et al 8 extended the algebraic method to general asymmetric systems with viscous damping recently chouchane et al 9 wrote an excellent review of the algebraic method for symmetric and asymmetric systems with viscous damping and extended their method to the second order and high order derivatives of eigensolutions in this note the algebraic method will be extended to symmetric and asymmetric systems with nonviscous damping the equations of motion describing free vibration of ann degreeof freedom dof linear system with nonviscous viscoelastic damping can be expressed by 3 6

design variable tolerance effects on the natural frequency variance of constrained multi body systems in dynamic equilibrium the design variable tolerance effects on the natural frequency variance of constrained multi body systems in dynamic equilibrium are investigated in this study monte carlo simulation is often employed for such investigations but it is known to have serious drawbacks excessive amount of computation time needs to be consumed since a large number of evaluations are usually required for the method furthermore the solution accuracy cannot be always guaranteed in spite of the excessive amount of computation time in order to overcome such drawbacks a method employing eigenvalue sensitivity information is proposed to obtain the variance of natural frequency in this study in order to verify the accuracy and the efficiency of the method some numerical examples of multi body systems in dynamic equilibrium are solved and the results are compared to those obtained by an analytical method and monte carlo simulation

minimizers of mass critical hartree energy functionals in bounded domains abstract we consider l 2 constraint minimizers of the mass critical hartree energy functional with a trapping potential v x in a bounded domain of r 4 we prove that minimizers exist if and only if the parameter a 0 satisfies a a q 2 2 where q 0 is the unique positive solution of u u r 4 u 2 y x y 2 d y u 0 in r 4 by investigating new analytic methods the refined limit behavior of minimizers as a a is analyzed for both cases where all the mass concentrates either at an inner point x 0 of or near the boundary of depending on whether v x attains its flattest global minimum at an inner point x 0 of or not as a byproduct we also establish two gagliardo nirenberg type inequalities which are of independent interest

simultaneous iterative method for eigenpair derivatives of damped systems a new iterative method is proposed to compute many eigenpair derivatives of damped systems with respect to system parameters the state space form of damped system is not used the proposed method simultaneously computes the derivatives of many eigenpairs and the solution of many large linear systems of equations with different coefficient matrices is avoided numerical experiments demonstrate that the proposed method can be efficiently used for the computation of the derivatives of the r eigenpairs with largest magnitude in large symmetric damped systems moreover when the size n of system and the number r of desired eigenpair derivatives become larger the proposed method has more advantage

extrapolation of optimum design based on sensitivity derivatives in many applications it is useful to have information regarding the effect of the perturbations of the design problem constants parameters on the optimum solution of the problem sobieszczanski sobieski et al 1981 have therefore proposed to utilize an optimum sensitivity analysis as a structural design tool to assess the parameter perturbation effects by extrapolations based on optimum sensitivity derivatives the present investigation is concerned with an example of optimum sensitivity analysis involving a composite panel the case was selected in connection with the nonlinearity of the constraints and the tendency for a set of active constraints to change when the parameters are disturbed under these adverse conditions sensitivity derivatives are used in linear extrapolations to extend optimization information over a limited range of variation for a typical parameter

calculation of sensitivity derivatives in thermal problems by finite differences the transient thermal analysis of large aerospace vehicles such as the space shuttle requires models with thousands of degrees of freedom the design of thermal protection systems for such vehicles requires the repetitive thermal analysis of the vehicle and the calculation of sensitivities derivatives of the transient response with respect to design parameters when the number of design parameters is large the cost of the sensitivity calculations may become prohibitive the present work is concerned with reducing the computational cost and errors in sensitivity calculations with one of the simplest available methods the finite difference approach a technique for calculating simultaneously the nominal and perturbed solutions required for the finite difference derivative is proposed an expression for the optimum value of the perturbation parameter is obtained which permits reducing the computational error an insulated cylinder is used as an example which demonstrates that significant improvements in accuracy and efficiency may be obtained

efficient and accurate computation of an annular slot on a dielectric half space an efficient and accurate method of analysis for an annular slot radiating on a dielectric half space is presented the result involves neither singularity nor any numerical integration and therefore is computationally very efficient and easy to implement numerically the method is very rigorous and general and should find applications in other related configurations

hybrid perturbation polynomial chaos approaches to the random algebraic eigenvalue problem the analysis of structures is affected by uncertainty in the structure s material properties geometric parameters boundary conditions and applied loads these uncertainties can be modelled by random variables and random fields amongst the various problems affected by uncertainty the random eigenvalue problem is specially important when analyzing the dynamic behavior or the buckling of a structure the methods that stand out in dealing with the random eigenvalue problem are the perturbation method and methods based on monte carlo simulation in the past few years methods based on polynomial chaos pc have been developed for this problem where each eigenvalue and eigenvector are represented by a pc expansion in this paper four variants of a method hybridizing perturbation and pc expansion approaches are proposed and compared the methods use rayleigh quotient the power method the inverse power method and the eigenvalue equation pc expansions of eigenvalues and eigenvectors are obtained with the proposed methods the new methods are applied to the problem of an euler bernoulli beam and a thin plate with stochastic properties

variational approach to first and second order sensitivity analysis of elastic structures for a linear elastic structure the first variations of arbitrary stress strain and displacement functionals were explicitly expressed in terms of variations of material stiffness parameters within specified domain in references 1 and 2 by using the solutions for primary and adjoint systems in this paper explicit expressions for second variations are derived for the case when the material stiffness or compliance variations depend on a set of parameters these expressions arc obtained in terms of variations of stiffness or compliance parameters by using the solutions of two additional sets of adjoint systems furthermore the variation of local stress strain or displacement is considered several illustrative examples are provided for the first and second order sensitivity analysis of beams

second order eigensensitivity analysis of asymmetric damped systems using nelson s method first order eigensensitivity analysis using nelson s method is first reviewed then nelson s approach is extended to the computation of the second order derivatives of the eigenvalues and eigenvectors of symmetric and asymmetric damped systems the computation of second order derivatives may be required for large variation of design parameters and for some optimization algorithms nelson s method has the advantage of requiring only the knowledge of the eigenvector to be differentiated only systems with distinct eigenvalues are considered in this paper two numerical examples have been selected to illustrate the application and the utility of the derived expressions of the first and second order eigensensitivities a four degree of freedom symmetric damped system is chosen for the first example whereas a finite element asymmetric 20 dof damped rotor model is considered in the second example

derivative of eigensolutions of nonviscously damped linear systems derivatives of eigenvalues and eigenvectors of multiple degree of freedom damped linear dynamic systems with respect to arbitrary design parameters are presented in contrast to the traditional viscous damping model a more general nonviscous damping model is considered the nonviscous damping model is such that the damping forces depend on the past history of velocities via convolution integrals over some kernel functions because of the general nature of the damping eigensolutions are generally complex valued and eigenvectors do not satisfy any orthogonality relationship it is shown that under such general conditions the derivative of eigensolutions can be expressed in a way similar to that of undamped or viscously damped systems numerical examples are provided to illustrate the derived results

study on an auto correlation function based damage index sensitivity analysis and structural damage detection abstract the damage index based on the auto correlation function to detect the damage of the structure under white noise excitation is studied in detail in this paper the maximum values of the auto correlation function of the vibration response signals displacement velocity and acceleration from different measurement points of the structure are collected and formulated as a vector called auto correlation function at maximum point value vector amv which is expressed as a weighted combination of the hadamard product of two mode shapes amv is normalized by its root mean square value so that the influence of the excitation can be eliminated sensitivity analysis for the different parts of the normalized amv shows that the sensitivity of the normalized amv to the local stiffness is dependent most on the sensitivity of the hadamard product of the two lower order mode shapes to the local stiffness which has a sudden change of the value around the local stiffness change position the sensitivity of the normalized amv has the similar shape and same trend that shows it is a very good damage indicator even for the very small damage the relative change of the normalized amv before and after damage occurs in the structure is adopted as the damage index to show the damage location several examples of the stiffness reduction detection of a 12 story shear frame structure are utilized to validate the results in sensitivity analysis illustrate the effectiveness and anti noise ability of the amv based damage detection method and compare the effect of the response type on the detectability of the normalized amv

calculation of eigenpair derivatives for asymmetric damped systems with distinct and repeated eigenvalues summary an algorithm is derived for the computation of eigenpair derivatives of asymmetric quadratic eigenvalue problem with distinct and repeated eigenvalues in the proposed method the eigenvector derivatives of the damped systems are divided into a particular solution and a homogeneous solution by introducing an additional normalization condition we construct two extended systems of linear equations with nonsingular coefficient matrices to calculate the particular solution the method is numerically stable and the homogeneous solutions are computed by the second order derivatives of the eigenequations two numerical examples are used to illustrate the validity of the proposed method copyright 2015 john wiley sons ltd

derivatives of complex eigenvectors with distinct and repeated eigenvalues in this paper we investigate the computation of the first order derivatives of complex eigenvectors for general non defective eigensystems a new normalization condition is proposed with which we can compute unique first order derivatives of arbitrary differentiable eigenvectors of systems with distinct and repeated eigenvalues we also present an efficient algorithm to compute the particular solutions to the governing equations of the first order derivatives of eigenvectors finally numerical examples are included to demonstrate the validity of the proposed method copyright 2007 john wiley sons ltd

sensitivity analysis of discrete structural systems methods for calculating sensitivity derivatives for discrete structural systems are surveyed primarily covering literature published during the past two decades methods are described for calculating derivatives of static displacements and stresses eigenvalues and eigenvectors transient structural response and derivatives of optimum structural designs with respect to problem parameters the survey is focused on publications developed in nonstructural fields such as electronics controls and physical chemistry which are directly applicable to structural problems most notable among the nonstructural based methods are the adjoint variable technique from control theory and the green s function and fast methods from physical chemistry

analysis of the zonal and rectangular slots on a conducting spherical cavity the zonal slot cut on a conducting spherical cavity is analyzed rigorously the slot is excited asymmetrically which excites higher order azimuthal modes the green s function approach is used to formulate an integral equation for the magnetic current in the slot which is solved using the moment method mom new recurrence formulas are derived so that within their stable region the admittance integrals can be evaluated without the need for any numerical integration in this case the solution is very easy to implement with extremely short computation time the effects of the latitude angle on the peak resistance and percentage bandwidth are investigated furthermore the cavity resonance modes and their degeneracy are examined by plotting the expansion coefficients of the magnetic current in addition the problem of a rectangular slot in the nonequatorial plane is also studied measurements are carried out and good agreement between theory and experiment is obtained

theory and experiment of the hemispherical cavity backed slot antenna the hemispherical cavity backed slot antenna is studied theoretically and experimentally the exact magnetic field green s function of the cavity is derived rigorously and expressed in a form convenient for numerical computation the moment method is used to find the equivalent magnetic current in the slot and hence the input impedance of the antenna configuration the effects of the cavity size of the slot length and of the slot offset on the input impedance are studied and very good agreement between theory and experiment is obtained the variation of the magnetic current around the slot and cavity resonances are discussed comparisons between the rigorous solution and the single mode theory are given and the limitation of the single mode theory is discussed

sensitivity calculations for iteratively solved problems the calculation of sensitivity derivatives of solutions of iteratively solved systems of algebraic equations is investigated a modified finite difference procedure is presented which improves the accuracy of the calculated derivatives the procedure is demonstrated for a simple algebraic example as well as an element by element preconditioned conjugate gradient iterative solution technique applied to truss examples

eigensensitivity analysis for asymmetric nonviscous systems t he eigensensitivities of mechanical systems with respect to structural design parameters have become an integral part of many engineering design methodologies including optimization structural health monitoring structural reliability model updating dynamic modification reanalysis techniques and many other applications fox and kapoor 1 computed the derivative of each eigenvector as a linear combination of all of the undamped eigenvectors later adhikari and friswell 2 and adhikari 3 extended the modal method to the more general asymmetric systems with viscous and nonviscous damping respectively nelson 4 presented a method which requires only the eigenvector of interest by expressing the derivative of each undamped eigenvector as a particular solution and a homogeneous solution later friswell and adhikari 5 extended nelson s method to symmetric and asymmetric systems with viscous damping recently adhikari and friswell 6 extendednelson s method to symmetric and asymmetric nonviscously damped systems fox and kapoor 1 also suggested a direct algebraic method to calculate the eigensensitivity for symmetric undamped systems by solving a nonsingular linear system of algebraic equations lee et al 7 derived an efficient algebraic method which has a compact linear system with a symmetric coefficient matrix for symmetric systems with viscous damping later guedria et al 8 extended the algebraic method to general asymmetric viscous damped systems chouchane et al 9 reviewed the algebraic method and extended their method to the second order and high order derivatives of eigensolutions li et al 10 extended the algebraic method to symmetric and asymmetric nonviscously damped systems xu andwu 11 proposed a new normalization and presented a method for the computation of eigensolution derivatives of asymmetric systemswith viscously damping recently mirzaeifar et al 12 proposed a new method based on a combination of algebraic and modal methods for generally asymmetric viscously damped systems more recently li et al 13 proposed a method of design sensitivity analysis of asymmetric viscously damped systems with distinct and repeated eigenvalues which can compute the left and right eigenvector derivatives separately and independently all of the methods mentioned previously compute the eigensensitivities of asymmetric damped systems by using the left eigenvector however these methods have disadvantages in computational cost and storage capacity for the left eigenvector should be calculated to avoid using the left eigenvector an algebraic method is presented 14 which does not require the left eigenvector for asymmetric damped systems but this method is restricted to the case of viscous damping it should be noted that the coefficientmatrices of the algebraicmethodmay be ill conditioned due to the components of the additional constraints and system matrices in the coefficient matrices are not all of the same order of magnitude in addition 15 the normalization adapted in 14 and 12 will fail in some cases because it can equal zero or a very small number this note will present a method which is well conditioned and can calculate the eigensensitivity of asymmetric nonviscous damped systems without using the left eigenvector considering an n degree of freedom linear system with nonviscous viscoelastic damping 3 6 10

direct method for second order sensitivity analysis of modal strain energy abstract this paper proposed a lagrange based method for calculation of the second order sensitivity of modal strain energy mse values according to scheme of the direct method a lagrange functional of the element mse which added the eigenproblem and the normalization augmented by lagrange multipliers as constraints is constructed firstly after that the lagrange multipliers can be decided by setting variations of the lagrange functional with respect to state variables to zeros and then the sensitivity of mse can be yielded by derivative of the lagrange functional easily the accuracy of the new method and the affection of normalization criterion are verified by the numerical instances of a simply supported beam and a truss structure a portal frame is employed to testify the prediction ability of both the first order and the second order sensitivities of mse values as the numerical example of the portal frame shows the computation complexity of the new method can be reduced significantly in comparison of indirect methods at similar accuracy

an efficient algebraic method for computing eigensolution sensitivity of asymmetric damped systems in this paper an efficient algebraic method for the computation of eigensolution derivatives of asymmetric damped systems with distinct eigenvalues is presented by introducing an additional and new normalization condition we construct two extended systems of linear equations with nonsingular coefficient matrices which are transpose to each other we can compute the derivatives of the eigenvalues and their associated right and left eigenvectors by solving the two systems respectively in this way the cpu computation time and the storage space are considerably reduced finally a numerical example is included to demonstrate the validity of the proposed method