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

identifying preferences in networks with bounded degree this paper provides a framework for identifying preferences in a large network under the assumption of pairwise stability of network links network data present difficulties for identification especially when links between nodes in a network can be interdependent e g where indirect connections matter given a preference specification we use the observed proportions of various possible payoff relevant local network structures to learn about the underlying parameters we show how one can map the observed proportions of these local structures to sets of parameters that are consistent with the model and the data our main result provides necessary conditions for parameters to belong to the identified set and this result holds for a wide class of models we also provide sufficient conditions and hence a characterization of the identified set for two empirically relevant classes of specifications an interesting feature of our approach is the use of the economic model under pairwise stability as a vehicle for effective dimension reduction the paper then provides a quadratic programming algorithm that can be used to construct the identified sets this algorithm is illustrated with a pair of simulation exercises

an econometric model of network formation with degree heterogeneity i introduce a model of undirected dyadic link formation which allows for assortative matching on observed agent characteristics homophily as well as unrestricted agent level heterogeneity in link surplus degree heterogeneity like in fixed effects panel data analyses the joint distribution of observed and unobserved agent level characteristics is left unrestricted two estimators for the common homophily parameter 0 are developed and their properties studied under an asymptotic sequence involving a single network growing large the first tetrad logit tl estimator conditions on a sufficient statistic for the degree heterogeneity the second joint maximum likelihood jml estimator treats the degree heterogeneity a i0 i 1 super n as additional incidental parameters to be estimated the tl estimate is consistent under both sparse and dense graph sequences whereas consistency of the jml estimate is shown only under dense graph sequences

kinetic roughening with power law waiting time distribution we introduce a surface growth model where the elementary events are characterized by a waiting time distribution p t exact relations to directed polymer statistics and to continuous time random walk problems are established for p t i the behaviour is similar to that of the zhang model where rare event dominated kinetic roughening occurs due to a power law noise in the surface increments a careful correction to scaling analysis of our numerical results in 1 1 dimensions indicates universality with the zhang model for fixed values of p

iterative computation of modal sensitivities the computation of frequency and mode shape sensitivities with respect to design parameters is essential to many structural optimization and finite element updating algorithms approximating the gradients may lead to poor estimates and loss of convergence but on the other hand the cost of exact methods such as nelson s nelson r simplified calculations of eigenvector derivatives a1aa journal vol 14 no 9 1976 pp 1201 1205 is often not acceptable for industrial size models the present study thus analyses existing approximation methods suggests a first order improvement of fox and kapoor s modal basis fox r and kapoor m rate of change of eigenvalues and eigenvectors a1aa journal vol 6 no 12 1968 pp 2426 242 9 and proposes a residual iteration technique allowing accurate computations of modeshape sensitivities an engine cover model and a model of the ariane 5 main cryogenic stage are used to analyze for models of realistic sizes the accuracy and computational costs of the proposed methods

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

roughness fluctuations roughness exponents and the universality class of ballistic deposition abstract in order to estimate roughness exponents of interface growth models we propose the calculation of effective exponents from the roughness fluctuation in the steady state we compare the finite size behavior of these exponents and the ones calculated from the average roughness w 2 for two models in the 2 1 dimensional kardar parisi zhang kpz class and for a model in the 1 1 dimensional villain lai das sarma vlds class the values obtained from provide consistent asymptotic estimates eventually with smaller finite size corrections for the vlds nonlinear molecular beam epitaxy class we obtain 0 93 0 01 improving previous estimates we also apply this method to two versions of the ballistic deposition model in two dimensional substrates in order to clarify the controversy in terms of its universality class raised by numerical results and a recent derivation of its continuous equation effective exponents calculated from suggest that both versions are in the kpz class additional support for this conclusion is obtained by a comparison of the full roughness distributions of these models and the distribution of other discrete kpz models

surface growth and crossover behaviour in a restricted solid on solid model the authors describe results of numerical simulations of growth in a restricted solid on solid model they first extract the exponent beta by comparing the surface width w t approximately tbeta with a spatially averaged height correlation function g t the latter is shown to give a more accurate estimate for beta even for relatively small systems the exponent x is obtained from the saturation of the interface fluctuations in the late time regime their results lead to a conjecture of dimension dependent exponents as beta d 1 d 1 x d 2 d 2 in addition they study time dependent crossover phenomena by relaxing the local height constraint and using a noise reduction method in the model they demonstrate how the effect of such crossover effects may lead to spurious values of the growth exponents finally they discuss their results in relation to other discrete models and the continuum growth equations

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

persistence in nonequilibrium surface growth persistence probabilities of the interface height in 1 1 and 2 1 dimensional atomistic solid on solid stochastic models of surface growth are studied using kinetic monte carlo simulations with emphasis on models that belong to the molecular beam epitaxy mbe universality class both the initial transient and the long time steady state regimes are investigated we show that for growth models in the mbe universality class the nonlinearity of the underlying dynamical equation is clearly reflected in the difference between the measured values of the positive and negative persistence exponents in both transient and steady state regimes for the mbe universality class the positive and negative persistence exponents in the steady state are found to be 0 66 0 02 and 0 78 0 02 respectively in 1 1 dimensions and 0 76 0 02 and 0 85 0 02 respectively in 2 1 dimensions the noise reduction technique is applied on some of the 1 1 dimensional models in order to obtain accurate values of the persistence exponents we show analytically that a relation between the steady state persistence exponent and the dynamic growth exponent found earlier to be valid for linear models should be satisfied by the smaller of the two steady state persistence exponents in the nonlinear models our numerical results for the persistence exponents are consistent with this prediction we also find that the steady state persistence exponents can be obtained from simulations over times that are much shorter than that required for the interface to reach the steady state the dependence of the persistence probability on the system size and the sampling time is shown to be described by a simple scaling form

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

design sensitivity analysis and optimization of dynamic response abstract the paper presents methods of design sensitivity analysis and optimization of dynamic response of mechanical and structural systems a key feature of the paper is the development of procedures to handle point wise state variable constraints difficulties with a previous treatment where such constraints were transformed to equivalent integral constraints are noted and explained from theoretical as well as engineering standpoints an alternate treatment of such constraints is proposed developed and evaluated in this treatment each point wise state variable constraint is replaced by several constraints that are imposed at all the local max points for the original constraint function the differential equations of motion are formulated in the first order form so as to handle more general problems the direct differentiation and adjoint variable methods of design sensitivity analysis to deal with the point wise constraints are presented with the adjoint variable methods there are two ways of calculating design sensitivity coefficients the first approach uses an impulse load and the second approach uses a step load for the corresponding adjoint equation since the adjoint variable methods are better for a large class of problems an efficient computational algorithm with these methods is presented in detail optimum results for several problems are obtained and compared with those available in the literature the new formulation works extremely well as precise optimum designs are obtained

phase separation in a single step growth model with surface diffusion we consider a one dimensional solid on solid growth model in which the nearest neighbour height differences are restricted to take the values deposition occurs at local minima with probability f while diffusion moves of single adatoms within a layer occur with probability 1 f interlayer transport and detachment from step edges is suppressed for the stationary distribution of the model is known hence the growth induced surface current can be computed analytically for small diffusion rates in the opposite diffusion dominated limit a description in terms of step flow is possible for slopes larger than a critical slope for smaller slopes the surface phase separates into regions of slope the stationary domain size diverges for as where we suggest that the large scale behaviour in this limit can be described by the noisy kuramoto sivashinsky equation in its noise dominated regime

a direct mapping between eden growth model and directed polymers in random media on the basis of preserving required symmetries and capturing the essential ingredients the scaling properties of the frontier of eden model clusters 1961 have been suggested by kardar et al 1986 to be governed in the continuum limit by a burger s equation with noise the authors show that such a correspondence is exact by mapping explicitly the eden model onto the problem of the conformation of a directed polymer in a random medium at zero temperature the latter model has been shown to be itself in correspondence with burgers equations in addition this mapping allows them to have access to the distribution of noise to be included in burgers equation to exactly reproduce the eden growth model finally this implies that the eden model always shows the same universal behaviour in contrast to a recently found breakdown of universality for the directed polymer problem in a very disordered medium

surface pattern formation and scaling described by conserved lattice gases we extend our 2 1 dimensional discrete growth model odor et al phys rev e 79 021125 2009 with conserved local exchange dynamics of octahedra describing surface diffusion a roughening process was realized by uphill diffusion and curvature dependence by mapping the slopes onto particles two dimensional nonequilibrium binary lattice model emerges in which the smoothing or roughening surface diffusion can be described by attracting or repelling motion of oriented dimers the binary representation allows simulations on very large size and time scales we provide numerical evidence for mullins herring or molecular beam epitaxy class scaling of the surface width the competition of inverse mullins herring diffusion with a smoothing deposition which corresponds to a kardar parisi zhang kpz process generates different patterns dots or ripples we analyze numerically the scaling and wavelength growth behavior in these models in particular we confirm by large size simulations that the kpz type of scaling is stable against the addition of this surface diffusion hence this is the asymptotic behavior of the kuramoto sivashinsky equation as conjectured by field theory in two dimensions but has been debated numerically if very strong normal surface diffusion is added to a kpz process we observe smooth surfaces with logarithmic growth which can describe the mean field behavior of the strong coupling kpz class we show that ripple coarsening occurs if parallel surface currents are present otherwise logarithmic behavior emerges

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

nonuniversal effects in mixing correlated growth processes with randomness interplay between bulk morphology and surface roughening to construct continuum stochastic growth equations for competitive nonequilibrium surface growth processes of the type rd x that mixes random deposition rd with a correlated growth process x we use a simplex decomposition of the height field a distinction between growth processes x that do and do not create voids in the bulk leads to the definition of the effective probability p eff of the process x that is a measurable property of the bulk morphology and depends on the activation probability p of x in the competitive process rd x the bulk morphology is reflected in the surface roughening via nonuniversal prefactors in the universal scaling of the surface width that scales in p eff the equation and the resulting scaling are derived for x in either a kardar parisi zhang or edwards wilkinson universality class in 1 1 dimensions and are illustrated by an example of x being a ballistic deposition we obtain full data collapse on its corresponding universal scaling function for all p 0 1 we outline the generalizations to 1 n dimensions and to many component competitive growth processes

dynamic scaling in thin film growth with irreversible step edge attachment we study dynamic scaling in a model with collective diffusion cd of isolated atoms in terraces and irreversible aggregation at step edges simulations are performed in two dimensional substrates with several diffusion to deposition ratios r identical with d f data collapse of scaled roughness distributions confirms that this model is in the class of the fourth order nonlinear growth equation by villain lai and das sarma vlds with negligible finite size effects while estimates of scaling exponents show some discrepancies this result is consistent with the prediction of a recent renormalization group approach and improves previous numerical works on related models the roughness follows dynamic scaling as w lalpha r1 2f xi l with correlation length xi rt 1 z where z is the dynamic exponent we also propose a limited mobility lm model where the incident atom executes up to g steps before a new atom is adsorbed and irreversibly aggregates at step edges this model is also shown to belong to the vlds class the size of the plateaus in the film surface increases as g1 2 and the lateral correlation scales as g1 2t1 z the time evolution of the roughness reproduces that of the cd model if an equivalent parameter g approximately r2 z is chosen this suggests the possibility of using lm models with tunable diffusion length to simulate processes with simultaneous diffusion of many atoms a scaling approach is used to justify exponent values and dynamic relations for both models including the significant decrease of surface roughness as r or g increases

cluster structure of the eden model and directed polymers in a random potential the cluster structures of the eden model and directed polymers in a random potential are described the heights hs and widths ws of each cluster grows as hs approximately sv and ws approximately sv perpendicular to with v sub z d 1 z and vperpendicular to 1 d 1 z in d dimensions where s is the size of the cluster and z is the dynamic critical exponent generalization to include the cluster size distribution is discussed

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

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

non equilibrium critical behaviour on fractal lattices a simple one component surface reaction model exhibiting a critical point has been studied on deterministic fractal lattices with fractal dimension 1 df 2 results from time dependent simulations show a change in the value of the critical exponents with df two of the exponents interpolate very nicely between the values in one and two dimensions

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

simplified calculation of eigenvector derivatives with repeated eigenvalues an efficient and practical procedure based on the generalized schmidt orthogonalization is presented to compute the eigenvector derivatives when repeated eigenvalues exist

bounds on parameters in panel dynamic discrete choice models identification of dynamic nonlinear panel data models is an important and delicate problem in econometrics in this paper we provide insights that shed light on the identification of parameters of some commonly used models using these insights we are able to show through simple calculations that point identification often fails in these models on the other hand these calculations also suggest that the model restricts the parameter to lie in a region that is very small in many cases and the failure of point identification may therefore be of little practical importance in those cases although the emphasis is on identification our techniques are constructive in that they can easily form the basis for consistent estimates of the identified sets

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

calculation of eigenvector derivatives for structures with repeated eigenvalues in structural optimization and system identification eigenvector derivatives provide important information for updating design model parameters when the current parameter values yield repeated eigenvalues it has not been possible previously to calculate unique eigenvector derivatives recent work has provided a method for determining unique eigenvalue derivatives for this case but methods for calculating the eigenvector sensitivities have been incomplete in this work a complete method for calculation of repeated root eigenvector derivatives is shown for the real symmetric structural eigenproblem the derivation is completed by using information from the second derivative of the eigen problem and is limited to the case of distinct eigenvalue sensitivities as an example the repeated root eigenvector sensitivities are calculated for a simple three degree of freedom beam grillage comparisons of linear approximations using these derivatives to the calculated eigenvectors support the accuracy of the formulation

mode traces in degenerate eigensystems and augmented assurance eigenpairs contextually indicative of the frequencies and oscillatory modes of structural systems are considered as functions of a single parameter undesired permutation of modal designations is noted to arise following events of transitory eigenvalue coalescence with respect to the parameter resulting in the definition of nonsmooth functions the tracing of eigenpairs across such events is outlined for the permanently degenerate general eigenproblem as the general case the cases of transitorily degenerate and distinct eigenvalues are thus accounted for the foundation for mode tracing is the assumption of eigenvector consistency across parameter intervals used as a means of eigenpair reconciliation the resulting traced modes are smooth and their variations physically pertinent which is necessary in iterative schemes in which convergence may otherwise be jeopardized the proposal of an augmented modal assurance routine that potentially augments the assurance of tracing and therefore the maximum permissible parameter perturbation is given a threefold increase in the insight to consistency entails the notion of the routine is to forward and backward cast eigenvectors utilizing their derivatives in affine modeling a numerical example of the modes of a cyclic frame as functions of a cyclic distribution of membrane forces demonstrates the concepts and utility of the proposal

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

conserved growth in a restricted solid on solid model the surface dynamics of conserved growth in a restricted solid on solid rsos model is described a randomly deposited particle is allowed to migrate to the nearest site satisfying the rsos condition the surface width the correlation function and the structure factor measurements are consistent with the lai and das sarma 1991 and villain 1991 equation delta h delta t nu del 4h lambda del 2 del h 2 eta the physical origin of the nonlinearity is also discussed

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

eigenmode computation of cavities with perturbed geometry using matrix perturbation methods applied on generalized eigenvalue problems abstract generalized eigenvalue problems are standard problems in computational sciences they may arise in electromagnetic fields from the discretization of the helmholtz equation by for example the finite element method fem geometrical perturbations of the structure under concern lead to a new generalized eigenvalue problems with different system matrices geometrical perturbations may arise by manufacturing tolerances harsh operating conditions or during shape optimization directly solving the eigenvalue problem for each perturbation is computationally costly the perturbed eigenpairs can be approximated using eigenpair derivatives two common approaches for the calculation of eigenpair derivatives namely modal superposition method and direct algebraic methods are discussed in this paper based on the direct algebraic methods an iterative algorithm is developed for efficiently calculating the eigenvalues and eigenvectors of the perturbed geometry from the eigenvalues and eigenvectors of the unperturbed geometry

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

maximal and minimal height distributions of fluctuating interfaces maximal and minimal height distributions mahd mihd of two dimensional interfaces grown with the nonlinear equations of kardar parisi zhang kpz second order and of villain lai das sarma vlds fourth order are shown to be different two universal curves may be mahd or mihd of each class depending on the sign of the relevant nonlinear term which is confirmed by results of several lattice models in the kpz and vlds classes the difference between mahd and midh is connected with the asymmetry of the local height distribution a simple exactly solvable deposition erosion model is introduced to illustrate this feature the average extremal heights scale with the same exponent of the average roughness in contrast to other correlated systems generalized gumbel distributions do not fit those mahd and mihd nor those of edwards wilkinson growth

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

new method for eigenvector sensitivity analysis with repeated eigenvalues and eigenvalue derivatives this paper proposes a new method of eigenvector sensitivity analysis for real symmetric systems with repeated eigenvalues and eigenvalue derivatives the derivation is completed by using information from the second and third derivatives of the eigenproblem and is applicable to the case of repeated eigenvalue derivatives the extended systems with a nonsingular coefficient matrix are constructed to calculate the particular solutions an efficient approach is developed to compute the homogeneous solutions associated with the repeated eigenvalue derivatives different constraints for calculating the eigenvector sensitivities are derived and discussed four numerical examples are presented to demonstrate the validity of the proposed method

the surface statistics of a granular aggregate the problem of the surface fluctuations in a settled granular material is posed a simple model is given which describes the process by which a particle settles and comes to rest on the existing surface of the packing and from this a set of langevin equations for the fourier modes of the surface are derived these equations imply that the fourier amplitudes behave like the velocities of a set of independent brownian particles we show that this results in logarithmically divergent surface fluctuations if the flux of particles onto the surface is random the divergence being removed by a more accurate description of the settling material for example by having the granules fall through a sieve

lack of knowledge in structural model validation this paper deals with the structural modeling of a family of similar actual structures taking into account uncertainties and modeling errors only errors of the structural stiffness type are considered we develop a new theory in which what we call the lack of knowledge lok is defined through an internal variable whose upper and lower bounds are stochastic associated with each substructure two main questions are discussed the impact of the basic loks on the predicted structural response and the reduction of the basic loks through the use of additional information

accelerated iterative procedure for calculating eigenvector derivatives an iterative method for the computation of eigenvector derivatives of real valued symmetric systems has been proposed the process is shown to converge to the exact solution with any initial values and it is numerically stable however since the convergence rate is determined by the ratios of the eigenvalues the convergence rate will be prohibitively slow when the ratio is close to 1 two ef cient accelerated algorithms are presented numerical examples show that when the eigenvalue ratio is greater than 0 70 the computational effort in the iterative process is drastically reduced the procedure incorporates the method of frequency shift to deal with a singular stiffness matrix that also provides an additional improvement for the convergence when the eigenvalue ratios are extremely close to 1 the second generalized stiffness matrix inverse is suggested to reestablish a high convergence rate the procedure can be used as an exact as well as an approximate method it requires no more eigenvectors than those whose derivatives are to be calculated and can be applied to systems with repeated eigenvalues

optimal inertia lumping from modal mass matrices for structural dynamics an optimal inertia lumping formulation using modal mass matrices obtained from structural analysis is presented enabling recovery of lumped inertia information even when substructuring has been employed the method is implemented with a commercial optimization program and uses reduced mass and stiffness formulations employed in commercial finite element analysis programs the result is lumped inertia models of complex flexible bodies for use in structural dynamics including flexible multibody dynamic simulation a method for dealing with nondifferentiability of repeated eigenvalues that arise in the optimization process is derived and implemented in the proposed formulation numerical examples are presented to validate the method relative to conventional mass lumping approaches

improved nelson s method for computing eigenvector derivatives with distinct and repeated eigenvalues t he computation of derivatives of eigenvalues and eigenvectors with respect to model parameters is crucial for many applications 1 nelson 2 proposed an approach for general real eigensystemswith distinct eigenvalueswhere only the eigenvector of interest was required ojalvo 3 mills curran 4 dailey 5 and shaw and jayasuriya 6 developed nelson s method for real symmetric eigensystems with repeated eigenvalues for computation of a particular solution themethods described in 3 5 may fail in some circumstances other methods have been suggested by millscurran 4 and song et al 7 but they are difficult to implement when a system of linear equations is solved one hopes that the calculated solution is a close representation of the true solution the computed solution for a system with a large condition number may be inexact due to extreme sensitivity of the solution to small changes in its coefficient matrix and right column vectors 8 because some diagonal elements in the coefficient matrix for particular solutions are set to units its coefficients are not all of the same order of magnitude so the coefficient matrix has a large condition number in this note nelson s method is first extended to the case of repeated eigenvalues for the symmetric real eigensystems and its improvements are then presented the condition number of the coefficient matrix of the improved method is considerably reduced