on the hughes model for pedestrian flow the one dimensional case in this paper we investigate the mathematical theory of hughes model for the ow of pedestrians cf 17 consisting of a nonlinear conservation law for the density of pedestrians coupled with an eikonal equation for a potential modelling the common sense of the task we rst consider an approximation of the original model in which the eikonal equation is replaced by an elliptic approximation for such an approximated system we prove existence and uniqueness of entropy solutions in one space dimension in the sense of kru zkov 21 in which the boundary conditions are posed following the approach of bardos et al 7 we use bv estimates on the density and stability estimates on the potential in order to prove uniqueness furthermore we analyse the evolution of characteristics for the original hughes model in one space dimension and study the behaviour of simple solutions in order to reproduce interesting phenomena related to the formation of shocks and rarefaction waves the characteristic calculus is supported by numerical simulations

optimum sensitivity derivatives of objective functions in nonlinear programming the feasibility of eliminating second derivatives from the input of optimum sensitivity analyses of optimization problems is demonstrated this elimination restricts the sensitivity analysis to the first order sensitivity derivatives of the objective function it is also shown that when a complete first order sensitivity analysis is performed second order sensitivity derivatives of the objective function are available at little additional cost an expression is derived whose application to linear programming is presented

second order design sensitivity analysis of structural systems esign sensitivity analysis methods for calculation of first derivatives of structural performance with respect to design parameters have become quite well developed and applied 1 2 it is the purpose of this note to extend the adjoint method of design sensitivity analysis to calculation of second derivatives of performance with respect to design parameters the value of second deriva tive information in design optimization is well known using only first order gradient information optimization algorithms must resort to onedimensional search or heuristic step size selection methods if second derivative information is available at a reasonable computing cost much more powerful iterative optimization algorithms are possible 3 finite dimensional matrix structural models are considered here even though the methods presented can be extended to distributed parameter boundary value problem structural models structural design is presumed to be described by a vector of design parameters b b1 b2 bk t structural response is described by a vector of nodal displacement coordinates z zi z2 zn 7

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

sensitivity of optimum solutions of problem parameters solution of the optimum sensitivity problem yields the values of derivatives of the optimal objective function and design variables with respect to those physical quantities which were kept constant as problem parameters during optimization examples of these sensitivity derivatives might include derivatives of cross section al area and structural mass with respect to allowable stress and derivatives of fuel consumed and wing aspect ratio with respect to aircraft range derivation of the sensitivity equations that yield the sensitivity derivatives directly which avoids the costly and inaccurate perturb and reoptimize approach is discussed and solvability of the equations is examined the equations apply to optimum solutions obtained by direct search methods as well as those generated by procedures of the sequential unconstrained minimization technique sumt class applications are discussed for the use of the sensitivity derivatives in extrapolation of the optimal objective function and design variable values for incremented parameters optimization with multiple objectives and decomposition of large optimization problems several aspects of these applications and verification of the sensitivity equations are presented through numerical examples

efficient calculation of optimum design sensitivity two methods for obtaining the sensitivity of an optimized design to changes in problem parameters are presented the first order method uses gradient information to estimate the required sensitivity the second order method uses second derivatives of the design objective and constraint functions to provide a quadratic approximation to the new design which results from changing the specified parameter it is shown that the sensitivity of the optimized design may be discontinuous with respect to the new parameter this discontinuity is accounted for in the present methods these methods are compared to a previous approach based on the kuhn tucker necessary conditions for optimality it is shown that the present methods provide the appropriate sensitivity information in an efficient manner the methods are demonstrated by a structural synthesis example

one sided finite difference approximations suitable for use with richardson extrapolation new expressions for one sided finite difference approximations are proposed in these approximations the odd order error terms are eliminated while the even order terms are left to be taken care of by richardson extrapolation the effective local truncation error is shown to be less than for higher order one sided finite difference approximations but the solutions for a test problem are shown to have comparable accuracy for both approximations

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

a model for the formation and evolution of traffic jams in this paper we establish and analyze a traffic flow model which describes the formation and dynamics of traffic jams it consists of a pressureless gas dynamics system under a maximal constraint on the density and is derived through a singular limit of the aw rascle model from this analysis we deduce the particular dynamical behavior of clusters or traffic jams defined as intervals where the density limit is reached an existence result for a generic class of initial data is proved by means of an approximation of the solution by a sequence of clusters finally numerical simulations are produced

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

eigensolutions sensitivity for nonsymmetric matrices with repeated eigenvalues a perturbation algorithm is developed for evaluating eigenvalue and eigenvector directional derivatives of nonsymmetric defective matrices some properties are recalled in particular chains of generalized right and left eigenvectors and their orthogonal properties are defined small perturbations of the matrix are then considered an asymptotic expansion of the eigensolutions of the perturbed problem is obtained in terms of noninteger powers of the perturbation parameter particular attention is devoted to the eigenvectors of the perturbed system and to the strong coupling that occurs between the chains

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

skew hermitian based iterations for nine point approximations of convection diffusion problems we study skew hermitian based splitting methods for the iterative solution of nonsymmetric linear systems arising from high order compact hoc approximations of two dimensional convection diffusion problems such discretisations lead to nine point system matrices with block tridiagonal structures the nonsymmetric discretisation matrix for a constant coefficient grid aligned flow problem is shown to be positive definite and thus hermitian and skew hermitian splitting hss methods can be considered for the solution of the corresponding linear system for the solution of this linear system using the hss iteration we derive an analytical expression for the optimal value for the upper bound of the contraction factor comparison of the hss iteration with triangular and skew hermitian splitting tss methods and their corresponding block variants btss are carried out for various test problems cpu timings for the different splitting methods using experimentally determined optimal acceleration parameters are also given our results show that the combination of high order compact discretisations and skew hermitian based iterative methods for solving the corresponding linear systems provide efficient procedures for approximating the solutions of two dimensional convection diffusion equations

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

higher order operator splitting methods via zassenhaus product formula theory and applications abstract in this paper we contribute higher order operator splitting methods improved by zassenhaus product we apply the contribution to classical and iterative splitting methods the underlying analysis to obtain higher order operator splitting methods is presented while applying the methods to partial differential equations the benefits of balancing time and spatial scales are discussed to accelerate the methods the verification of the improved splitting methods are done with numerical examples an individual handling of each operator with adapted standard higher order time integrators is discussed finally we conclude the higher order operator splitting methods

a collocation method for the gurtin maccamy equation with finite life span a collocation method along the characteristics for a stiff problem arising from population dynamics is proposed and analyzed it is a fourth order implicit runge kutta method of two stage to the integration of the ode along the characteristics whose collocation points are zeros of the linearly transformed legendre monic polynomial nonnegativity of the numerical solutions is shown the stability of the method is discussed it is proven that the scheme is convergent at a fourth order rate in the maximum norm several numerical examples are presented

structural design sensitivity analysis with general boundary conditions dynamic problem this paper presents and analyses methods for design sensitivity analysis of static response for structural systems when general boundary conditions are used during the analysis phase the design sensitivity analysis methods where the boundary conditions are used to reduce the equilibrium equation have been presented earlier in such procedures the reduced stiffness matrix is non singular this fact is used to show differentiability of the displacement vector when reduction procedures are not used the global structural stiffness matrix may be singular in such a case differentiability of the displacement vector may be questionable however it is shown that the displacement vector is still differentiable several forms of the structural design sensitivity equations are presented although these equations are derived with lagrange multiplier treatment of general boundary conditions it is shown that they are applicable with any treatment of general boundary conditions for example they can also be used with co ordinate reduction schemes therefore design sensitivity analysis can be carried out with any existing finite element analysis computer code moreover the procedures presented can be used to calculate sensitivity of a force of constraint can that is desired the formulations are kept quite general such that design dependent boundary conditions can be routinely treated an example is used to illustrate the procedures

numerical solutions of a hyperbolic differential integral equation abstract investigated is the method of lines and a corresponding numerical method for the solution of a hyperbolic partial differential integral equation convergence theorems and results of numerical experiments are presented

an upwind scheme for a nonlinear hyperbolic integro differential equation with integral boundary condition abstract we analyze an upwind method for a nonlinear hyperbolic integro differential equation with an integral boundary condition the problem considered describes the evolution of the age structure of a population the analysis is carried out employing an abstract theory of discretizations based on the notion of stability thresholds and on a result due to stetter

an upwind scheme for a nonlinear model in age structured population dynamics abstract the large time behavior of numerical solutions for a model describing age structured population dynamics is investigated an upwind scheme is considered for the approximation the optimal rate of convergence of the scheme is demonstrated for the maximum norm and the stability of the method is discussed numerical experiments are carried out

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

second order characteristic schemes in time and age for a nonlinear age structured population model in this paper we analyze two new second order characteristic schemes in time and age for an age structured population model with nonlinear diffusion and reaction by using the characteristic difference to approximate the transport term and the average along the characteristics to treat the nonlinear spatial diffusion and reaction terms an implicit second order characteristic scheme is proposed to compute the nonlinear approximation system an explicit second order characteristic scheme in time and age is further proposed by using the extrapolation technique the global existence and uniqueness of the solution of the nonlinear approximation scheme are established by using the theory of variation methods schauder s fixed point theorem and the technique of prior estimates the optimal error estimates of second order in time and age are strictly proved for both the implicit and the explicit characteristic schemes numerical examples are given to illustrate the performance of the methods

a comparison of uniformly convergent difference schemes for two dimensional convection diffusion problems galerkin and petrov galerkin finite element methods are used to obtain new finite difference schemes for the solution of linear two dimensional convection diffusion problems numerical estimates are made of the rates of convergence of these schemes uniformly with respect to the perturbation parameter and these uniform rates are shown to compare favourably with those of established methods

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

mixed approximation of a population diffusion equation abstract a numerical method is proposed to approximate the solution of a nonlinear and nonlocal system of integro differential equations describing age dependent population dynamics with spatial diffusion a finite difference method along the characteristic age time direction combined with mixed finite elements in the spatial variable is used for the approximation optimal order error estimates are derived for the relevant variables using nonnegativity of the discrete solution a stability of the method is also proved

age time continuous galerkin methods for a model of population dynamics continuous galerkin finite element methods in the age time domain are proposed to approximate the solution to the model of population dynamics with unbounded mortality coefficient function stability of the method is established and a priori l 2 error estimates are obtained treatment of the nonlocal boundary condition is straightforward in this framework the approximate solution is computed strip by strip marching in time some numerical examples are presented

derivation and significance of second order modal design sensitivities if the system is considered to be unsatisfactory in some way e g a natural frequency to the system coinciding with a constant shaft speed then it is of interest to find the design change which will give maximum benefit with minimum disturbance to the system this may be quantified in terms of design sensitivities of the structure fox and kapoor derived an expression for the sensitivity of an eigenvalue for self adjoint systems i e where the left and right eigenvectors are simple transposes typical of conservative systems and duncan forms garg extended this result to apply to non self adjoint systems e g where dissipative and gyroscopic effects are present the analysis for eigenvectors was presented by plaut and huseyin for the general case most recently nelson has provided an alternative approach to eigenvector sensitivity the sensitivities derived by all of the above methods are expressed in the form of first derivatives and hence are properties local to the original solution of the system in a recent paper by vanhonacker there is a clear implication that these first derivatives may be extrapolated for gross system changes clearly this assumption can be valid only if the value of the second derivative is negligible

structural design sensitivity analysis with generalized global stiffness and mass matrices first and second order derivatives of measures of structural response with respect to design variables are calculated using the generalized global stiffness and mass matrix formulation of structural equations the adjoint variable method is extended for first order design sensitivity calculation and a mixed direct differentiationadjoint variable method is employed to calculate second derivatives of static structural response measures the variational or virtual work form of the structural equations is shown to be well suited for design sensitivity analysis with generalized global stiffness and mass matrices avoiding the requirement for explicit reduction of the matrices the method is illustrated using a simple structure with a multipoint constraint

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

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

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

discrete kinetic theory for 2d modeling of a moving crowd application to the evacuation of a non connected bounded domain abstract this paper concerns the mathematical modeling of the motion of a crowd in a non connected bounded domain based on kinetic and stochastic game theories the proposed model is a mesoscopic probabilistic approach that retains features obtained from both micro and macro scale representations pedestrian interactions with various obstacles being managed from a probabilistic perspective a proof of the existence and uniqueness of the proposed mathematical model s solution is given for large times a numerical resolution scheme based on the splitting method is implemented and then applied to crowd evacuation in a non connected bounded domain with one rectangular obstacle the evacuation time of the room is then calculated by our technique according to the dimensions and position of a square shaped obstacle and finally compared to the time obtained by a deterministic approach by means of randomly varying some of its parameters

a semi analytical fourier spectral method for the swift hohenberg equation abstract the swift hohenberg sh equation has been widely used as a model for the study of pattern formation the sh equation is a fourth order nonlinear partial differential equation and cannot generally be solved analytically therefore computer simulations play an essential role in understanding of nonequilibrium processing the aim of this paper is to present first and second order semi analytical fourier spectral methods as an accurate and efficient approach for solving the sh equation the methods are based on the operator splitting method and are to split the sh equation into linear and nonlinear subequations the linear and nonlinear subequations have closed form solutions in the fourier and physical spaces respectively the methods are simple to implement and computationally cheap to achieve high order time accuracy numerical experiments are presented demonstrating the accuracy and efficiency of proposed methods

on the approximation of the lotka mckendrick equation with finite life span abstract we consider the linear lotka mckendrick equation and discuss in detail how to solve the problem of the breakdown of the usual finite difference methods when the mortality is unbounded usual error bounds require some derivative of the mortality rate to be bounded across all ages our approach works for a model class of mortality rates and we show that not all methods are compatible with any mortality function

a variable time step method for an age dependent population model with nonlinear diffusion we propose a method for solving a model of age dependent population diffusion with random dispersal this method unlike previous methods allows for variable time steps and independent age and time discretizations we use a moving age discretization that transforms the problem to a coupled system of parabolic equations the system is then solved by backward differences in time and a galerkin approximation in space the equations that need to be solved at each step treat each age group separately a priori l2 error estimates are obtained by an energy analysis these estimates are superconvergent in the age variable we present a postprocessing technique which capitalizes on the superconvergence

operator splitting methods via the zassenhaus product formula abstract in this paper we contribute an operator splitting method improved by the zassenhaus product zassenhaus products are of fundamental importance for the theory of lie groups and lie algebras while their applications in physics and physical chemistry are important novel applications in cfd computational fluid dynamics arose based on the fact that their sparse matrices can be seen as generators of an underlying lie algebra we apply this to classical splitting and the novel zassenhaus product formula the underlying analysis for obtaining higher order operator splitting methods based on the zassenhaus product is presented the benefits of dealing with sparse matrices given by spatial discretization of the underlying partial differential equations are due to the fact that the higher order commutators are very quickly computable their matrix structures thin out and become nilpotent when applying these methods to convection diffusion reaction equations the benefits of balancing time and spatial scales can be used to accelerate these methods and take into account these sparse matrix structures the verification of the improved splitting methods is done with numerical examples finally we conclude with higher order operator splitting methods

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

the baker campbell hausdorff formula and nested commutator identities the coefficients of the baker campbell hausdorff expansion are calculated by using various methods comparison of the results yields several remarkable identities satisfied by multiple commutators which in turn allow us to greatly simplify the form of the expansion up to order eight

endemic thresholds and stability in a class of age structured epidemics an age structured epidemic model is analyzed when the fertility mortality and removal rates depend on age for certain general forms of the force of infection terms endemic threshold criteria are derived and the stability of steady state solutions is determined the relation between age structured models of this type and catalytic curve models of epidemics is derived the possibility of identifying vertically transmitted diseases from the catalytic curve is demonstrated

convergence of a splitting method of high order for reaction diffusion systems in this article we prove the convergence of a splitting scheme of high order for a reaction diffusion system of the form u t m u f u 0 where m is an m m matrix whose spectrum is included in rz 0 this scheme is obtained by applying a richardson extrapolation to a strang formula

global solution to the cauchy problem for discrete velocity models of vehicular traffic abstract this paper offers the proof of the existence for large times of the cauchy problem for a class of vehicular traffic models with discrete velocities derived from the mathematical tools of the kinetic theory for active particles this approach has the ability to capture some basic aspects concerning the complexity of the systems under consideration through an appropriate modeling via stochastic games of the interactions at the microscopic scale the proof is related to specific aspects of the real system and hopefully can contribute towards improving the modeling approach

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

long time stability of numerical solutions to gurtin maccamy equations by method of characteristics a backward finite difference scheme along the characteristics is considered to approximate the solution of a nonlinear and nonlocal system of integro differential equations that models the dynamics of a single population the long time stability of the numerical solution is proved the optimal rate of convergence of the scheme is demonstrated in the maximum norm results from numerical experiments are presented

second order accurate difference methods for a one sex model of population dynamics a first order hyperbolic equation of age and time variables describes a one sex model of population dynamics the second order finite difference method in the characteristic age time direction is used to approximate simultaneously both the age distribution u x t of the solution and the total population p t algorithms and error analysis are given for the second order convergence rate

incremental unknowns for solving partial differential equations incremental unknowns have been proposed in t as a method to approximate fractal attractors by using finite difference approximations of evolution equations in the case of linear elliptic problems the utilization of incremental unknown methods provides a new way for solving such problems using several levels of discretization the method is similar but different from the classical multigrid method in this article we describe the application of incremental unknowns for solving laplace equations in dimensions one and two we provide theoretical results concerning two level approximations and we report on numerical tests done with multi level approximations

an age structured population model with state dependent delay derivation and numerical integration c and j wu abstract we present an age structured population model that accounts for complex life cycles and competition for resources limiting the transition to maturity taking all of these into account leads to a new mathematical model with a state dependent delay that cannot be analyzed using the established framework of functional differential equations or directly simulated by standard numerical schemes for age structured populations in this paper we present the derivation of the model and a numerical scheme to integrate the equations convergence of the method is proven by verifying first consistency and then stability the difficulties involved in this proof as well as in the implementation of the scheme stem from the state dependent delay term the numerical scheme is shown to be of order 2 in the given norm and at least of order 3 in the supremum norm over the mesh point values

sensitivity analysis of distributed parameter systems a method is developed for the calculation of sensitivity coefficients of general distributed parameter systems to the variation of spatially and temporally varying parameters appearing in the system equations and initial and boundary conditions

discontinuous galerkin methods for a model of population dynamics with unbounded mortality discontinuous galerkin finite element methods in age time domain are proposed to approximate a possibly nonsmooth solution to the model of population dynamics with unbounded mortality coefficient function the strong stability is established and a priori l 2 error estimates are obtained we show that the error estimates are optimal both in rate and in regularity away from the maximum age nonnegativity of the solution is also shown the scheme is explicit and the solution is computed strip by strip marching in time several numerical examples are presented

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

sensitivity analysis and optimization of aeroelastic stability abstract the present paper deals with problems concerning sensitivity analysis and optimization of aeroelastic stability of distributed systems in sections 1 3 the optimization problem of aeroelastic stability of a slender wing in incompressible flow is formulated and solved the optimal control function h y to be determined represents the mass and stiffness distribution along a wing span sensitivity analysis of flutter systems is developed first the gradients of flutter and divergence critical speeds are derived and necessary optimality conditions are obtained then the solution technique is described and numerical results are presented in section 4 the problem of determining the optimal distribution of nonstructural mass along the wing span is considered the optimality conditions are established and the bang bang optimal distributions are obtained

nonlinear singular perturbation problems and one sided difference schemes we consider the boundary value problem varepsilon y a y y b x y f x 1 leqq x leqq 1 for y 1 and y 1 given and b x 0 equiv 0 b y x y geqq 0 we construct a simple difference scheme approximating varepsilon u xx a u u x b x u f x u t for t geqq 0 1 leqq x leqq 1 with the same boundary conditions and an arbitrary initial guess as the numerical artificial time approaches infinity we prove convergence to a unique steady state solution as the mesh size delta x and the parameter varepsilon approach zero in any order we prove convergence to the solution of the corresponding o d e we use one sided differences to approximate the transport term the method applies equally well to multidimensional analogues we present numerical results verifying the theory

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

weighted sequential splittings and their analysis operator splitting is a widely used procedure in the numerical solution of parabolicproblems several splitting methods have been constructed and used in the fields of applied mathematics in this paper the weighted sequential splittings are introduced and their local splitting error is analyzed new results are presented for the case where the subproblems of the weighted splitting schemes are solved numerically the results are illustrated with test examples for ordinary differential systems

local rbf method for multi dimensional partial differential equations abstract in this paper a local meshless differential quadrature collocation method is utilized to solve multi dimensional reaction convection diffusion pdes numerically in some cases global version of the meshless method is considered as well the meshless methods approximate solution on scattered and uniform nodes in both local and global sense in the case of convection dominated pdes the local meshless method is coupled with an upwind technique to avoid spurious oscillations for this purpose a physically motivated local domain is utilized in the flow direction both regular and irregular geometries are taken into consideration numerical experiments are performed to demonstrate effective applications and accuracy of the meshless method on regular and irregular domains

high order sensitivity analysis of complex coupled systems the sobieszczanski sobieski 1988 algorithm is extended to include second and higher order derivatives while retaining the obviation of finite differencing of the system analysis this is accomplished by means of a recursive application of the same implicit function theorem as in the original algorithm in optimization the computational cost of the higher order derivatives is relative to the aggregate cost of analysis together with a repetition of the first order sensitivity analysis as often as is required to produce the equivalent information by successive linearizations within move limits

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

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

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

variational iteration method for nonlinear age structured population models this paper investigates nonlinear age structured population models the numerical solutions are obtained by using the variational iteration method and are of high accuracy two numerical examples are presented to illustrate the strength of the method

splitting methods for the numerical approximation of some models of age structured population dynamics and epidemiology splitting methods for nonlinear models arising from population dynamics and epidemiology are described and analyzed a backward finite differencing along the characteristic is used for the approximation it is shown that the schemes are convergent at first order rate in the maximum norm the stability of the methods is discussed several numerical examples are presented

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

a modified crank nicolson scheme with incremental unknowns for convection dominated diffusion equations a modified crank nicolson scheme based on one sided difference approximations is proposed for solving time dependent convection dominated diffusion equations in two dimensional space the modified scheme is consistent and unconditionally stable a priori l 2 error estimate for the fully discrete modified scheme is derived with the use of the incremental unknowns preconditioner at each time step a comparison among several classical numerical schemes has been made and numerical results confirm stability and efficiency of the modified crank nicolson scheme

high order discontinuous galerkin methods for a class of transport equations with structured populations in this paper we analyze a discontinuous galerkin finite element method for approximating solutions to transport equations with certain nonlinearities we consider models for age structured populations allowing for a nonlinear removal rate with non local boundary conditions on the in flow boundary the method employs a stabilizing term over the interior edges allowing for convergence in a stronger than usual norm we establish convergence rates for general higher order basis functions and provide numerical examples consistent with this result

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

an explicit extrapolated box scheme for the gurtin maccamy equation abstract an explicit extrapolated box method is considered for a nonlinear partial integrodifferential equation the gurtin maccamy equation subject to a nonlocal boundary condition this problem describes the evolution in time of the age structure of a population the consistency stability and convergence properties of the method are studied and its second order accuracy is demonstrated both analytically and numerically

spatial filter averaging approach of probabilistic method to linear second order partial differential equations of the parabolic type a backward in time probabilistic method with spatial filter averaging is presented to solve linear second order partial differential equations of the parabolic type an advantage of this methodology is that while forward methods are subject to region with loss of density of particles and hence loss of spatial resolution of the solution the solution given by backward methods is given on any desired grid however traditional backward time probabilistic method using monte carlo averaging are computationally expensive we prove a convergence result and present several examples the method leads to important improvement in computational efficiency and is expected to perform well to solve high dimensional problems where a solution is needed on a large grid

generation of finite difference formulas on arbitrarily spaced grids on etablit des recurrences simples pour le calcul des poids dans les formules aux differences finies compactes pour les derivees de tous ordres avec une precision d ordre arbitraire sur des grilles a une dimension d espacement arbitraire

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

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

a note on the zassenhaus product formula we provide a simple method for the calculation of the terms cn in the zassenhaus product ea b ea eb n 2 ecn for noncommuting a and b this method has been implemented in a computer program furthermore we formulate a conjecture on how to translate these results into nested commutators this conjecture was checked up to order n 17 using a computer

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

time evolution of continuous time quantum walks on dynamical percolation graphs we study the time evolution of continuous time quantum walks on randomly changing graphs at certain moments edges of the graph appear or disappear with a given probability as in percolation we treat this problem in a strong noise limit we focus on the case when the time interval between the subsequent changes of the graph tends to zero we derive explicit formulae for the general evolution in this limit we find that the percolation in this limit causes an effective time rescaling independently of the graph and the initial state of the walk the time is rescaled by the probability of keeping an edge both the individual trajectories for a single system and average properties with a superoperator formalism are discussed we give an analytical proof for our theorem and we also present results from the numerical simulations of the phenomena for different graphs we also analyze the effect of finite step size on the evolution

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