topology optimization for effective energy propagation in rate independent elastoplastic material systems abstract a topology optimization framework for effective energy management in dynamically loaded structures with rate independent elastoplastic material behavior is presented the elastoplastic constitutive equations are integrated using an implicit numerical scheme based on a multi level newton procedure this primal analysis procedure requires the computation of the incremental algorithmic consistent tangent operator derived from the mappings that enforce the evolving constitutive relations for our gradient based optimization we develop discrete adjoint sensitivity expressions finite element implementation is discussed and the optimization framework is exemplified via the design of three dimensional structures subjected to dynamic loads

sensitivity analysis of nonlinear material and its application to shape optimization a direct sensitivity analysis technique for the path dependent nonlinear problem based on the updated lagrangian formulation is suggested in which the elastoplastic material of bilinear constitutive law is assumed for such a material property the design sensitivities are discontinuous at the material transition points the incremental equilibirum equation is differentiated with respect to the design variables to derive the sensitivity equations which ar solved during the newton raphson iterative scheme of response analysis

shape design sensitivity analysis for geometrically and materially nonlinear problems by the boundary element method abstract this paper is concerned with the determination of design sensitivity coefficients dscs for fully nonlinear problems i e problems with both material and geometric nonlincarities in solid mechanics the clastic strains arc assumed to be small but the nonclastic plastic or viscoplustic strains as well as rotations can be arbitrarily large dscs here refer to rates of change of response variables such as displacements or stresses in a deforming solid body with respect to parameters called design variables that define the initial undeformed shape of the body these dscs are history dependent the direct differentiation approach dda of the relevant derivative boundary element method dbem formulation is employed to obtain the dscs a general computer program is developed for the analysis of two dimensional plane strain and plane stress problems numerical results are obtained for illustrative problems and arc compared against direct solutions the results from the two methods agree almost perfectly for these problems this research has potential applications in the optimization of manufacturing processes such as extrusion rolling or forging

homoclinic tangencies and hyperbolicity for surface diffeomorphisms we prove here that in the complement of the closure of the hyperbolic surface diffeomorphisms the ones exhibiting a homoclinic tangency are c 1 dense this represents a step towards the global understanding of dynamics of surface diffeomorphisms

formulation methods of geometric and material nonlinearity problems a concise survey of formulation methods of geometric and material nonlinearity problems is given differences between existing formulations in the literature are discussed a consistent lagrangian and updated lagrangian formulations are derived from the energy balance equation transformed to the proper reference configuration differences between the existing formulations and similar ones in the literature are found to be in specific geometric nonlinear terms in the final incremental equation as well as in the definition of the load increment vector specific forms of constitutive equations for elastic and elastoplastic materials are presented some of the forms used by other authors in application to elastoplastic materials are shown to be inconsistent sample numerical examples intended for checking some particular points in the formulation are presented

design sensitivity analysis and optimization of non linear transient dynamics part ii configuration design for con guration design of non linear dynamic structures such as crashworthiness design a continuumbased con guration design sensitivity analysis dsa and optimization methods are developed the same transient dynamic analysis method used in part i of this paper is employed here the elastic plastic material and nite strain and rotation e ect are considered the rst order variations of the energy forms load form and kinematic and structural responses with respect to con guration design variables are derived for the con guration design both the shape and orientation variations contribute to the rst order variation of the equations of motion the angular design velocity eld associated with euler angles is used to represent the orientation variation to obtain accurate design sensitivity results like part i of the paper the updated lagrangian formulation and direct di erentiation method are used for dsa for the path dependent problem for the updated lagrangian formulation the design velocity eld that de nes the mapping between the initial and perturbed designs should also be updated at each con guration numerical implementation of con guration dsa and optimization is carried out for xed time steps using dyna3d and the modi ed feasible direction mfd method it is observed that the proposed dsa method yields better sensitivity results than the nite di erence method for the highly non linear problem design optimization is carried out using the design sensitivity information copyright 2000 john wiley sons ltd

design sensitivity analysis for rate independent elastoplasticity abstract a new incremental direct differentiation method for design sensitivity analysis of structures with rate independent elastoplastic behavior is presented we formulate analytical sensitivity expressions that are consistent with numerical algorithms for elastoplasticity that use implicit methods to integrate the constitutive equations and return mappings to enforce the consistency conditions the sensitivity expressions can be evaluated with only a modest increase in computational expense beyond the cost of simulation combined with the inherent advantages of implicit integration strategies this represents a significant improvement over previous sensitivity formulations for history dependent materials first order sensitivity expressions involving the complete set of design variables including shape design variables are derived for a generic response functional the reduced form of the consistent tangent stiffness matrix obtained at the end of each time or load step in the finite element procedure is used to update the response sensitivities for that time step no iterations are needed in the sensitivity computations a numerical example demonstrates the accuracy and efficiency of the new sensitivity analysis method for an elastoplastic analysis problem explicit sensitivities from the new method are confirmed by finite difference estimates

sensitivity analysis in plane stress elasto plasticity and elasto viscoplasticity abstract development of computational models for plane stress elasto plasticity is known to make some difficulties non existent in 3d and plane strain formulations tedious derivation of plane stress algorithmic tangent matrices which cannot be directly obtained as a special case of 3d matrices may serve here as an example in this paper specific features of plane stress computations are discussed in the context of parameter sensitivity analysis elasto plastic material model with isotropic kinematic hardening as well as two viscoplastic models are considered explicit expressions for sensitivity gradients are derived instructive examples solved by fem illustrate the paper

design sensitivity analysis of coupled thermoviscoelastic systems abstract both the direct differentiation and adjoint variable methods for design sensitivity analysis of transient dynamic arbitrarily nonlinear thermoviscoelastic coupled systems are presented in this paper the approach is based on the thermodynamic description of simple materials due to coleman large strains as well as arbitrary material nonlinearities are accounted for in the derivations the domain parametrization or reference volume concept is used allowing a uniform treatment of both the shape and sizing design sensitivity analysis analytical examples demonstrating the use of the derived equations are given this approach provides a solid basis for discretization of the developed equations for subsequent use in numerical computations

sensitivity of forming processes to shape parameters abstract shape sensitivity analysis is presented for metal forming processes described in terms of the flow approach 1 the direct differentiation method is employed to derive the sensitivity expressions the continuum approach is adopted so that both the equilibrium equations and response functional are differentiated before discretization the necessary derivatives with respect to shape design variables are calculated making use of the framework already available for isoparametric elements control volume approach in finite element implementation a system of equations is obtained which has the same system matrix as the equilibrium problem the right hand side is calculated by partially differentiating the internal and external forces with respect to the design parameters the procedure is illustrated by calculating sensitivities of some independent and dependent variables with respect to the die angle in an extrusion problem and to the roll radius in a plane rolling

adjoint sensitivity analysis for nonlinear dynamic thermoelastic systems an adjoint approach is presented to formulate explicit sensitivities for a general functional associated with a nonlinear dynamic thermoelastic system uncoupled thermoelastic response and small deformation behavior are assumed the formulation uses the lagrange multiplier method to define an adjoint system the convolution operator to incorporate transient effects and domain parameterization to describe shape variations variations of the functional are expressed in explicit form with respect to perturbations of the design fields structural shape material properties applied loads prescribed boundary conditions and initial conditions the functional is defined in terms of the design fields and the implicit response fields displacement temperature strain temperature gradient stress heat flux vector reaction force and reaction surface flux in an example problem the finite element method is used to evaluate the real and adjoint responses and the shape sensitivities for a tank design problem the tank is modeled as a nonlinear quasistatic uncoupled thermoelastic system

fluid queues with heavy tailed m g input we consider a fluid queue fed by several heterogeneous m g input processes with regularly varying session lengths under fairly mild assumptions we derive the exact asymptotic behavior of the stationary workload distribution in addition we obtain several asymptotic results for the transient workload distribution which are applied to obtain a conditional limit theorem for the most probable time to overflow the results are strongly inspired by the large deviations idea that overflow is typically due to some minimal combination of extremely long concurrent sessions causing positive drift the typical configuration of long sessions is identified through a simple integer program paving the way for the exact computation of the asymptotic workload behavior the calculations provide crucial insight in the typical overflow scenario

first and second order sensitivity analysis of linear and nonlinear structures this paper employs the principle of virtual work to derive sensitivity derivatives of structural response with respect to stiffness parameters using both direct and adjoint approaches the computations required are based on additional load conditions characterized by imposed initial strains body forces or surface tractions as such they are equally applicable to numerical or analytical solution techniques the relative efficiency of various approaches for calculating first and second derivatives is assessed it is shown that for the evaluation of second derivatives the most efficient approach is one that makes use of both the first order sensitivities and adjoint vectors two example problems are used for demonstrating the various approaches

implicit bem formulations for usual and sensitivity problems in elasto plasticity using the consistent tangent operator concept this paper presents boundary element method bem formulations for usual and sensitivity problems in small strain elasto plasticity using the concept of the local consistent tangent operator cto usual problems here refer to analysis of nonlinear problems in structural and solid continua for which simo and taylor first proposed the use of the cto within the context of the finite element method fem a new implicit bem scheme for such problems using the cto is presented first a formulation for sensitivity analysis follows it is shown that the sensitivity of the strain increment associated with an infinitesimal variation of some design parameter solves a linear problem which is governed by the converged value of the same global cto as the one that appears in the usual problem numerical results for both usual and sensitivity problems are shown for a one dimensional example they demonstrate the effectiveness of the present approach in particular accurate sensitivities with respect to material parameters e g exponent of the power type hardening law are obtained even with few integration cells and for large load increments

consistent tangent operators for rate independent elastoplasticity abstract it is shown that for problems involving rate constitutive equations such as rate independent elastoplasticity the notion of consistency between the tangent stiffness operator and the integration algorithm employed in the solution of the incremental problem plays a crucial role in preserving the quadratic rate of asymptotic convergence of iterative solution schemes based upon newton s method within the framework of closest point projection algorithms a methodology is presented whereby tangent operators consistent with this class of algorithms may be systematically developed to wit associative j 2 flow rules with general nonlinear kinematic and isotropic hardening rules as well as a class of non associative flow rules are considered the resulting iterative solution scheme preserves the asymptotic quadratic convergence characteristic of newton s method whereas use of the socalled elastoplastic tangent in conjunction with a radial return integration algorithm a procedure often employed results in newton type of algorithms with suboptimal rate of convergence application is made to a set of numerical examples which include saturation hardening laws of exponential type

robust criterion for the existence of nonhyperbolic ergodic measures we give explicit c 1 open conditions that ensure that a diffeomorphism possesses a nonhyperbolic ergodic measure with positive entropy actually our criterion provides the existence of a partially hyperbolic compact set with one dimensional center and positive topological entropy on which the center lyapunov exponent vanishes uniformly

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

shape sensitivity in elasto plastic computations an algorithm for design sensitivity analysis of elasto plastic response for the case of design dependent geometry is discussed the domain parametrization approach is employed some important implications for the concept of isoparametric element formulation are indicated the idea of shape sensitivity computations is similar to that of sensitivity with respect to material parameters however derivation of additional geometric matrices and integral expressions is necessary a possible discontinuity of sensitivity solutions is indicated and it is shown that the algorithm handles it correctly a plane stress computational example illustrates the application of the method

comparison between the variational and implicit differentiation approaches to shape design sensitivities the most commonly used approach to the shape design sensitivity problem results from the implicit differentiation of the discretized equilibrium equations the most general implementation of this technique requires that finite differences be used to differentiate the element stiffness matrices proper choice of the step size is necessary to obtain high levels of accuracy and to avoid round off errors furthermore since it is necessary to operate on the element matrices this method is difficult to implement into a general purpose finite element program a more recent shape design sensitivity formulation based upon variational calculus avoids having to differentiate the discretized equations and results in an analytical expression for the derivative the approach is based upon the total derivative of the variational state equation and uses an adjoint variable technique for design sensitivity analysis only structural response data on the boundary of the structure is necessary thereby making implementation into a general purpose program less difficult this paper attempts to compare the two different techniques and point out the similarities two test problems are shown to demonstrate the accuracy of the variational approach

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

nonlinear structural design sensivitity analysis for path dependent problems part 1 general theory abstract two procedures direct differentiation and adjoint structure for design sensitivity analysis of nonlinear structural systems with path dependent effects are described in this paper continuum formulation is used and both geometric and material nonlinearities are included in the derivations incremental nonlinear analysis based on the total lagrangian concept is adopted the reference volume concept is used to unify the shape and nonshape design problems to show the generality of the theory the derived expressions are reduced to some special cases that have been treated previously in the literature in part 2 of the paper a few analytical example problems will be solved to demonstrate the use of the design sensitivity analysis theory the theory has also beendiscretized elsewhere using finite element approximations for implementation into structural analysis programs these developments offer a firm foundation for design optimization of complex structural systems

topology and shape optimization for elastoplastic structural response abstract it is a common practice to base both material topology optimization as well as a subsequent shape optimization on linear elastic response however in order to obtain a realistic design it might be essential to base the optimization on a more realistic physical behavior i e to consider geometrically or and materially nonlinear effects in the present paper an elastoplastic von mises material model with linear isotropic hardening softening for small strains is used the objective of the design problem is to maximize the structural ductility in the elastoplastic range while the mass in the design space is prescribed with respect to the specific features of either topology or shape optimization for example the number of optimization variables or their local global influence on the structural response different methods are applied for topology optimization problems the gradient of the ductility is determined by the variational adjoint approach in shape optimization the derivatives of the state variables with respect to the optimization variables are evaluated analytically by a variational direct approach the topology optimization problem is solved by an optimality criteria oc method and the shape optimization problem by a mathematical programming mp method in topology optimization a geometrically adaptive procedure is additionally applied in order to increase the efficiency and to avoid artificial stress singularities the procedures are verified by 2d design problems under plane stress conditions

reduced basis technique for calculating sensitivity coefficients of nonlinear structural response an efficient reduced basis technique is presented for calculating the sensitivity of nonlinear structural response to variations in the design variables the structure is discretized by using two field mixed finite element models the vector of structural response and its sensitivity coefficients derivatives with respect to design variables are each expressed as a linear combination of a small number of basis or global approximation vectors the bubnov galerkin technique is then used to approximate each of the finite element equations governing the response and the sensitivity coefficients by a small number of algebraic equations in the amplitudes of these vectors the path derivatives derivatives of the response vector with respect to path parameters e g load parameters are used as basis vectors for approximating the response a combination of the path derivatives and their derivatives with respect to the design variables is used for approximating the sensitivity coefficients the potential of the proposed technique is discussed and its effectiveness is demonstrated by means of numerical examples of laminated composite plates subjected to mechanical and thermal loads

design variations of nonlinear elastic structures subjected to follower forces abstract the effect of incorporating the load stiffness matrix due to follower forces on design variations of nonlinear elastic structures is studied for the first time in this paper in general presence of the follower forces leads to unsymmetric equations for analysis and design variation calculations which are expensive to solve however the case of uniform pressure that remains normal to the deformed surface can be formulated to give only symmetric contribution to the tangent stiffness matrix analysis is focused on this case and an explicit expression for the load stiffness matrix is derived numerical examples are used to calculate design variations with and without the load stiffness matrix included in calculations it is shown that inclusion of the load stiffness matrix gives much more accurate design variations therefore it is recommended to include it in the design optimization process for better numerical performance and optimum solutions

shape and non shape structural sensitivity analysis for problems with any material and kinematic non linearity abstract the general problem of sizing material selection load parameter and shape sensitivity of non linear structural response is considered the non linear sensitivity path is followed by an incremental strategy employing the continuum based updated lagrangian description the resulting finite element equations are derived both the direct differentiation and adjoint structure strategies are considered the merits of using the consistent tangent matrix for sensitivity calculations are indicated the formulation employs the so called reference volume concept and may be seen as a further attempt to unify the shape and non shape sensitivity analysis procedures

variational approach by means of adjoint systems to structural optimization and sensitivity analysis i variation of material parameters within fixed domain abstract for a linear elastic structure the first variation of an arbitrary stress strain and displacement functionals corresponding to variation of material parameters within specified domain is derived by using the solution for primary and adjoint systems this variation is of fundamental importance in sensitivity analysis optimal design and identification problems simple examples of optimal stiffness design and identification of stiffness parameters in beams are presented

nonlinear dynamic response structural optimization using equivalent static loads it is well known that nonlinear dynamic response optimization using a conventional optimization algorithm is fairly difficult and expensive for the gradient or non gradient based optimization methods because many nonlinear dynamic analyses are required therefore it is quite difficult to find practical large scale examples with many design variables and constraints for nonlinear dynamic response structural optimization the equivalent static loads esls method is newly proposed and applied to nonlinear dynamic response optimization the equivalent static loads are defined as the linear static load sets which generate the same response field in linear static analysis as that from nonlinear dynamic analysis the esls are made from the results of nonlinear dynamic analysis and used as external forces in linear static response optimization then the same response from nonlinear dynamic analysis can be considered throughout linear static response optimization the updated design from linear response optimization is used again in nonlinear dynamic analysis and the process proceeds in a cyclic manner until the convergence criteria are satisfied several examples are solved to validate the method the results are compared to those of the conventional method with sensitivity analysis using the finite difference method

on large delays in multi server queues with heavy tails we present upper and lower bounds for the tail distribution of the stationary waiting time d in the stable gi gi s first come first served fcfs queue these bounds depend on the value of the traffic load which is the ratio of mean service and mean interarrival times for service times with intermediate regularly varying tail distribution the bounds are exact up to a constant and we are able to establish a principle of s k big jumps in this case here k is the integer part of which gives the most probable way for the stationary waiting time to be large another corollary of the bounds obtained is to provide a new proof of necessity and sufficiency of conditions for the existence of moments of the stationary waiting time

implementation of design sensitivity analysis for nonlinear elastic structures implementation of design sensitivity analysis into marc a general nonlinear finite element code is described follower forces and constraints treated via lagrange multipliers are also included in the implementation the sensitivity analysis is implemented in the most general way i e arbitrary design criteria and design parameters including shape sizing and material property variables can be defined through special user supplied subroutines local derivatives of all of the quantities needed for sensitivity analysis are calculated using the central differences allowing a very general implementation without sacrificing the accuracy design sensitivities to multiple parameters are implemented even though an option for multiple loading does not apply to nonlinear finite element analysis also design sensitivities can be calculated at any load level whereas the analysis calculations march on as if there was no interruption derivation of design sensitivities with constraints such as incompressibility or unilateral frictionless contact indicates that the same steps as in the implementation of regular displacement based nonlinear finite element method can be followed without any modifications a numerical example using the modified finite element code is presented to demonstrate the new capability

parameter sensitivity of inelastic buckling and post buckling response this paper is concerned with recent sensitivity results for buckling and post buckling problems the article starts with an overview of the fem based approach to inelastic structural mechanics systems with singular points along the equilibrium path the existing sensitivity techniques are reviewed their inherent drawbacks are indicated and a new methodology is suggested the discretized formalism is mainly employed throughout but a continuum formulation appropriate for this class of problems is also given numerical studies illustrate the presentation

topology optimization of material nonlinear continuum structures by the element connectivity parameterization the application of the element density based topology optimization method to nonlinear continuum structures is limited to relatively simple problems such as bilinear elastoplastic material problems furthermore it is very difficult to use analytic sensitivity when a commercial nonlinear finite element code is used as an alternative to the element density formulation the element connectivity parameterization ecp formulation is developed for the topology optimization of isotropic hardening elastoplastic or hyperelastic continua by using commercial software ecp varies the stiffness of zero length linear elastic links that connect design domain discretizing finite elements unloading was not considered but the advantages of ecp in material nonlinear problems were demonstrated considerably simple analytic sensitivity calculation using a commercial code and simple link stiffness penalization regardless of nonlinear material behaviour copyright 2006 john wiley sons ltd

delay analysis of the max weight policy under heavy tailed traffic via fluid approximations we consider switched queueing networks with a mix of heavy tailed i e arrival processes with infinite variance and exponential type traffic and study the delay performance of the max weight policy known for its throughput optimality and asymptotic delay optimality properties our focus is on the impact of heavy tailed traffic on exponential type queues flows which may manifest itself in the form of subtle rate dependent phenomena we introduce a novel class of lyapunov functions piecewise linear and nonincreasing in the length of heavy tailed queues whose drift analysis provides exponentially decaying upper bounds to queue length tail asymptotics despite the presence of heavy tails to facilitate a drift analysis we employ fluid approximations proving that if a continuous and piecewise linear function is also a lyapunov function for the fluid model then the same function is a lyapunov function for the original stochastic system furthermore we use fluid approximations and renewal theory i

an application of trefftz method to the sensitivity analysis of two dimensional potential problem this paper presents a new formulation for the sensitivity analysis with the help of the trefftz boundary type solution procedure direct differentiation of the expressions of the field variables leads to the expressions of the sensitivities since the field variables are approximated by linear combination of regular t complete functions the analytical expressions of the sensitivities are also regular sensitivity analysis schemes with respect to variations of dimensions of the object and boundary conditions are explained in detail two dimensional potential problems in the bounded region are considered as a numerical example in order to confirm its validity

design sensitivity coefficients for elasto viscoplastic problems by boundary element methods this paper is concerned with accurate and efficient determination of design sensitivity coefficients dscs for solid mechanics problems with small strains and rotations but with material non linearities present elasto plastic or elasto viscoplastic problems this approach is based on direct differentiation dda of the relevant derivative boundary element method dbem formulation of the problem analytical differentiation of the dbem equations leads to singular integral equations for the dscs with weakly logarithmically for 2 d singular kernels which are easy to deal with also stress components and their sensitivities are obtained on the boundary of a body with great accuracy these quantities are typically difficult to obtain accurately from finite element methods fem a computer program for general two dimensional plane strain and plane stress problems has been developed based on the above formulation numerical results are presented for some sample problems and these are compared against direct solutions the agreement between the dbem and direct solutions is excellent for these examples

essential hyperbolicity and homoclinic bifurcations a dichotomy phenomenon mechanism for diffeomorphisms we prove that any diffeomorphism of a compact manifold can be approximated in the c 1 topology by another diffeomorphism exhibiting a homoclinic bifurcation a homoclinic tangency or a heterodimensional cycle or by one which is essentially hyperbolic has a finite number of transitive hyperbolic attractors with open and dense basin of attraction

shape sensitivity analysis for structural problems with non linear material behaviour this paper describes some considerations around the analytical structural shape sensitivity analysis when the structural behaviour is computed using the nite element method with a non linear constitutive material model depending on the type of non linear behaviour two di erent approaches are proposed first a new direct non incremental formulation is proposed for material models characterized by the fact that the stresses at any time t can be expressed in terms of the strains at the same time t and in some cases the strains at a speci c past time t6 t6 t this is the case of elasticity linear as well as non linear perfect plasticity and damage models second a more classical incremental approach is proposed for general plasticity cases a special strategy is also proposed for material models with strain softening the quality and reliability of the proposed approaches are assessed through their application in di erent examples copyright 1999 john wiley sons ltd

tangent operators sensitivity expressions and optimal design of non linear elastica in contact with applications to beams the tangent operator and design sensitivity expressions for non linear elastica subject to frictionless unilateral contact are derived and computed via the finite element method the sensitivity computations are then combined with a numerical programming package to create an optimal design environment to exemplify the optimal design environment a beam s contact surface is contoured to minimize bending stress this work combines the research on the sensitivity analysis of non linear elastic bodies subject to constraints finite strain non linear elastic beam analysis and contact analyses the analysis is valid for any smooth contact surface and specialized for the case in which the surface is represented by a cubic spline the direct differentiation method is utilized to perform the sensitivity analysis in an example problem the sensitivity analysis is verified by finite difference computations and then combined with a numerical optimization program to design the stop profile of a valve

design sensitivity analysis of elastoplastic structures an iterative and incremental algorithm is presented for the design sensitivity analysis dsa of elastoplastic structures geometrical non linearity is included in the formulation and the structure may be subjected to a cyclic loading the key concept is discussed using an elastoplastic truss it is shown that the design sensitivity coefficients dscs are path dependent and discontinuous at the time at which yielding takes place in a member yield time in the proposed algorithm the yield time is considered to be a function of the design variables in this way the discontinuity in the dscs at the material transition points can be overcome in a simple and routine way incremental response analysis and dsa are carried out simultaneously using the newton raphson type iterative procedure since the proposed algorithm is completely consistent with the analysis procedure it can be implemented into an existing code for structural analysis application to distributed parameter structures with kinematic and or isotropic hardening is discussed using the von mises yield condition and the elastic predictor radial return method dsa is carried out for a ten bar truss with a piecewise linear constitutive relation it is shown that the dscs found using the proposed method agree quite well with those calculated by the central difference method finally an alternate method without any iteration for dsa is proposed and the results from the two methods are compared

variational principle for shape design sensitivity analysis in this paper the adjoint method of design sensitivity analysis is stated as a general variational principle that is applicable to a wide range of problems the principle gives a very simple and straightforward method of obtaining the design sensitivity expression for a functional dependent on the state fields the sensitivity expression involves certain adjoint fields and explicit design variations of the functional and the governing equations for the primary state fields the principle is stated in terms of an augmented functional that is defined by adding to the response functional whose sensitivity is desired the equilibrium equation of the primary problem then explicit design variation of the augmented functional gives the total design variation of the response functional stationarity of the augmented functional with respect to the state fields defines the adjoint problem for the response functional the principle is proved and applied to several classes of linear and nonlinear problems such as field problems structural problems and dynamic response problems it is shown that the design sensitivity expressions derived in the literature using long procedures are obtained quite routinely by use of the principle

a direct application of higher order parametric programming techniques to structural optimization computational methods based on a sequence of parametric programming problems are presented for solving constrained optimization problems cop without any parameter an auxiliary parametric programming problem appp is formulated in order to solve cop the procedure is started with an arbitrary initial solution which is the trivial solution of appp corresponding to the initial value of the parameter then the optimal solution for the final value of the parameter which is the optimal solution of cop is estimated by taylor s expansion with respect to the parameter where higher order terms are incorporated it is shown that the incorporation of the higher order terms indeed leads to a faster convergence of the solution as an extension of the method a general algorithm is presented for optimum design problems with state variable constraints which are implicit functions of the design variables logarithmic penalty functions are incorporated and the weight coefficients for the penalty terms are updated continuously the derivatives of the state variables with respect to the parameter and their sensitivity coefficients are expressed explicitly in terms of those of the design variables finally a method of simultaneous analysis and optimization is developed for trusses with geometrical non linearity

a computational method for design sensitivty analysis of elastoplastic structures abstract design sensitivity analysis of structural systems having elastoplastic material behavior is developed using the continuum formulation and its implementation into a finite element program is described the main idea is to develop a computational procedure for design sensitivity analysis based on the response obtained by an incremental load approach for non linear response analysis where the elastoplatic time independent constitutive model is employed to account for the plastic material behavior i e kinematic hardening and isotropic hardening discontinuities in the design sensitivity coefficients are investigated and a procedure is proposed to alleviate them the control reference domain approach is used to treat shape and non shape design variables simultaneously the direct variation method is adopted for this path dependent problem to obtain the design sensitivity expression for the response variables analytical examples are used to demonstrate the developed sensitivity procedure and gain insights for numerical implementation the design sensitivity expression is discretized to implement it into a computer program using the isoparametric formulation of finite element analysis numerical examples are presented to calculate the design sensitivity coefficients and compare them with those using the central finite difference method

sensitivity analysis of linear elastic systems using domain parameterization and a mixed mutual energy principle abstract a unified approach for explicit structural design sensitivity analysis of linear elastic systems is presented first order sensitivity expression involving the complete set of design variable including shape design parameters are derived for general response functional both direct differentiation and adjoint variable methods are developed in a general variational context a domain parameterization method related to the mixed eulerian lagrangian kinematic description is introduced for the treatment of shape variation problems this approach is an alternative to the material derivative methods presented elsewhere in the literature the adjoint variable derivation is based on a mixed energy principle the mutual hu washizu energy principle which is formally stated and proved the design sensitivity derivation based on this principle unifies and extends several of the adjoint variable methods found in the literature the derivations presented here are extended to nonlinear systems in a subsequent paper the sensitivity expressions are valid for use with force displacement or mixed approximate solutions of the elasticity equations other formulations are generally only consistent with displacement solution methods numerical implementation of the design sensitivity methods is discussed and illustrative examples are included

a sensitivity interpretation of adjoint variables in optimal design abstract the adjoint method of computing derivatives of cost and constraint functions with respect to design variables requires the calculation of certain adjoint variables until now the adjoint variables have been looked upon only as some intermediate vectors needed to calculate design derivatives in this paper they are shown to have an important significance they represent the sensitivity of the cost and constraint functions with respect to the loading or forcing function in the design problem a sensitivity theorem for the adjoint variables is presented for structural mechanical dynamic and distributed parameter systems these results offer some immediate practical advantages such as a method for computing influence coefficients for structural systems and a method for verifying debugging the analytical calculation of adjoint variables in development of a computer code

partial hyperbolicity far from homoclinic bifurcations abstract we prove that any diffeomorphism of a compact manifold can be c 1 approximated by a diffeomorphism which exhibits a homoclinic bifurcation a homoclinic tangency or a heterodimensional cycle or by a diffeomorphism which is partially hyperbolic its chain recurrent set splits into partially hyperbolic pieces whose centre bundles have dimensions less or equal to two we also study in a more systematic way the central models introduced in crovisier in press 10

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

computational method for optimization of structural shapes in the direct method of shape optimum design mesh regridding is a necessary step in avoiding the distortion of finite element grids the effect of internal nodal movements on the shape design sensitivity associated with the regridding process has been carefully examined in this paper it shows that to maintain computational accuracy the effects of internal nodal movement should be considered in shape design sensitivity analysis to take the internal nodal movement into account the numerical accuracy of the speed method is investigated a new approach which constructs the velocity field over the element domain is presented although it takes more computational time the new approach does improve the accuracy of shape design sensitivity

design sensitivity analysis and optimization of nonlinear structuralresponse using incremental procedure this paper describes procedures for design sensitivity analysis of nonlinear structural response using the incremental solution scheme the total and updated lagrangian approaches are discussed sensitivity analyses for the stress strain and displacement constraints are developed in the optimization process the buckling load constraint is also imposed numerical aspects of design sensitivity analysis are discussed three numerical examples are solved to illustrate the design sensitivity analysis and optimization process the numerical procedures are stable and work well they can be applied to more complex structures to optimize them

sensitivity analysis for non linear constrained elastostatic systems adjoint and direct differentiation methods are used to formulate design sensitivities for non linear constrained elastostatic systems variations of a general response functional are expressed in explicit form with respect to all design field variations including shape the response functional is expressed over the undeformed configuration however the displacement gradient is included so that field quantities that are defined over the deformed configuration such as cauchy stress may be modelled the finite element method is used to evaluate the sensitivities in an example problem

shape optimization in elasticity and elasto viscoplasticity by the boundary element method abstract this paper is concerned with the optimal shape design of solids undergoing small strain small rotation elasto viscoplastic deformation shape sensitivities for this class of problems are determined by using a direct differentiation approach dda to the governing boundary element method bem equations of the problem the standard bem and the sensitivity equations are discretized and solved numerically shape optimization is carried out by coupling the standard and sensitivity analyses with an optimizer the optimization algorithm chosen here uses sequential quadratic programming to obtain the desired optimal shape of a body in an iterative manner numerical solutions for optimal shapes of cutouts in plates two dimensional plane stress are presented the difference between optimal shapes of solids undergoing purely elastic and elasto viscoplastic deformation is shown clearly in these examples

a systematic approach to shape sensitivity analysis abstract a fully detailed development of the domain parameterization method is presented for shape sensitivity analysis it is shown that equivalent results are obtained from the material derivative method in fact the material derivative method may be viewed as a special case of the domain parameterization method which occurs when the reference configuration coincides with the body configuration the method is illustrated for the laplace problem in which explicit shape sensitivities are derived by the adjoint and direct differentiation methods both finite element and boundary element applications are discussed the similarities between this approach and the isoparametric finite boundary element method are transparent in the finite element approach it is shown that the sensitivity integrals may be transformed to the boundary as is commonly done in the material derivative method for the adjoint method however this does not seem possible for the direct differentiation method finally in the boundary element approach the sensitivities do not require the differentiation of the fundamental solutions

an exposition of the material derivative approach for structural shape sensitivity analysis abstract the material derivative approach for structural shape design sensitivity analysis is described in a unified manner using the continuum formulation both the direct differentiation and adjoint methods are derived and computer implementation aspects are discussed the basic purpose of the paper is to present the shape design sensitivity analysis as an application of the material derivative concept of continuum mechanics and the calculus of variation procedures for taking variations of integrals over the variable domains it is also shown that the use of a recently developed variational principle for design sensitivity analysis greatly simplifies derivation of the adjoint method the indicial notation is used and expressions derived by different authors are related to each other a basic understanding of the material derivative approach of shape sensitivity analysis has been gained which can lead to insights for computer implementations of the method

on the ergodicity of partially hyperbolic systems pugh and shub have conjectured that essential accessibility implies ergodicity for a c partially hyperbolic volume preserving diffeomorphism we prove this conjecture under a mild center bunching assumption which is satisfied in particular by all partially hyperbolic systems with 1 dimensional center bundle we also obtain ergodicity results for c 1ci partially hyperbolic systems

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

lyapunov stability and sectional hyperbolicity for higher dimensional flows we study c 1 generic vector fields on closed manifolds without points accumulated by periodic orbits of different indices we prove that these flows exhibit finitely many sinks and sectional hyperbolic transitive lyapunov stable sets whose basins form a residual subset of the ambient manifold this represents a partial positive answer to conjectures in arbieto and morales proc am math soc 141 2817 2827 2013 the palis conjecture palis nonlinearity 21 t37 t43 2008 and gives a flow version of crovisier and pujals essential hyperbolicity and homoclinic bifurcations a dichotomy phenomenon mechanism for diffeomorphisms 2010

variational method for design sensitivity analysis in nonlinear structural mechanics this paper describes a unified variational theory of design sensitivity analysis of linear and nonlinear structures for shape nonshape and material selection problems the concepts of reference volume and adjoint structure are used to develop the unified viewpoint a general formula for design sensitivity analysis is derived several analytical linear and nonlinear examples are used to interpret various terms of the formula and demonstrate its use such analytical examples give insight into the application of the general formula to more complex problems that must be treated numerically adjoint state fields are also interpreted as sensitivities for the functional related to the primary structure

optimization of structures with nonlinear behavior using equivalent loads abstract nonlinear response optimization using the equivalent loads nroel method algorithm is proposed to perform optimization of non linear response structures the conventional method spends most of the design time on nonlinear analysis the nroel algorithm makes the equivalent load cases for each response and repeatedly performs linear response optimization and uses them as multiple loading conditions the equivalent loads are defined as the loads in linear analysis which generate the same response fields as those in nonlinear analysis the algorithm is validated for convergence and optimality the proposed algorithm is applied to a simple mathematical problem to verify the convergence and the optimality the nroel algorithm is applied to several structural problems with geometric and or material nonlinearity conventional optimization with sensitivity analysis using the finite difference method is also applied to the same examples the results of the optimizations are compared the proposed nroel method is found to be very efficient and good solutions are derived

on the computation of design derivatives for huber mises plasticity with non linear hardening this paper concerns design sensitivity analysis dsa for an elasto plastic material with material parameters depending on or serving as design variables the considered constitutive model is huber mises deviatoric plasticity with non linear isotropic kinematic hardening one which is applicable to metals the standard radial return algorithm for linear hardening is generalized to account for non linear hardening functions two generalizations are presented in both the non linearity is treated iteratively but the iteration loop contains either a scalar equation or a group of tensorial equations it is proven that the second formulation which is the one used in some parallel codes can be equivalently brought to a scalar form more suitable for design differentiation the design derivatives of both the algorithms are given explicitly enabling thus calculation of the explicit design derivative of stresses entering the global sensitivity equation the paper addresses several issues related to the implementation and testing of the dsa module among them the concept of verification tests both outside and inside a fe code as well as the data handling implied by the algorithm the numerical tests which are used for verification of the dsa module are described they shed light on a the accuracy of the design derivatives by comparison with finite difference computations and b the effect of the finite element formulation on the design derivatives for an isochoric plastic flow copyright 2003 john wiley sons ltd

structural shape sensitivity analysis relationship between material derivative and control volume approaches the material derivative and control volume approaches for structural shape design sensitivity analysis are presented analyzed and compared starting with a continuum formulation and a general response functional needing sensitivity analysis the two approaches are derived it is shown that the final design sensitivity expression for one approach can be obtained from the final expression for the other thus the two approaches are theoretically equivalent discretizations of the continuum expressions are presented and it is shown that the two approaches can lead to different implementations for numerical calculations a unified interpretation is developed in which the explicit design variations partial derivatives with respect to the design variables of the internal and external forces are the major calculations needed to implement the design sensitivity analysis the discretized forms of the continuum sensitivity expressions are also compared with the ones obtained by starting with the discretized model ab initio this comparison shows that these two approaches lead to similar expressions for numerical calculations therefore the exact same procedures can be used for computer implementations for both of the approaches the presented analyses give insights for implementation of the design sensitivity analysis theory with the finite element analysis programs

a c 2 generic trichotomy for diffeomorphisms hyperbolicity or zero lyapunov exponents or the c 1 creation of homoclinic bifurcations palis conjectured that densely in diff m r 1 diffeomorphisms are either hyperbolic or exhibit homoclinic bifurcations we prove a generic trichotomy for c2 diffeomorphisms an axiom a diffeomorphism with no cycles or kupka smale ones admitting zero lyapunov exponents or the c1 creation of homoclinic bifurcations i e the creation of homoclinic tangencies or heterodimensional cycles by some c1 small perturbations

optimal design of nonlinear parabolic systems part ii variable spatial domain with applications to casting optimization abstract in part i design sensitivity expressions for nonlinear parabolic systems were derived and then specialized for the analysis for thermal systems which govern many processing operations however that presentation did not account for shape variations here we present the shape sensitivity analysis for the thermal system the sensitivities are derived via the domain parameterization method and computed with the finite element method various means for parameterizing the shape are also discussed finally the sensitivity analysis is coupled with nonlinear programming to optimize the design of a sand casting

shape design sensitivity analysis and optimization of elasto plasticity with frictional contact a shape design sensitivity analysis and optimization are proposed for the ine nitesimal elasto plasticity with a frictional contact condition rate independent plasticity is considered with a return mapping algorithm and a von mises yield criterion the contact condition is formulated using the penalty method and the modie ed coulomb friction law a continuum based shape design sensitivity formulation is developed for structural and frictional contact variational equations the direct differentiation method is used to compute the displacement sensitivity and the sensitivities of various performance measures are computed from the displacement sensitivity path dependency of the sensitivity equation due to the constitutive relation and friction is discussed it is shown that no iteration is required to solve the sensitivity equation response analysis and the proposed sensitivity formulation are implemented using the mesh free method where the mesh distortion problem can be resolved numerical examples show accurate results of the proposed method compared to the e nite difference method dife culties in the sensitivity formulation for the e nite deformation problem are discussed i introduction b ecauseoftherecentdevelopmentofcomputationalmechanics it is now possible to analyze practical examples of complicated structural problems many design engineers who are not satise ed with response analysis alone have keen interests in the methodology of the design for more than two decades signie cant research efforthasbeen focused on the rate of response with respect to the changes in structural shape under shape design sensitivity analysis dsa analysis of the design sensitivity information is the most important and costly procedure in the automated optimum design process it supplies useful quantitative information to the design engineer about the direction of the desired design change in a classical linear problem dsa research has proved the differentiability of the solution of the response analysis using the linear operator theory and has derived specie c sensitivity expressions for variousproblems 1 aresultworthy of attention in linear dsais that theoriginalresponseandthesensitivityoftheresponsebelongtothe samekinematicallyadmissiblespaceandhavethesameregularities owing to the development of the response analysis capability engineers have directed their interest to the nonlinear problems that are dealt with efe ciently because many design application problems are accompanied by plastic deformation the design sensitivity of nonlinear problems has been actively developed and many research results are reported in the procedure of nonlinear response analysis the projection called a return mapping of the elastic trial stress is carried out to satisfy the variational inequality vi through an iteration in the stress space 2 the dsa on the other hand computes the rate of change of the projected response in the tangential direction of the constraint set without iteration note that the sensitivity analysis is linear and can be computed without iteration even thoughtheresponseanalysisisnonlinear 3 unlikethenonlinearelastic problem the sensitivity equation of the plastic problem requires sensitivity of the stress and internal evolution variables at the previous time the sensitivity equation is solved at each time without iteration and the sensitivity of the stress and evolution variables are

shape design sensitivity analysis of viscoplastic structures abstract design sensitivity analysis of nonlinear viscoplastic structures is developed in the continuum form starting from hamilton s principle constitutive models based on internal variable theory are incorporated in response analysis such models are described by a set of first order nonlinear differential equations with respect to time and some of them allow development of the yield surface the direct variation method is employed to obtain the sensitivity of the response discretization of the analysis and sensitivity equations and computer implementation aspects are discussed the discretized equations are implemented into a computer program and two numerical examples are solved to demonstrate the theory in the numerical examples only material nonlinearity under quasi static loading history is considered and the explicit integration method is employed to integrate the constitutive equations and their design variations the accuracy and efficiency of the developed sensitivity analysis are compared with the central finite difference method cyclic loading and loading rate effects are studied and discussed to show the general applicability of the developed sensitivity formulation

birth of homoclinic intersections a model for the central dynamics of partially hyperbolic systems creation d intersection homoclines un modele pour la dynamique centrale des systemes partiellement hyperboliques nous montrons une conjecture de j palis tout diffeomorphisme d une variete compacte peut etre approche en topologie c 1 par un diffeomorphisme morse smale ou par un diffeomorphisme ayant une intersection homocline transverse

a c 1 generic dichotomy for diffeomorphisms weak forms of hyperbolicity or infinitely many sinks or sources we show that for every compact n dimensional manifold n 1 there is a residual subset of diff m of diffeomorphisms for which the homoclinic class of any periodic saddle of f verifies one of the following two possibilities either it is contained in the closure of an infinite set of sinks or sources newhouse phenomenon or it presents some weak form of hyperbolicity called dominated splitting this is a generalization of a bidimensional result of mafine ma3 in particular we show that any cl robustly transitive diffeomorphism admits a dominated splitting