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

review of options for structural design sensitivity analysis part 1 linear systems the paper reviews options for structural design sensitivity analysis including global finite differences continuum derivatives discrete derivatives and computational or automated differentiation the objective is to put these different approaches to design sensitivity analysis in the context of accuracy and consistency computational cost and implementation options and effort linear static analysis and transient dynamic analysis are reviewed in a separate appendix special attention is paid to the semi analytical method a future paper will address design sensitivity analysis in nonlinear structural problems

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

continuous methods for extreme and interior eigenvalue problems in this paper continuous methods are introduced to compute both the extreme and interior eigenvalues and their corresponding eigenvectors for real symmetric matrices the main idea is to convert the extreme and interior eigenvalue problems into some optimization problems then a continuous method which includes both a merit function and an ordinary differential equation ode is introduced for each resulting optimization problem the convergence of each ode solution is proved for any starting point the limit of each ode solution for any starting point is fully studied both the extreme and the interior eigenvalues and their corresponding eigenvectors can be easily obtained under a very mild condition promising numerical results are also presented

an approach to optimal design of structures with non linear response a commonly adopted procedure to solve a structural optimum design problem is to improve the design iteratively by solving a sequence of simpler problems where each of these is obtained as an approximation of the original problem considering structures with linear response this procedure works very well and can be implemented quite straightforwardly however considering structures with non linear response the situation is more sophisticated since structural stability properties have to be taken into account the paper presents an approach to optimal design of statically loaded structures with non linear response it is proposed to replace the original problem with a sequence of similar problems from which corresponding approximate problems can easily be obtained these similar problems additionally contain an estimation of the critical load constraint while other constraints are imposed at some properly chosen load level which is lower or at most equal to the critical load level by this approach the structural stability requirements are taken into account in an effective manner the efficiency is illustrated with several examples involving kinematic as well as material non linear effects

on the hausdorff dimension of fractal attractors we consider such mappingsx n 1 f xn of an interval into itself for which the attractor is a cantor set for the same class of mappings for which the feigenbaum scaling laws hold we show that the hausdorff dimension of the attractor is universally equal tod 0 538

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

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

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

minimum weight design of non linear elastic structures with multimodal buckling constraints it well known that multimodal instability is an event particularly relevant in structural optimization here in the context of non linear stability theory an exact method is developed for minimum weight design of elastic structures with multimodal buckling constraints given an initial design the method generates a sequence of improved designs by determining a sequence of critical equilibrium points related to decreasing values of the structural weight multimodal buckling constraints are imposed without repeatedly solving an eigenvalue problem and the difficulties related to the non differentiability in the common sense of state variables in multimodal critical states are overcome by means of the lagrange multiplier method further constraints impose that only the first critical equilibrium states local maxima or bifurcation points on the initial equilibrium path of the actual designs are taken into account copyright 2002 john wiley sons ltd

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

dimensions and entropies of strange attractors from a fluctuating dynamics approach abstract it is shown that the fluctuations in the divergence of near by trajectories on strictly deterministic strange attractors can be modelled by stochastic concepts in particular we propose kramers moyal type equations for correlation functions between points on the attractor the drift terms are the lyapunov exponents the diffusion terms depend on the above fluctuations from this we obtain bounds on generalized dimensions and entropies numerical results show that in nearly all studied cases henon map zaslavskii map mackey glass eq the attractors are fractal measures in the sense of farmer information dimension hausdorff dimension metric entropy topological entropy

minimum weight design of truss structures with geometric nonlinear behavior the paper presents an optimization method based on optimality criterion for minimum weight design of structures with geometric nonlinear behavior the nonlinear critical load is determined by finding the load level at which the hessian of the potential energy ceases to be positive definite a recurrence relation based on the criterion that at optimum the nonlinear strain energy density be equal in all the members is used to develop an algorithm sample problems are given to illustrate the application of the method to truss type structures with a large number of design variables number of algorithms based on the optimality criterion approach have been developed to design a minimum weight structure with specified constraints on nodal displacements element stresses system stability etc 1 2 the optimization algorithm consisted of analyzing the structure by the finite element method and using a recurrence relation derived from the appropriate optimality criterion to modify the design variables the optimality criterion was derived by differentiating the lagrangian with respect to the design variables the displacements were assumed to be small and in the finite element analysis linear equilibrium equations were solved to determine the response of the structure to the ap plied loads in the case of system stability the constraints were defined see refs 3 6 by the associated linear eigenvalue problem this definition of system stability for some structures may not be valid because of the nonlinear behavior of the structure which may be due to the geometry of the structure or the presence of geometric imperfections the correct procedure for these structures would be to analyze the structure by using nonlinear equilibrium equations this will be particularly true for large space structures lss in this paper an optimization method based on the optimality criterion approach is presented for structures with geometric nonlinear behavior in the case of a structure optimized with constraints on linear stability the optimum structure can have more than one critical buckling mode this design tends to become im perfection sensitive and a small deviation in the geometry of the structure can reduce its load carrying capacity sub stantially the imperfection sensitivity of the optimized structure can be reduced by designing the structure so that buckling loads associated with the critical buckling modes are not equal see ref 7 a real structure has geometric im perfections and these structures tend to have nonlinear

the dimension of chaotic attractors dimension is perhaps the most basic property of an attractor in this paper we discuss a variety of different definitions of dimension compute their values for a typical example and review previous work on the dimension of chaotic attractors the relevant definitions of dimension are of two general types those that depend only on metric properties and those that depend on the frequency with which a typical trajectory visits different regions of the attractor both our example and the previous work that we review support the conclusion that all of the frequency dependent dimensions take on the same value which we call the dimension of the natural measure and all of the metric dimensions take on a common value which we call the fractal dimension furthermore the dimension of the natural measure is typically equal to the lyapunov dimension which is defined in terms of lyapunov numbers and thus is usually far easier to calculate than any other definition because it is computable and more physically relevant we feel that the dimension of the natural measure is more important than the fractal dimension

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

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

geometric properties of runge kutta discretizations for index 2 differential algebraic equations we analyze runge kutta discretizations applied to index 2 differential algebraic equations daes the asymptotic features of the numerical and the exact solutions are compared it is shown that runge kutta methods satisfying the first order constraint condition of the dae correctly reproduce the geometric properties of the continuous system the proof combines embedding techniques of index 2 daes and ordinary differential equations odes with some invariant manifolds results of nipp and stoffer attractive invariant manifolds for maps sam research report 92 11 eth zurich switzerland 1992 the results support the favorable behavior of these runge kutta methods applied to index 2 daes for t ge 0

one and multistep discretizations of index 2 differential algebraic systems and their use in optimization an approach to solve constrained minimization problems is to integrate a corresponding index 2 differential algebraic equation dae here corresponding means that the limit sets of the dae dynamics are local solutions of the minimization problem in order to obtain an efficient optimization code we analyze the behavior of certain runge kutta and linear multistep discretizations applied to these daes it is shown that the discrete dynamics reproduces the geometric properties and the long time behavior of the continuous system correctly finally we compare the dae approach with a classical sqp method

structural optimization of guyed trusses here is a strong trend in the design of skeletal structures to wherever practical make increasing use of tensioned cables one of the best examples of the convergence of structural efficiency to a best design can probably be found when studying the evolution of the configurations of power transmission towers from the early days of cantilever truss structures these have now evolved to the much lighter cross rope type in which two separate vertical cable stayed lattice struts support the power conductors through suspension from a cross rope apart from the savings in material significant savings in transportation cost can also be obtained especially since erection often has to occur in remote areas although the cable stayed structures are easy to analyze the determination of a best or least weight design can be complex due to the variety of design variables available these are typically configuration geometry member sizing and prestraining the optimization of a configuration is complex and is best accomplished manually perhaps with the aid of layout optimization methods on the contrary geometric and prestrain optimization and sizing can be treated by direct sensitivity analysis of a chosen topology this paper describes the sizing and prestrain optimization of guyed trusses for the nonlinear analysis the total lagrangian method is used in combination with the pure newton method and adaptive arc length procedures because of the lack of stiffness of a cable structure in its undeformed state soft spring elements are used to stabilize the structure temporarily during the load stepping phase the choice of prestrain instead of prestress variables is justified by both the simplicity of inclusion of prestrain in the sensitivity analysis and by the independence of the prestrain quantity with respect to the sequence of pretensioning of the structure the prestraining of a member can also simply be seen as the lack of fit of a member in a fixed geometry and can in practice be accomplished by specifying member lengths to create a predetermined lack of fit or by specifying turns of a turnbuckle included in the member for the optimization process two optimization methods are used namely the sequential quadratic programming sqp method idesign plba authored by arora1 and the successive approximation method sam algorithm of snyman and stander2 and stander et al 3 using two methods gives optima in which greater confidence can be placed additionally some idea of the degree of difficulty of the newly formulated optimization problems is gained five examples are given to demonstrate the analysis and optimization methodology the optimization of example structures subjected to second order effects show that unexpected results can be obtained a number of papers 4 8 related to the general topic of the sensitivity analysis and optimization of structures with nonlinear geometric

efficient adjoint sensitivity analysis of isotropic hardening elastoplasticity via load steps reduction approximation abstract sensitivity analysis plays a key role in gradient based shape optimization in this paper a highly efficient procedure for design sensitivity analysis of elastoplastic material with isotropic hardening rule is presented here an adjoint variable method is employed where the adjoint sensitivity formulation is derived from the discrete master equation yield condition and consistency condition due to history dependence of plasticity adjoint variables must be solved via a stepwise backward procedure therefore the computational cost increases in proportion to the number of load steps to address this cost and improve efficiency condensation rules are proposed to reduce the number of load steps in the sensitivity analysis based on properties of adjoint variables the properties of adjoint variables and proposed condensation rules are all theoretically proven in this paper the accuracy and efficiency of the proposed techniques are demonstrated using bar truss structures and solid structures in various complex load cases the results show that the techniques significantly reduce the number of load steps required in the sensitivity analysis while keeping satisfying accuracy

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

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

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

ssc eke semi supervised classification with extensive knowledge exploitation we introduce a new semi supervised classification method that extensively exploits knowledge the method has three steps first the manifold regularization mechanism adapted from the laplacian support vector machine lapsvm is adopted to mine the manifold structure embedded in all training data especially in numerous label unknown data meanwhile by converting the labels into pairwise constraints the pairwise constraint regularization formula pcrf is designed to compensate for the few but valuable labelled data second by further combining the pcrf with the manifold regularization the precise manifold and pairwise constraint jointly regularized formula mpcjrf is achieved third by incorporating the mpcjrf into the framework of the conventional svm our approach referred to as semi supervised classification with extensive knowledge exploitation ssc eke is developed the significance of our research is fourfold 1 the mpcjrf is an underlying adjustment with respect to the pairwise constraints to the graph laplacian enlisted for approximating the potential data manifold this type of adjustment plays the correction role as an unbiased estimation of the data manifold is difficult to obtain whereas the pairwise constraints converted from the given labels have an overall high confidence level 2 by transforming the values of the two terms in the mpcjrf such that they have the same range with a trade off factor varying within the invariant interval 0 1 the appropriate impact of the pairwise constraints to the graph laplacian can be self adaptively determined 3 the implication regarding extensive knowledge exploitation is embodied in ssc eke that is the labelled examples are used not only to control the empirical risk but also to constitute the mpcjrf moreover all data both labelled and unlabelled are recruited for the model smoothness and manifold regularization 4 the complete framework of ssc eke organically incorporates multiple theories such as joint manifold and pairwise constraint based regularization smoothness in the reproducing kernel hilbert space empirical risk minimization and spectral methods which facilitates the preferable classification accuracy as well as the generalizability of ssc eke

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 of bifurcation load of finite dimensional symmetric systems three methods are presented for sensitivity analysis of bifurcation load factor of finite dimensional conservative symmetric systems subjected to a set of symmetric proportional loads in the first method a conventional method with diagonalization is utilized to derive an explicit formula of sensitivity coefficients corresponding to a minor imperfection next a new concept is introduced to find the sensitivity coefficients of the load factor displacements and the eigenmodes under fixed lowest eigenvalue of the tangent stiffness matrix based on this concept a method is presented for finding approximate sensitivity coefficients of the buckling load factor finally a direct method is presented to find the accurate sensitivity coefficients of the bifurcation load factor displacements at buckling and the buckling mode of a symmetric system note that different formula should be used for sensitivity analysis of a limit point load factor in the examples the proposed three methods are compared in view of accuracy of the results and simplicity in coding

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

shaping optimal design of elastic planar frames with frequency constraints an approach to shaping optimal design of planar frames subject to eigenfrequency constraints displacement constraints and stress constraints simultaneously is described the eigenfrequency constraints may be imposed at the deformed due to the static load cone guration of the structure for the shape representation of the structure a design element technique is used a frame structure is treated as assembled from several design elements each of them being dee ned as a bezier curve the design variables may ine uence the position and the shape of each design element plus the cross sectional properties of beam elements the optimal design problem is dee ned in a general form and its solution by employing gradient based methods of mathematical programming is discussed briee y the structural response analysis is based on a nonlinear model that can handle virtually arbitrary large structural displacements two numerical examples illustrate the theory nomenclature ai area of the cross section asi shear area of the cross section b vector of design variables ci centroid curve e elastic modulus of the material f vector of internal forces fi eigenfrequencies g shear modulus of the material hi constrained quantitites ii moment of inertia of the cross section k stiffness matrix m mass matrix pi lagrange interpolation polynomial p external distributed force per unit length qi position vector of a control point r vector of external forces ri position vector u vector of structural displacements v vector of structural response variables w external distributed force per unit length shear strain extensional strain i eigenvalues angle of inclination at the undeformed cone guration i eigenvector a angle of rotation

on the metric properties of the feigenbaum attractor the two standard literature definitions of the function associated with the feigenbaum attractor are not equivalent the method due to vulet al and feigenbaum is used to calculate the haussdorff dimension of the feigenbaum attractor using as input the trajectory scaling functions the two calculations yield the same hausdorff dimensiond 0 5380451435 to within the accuracy of the computation

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

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

trust region algorithms and timestep selection unconstrained optimization problems are closely related to systems of ordinary differential equations odes with gradient structure in this work we prove results that apply to both areas we analyze the convergence properties of a trust region or levenberg marquardt algorithm for optimization the algorithm may also be regarded as a linearized implicit euler method with adaptive timestep for gradient odes from the optimization viewpoint the algorithm is driven directly by the levenberg marquardt parameter rather than the trust region radius this approach is discussed for example in r fletcher practical methods of optimization 2nd ed john wiley new york 1987 but no convergence theory is developed we give a rigorous error analysis for the algorithm establishing global convergence and an unusual extremely rapid type of superlinear convergence the precise form of superlinear convergence is exhibited the ratio of successive displacements from the limit point is bounded above and below by geometrically decreasing sequences we also show how an inexpensive change to the algorithm leads to quadratic convergence from the ode viewpoint this work contributes to the theory of gradient stability by presenting an algorithm that reproduces the correct global dynamics and gives very rapid local convergence to a stable steady state

efficient handling of stability problems in shell optimization by asymmetric worst case shape imperfection the paper presents an approach to shape optimization of proportionally loaded elastic shell structures under stability constraints to reduce the stability related problems a special technique is utilized by which the response analysis is always terminated before the first critical point is reached in this way the optimization is always related to a precritical structural state the necessary load carrying capability of the optimal structure is assured by extending the usual formulation of the optimization problem by a constraint on an estimated critical load factor since limit points are easier to handle the possible presence of bifurcation points is avoided by introducing imperfection parameters they are related to an asymmetric shape perturbation of the structure during the optimization the imperfection parameters are updated to get automatically the worst case pattern and amplitude of the imperfection both the imperfection parameters and the design variables are related to the structural shape via the design element technique a gradient based optimizer is employed to solve the optimization problem three examples illustrate the proposed approach copyright 2007 john wiley sons ltd

fractal properties of critical invariant curves we examine the dimension of the invariant measure for some singular circle homeomorphisms for a variety of rotation numbers through both the thermodynamic formalism and numerical computation the maps we consider include those induced by the action of the standard map on an invariant curve at the critical parameter value beyond which the curve is destroyed our results indicate that the dimension is universal for a given type of singularity and rotation number and that among all rotation numbers the golden mean produces the largest dimension

a method based on rayleigh quotient gradient flow for extreme and interior eigenvalue problems abstract recently a continuous method has been proposed by golub and liao as an alternative way to solve the minimum and interior eigenvalue problems according to their numerical results their method seems promising this article is an extension along this line in this article firstly we convert an eigenvalue problem to an equivalent constrained optimization problem secondly using the karush kuhn tucker conditions of this equivalent optimization problem we obtain a variant of the rayleigh quotient gradient flow which is formulated by a system of differential algebraic equations thirdly based on the rayleigh quotient gradient flow we give a practical numerical method for the minimum and interior eigenvalue problems finally we also give some numerical experiments of our method the golub and liao method and eigs a matlab implementation for computing eigenvalues using restarted arnoldi s method for some typical eigenvalue problems our numerical experiments indicate that our method seems promising for most test problems