complexity and multiple complexes in this chapter we introduce the notion of the complexity of a module and explore some related ideas complexity was first defined by jon alperin in the late 1970 s and it helped to motivate much of the development of the homological properties of modules in fact the complexity of a finitely generated kg module m is a rather crude invariant in that it only measures the polynomial rate of growth of a minimal projective resolution of the module the original theorem of alperin and evens which extended quillen s dimension theorem to modules was proved only in terms of complexity later it was extended to support varieties as well the theorem computes the complexity of a kg module m as the maximum of the complexities of the restrictions of m to the elementary abelian subgroups of the group g

control design variable linking for optimization of structural control systems a method is presented to integrate the design space of structural control system optimization problems in the case of linear state feedback control conventional structural sizing variables and elements of the feedback gain matrix are both treated as strictly independent design variables in optimization by extending design variable linking concepts to the control gains several approximation concepts including new control design variable linking schemes are used to formulate the integrated structural control optimization problem as a sequence of explicit nonlinear mathematical programming problems examples which involve a variety of behavior constraints including constraints on dynamic stability damped frequencies control effort peak transient displacement acceleration and control force limits are effectively solved by using the method presented

new dual type decomposition algorithm for nonconvex separable optimization problems abstract the aim of the paper is to present a new dual type decomposition algorithm for large scale nonconvex optimization problems of general separable structure after reformulation of the augmented lagrange function and introduction of auxiliary variables called approximation points separable local optimization problems are created the lower level of the method consists of a few optimization runs of these problems for subsequently improved approximation points whereas standard updating of lagrange multipliers constitutes the highest level simple rules for adjusting the approximation points and the lagrange multipliers are given and thoroughly analysed applicability conditions and example numerical results indicate that the presented algorithm eliminates to a great extent drawbacks of the previous approaches

dynamic response optimization using an active set rqp algorithm a recently developed recursive quadratic programming rqp algorithm is applied to dynamic response design problems the algorithm incorporates updated hessians of the lagrange function and uses an active set strategy in which only a subset of the constraints is included in the direction finding qp subproblem hessian updating with dynamic response constraints presents some difficulties the primary difficulty is that the total number of constraints and the location of time points where they have to be imposed can change from iteration to iteration this can cause inconsistencies in hessian updating if proper numerical procedures are not used a numerical procedure to handle the situation is developed implemented and evaluated automatic restarting procedures are necessary for proper convergence of the algorithm the new algorithm is robust as well as more efficient than the purely linear algorithm the active set strategy plays an important role for application to dynamic response problems the rqp algorithms that do not use such a strategy are not applicable to this class of problems

on the integral cohomology of wreath products we use the techniques pioneered by nakaoka to study the integral cohomology of wreath products the situation is considerably more complicated than for field coefficients we describe the cohomology of h wr c for c cyclic of prime order and any h whose cohomology is finitely generated in each degree we discuss various related results and conjectures concerning the exponent of integral cohomology of finite groups these include a conjecture of a adem recently solved by j pakianathan conjectures of j carlson one solved in this work and some of our own one example given is a 2 group whose index four subgroups have non trivial intersection but whose integral cohomology has exponent four answering a question posed in 8

a lanczos type method for multiple starting vectors given a square matrix and single right and left starting vectors the classical nonsymmetric lanczos process generates two sequences of bior thogonal basis vectors for the right and left krylov subspaces induced by the given matrix and vectors in this paper we propose a lanczos type algorithm that extends the classical lanczos process for single starting vectors to mul tiple starting vectors given a square matrix and two blocks of right and left starting vectors the algorithm generates two sequences of biorthogonal basis vectors for the right and left block krylov subspaces induced by the given data the algorithm can handle the most general case of right and left start ing blocks of arbitrary sizes while all previously proposed extensions of the lanczos process are restricted to right and left starting blocks of identical sizes other features of our algorithm include a built in deation procedure to detect and delete linearly dependent vectors in the block krylov sequences and the option to employ look ahead to remedy the potential breakdowns that may occur in nonsymmetric lanczos type methods

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

the varieties and the cohomology ring of a module let g be a finite group and let k be a field of characteristic p 0 recently several studies have focused on the homological invariants of finitely generated kg modules in l alperin proposed the study of the complexity of a module the complexity is related to the degree of the polynomial rate of growth of the terms in a projective resolution of the module see definition 2 4 in 2 alperin and evens proved that the complexity of a kg module m is equal to the maximum of the complexities of the restrictions of m to elementary abelian p subgroups of g one of the roots of the alperin evens theorem is quillen s dimension theorem see 21 or 22 which with some difficulty can be interpreted as saying the same thing for the special case of the trivial kg module alperin and evens 3 and avrunin 4 have further studied the annihilator in ext k k of the cohomology of m and have produced theorems of a similar nature these are related to the work in 9 on the structure of the cohomology ring ext m m the purpose of this paper is to investigate the relationship between these homological invariants and the structure of the module we concentrate on the case in which g is an elementary abelian p group however using such results as those mentioned above we can apply the present work to more general groups even as the earlier results indicate that much of the structure and cohomology of a module is revealed in the restrictions to elementary abelian p subgroups we show here that in the case of an elementary abelian group much of this information can be found by looking at the restrictions to cyclic p subgroups of the group of units of kg some of the results in this paper were announced in s let g xl xn be an elementary abelian group of order p suppose that k is algebraically closed to any finitely generated kg module m we

lanczos algorithm for the quadratic eigenvalue problem in engineering applications abstract a new approach of employing the lanczos two sided recursion to solve the quadratic eigenvalue problem is presented the methodology employed retains the n order quadratic problems as posed without the need to use the method of matrix augmentation traditionally used to cast the problem as a linear eigenvalue problem of order 2n pursuing the concept of finding a characteristic matrix to yield an eigenvalue square matrix the lanczos recursion has been devised to solve the quadratic eigenvalue problem appropriate proofs showing the lanczos biorthogonal transformation are included the example problems presented validate the algorithm and show the effectiveness of the method for large quadratic eigenproblems

multiplier methods for engineering optimization multiplier methods used to solve the constrained engineering optimization problem are described these methods solve the problem by minimizing a sequence of unconstrained problems defined using the cost and constraint functions the methods proposed in 1969 have been determined to be quite robust although not as efficient as other algorithms they can be more effective for some engineering applications such as optimum design and control oflarge scale dynamic systems since 1969 several modifications and extensions of the methods have been developed therefore it is important to review the theory and computational procedures of these methods so that more efficient and effective ones can be developed for engineering applications recent methods that are similar to the multiplier methods are also discussed these are continuous multiplier update exact penalty and exponential penalty methods

a linear programming formulation for incremental contact analysis a novel algorithm for the analysis of contact problems in elasticity has been presented in this paper the algorithm is based on the boundary element method and a direct approximation of the contact complementarity conditions using linear programming an incremental loading scheme has been developed to ensure an accurate approximation of the deformation path that the object experiences during the process of contact several numerical examples have been analysed to illustrate the validity of the proposed formulations

a continuously differentiable exact penalty function for nonlinear programming problems with inequality constraints in this paper it is shown that given a nonlinear programming problem with inequality constraints it is possible to construct a continuously differentiable exact penalty function whose global or local unconstrained minimizers correspond to global or local solutions of the constrained problem

design oriented identification of critical times in transient response on decrit 3 techniques de reduction de l effort de calcul implique dans l identification des temps critiques dans les problemes d optimisation des structures

optimum design of systems for dynamics and controls using sequentialquadratic programming this paper describes application of a sequential quadratic programming method for dynamic response optimization and optimal control problems several methods to treat time dependent constraints are described

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

topology optimization for linear thermo mechanical transient problems modal reduction and adjoint sensitivities summary this paper focuses on topology optimization for linear transient thermo mechanical problems the latter are for example encountered for extreme precision tools and instrumentation due to the transient nature a standard adjoint sensitivity analysis will result in a backward transient analysis for the adjoint variables leading to both storage and computational inefficiencies a method is proposed which rigorously eliminates the backward transient integration for the adjoint sensitivity analysis at the basis is a model order reduction technique which relies on a reduced thermal modal basis combined with static correction the modal amplitudes can be readily obtained semi analytically using simple convolutions this accurate but reduced order model is the starting point for an adjoint sensitivity analysis via a tactic selection of adjoint variables the backward transient analysis for the adjoints is completely eliminated whereas computational efficiency and consistency are maintained the effectiveness of the resulting adjoint sensitivities and their application in topology optimization are demonstrated on the basis of several test examples this article is protected by copyright all rights reserved

sensitivity of control augmented structure obtained by a system decomposition method the verification of a method for computing sensitivity derivatives of a coupled system is presented the method deals with a system whose analysis can be partitioned into subsets that correspond to disciplines and or physical subsystems that exchange input output data with each other the method uses the partial sensitivity derivatives of the output with respect to input obtained for each subset separately to assemble a set of linear simultaneous algebraic equations that are solved for the derivatives of the coupled system response this sensitivity analysis is verified using an example of a cantilever beam augmented with an active control system to limit the beam s dynamic displacements under an excitation force the verification shows good agreement of the method with reference data obtained by a finite difference technique involving entire system analysis the usefulness of a system sensitivity method in optimization applications by employing a piecewise linear approach to the same numerical example is demonstrated the method s principal merits are its intrinsically superior accuracy in comparison with the finite difference technique and its compatibility with the traditional division of work in complex engineering tasks among specialty groups

structural optimization using equivalent static loads at all time intervals a quasi static structural optimization for elastic structures under dynamic loads is presented an equivalent static load esl set is defined as a static load set which generates the same displacement field as that from a dynamic load at a certain time multiple esl sets calculated at all the time intervals are employed to represent the various states of the structure under the dynamic load they can cover all the critical states that might happen at arbitrary times the continuous characteristics of a dynamic load are considered by multiple static load sets the calculated sets of esls are utilized as a multiple loading condition in the optimization process a design cycle is defined as a circulated process between an analysis domain and a design domain the analysis domain gives the loading condition needed in the design domain the design domain gives a new updated design to be verified by the analysis domain in the next design cycle the design cycles are iterated until the design converges structural optimization with dynamic loads is tangible by the proposed method standard example problems are solved to verify the validity of the method

a hybrid formulation for treatment of point wise state variable constraints in dynamic response optimization abstract this paper presents methods of design sensitivity analysis and optimization of dynamic response of mechanical and structural systems the point wise state variable constraint function is divided into time sub domains such that each sub domain contains only one local maximum point then the original constraint is replaced by a number of equivalent functional constraints each functional constraint is the integration of the positive value of the original constraint over its own time sub domain a direct differentiation method and three adjoint variable methods of design sensitivity analysis are presented all of these methods are discussed and compared it turns out that two of the adjoint variable methods are more efficient than others a hybrid optimization algorithm based on these methods is proposed in detail two problems are solved for optimal design comparisons of results with those available in the literature are made numerical experience with the proposed method is discussed in detail it is concluded that the new formulation is extremely efficient and converges to either optimal or near optimal solutions without any difficulty

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

structural control system optimization with variable actuator masses a method is presented to integrate the design space for structural control system optimization problems in the case of linear state feedback control nonstructural lumped masses and control system design variables as well as structural sizing variables are all treated equally as independent design variables in the optimization process structural and control design variable linking schemes are used in order to avoid a prohibitively large increase in the total number of independent design variables when actuator masses are treated as nonstructural lumped mass design variables special consideration is given to the relation between the transient peak responses and the required actuator masses that is formulated as a behavior constraint form a method to prevent instability of uncontrolled higher modes by modifying the feedback gain matrix without information about higher modes is also presented the original nonlinear mathematical programming problem based on a finite element formulation and linear state feedback is replaced by a sequence of explicit approximate problems exploiting various approximation concepts such as design variable linkings temporary constraint deletion and first order taylor series expansion of nonlinear behavior constraints in terms of intermediate design variables examples which involve a variety of dynamic behavior constraints including constraints on closed loop eigenvalues peak transient displacements peak actuator forces and relations between the peak responses and the actuator masses are effectively solved by using the method presented

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

a computational study of transformation methods for optimal design in this paper computational aspects of transformatio n methods are studied transformation methods which include sequential unconstrained minimization techniques sumts and multiplier methods are based on solving a sequence of unconstrained minimization problems an efficient technique is given to compute gradients of the transformation function an operations count is given to demonstrate savings of the suggested technique over two other techniques used in the literature computer programs implementing the use of this technique in sumts and multiplier algorithms are developed applications of these programs on a set of structural design problems are given multiplier methods are found to be very stable and reliable even on some relatively difficult problems and they perform better than sumts some ways of improving the efficiency of transformation methods are given together with possible extensions of using the suggested approach to compute gradients of implicit functions

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

structural optimization by multilevel decomposition a method for decomposing an optimization problem into a set of subproblems and a coordination problem that preserves coupling between the subproblems is described the decomposition is achieved by separating the structural element optimization subproblems from the assembled structural optimization problem each element optimization and optimum sensitivity analysis yields the cross sectional dimensions that minimize a cumulative measure of the element constraint violation as a function of the elemental forces and stiffness the assembled structural optimization produces the overall mass and stiffness distributions optimized for minimum total mass subject to constraints that include the cumulative measures of the element constraint violations extrapolated linearly with respect to the element forces and stiffnesses the method is introduced as a special case of a multilevel multidisciplinary system optimization and its algorithm is fully described for two level optimization for structures assembled of finite elements of arbitrary type numerical results are given as an example of a framework to show that the decomposition method converges and yields results comparable to those obtained without decomposition it is pointed out that optimization by decomposition should reduce the design time by allowing groups of engineers using different computers to work concurrently on the same large problem

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

quillen stratification for modules let g be a finite group and k a fixed algebraically closed field of characteristic p o if p is odd let h be the subring of h g k consisting of elements of even degree following 20 221 we take h h g k if p 2 though one could just as well use the subring of elements of even degree for all p h is a finitely generated commutative k algebra 13 and we let v denote its associated affine variety max h if m is any finitely generated kg module then the cohomology variety v m of m may be defined as the support in v of the h module h g m if g is a p group and in general as the largest support of h g l m where l is any kg module 4 91 a module l with each irreducible kc module as a direct summand will serve d quillen 20 221 proved a number of beautiful results relating k to the varieties v associated with the various elementary abelian p subgroups e of g culminating in his stratification theorem 20 221 this theorem gives a piecewise description of v almost explicitly in terms of the subgroups e and their normalizers in g a well known corollary is that dim v max dim v where e

the variety of an indecomposable module is connected let g be a finite group and let k be an algebraically closed field of characteristic p 0 the connections between the modular representation theory of finite groups and cohomology theory is currently a topic of great interest in particular cohomology theory has been used to attach to each finitely generated kgmodule m a homogeneous affine variety v m if we let 17 m be the corresponding projective variety then our main result is the following

continuum design sensitivity of transient responses using ritz and mode acceleration methods in this paper a unified continuum based sizing design sensitivity analysis method is developed for the transient dynamic response of built up structures using the direct differentiation method the variational equation of the built up structure is differentiate d with respect to the design variable to obtain a variational design sensitivity equation of the transient response the same finite element analysis model that is used to obtain an approximate solution of the variational equation of the built up structure is also used to obtain an approximate solution of the variational design sensitivity equation for large size built up structures the same superposition method is used both to reduce the dimension of the matrix equation of the built up structure and also to reduce the dimension of the design sensitivity equation the continuum based design sensitivity analysis method can be implemented outside established finite element analysis codes using postprocessing data since it does not require derivatives of the stiffness damping and mass matrices moreover the method is efficient since it does not require derivatives of basis vectors either the ritz or mode acceleration method is used to further improve accuracy and efficiency of both analysis and sensitivity results an example with damping shows that accurate design sensitivity of pointwise transient displacements is obtained using only 10 of basis vectors this example indicates that in the continuum based design sensitivity analysis method the same basis vectors that are used for analysis of the built up structure are suitable for analysis of the design sensitivity equation

steady flow for shear thickening fluids with arbitrary fluxes abstract we solve the stationary navier stokes equations for non newtonian incompressible fluids with shear dependent viscosity in domains with unbounded outlets in the case of shear thickening viscosity i e the viscosity is given by the shear rate raised to the power p 2 where p 2 the flux assumes arbitrary given values and the dirichlet integral of the velocity field grows at most linearly in the outlets of the domain under some smallness conditions on the energy dispersion we also show that the solution of this problem is unique our results are an extension of those obtained by o a ladyzhenskaya and v a solonnikov o a ladyzhenskaya v a solonnikov determination of the solutions of boundary value problems for steady state stokes and navier stokes equations in domains having an unbounded dirichlet integral j soviet math 21 1983 728 761 for newtonian fluids p 2

new adaptive move limit management strategy for approximate optimization part 2 approximations play an important role in multidisciplinary design optimization by offering system behavior information at a relatively low cost most approximate optimization strategies are sequential in which an optimization of an approximate problem subject to design variable move limits is iteratively repeated until convergence the move limits are imposed to restrict the optimization to regions of the design space in which the approximations provide meaningful information to ensure convergence of the sequence of approximate optimizations to a karush kuhn tucker solution a move limit management strategy is required in a companion paper issues of move limit management are reviewed and a new adaptive strategy for move limit management is developed wujek b a and renaud j e new adaptive move limit management strategy for approximate optimization part 1 aiaa journal vol 36 no 10 1998 pp 1911 1921 with its basis in the provably convergent trust region methodology the trust region ratio approximation method tram strategy utilizes available gradient information and employs a backtracking process using various two point approximation techniques to provide a flexible move limit adjustment factor the tram strategy is successfully implemented in application to several multidisciplinary design optimization test problems in addition a comprehensive study comparing the performance of the tram strategy to existing move limit strategies is conducted results indicate that application of the tram strategy results in increased efficiency for approximate optimization processes these implementation studies highlight the ability of the tram strategy to control the amount of approximation error and efficiently manage the convergence to a karush kuhn tucker solution

non conforming mesh gluing in the finite elements method a methodology to connect two or more non conforming meshes generated from a geometric model is presented the problem leads to a variational problem subject to constraints the emphasis of the work is on constraints construction two methods namely discrete connection and continuous connection are developed the relation with hybrid formulations is considered and two application examples are presented

isogeometric shape optimization for quasi static processes summary a framework to solve shape optimization problems for quasi static processes is developed and implemented numerically within the context of isogeometric analysis iga recent contributions in shape optimization within iga have been limited to static or steady state loading conditions in the present contribution the formulation of shape optimization is extended to include time dependent loads and responses a general objective functional is used to accommodate both structural shape optimization and passive control for mechanical problems an adjoint sensitivity analysis is performed at the continuous level and subsequently discretized within the context of iga the methodology and its numerical implementation are tested using benchmark static problems of optimal shapes of orifices in plates under remote bi axial tension and pure shear under quasi static loading conditions the method is validated using a passive control approach with an a priori known solution several applications of time dependent mechanical problems are shown to illustrate the capabilities of this approach in particular a problem is considered where an external load is allowed to move along the surface of a structure the shape of the structure is modified in order to control the time dependent displacement of the point where the load is applied according to a pre specified target copyright 2015 john wiley sons ltd

improved simulated annealing search for structural optimization an optimization algorithm based on simulated annealing is proposed in which the domain of the search is successively reduced based on a probability concept until the stopping criteria are satise ed by introducing the ideas of probability cumulative distribution function and stable energy the selection of initial temperature and equilibrium criterion in the process of simulated annealing becomes easy and effective numerical studies using a set of standard test functions and an example of a 10 bar truss show that the approach is effective and robust in solving both functional and structural optimization problems he achievement of optimum design is a goal naturally attractive to the designer the e eld of structural design is generally characterized by a large number of variables which are usually discrete in dimensions or properties and these variables should satisfy all of the constraints to be a feasible structural design some structuraloptimizationproblemsarefairlyamenableto amathematical programming approach in such instances the design space is continuousand convex the searchmay be deterministic and methods are employed using for example gradient concepts however there is a large class of structural optimization problems with nonconvexities in the design space and with a mix of continuous and discrete variables under such circumstances standard mathematical programming techniques are usually inefe cientbecause they are computationally expensive and are almost assured of locating the relative optimum close to the starting design to overcome these dife culties the stochastic search in structural optimization is considered many methods have become possible with the powerful computing facilities available in recent years among the stochastic algorithms pure random search 1 is the simplest strategy for optimal design some modie ed versions have been suggested such as single start multistart and random directions 2 3 methods in this class generally are quite simple to implement but the appropriate stopping rules are very dife cult to derive recently two classes of powerful search methods which have their philosophical basis in processes found in nature have been widely used in structural optimization the e rst class of methods including genetic algorithms 4 and simulated evolution 5 is based on the spirit of darwinian theory of evolution the second class of methods is generally referred to as simulated annealing techniques 6 because they are qualitatively derived from a simulation of the behaviorofparticlesinthermalequilibriumatagiventemperature because of their global capabilities research on the utilization of these search methods in design optimization has been undertaken 7i 11 somehybridtechniqueshavealsobeendevelopedbycombiningfeaturesofthesetwoalgorithms suchasusingageneticalgorithmtodetermine better annealing schedules 12 and introducing a boltzmanntype mutation or selection process into simulated evolution 13 14 in this paper we propose a method based on simulated annealing that searches from a population as in the method of simulated evolution instead of from a single point the algorithm is called the region reduction simulated annealing rrsa method because it locates the optimum by successively eliminating the regions with low probability of containing the optimum a brief review of basic simulated annealing is given in sec ii which is helpful for development and explanation of the proposed

computational enhancement of an unsymmetric block lanczos algorithm an unsymmetric block lanczos algorithm has been employed for the dynamic analysis of a large system which has arbitrary damping and or repeated or closely spaced eigenvalues in the algorithm development the right and left lanczos vectors are all theoretically biorthogonal to each other however these vectors may lose the biorthogonality owing to cancellation and roundoff errors for the unsymmetric case there can be a breakdown even without numerical errors this paper describes computational techniques which have led to a robust unsymmetric block lanczos algorithm

trust region methods for structural optimization using exact second order sensitivity the performance of multiplier algorithms for structural optimization has been significantly improved by using trust regions the trust regions are constructed using analytical second order sensitivity and within this region the augmented lagrangian is minimized subject to bounds evaluation of first and second derivatives of by the adjoint method does not require derivations of individual implicit constraint functions which makes the method economical eight test problems are considered and a vast improvement over previously used multiplier algorithms has been noted also the algorithm is robust with respect to scaling input parameters and starting designs

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

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

structural dynamics analysis using an unsymmetric block lanczos algorithm this paper describes the way in which an unsymmetric block lanczos algorithm can be employed for approximate eigensolutions or dynamic response solutions for large systems having arbitrary damping and or repeated or closely spaced eigenvalues to reduce a square unsymmetric matrix which may have repeated eigenvalues to block tridiagonal form an unsymmetric block lanczos algorithm is developed right and left lanczos vectors generated by the reduction process are employed to define the reduced order model based on a discrete analytical model of general linear time invariant dynamic systems a special type of starting vector is also introduced which automatically includes the static displacement this reduced order model is employed to compute approximate eigenvalues vectors and dynamic responses of the original system to verify the proposed algorithm examples of an 8 dof beam rotor system are provided for eigensolutions and dynamic responses due to step and random external forces these examples showed promising results indicating that the proposed algorithm is worthy of further study

a recursive quadratic programming method with active set strategy for optimal design recursive quadratic programming methods have become popular in the field of mathematical programming owing to their excellent convergence characteristics there are two recursive quadratic programming methods that have been published in the literature one is by han and the other is by pshenichny published in 1977 and 1970 respectively the algorithm of pshenichny had been undiscovered until now and is examined here for the first time it is found that the proof of global convergence by han requires computing sensitivity coefficients derivatives of all constraint functions of the problem at every iteration this is prohibitively expensive for large scale applications in optimal design in contrast pshenichny has proved global convergence of his algorithm using only an active set strategy this is clearly preferable for large scale applications the method of pshenichny has been coded into a fortran program applications of this method to four example problems are presented the method is found to be very reliable however the method is found to be very sensitive to local minima i e it converges to a local minimum nearest to the starting design thus for optimal design problems which usually possess multiple local minima it is suggested that pshenichny s method be used as part of a hybrid method

improved transient response approximation for general damped systems a new approximation for peak transient dynamic displacements is presented this approximation is based on modal analysis and is valid for general damped structures which includes the case of actively controlled structures intermediate design variable and intermediate response quantity concepts are used to enhance the accuracy of the approximation when this high quality approximation is used within the context of the approximation concepts approach to design synthesis the overall efficiency of the process expected to be improved by reducing the number of complete analyses required for convergence numerical results that illustrate the effectiveness of the method are presented

several simultaneous formulations for transient dynamic response optimization an evaluation several simultaneous formulations for transient dynamic response optimization are described analyzed and compared a key feature of the formulations is that the state variables in addition to the real design variables are treated as independent variables in the optimization process the state variables include different combinations of generalized displacements velocities and accelerations formulations based on different discretization techniques for first order and second order differential equations are presented finite difference newmark s method and methods based on collocation are all discussed similar to the simultaneous analysis and design approach used for the optimization of structures subjected to static loads and the direct collocation transcription method for optimal control the equations of motion are treated as equality constraints a major advantage of these formulations is that special design sensitivity analysis methods are no longer needed however the formulations have larger numbers of variables and constraints therefore sparsity of the problem functions must be exploited in all the calculations advantages and disadvantages of the formulations are discussed the numerical results for an example problem and performance features of the formulations are compared all the formulations converge to the optimum solution for the example problem they are quite efficient for a smaller number of time grid points however the computational times increase as the number of grid points is increased copyright 2009 john wiley sons ltd

a subspace iteration algorithm for the eigensolution of large structures subject to non classical viscous damping reference lmaf article 2000 002view record in web of science record created on 2005 09 14 modified on 2017 05 10

alternative formulations for transient dynamic response optimization three alternative formulations for transient dynamic response optimization of mechanical systems are presented analyzed and evaluated the basic idea of these formulations is to treat various state variables as independent variables in the optimization process in addition to the real design variables that is generalized displacements velocities and accelerations because all functions of the optimization problem become explicit in terms of the variables in these formulations their gradients can be calculated easily as compared to the conventional approach where special design sensitivity analysis methods need to be used also the equations of motion are not integrated explicitly they are imposed as equality constraints for the alternative formulations the optimization problem is quite large in terms of the numbers of variables and constraints however the problem functions are quite sparse which is exploited in the optimization process advantages and disadvantages of the formulations are discussed several cases of two example problems are solved to study the performance of the formulations based on the extensive numerical experiments it is concluded that the proposed formulations work well for the example problems and have potential for further development for practical applications

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

structural shape optimization using equivalent static loads transformed from dynamic loads in structural optimization static loads are generally utilized although real external forces are dynamic dynamic loads have been considered only in small scale problems recently an algorithm for dynamic response optimization using transformation of dynamic loads into equivalent static loads has been proposed the transformation is conducted to match the displacement fields from dynamic and static analyses this algorithm can be applied to large scale problems however the application has been limited to size optimization the present study applies the algorithm to shape optimization because the number of degrees of freedom of finite element models is usually very large in shape optimization it is difficult to conduct dynamic response optimization with conventional methods that directly treat dynamic response in the time domain the optimization process is carried out by interfacing an optimization system and an analysis system for structural dynamics various examples are solved to verify the algorithm the results are compared to the results from static loads it is found that the algorithm using static loads transformed from dynamic loads based on displacement is valid for very large scale shape optimization problems copyright 2005 john wiley sons ltd

an exact penalty function method with global convergence properties for nonlinear programming problems in this paper a new continuously differentiable exact penalty function is introduced for the solution of nonlinear programming problems with compact feasible set a distinguishing feature of the penalty function is that it is defined on a suitable bounded open set containing the feasible region and that it goes to infinity on the boundary of this set this allows the construction of an implementable unconstrained minimization algorithm whose global convergence towards kuhn tucker points of the constrained problem can be established

an iterative svd krylov based method for model reduction of large scale dynamical systems abstract in this paper we propose a model reduction algorithm for approximation of large scale linear time invariant dynamical systems the method is a two sided projection combining features of the singular value decomposition svd based and the krylov based model reduction techniques while the svd side of the projection depends on the observability gramian the krylov side is obtained via iterative rational krylov steps the reduced model is asymptotically stable matches certain moments and solves a restricted h 2 minimization problem we present modifications to the proposed approach for employing low rank gramians in the reduction step and also for reducing discrete time systems several numerical examples from various disciplines verify the effectiveness of the proposed approach it performs significantly better than the q cover a yousouff r e skelton covariance equivalent realizations with applications to model reduction of large scale systems in c t leondes ed control and dynamic systems vol 22 academic press 1985 pp 273 348 a yousouff d a wagie r e skelton linear system approximation via covariance equivalent realizations journal of math anal and appl 196 1985 91 115 and the least squares s gugercin a c antoulas model reduction of large scale systems by least squares linear algebra appl 415 2 3 2006 290 321 methods that have a similar projection structure to the proposed method also in terms of both the h 2 and h error measures its performance is comparable to or sometimes better than that of balanced truncation moreover the method proves to be robust with respect to the perturbations due to usage of approximate gramians

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

design optimization of non linear structures with rate dependent and rate independent constitutive models a design optimization process for non linear structures with history dependent materials modelled using the endochronic constitutive theory is described this constitutive model does not use the concept of a yield surface and describes plastic viscoplastic and viscoelastic behaviour with one set of equations therefore development of the yield surface is not traced in numerical calculations simplifying the implementation of response and sensitivity analyses the total lagrangian formulation incorporating both geometric and material non linear effects is used shape non shape and material design parameters are treated simultaneously using the control volume concept and static and dynamic problems are treated a simple structure is optimized for several cases of static and impact loading conditions to demonstrate the procedure plastic and viscoplastic material behaviour as well as shape and non shape design parameters are treated in the example problems

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

the lanczos biorthogonalization algorithm and other oblique projection methods for solving large unsymmetric systems many powerful methods for solving systems of equations can be regarded as projection methods most of the projection methods known for solving linear systems are orthogonal projection methods but little attention has been given to the class of nonorthogonal or oblique projection methods which is particularly attractive for large nonsymmetric systems the purpose of the paper is to present some methods in the general setting of oblique projection methods and to give some theoretical results some experiments comparing the various algorithms are reported

block krylov subspace methods for approximating the linear combination of ź functions arising in exponential integrators in this paper we present numerical methods for computing the matrix functions arising in exponential integrators the matrix functions are the linear combination of the form j 0 p j a b j where j is related to exponential function by reducing the corresponding systems of linear odes with polynomial inhomogeneity based on block krylov subspace methods two effective exponentials of low dimension matrix are derived to approximate the objective functions some useful a posteriori error estimates are established the algorithms can be combined with time stepping strategy to satisfy the accuracy requirements under a moderate dimension of the block krylov subspace several numerical experiments are presented to demonstrate the efficiency of the methods

sensitivity analysis of the non linear dynamic viscoplastic response of 2 d structures with respect to material parameters a computational procedure is presented for evaluating the sensitivity coefficients of the viscoplastic response of structures subjected to dynamic loading a state of plane stress is assumed to exist in the structure a velocity strain cauchy stress formulation is used and the geometric non linearities arising from large strains are incorporated the jaumann rate is used as a frame indifferent stress rate the material model is chosen to be isothermal viscoplasticity and an associated flow rule is used with a von mises effective stress the equations of motion emanating from a finite element semi discretization are integrated using an explicit central difference scheme with an implicit stress update the sensitivity coefficients are evaluated using a direct differentiation approach since the domain of integration is the current configuration the sensitivity coefficients of the spatial derivatives of the shape functions must be included numerical results are presented for a thin plate with a central circular cutout subjected to an in plane compressive loading the sensitivity coefficients are generated by evaluating the derivatives of the response quantities with respect to young s modulus and two of the material parameters characterizing the viscoplastic response time histories of the response and sensitivity coefficients and spatial distributions at selected times are presented

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

structural redesign for forced response with proportional damping by large admissible perturbations a method to solve redesign inverse design problems of complex structures with forced response amplitude constraints is developed the assumption is made that a structure is excited by harmonic external forces at a given frequency the problem is to find optimum values of structural characteristics in order to achieve a desired level of forced response at one or several locations on the structure the method of large admissible perturbations leap is used the main advantage of leap is that a solution of the inverse problem can be found automatically without trial and error or repetitive finite element analyses using this method algorithms with modal dynamic constraints and static displacement constraints have been developed and tested in the past in this paper the redesign problem with forced response amplitude constraints is formulated the method consists of two distinctive parts first the general perturbation equations are derived they provide relations between the original structure s1 which is known and has undesired forced response amplitudes and the unknown objective structure s2 second the redesign problem is solved by an incremental prediction correction scheme which permits large changes in redesign structural damping is considered in the form of rayleigh damping under this formulation the damping matrix can be diagonalized by use of the real mode shapes of the undamped structure which leads to the derivation of an exact perturbation equation with no loss of accuracy modal dynamic and static constraints may also be imposed simultaneously by the designer the algorithm produces accurate results for large changes in response amplitude without additional finite element analyses the number of extracted modes and the increment size control the accuracy of the results and the computational time the importance of the choice of the objective function is discussed and examples are presented

leray s problem for a viscous incompressible micropolar fluid this work concerns the steady motion of a viscous incompressible micropolar fluid in unbounded domains having cylindrical outlets to infinity we prove the existence of a solution that approaches prescribed parallel solutions along the outlets of the domain we also study the uniqueness the regularity and the asymptotic behavior of the solution

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

on a lemma due to ladyzhenskaya and solonnikov and some applications abstract we present some applications of a lemma by ladyzhenskaya and solonnikov determination of solutions of boundary value problems for stationary stokes and navier stokes equations having an unbounded dirichlet integral zap nauchn sem leningrad otdel mat inst steklov lomi 96 1980 117 160 english transl j soviet math 21 1983 728 761 some other results in that paper referring to stationary navier stokes equations are extended to a non newtonian fluid the so called micropolar fluid this model depends on the microrotational viscosity r which vanishes for a navier stokes fluid we use the lemma in full to show that as r tends to zero the solutions of the ladyzhenskaya solonnikov problem converge to the solutions of the corresponding problem for navier stokes equations in addition we obtain a similar convergence regarding the leray problem for micropolar fluids

methods for investigating the sensitivity of joint resultants to body segment parameter variations two methods are presented for determining the sensitivity of joint resultants to variations in body segment parameters bsps when solving inverse dynamics problems the exact sampling method esm and the approximate variational method avm these methods are illustrated by applying them to a single segment in fixed axis rotation the results indicate that a the avm provides first order linear approximations to the total variations in joint resultants obtained when using the esm b the esm should be used when bsp variations become large and higher order terms in these variations can no longer be ignored as negligible in the bsp dependent expansions for the joint resultants c small percent variations in bsps can propagate into considerably larger percent variations in the joint resultants obtained when solving inverse dynamics problems and d if bsp variations are sufficiently small to allow nonlinear terms in these variations to be ignored as negligible the avm is much simpler and easier to use than the esm

the lanczos algorithm applied to unsymmetric generalized eigenvalue problem application of the two sided lanczos recursion to the unsymmetric generalized eigenvalue problem is presented the system matrices are real and unsymmetric therefore the recursions are performed in real arithmetic and complex arithmetic is employed in the qr algorithm used to extract the eigenvalues of the transformed tridiagonal matrix the biorthonormal transformation of the unsymmetric generalized eigenvalue problem is considered in detail with appropriate proofs presented in appendices issues relating to the computer implementation of the unsymmetric generalized eigenvalue problem are discussed the example problems solved demonstrate the working of the algorithm in extracting the complex and or real eignevalues of an unsymmetric system of matrices also the algorithm is applied to extract a few of the eigenvalues of a large fluid structure interaction problem and the results are compared with the eigenfrequencies extracted by an unsymmetric subspace iteration procedure presented in the literature

nonlinear dynamic response topology optimization using the equivalent static loads method abstract a novel method for nonlinear dynamic response topology optimization is proposed using the equivalent static loads esls method the esls are the loads that generate the same response field of linear static analysis as that of nonlinear dynamic analysis at each time step in the proposed procedure nonlinear dynamic analysis is performed esls are made and linear static topology optimization is carried out with the esls the process cyclically proceeds until the convergence criterion which is specifically defined for this problem is satisfied since the density method for topology optimization is utilized the low density finite elements can cause mesh distortion in nonlinear dynamic analysis transformation variables are introduced for a new update method for the incorporating process of the topology results into nonlinear dynamic analysis also a new objective function is proposed to minimize the peaks of the time dependent transient responses a couple of standard problems and a practical problem are solved to validate the proposed method

adjoint design sensitivity analysis of reduced atomic systems using generalized langevin equation for lattice structures an efficient adjoint design sensitivity analysis method is developed for reduced atomic systems a reduced atomic system and the adjoint system are constructed in a locally confined region utilizing generalized langevin equation gle for periodic lattice structures due to the translational symmetry of lattice structures the size of time history kernel function that accounts for the boundary effects of the reduced atomic systems could be reduced to a single atom s degrees of freedom for the problems of highly nonlinear design variables the finite difference method is impractical for its inefficiency and inaccuracy however the adjoint method is very efficient regardless of the number of design variables since one additional time integration is required for the adjoint gle through numerical examples the derived adjoint sensitivity turns out to be accurate and efficient through the comparison with finite difference sensitivity

optimization of large structures subjected to dynamic loads with the multiplier method the multiplier method for optimization of large scale mechanical and structural systems subjected to dynamic loads is investigated a large scale dynamic response optimization problem is formulated and solved in several alternate ways using the first and second order forms of the equations of motion results are compared with those obtained with the sequential quadratic programming algorithm a primal method in all the cases investigated the multiplier algorithm is more efficient than the primal method therefore it is concluded that the multiplier method is more appropriate for dynamic response optimization of large scale problems

a basis change strategy for the reduced gradient method and the optimum design of large structures this paper proposes a basis change strategy within the reduced gradient method for optimization under linear constraints it ensures a non singular basis matrix at every iteration the same strategy can reliably be used within the generalized reduced gradient method for optimization under non linear constraints this method is applied to the minimum weight design of large structures under displacement and stress constraints exploiting the sparsity of the constraint jacobian matrix

new lagrangian function for nonconvex primal dual decomposition abstract in this paper a new lagrangian function is reported which is particularly suited for large scale nonconvex optimization problems with separable structure our modification convexifies the standard lagrangian function without destroying its separable structure so that the primal dual decomposition technique can be applied even to nonconvex optimization problems furthermore the proposed lagrangian results in two levels of iterative optimization as compared with the three levels needed for techniques recently proposed for nonconvex primal dual decomposition