global stability ofxn 1 a xpn 1 x1 pn 1 we obtain a general global attractivity result for a difference equation of the formxn 1 f xn xn 1 n 0 1 wheref x y is strictly decreasing in both arguments the result applies to the difference equationxn 1 a xpn 1 x1 pn 1 n 0 1 wherex 1 x0 a 0 andp 1

achieving minimum length scale in topology optimization using nodal design variables and projection functions a methodology for imposing a minimum length scale on structural members in discretized topology optimization problems is described nodal variables are implemented as the design variables and are projected onto element space to determine the element volume fractions that traditionally define topology the projection is made via mesh independent functions that are based upon the minimum length scale a simple linear projection scheme and a non linear scheme using a regularized heaviside step function to achieve nearly 0 1 solutions are examined the new approach is demonstrated on the minimum compliance problem and the popular simp method is used to penalize the stiffness of intermediate volume fraction elements solutions are shown to meet user defined length scale criterion without additional constraints penalty functions or sensitivity filters no instances of mesh dependence or checkerboard patterns have been observed copyright 2004 john wiley sons ltd

topology optimization of multiscale elastoviscoplastic structures in this chapter we take a step further toward the design of multiscale elastoviscoplastic structures using the multiscale design framework discussed in chapter 1 this subject is extremely challenging from both aspects of topology optimization and multiscale modeling first unlike linear designs topology optimization of elastovisoplastic structures encounters instability issues during the iterative solution process and the evaluation of sensitivities is more demanding second the consideration of path dependent plastic behavior at the microscopic scale results in significantly augmented computational burden in terms of computing time and storage requirement when using the fe 2 method

multiscale finite element methods for high contrast problems using local spectral basis functions in this paper we study multiscale finite element methods msfems using spectral multiscale basis functions that are designed for high contrast problems multiscale basis functions are constructed using eigenvectors of a carefully selected local spectral problem this local spectral problem strongly depends on the choice of initial partition of unity functions the resulting space enriches the initial multiscale space using eigenvectors of local spectral problem the eigenvectors corresponding to small asymptotically vanishing eigenvalues detect important features of the solutions that are not captured by initial multiscale basis functions multiscale basis functions are constructed such that they span these eigenfunctions that correspond to small asymptotically vanishing eigenvalues we present a convergence study that shows that the convergence rate in energy norm is proportional to h 1 2 where is proportional to the minimum of the eigenvalues that the corresponding eigenvectors are not included in the coarse space thus we would like to reach to a larger eigenvalue with a smaller coarse space this is accomplished with a careful choice of initial multiscale basis functions and the setup of the eigenvalue problems numerical results are presented to back up our theoretical results and to show higher accuracy of msfems with spectral multiscale basis functions we also present a hierarchical construction of the eigenvectors that provides cpu savings

topology optimisation of manufacturable microstructural details without length scale separation using a spectral coarse basis preconditioner abstract this paper applies topology optimisation to the design of structures with periodic and layered microstructural details without length scale separation i e considering the complete macroscopic structure and its response while resolving all microstructural details as compared to the often used homogenisation approach the approach takes boundary conditions into account and ensures connected and macroscopically optimised microstructures regardless of the difference in micro and macroscopic length scales this results in microstructures tailored for specific applications rather than specific properties manufacturability is further ensured by the use of robust topology optimisation dealing with the complete macroscopic structure and its response is computationally challenging as very fine discretisations are needed in order to resolve all microstructural details therefore this paper shows the benefits of applying a contrast independent spectral preconditioner based on the multiscale finite element method msfem to large structures with fully resolved microstructural details it is shown that a single preconditioner can be reused for many design iterations and used for several design realisations in turn leading to massive savings in computational cost the density based topology optimisation approach combined with a heaviside projection filter and a stochastic robust formulation is used on various problems with both periodic and layered microstructures the presented approach is shown to allow for the topology optimisation of very large problems in matlab specifically a problem with 26 million displacement degrees of freedom in 26 hours using a single computational thread

filters in topology optimization in this article a modified filtered version of the minimum compliance topology optimization problem is studied the direct dependence of the material properties on its pointwise density is replaced by a regularization of the density field by the mean of a convolution operator in this setting it is possible to establish the existence of solutions moreover convergence of an approximation by means of finite elements can be obtained this is illustrated through some numerical experiments the filtering technique is also shown to cope with two important numerical problems in topology optimization checkerboards and mesh dependent designs copyright 2001 john wiley sons ltd

necessary conditions for constant solutions of coupled riccati equations in nash games abstract the asymptotic behaviour of solutions as t for coupled matrix riccati equations occurring in open loop linear quadratic nash games is studied in this paper a general formula representing all possible solutions is given necessary conditions for constant real solutions are derived and an estimate for the rate of convergence is obtained two examples illustrate these results

generalized multiscale finite element methods gmsfem in this paper we propose a general approach called generalized multiscale finite element method gmsfem for performing multiscale simulations for problems without scale separation over a complex input space as in multiscale finite element methods msfems the main idea of the proposed approach is to construct a small dimensional local solution space that can be used to generate an efficient and accurate approximation to the multiscale solution with a potentially high dimensional input parameter space in the proposed approach we present a general procedure to construct the offline space that is used for a systematic enrichment of the coarse solution space in the online stage the enrichment in the online stage is performed based on a spectral decomposition of the offline space in the online stage for any input parameter a multiscale space is constructed to solve the global problem on a coarse grid the online space is constructed via a spectral decomposition of the offline space and by choosing the eigenvectors corresponding to the largest eigenvalues the computational saving is due to the fact that the construction of the online multiscale space for any input parameter is fast and this space can be re used for solving the forward problem with any forcing and boundary condition compared with the other approaches where global snapshots are used the local approach that we present in this paper allows us to eliminate unnecessary degrees of freedom on a coarse grid level we present various examples in the paper and some numerical results to demonstrate the effectiveness of our method

filters in topology optimization based on helmholtz type differential equations the aim of this paper is to apply a helmholtz type partial differential equation as an alternative to standard density filtering in topology optimization problems previously this approach has been successfully applied as a sensitivity filter the usual filtering techniques in topology optimization require information about the neighbor cells which is difficult to obtain for fine meshes or complex domains and geometries the complexity of the problem increases further in parallel computing when the design domain is decomposed into multiple non overlapping partitions obtaining information from the neighbor subdomains is an expensive operation the proposed filter technique requires only mesh information necessary for the finite element discretization of the problem the main idea is to define the filtered variable implicitly as a solution of a helmholtz type differential equation with homogeneous neumann boundary conditions the properties of the filter are demonstrated for various 2d and 3d topology optimization problems in linear elasticity solved on serial and parallel computers copyright 2010 john wiley sons ltd

special finite element methods for a class of second order elliptic problems with rough coefficients in this paper the approximate solution of a class of second order elliptic equations with rough coefficients is considered problems of the type considered arise in the analysis of unidirectional composites where the coefficients represent the properties of the material several methods for this class of problems are presented and it is shown that they have the same accuracy as usual methods have for problems with smooth coefficients the methods are referred to as special finite element methods because they are of finite element type but employ special shape functions chosen to accurately model the unknown solution

global attractivity in a nonlinear difference equation abstract we consider the nonlinear difference equation x n a k 1 m b k x n k n 0 1 2 where a and bk k 1 2 m are nonnegative numbers with b mk 1bk 0 and we are interested in whether all positive solutions are attracted by the positive equilibrium l a 2 a 2 2 b

generalized finite element methods their performance and their relation to mixed methods the notion of a generalized finite element method is introduced this class of methods is analyzed and their relation to mixed methods is discussed the class of generalized finite element methods offers a wide variety of computational procedures from which particular procedures can be selected for particular problems a particular generalized finite element method which is very effective for problems with rough coefficients is discussed in detail

nonlinear structural design using multiscale topology optimization part i static formulation we present a hierarchical multiscale design framework that couples computational homogenization with topology optimization to design a composite structure s microstructure to optimize its nonlinear elastostatic behavior to generate a well posed macroscopic topology optimization problem we use relaxation which requires homogenization to relate the macroscopic homogenized response to its microstructure and because closed form expressions for homogenized properties generally do not exist for materials with nonlinear response we rely on computational homogenization to evaluate them to optimize the homogenized properties of the unit cell we again use topology optimization and to make this unit cell optimization problem well posed we use restriction and thereby obtain a minimum microstructural length scale the coupled nonlinear analyzes and optimization problems are computationally intensive tasks that we resolve with a scalable parallel framework based on a single program multiple data programming paradigm numerical implementation is discussed and examples are provided

scale related topology optimization of cellular materials and structures the integrated optimization of lightweight cellular materials and structures are discussed in this paper by analysing the basic features of such a two scale problem it is shown that the optimal solution strongly depends upon the scale effect modelling of the periodic microstructure of material unit cell muc i e the so called representative volume element rve however with the asymptotic homogenization method used widely in actual topology optimization procedure effective material properties predicted can give rise to limit values depending upon only volume fractions of solid phases properties and spatial distribution of constituents in the microstructure regardless of scale effect from this consideration we propose the design element de concept being able to deal with conventional designs of materials and structures in a unified way by changing the scale and aspect ratio of the de scale related effects of materials and structures are well revealed and distinguished in the final results of optimal design patterns to illustrate the proposed approach numerical design problems of 2d layered structures with cellular core are investigated

global behavior of solutions of a nonlinear second order difference equation abstract our aim in this paper is to investigate the boundedness the persistence the global attractivity the existence of periodic solutions of period 2 and the oscillatory nature of solutions of the equation x n 1 f x n x n 1 n 0 1 where f is a continuous positive and decreasing function on 0 2 0 0 the results are applied to the difference equation y n 1 a y n p 1 y n 1 q n 0 1 where a p q 0

homogenization based topology optimization for high resolution manufacturable micro structures summary this paper presents a projection method to obtain high resolution manufacturable structures from efficient and coarse scale homogenization based topology optimization results the presented approach bridges coarse and fine scale such that the complex periodic microstructures can be represented by a smooth and continuous lattice on the fine mesh a heuristic methodology allows control of the projected topology such that a minimum length scale on both solid and void features is ensured in the final result numerical examples show excellent behavior of the method where performances of the projected designs are almost equal to the homogenization based solutions a significant reduction in computational cost is observed compared to conventional topology optimization approaches

stability and periodicity in discrete delay equations abstract in this research we study the stability of the zero solution of delay difference equations and the existence of unique periodic solution by utilizing fixed point theory mainly we are interested in the qualitative analysis of the complete delay difference equation x t a t x t

concurrent topology optimization design of material and structure within fe2 nonlinear multiscale analysis framework abstract this paper revisits concurrent design of material and structure within fe2 nonlinear multiscale analysis framework for structural stiffness maximization at macroscopic scale design variables are defined at the both scales cellular material models are defined at microscopic scale in a pointwise manner for the considered macroscopic structure they are optimized to adapt the macroscopic structural physical response though linear models are assumed at both scales the macroscopic structural equilibrium is in general nonlinear due to the adaptation of cellular material microstructures for this reason an iterative resolution based on fe2 scheme is developed to address this nonlinearity discrete topology optimization algorithm bi directional evolutionary structural optimization beso is used at the both scales it is shown by means of numerical tests that fe2 scheme can well bridge the two scales and address the nonlinearity reasonable design solutions of the macroscopic structure and its corresponding cellular materials have been obtained by the developed concurrent design framework

efficient two scale optimization of manufacturable graded structures a new method for the optimal design of inhomogeneous meso structured components is presented the method is based on a variant of the free material optimization fmo method instead of coupling a macroscopic model directly with the meso scale in the framework of a two scale approach information from the meso scale is incorporated into the fmo model by additional constraints the fmo result given in terms of optimized tensors is interpreted by the inverse homogenization approach rather than only computing the meso structure for each fmo cell independently approximate continuity of the meso structure is enforced thus the final result is a manufacturable structurally graded material the feasibility of the whole method is demonstrated by means of an academic example

necessary and sufficient condition for two player stackelberg strategy this technical note revisits the open loop stackelberg strategy for a two player game by introducing a new costate which captures the future information of the control input we present a necessary and sufficient condition for the existence and uniqueness of the two player game the optimal strategy is designed in terms of three decoupled and symmetric riccati equations which improves the existing results greatly on computation

an open loop stackelberg strategy for the linear quadratic mean field stochastic differential game this paper is concerned with the open loop linear quadratic lq stackelberg game of the mean field stochastic systems in finite horizon by means of two generalized differential riccati equations the follower first solves a mean field stochastic lq optimal control problem then the leader turns to solve an optimization problem for a linear mean field forward backward stochastic differential equation by introducing new state and costate variables we present a sufficient condition for the existence and uniqueness of the stackelberg strategy in terms of the solvability of some riccati equations and a convexity condition furthermore it is shown that the open loop stackelberg equilibrium admits a feedback representation involving the new state and its mean finally two examples are given to show the effectiveness of the proposed results

topology optimization of hierarchical lattice structures with substructuring abstract this work presents a generalized topology optimization approach for the design of hierarchical lattice structures with the development of an approximation of reduced substructure with penalization arsp model the structure is assumed to be composed of substructures with a common lattice geometry pattern unlike conventional homogenization based designs assuming the separation of scales this work considers two different yet connected scales each substructure is condensed first into a super element with a reduced degrees of freedom and is associated with a density design variable indicating the material volume fraction the density design variable is linked to a lattice geometry feature parameter a surrogate model is particularly built with the aid of proper orthogonal decomposition and diffuse approximation mapping the density to super element stiffness matrix the derivative of super element matrix with respect to the associated density can therefore be evaluated efficiently and explicitly the super element matrix is further augmented with a penalized density to control the structural complexity the optimality criteria method is used for the update of design variables numerical examples show that both the size and the lattice pattern of substructure have essential influences on the design indicating the necessity of performing connected hierarchical modeling and design

discrete time robust hierarchical linear quadratic dynamic games in this paper the theory of robust min max control is extended to hierarchical and multiplayer dynamic games for linear quadratic discrete time systems the proposed game model consists of one leader and many followers while the performance of all players is affected by disturbance the stackelberg nash saddle equilibrium point of the game is derived and a necessary and sufficient condition for the existence and uniqueness of such a solution is obtained in the infinite time horizon it is shown that the solution of the riccati equation is upper bounded under a condition that is called individual controllability by assuming such a condition and using a time varying lyapunov function the input to state stability of the hierarchical dynamic game is achieved considering the optimal feedback strategies of the players and an arbitrary disturbance as the input

infinite horizon linear quadratic stackelberg games for discrete time stochastic systems abstract in this paper we consider infinite horizon linear quadratic stackelberg games for a discrete time stochastic system with multiple decision makers necessary conditions for the existence of the stackelberg strategy set are derived in terms of the solvability of cross coupled stochastic algebraic equations csaes as an important application the hierarchical h constraint control problem for the infinite horizon discrete time stochastic system with multiple channel inputs is solved using the stackelberg game approach computational methods for solving the csaes are also discussed a numerical example is provided to demonstrate the usefulness of the proposed algorithms

tangent operators and design sensitivity formulations for transient non linear coupled problems with applications to elastoplasticity tangent operators and design sensitivities are derived for transient non linear coupled problems the solution process and the formation of tangent operators are presented in a systematic manner and sensitivities for a generalized response functional are formulated via both the direct differentiation and adjoint methods the derived formulations are suitable for finite element implementations analyses of systems with materials that exhibit history dependent response may be obtained directly by applying the analyses of transient non linear coupled systems rate independent elastoplasticity is investigated as a case study and a problem with an analytical solution is analysed for demonstration purposes

h sup spl infin control and estimation with preview part ii fixed size are solutions in discrete time h sup spl infin preview control and fixed lag smoothing problems are solved in general discrete time linear systems via a reduction to equivalent open loop differential games to prevent high order riccati equations found in some solutions the state of the hamilton jacobi system resides in a quotient space of an auxiliary extended state space system the dimension of that auxiliary space is equal to the state space dimension of the original system ignoring the delay

materials with prescribed constitutive parameters an inverse homogenization problem abstract this paper deals with the construction of materials with arbitrary prescribed positive semi definite constitutive tensors the construction problem can be called an inverse problem of finding a material with given homogenized coefficients the inverse problem is formulated as a topology optimization problem i e finding the interior topology of a base cell such that cost is minimized and the constraints are defined by the prescribed constitutive parameters numerical values of the constitutive parameters of a given material are found using a numerical homogenization method expressed in terms of element mutual energies numerical results show that arbitrary materials including materials with poisson s ratio 1 0 and other extreme materials can be obtained by modelling the base cell as a truss structure furthermore a wide spectrum of materials can be constructed from base cells modelled as continuous discs of varying thickness only the two dimensional case is considered in this paper but formulation and numerical procedures can easily be extended to the three dimensional case

the method of moving asymptotes a new method for structural optimization a new method for non linear programming in general and structural optimization in particular is presented in each step of the iterative process a strictly convex approximating subproblem is generated and solved the generation of these subproblems is controlled by so called moving asymptotes which may both stabilize and speed up the convergence of the general process

on linear quadratic stackelberg games with time preference rates this note deals with linear quadratic stackelberg differential games including time preference rates with an open loop information structure the properties of the characteristic matrix associated with the necessary conditions for a stackelberg strategy are pointed out it is shown that such a matrix exhibits a special symmetry property of its eigenvalues sufficient conditions to guarantee a predefined degree of stability are given based on the distribution of the eigenvalues in the complex plane

global behavior of solutions of a nonlinear difference equation our aim in this paper is to investigate the persistence and the global attractivity of solutions of the equationx n 1 f x n x n 1 x n k forn 0 1 where f is a continuous positive and decreasing function on 0 k 1 0 the results are applied to the difference equationy n 1 ay p n 1y q n kforn 0 1 where a p email protected 0 email protected 2 3