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

strain driven homogenization of inelastic microstructures and composites based on an incremental variational formulation the paper investigates computational procedures for the treatment of a homogenized macro continuum with locally attached micro structures of inelastic constituents undergoing small strains the point of departure is a general internal variable formulation that determines the inelastic response of the constituents of a typical micro structure as a generalized standard medium in terms of an energy storage and a dissipation function consistent with this type of inelasticity we develop a new incremental variational formulation of the local constitutive response where a quasi hyperelastic micro stress potential is obtained from a local minimization problem with respect to the internal variables it is shown that this local minimization problem determines the internal state of the material for finite increments of time we specify the local variational formulation for a setting of smooth single surface inelasticity and discuss its numerical solution based on a time discretization of the internal variables the existence of the quasi hyperelastic stress potential allows the extension of homogenization approaches of elasticity to the incremental setting of inelasticity focusing on macro strain driven micro structures we develop a new incremental variational formulation of the global homogenization problem where a quasi hyperelastic macro stress potential is obtained from a global minimization problem with respect to the fine scale displacement fluctuation field it is shown that this global minimization problem determines the state of the micro structure for finite increments of time we consider three different settings of the global variational problem for prescribed linear displacements periodic fluctuations and constant stresses on the boundary of the micro structure and discuss their numerical solutions based on a spatial discretization of the fine scale displacement fluctuation field the performance of the proposed methods is demonstrated for the model problem of von mises type elasto visco plasticity of the constituents and applied to a comparative study of micro to macro transitions of inelastic composites copyright 2002 john wiley sons ltd

a reduced multiscale model for nonlinear structural topology optimization abstract this paper presents a reduced multiscale model for macroscopic structural design considering microscopic material nonlinear microstructures this work introduces reduced order model rom to alleviate the heavy computational demand of nonlinear nested multiscale procedures particularly in an optimization framework which requires multiple loops involving similar computations the surrogate model constructed using proper orthogonal decomposition pod and diffuse approximation reduces the computational effort for solving the microscopic boundary value problems multiscale analysis model fe 2 is applied to link structure and microstructures in the two scales maximum stiffness design of the macroscopic structure is realized using a discrete level set topology optimization model it is shown by means of numerical tests that the reduced multiscale model provides reasonable designs as compared to those obtained by the unreduced model while with a significantly reduced computational effort

elliptic solutions to difference non linear equations and related many body problems we study algebro geometric finite gap and elliptic solutions of fully discretized kp or 2d toda equations in bilinear form they are hirota s difference equation for functions starting from a given algebraic curve we express the function and the baker akhiezer function in terms of the riemann theta function we show that the elliptic solutions when the function is an elliptic polynomial form a subclass of the general algebro geometric solutions we construct the algebraic curves of the elliptic solutions the evolution of zeros of the elliptic solutions is governed by the discrete time generalization of the ruijsenaars schneider many body system the zeros obey equations which have the form of nested bethe ansatz equations known from integrable quantum field theories we discuss the lax representation and the action angle type variables for the many body system we also discuss elliptic solutions to discrete analogues of kdv sine gordon and 1d toda equations and describe the loci of the zeros

a strong version of the redundancy capacity theorem of universal coding the capacity of the channel induced by a given class of sources is well known to be an attainable lower bound on the redundancy of universal codes with respect to this class both in the minimax sense and in the bayesian maximin sense we show that this capacity is essentially a lower bound also in a stronger sense that is for most sources in the class this result extends rissanen s 1984 1986 lower bound for parametric families we demonstrate the applicability of this result in several examples e g parametric families with growing dimensionality piecewise fixed sources arbitrarily varying sources and noisy samples of learnable functions finally we discuss implications of our results to statistical inference

constructing solutions to the ultradiscrete painleve equations we investigate the nature of particular solutions to the ultradiscrete painleve equations we start by analysing the autonomous limit and show that the equations possess an explicit invariant which leads naturally to the ultradiscrete analogue of elliptic functions for the ultradiscrete painleve equations ii and iii we present special solutions reminiscent of the casorati determinant ones which exist in the continuous and discrete cases finally we analyse the discrete painleve equation i and show how it contains both the continuous and the ultradiscrete ones as particular limits

general entropy criteria for inverse problems with applications to data compression pattern classification and cluster analysis minimum distance approaches are considered for the reconstruction of a real function from finitely many linear functional values an optimal class of distances satisfying an orthogonality condition analogous to that enjoyed by linear projections in hilbert space is derived these optimal distances are related to measures of distances between probability distributions recently introduced by c r rao and t k nayak 1985 and possess the geometric properties of cross entropy useful in speech and image compression pattern classification and cluster analysis several examples from spectrum estimation and image processing are discussed

element connectivity parameterization for topology optimization of geometrically nonlinear structures in the current element density based topology optimization element stiffness is penalized to yield either solid or void or very weak materials but low density elements appear during and even after optimization iterations especially when nonlinear problems involving large deformation are considered low density finite elements cause serious numerical problems as their tangent stiffness matrices lose positive definiteness to completely eliminate the problem caused by low density elements we propose a new method all finite elements are kept solid throughout the optimization process zero length elastic links are introduced to parameterize inter element connectivity the link stiffness is penalized in this approach the design variables are defined on the links and vary from 0 and 1 corresponding to the unconnected and rigidly connected states since the finite elements used to discretize the analysis domain always remain solid typical numerical problems encountered by the standard element density based formulation disappear to implement the present method several issues such as the handling of the mass constraint and raster imaging are also investigated

an algorithm for calculating γ minimax decision rules under generalized moment conditions we present an algorithm for calculating a minimax decision rule when r is given by a finite number of generalized moment conditions such a decision rule minimizes the maximum of the integrals of the risk function with respect to all distributions in r the inner maximization problem is approximated by a sequence of linear programs this approximation is combined with an elimination technique which quickly reduces the domain of the variables of the outer minimization problem to test for convergence in a final step the inner maximization problem has to be completely solved once for the candidate of the r minimax rule found by the algorithm for an infinite compact parameter space this is done by semi infinite programming the algorithm is applied to calculate robustified bayesian designs in a logistic regression model and r minimax tests in monotone decision problems

the coc algorithm part ii topological geometrical and generalized shape optimization abstract after outlining analytical methods for layout optimization and illustrating them with examples the coc algorithm is applied to the simultaneous optimization of the topology and geometry of trusses with many thousand potential members the numerical results obtained are shown to be in close agreement up to twelve significant digits with analytical results finally the problem of generalized shape optimization finding the best boundary topology and shape is discussed

numerically explicit potentials for the homogenization of nonlinear elastic heterogeneous materials the homogenization of nonlinear heterogeneous materials is much more difficult than the homogenization of linear ones this is mainly due to the fact that the general form of the homogenized behavior of nonlinear heterogeneous materials is unknown at the same time the prevailing numerical methods such as concurrent methods require extensive computational efforts a simple numerical approach is proposed to compute the effective behavior of nonlinearly elastic heterogeneous ma terials at small strains the proposed numerical approach comprises three steps at the rst step a representative volume element rve for a given nonlinear heterogeneous material is de ned and a loading space consisting of all the boundary conditions to be imposed on the rve is discretized into a sufficiently large number of points called nodes at the second step the boundary condition corresponding each node is prescribed on the surface of the rve and the resulting nonlinear boundary value problem is solved by the nite element method fem so as to determine the effective response of the heterogeneous material to the loading as sociated to each node of the loading space at the third step the nodal effective responses are interpolated via appropriate interpolation functions so that the effective strain energy stress strain relation and tangent stiffness tensor of the nonlinear heterogeneous material are provided in a numerically explicit way this leads to a non concurrent nonlinear multiscale approach to the computation of structures made of nonlinearly heterogeneous materials the rst version of the proposed approach uses mutidimensional cubic splines to interpolate effective nodal responses while the second version of the proposed approach takes advantage of an outer product decomposition of multidimensional data into rank one tensors to interpolate effective nodal responses and avoid high rank data these two versions of the proposed approach are applied to a few examples where nonlinear composites whose phases are characterized by the power law model are involved the numerical results given by our approach are compared with available analytical estimates exact results and full fem or concurrent multilevel fem solutions

towards surrogate modeling of material microstructures through the processing variables in order to obtain high performance materials it is of significant importance to be able to depict the material microstructure corresponding to given values of processing variables in the manufacturing process conventional approaches require a knowledge of the internal mechanisms of the evolution in order to numerically simulate the microstructures this work focuses instead on establishing a surrogate model in order to parameterize microstructures of representative volume elements rve using processing variables the surrogate model requires a set of rve microstructure snapshots generated experimentally or numerically by using the proper orthogonal decomposition pod method the parametric space is developed using a series of approximated response surfaces of the pod projection coefficients thereafter rve microstructures may be parameterized for any given value of the processing variables in addition for the purpose of scaling down the storage requirement due to a high quality digital representation the snapshots are given a bi level reduced order epresentation in terms of the extracted common spatial and parametric bases we showcase the approach by parameterizing voronoi simulated rve microstructures under both uniaxial and biaxial conceptional compressions

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

maximum entropy versus minimum risk and applications to some classical discrete distributions the game which can be taken to lie behind the maximum entropy principle is studied refining previous techniques new theoretical results are obtained these results are illustrated by concrete examples pertaining to well known classical models

a pgd based homogenization technique for the resolution of nonlinear multiscale problems abstract this paper deals with the offline resolution of nonlinear multiscale problems leading to a pgd reduced model from which one can derive microinformation which is suitable for design here we focus on an improvement to the proper generalized decomposition pgd technique which is used for the calculation of the homogenized operator and which plays a central role in our multiscale computational strategy this homogenized behavior is calculated offline using the latin multiscale strategy and pgd leads to a drastic reduction in cpu cost indeed the multiscale aspect of the latin method is based on splitting the unknowns between a macroscale and a microcomplement this macroscale is used to capture the homogenized behavior of the structure and then to accelerate the convergence of the iterative algorithm the novelty of this work lies in the reduction of the computation cost of building the homogenized operator in order to do that the problems defined within each subdomain are solved using a new pgd representation of the unknowns in which the separation of the unknowns is performed not only in time and in space but also in terms of the interface macrodisplacements the numerical example presented shows that the convergence rate of the approach is not affected by this new representation and that a significant reduction in computation cost can be achieved particularly in the case of nonlinear behavior the technique used herein is then particularly important to enhance the performances of the latin strategy but is not limited to this method the more general path which is followed in this paper is to solve the microproblems arising in the homogenization for all the configurations that can be encountered in the calculation an additional and novel benefit of this approach is that it includes the calculation of the homogenized tangent operator which gives over the whole time interval and for any cell the relation between the perturbation in terms of macroforces and the perturbation in terms of macrodisplacements in linear viscoelasticity this operator represents the homogenized behavior of the structure exactly

box and ball system as a realization of ultradiscrete nonautonomous kp equation cellular automata which are realized by dynamics of several kinds of balls in an infinite array of boxes are investigated they show soliton patterns even in the case when each box has arbitrary capacity the analytical expression for the soliton patterns are obtained using ultradiscretization of the nonautonomous discrete kp equation

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

three dimensional finite element implementation of the nonuniform transformation field analysis in this paper aspects of the nonuniform transformation field analysis ntfa introduced by michel and suquet int j solids struct 2003 40 6937 6955 are investigated for materials with three dimensional microtopology a novel implementation of the ntfa into the finite element method fem is described in detail whereas the ntfa was originally used in combination with the fast fourier transformation fft in particular the discrete equivalents of the averaging operators required for the preprocessing steps and an algorithm for the implicit time integration and linearization of the constitutive equations of the homogenized material are provided to the authors knowledge this is the first implementation of the method for three dimensional problems further an alternative mode identification strategy is proposed with the aim of small computational cost in combination with good efficiency the new identification strategy is applied to three dimensional metal matrix composites in order to investigate its effective non linear behaviour the homogenized material model is implemented into abaqus standard numerical examples at integration point level and in terms of structural problems highlight the efficiency of the method for three dimensional problems copyright 2010 john wiley sons ltd

multiple addition theorem for discrete and continuous nonlinear problems the addition relation for the riemann theta functions and for its limits which leads to the appearance of exponential functions in soliton type equations is discussed the form of addition property presented resolves itself to the factorization of the n tuple product of the shifted functions and it seems to be useful for analysis of soliton type continuous and discrete processes in the n 1 space time a close relation with the natural generalization of bi and tri linear operators into multiple linear operators concludes the paper

ultradiscretization of elliptic functions and its applications to integrable systems it is shown that there exist three kinds of ultradiscrete analogues of jacobi s elliptic functions in this process the asymptotic behaviour of the poles and the zeros of the functions plays a crucial role using the ultradiscrete analogues and an addition formula exact solutions to the ultradiscrete kp equation are constructed and their relation to the ultradiscrete qrt system is discussed

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

multiscale structural topology optimization with an approximate constitutive model for local material microstructure abstract this paper builds on our recent work xia and breitkopf 2014 on multiscale structural topology optimization where at the microscopic scale local materials are optimized concurrently according to current loading status the former design framework requires intensive computational cost due to large number of repetitive local material optimizations to circumvent this limitation in the present work we construct a reduced database model viewing the local material optimization process as a generalized constitutive behavior using separated representations in this model the database is built from a set of numerical experiments of local material optimizations in the macroscopic strain tensor space each value in the database corresponds to the strain energy density evaluated on a material microstructure optimized according to the imposed macroscopic strain by tensor decomposition a continuous representation of the strain energy density is built as a sum of products of one dimensional interpolation functions as a result of this a priori off line step the effective strain energy and stress strain relations required for macroscopic structural evaluation and optimization are provided in a numerically explicit manner the results given by the reduced database model are compared with full scale results it is also shown that this explicit constitutive behavior representation can well serve multiscale structural design at a significantly reduced computational cost

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

periodic multiwave solutions to the toda type cellular automaton an ultradiscretization of the riemann theta function is proposed the ultradiscretization satisfies an addition formula which is an ultradiscrete analogue of an addition formula for the riemann theta function using the addition formula periodic multiwave solutions to the toda type cellular automaton are obtained

a homogenization method for shape and topology optimization abstract shape and topology optimization of a linearly elastic structure is discussed using a modification of the homogenization method introduced by bendsoe and kikuchi together with various examples which may justify validity and strength of the present approach for plane structures

game theory maximum entropy minimum discrepancy and robust bayesian decision theory we describe and develop a close relationship between two problems that have customarily been regarded as distinct that of maximizing entropy and that of minimizing worst case expected loss using a formulation grounded in the equilibrium theory of zero sum games between decision maker and nature these two problems are shown to be dual to each other the solution to each providing that to the other although topsoe described this connection for the shannon entropy over 20 years ago it does not appear to be widely known even in that important special case we here generalize this theory to apply to arbitrary decision problems and loss functions we indicate how an appropriate generalized definition of entropy can be associated with such a problem and we show that subject to certain regularity conditions the above mentioned duality continues to apply in this extended context this simultaneously provides a possible rationale for maximizing entropy and a tool for finding robust bayes acts we also describe the essential identity between the problem of maximizing entropy and that of minimizing a related discrepancy or divergence between distributions this leads to an extension to arbitrary discrepancies of a well known minimax theorem for the case of kullback leibler divergence the redundancy capacity theorem of information theory for the important case of families of distributions having certain mean values specified we develop simple sufficient conditions and methods for identifying the desired solutions we use this theory to introduce a new concept of generalized exponential family linked to the specific decision problem under consideration and we demonstrate that this shares many of the properties of standard exponential families finally we show that the existence of an equilibrium in our game can be rephrased in terms of a pythagorean property of the related divergence thus generalizing previously announced results for kullback leibler and bregman divergences

the reduced model multiscale method r3m for the non linear homogenization of hyperelastic media at finite strains this paper presents a new multi scale method for the homogenization analysis of hyperelastic solids undergoing finite strains the key contribution is to use an incremental nonlinear homogenization technique in tandem with a model reduction method in order to alleviate the complexity of multiscale procedures which usually involve a large number of nonlinear nested problems to be solved the problem associated with the representative volume element rve is solved via a model reduction method proper orthogonal decomposition the reduced basis is obtained through pre computations on the rve the technique coined as reduced model multiscale method r3m allows reducing significantly the computation times as no large matrix needs to be inverted and as the convergence of both macro and micro problems is enhanced furthermore the r3m drastically reduces the size of the data base describing the history of the micro problems in order to validate the technique in the context of porous elastomers at finite strains a comparison between a full and a reduced multiscale analysis is performed through numerical examples involving different micro and macro structures as well as different nonlinear models neo hookean mooney rivlin it is shown that the r3m gives good agreement with the full simulations at lower computational and data storage requirements

interpolation scheme for fictitious domain techniques and topology optimization of finite strain elastic problems abstract the focus of this paper is on interpolation schemes for fictitious domain and topology optimization approaches with structures undergoing large displacements numerical instability in the finite element simulations can often be observed due to excessive distortion in low stiffness regions a new energy interpolation scheme is proposed in order to stabilize the numerical simulations the elastic energy density in the solid and void regions is interpolated using the elastic energy densities for large and small deformation theory respectively the performance of the proposed method is demonstrated for a challenging test geometry as well as for topology optimization of minimum compliance and compliant mechanisms the effect of combining the proposed interpolation scheme with different hyperelastic material models is investigated as well numerical results show that the proposed approach alleviates the problems in the low stiffness regions and for the simulated cases results in stable topology optimization of structures undergoing large displacements

optimal design and relaxation of variational problems iii abstract this paper has attempted a synthesis of three subjects that have developed separately optimal design homogenization and relaxation in the calculus of variations the underlying theme is that of oscillations they occur with certain definite patterns in minimizing sequences for the variational problem the same patterns recur in nearly optimal designs and in the coefficients of the equation being homogenized each problem is resolved by determining the structure of those oscillations and above all their macroscopic averages it is those averages that solve the relaxed problem they also determine the optimal design and the solution of the homogenized equation we emphasize also the role of optimization the homogenized equations and composite materials of interest here come not from arbitrary mixtures but from those that are extremal since they correspond to minimizing sequences for the variational problem the effective properties of these composites attain the optimal bounds for given materials in fixed proportions

a tropical analogue of fay s trisecant identity and the ultra discrete periodic toda lattice we introduce a tropical analogue of fay s trisecant identity for a special family of hyperelliptic tropical curves we apply it to obtain the general solution of the ultra discrete toda lattice with periodic boundary conditions in terms of the tropical riemann s theta function

computational analysis of nonlinear composite structures using the nonuniform transformation field analysis abstract a reduced model for the effective behavior of nonlinear composites such as metal matrix composite materials has been recently proposed by the authors 1 it extends and improves on the transformation field analysis of dvorak 2 by considering nonuniform transformation strains also called plastic modes and is referred to as the nonuniform transformation field analysis the present study is devoted to the implementation of this new homogenized model into a structural computation a brief account on the reduction procedure is given first then the time integration of the model which is required at each integration point of the structural problem is performed by means of an implicit scheme two examples are discussed the response of the homogenized structure is compared to the exact response of the actual heterogeneous structure computed with a very fine mesh it is seen that not only the structural response is accurately captured by the ntfa model but also the local stress field is correctly approximated

eigendeformation based reduced order homogenization for failure analysis of heterogeneous materials the manuscript presents a novel model reduction approach for periodic heterogeneous media which combines the multiple scale asymptotic msa expansion method with the transformation field analysis tfa to reduce the computational cost of a direct homogenization approach without significantly compromising on solution accuracy the evolution of failure in micro phases and interfaces is modeled using eigendeformation adaptive model improvement strategy incorporating a hierarchical sequence of computational homogenization models is employed to control the accuracy of the model we present the model formulation and the computational details along with verification with respect to direct homogenization and validation with respect to physical experiments studies

reduced basis hybrid computational homogenization based on a mixed incremental formulation a new reduced basis method for the computer assisted homogenization of microheterogeneous materials is proposed key features of the nonuniform transformation field analysis ntfa 1 2 are taken as a point of departure a short coming of the ntfa method is its limitation to simple constitutive models on the microscale a second disadvantage is the possible loss of accuracy when the reduced basis approximating the plastic strains has increasing dimension both issues are related to the simple evolution law for the mode activity coefficients which are the new macroscopic internal variables of the homogenized material in the present contribution a generalization of the ntfa is proposed in which the evolution of the new internal variables is derived from a mixed incremental variational formulation the derivation is based on purely micro mechanical considerations the modification allows for arbitrary generalized standard materials 3 on the microscopic scale including e g crystal visco plasticity additionally the fidelity of the homogenized response is now directly linked to the reduced basis approximating the plastic strain field numerical examples for nonlinear viscous materials and single crystal plasticity outline the accuracy of the proposed multi scale method

design sensitivity analysis and topology optimization of displacement loaded non linear structures abstract a continuum based design sensitivity analysis dsa method for geometrically nonlinear systems with nonhomogeneous boundary conditions is developed to topologically optimize the displacement loaded nonlinear structures in the adjoint variable method the solution space requires just homogeneous boundary conditions even if the original system has nonhomogeneous ones a design sensitivity expression for the instantaneous rigidity functional is derived for the displacement loaded nonlinear topology optimization the tangent stiffness is obtained at the end of the equilibrium iterations in the nonlinear analysis of the original system this stiffness is used in the dsa so that no iteration would be necessary to evaluate the design sensitivity expressions in force loaded systems the solution dose not converge easily because the material distributes sparsely sometimes during optimization however when the displacement loaded system is used there is no convergence difficulty

a general minimax result for relative entropy suppose nature picks a probability measure p sub spl theta on a complete separable metric space x at random from a measurable set p sub spl theta p spl theta spl theta spl isin spl theta then without knowing spl theta a statistician picks a measure q on s finally the statistician suffers a loss d p sub 0 spl par q the relative entropy between p sub spl theta and q we show that the minimax and maximin values of this game are always equal and there is always a minimax strategy in the closure of the set of all bayes strategies this generalizes previous results of gallager 1979 and davisson and leon garcia 1980

topology optimization of geometrically nonlinear structures based on an additive hyperelasticity technique abstract this paper presents a simple but effective additive hyperelasticity technique to circumvent numerical difficulties in solving the material density based topology optimization of elastic structures undergoing large displacements by adding a special hyperelastic material to the design domain excessive distortion and numerical instability occurred in the low density or intermediate density elements are thus effectively alleviated during the optimization process the properties of the additional hyperelastic material are established based on a new interpolation scheme which allows the nonlinear mechanical behaviour of the remodelled structure to achieve an acceptable approximation to the original structure in conjunction with the adjoint variable scheme for sensitivity analysis the topology optimization problem is solved by a gradient based mathematical programming algorithm numerical examples are given to demonstrate the effectiveness of the proposed method

nonuniform transformation field analysis the exact description of the overall behavior of composites with nonlinear dissipative phases requires an infinity of internal variables approximate models involving only a finite number of those can be obtained by considering a decomposition of the microscopic anelastic strain field on a finite set of transformation fields the transformation field analysis of dvorak proc r soc lond a 437 1992 311 corresponds to piecewise uniform transformation fields the present theory considers nonuniform transformation fields comparison with numerical simulations shows the accuracy of the proposed model

coherent dispersion criteria for optimal experimental design we characterize those coherent design criteria which depend only on the dispersion matrix assumed proper and nonsingular of the state of nature which may be a parameter vector or a set of future observables and describe the associated decision problems connections are established with the classical approach to optimal design theory for the normal linear model based on concave functions of the information matrix implications of the theory for more general models are also considered

maximum entropy reconstruction using derivative information part 1 fisher information and convex duality maximum entropy spectral density estimation is a technique for reconstructing an unknown density function from some known measurements by maximizing a given measure of entropy of the estimate here we present a variety of new entropy measures which attempt to control derivative values of the densities our models apply among others to the inference problem based on the averaged fisher information measure the duality theory we develop resembles models used in convex optimal control problems we present a variety of examples including relaxed moment matching with fisher information and best interpolation on a strip

box and ball system with a carrier and ultradiscrete modified kdv equation a new soliton cellular automaton is proposed it is defined by an array of an infinite number of boxes a finite number of balls and a carrier of balls moreover it reduces to a discrete equation obtained from the discrete modified korteweg de vries equation through a limit an algebraic expression of soliton solutions is also proposed

structural boundary design via level set and immersed interface methods we develop and test an algorithmic approach to the boundary design of elastic structures the goal of our approach is two fold first to develop a method which allows one to rapidly solve the two dimensional lam equations in arbitrary domains and compute for example the stresses and second to develop a systematic way of modifying the design to optimize chosen properties at the core our approach relies on two distinct steps given a design we first apply an explicit jump immersed interface method to compute the stresses for a given design shape we then use a narrow band level set method to perturb this shape and progress towards an improved design the equations of 2d linear elastostatics in the displacement formulation on arbitrary domains are solved quickly by domain embedding and the use of fast elastostatic solvers this effectively reduces the dimensionality of the problem by one once the stresses are found the level set method which represents the design structure through an embedded implicit function is used in the second step to alter the shape with velocities depending on the stresses in the current design criteria are provided for advancing the shape in an appropriate direction and for correcting the evolving shape when given constraints are violated

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

information theoretic asymptotics of bayes methods in the absence of knowledge of the true density function bayesian models take the joint density function for a sequence of n random variables to be an average of densities with respect to a prior the authors examine the relative entropy distance d sub n between the true density and the bayesian density and show that the asymptotic distance is d 2 log n c where d is the dimension of the parameter vector therefore the relative entropy rate d sub n n converges to zero at rate log n n the constant c which the authors explicitly identify depends only on the prior density function and the fisher information matrix evaluated at the true parameter value consequences are given for density estimation universal data compression composite hypothesis testing and stock market portfolio selection

high performance model reduction techniques in computational multiscale homogenization a novel model order reduction technique for the solution of the fine scale equilibrium problem appearing in computational homogenization is presented the reduced set of empirical shape functions is obtained using a partitioned version that accounts for the elastic inelastic character of the solution of the proper orthogonal decomposition pod on the other hand it is shown that the standard approach of replacing the nonaffine termby an interpolant constructed using only pod modes leads to ill posed formulations we demonstrate that this ill posedness can be avoided by enriching the approximation space with the span of the gradient of the empirical shape functions furthermore interpolation points are chosen guided not only by accuracy requirements but also by stability considerations the approach is assessed in the homogenization of a highly complex porous metal material computed results show that computational complexity is independent of the size and geometrical complexity of the representative volume element the speedup factor is over three orders of magnitude as compared with finite element analysis whereas the maximum error in stresses is less than 10