a new reiterated structure with optimal macroscopic behavior we consider homogenization of sequences of integral functionals of the type mathcal f h u int omega f h x du x dx omega subset mathbf r n the functions fh are defined in the first variable via a new reiterated structure with m microlevels in particular our results imply that the corresponding homogenized functional becomes optimal as m turns to infinity both the scalar case the conductivity problem and the vector valued case the elasticity problem are considered

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

general topology optimization method with continuous and discrete orientation design using isoparametric projection a general topology optimization method which is capable of simultaneous design of density and orientation of anisotropic material is proposed by introducing orientation design variables in addition to the density design variable in this work the cartesian components of the orientation vector are utilized as the orientation design variables the proposed method supports continuous orientation design which is out of the scope of discrete material optimization approaches as well as design using discrete angle sets the advantage of this approach is that vector element representation is less likely to fail into local optima because it depends less on designs of former steps especially compared with using the angle as a design variable continuous fiber angle optimization by providing a flexible path from one angle to another with relaxation of orientation design space an additional advantage is that it is compatible with various projection or filtering methods such as sensitivity filters and density filters because it is free from unphysical bound or discontinuity such as the one at theta 2 pi and theta 0 seen with direct angle representation one complication of cartesian component representation is the point wise quadratic bound of the design variables that is each pair of element values has to reside in a given circular bound to overcome this issue we propose an isoparametric projection method which transforms box bounds into circular bounds by a coordinate transformation with isoparametric shape functions without having the singular point that is seen at the origin with polar coordinate representation a new topology optimization method is built by taking advantage of the aforementioned features and modern topology optimization techniques several numerical examples are provided to demonstrate its capability copyright c 2014 john wiley sons ltd

the blow up rate for a system of heat equations with nonlinear boundary conditions this paper deals with the blow up rate estimates of positive solutions for systems of heat equations with nonlinear boundary conditions the upper and lower bounds of blow up rate are obtained

blowup rate for heat equation in lipschitz domains with nonlinear heat source terms on the boundary abstract this paper establishes the blowup rate estimate near the blowup time for the heat equation ut u with the nonlinear boundary condition un up on the boundary 0 t for a bounded lipschitz domain and for the nonlinear boundary condition un exp u on the boundary for a two dimensional disc with nonradially symmetric data

composites with symmetry and their extremal properties abstract we identify microstructures that extremize sums of strain energy densities for one energy we exhibit optimal laminar composites that are orthotropic with axes given by the eigenbasis of the homogeneous strain some partial results on the uniqueness of the optimal microstructures are obtained

computational design and additive manufacturing of periodic conformal metasurfaces by synthesizing topology optimization with conformal mapping abstract in this paper we present a computational framework for computational design and additive manufacturing of free form periodic metasurfaces the proposed scheme rests on the level set based topology approach and the conformal mapping theory a metamaterial with pre specified performance is created using a level set based topology optimization method the achieved unit cell is further mapped to the 3d quad meshes on a free form surface by applying the conformal mapping method which can preserve the local shape and angle during the mapping with embedded geometric information the proposed level set based optimization methods not only can act as a motivator for design synthesis but also can be seamlessly hooked with additive manufacturing without the need of cad reconstructions the current computational framework provides a solution to increasing applications involving innovative metamaterial designs on free form surfaces for different fields of interests the performance of the proposed scheme is illustrated through two benchmark examples where a negative poisson s ratio unit cell pattern and a stiff and light inner structure are mapped to 3d free form surfaces and fabricated through additive manufacturing

numerical minimization of dirichlet laplacian eigenvalues of four dimensional geometries we develop the first numerical study in four dimensions of optimal eigenmodes associated with the dirichlet laplacian we describe an extension of the method of fundamental solutions adapted to the four dimensional context based on our numerical simulation and a postprocessing adapted to the identification of relevant symmetries we provide and discuss the numerical description of the eighth first optimal domains

second order sensitivity derivatives in structural analysis haug s 1981 expressions for the calculation of static constraint function second derivatives have resulted in an order of magnitude saving over conventional methods for the calculation of such derivatives a similar but simplified procedure is presented which results in a reduction of computational cost by an additional factor of two the results are presented in a notation that is different from that of haug but more common in other work on structural synthesis and analysis

an investigation concerning optimal design of solid elastic plates abstract we consider the problem of maximizing the integral stiffness of solid elastic plates described by thin plate theory assuming the material volume and plate domain to be given we use the plate thickness function as the design variable and take both maximum and minimum allowable thickness values into account on the basis of a convenient tensorial formulation of the problem where the governing equations are derived by variational analysis and constitute necessary conditions for stationarity we develop an efficient and quite general numerical algorithm by means of which a number of stationary solutions for rectangular and axisymmetric annular plates with various boundary conditions are obtained these numerical results enable us to investigate the optimization problem itself in terms of its major parameters particularly the maximum and minimum values specified for the plate thickness for problems associated with large ratios between these constraint values plate designs with significant integral stiffeners are obtained we find however that these designs are only local optima and that a global optimal plate thickness function does generally neither exist within the class of smooth functions nor within the class of smooth functions with a finite number of discontinuities in order to determine a global optimal solution associated with given thickness constraint values it is therefore necessary to change the optimal design formulation with implications for a number of similar two dimensional optimization problems our results offer valuable indications of the lines along which such changes should be performed

h measures a new approach for studying homogenisation oscillations and concentration effects in partial differential equations new mathematical objects called h measures are introduced for studying oscillations and concentration effects in partial differential equations applications to transport properties and to homogenisation are given as an example of the new results which can be obtained by this approach

minimizers for a double well problem with affine boundary conditions this paper is concerned with the existence of minimizers for functionals having a double well integrand with affine boundary conditions such functionals are related to the so called kohn strang functional which arises in optimal shape design problems in electrostatics or elasticity they are known to be not quasi convex and therefore existence of minimizers is in general guaranteed only for their quasi convex envelopes we generalize previous results in 1 and give necessary and sufficient conditions on the affine boundary conditions for existence of minimizers our method relies on the computation of the quasiconvexification of these functionals by using homogenization theory we also prove by a general argument that their rank one convexifications coincide with their quasi convexifications

parabolic evolution equations and nonlinear boundary conditions on etudie le probleme aux valeurs limites et initiales u 7b a t u f t u 7b b t u g t u 0 t t u 0 u 0 qui peut etre considere comme une contrepartie abstraite a des problemes paraboliques semilineaires

optimal structural design considering flexibility a topology optimization method based on the homogenization method the homogenization design method has been successfully applied to a variety of optimization problems such as the stiffness maximization problem and the eigen frequency maximization problem in this study a methodology to obtain the optimal structure design considering flexibility is developed as a new extension of the homogenization design method first flexibility is formulated using the mutual energy concept second a new multi objective function is proposed to obtain optimal solutions incorporating flexibility and stiffness next the topology optimization procedure is constructed using the homogenization method and sequential linear programming slp finally some examples are presented to confirm that the methodology presented here can provide flexible structures satisfying the problem specifications

sensitivity analysis and optimization of vibration modes in continuum systems abstract the sensitivity analysis of objective functions including the eigenmodes of continuum systems governed by scalar helmholtz equations is carried out in continuum form in addition based on the sensitivity the mode shapes are specified through numerical optimization using the continuum sensitivity and adjoint equation the physical nature of them can be analyzed which helps to explain the nature of the target optimization problem moreover the continuum sensitivity and adjoint equation contribute to the quick numerical implementation of sensitivity analysis using software that can solve an arbitrary partial differential equation directly a scalar helmholtz equation in 1d or 2d domain is considered the sensitivity analysis is performed for the general objective function formulated as a function of the eigenmode in continuum form a minimization problem using the least squared error i e difference between the eigenvector and target mode shape is set as a sample objective function for both the first and second eigenmodes the sensitivity and the adjoint equation are derived for this objective function 1d and 2d numerical sensitivity analysis and optimization examples are studied to illustrate the validity of the derived sensitivity

optimal bounds on the in plane shear moduli for orthotropic elastic composites composite and porous materials often appear in nature many composites may be considered orthotropic such as wood or bone the elastic behaviour of these composites under shear stresses is characterized by three independent shear moduli the authors consider the totality of orthotropic composites made from two isotropic linearly elastic components in fixed proportion for a prescribed triple of shear stresses optimal bounds are found on the strongest and weakest orthotropic composites mathematically this problem is one of constrained optimization the set of constraints is related to the convex hull of a surface in three dimensions for given values of the component elasticities the bounds are computed numerically

on the effective viscoelastic moduli of two phase media i rigorous bounds on the complex bulk modulus the dynamic response of isotropic composites of two viscoelastic isotropic phases mixed in fixed proportions is considered in the frequency range where the acoustic wavelength is much larger than the inhomogeneities the effective bulk modulus bounds of hashin shtrikman walpole are extended to viscoelasticity in this quasi static regime where the properties of the isotropic composite can be described by complex bulk and shear moduli the effective bulk modulus is shown to be constrained to a lens shaped region of the complex plane bounded by the outermost pair of four circular arcs three circular arcs in the case of two dimensional elasticity this is proved using a new variational principle for viscoelasticity together with two established techniques for deriving bounds on effective moduli namely the translation method and the hashin shtrikman method in this application the hashin shtrikman method needs to be generalized to allow the reference tensor to have an associated quasiconvex energy microstructures are identified which have bulk moduli that correspond to various points on each of the circular arcs thus these microstructures have extremal viscoelastic behaviour when the associated arc forms one of the outermost pair the bounds and the extremal microstructures are similar to those obtained for the complex dielectric constant but the methods used here are entirely different

simultaneous isogeometrical shape and material design of functionally graded structures for optimal eigenfrequencies abstract a new optimization strategy for eigenfrequency optimization of functionally graded fg structures within the framework of isogeometric analysis iga approach is introduced the proposed methodology which utilizes a concurrent procedure by combining the shape and material composition optimization of these structures employs an extended form of the standard iga method by allowing for gradation of material properties through patches the distributions of the graded material properties are considered as imaginary surfaces over the computational domain and captured in a fully isogeometric formulation using the same nurbs based parameterization which is employed for the geometry modeling as well as the solution approximation considering the in plane coordinates of the control points defining the design boundary surfaces as well as the applicates of all the control points describing the variations of material properties as design variables we subsequently adopt a mathematical programming algorithm to simultaneously find the optimum shape and material composition of fg structures a couple of illustrative numerical examples in 2d elasticity with eigenfrequencies as their either constraints or objective functions are presented to demonstrate the high performance of the proposed methodology it will be seen that the obtained results by this concurrent optimization procedure have much better dynamic performance compared to the optimal results of the simple shape or material composition design

least weight design of perforated elastic plates ii abstract the paper investigates implications of some recent mathematical developments in the fields of shape optimization and relaxation of variational problems considering the least weight design of perforated elastic plates in either flexure or plane stress for a prescribed compliance it is shown that at low rib densities microstructures consisting of a combination of first and second order infinitesimal ribs is superior to those consisting of purely first order infinitesimal ribs moreover it is indicated that thin ribs of infinite length thickness ratio do not contribute significantly to the stiffness in a direction normal to their plane on the basis of this conclusion a simple specific cost function is derived and then it is used in the design of circular uniformly loaded perforated plates with zero value of poisson s ratio as a basis for comparison several intuitively selected topographies are optimized for the case of simply supported plates and in part ii of this study a variational analysis is used to obtain the optimal solutions for plates with simply supported clamped or loaded edges

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

a multi resolution approach for multi material topology optimization based on isogeometric analysis abstract a novel multi resolution scheme to perform the multi material topology optimization in the framework of isogeometric analysis iga is proposed to create high resolution optimized designs with a lower computational cost a new variable parameter space is supplemented to define design density variables and represent the optimal material distribution utilizing the bivariate b spline basis functions these functions are simply obtained by k refinement strategy in the iga the non uniform rational b spline nurbs basis functions are employed to exactly describe geometric domains and approximate unknown solutions in finite element analysis fea as well an alternating active phase algorithm associated with the block gauss seidel method is employed to convert a multiphase topology optimization problem under multiple volume fraction constraints into many binary phase topology optimization sub problems with only one volume fraction constraint accordingly the number of design variables depends only on one active phase in each of those sub problems regardless of the number of material phases and is significantly decreased in comparison with the original problem the optimality criteria oc method is used as an optimizer to update solutions for such sub problems the effectiveness and robustness of the proposed method are verified via testing various benchmark examples

probability distribution function inspired structural optimization for frequency response problems abstract this study introduces an innovative topological optimization scheme for frequency response problems using the phase field design method in the proposed approach the frequency response function is mapped onto a probability distribution function where the mean value of the probability distribution function becomes the resonance frequency and the variance corresponds to its bandwidth using the mapping concept the frequency characteristics such as the resonance frequency and bandwidth can be directly included in the formulation of the design objective function therefore the mapping process enables to handle frequency characteristics in topological design for frequency response problems as numerical examples the resonance frequency maximization problem of a cantilever beam bandwidth maximization of a clamped end beam and the bandwidth minimization problem of a plasmonic grating coupler are presented to demonstrate the validity of the proposed method

reinforced mindlin plates with extremal stiffness for a prescribed area fraction of stiffeners we characterize the set of stiffener reinforced mindlin plates with extremal overall stiffness the method rests upon the derivation of optimal bounds of the hashin shtrikman type our method is distinct from the usual hashin shtrikman approach we make use of the underlying variational structure behind the hashin shtrikman method to show that the use of a comparison material is redundant we do this by proceeding directly and express the equilibrium equations in terms of positive definite integral operators the positivity of the operators is used to obtain a new hilbert space variational principle for the effective stiffness the associated bounds are shown to be realized by effective rigidities associated with hierarchical laminar arrangements of stiffeners

blow up analysis for a system of heat equations with nonlinear flux which obey different laws abstract we consider a system of heat equations u t u and v t v in 0 t completely coupled by nonlinear boundary conditions u e p v u v u q e v on 0 t we prove that the solutions always blow up in finite time for non zero and non negative initial values also the blow up only occurs on with c 1 t t p 2 p q max u x t c 2 t t p 2 p q log c 3 t t q 1 2 p q max v x t log c 4 t t q 1 2 p q for p q 0 0 1 and 0 p

the blow up behavior of the heat equation with neumann boundary conditions abstract we establish the blow up rate for the solution of the heat equation u t u xx 0 x t 0 subject to neumann boundary conditions u x 0 t 0 u x 1 t e u 1 t with convex initial data

risk averse elastic shape optimization with parametrized fine scale geometry shape optimization of the fine scale geometry of elastic objects is investigated under stochastic loading thus the object geometry is described via parametrized geometric details placed on a regular lattice here in a two dimensional set up we focus on ellipsoidal holes as the fine scale geometric details described by the semiaxes and their orientation optimization of a deterministic cost functional as well as stochastic loading with risk neutral and risk averse stochastic cost functionals are discussed under the assumption of linear elasticity and quadratic objective functions the computational cost scales linearly in the number of basis loads spanning the possibly large set of all realizations of the stochastic loading the resulting shape optimization algorithm consists of a finite dimensional constraint optimization scheme where the cost functional and its gradient are evaluated applying a boundary element method on the fine scale geometry various numerical results show the spatial variation of the geometric domain structures and the appearance of strongly anisotropic patterns

structural topology optimization of vibrating structures with specified eigenfrequencies and eigenmode shapes in vibration optimization problems eigenfrequencies are usually maximized in the optimization since resonance phenomena in a mechanical structure must be avoided and maximizing eigenfrequencies can provide a high probability of dynamic stability however vibrating mechanical structures can provide additional useful dynamic functions or performance if desired eigenfrequencies and eigenmode shapes in the structures can be implemented in this research we propose a new topology optimization method for designing vibrating structures that targets desired eigenfrequencies and eigenmode shapes several numerical examples are presented to confirm that the method presented here can provide optimized vibrating structures applicable to the design of mechanical resonators and actuators

a level set based topology optimization method targeting metallic waveguide design problems in this paper we propose a level set based topology optimization method targeting metallic waveguide design problems where the skin effect must be taken into account since the metallic waveguides are generally used in the high frequency range where this effect critically affects performance one of the most reasonable approaches to represent the skin effect is to impose an electric field constraint condition on the surface of the metal to implement this approach we develop a boundary tracking scheme for the arbitrary lagrangian eulerian ale mesh pertaining to the zero iso contour of the level set function that is given in an eulerian mesh and impose dirichlet boundary conditions at the nodes on the zero iso contour in the ale mesh to compute the electric field since the ale mesh accurately tracks the zero iso contour at every optimization iteration the electric field is always appropriately computed during optimization for the sensitivity analysis we compute the nodal coordinate sensitivities in the ale mesh and smooth them by solving a helmholtz type partial differential equation the obtained smoothed sensitivities are used to compute the normal velocity in the level set equation that is solved using the eulerian mesh and the level set function is updated based on the computed normal velocity finally the utility of the proposed method is discussed through several numerical examples copyright 2011 john wiley sons ltd

probabilistic effective characteristics of cables for superconducting coils the article is devoted to computational estimation of the probabilistic effective properties of the superconducting coil cable 4 component composite for this purpose the well known deterministic effective modules method has been used and reformulated for the case where the composite constituents elastic characteristics in the form of the young moduli and poisson coefficients have random elastic character the variational formulation being a modified virtual work principle has been implemented into the mcceff system which is a probabilistic homogenization oriented finite element program the probabilistic approach implemented in the program is the monte carlo simulation mcs technique based on the random sampling and statistical estimation method by using the program the first four probabilistic moments of the effective elasticity tensor have been computed for the superconductors for coils with periodic cross section and compared against the statistical estimators of the upper and lower bounds of the tensor

mining spatial association rules in image databases in this paper we propose a novel spatial mining algorithm called 9dlt miner to mine the spatial association rules from an image database where every image is represented by the 9dlt representation the proposed method consists of two phases first we find all frequent patterns of length one next we use frequent k patterns k 1 to generate all candidate k 1 patterns for each candidate pattern generated we scan the database to count the pattern s support and check if it is frequent the steps in the second phase are repeated until no more frequent patterns can be found since our proposed algorithm prunes most of impossible candidates it is more efficient than the apriori algorithm the experiment results show that 9dlt miner runs 2 5 times faster than the apriori algorithm

topology design with optimized self adaptive materials significant performance improvements can be obtained if the topology of an elastic structure is allowed to vary in shape optimization problems we study the optimal shape design of a two dimensional elastic continuum for minimum compliance subject to a constraint on the total volume of material the macroscopic version of this problem is not well posed if no restrictions are placed on the structure topology relaxation of the optimization problem via quasiconvexification or homogenization methods is required the effect of relaxation is to introduce a perforated microstructure that must be optimized simultaneously with the macroscopic distribution of material

on accuracy of approximate boundary and distributed h1 shape gradient flows for eigenvalue optimization abstract the boundary and distributed shape gradients of elliptic eigenvalues in shape optimization are approximated by the finite element method we show a priori error estimates for the two approximate shape gradients in h 1 shape gradient flows the convergence analysis shows that the volume integral formula converges faster and offers higher accuracy when the finite element method is used for discretization numerical results verify the theory for the dirichlet case shape optimization examples solved by algorithms illustrate the more effectiveness of distributed shape gradients for the dirichlet case for optimizing a neumann eigenvalue the boundary and volume h 1 flows have the same efficiency moreover we observe that the distributed h 1 shape gradient flow is more efficient than the boundary l 2 shape gradient flow in literature

on the effective elasticity of a two dimensional homogenised incompressible elastic composite the set of effective elasticity tensors for all two dimensional mixtures of two isotropic incompressible elastic materials taken in prescribed proportion is described in two dimensions the effective tensors are completely characterised by bounds on their eigenvalues

a level set method for topology optimization of continuum structures with bounded design domains in practice a continuum structure is usually designed to carry the traction applied to the boundary of the structure subject to prescribed displacements imposed on its boundary the design domains of practical structures are often limited and significantly affect the final optimal design of the structures structural boundaries under traction and prescribed displacements should be treated as a zero level set in the level set methods firstly to overcome the limitations of current level set methods and the stopping issue of structural boundary movements this paper presents a set of new level set based optimization formulae for the optimal design of continuum structures with bounded design domains a set of new normal velocities required by level set movements is given a level set based optimization algorithm is developed and implemented with several robust and efficient numerical techniques which include the level set regularization algorithm gradient projection algorithm nonlinear velocity mapping algorithm and return mapping algorithm secondly in order to overcome the difficulty in nucleating holes in the design domain in the level set based optimization method this paper introduces a new optimization strategy with a small possibility random topology mutations and crossovers a mixed topology optimization algorithm is implemented and presented for the compliance minimization problems of continuum structures with material volume constraints the validity and effectiveness of the proposed method are demonstrated with two examples

blow up properties for heat equations coupled via different nonlinearities this paper deals with heat equations coupled via exponential and power nonlinearities subject to null dirichlet boundary conditions the complete and optimal classification on non simultaneous and simultaneous blow ups is proposed by four sufficient and necessary conditions we find out that in some exponent region the blow up properties of the solutions depend much on the choosing of initial data moreover all kinds of non simultaneous and simultaneous blow up rates are obtained

topology optimization using a reaction diffusion equation abstract this paper presents a structural topology optimization method based on a reaction diffusion equation in our approach the design sensitivity for the topology optimization is directly employed as the reaction term of the reaction diffusion equation the distribution of material properties in the design domain is interpolated as the density field which is the solution of the reaction diffusion equation so free generation of new holes is allowed without the use of the topological gradient method our proposed method is intuitive and its implementation is simple compared with optimization methods using the level set method or phase field model the evolution of the density field is based on the implicit finite element method as numerical examples compliance minimization problems of cantilever beams and force maximization problems of magnetic actuators are presented to demonstrate the method s effectiveness and utility

finite element approximation to the extremal eigenvalue problem for inhomogenous materials in this paper an extremal eigenvalue problem corresponding to an inhomogeneous membrane which is composed of two different materials with different densities is investigated the convergence of the finite element discretization and the error order for the smallest eigenvalue are obtained a monotonic decreasing algorithm is presented to solve the discretized problem and numerical examples are given to demonstrate the error estimation as well as the efficiency of the method

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

eulerian shape design sensitivity analysis and optimization with a fixed grid conventional shape optimization based on the finite element method uses lagrangian representation in which the finite element mesh moves according to shape change while modern topology optimization uses eulerian representation in this paper an approach to shape optimization using eulerian representation such that the mesh distortion problem in the conventional approach can be resolved is proposed a continuum geometric model is defined on the fixed grid of finite elements an active set of finite elements that defines the discrete domain is determined using a procedure similar to topology optimization in which each element has a unique shape density the shape design parameter that is defined on the geometric model is transformed into the corresponding shape density variation of the boundary elements using this transformation it has been shown that the shape design problem can be treated as a parameter design problem which is a much easier method than the former a detailed derivation of how the shape design velocity field can be converted into the shape density variation is presented along with sensitivity calculation very efficient sensitivity coefficients are calculated by integrating only those elements that belong to the structural boundary the accuracy of the sensitivity information is compared with that derived by the finite difference method with excellent agreement two design optimization problems are presented to show the feasibility of the proposed design approach

blow up phenomena for a system of semilinear heat equations with nonlinear boundary flux abstract this paper deals with the blow up for a system of semilinear r laplace heat equations with nonlinear boundary flux it is shown that under certain conditions on the nonlinearities and data blow up will occur at some finite time and when blow up does occur upper and lower bounds for the blow up time 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

a consistent relaxation of optimal design problems for coupling shape and topological derivatives in this article we introduce and analyze a general procedure for approximating a black and white shape and topology optimization problem with a density optimization problem allowing for the presence of grayscale regions our construction relies on a regularizing operator for smearing the characteristic functions involved in the exact optimization problem and on an interpolation scheme which endows the intermediate density regions with fictitious material properties under mild hypotheses on the smoothing operator and on the interpolation scheme we prove that the features of the approximate density optimization problem material properties objective function etc converge to their exact counterparts as the smoothing parameter vanishes in particular the gradient of the approximate objective functional with respect to the density function converges to either the shape or the topological derivative of the exact objective these results shed new light on the connections between these two different notions of sensitivities for functions of the domain and they give rise to different numerical algorithms which are illustrated by several experiments

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

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

eigenfrequency optimization in optimal design we maximize the first eigenfrequency or a sum of the first ones of a bounded domain occupied by two elastic materials with a volume constraint for the most rigid one a relaxed formulation of this problem is introduced which allows for composite materials as admissible designs these composites are obtained by homogenization of fine mixtures of the two original materials we prove a saddle point theorem that permits to reduce the full unknown set of admissible composite designs to the smaller set of sequential laminates which is explicitly known although our relaxation theorem is valid only for two non degenerate materials we deduce from it a numerical algorithm for eigenfrequency optimization in the context of optimal shape design i e when one of the two materials is void as is the case with all homogenization methods our algorithm can be seen as a topology optimizer numerical results are presented for various two and three dimensional problems

composite material design of two dimensional structures using the homogenization design method summary composite materials of two dimensional structures are designed using the homogenization design method the composite material is made of two or three dierent material phases designing the composite material consists of nding a distribution of material phases that minimizes the mean compliance of the macrostructure subject to volume fraction constraints of the constituent phases within a unit cell of periodic microstructures at the start of the computational solution the material distribution of the microstructure is represented as a pure mixture of the constituent phases as the iteration procedure unfolds the component phases separate themselves out to form distinctive interfaces the eective material properties of the articially mixed materials are dened by the interpolation of the constituents the optimization problem is solved using the sequential linear programming method both the macrostructure and the microstructures are analysed using the nite element method in each iteration step several examples of optimal topology design of composite material are presented to demonstrate the validity of the present numerical algorithm copyright 2001 john wiley sons ltd

optimal bounds on effective elastic tensors for orthotropic composites we consider the totality of orthotropic composites made from two isotropic linearly elastic components in fixed proportion the elastic properties of orthotropic composites are characterized by nine independent moduli we provide bounds for six of these namely the three youngs moduli and three in plane shear moduli the bounds are optimal and correlate the six moduli

a material based model for topology optimization of thermoelastic structures this paper presents the development of a computational model for the topology optimization problem using a material distribution approach of a 2 d linear elastic solid subjected to thermal loads with a compliance objective function and an isoperimetric constraint on volume defining formally the augmented lagrangian associated with the optimization problem the optimality conditions are derived analytically the results of analysis are implemented in a computer code to produce numerical solutions for the optimal topology considering the temperature distribution independent of design the design optimization problem is solved via a sequence of linearized subproblems the computational model developed is tested in example problems the influence of both the temperature and the finite element model on the optimal solution obtained is analysed

determination of the closure of the set of elasticity functionals we determine the closure for mosco convergence in l2 3 of the set of elasticity functionals we prove that this closure coincides with the set of all non negative lower semicontinuous quadratic functionals which are objective i e which vanish for rigid motions the result is still valid if we consider only the set of isotropic elasticity functionals which have a prescribed poisson coefficient this shows that a very large family of materials can be reached when homogenizing a composite material with highly contrasted rigidity coefficients

topological design for vibrating structures the topological optimization technique using micro scale voids with the homogenization method for stress analysis of perforated structures has been applied to solve stiffness maximization problem with success here the homogenization design technique is extended to design problems concerning vibrating structures an extended optimization algorithm is also derived to maximize a set of eigenvalues as well as to identify the topology design for specified eigenvalues to characterize forced vibration of a structure

on characterizing the set of possible effective tensors of composites the variational method and the translation method a general algebraic framework is developed for characterizing the set of possible effective tensors of composites a transformation to the polarization problem simplifies the derivation of the hashin shtrikman variational principles and simplifies the calculation of the effective tensors of laminate materials a general connection is established between two methods for bounding effective tensors of composites the first method is based on the variational principles of hashin and shtrikman the second method due to tartar murat lurie and cherkaev uses translation operators or equivalently quadratic quasiconvex functions a correspondence is established between these translation operators and bounding operators on the relevant non local projection operator t1 an important class of bounds namely trace bounds on the effective tensors of two component media are given a geometrical interpretation after a suitable fractional linear transformation of the tensor space each bound corresponds to a tangent plane to the set of possible tensors a wide class of translation operators that generate these bounds is found the extremal translation operators in this class incorporate projections onto spaces of antisymmetric tensors these extremal translations generate attainable trace bounds even when the tensors of the two components are not well ordered in particular they generate the bounds of walpole on the effective bulk modulus the variational principles of gibiansky and cherkaev for bounding complex effective tensors are reviewed and used to derive some rigorous bounds that generalize the bounds conjectured by golden and papanicolaou an isomorphism is shown to underlie their variational principles this isomorphism is used to obtain dirichlet type variational principles and bounds for the effective tensors of general non selfadjoint problems it is anticipated that these variational principles which stem from the work of gibiansky and cherkaev will have applications in many fields of science

optimal bounds and microgeometries for elastic two phase composites consider two isotropic linearly elastic materials with elasticity tensors c 1 c 2 and assume c 1 c 2 we prove that the class of effective tensors of composites built from these materials with given volume fractions that are invariant under a given symmetry group is such that each of its elements is bounded above and below by tensors in the same class corresponding to finite rank laminates this implies that for any imposed uniform strain or stress field optimal bounds on the effective strain or stress energy per unit volume are attained by finite rank laminates explicit bounds on the strain energy with no symmetry assumptions are given in dimensions 2 and 3 these bounds are more stringent than the classical voigt reuss bounds finally explicit bounds and microgeometries are given for the effective moduli of composites with cubic symmetry

a non local variational approach to the elastic energy minimalization of martensitic polycrystals solid solid phase transformations in polycrystals are considered in the context of energy minimization the energy of a single crystal is specified by a non convex multi well energy function in the approximation of infinitesimal deformations the number of wells corresponds to the number of distinct phases and each is assumed to have the same isotropic elastic modulus the polycrystal s energy is defined a priori via minimization of the energy functional with proper account of the orientation distribution with respect to all kinematically admissible displacement fields a variational principle of hashin shtrikman type is derived for the polycrystal s energy by developing and generalizing the approach of bruno and co workers the variational principle involves a non local functional with a green s function related kernel operating on trial transformation fields which are appropriately constrained to accommodate both the single crystal s constitutive law and the polycrystal s texture for a statistically uniform polycrystal the variational principle is reformulated to require minimization with respect to all possible two point correlation functions of submicrostructure compatible with the texture this variational principle is applied to derive upper bounds for a statistically uniform polycrystal by employing a separation of scales i e by constraining the set of trial fields to those with the property that the scale of the trial submicrostructure is much finer than the scale of the polycrystal s texture subsequent optimization with respect to this submicrostructure for each particular orientation reveals a connection with relaxation of a single crystal with fixed volume fractions and associated h measures as discussed by kohn the resulting upper bound is developed and compared with a bound derived by bruno et al for some examples the new bound is demonstrated to be sharper than the latter as a result of an improved optimization procedure the present approach also extends that of bruno et al to more general orientation distribution statistics and clarifies the effect of incompatibility of transformation strains in the single crystals for the overall performance of polycrystals

homogenization and optimal bounds in linear elasticity in their celebrated paper of 1963 hashin shtrikman 1 addressed the problem of determining optimal bounds for the bulk and shear moduli of a statistically isotropic elastic composite material with arbitrary isotropic phase geometry they derived a set of bounds for these moduli in the physically meaningful case of three dimensional elasticity in the case of a two phase composite let k 1 1 and k 2 2 respectively denote the bulk and shear moduli for the first and the second phase let k denote their analogues for the composite and let 0 stand for the volume fraction of the first phase in the composite under the ordering restriction that both

thermal avalanche for blowup solutions of semilinear heat equations we consider the semilinear heat equation ut u up both in n and in a bounded domain with homogeneous dirichlet boundary conditions with 1 t we consider a natural approximation by reaction problems with global solution and pass to the limit as is well known the limit solution undergoes complete blowup after it blows up at t t the continuation is identically infinite for all t t we perform here a deeper analysis of how complete blowup occurs actually the singularity set of a solution that blows up as t t propagates instantaneously at time t t to cover the whole space producing a catastrophic discontinuity between t t and t t this is called the avalanche we describe its formation as a layer appearing in the limit of the natural approximate problems after a suitable scaling the initial structure of the layer is given by the solution of a limit problem described by a simple ordinary differential equation as t proceeds past t the solutions of the approximate problems have a traveling wave behavior with a speed that we compute the situation in the inner region depends on the type of approximation a conical pattern is formed with time when we approximate the power up by a flat truncation at level n while for tangent truncations we get an exponential increase in time and a diffusive spatial pattern 2003 wiley periodicals inc

binary level set methods for topology and shape optimization of a two density inhomogeneous drum abstract we optimize eigenvalues in optimal shape design using binary level set methods the interfaces of subregions are represented implicitly by the discontinuities of binary level set functions taking two values 1 or 1 at convergence a binary constraint is added to the original model problems we propose two variational algorithms to solve the constrained optimization problems one is a hybrid type by coupling the lagrange multiplier approach for the geometry constraint with the augmented lagrangian method for the binary constraint the other is devised using the lagrange multiplier method for both constraints the two iterative algorithms are both largely independent of the initial guess and can satisfy the geometry constraint very accurately during the iterations intensive numerical results are presented and compared with those obtained by level set methods which demonstrate the effectiveness and robustness of our algorithms

solutions to shape and topology eigenvalue optimization problems using a homogenization method a solution strategy to find the shape and topology of structures that maximize a natural frequency is presented the methodology is based on a homogenization method and the representation of the shape of the structure as a material property the problem is formulated as a reinforcement problem in which a given structure is reinforced using a prescribed amount of material two dimensional plane elasticity problems are considered examples are presented for illustration

fourth order moments of nonnegative measures on s2 and applications this paper is mainly devoted to a study of the fifteen dimensional set of all fourth order moments of a nonnegative radon borel regular measure on s 2 we seek a complete characterization of the set a closed cone with the help of the hilbert decomposition theorem for nonnegative polynomials of degree four in two variables special emphasis is placed on the boundary of this set which is shown to be generated by atomic measures made of five dirac masses or fewer which are located on the intersection of s 2 with the zero set of a quadratic form as a consequence every point of the moment set is generated by atomic measures made of six dirac masses or less this study may be viewed as a contribution to the moment problem note that the analogous two dimensional case has been analyzed in avellaneda milton am potential applications for this result are however manifold especially in the field of homogenization which is of especial interest to us in the setting of linearized elasticity effective properties of fine mixtures of two phases are investigated specifically the goal is to analyze the coefficients of the linear second order elliptic system satisfied by the weak limit u of the solution fields w to a sequence of elasticity problems of the form

an order clique based approach for mining maximal co locations most algorithms for mining spatial co locations adopt an apriori like approach to generate size k prevalence co locations after size k 1 prevalence co locations however generating and storing the co locations and table instances is costly a novel order clique based approach for mining maximal co locations is proposed in this paper the efficiency of the approach is achieved by two techniques 1 the spatial neighbor relationships and the size 2 prevalence co locations are compressed into extended prefix tree structures which allows the order clique based approach to mine candidate maximal co locations and co location instances and 2 the co location instances do not need to be stored after computing some characteristics of the corresponding co location which significantly reduces the execution time and space required for mining maximal co locations the performance study shows that the new method is efficient for mining both long and short co location patterns and is faster than some other methods in particular the join based method and the join less method

a multilevel level set method for optimizing eigenvalues in shape design problems in this paper we consider optimal design problems that involve shape optimization the goal is to determine the shape of a certain structure such that it is either as rigid or as soft as possible to achieve this goal we combine two new ideas for an efficient solution of the problem first we replace the eigenvalue problem with an approximation by using inverse iteration second we use a level set method but rather than propagating the front we use constrained optimization methods combined with multilevel continuation techniques combining these two ideas we obtain a robust and rapid method for the solution of the optimal design problem

blow up estimates for system of heat equations coupled via nonlinear boundary flux abstract this paper deals with a system of heat equations coupled via nonlinear boundary flux the precise blow up rate estimates are established together with the blow up set it is observed that there is some quantitative relationship regarding the blow up properties between the heat system with coupled nonlinear boundary flux terms and the corresponding reaction diffusion system with the same nonlinear terms as the source

blow up estimates of the positive solution of a parabolic system this paper establishes the blow up estimates for the systems ut u 0 vt v 0 in br 0 t br rn with the nonlinear boundary conditions u um1vn1 and v um2vn2 on sr 0 t where 0 m1 1 m2 and 0 n2 1 n1 we prove that c t t 2 max u x t c t t 2 and c t t 2 max v x t c t t 2 under some monotonicity assumptions on the initial values where n1 n2 1 m2 m1 1 and n1m2 1 m1 1 n2

a topology optimization method based on the level set method incorporating a fictitious interface energy this paper proposes a new topology optimization method which can adjust the geometrical complexity of optimal configurations using the level set method and incorporating a fictitious interface energy derived from the phase field method first a topology optimization problem is formulated based on the level set method and the method of regularizing the optimization problem by introducing fictitious interface energy is explained next the reaction diffusion equation that updates the level set function is derived and an optimization algorithm is then constructed which uses the finite element method to solve the equilibrium equations and the reaction diffusion equation when updating the level set function finally several optimum design examples are shown to confirm the validity and utility of the proposed topology optimization method

mining maximal hyperclique pattern a hybrid search strategy a hyperclique pattern is a new type of association pattern that contains items which are highly affiliated with each other specifically the presence of an item in one transaction strongly implies the presence of every other item that belongs to the same hyperclique pattern in this paper we present an algorithm for mining maximal hyperclique patterns which specifies a more compact representation of hyperclique patterns and are desirable for many applications such as pattern based clustering our algorithm exploits key advantages of both the depth first search dfs strategy and the breadth first search bfs strategy indeed we adapt the equivalence pruning method one of the most efficient pruning methods of the dfs strategy into the process of the bfs strategy our experimental results show that the performance of our algorithm can be orders of magnitude faster than standard maximal frequent pattern mining algorithms particularly at low levels of support

optimal design of multiphase composites under elastodynamic loading abstract an algorithm is proposed to optimize the performance of multiphase structures composites under elastodynamic loading conditions the goal is to determine the distribution of material in the structure such that the time averaged total stored energy of structure is minimized a penalization strategy is suggested to avoid the checkerboard instability simultaneously to generate near 0 1 topologies as a result of this strategy the solutions of presented algorithm are sufficiently smooth and possess the regularity of h 1 function space a simple method for the continuum adjoint sensitivity analysis of the corresponding pde constrained optimization problem is presented it is general and can be easily applied to a wide range of alternative problems the success of the introduced algorithm is studied by numerical experiments on two dimensional model problems for different numbers of phases according to numerical results the objective functional is reduced monotonically with iterations moreover the final topologies at the optimal solutions are near 0 1 the dynamic behavior of optimal designs is compared to that of initial ones to show the impact of optimization on the performance of structures

bounds for effective parameters of heterogeneous media by analytic continuation we give a mathematical formulation of a method for obtaining bounds on effective parameters developed by d bergman and g w milton this method in contrast to others used before does not rely on a variational principle but exploits the properties of the effective parameter as an analytic function of the component parameters the method is at present restricted to two component media