a level set method for shape and topology optimization of large displacement compliant mechanisms a parameterization level set method is presented for structural shape and topology optimization of compliant mechanisms involving large displacements a level set model is established mathematically as the hamilton jacobi equation to capture the motion of the free boundary of a continuum structure the structural design boundary is thus described implicitly as the zero level set of a level set scalar function of higher dimension the radial basis function with compact support is then applied to interpolate the level set function leading to a relaxation and separation of the temporal and spatial discretizations related to the original partial differential equation in doing so the more difficult shape and topology optimization problem is now fully parameterized into a relatively easier size optimization of generalized expansion coefficients as a result the optimization is changed into a numerical process of implementing a series of motions of the implicit level set function via an existing efficient convex programming method with the concept of the shape derivative the geometrical non linearity is included in the rigorous design sensitivity analysis to appropriately capture the large displacements of compliant mechanisms several numerical benchmark examples illustrate the effectiveness of the present level set method in particular its capability of generating new holes inside the material domain the proposed method not only retains the favorable features of the implicit free boundary representation but also overcomes several unfavorable numerical considerations relevant to the explicit scheme the reinitialization procedure and the velocity extension algorithm in the conventional level set method

multiscale approach for optimal design in conductivity of composite materials this paper discusses the multiscale approach for optimal design in conductivity of composite materials the homogenization method and the multiscale asymptotic method are presented the associated numerical algorithms and the convergence analysis are provided finally numerical examples are carried out to confirm the validity of the algorithm

minimum length scale control in structural topology optimization based on the moving morphable components mmc approach abstract a novel approach which can control the minimum length scale in topology optimization in a straightforward and explicit way is proposed in the present paper this approach is constructed under the so called moving morphable components mmc based solution framework where optimized structural topology can be found by changing the shapes and layout of a set of trapezoid shaped structural components on a fixed finite element mesh the extended finite element method xfem is employed for structural response analysis and shape sensitivity information is obtained by carrying out numerical integration along the structural boundary a precision definition of the minimum length scale of a structure is also suggested based on the proposed solution framework compared with existing approaches the proposed approach can achieve minimum length scale control by simply setting lower bounds on a set of geometric design variables only a small number of purely geometric constraints imposed on the sizes of intersection regions should be considered explicitly in order to get a complete minimum length scale control numerical examples demonstrate the effectiveness of the proposed method

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

color level sets a multi phase method for structural topology optimization with multiple materials in this paper we address the problem of structural shape and topology optimization in a multi material domain a level set method is employed as an alternative approach to the popular homogenization based methods of rule of mixtures for multi material modeling a multi phase level set model is adapted for material and topology representation this model eliminates the need for a material interpolation or phase mixing scheme it only requires m level set functions to represent a structure of n 2m different material phases in a principle similar to combining colors from the three primary colors therefore this multi phase model may be referred to as a color level set representation which has its unique benefits it is flexible to handle complex topologies it substantially reduces the number of model functions when n 3 it automatically avoids the problem of overlap between material phases of a conventional partitioning approach we describe numerical techniques for efficient and robust implementation of the method by embedding a rectilinear grid in a fixed finite element mesh defined on a reference design domain this would separate the issues of accuracy in numerical calculations of the physical equation and in the level set model propagation a gradient projection method is described for incorporating multiple constraints in the problem finally the benefits and the advantages of the developed method are illustrated with several 2d examples of mean compliance minimization of multi material structures

static and vibrational shape and topology optimization using homogenization and mathematical programming abstract the homogenization method together with mathematical optimization concepts were combined to perform two dimensional shape and topology optimization of structures under volume constraint static as well as single or multi eigenvalue vibrational optimization of structures are considered the basic assumption of the method is that the structure under consideration is formed by the spatial repetition of a non homogeneous microstructure composed of solid material and void regions the method upon convergence allocates material or voids at different parts of the structure and thus an optimum shape is achieved a detailed study was made to assess the effect of various microstructural models on the final optimum shape good convergence characteristics were obtained for all examples considered comparison of the algorithm with published results using the optimality criteria method was attempted and similarities and differences are discussed the results show that the proposed method can perform accurate static and vibrational optimization furthermore its generality allows the implementation of more than one constraint in a straightforward manner a very important feature for the design engineer

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

optimal design of elastic plates with a constraint on the slope of the thickness function abstract a constraint on the maximum absolute value of the slope of the thickness function is found to have a profound influence on the problem of optimal design of axisymmetric elastic plates by securing a unique regular solution it is also shown that except for the case of axisymmetric deformation the constraint is active everywhere and the optimal design is a plate subdivided into annular regions of constant slope equal to the maximum value permitted

new results on stability analysis for systems with discrete distributed delay the integral inequality technique is widely used to derive delay dependent conditions and various integral inequalities have been developed to reduce the conservatism of the conditions derived in this study a new integral inequality was devised that is tighter than existing ones it was used to investigate the stability of linear systems with a discrete distributed delay and a new stability condition was established the results can be applied to systems with a delay belonging to an interval which may be unstable when the delay is small or nonexistent three numerical examples demonstrate the effectiveness and the smaller conservatism of the method

faster algorithms for all pairs approximate shortest paths in undirected graphs let g v e be a weighted undirected graph having nonnegative edge weights an estimate hat delta u v of the actual distance delta u v between u v in v is said to be of stretch t if and only if delta u v leq hat delta u v leq t cdot delta u v computing all pairs small stretch distances efficiently both in terms of time and space is a well studied problem in graph algorithms we present a simple novel and generic scheme for all pairs approximate shortest paths using this scheme and some new ideas and tools we design faster algorithms for all pairs t stretch distances for a whole range of stretch t and we also answer an open question posed by thorup and zwick in their seminal paper j acm 52 2005 pp 1 24

effective stiffnesses of transversely non homogeneous plates with unidirectional periodic structure abstract this paper presents a derivation and analysis of closed formulae for the membrane bending and reciprocal effective stiffnesses of elastic orthotropic and transversely asymmetric plates with microstructures periodic in one direction the derivation is based on the concept of imposing hencky type displacement constraints on the solutions to the caillerie kohn vogelius local homogenization problems reduction of the dimension of the problems by one makes it possible to solve the new approximate local problems exactly thus enabling one to find the relevant formulae in closed form ready for engineering applications as well as for optimization and sensitivity studies this paper generalizes the previous results of the author int j solids structures 29 309 326 1992 concerning the bending of transversely symmetric periodic plates

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

some convergence results in perimeter controlled topology optimization abstract the computation of optimal topologies of elastic continuum structures using a constraint on the perimeter is investigated predicting macroscopic black white topologies without the use of homogenization techniques this approach is presently one of the most attractive approaches in topology optimization mathematical justifications are given for both the relaxation of the discrete value constraint on the design variable and for the finite element discretizations it turns out that the way in which the perimeter has been calculated to date the numerical results will not approximate the intended original problem but one with a taxi cab perimeter which measures lengths of structural edges after projection onto the coordinate axes

a review of an optimal design problem for a plate of variable thickness we revisit a classic design problem for a plate of variable thickness under the model of kirchhoff our main contribution has two goals one is to provide a rather general existence result under a main assumption on the structure of the tensor of material constants the other focuses on providing a minimal number of additional design variables for a relaxation of the problem when that assumption on the tensor of elastic constants does not hold in both situations the cost functional can be pretty general

necessary stability conditions for delay systems with multiple pointwise and distributed delays necessary stability conditions for linear systems with multiple pointwise and distributed delays are presented new properties that connect the lyapunov delay matrix and the fundamental matrix of these systems are instrumental in proving that these conditions depend exclusively on the lyapunov delay matrix the result is illustrated by some examples

explicit feature control in structural topology optimization via level set method abstract the present paper aims to address a long standing and challenging problem in structural topology optimization explicit feature control of the optimal topology the basic idea is to resort to the level set solution framework and impose constraints on the extreme values of the signed distance level set function used for describing the topology of the structure numerical examples are also presented and discussed to illustrate the effectiveness of the proposed approach

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

structural acoustic optimum design of shell structures in open closed space based on a free form optimization method abstract noise reduction by structural geometry optimization has attracted much attention among designers in the present work we propose a free form optimization method for the structural acoustic design optimization of shell structures to reduce the noise of a targeted frequency or frequency range in an open or closed space the objective of the design optimization is to minimize the average structural vibration induced sound pressure at the evaluation points in the acoustic field under a volume constraint for the shape design optimization we carry out structural acoustic coupling analysis and adjoint analysis to calculate the shape gradient functions then we use the shape gradient functions in velocity analysis to update the shape of shell structures we repeat this process until convergence is confirmed to obtain the optimum shape of the shell structures in a structural acoustic coupling system the numerical results for the considered examples showed that the proposed design optimization process can significantly reduce the noise in both open and closed spaces

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

thickness optimization of a multilayered structure on the coupling surface between a structure and an acoustic cavity this paper describes a new design method to optimize thickness distribution of a multilayered structure which is located on the coupling surface between a structure and an acoustic cavity the design method is based on the concept of the density approach in topology optimization incorporating a transfer matrix for a multilayered structure that includes a poroelastic media layer the one dimensional transfer matrix adopted here is an approximate representation addressing vibro acoustic effects inherent in a multilayered structure and balances calculation resources and desired accuracy applying the transfer matrix representation as boundary conditions on the coupling surface between a structure and an acoustic cavity the modified equilibrium equation of the vibro acoustic system is derived which is approximately but efficiently solved by the modal approach in this study the problem of minimizing the acoustic pressure within the cavity over the prescribed frequency range is formulated under the volume constraint of the poroelastic media layer the continuous approximation of thickness distribution is assumed and the thickness of the poroelastic media layer at each nodal point is chosen as design variables numerical results show that an acoustic response is significantly reduced by the optimal thickness distribution having a total weight equal to or less than that in the initial uniform thickness these demonstrate that the proposed method is effective to design the optimal thickness distribution of a multilayered structure

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

sliding regimes and anisotropy in optimal design of vibrating axisymmetric plates abstract this paper deals with optimal design of solid elastic axisymmetric plates performing free transverse vibrations it is the objective to determine the plate thickness distribution from the condition that the plate volume is minimized for a given value of the fundamental natural frequency or for a given higher order natural frequency that corresponds to a vibration mode with a prescribed number of nodal diameters it is found that the weierstrass necessary condition for optimality is generally not satisfied for a traditional formulation of this problem and that the optimal design is characterized by a sliding regime of control where the plate thickness exhibits an infinite number of discontinuities as a system of infinitely thin circumferential stiffeners are formed on the optimal axisymmetric plate this inherent anisotropy of the optimal design is taken into account in a regularization of the initial optimization problem by establishing the tensorial character of the plate bending rigidity and using the concentration of thin circumferential stiffeners as a new design variable control it is shown that the new formulation of the problem can be solved numerically and examples of optimal designs are presented in the paper

stress energy minimization as a tool in the material layout design of shallow shells abstract the present research deals with the compliance minimization problem of an elastic thin shallow shell subjected to simultaneous in plane and bending loads in this context our goal is to lay out a given amount of material in the volume of a shell assuming that the distribution in the direction transversal to its middle surface s is homogeneous the discussion hence reduces to the question of finding the optimal material arrangement on s similar problems were solved in the framework of two dimensional elasticity or kirchhoff plate theory and the present research attempts to generalize these results following the pattern emerging from the above mentioned considerations our research starts from the minimum compliance problem of a structure made of two elastic materials whose volumetric fractions are fixed the existence of a solution to thus posed optimization task is guaranteed if the fine scale microstructural composites are admitted in the analysis their constitutive tensors can be obtained by certain averaging ensuing from the theory of homogenization for periodic media additionally by the castigliano theorem the compliance minimization problem is equivalent to the one for structural stress energy in turn the lower estimation of the energy is achieved in two steps i its modification by a certain energy like functional and ii utilizing the quasiconvexity property of thus obtained expression as a result formulae describing the effective stress energy of one material shallow shell and the material distribution function are explicitly derived

fast diameter and radius bfs based computation in weakly connected real world graphs in this paper we propose a new algorithm that computes the radius and the diameter of a weakly connected digraph g v e by finding bounds through heuristics and improving them until they are validated although the worst case running time is o v e we will experimentally show that it performs much better in the case of real world networks finding the radius and diameter values after 10 100 bfss instead of v bfss independently of the value of v and thus having running time o e in practice as far as we know this is the first algorithm able to compute the diameter of weakly connected digraphs apart from the naive algorithm which runs in time v e performing a bfs from each node in the particular cases of strongly connected directed or connected undirected graphs we will compare our algorithm with known approaches by performing experiments on a dataset composed by several real world networks of different kinds these experiments will show that despite its generality the new algorithm outperforms all previous methods both in the radius and in the diameter computation both in the directed and in the undirected case both in average running time and in robustness finally as an application example we will use the new algorithm to determine the solvability over time of the six degrees of kevin bacon game and of the six degrees of wikipedia game as a consequence we will compute for the first time the exact value of the radius and the diameter of the whole wikipedia digraph

spanners for bounded tree length graphs this paper concerns construction of additive stretched spanners with few edges for n vertex graphs having a tree decomposition into bags of diameter at most d i e the tree length d graphs for such graphs we construct additive 2 d spanners with o dn nlogn edges and additive 4 d spanners with o dn edges this provides new upper bounds for chordal graphs for which d 1 we also show a lower bound and prove that there are graphs of tree length d for which every multiplicative d spanner and thus every additive d 1 spanner requires w n 1 1 q d edges

distributed exact shortest paths in sublinear time the distributed single source shortest paths problem is one of the most fundamental and central problems in the message passing distributed computing classical bellman ford algorithm solves it in o n time where n is the number of vertices in the input graph g peleg and rubinovich 49 showed a lower bound of d s n for this problem where d is the hop diameter of g whether or not this problem can be solved in o n time when d is relatively small is a major open question despite intensive research 10 17 33 41 45 that yielded near optimal algorithms for the approximate variant of this problem no progress was reported for the original problem in this article we answer this question in the affirmative we devise an algorithm that requires o n log n 5 6 time for d e o s n log n and o d1 3 n log n 2 3 time for larger d this running time is sublinear in n in almost the entire range of parameters specifically for d e o n log2 n we also generalize our result in two directions one is when edges have bandwidth b 1 and the other is the s sources shortest paths problem for both problems our algorithm provides bounds that improve upon the previous state of the art in almost the entire range of parameters in particular we provide an all pairs shortest paths algorithm that requires o n5 3 log 2 3 n time even for b e 1 for all values of d we also devise the first algorithm with non trivial complexity guarantees for computing exact shortest paths in the multipass semi streaming model of computation from the technical viewpoint our distributed algorithm computes a hopset g of a skeleton graph g of g without first computing g itself we then conduct a bellman ford exploration in g g while computing the required edges of g on the fly as a result our distributed algorithm computes exactly those edges of g that it really needs rather than computing approximately the entire g

necessary stability conditions for linear delay systems we present necessary conditions for the exponential stability of linear systems with multiple delays they are expressed in terms of the delay lyapunov matrix of the lyapunov krasovskii functionals of complete type approach new properties of independent interest that establish connections of the system fundamental matrix with its lyapunov matrix are crucial elements of our proof we illustrate our work with a number of examples

improved approximation algorithms for maximum lifetime problems in wireless networks a wireless ad hoc network consists of a collection of transceivers positioned in the plane each transceiver is equipped with a limited battery charge the battery charge is reduced after each transmission depending on the transmission distance one of the major problems in wireless network design is to route network traffic efficiently so as to maximize the network lifetime i e the number of successful transmission rounds in this paper we consider rooted maximum lifetime broadcast convergecast problems in wireless settings the instance consists of a directed graph g v e with edge weight w e the power needed to transmit a message along e for every email protected e node capacity b v the battery charge of v for every email protected v and a root r the goal is to find a maximum size collection t 1 t k of broadcast convergecast trees rooted at r such that i 1 kw d t i v b v where d t v is the set of edges leaving v in t in the single topology version the same tree is used to transmit all the messages namely all the broadcast convergecast trees t i are identical using recent work on degree constrained network design problems nutov 2008 26 we give constant ratio approximation algorithms for various broadcast and convergecast problems improving the previously best known approximation w 1 email protected by elkin et al 2011 12 similar results are shown for the more general rooted maximum lifetime mixedcast problem where in addition we are given an integer c 0 and the goal is to find the maximum integer k so that k broadcast and ck convergecast rounds can be performed we also consider the model with partial level aggregation

topology optimization of acoustic structure interaction problems using a mixed finite element formulation the paper presents a gradient based topology optimization formulation that allows to solve acoustic structure vibro acoustic interaction problems without explicit boundary interface representation in acoustic structure interaction problems the pressure and displacement fields are governed by helmholtz equation and the elasticity equation respectively normally the two separate fields are coupled by surface coupling integrals however such a formulation does not allow for free material re distribution in connection with topology optimization schemes since the boundaries are not explicitly given during the optimization process in this paper we circumvent the explicit boundary representation by using a mixed finite element formulation with displacements and pressure as primary variables a u p formulation the helmholtz equation is obtained as a special case of the mixed formulation for the elastic shear modulus equating to zero hence by spatial variation of the mass density shear and bulk moduli we are able to solve the coupled problem by the mixed formulation using this modelling approach the topology optimization procedure is simply implemented as a standard density approach several two dimensional acoustic structure problems are optimized in order to verify the proposed method copyright 2006 john wiley sons ltd

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

a new formulation of the problem of optimum reinforcement of reissner mindlin plates abstract a new formulation of the problem of computing the layout of reinforcement of reissner mindlin plates for minimum compliance is presented bending and transverse shear stiffness properties of plates with an arbitrary but finite number of local rib directions and widths are computed using standard homogenization techniques these properties are then expressed in terms of only four variables that fully describe the anisotropy of the plate the optimization problem is solved in two stages consisting of a local four dimensional maximization problem where a given amount of material is optimally allocated into an arbitrary number of fine scale stiffeners of different widths and orientations and a global one dimensional optimization problem where the optimum spatial layout of material is determined a discussion of implementation issues is included and an example is solved to illustrate the approach

free matrix based integral inequality for stability analysis of systems with time varying delay the free weighting matrix and integral inequality methods are widely used to derive delay dependent criteria for the stability analysis of time varying delay systems because they avoid both the use of a model transformation and the technique of bounding cross terms this technical note presents a new integral inequality called a free matrix based integral inequality that further reduces the conservativeness in those methods it includes well known integral inequalities as special cases using it to investigate the stability of systems with time varying delays yields less conservative delay dependent stability criteria which are given in terms of linear matrix inequalities two numerical examples demonstrate the effectiveness and superiority of the method

all pairs shortest paths using bridging sets and rectangular matrix multiplication we present two new algorithms for solving the all pairs shortest paths apsp problem for weighted directed graphs both algorithms use fast matrix multiplication algorithms the first algorithm solves the apsp problem for weighted directed graphs in which the edge weights are integers of small absolute value in o n2 time where satisfies the equation 1 1 1 2 and 1 1 is the exponent of the multiplication of an n n matrix by an n n matrix currently the best available bounds on 1 1 obtained by coppersmith imply that 0 is an error parameter and w is the largest edge weight in the graph after the edge weights are scaled so that the smallest non zero edge weight in the graph is 1 it returns estimates of all the distances in the graph with a stretch of at most 1 e corresponding paths can also be found efficiently

optimization of material distribution within isotropic and anisotropic plates using homogenization abstract the creation of optimum isotropic single layered orthotropic and multilayered anisotropic plate topologies was attempted in the present study our objective was the minimization of the structural strain energy under volume constraint the concept of optimal material distribution based on homogenization of a unit microstructure together with sequential linear programming was used to perform the optimization task the structural response computed via a finite element analysis was calculated using first order shear deformation theory in conjunction with a laminated plate finite element isotropic and single layered orthotropic plates were treated as special cases of multilayered anisotropic laminates the accuracy of the algorithm developed is demonstrated by means of numerical examples the optimality of the results obtained was verified by performing structural analysis on different same volume topologies created according to the algorithmic output and comparing the resulting displacement and stress fields

approximate proof labeling schemes abstract we study a new model of verification of boolean predicates over distributed networks given a network configuration the proof labeling scheme pls model defines a distributed proof in the form of a label that is given to each node and all nodes locally verify that the network configuration satisfies the desired boolean predicate by exchanging labels with their neighbors the proof size of the scheme is defined to be the maximum size of a label in this work we extend this model by defining the approximate proof labeling scheme apls model in this new model the predicates for verification are of the form where f n for a family of configurations f informally the predicates considered in this model are a comparison between two values of the configuration as in the pls model nodes exchange labels in order to locally verify the predicate and all must accept if the network satisfies the predicate the soundness condition is relaxed with an approximation ration so that only if some node must reject we focus on two verification problems upper and lower bounds on the diameter of the graph and the maximality of a given matching for these problems we present the first results that apply to all graph structures in our new apls model we show that the proof size can be much smaller than the proof size of the same predicate in the pls model moreover we prove that there is a tradeoff between the approximation ratio and the proof size finally we present the first general result for maximum cardinality matching in the pls model

material cloud method for topology optimization material cloud method mcm a new approach for topology optimization is presented in mcm an optimal structure can be obtained by manipulating the sizes and positions of material clouds which are material patches with finite sizes and constant material densities the optimal distributions of material clouds can be obtained by mcm using fixed background finite element meshes in the numerical analysis procedure only active elements where more than one material cloud is contained are treated optimal material distribution can be element wise extracted from the distribution of material clouds with mcm an expansion reduction procedure of design domain can be naturally realized through movements of material clouds so that a true optimal solution can be found without any significant increase of computational costs it is also shown that a clear material distribution with narrow region of intermediate density can be obtained with relatively fast convergence several numerical examples are shown some of the results are compared with those of the traditional density distribution method ddm copyright 2005 john wiley sons ltd

eigenfrequency optimized 3d continua with possibility for cavities abstract eigenfrequency optimization for 3d continua is formulated and exemplified by the geometry and boundary conditions of a thick plate numerical finite element models are based on four node tetrahedra and results from subspace iterations give directly the basis for the continuum redesign the 3d modeling with a large number of elements has the possibility in optimal design to obtain as found not only holes but also cavities in the continuum sensitivity analysis is presented on the element level with simple physical interpretation of the involved terms this general result has general value for control of eigenfrequencies it is found that in the combination of partial differentiation with the chain rule of differentiation a specific notation is needed and a suggestion is presented the optimization method is based on a derived optimality criterion and as such the maximization problem change to a problem of determining a design with uniform values of this criterion nonlinear stiffness interpolation may be a physical reality a two parameter interpolation function is incorporated analytical also in the sensitivity analysis and the optimality criterion but without focusing on 1 0 optimal solutions two cases of boundary conditions two cases of total amount of material and cases of linear and nonlinear stiffness interpolation are studied

all pairs nearly 2 approximate shortest paths in o n2polylogn time let g v e be an unweighted undirected graph on v n vertices and e m edges let d u v denote the distance between vertices u v v an algorithm is said to compute all pairs t approximate shortest paths distances for some t 1 if for each pair of vertices u v v the path distance reported by the algorithm is not longer greater than t d u v this paper presents two extremely simple randomized algorithms for computing all pairs nearly 2 approximate distances the first algorithm requires an expected o m 2 3nlogn n 2 time and for any u v v reports a distance no greater than 2 d u v 1 our second algorithm requires an expected o n 2log 3 2n time and for any u v v reports a distance bounded by 2 d u v 3

lur e lyapunov function and absolute stability criterion for lur e singularly perturbed systems this technical note investigates the absolute stability problem for lur e singularly perturbed systems with multiple nonlinearities the objective is to determine if the system is absolutely stable for any e 0 e0 where e denotes the perturbation parameter and e0 is a pre defined positive scalar first an e dependent lur e lyapunov function is constructed that facilitates the stability analysis of the singularly perturbed system then a stability criterion expressed in terms of e independent linear matrix inequalities lmis is derived based on the stability criterion an algorithm is proposed to compute the stability bound that is shown to be less conservative than those computed using other existing methods finally examples are given to show the feasibility and effectiveness of the obtained method

design space optimization using a numerical design continuation method summary a generalized optimization problem in which design space is also a design to be found is de9ned and a numerical implementation method is proposed in conventional optimization only a portion of structural parameters is designated as design variables while the remaining set of other parameters related to the design space are often taken for granted a design space is speci9ed by the number of design variables and the layout or con9guration to solve this type of design space problems a simple initial design space is selected and gradually improved while the usual design variables are being optimized to make the design space evolve into a better one one may increase the number of design variables but in this transition there are discontinuities in the objective and constraint functions accordingly the sensitivity analysis methods based on continuity will not apply to this discontinuous stage to overcome the di culties a numerical continuation scheme is proposed based on a new concept of a pivot phase design space two new categories of structural optimization problems are formulated and concrete examples shown copyright 2001 john wiley sons ltd

fault tolerant additive and µ α spanners graph spanners are sparse subgraphs that preserve the distances of the original graph up to some approximation ratio the spanner s stretch a number of algorithms are known for constructing sparse spanners with small multiplicative or additive stretch recently algorithms were introduced for constructing fault tolerant multiplicative spanners of general graphs this paper addresses the analogous problem of constructing fault tolerant additive and spanners for general graphs and presents a number of deterministic and randomized construction algorithms for such spanners

on the k shortest simple paths problem in weighted directed graphs we present the first approximation algorithm for finding the k shortest simple paths connecting a pair of vertices in a weighted directed graph that breaks the barrier of mn it is deterministic and has a running time of o k m sqrt n n 3 2 log n where m is the number of edges in the graph and n is the number of vertices let s t in v the length of the i th simple path from s to t computed by our algorithm is at most frac 3 2 times the length of the i th shortest simple path from s to t the best algorithms for computing the exact k shortest simple paths connecting a pair of vertices in a weighted directed graph are due to yen management sci 17 1970 1971 pp 712 716 and lawler management sci 18 1971 1972 pp 401 405 the running time of their algorithms using modern data structures is o k mn n 2 log n both algorithms are from the early 70s although this problem and other variants of the k shortest path problem has drawn a lot of attention during the last three and a half decades the o k mn n 2 log n bound is still unbeaten

regularized formulation for optimal design of axisymmetric plates abstract this paper presents a regularized mathematical formulation necessary conditions of optimality and solutions of problems of optimal design of solid elastic axisymmetric plates of prescribed total material volume single purpose design criteria of minimum static compliance maximum stiffness and maximum frequency of free transverse vibrations are considered the regularization which alleviates some anomalies and difficulties encountered earlier in plate optimization problems is based on a new compound plate model with two simultaneous design variables namely variable thickness of a solid part of the plate and variable concentration of a dense system of thin integral stiffeners attached to the solid plate part a numerical optimality criterion based method of solution is developed for the problem and several optimal designs are presented the results are compared with results obtained from optimal design formulations applied heretofore and substantiate the superiority of the new regularized formulation

continuous approximation of material distribution for topology optimization in this paper we propose a checkerboard free topology optimization method without introducing any additional constraint parameter this aim is accomplished by the introduction of finite element approximation for continuous material distribution in a fixed design domain that is the continuous distribution of microstructures or equivalently design variables is realized in the whole design domain in the context of the homogenization design method hdm by the discretization with finite element interpolations by virtue of this continuous fe approximation of design variables discontinuous distribution like checkerboard patterns disappear without any filtering schemes we call this proposed method the method of continuous approximation of material distribution camd to emphasize the continuity imposed on the material field two representative numerical examples are presented to demonstrate the capability and the efficiency of the proposed approach against some classes of numerical instabilities copyright 2004 john wiley sons ltd

a duality approach in the optimization of beams and plates we introduce a class of nonlinear transformations called resizing rules which associate with optimal shape design problems certain equivalent distributed control problems while preserving the state of the system this puts into evidence the duality principle that the class of system states that can be achieved under a prescribed force via modifications of the structure shape of the system can be obtained as well via the modifications of the force action under a prescribed structure we apply such transformations to the optimization of beams and plates and in the simply supported or cantilevered cases the obtained control problems are even convex in all cases we establish existence theorems for optimal pairs by assuming only boundedness conditions moreover in the simply supported case we also prove the uniqueness of the global minimizer a general algorithm that iterates between the original transformed problems is introduced and studied the applications also include the case of variational inequalities

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

on non smoothness in optimal design of solid elastic plates abstract in the present paper minimum compliance design problems of thin solid elastic plates are discussed we follow up indications provided by numerical results obtained in a recent paper 1 and propose a new plate model where a uniform plate is equipped with an infinite number of infinitely thin integral stiffeners on the basis of this new plate model numerical and analytical investigations of a new formulation of the minimum compliance design problem are carried out it is demonstrated that the new formulation is under certain conditions superior to the corresponding traditional formulation for determining a possible global optimal solution for solid plates since both the new formulation and the traditional formulation are based on classical thin plate theory the superiority of the new plate model over the corresponding smooth one directly implies that optimal thickness distributions for thin solid elastic plates are generally not smooth functions but functions with an infinite number of discontinuities

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

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

comments on the validity of transfer matrix based models for the prediction of the effect of curved sound packages abstract in this paper the validity of using the transfer matrix method tmm to account for the effect of practical curved sound packages i e spring mass treatment in vibroacoustic systems is assessed for this purpose a sub structuring method that employs a patch transfer functions ptf approach is used to efficiently couple standard finite element method fem of the structure and the acoustic cavity with the sound package model three models of the noise control treatment are compared namely i a locally reacting model ii a non locally reacting nlr model based on the tmm and iii a fem the fem model considers the actual curved geometry of the sp in order to estimate its dynamic behavior while the simplified tmm based models assume the sp to be flat i e unwrapped if curvature is present homogeneous and of infinite lateral extent in addition these results are systematically compared to a full finite element boundary element methodology the accuracy and the limitations of adopting the unwrapped approximation in tmm based models are shown through two parametric studies the first considers the radius of curvature of the system as a parameter while the sp thickness is constant the second considers the thickness of the sp as a parameter while the radius of curvature of the system is held fixed for the considered configurations the first parametric study allows to conclude that the curvature of the sp does not affect the dynamic behavior of the system due to its large dissipation and softness and it can be practically assumed flat if the nlr model is used to model the sp however the second parametric study shows that the thickness of the curved sp may affect the efficiency of the nlr model moreover the limitations of locally reacting models are confirmed

fast estimation of diameter and shortest paths without matrix multiplication in the recent past there has been considerable progress in devising algorithms for the all pairs shortest paths apsp problem running in time significantly smaller than the obvious time bound of o n3 unfortunately all the new algorithms are based on fast matrix multiplication algorithms that are notoriously impractical our work is motivated by the goal of devising purely combinatorial algorithms that match these improved running times our results come close to achieving this goal in that we present algorithms with a small additive error in the length of the paths obtained our algorithms are easy to implement have the desired property of being combinatorial in nature and the hidden constants in the running time bound are fairly small our main result is an algorithm which solves the apsp problem in unweighted undirected graphs with an additive error of 2 in time o n 2 5 sqrt log n this algorithm returns actual paths and not just the distances in addition we give more efficient algorithms with running time footnotesize o n 1 5 sqrt k log n n 2 log 2 n for the case where we are only required to determine shortest paths between k specified pairs of vertices rather than all pairs of vertices the starting point for all our results is an o m sqrt n log n algorithm for distinguishing between graphs of diameter 2 and 4 and this is later extended to obtaining a ratio 2 3 approximation to the diameter in time o m sqrt n log n n 2 log n unlike in the case of apsp our results for approximate diameter computation can be extended to the case of directed graphs with arbitrary positive real weights on the edges

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

necessary conditions of exponential stability for a class of linear neutral type time delay systems for a class of linear neutral type time delay systems nttdss this paper will present necessary exponential stability conditions by employing the lyapunov krasovskii functional approach since t

the 4 3 additive spanner exponent is tight a spanner is a sparse subgraph that approximately preserves the pairwise distances of the original graph it is well known that there is a smooth tradeoff between the sparsity of a spanner and the quality of its approximation so long as distance error is measured multiplicatively a central open question in the field is to prove or disprove whether such a tradeoff exists also in the regime of additive error that is is it true that for all v 0 there is a constant kv such that every graph has a spanner on o n1 v edges that preserves its pairwise distances up to kv previous lower bounds are consistent with a positive resolution to this question while previous upper bounds exhibit the beginning of a tradeoff curve all graphs have 2 spanners on o n3 2 edges 4 spanners on o n7 5 edges and 6 spanners on o n4 3 edges however progress has mysteriously halted at the n4 3 bound and despite significant effort from the community the question has remained open for all 0 our main result is a surprising negative resolution of the open question even in a highly generalized setting we show a new information theoretic incompressibility bound there is no function that compresses graphs into o n4 3 v bits so distance information can be recovered within no 1 error as a special case of our theorem we get a tight lower bound on the sparsity of additive spanners the 6 spanner on o n4 3 edges cannot be improved in the exponent even if any subpolynomial amount of additive error is allowed our theorem implies new lower bounds for related objects as well for example the 20 year old 4 emulator on o n4 3 edges also cannot be improved in the exponent unless the error allowance is polynomial central to our construction is a new type of graph product which we call the obstacle product intuitively it takes two graphs g h and produces a new graph g h whose shortest paths structure looks locally like h but globally like g

explicit three dimensional topology optimization via moving morphable void mmv approach abstract three dimensional 3d topology optimization problems always involve huge numbers of degrees of freedom dofs in finite element analysis fea and design variables in numerical optimization respectively this will inevitably lead to large computational efforts in the solution process in the present paper an efficient and explicit topology optimization approach which can reduce not only the number of design variables but also the number of degrees of freedom in fea is proposed based on the moving morphable voids mmvs solution framework this is achieved by introducing a set of geometry parameters e g control points of b spline surfaces to describe the boundary of a structure explicitly and removing the unnecessary dofs from the fe model at every step of numerical optimization numerical examples demonstrate that the proposed approach does can overcome the bottleneck problems associated with a 3d topology optimization problem in a straightforward way and enhance the solution efficiency significantly

a gradient based parameter free approach to shape optimization in shape optimization the independent node movement approach wherein finite element node coordinates are used directly as design variables allows the most freedom for shape change and avoids the time consuming parameterization preprocess however this approach lacks a length scale control that is necessary to ensure a well posed shape optimization problem and avoid numerical instability motivated by the success of filtering techniques that impose minimum length scales in topology optimization we propose a scheme with consistent filtering to introduce a length scale and thereby ensure smoothness in shape optimization while preserving the advantages of the independent node movement approach

spatial frequency control for the systematic generation of optimal candidate designs having various topological complexities abstract in practical engineering design problems there are some design and manufacturing considerations that are difficult or infeasible to express in terms of an objective function or a constraint among them we pay attention to the topological complexity of optimized designs which are difficult to quantify in this situation a set of optimal candidate designs having different topological complexities not just a single optimal design is preferred in this work the mean spatial frequency is proposed as a means to measure the topological complexity of an optimized design to maximize the system performance and to control the mean spatial frequency the topology design optimization in the m band wavelet transformed multiscale design space is proposed in the proposed scheme the mean frequency is controlled effectively by choosing different design space resolution in the multiscale design space and the performance maximization is treated as the objective function of the topology optimization the procedure to transform the original design space into the multiscale design space by the m band daubechies wavelet is also explained in some details

a new model for thin plates with rapidly varying thickness abstract we study the bending of a thin plate with rapidly varying thickness for example one with rib like stiffeners or perforated by small holes we obtain a fourth order equation for the midplanc displacement using an asymptotic analysis based on three dimensional linear elasticity the coefficients of this equation represent the constitutive law relating bending moments to midplane curvature they are explicitly determined by the plate geometry our analysis distinguishes between three different cases in which the thickness varies on a length scale longer than on the order of or shorter than the mean thickness

brief a practical method for analyzing the stability of neutral type lti time delayed systems a new paradigm is presented for assessing the stability posture of a general class of linear time invariant neutral time delayed systems lti ntds the ensuing method which we name the direct method dm offers several unique features it returns the number of unstable characteristic roots of the system in an explicit and non sequentially evaluated function of time delay t consequently the direct method creates exclusively all possible stability intervals of t furthermore it is shown that this method inherently verifies a widely accepted necessary condition for the t stabilizability of a lti ntds in the core of the dm lie a root clustering paradigm and the strength of rekasius transformation in mapping a transcendental characteristic equation into an equivalent rational polynomial in addition we also demonstrate by an example that dm can tackle systems with unstable starting posture for t 0 only to stabilize for higher values of delay which is rather unique in the literature

optimal rectangular plate and shallow shell topologies using thickness distribution or homogenization abstract topological optimization of plates as well as shallow cylindrical and curved spherical shells are attempted in the present study for all structures examined our objective is the minimization of the strain energy function under a volume constraint an optimum distribution of thickness or microstructural density is sought under the hypothesis that the design variables can only be assigned their extreme allowable bounds or values very near them so that material can be removed from low density areas and thus an optimum topology can be determined the structural response is computed via a finite element analysis the analytical formulation is based on a form of linear shallow shell theory with the effects of transverse shear deformation and bending extensional coupling included the method of feasible directions is used to perform the optimization task numerical examples for various boundary conditions showing similarities or differences of the two methods are presented and discussed for all structures examined it is found that the assumption of a repetitious microstructure based on homogenization theory resulted in stronger optima for clamped plates and shells both methods converged to nearly identical topologies an indication of possible global optimal layouts

approximate distance oracles let g v e be an undirected weighted graph with vvv n and vev m let k 1 be an integer we show that g v e can be preprocessed in o kmn1 k expected time constructing a data structure of size o kn1p1 k such that any subsequent distance query can be answered approximately in o k time the approximate distance returned is of stretch at most 2k 1 that is the quotient obtained by dividing the estimated distance by the actual distance lies between 1 and 2k 1 a 1963 girth conjecture of erdos implies that n1p1 k space is needed in the worst case for any real stretch strictly smaller than 2kp1 the space requirement of our algorithm is therefore essentially optimal the most impressive feature of our data structure is its constant query time hence the name oracle previously data structures that used only o n1p1 k space had a query time of n1 k our algorithms are extremely simple and easy to implement efficiently they also provide faster constructions of sparse spanners of weighted graphs and improved tree covers and distance labelings of weighted or unweighted graphs

optimality conditions for mass design problems and applications to thin plates we derive necessary and sufficient optimality conditions for a quite large class of structural design problems which can be formulated as follows under a given load and a total volume constraint minimize a suitable notion of compliance among all admissible mass distributions represented by positive measures with prescribed integral mean as a special case we focus attention on the optimization of thin plates we detail the corresponding optimality conditions and we show how they can be handled in order to determine analytically some optimal plates