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

on asymptotic behaviour and w 2 p regularity of potentials in optimal transportation in this paper we study local properties of cost and potential functions in optimal transportation we prove that in a proper normalization process the cost function is uniformly smooth and converges locally smoothly to a quadratic cost x y while the potential function converges to a quadratic function as applications we obtain the interior w2 p estimates and sharp c1 estimates for the potentials which satisfy a monge ampere type equation the w2 p estimate was previously proved by caffarelli for the quadratic transport cost and the associated standard monge ampere equation

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

regularity of potential functions of the optimal transportation problem the potential function of the optimal transportation problem satisfies a partial differential equation of monge ampere type in this paper we introduce the notion of a generalized solution and prove the existence and uniqueness of generalized solutions of the problem we also prove the solution is smooth under certain structural conditions on the cost function

nearly round spheres look convex we prove that a riemannian manifold m g close enough to the round sphere in the c 4 topology has uniformly convex injectivity domains so m appears uniformly convex in any exponential chart the proof is based on the ma trudinger wang nonlocal curvature tensor which originates from the regularity theory of optimal transport

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

topology optimization with closed b splines and boolean operations abstract in this work free form curves of closed b splines cbs are introduced as basic design primitives to realize structural topology optimization in the way of shape optimization unlike the existing density and level set methods the cbs takes full advantage of its parametric form for the design deformability and flexibility with a few number of design variables and its implicit form for boolean operations of the structural topology accounting for any overlapping merging and separation of different cbs no matter how geometric complexities of the cbs and the design domain are the cbs is implemented within the computing framework of finite cell method fcm so that structural analysis and topology optimization can efficiently be carried out using the fixed mesh typical examples are tested to demonstrate the effectiveness and generality of the proposed method in dealing with engineering design problems

towards the smoothness of optimal maps on riemannian submersions and riemannian products of round spheres in particular the variant a3w of ma trudinger and wang s condition for regularity of optimal transportation maps is implied by the non negativity of a pseudo riemannian curvature which we call cross curvature induced by the transportation cost for the riemannian distance squared cost it is shown that 1 cross curvature non negativity is preserved for products of two manifolds 2 both a3w and cross curvature non negativity are inherited by riemannian submersions as is domain convexity for the exponential maps and 3 the n dimensional round sphere satisfies cross curvature non negativity from these results a large new class of riemannian manifolds satisfying cross curvature non negativity thus a3w is obtained including many whose sectional curvature is far from constant all known obstructions to the regularity of optimal maps are absent from these manifolds making them a class for which it is natural to conjecture that regularity holds this conjecture is confirmed for certain riemannian submersions of the sphere such as the complex projective spaces cp n

stability of a 4th order curvature condition arising in optimal transport theory abstract a certain curvature condition introduced by ma trudinger and wang in relation with the regularity of optimal transport is shown to be stable under gromov hausdorff limits even though the condition implicitly involves fourth order derivatives of the riemannian metric two lines of reasoning are presented with slightly different assumptions one purely geometric and another one combining geometry and probability then a converse problem is studied prove some partial regularity for the optimal transport on a perturbation of a riemannian manifold satisfying a strong form of the ma trudinger wang condition

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

monge kantorovich depth quantiles ranks and signs we propose new concepts of statistical depth multivariate quantiles ranks and signs based on canonical transportation maps between a distributionof interest on ird and a reference distribution on the d dimensionalunit ball the new depth concept called monge kantorovich depth specializesto halfspace depth in the case of elliptical distributions but for more generaldistributions differs from the latter in the ability for its contours to account fornon convex features of the distribution of interest we propose empirical counterpartsto the population versions of those monge kantorovich depth contours quantiles ranks and signs and show their consistency by establishing a uniformconvergence property for transport maps which is of independent interest

some parseval goldstein type identities involving the fs 2 transform the fc 2 transform and the p4 transform and their applications abstract in the present paper the authors consider several new integral transforms including the f s 2 transform the f c 2 transform and the p 4 transform as generalizations of the classical fourier sine transform the classical fourier cosine transform and the classical stieltjes transform respectively many identities involving these transforms are given by making use of these identities a number of new parseval goldstein type identities are obtained for these and many other well known integral transforms the identities proven in this paper are shown to give rise to useful corollaries for evaluating infinite integrals of special functions some examples are also given as illustrations of the results presented here

topology optimization with implicit functions and regularization summary topology optimization is formulated in terms of the nodal variables that control an implicit function description of the shape the implicit function is constrained by upper and lower bounds so that only a band of nodal variables needs to be considered in each step of the optimization the weak form of the equilibrium equation is expressed as a heaviside function of the implicit function the heaviside function is regularized to permit the evaluation of sensitivities we show that the method is a dual of the bendshe kikuchi method the method is applied both to problems of optimizing single material and

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

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

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

an alternative truly mixed formulation to solve pressure load problems in topology optimization abstract the paper deals with an alternative formulation for the topology optimization of structures acted upon by pressure loads exploiting finite elements techniques that are able to handle incompressible materials the method firstly presented in o sigmund p m clausen topology optimization using a mixed formulation an alternative way to solve pressure load problems comput methods appl mech engrg 196 2007 1874 1889 33 consists in exploiting the modeling of fluid incompressibility within a topology optimization framework the implementation of a fluid phase enables to transfer pressure loads from the domain boundaries to the evolving edges of optimal design without relying on more complex techniques traditionally employed to recover the load application surfaces at each step of the minimization process in this context the main numerical trouble is therefore the application of finite elements techniques to solve incompressible materials analysis this topic in fact cannot be tackled using most of the approaches of the current literature that are mainly based on displacement finite elements which are well known to be affected by the locking phenomenon while sigmund and clausen 2007 uses a displacement pressure finite element discretization to solve the problem the approach herein presented consists in the adoption of a truly mixed variational formulation coupled to a discretization based on the johnson and mercier finite element that both pass the inf sup conditions of the problem even in the presence of incompressible materials the well known method of moving asymptotes mma is adopted in the numerical studies presented along with a particular density interpolation to model the presence of a fluid and solid phase within the same design the adopted scheme is especially conceived to avoid the arising of numerical instabilities that may arise within the optimization procedure when handling incompressible material moreover the accuracy and stability in stress evaluation provided by the truly mixed setting are herein exploited to introduce an alternative procedure that implements pressure constraints to avoid optimal designs that present cavities filled by fluid

optimum shape and topology design using the boundary element method a procedure is developed for simultaneous shape and topology design optimization of linear elastic two dimensional continuum structures an intuitive approach is presented to treat such topological problems whereby material is eliminated from within the structure by introducing holes at regions of low stress through a sequence of shape optimization processes a mathematical programming technique coupled with the boundary element be method of response and sensitivity analyses enables the optimal positioning of these holes plus optimization of the overall structural shape the analytical derivative be formulation is explained together with the use of appropriate design velocity fields and example problems are solved to demonstrate the optimization procedure

on the regularity of solutions of optimal transportation problems we give a necessary and sufficient condition on the cost function so that the map solution of monge s optimal transportation problem is continuous for arbitrary smooth positive data this condition was first introduced by ma trudinger and wang 24 30 for a priori estimates of the corresponding monge ampere equation it is expressed by a socalled cost sectional curvature being non negative we show that when the cost function is the squared distance of a riemannian manifold the cost sectional curvature yields the sectional curvature as a consequence if the manifold does not have non negative sectional curvature everywhere the optimal transport map cannot be continuous for arbitrary smooth positive data the non negativity of the cost sectional curvature is shown to be equivalent to the connectedness of the contact set between any cost convex function the proper generalization of a convex function and any of its supporting functions when the cost sectional curvature is uniformly positive we obtain that optimal maps are continuous or holder continuous under quite weak assumptions on the data compared to what is needed in the euclidean case this case includes the quadratic cost on the round sphere

regularity of optimal transport on compact locally nearly spherical manifolds given a couple of smooth positive measures of same total mass on a compact connected riemannian manifold m we look for a smooth optimal transportation map g pushing one measure to the other at a least total squared distance cost directly by using the continuity method to produce a classical solution of the elliptic equation of monge ampere type satisfied by the potential function u such that g exp grad u this approach boils down to proving an textit a priori upper bound on the hessian of u which was done on the flat torus by the first author the recent local c 2 estimate of ma trudinger wang enabled loeper to treat the standard sphere case by overcoming two difficulties namely in collaboration with the first author he kept the image g m of a generic point m in m uniformly away from the cut locus of m he checked a fourth order inequality satisfied by the squared distance cost function proving the uniform positivity of the so called c curvature of m in the present paper we treat along the same lines the case of manifolds with curvature sufficiently close to 1 in c 2 norm specifying and proving a conjecture stated by trudinger

topology optimization of continuum structures with local stress constraints we introduce an extension of current technologies for topology optimization of continuum structures which allows for treating local stress criteria we first consider relevant stress criteria for porous composite materials initially by studying the stress states of the so called rank 2 layered materials then on the basis of the theoretical study of the rank 2 microstructures we propose an empirical model that extends the power penalized stiffness model also called simp for solid isotropic microstructure with penalization for inter mediate densities in a second part solution aspects of topology problems are considered to deal with the so called singularity phenomenon of stress constraints in topology design an constraint relaxation of the stress constraints is used we describe the mathematical programming approach that is used to solve the numerical optimization problems and show results for a number of example applications 1998 john wiley sons ltd

a new method of solving bessel s differential equation using the l 2 transform in the present paper a new method of solving bessel s differential equation is given using the l 2 transform

continuity of optimal transport maps and convexity of injectivity domains on small deformations of 2 given a compact riemannian manifold we study the regularity of the optimal transport map between two probability measures with cost given by the squared riemannian distance our strategy is to define a new form of the so called ma trudinger wang condition and to show that this condition together with the strict convexity on the nonfocal domains implies the continuity of the optimal transport map moreover our new condition again combined with the strict convexity of the nonfocal domains allows us to prove that all injectivity domains are strictly convex too these results apply for instance on any small c4 deformation of the 2 sphere 2009 wiley periodicals inc

identities for the e2 1 transform and their applications in the present paper the authors introduce the e 2 1 transform with kernel the exponential integral function it is shown that the third iterate of the l 2 transform is the exponential integral transform and some identities involving the new transform the l 2 transform and the widder potential transform are given using the identities parseval goldstein type results involving these transforms are proved some illustrative examples are also given

shape and topology optimization of compliant mechanisms using a parameterization level set method in this paper a parameterization level set method is presented to simultaneously perform shape and topology optimization of compliant mechanisms the structural shape boundary is implicitly embedded into a higher dimensional scalar function as its zero level set resultantly establishing the level set model by applying the compactly supported radial basis function with favorable smoothness and accuracy to interpolate the level set function the temporal and spatial hamilton jacobi equation from the conventional level set method is then discretized into a series of algebraic equations accordingly the original shape and topology optimization is now fully transformed into a parameterization problem namely size optimization with the expansion coefficients of interpolants as a limited number of design variables design of compliant mechanisms is mathematically formulated as a general optimization problem with a nonconvex objective function and two additionally specified constraints the structural shape boundary is then advanced as a process of renewing the level set function by iteratively finding the expansion coefficients of the size optimization with a sequential convex programming method it is highlighted that the present method can not only inherit the merits of the implicit boundary representation but also avoid some unfavorable features of the conventional discrete level set method such as the cfl condition restriction the re initialization procedure and the velocity extension algorithm finally an extensively investigated example is presented to demonstrate the benefits and advantages of the present method especially its capability of creating new holes inside the design domain

the coc algorithm part i cross section optimization or sizing abstract in iterative continuum based optimality criteria coc methods necessary conditions of optimality are first generated analytically using variational methods and then reinterpreted in terms of a fictitious or adjoint structure finally the real and adjoint structures are discretized and the problem is solved by an iterative procedure using finite element fe methods it is shown by means of fe simulators for special classes of problems that a coc optimizer could potentially handle millions of variables in cross section optimization or sizing thereby not only eliminating but also reversing the present discrepancy between analysis capability and optimization capability

identities for the hankel transform and their applications abstract in the present paper the authors show that iterations of the hankel transform with k transform is a constant multiple of the widder transform using these iteration identities several parseval goldstein type theorems for these transforms are given by making use of these results a number of new goldstein type exchange identities are obtained for these and the laplace transform the identities proven in this paper are shown to give rise to useful corollaries for evaluating infinite integrals of special functions some examples are also given as illustration of the results presented here

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

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

regularity of optimal transport in curved geometry the nonfocal case we explore some geometric and analytic consequences of a curvature condition introduced by ma trudinger and wang in relation to the smoothness of optimal transport in curved geometry we discuss a conjecture according to which a strict version of the ma trudinger wang condition is sufficient to prove regularity of optimal transport on a riemannian manifold we prove this conjecture under a somewhat restrictive additional assumption of nonfocality at the same time we establish the striking geometric property that the tangent cut locus is the boundary of a convex set partial extensions are presented to the case when there is no pure focalization on the tangent cut locus

topology optimization with mixed finite elements on regular grids abstract recently new families of mixed finite elements have been proposed to address the analysis of linear elastic bodies on regular grids adopting a limited number of degrees of freedom per element a two dimensional mixed discretization is implemented to formulate an alternative topology optimization problem where stresses play the role of main variables and both compressible and incompressible materials can be dealt with the structural compliance is computed through the evaluation of the complementary energy whereas the enforcement of stress constraints is straightforward numerical simulations investigate the features of the proposed approach comparisons with a conventional displacement based scheme are provided for compressible materials stress constrained solutions for structures made of incompressible media are introduced

on strict convexity and continuous differentiability of potential functions in optimal transportation this note concerns the relationship between conditions on cost functions and domains the convexity properties of potentials in optimal transportation and the continuity of the associated optimal mappings in particular we prove that if the cost function satisfies the condition a3 introduced in our previous work with xinan ma the densities and their reciprocals are bounded and the target domain is convex with respect to the cost function then the potential is continuously differentiable and its dual potential strictly concave with respect to the cost function our results extend by different and more direct proof similar results of loeper proved by approximation from our earlier work on regularity

a new method of solving hermite s differential equation using the l2 transform in the present paper a new method of solving hermite s differential equation is given using the l 2 transform

efficient topology optimization of large dynamic finite element systems using fatigue the present paper modifies and extends the recently developed equivalent static load method for the optimization of dynamically loaded linear elastic finite element systems with a huge number of degrees of freedom in the equivalent static load method dynamic loads have been transformed into equivalent static loads this leads to an equivalent static response optimization with multiple loading conditions instead of a dynamic optimization problem in the present paper the equivalent static load algorithm is modified and extended by introducing a fatigue analysis in the iterative optimization procedure where damage is used as suitable termination criterion of the iteration as well as for the determination of a single and meaningful equivalent static load that leads to maximal damage in the structure during the evolution process the structure is systematically stiffened by using the solid isotropic microstructure with penalization approach until a user defined damage level is reached three standard examples from literature and an industrial application with a large number of degrees of freedom 600 000 demonstrate the computational efficiency of the proposed method

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 density gradient approach to topology optimization under design dependent boundary loading abstract the paper proposes a density gradient based approach to topology optimization under design dependent boundary loading since there is no explicit boundary representation in density based topology optimization the design dependent boundary loads are implicitly imposed through a domain integration of spatial gradient of the density field the density gradient based loading is formally derived and justified based on the smoothed boundary method for numerical solution of pde the heaviside projection density filtering and local refinement are combined to efficiently control the thickness of the loading interface and the number of elements within it for problems with both loading and non loading boundary an auxiliary density field is introduced to keep track of the loading boundary both 2d and 3d heat conduction problems linear elasticity problems and coupled thermoelastic problems under design dependent boundary loading are presented to illustrate the effectiveness and efficiency of the method

eigenvalue based optimization of incompressible media using mixed finite elements with application to isolation devices abstract an alternative formulation for the eigenvalue optimization of structures made of incompressible material is presented the proposed methodology is based on the adoption of a truly mixed variational formulation that descends from the principle of hellinger reissner and passes the inf sup condition f brezzi m fortin mixed and hybrid finite element methods springer verlag new york 1991 even for incompressible materials this therefore allows to avoid the locking phenomenon that is a difficulty often found handling rubber material by means of commonly used displacement based finite elements the approach implemented in the work consists in the adoption of the discretized form of the truly mixed formulation that uses the composite element of johnson and mercier c johnson b mercier some equilibrium finite elements methods for two dimensional elasticity problems numer math 30 1978 103 116 to address eigenvalue based optimization problems for rubber like materials firstly classical eigenvalue designs for in plane vibrating bidimensional structures are studied relying on mma k svamberg method of moving asymptotes a new method for structural optimization int j numer methods engrg 24 3 1987 359 373 as minimization algorithm peculiar attention is paid to numerical troubles that may arise managing such kind of problems as the appearance of localized modes preventing an adequate convergence pattern and the difficulties in getting pure 0 1 designs typical of plane strain conditions suitable mass and stiffness interpolations are presented and tested to overcome these problems afterwards the proposed framework is used to approach the design of rubber like materials in a minimum compliance setting with the imposition of an upper limit on the first eigenvalue a bi phase stiffness interpolation is moreover implemented to deal with steel rubber design of aseismic bearing devices in civil engineering some forthcoming investigations are eventually highlighted including the implementation of stress constraints to improve the presented designs for manufacturing

on the geometry of metric measure spaces ii we introduce a curvature dimension condition cd k n for metric measure spaces it is more restrictive than the curvature bound underline text curv left m text d m right k introduced in sturm k t 2006 on the geometry of metric measure spaces i acta math 196 65 131 which is recovered as the borderline case cd k the additional real parameter n plays the role of a generalized upper bound for the dimension for riemannian manifolds cd k n is equivalent to text ric m left xi xi right k left xi right 2 and dim m n the curvature dimension condition cd k n is stable under convergence for any triple of real numbers k n l the family of normalized metric measure spaces m d m with cd k n and diameter l is compact condition cd k n implies sharp version of the brunn minkowski inequality of the bishop gromov volume comparison theorem and of the bonnet myers theorem moreover it implies the doubling property and local scale invariant poincare inequalities on balls in particular it allows to construct canonical dirichlet forms with gaussian upper and lower bounds for the corresponding heat kernels

new identities involving the laplace and the ℒ 2 transforms and their applications in the present paper parseval goldstein type theorems involving the 2 transforms and the laplace transform are proved the theorems are then shown to yield a number of new identities involving several well known integral transforms and special functions using the theorem and its corollaries a number of interesting infinite integrals of elementary and special functions are presented some illustrative examples are also given

short note shape and topology optimization for electrothermomechanical microactuators using level set methods in this short note a shape and topology optimization method is presented for multiphysics actuators including geometrically nonlinear modeling based on an implicit free boundary parameterization method a level set model is established to describe structural design boundary by embedding it into the zero level set of a higher dimensional level set function the compactly supported radial basis functions csrbf are introduced to parameterize the implicit level set surface with a high level of accuracy and smoothness the original more difficult shape and topology optimization driven by the hamilton jacobi partial differential equation pde is transferred into a relatively easier parametric size optimization to which many well founded optimization algorithms can be applied thus the structural optimization is transformed to a numerical process that describes the design as a sequence of motions of the design boundaries by updating the expansion coefficients of the size optimization two widely studied examples are chosen to demonstrate the effectiveness of the proposed method

new identities and parseval type relations for the generalized integral transforms in the present paper the authors consider several new integral transforms including the l 4 n transform the p 4 n transform the f s 2 n transform and the f c 2 n transform as generalizations of the classical laplace transform the classical stieltjes transform the classical fourier sine transform and the classical fourier cosine transform respectively identities involving these transforms are given using this identities a number of new parseval goldstein type identities are obtained some examples are also given as illustrations of the results presented here

ricci curvature for metric measure spaces via optimal transport we dene a notion of a measured length space x having nonnegative n ricci curvature for n 2 1 1 or having1 ricci curvature bounded below byk fork2 r the denitions are in terms of the displacement convexity of certain functions on the associated wasserstein metric space p2 x of probability measures we show that these properties are preserved under measured gromov hausdor limits we give geometric and analytic consequences this paper has dual goals one goal is to extend results about optimal transport from the setting of smooth riemannian manifolds to the setting of length spaces a second goal is to use optimal transport to give a notion for a measured length space to have ricci curvature bounded below we refer to 11 and 44 for background material on length spaces and optimal transport respectively further bibliographic notes on optimal transport are in appendix f in the present introduction we motivate the questions that we address and we state the main results to start on the geometric side there are various reasons to try to extend notions of curvature from smooth riemannian manifolds to more general spaces a fairly general setting is that of length spaces meaning metric spaces x d in which the distance between two points equals the inmum of the lengths of curves joining the points in the rest of this introduction we assume that x is a compact length space alexandrov gave a good notion of a length space having curvature bounded below by k with k a real number in terms of the geodesic triangles in x in the case of a riemannian manifold m with the induced length structure one recovers the riemannian notion of having sectional curvature bounded below by k length spaces with alexandrov curvature bounded below by k behave nicely with respect to the gromovhausdor topology on compact metric spaces modulo isometries they form

improving the efficiency of large scale topology optimization through on the fly reduced order model construction summarytopology optimization of large scale structures is computationally expensive notably because of the cost of solving the equilibrium equations at each iteration reduced order models by projection also known as reduced basis models have been proposed in the past for alleviating this cost we propose here a new method for coupling reduced basis models with topology optimization to improve the efficiency of topology optimization of large scale structures the novel approach is based on constructing the reduced basis on the fly using previously calculated solutions of the equilibrium equations the reduced basis is thus adaptively constructed and enriched based on the convergence behavior of the topology optimization a direct approach and an approach with adjusted sensitivities are described and their algorithms provided the approaches are tested and compared on various 2d and 3d minimum compliance topology optimization benchmark problems computational cost savings by up to a factor of 12 are demonstrated using the proposed methods copyright textcopyright 2014 john wiley sons ltd