step fluctuations for a faceted crystal a statistical mechanics model for a faceted crystal is the 3d ising model at zero temperature it is assumed that in one octant all sites are occupied by atoms the remaining ones being empty allowed atom configurations are such that they can be obtained from the filled octant through successive removals of atoms with breaking of precisely three bonds if v denotes the number of atoms removed then the grand canonical boltzmann weight is q v 0 q 1 as shown by cerf and kenyon in the limit q 1 a deterministic shape is attained which has the three facets 100 010 001 and a rounded piece interpolating between them we analyse the step statistics as q 1 in the rounded piece it is given by a determinantal process based on the discrete sine kernel exactly at the facet edge the steps have more space to meander their statistics is again determinantal but this time based on the airy kernel in particular the border step is well approximated by the airy process which has been obtained previously in the context of growth models our results are based on the asymptotic analysis for space time inhomogeneous transfer matrices

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

structural topology optimization based on non local shepard interpolation of density field abstract this paper presents a non local density interpolation strategy for topology optimization based on nodal design variables in this method design variable points can be positioned at any locations in the design domain and may not necessarily coincide with elemental nodes by using the shepard family of interpolants the density value of any given computational point is interpolated by design variable values within a certain circular influence domain of the point the employed interpolation scheme has an explicit form and satisfies non negative and range restricted properties required by a physically significant density interpolation since the discretizations of the density field and the displacement field are implemented on two independent sets of points the method is well suited for a topology optimization problem with a design domain containing higher order elements or non quadrilateral elements moreover it has the ability to yield mesh independent solutions if the radius of the influence domain is reasonably specified numerical examples demonstrate the validity of the proposed formulation and numerical techniques it is also confirmed that the method can successfully avoid checkerboard patterns as well as islanding phenomenon

cbs based topology optimization including design dependent body loads abstract in this paper the inherent problem of the so called parasitic effect of low density region caused by design dependent loads is investigated for topology optimization a cbs closed b splines based method is developed to solve efficiently the problem in the way of shape optimization compared to the standard density method design variables are unattached to the finite element model and defined by control parameters dominating the parametric equation of the cbs as a result design dependent loads associated with the material layout of a structure are made change indirectly by the boundary variation of the structure to favour structural reanalyses and sensitivity analysis the implicit form of the cbs i e level set function lsf is used in conjunction with the fixed computing grid of finite cell method fcm a variety of design dependent body loads is considered for the mathematical formulation of the cbs based method typical examples are given to illustrate that the cbs based method is free of low density regions in the optimized topology owing to the proper definitions of design variables

scale invariance of the png droplet and the airy process we establish that the static height fluctuations of a particular growth model the png droplet converges upon proper rescaling to a limit process which we call the airy process a y the airy process is stationary it has continuous sample paths its single time fixed y distribution is the tracy widom distribution of the largest eigenvalue of a gue random matrix and the airy process has a slow decay of correlations as y 2 roughly the airy process describes the last line of dyson s brownian motion model for random matrices our construction uses a multi layer version of the png model which can be analyzed through fermionic techniques specializing our result to a fixed value of y one reobtains the celebrated result of baik deift and johansson on the length of the longest increasing subsequence of a random permutation

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

topology optimization of beam cross sections abstract perhaps this paper reports the first successful applications of the topology optimization in the design of thin walled beam sections in particular topologically different thin walled beam cross sections can be obtained by the present approach which is very useful in identifying the direction and location of stiffeners in formulating the topology optimization problems a simple power law is used for the relation between the density of an element with a hole and the mechanical properties of the element the sensitivity of the torsional rigidity is obtained by developing a finite element model of a st venant torsion problem and the euler beam theory is used for the sensitivity analysis of the bending rigidities

the finite cell method for three dimensional problems of solid mechanics this article presents a generalization of the recently proposed finite cell method to three dimensional problems of linear elasticity the finite cell method combines ideas from embedding or fictitious domain methods with the p version of the finite element method besides supporting a fast simple generation of meshes it also provides high convergence rates mesh generation for a boundary representation of solids and for voxel based data obtained from ct scans is addressed in detail in addition the implementation of non homogeneous neumann boundary conditions and the computation of cell matrices based on a composed integration is presented the performance of the proposed method is demonstrated by three numerical examples including the elastostatic computation of a human bone biopsy

a level set method for structural topology optimization this paper presents a new approach to structural topology optimization we represent the structural boundary by a level set model that is embedded in a scalar function of a higher dimension such level set models are flexible in handling complex topological changes and are concise in describing the boundary shape of the structure furthermore a well founded mathematical procedure leads to a numerical algorithm that describes a structural optimization as a sequence of motions of the implicit boundaries converging to an optimum solution and satisfying specified constraints the result is a 3d topology optimization technique that demonstrates outstanding flexibility of handling topological changes fidelity of boundary representation and degree of automation we have implemented the algorithm with the use of several robust and efficient numerical techniques of level set methods the benefit and the advantages of the proposed method are illustrated with several 2d examples that are widely used in the recent literature of topology optimization especially in the homogenization based methods

stress constrained topology optimization with free form design domains abstract this paper aims at dealing with realistic and challenging design problems of stress constrained topology optimization with free form design domains first the concept of level set function lsf based modelers is introduced to transform this kind of problems into the boolean conjunction operation of a topology variation modeler tvm onto a free form design domain modeler fddm such an operation is mathematically realized by means of the so called r functions in the form of implicit lsfs within this framework topology optimization problems are classified into two general cases depending upon the existence of non designable solid feature analytical sensitivity analysis formulas are further derived compared with the existing level set based method the important sensitivity property of design domain preserving makes it possible to avoid automatically the boundary violation of the design domain caused by the zero level set movement and both the topology and boundary shape of the free form design domain can be simultaneously optimized second the implementation of the finite cell method fcm ensures the stress computing accuracy in the fixed mesh due to the use of high order shape functions and adaptive integration scheme the combination of the active set strategy and the dynamic aggregation technique also reduces the number of local stress constraints greatly finally representative examples are presented to illustrate the conveniences and effectiveness of the proposed method

shape optimization of dirichlet boundaries based on weighted b spline finite cell method and level set function abstract this paper addresses extended shape optimization problems where structural supports i e dirichlet boundaries and free boundaries are simultaneously optimized unlike traditional fem the weighted b spline finite cell method fcm is applied as structural analysis tool and combined with the level set function lsf to take into account dirichlet boundary condition dbc automatically through penalization of the displacement field the proposed shape optimization method achieves a comprehensive integration of fixed grid b spline shape function and level set function as both the structure and dirichlet boundaries are described in the form of lsf any modification of dirichlet boundaries can be made in a straightforward way as easily as free boundaries by changing continuous design variables meanwhile the computing accuracy is ensured within the framework of fixed grid owing to the quadtree refinements of boundary cells stress related shape optimization problems are finally solved to demonstrate the merit and validity of the proposed optimization method in dealing with shape optimization of dirichlet boundaries

the low temperature expansion of the wulff crystal in the 3d ising model we compute the expansion of the surface tension of the 3d random cluster model for q 1 in the limit where p goes to 1 we also compute the asymptotic shape of a plane partition of n as n goes to this same shape determines the wulff crystal to order o in the 3d ising model and more generally in the 3d random cluster model for q 1 at temperature

a robust evolutionary algorithm for the recovery of rational gielis curves gielis curves gc can represent a wide range of shapes and patterns ranging from star shapes to symmetric and asymmetric polygons and even self intersecting curves such patterns appear in natural objects or phenomena such as flowers crystals pollen structures animals or even wave propagation gielis curves and surfaces are an extension of lame curves and surfaces superquadrics which have benefited in the last two decades of extensive researches to retrieve their parameters from various data types such as range images 2d and 3d point clouds etc unfortunately the most efficient techniques for superquadrics recovery based on deterministic methods cannot directly be adapted to gielis curves indeed the different nature of their parameters forbids the use of a unified gradient descent approach which requires initial pre processings such as the symmetry detection and a reliable pose and scale estimation furthermore even the most recent algorithms in the literature remain extremely sensitive to initialization and often fall into local minima in the presence of large missing data we present a simple evolutionary algorithm which overcomes most of these issues and unifies all of the required operations into a single though efficient approach the key ideas in this paper are the replacement of the potential fields used for the cost function closed form by the shortest euclidean distance sed iterative approach the construction of cost functions which minimize the shortest distance as well as the curve length using r functions and slight modifications of the evolutionary operators we show that the proposed cost function based on sed and r function offers the best compromise in terms of accuracy robustness to noise and missing data

domino tilings and the six vertex model at its free fermion point at the free fermion point the six vertex model with domain wall boundary conditions dwbc can be related to the aztec diamond a domino tiling problem we study the mapping on the level of complete statistics for general domains and boundary conditions this is obtained by associating with both models a set of non intersecting lines in the lindstrom gessel viennot lgv scheme one of the consequences for dwbc is that the boundaries of the ordered phases are described by the airy process in the thermodynamic limit

on the topological derivative in shape optimization in this paper the topological derivative for an arbitrary shape functional is defined examples are provided for elliptic equations and the elasticity system in the plane the topological derivative can be used for solving shape optimization problems in structural mechanics

fluctuations of an atomic ledge bordering a crystalline facet when a high symmetry facet joins the rounded part of a crystal the step line density vanishes as square root of r with r denoting the distance from the facet edge this means that the ledge bordering the facet has a lot of space to meander as caused by thermal activation we investigate the statistical properties of the border ledge fluctuations in the scaling regime they turn out to be non gaussian and related to the edge statistics of gaussian unitary ensemble multimatrix models

plurisubharmonic functions in calibrated geometry and q convexity let m be a kahler manifold an integrable function varphi on m is called q plurisubharmonic if the current dd c varphi wedge omega q 1 is positive we prove that varphi is q plurisubharmonic if and only if varphi is subharmonic on all q dimensional complex subvarieties we prove that a q plurisubharmonic function is q convex and admits a local approximation by smooth q plurisubharmonic functions for any closed subvariety z subset m dim mathbb c z leq q 1 there exists a strictly q plurisubharmonic function in a neighbourhood of z this result is known for q convex functions this theorem is used to give a new proof of sibony s lemma on integrability of positive closed p p forms which are integrable outside of a complex subvariety of codimension p 1

structural optimization using sensitivity analysis and a level set method in the context of structural optimization we propose a new numerical method based on a combination of the classical shape derivative and of the level set method for front propagation we implement this method in two and three space dimensions for a model of linear or nonlinear elasticity we consider various objective functions with weight and perimeter constraints the shape derivative is computed by an adjoint method the cost of our numerical algorithm is moderate since the shape is captured on a fixed eulerian mesh although this method is not specifically designed for topology optimization it can easily handle topology changes however the resulting optimal shape is strongly dependent on the initial guess

a unified parametric design approach to structural shape optimization this paper presents a general parametric design approach for 2 d shape optimization problems this approach has been achieved by integrating practical design methodologies into numerical procedures it is characterized by three features i automatic selection of a minimum number of shape design variables based on the cad geometric model ii integration of sequential convex programming algorithms to solve equality constrained optimization problems iii efficient sensitivity analysis by means of the improved semi analytical method it is shown that shape design variables can be either manually or systematically identified with the help of equality constraints describing the relationship between geometric entities numerical solutions are performed to demonstrate the applicability of the proposed approach a discussion of the results is also given

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

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

structural topology optimization with minimum distance control of multiphase embedded components by level set method abstract this paper presents a novel topology optimization method for designing structures with multiphase embedded components under minimum distance constraints in the level set framework by using the level set representation for both the component layout and the host structure topology the shapes of the components can be easily preserved and optimal structural topologies with smooth boundary material interface can be obtained with the purpose of preventing the components moving too close to each other a minimum distance constraint based on virtual boundary offset is proposed different from existing distance detection methods relying on explicit topology representation the proposed constraint is imposed as a unified integral form for which the design sensitivity can be readily obtained moreover this constraint is effective for detecting the distance between any complex shaped components several numerical examples are presented to demonstrate the validity and effectiveness of the proposed method

feature driven topology optimization method with signed distance function abstract in this paper a feature driven topology optimization method is developed this is the first study on layout design of multiple engineering features using level set functions lsfs and boolean operations the novelty of this work is threefold first multiple engineering features of arbitrary shape are considered as basic design primitives and topology variation is achieved via the layout and shape optimization of the features kreisselmeier steinhauser ks function constructed by means of boolean operations is adopted as the lsf which uses an implicit function to ensure a smooth description and topological changes of basic features and the whole structure second using a modified heaviside function to smooth the void solid material transition over a fixed computing mesh a narrow band domain integral scheme is developed for the efficient sensitivity analysis third the gray material distribution regions at the feature connecting portions are analyzed and the underlying reason for that is traced to the non equidistant distribution of level set contours of specific features to avoid the gray regions an approximated signed distance function is proposed to regularize the lsf and ks function the bounded normalization property of the ks function is highlighted for its construction with the signed distance functions or normalized first order approximations numerical examples are finally tested to demonstrate the validity and merits of the proposed feature driven topology optimization for complicated design problems

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

integrated layout design of multi component systems using xfem and analytical sensitivity analysis this study presents the integrated layout optimization of multi component systems using a fixed mesh the optimization formulation is established under the framework of the extended finite element method xfem the level set method is used to represent components and is combined with the xfem to describe material discontinuities across elements sensitivity analysis is proposed with respect to geometric variables of components and pseudo densities of the basic structure an analytical shape sensitivity analysis method with respect to positions and shapes of components is developed both solid and void components are considered to show the efficiency and accuracy of the proposed shape sensitivity analysis method furthermore a revised finite circle method that adapts shape changes of elliptical components is proposed for the definition of non overlapping constraints finally numerical examples of maximizing the structural stiffness are tested to demonstrate the proposed method

an implicit model for the integrated optimization of component layout and structure topology abstract integrated design of the structure topology and involved component layout is a challenging design issue when compared with traditional topology optimization in this paper we propose an implicit modeling approach that works completely on an eulerian finite element mesh throughout the whole optimization process to this aim implicit level set functions and r functions are employed to describe geometrical shapes of movable components in particular a modified arctan function is adopted to depict the material discontinuity along the interface between the structure domain and each component domain they are then used for material interpolations and analytical sensitivity analysis w r t both pseudo density design variables and location design variables related to the host structure and components respectively based on a variety of numerical tests it is demonstrated that considered design problems with movable components can easily be solved by extending the simp material model based topology optimization approach using an eulerian mesh and the gradient based optimization algorithm