full analytical sensitivities in nurbs based isogeometric shape optimization abstract non uniform rational b spline nurbs has been widely used as an effective shape parameterization technique for structural optimization due to its compact and powerful shape representation capability and its popularity among cad systems the advent of nurbs based isogeometric analysis has made it even more advantageous to use nurbs in shape optimization since it can potentially avoid the inaccuracy and labor tediousness in geometric model conversion from the design model to the analysis model although both positions and weights of nurbs control points affect the shape until very recently usually only control point positions are used as design variables in shape optimization thus restricting the design space and limiting the shape representation flexibility this paper presents an approach for analytically computing the full sensitivities of both the positions and weights of nurbs control points in structural shape optimization such analytical formulation allows accurate calculation of sensitivity and has been successfully used in gradient based shape optimization the analytical sensitivity for both positions and weights of nurbs control points is especially beneficial for recovering optimal shapes that are conical e g ellipses and circles in 2d cylinders ellipsoids and spheres in 3d that are otherwise not possible without the weights as design variables

shape optimization of shell structures based on nurbs description using automatic differentiation the consolidation of the link among four fields in computational mathematics and mechanics is the main objective of this work surfaces based on nurbs non uniform rational b spline mathematical optimization the finite element method fem in structural analysis and automatic differentiation ad are applied in shape optimization of shells this problem is performed taking into account the fact that material and mechanical characteristics influence both the structural shape and the thickness variation in order to obtain the best performance with respect to a specific criterium some techniques were implemented to modify the shell geometry conserving the same parameterization without a new finite element mesh generation the shape modification is carried out using an optimization code based on the data obtained by a finite element analysis and gradients evaluation in this work the optimization procedure is performed using an sqp sequential quadratic programming algorithm where the variables are the control points in homogeneous coordinates the knot vectors and the thickness the functional value is determined by the fem and gradients are evaluated using ad some examples are analyzed and discussed as a consequence of the shape optimization shells with high structural performance and aesthetically beautiful shapes can be obtained copyright 2011 john wiley sons ltd

a probabilistic approach to the geometry of the lᵨⁿ ball this article investigates by probabilistic methods various geometric questions on b n p the unit ball of ln p we propose realizations in terms of independent random variables of several distributions on b n p including the normalized volume measure these representations allow us to unify and extend the known results of the sub independence of coordinate slabs in b n p as another application we compute moments of linear functionals on b n p which gives sharp constants in khinchine s inequalities on b n p and determines the 2 constant of all directions on b n p we also study the extremal values of several gaussian averages on sections of b n p including mean width and l norm and derive several monotonicity results as p varies applications to balancing vectors in l 2 and to covering numbers of polyhedra complete the exposition

approximate independence of distributions on spheres and their stability properties let zeta be chosen at random on the surface of the p sphere in mathbb r n 0 p n x in mathbb r n sum n i 1 x i p n if p 2 then the first k components zeta 1 ldots zeta k are for k fixed in the limit as n rightarrow infty independent standard normal considering the general case p 0 the same phenomenon appears with a distribution f p in an exponential class mathscr e f p can be characterized by the distribution of quotients of sums by conditional distributions and by a maximum entropy condition these characterizations have some interesting stability properties some discrete versions of this problem and some applications to de finetti type theorems are discussed

rates of convergence for densities in extreme value theory on donne un resultat de taux de convergence pour la densite de maxima d echantillons normalises vers la densite limite appropriee

diffusion kernels on statistical manifolds a family of kernels for statistical learning is introduced that exploits the geometric structure of statistical models the kernels are based on the heat equation on the riemannian manifold defined by the fisher information metric associated with a statistical family and generalize the gaussian kernel of euclidean space as an important special case kernels based on the geometry of multinomial families are derived leading to kernel based learning algorithms that apply naturally to discrete data bounds on covering numbers and rademacher averages for the kernels are proved using bounds on the eigenvalues of the laplacian on riemannian manifolds experimental results are presented for document classification for which the use of multinomial geometry is natural and well motivated and improvements are obtained over the standard use of gaussian or linear kernels which have been the standard for text classification

stress adapted numerical form finding of pre stressed surfaces by the updated reference strategy in this paper we present the updated reference strategy for numerical formfinding of pre stressed membranes which is based on standard finite elementdiscretization the singularities of the inverse problem are regularizedby a homotopy mapping a projection scheme is proposed where anisotropicpre stress is defined with respect to an additional reference plane whichreflects the initially developable surface of membrane strips in the productionprocess physically problematic combinations of edge geometry and surfacestress are solved by a self adaptive stress correction scheme the algorithmis based on a local criterion derived from differential geometry severalexamples illustrate the success of each idea and implementation

a fictitious energy approach for shape optimization this paper deals with shape optimization of continuous structures as in early works on shape optimization coordinates of boundary nodes of the fe domain are directly chosen as design variables convergence problems and problems with jagged shapes are eliminated by a new regularization technique an artificial inequality constraint added to the optimization problem limits a fictitious total strain energy that measures the shape change of the design with respect to a reference design the energy constraint defines a feasible design space whose size can be varied by one parameter the upper energy limit by construction the proposed regularization is applicable to a wide range of problems although in this paper the application is restricted to linear elastostatic problems

form finding of membrane structures by the updated reference method with minimum mesh distortion this paper presents a new technique for the solution of the well known form finding problem in membrane structures the technique proposed is based on the updated reference configuration method proposed by bletzinger proceedings of the iass colloquium on structural morphology towards the new millenium nottingham 1997 in which an area functional is minimised within the context of a finite element discretisation in this paper an additional functional term is introduced with the aim of minimising the mesh distortion during the form finding process this new term provides in plane stiffness which prevents the emergence of mechanisms without the need for ad hoc changes of the tangent matrix as a consequence of this additional strain energy term the form finding process can be viewed as a standard large strain membrane analysis with a specific choice of hyperelastic strain energy function this implies that it can be implemented in any standard code as an additional constitutive model two well known simple examples of form finding will be presented to illustrate the method proposed

rates for the clt via new ideal metrics let b be a separable banach space and mathscr x mathscr x b the vector space of all random variables defined on a probability space omega mathscr a p and taking values in b it is shown that new ideal metrics for mathscr x may be used to obtain refined rates of convergence of normalized sums to a stable limit law the rates hold uniformly in n and are expressed in terms of a variety of uniform metrics on mathscr x in the banach space setting the rates hold with respect to the total variation metric and in the euclidean space setting the rates hold with respect to uniform metrics between density and characteristic functions the main result provides a sharp order estimate of the rate of convergence in local limit theorems with respect to the uniform distance between densities the method is based on the theory of probability metrics especially those of convolution type

shape optimization of thin walled structures governed by geometrically nonlinear mechanics this paper introduces a structural optimization strategy that combines fe based parametrization with nonlinear kinematics in order to optimize the shape of thin shell structures the optimization is based on gradient based strategies where the required derivatives are formulated by the adjoint approach the applied solution algorithm combines the well known nonlinear path following strategies with the design update procedure of the shape optimization this results in robust and flexible methods that require only a minimum of system evaluations the proposed optimization goals improve the load carrying behavior of the structure and minimize displacements and stresses it is shown that such efficient designs also exhibit an improved limit load this contribution illustrates the application of the proposed method by several shape optimization problems the presented results prove the exceptional performance of the optimized designs even if the optimal design is disturbed by unavoidable imperfections it is shown that the application of nonlinear kinematics in the shape optimization of thin shell structures allows for a much more realistic system response and gradient data the proposed approach is applicable to all kind of optimization strategies like topology sizing or material optimization respectively

eigenfunctions of the edge based laplacian on a graph abstract in this paper we analyze the eigenfunctions of the edge based laplacian on a graph and the relationship of these functions to random walks on the graph we commence by discussing the set of eigenfunctions supported at the vertices and demonstrate the relationship of these eigenfunctions to the classical random walk on the graph then from an analysis of functions supported only on the interior of edges we develop a method for explicitly calculating the edge interior eigenfunctions of the edge based laplacian this reveals a connection between the edge based laplacian and the adjacency matrix of backtrackless random walk on the graph the edge based eigenfunctions therefore correspond to some eigenfunctions of the normalized hashimoto matrix

an overview and comparison of structural form finding methods for general networks abstract this paper discusses and compares existing form finding methods for discrete networks well known methods such as the force density method dynamic relaxation updated reference strategy and others are discussed by mathematically structuring and presenting them in the same way using the same notation and combining terminology based on this a single computational framework using a sparse branch node data structure is presented it is shown how each method approaches the initial equilibrium problem defines and linearizes the equilibrium equations applied to linear elements and uses particular solving strategies this framework marginalizes any differences related to operating platforms programming language and style offering a better baseline for independent comparison of performance and results as a consequence it is possible to more clearly relate distinguish and compare existing methods allow for hybrid methods and identify new avenues for research

a gradient based shape optimization scheme using an interface enriched generalized fem abstract a gradient based shape optimization scheme using an interface enriched generalized finite element method igfem is presented wherein the design geometry is projected onto a fixed mesh and the igfem is used for analysis this approach eliminates the mesh distortion present in conventional lagrangian shape optimization methods as well as the need for remeshing an analytical sensitivity analysis using both the adjoint or direct approaches is presented to compute derivatives of the objective and constraint functions due to the fixed nature of the mesh the so called design velocity field only needs to be computed on the structure boundary interface a comparison between igfem and conventional fem based shape optimization schemes is presented showing an improved precision for the igfem approach finally we solve various numerical examples to demonstrate the capability of the method including the computational design of particulate and microvascular composites

optimal shapes of mechanically motivated surfaces abstract subject of this contribution is form finding of optimal structural shapes with regard to the load carrying behaviour of surface structures under certain load cases in general those optimal shapes prefer a membrane state of stress to transfer loading bending is omitted as much as possible it will be focused on two different disciplines and related numerical approaches which deal with solutions of the mentioned task form finding of prestressed membranes and general shape optimization as design is an inverse problem both approaches share similar problematic properties as e g indeterminate in plane location of surface discretization or necessary regularization and filtering of sensitivity and other data as it will turn out those remedies found for the very special methods of membrane design can be abstracted and transferred to general optimization procedures that merges into elegant numerical shape optimal design techniques which combine advantages of both approaches and allow for effective and efficient shape optimization of free formed surfaces directly on the finite element mesh and for a large number of variables typical applications are for example membrane design free form architecture and structural engineering and metal sheet design

shape optimisation with multiresolution subdivision surfaces and immersed finite elements abstract we develop a new optimisation technique that combines multiresolution subdivision surfaces for boundary description with immersed finite elements for the discretisation of the primal and adjoint problems of optimisation similar to wavelets multiresolution surfaces represent the domain boundary using a coarse control mesh and a sequence of detail vectors based on the multiresolution decomposition efficient and fast algorithms are available for reconstructing control meshes of varying fineness during shape optimisation the vertex coordinates of control meshes are updated using the computed shape gradient information by virtue of the multiresolution editing semantics updating the coarse control mesh vertex coordinates leads to large scale geometry changes and conversely updating the fine control mesh coordinates leads to small scale geometry changes in our computations we start by optimising the coarsest control mesh and refine it each time the cost function reaches a minimum this approach effectively prevents the appearance of non physical boundary geometry oscillations and control mesh pathologies like inverted elements independent of the fineness of the control mesh used for optimisation on the immersed finite element grid the domain boundary is always represented with a relatively fine control mesh of fixed resolution with the immersed finite element method there is no need to maintain an analysis suitable domain mesh in some of the presented two and three dimensional elasticity examples the topology derivative is used for introducing new holes inside the domain the merging or removing of holes is not considered

on the geometry of random convex sets between polytopes and zonotopes abstract in this work we study a class of random convex sets that interpolate between polytopes and zonotopes these sets arise from considering a q t h moment q 1 of an average of order statistics of 1 dimensional marginals of a sequence of n n independent random vectors in r n we consider the random model of isotropic log concave distributions as well as the uniform distribution on an l p n sphere 1 p with respect to the cone probability measure and study the geometry of these sets in terms of the support function and mean width we provide asymptotic formulas for the expectation of these geometric functionals which are sharp up to absolute constants our model includes and generalizes the standard one for random polytopes

isogeometric shape optimization of shells using semi analytical sensitivity analysis and sensitivity weighting abstract we present isogeometric shape optimization for shell structures applying sensitivity weighting and semi analytical analysis we use a rotation free shell formulation and all involved geometry models i e initial design analysis model optimization model and final design use the same geometric basis in particular nurbs a sensitivity weighting scheme is presented which eliminates certain effects of the chosen discretization on the design update a multilevel design approach is applied such that the design space can be chosen independently from the analysis space the use of semi analytical sensitivities allows having different polynomial degrees for design and analysis model different numerical examples are performed which confirm the applicability of the proposed method furthermore a shape optimization example with an exact solution is presented which can serve as general benchmark for shape optimization methods

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

on the discrete variant of the traction method in parameter free shape optimization abstract one of the major challenges in the parameter free approach to computational shape optimization is the avoidance of oscillating i e non smooth boundaries in the optimal design trials in order to achieve this we investigate a method for regularization that corresponds to the discrete variant of the so called traction method in this approach the design updates are generated in terms of a displacement field which is obtained as the solution to an auxiliary boundary value problem that is defined on the actual design domain the main idea herein is to apply fictitious nodal forces corresponding to the discrete sensitivity of the objective function we propose an algorithm in which constraint functions will be taken into account by using an augmented lagrangian formulation and a step length control ensures a sufficient decrease condition in terms of the objective function within each iteration we examine the benefits of the proposed regularization method on the basis of some numerical examples in comparison to an unregularized steepest descent algorithm

convergence of spectra of mesoscopic systems collapsing onto a graph abstract let m be a finite graph in the plane and let me be a domain that looks like the e fattened graph m exact conditions on the domain are given it is shown that the spectrum of the neumann laplacian on me converges when e 0 to the spectrum of an ode problem on m the presence of an electromagnetic field is also allowed considerations of this kind arise naturally in mesoscopic physics and other areas of physics and chemistry the results of the paper extend the ones previously obtained by j rubinstein and m schatzman

in plane mesh regularization for node based shape optimization problems abstract in this contribution a global and linear method for controlling the surface mesh quality during node based shape optimization is presented in this method an artificial stress is applied on the surface mesh and the global linear system of equations is solved for the equilibrium state by finite elements the applied stress adapts each element towards a desired predefined template geometry and at the end a globally smooth mesh is achieved in this way both the shape and the size of each element is controlled some examples as well as possible fields of application are shown at the end the strength of the method in shape optimization of computational fluid dynamics problems is described

the surface measure and cone measure on the sphere of ell_p n we prove a concentration inequality for the l n p norm on the l n p sphere for p q 0 this inequality which generalizes results of schechtman and zinn 2000 is used to study the distance between the cone measure and surface measure on the sphere of l n p in particular we obtain a significant strengthening of the inequality derived by naor and romik 2003 and calculate the precise dependence of the constants that appeared there on p

variational problems on multiply connected thin strips i basic estimates and convergence of the laplacian spectrum let m be a planar embedded graph whose arcs meet transversally at the vertices let be a strip shaped domain around m of width except in a neighborhood of the singular points assume that the boundary of is smooth we define comparison operators between functions on and on m and we derive energy estimates for the compared functions we define a laplace operator on m which is in a certain sense the limit of the laplace operator on with neumann boundary conditions in particular we show that the p th eigenvalue of the laplacian on converges to the p th eigenvalue of the laplacian on m as tends to 0 a similar result holds for the magnetic schrodinger operator

coined quantum walks lift the cospectrality of graphs and trees in this paper we explore how a spectral technique suggested by coined quantum walks can be used to distinguish between graphs that are cospectral with respect to standard matrix representations the algorithm runs in polynomial time and moreover can distinguish many graphs for which there is no subexponential time algorithm that is proven to be able to distinguish between them in the paper we give a description of the coined quantum walk from the field of quantum computing the evolution of the walk is governed by a unitary matrix we show how the spectrum of this matrix is related to the spectrum of the transition matrix of the classical random walk however despite this relationship the behaviour of the quantum walk is vastly different from the classical walk this leads us to define a new matrix based on the amplitudes of paths of the walk whose spectrum we use to characterise graphs we carry out three sets of experiments using this matrix representation firstly we test the ability of the spectrum to distinguish between sets of graphs that are cospectral with respect to standard matrix representation these include strongly regular graphs and incidence graphs of balanced incomplete block designs bibds secondly we test our method on all regular graphs on up to 14 vertices and all trees on up to 24 vertices this demonstrates that the problem of cospectrality is often encountered with conventional algorithms and tests the ability of our method to resolve this problem thirdly we use distances obtained from the spectra of s u 3 to cluster graphs derived from real world image data and these are qualitatively better than those obtained with the spectra of the adjacency matrix thus we provide a spectral representation of graphs that can be used in place of standard spectral representations far less prone to the problems of cospectrality