isogeometric structural shape optimization an old dilemma in structural shape optimization is the needed tight link between design model or geometric description and analysis model the intention of this paper is to show that isogeometric analysis offers a potential and promising way out of this dilemma to this end we show a structural shape optimization framework based on the isogeometric analysis approach with the discretization based on non uniform rational b splines nurbs the analysis model represents the structural geometry exactly furthermore nurbs enable efficient geometry control together with smooth boundaries they are the de facto standard in cad systems but are also widely used in a shape optimal design context to define the geometry representation and the design variables with the presented isogeometric approach to shape optimization the analysis model is inherently merged with the design model omitting the typically involved interplay between both we derive analytical sensitivities for nurbs discretizations which allow application of efficient gradient based optimization algorithms the present contribution is restricted to two dimensional problems of linear elasticity but the extension to three dimensions and other problem classes is straightforward some representative examples demonstrate and validate the methodology further the potential of boundary continuity control within isogeometric structural shape optimization is explored to trigger smooth or less smooth angular designs

kullback leibler divergence measure of intermittency application to turbulence for generic systems exhibiting power law behaviors and hence multiscale dependencies we propose a simple tool to analyze multifractality and intermittency after noticing that these concepts are directly related to the deformation of a probability density function from gaussian at large scales to non gaussian at smaller scales our framework is based on information theory and uses shannon entropy and kullback leibler divergence we provide an extensive application to three dimensional fully developed turbulence seen here as a paradigmatic complex system where intermittency was historically defined and the concepts of scale invariance and multifractality were extensively studied and benchmarked we compute our quantity on experimental eulerian velocity measurements as well as on synthetic processes and phenomenological models of fluid turbulence our approach is very general and does not require any underlying model of the system although it can probe the relevance of such a model

topology optimization in b spline space abstract in this paper we present a new form of density based topology optimization where the design space is restricted to the b spline space an arbitrarily shaped design domain is embedded into a rectangular domain in which tensor product b splines are used to represent the density field we show that with proper choice of b spline degrees and knot spans the b spline design space is free from checkerboards without extraneous filtering or penalty we further reveal that the b spline representation provides an intrinsic filter for topology optimization where the filter size is controlled by b spline degrees and knot spans this b spline filter is effective in removing numerical artifacts and controlling minimal feature length in optimized structures when the b spline basis functions span multiple analysis elements we demonstrate that the b spline filter is linear in storage cost and does not require neighboring element information further this b spline based density representation decouples the design representation of density distribution from the finite element mesh thus multi resolution designs can be obtained without re meshing the design domain in particular successive optimization with respect to design resolutions leads to topologically simple features obtainable in either coarse or fine design resolutions thus achieving a form of mesh independency with respect to design representation this approach is versatile in the sense a variety of finite element and isogeometric analysis techniques can be used for solution of equilibrium equations and a variety of projection methods can be used to approximate b spline density in analysis numerical studies have been conducted over several representative topology optimization problems including minimal compliance of mbb beams compliant mechanism inverters and heat conductions

route to chaos and some properties in the boundary crisis of a generalized logistic mapping abstract a generalization of the logistic map is considered showing two control parameters and that can reproduce different logistic mappings including the traditional second degree logistic map cubic quartic and all other degrees we introduce a parametric perturbation such that the original logistic map control parameter r changes its value periodically according an additional parameter 2 q the value of q gives this period for this system an analytical expression is obtained for the first bifurcation that starts a period doubling cascade and using the feigenbaum universality we found numerically the accumulation point r c where the cascade finishes giving place to chaos in the second part of the paper we study the death of this chaotic behavior due to a boundary crisis at the boundary crisis orbits can reach a maximum value x x max 1 when it occurs the trajectory is mapped to a fixed point at x 0 we show that there exist a general recursive formula for initial conditions that lead to x x max

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

optimal tailoring of 2d volume fraction distributions for heat resisting functionally graded materials using fdm thermoelastic behavior of heat resisting functionally graded materials fgms under given thermal loading and boundary conditions is definitely characterized by the spatial distribution of volume fractions of constituent particles hence the determination of volume fraction distribution becomes a crucial step in tailoring an optimal heat resisting fgm in this paper we address a two dimensional volume fraction optimization procedure for relaxing the effective thermal stress distribution in order for the optimization efficiency and the volume fraction continuity we approximate the volume fraction field with bilinear elements of n times larger size than for the thermoelastic analysis as well the refined material property estimate is employed to assure the design quality numerical experiments illustrating our theoretical work are also presented

contact between plates by a new direct boundary integral equation formulation abstract the unilateral contact between two plates according to kirchhoff s theory is solved by using a boundary integral equation method a discretization leading to a matrix formulation is proposed to solve this problem an elimination of boundary unknowns is performed an iterative method proves successful with the example treated

the local structure of turbulence in incompressible viscous fluid for very large reynolds numbers 1 we shall denote by u p u x 1 x 2 x 3 t 1 2 3 the components of velocity at the moment t at the point with rectangular cartesian coordinates x 1 x 2 x 3 in considering the turbulence it is natural to assume the components of the velocity u p at every point p x 1 x 2 x 3 t of the considered domain g of the four dimensional space x 1 x 2 x 3 t are random variables in the sense of the theory of probabilities cf for this approach to the problem millionshtchikov 1939 denoting by the mathematical expectation of the random variable a we suppose that 2 and d u d x 2 are finite and bounded in every bounded subdomain of the domain g

the first digit frequencies in data of turbulent flows abstract considering the first significant digits noted d in data sets of dissipation for turbulent flows the probability to find a given number d 1 or 2 or 9 would be 1 9 for a uniform distribution instead the probability closely follows newcomb benford s law namely p d log 1 1 d the discrepancies between newcomb benford s law and first digits frequencies in turbulent data are analysed through shannon s entropy the data sets are obtained with direct numerical simulations for two types of fluid flow an isotropic case initialized with a taylor green vortex and a channel flow results are in agreement with newcomb benford s law in nearly homogeneous cases and the discrepancies are related to intermittent events thus the scale invariance for the first significant digits which supports newcomb benford s law seems to be related to an equilibrium turbulent state namely with a significant inertial range a matlab octave program provided in appendix is such that part of the presented results can easily be replicated

isogeometric shape optimization of photonic crystals via coons patches abstract in this paper we present an approach that extends isogeometric shape optimization from optimization of rectangular like nurbs patches to the optimization of topologically complex geometries we have successfully applied this approach in designing photonic crystals where complex geometries have been optimized to maximize the band gaps salient features of this approach include the following 1 multi patch coons representation of design geometry the design geometry is represented as a collection of coons patches where the four boundaries of each patch are represented as nurbs curves the use of multiple patches is motivated by the need for representing topologically complex geometries the coons patches are used as a design representation so that designers do not need to specify interior control points and they provide a mechanism to compute analytical sensitivities for internal nodes in shape optimization 2 exact boundary conversion to the analysis geometry with guaranteed mesh injectivity the analysis geometry is a collection of nurbs patches that are converted from the multi patch coons representation with geometric exactness in patch boundaries the internal nurbs control points are embedded in the parametric domain of the coons patches with a built in mesh rectifier to ensure the injectivity of the resulting b spline geometry i e every point in the physical domain is mapped to one point in the parametric domain 3 analytical sensitivities sensitivities of objective functions and constraints with respect to design variables are derived through nodal sensitivities the nodal sensitivities for the boundary control points are directly determined by the design parameters and those for internal nodes are obtained via the corresponding coons patches

mutual information functions versus correlation functions this paper studies one application of mutual information to symbolic sequences the mutual information functionm d this function is compared with the more frequently used correlation function d an exact relation betweenm d and d is derived for binary sequences for sequences with more than two symbols no such general relation exists in particular d 0 may or may not lead tom d 0 this linear but not general independence between symbols separated by a distance is studied for ternary sequences also included is the estimation of the finite size effect on calculating mutual information finally the concept of symbolic noise is discussed

the impact problem for two strips abstract this paper considers the dynamic problem of contact interaction of two infinite strips arranged in parallel with an initial gap contact interaction of strips occurs due to impulse loading equations of the 2 dimensional generalized theory of plates and the theory of elasticity are used to describe the dynamic behavior of strips the contact problem is solved using the collocation method with iterative refining of the contact area sizes at each time step

large volume limit of the distribution of characteristic exponents in turbulence for spatially extended conservative or dissipative physical systems it appears natural that a density of characteristic exponents per unit volume should exist when the volume tends to infinity in the case of a turbulent viscous fluid however this simple idea is complicated by the phenomenon of intermittency in the present paper we obtain rigorous upper bounds on the distribution of characteristic exponents in terms of dissipation these bounds have a reasonable large volume behavior for two dimensional fluids a particularly striking result is obtained the total information creation is bounded above by a fixed multiple of the total energy dissipation at fixed viscosity the distribution of characteristic exponents is estimated in an intermittent model of turbulence see 7 and it is found that a change of behavior occurs at the valued 2 6 of the self similarity dimension

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

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

parallelized structural topology optimization for eigenvalue problems a parallel topology optimization method is proposed to deal with large scale structural eigenvalue related design problems while existing parallel methods have been employed mainly for finite element analysis the present work deals with the entire parallel process of analysis and topology optimization using a parallel processing technique in the framework of the physical domain decomposition method the sensitivity analysis and design variable update are performed independently in each subdomain with minimum data communication among subdomains a successive estimation strategy over design iterations is proposed to expedite the design optimization process the preconditioned conjugate gradient method and the subspace iteration method are used as parallel solvers several numerical examples are considered to show the scalability and effectiveness of the present parallel approach in handling large scale design problems

a boundary integral equation method for plate flexure with conditions inside the domain a method for solving boundary value problems for thin plate flexure as described by kirchhoff s theory is proposed an integral formulation leads to a system of boundary integral equations involving values of deflection slope bending moment and transverse shear force along the edge a discretization leading to a matrix formulation is proposed to solve problems with inner conditions in the plate domain an elimination of boundary unknowns proves successful the degenerate case where the boundary is free which leads to a non invertible matrix is investigated three examples illustrate the efficiency of the method

structural information in two dimensional patterns entropy convergence and excess entropy we develop information theoretic measures of spatial structure and pattern in more than one dimension as is well known the entropy density of a two dimensional configuration can be efficiently and accurately estimated via a converging sequence of conditional entropies we show that the manner in which these conditional entropies converge to their asymptotic value serves as a measure of global correlation and structure for spatial systems in any dimension we compare and contrast entropy convergence with mutual information and structure factor techniques for quantifying and detecting spatial structure

structural topology optimization for frequency response problem using model reduction schemes abstract this study uses model reduction mr schemes such as the mode superposition ms ritz vector rv and quasi static ritz vector qsrv methods which reduce the size of the dynamic stiffness matrix of dynamic structures to calculate dynamic responses and sensitivity values with adequate efficiency and accuracy for topology optimization in the frequency domain the calculation of structural responses to dynamic excitation using the framework of the finite element fe procedure usually requires a significant amount of computation time that is mainly attributable to repeated inversions of dynamic stiffness matrices depending on time or frequency intervals which hastens the dissemination of the mr schemes in the analysis however using well established mr schemes in topology optimization has not been prevalent therefore this study conducted a comprehensive investigation to highlight the drawbacks and advantages of these mr schemes for topology optimization in the results the ms method which generates reduction bases by considering some of the lowest eigenmodes can lose the accuracy in both approximated structural responses and sensitivity values due to locally vibrating eigenmodes and higher mode truncation in the solid isotropic material with penalization simp approach in addition the rv and qsrv methods which generate reduction bases by considering the external force mass and stiffness matrices of a structure can be used as alterative model reduction schemes for stable optimization through several analysis and design examples the efficiency and reliability of the model reduction schemes for topology optimization are compared and validated

information content of turbulence we treat a turbulent velocity field as a message in the same way as a book or a picture all messages can be described by their entropy per symbol h defined as in shannon s theory of communication in a turbulent flow as the reynolds number re increases more correlated degrees of freedom are excited and participate in the turbulent cascade experiments in a turbulent soap film suggest that the spatial entropy density h is a decreasing function of re namely h proportionality logre const in the logistic map also analyzed here increasing the control parameter r increases h a modified logistic map with additional coupling to past iterations suggests the significance of correlations