sensitivity analysis with the modified heaviside function for the optimal layout design of multi component systems two kinds of design variables i e pseudo density variables associated with the framework structure and location design variables associated with connected components are involved in the layout design of multi component systems although sensitivities with respect to the first ones can easily be carried out as in topology optimization the semi analytical method sam is often used for sensitivity analysis with respect to the location design variables due to the geometric perturbation of the finite element mesh the latter can then be regarded as a geometric perturbation model gpm in this paper we propose a material perturbation model mpm using fixed finite element fe mesh for sensitivity analysis with respect to location design variables the material discontinuity across the boundary between each component and the framework structure is smoothed approximately by means of a modified heaviside function when a location design variable of a certain component is perturbed attached finite elements to the component boundary are assumed to undertake only a shift of material properties while the finite element mesh itself remains geometrically unchanged as a result analytical sensitivities with respect to location design variables are achieved as easily as for pseudo density variables the computing efficiency is thus improved because the velocity field for the mesh perturbation in the semi analytical scheme is no longer needed the mpm is illustrated by means of numerical tests especially the design optimization of 3d multi component systems

cutfem topology optimization of 3d laminar incompressible flow problems abstract this paper studies the characteristics and applicability of the cutfem approach of burman et al 2014 as the core of a robust topology optimization framework for 3d laminar incompressible flow and species transport problems at low reynolds numbers r e 200 cutfem is a methodology for discretizing partial differential equations on complex geometries by immersed boundary techniques in this study the geometry of the fluid domain is described by an explicit level set method where the parameters of a discretized level set function are defined as functions of the optimization variables the fluid behavior is modeled by the incompressible navier stokes equations species transport is modeled by an advection diffusion equation the governing equations are discretized in space by a generalized extended finite element method face oriented ghost penalty terms are added for stability reasons and to improve the conditioning of the system the boundary conditions are enforced weakly via nitsche s method the emergence of isolated volumes of fluid surrounded by solid during the optimization process leads to a singular analysis problem an auxiliary indicator field is modeled to identify these volumes and to impose a constraint on the average pressure numerical results for 3d steady state and transient problems demonstrate that the cutfem analyses are sufficiently accurate and the optimized designs agree well with results from prior studies solved in 2d or by density approaches

topology optimization with pressure load through a level set method abstract the topology optimization problem with pressure load is solved by using a level set method the free boundary and the pressure boundary of a structure are represented separately as two zero level sets of two level set functions and they are independently propagated during the optimization by solving two hamilton jacobi equations in order to prevent the two boundaries from touching or crossing each other the design velocities of the two boundaries that amount to the steepest descent directions are modified the optimization problem of minimum compliance with perimeter regularization is considered the shape derivatives of the two boundaries are derived by using the material derivative approach and the adjoint method the finite element analysis is done through an eulerian method by employing a fixed mesh and an artificial weak material that represents void numerical examples in two dimensions are investigated

level set based structural topology optimization for minimizing frequency response abstract for the purpose of structure vibration reduction a structural topology optimization for minimizing frequency response is proposed based on the level set method the objective of the present study is to minimize the frequency response at the specified points or surfaces on the structure with an excitation frequency or a frequency range subject to the given amount of the material over the admissible design domain the sensitivity analysis with respect to the structural boundaries is carried out while the extended finite element method x fem is employed for solving the state equation and the adjoint equation the optimal structure with smooth boundaries is obtained by the level set evolution with advection velocity derived from the sensitivity analysis and the optimization algorithm a number of numerical examples in the frameworks of two dimension 2d and three dimension 3d are presented to demonstrate the feasibility and effectiveness of the proposed approach

quasi shadowing for partially hyperbolic dynamics on banach spaces abstract a partially hyperbolic dynamical system is said to have the quasi shadowing property if every pseudotrajectory can be shadowed by a sequence of points x n n z such that x n 1 is obtained from the image of x n by moving it by a small factor in the central direction in the present paper we prove that a small nonlinear perturbation of a partially dichotomic sequence of not necessarily invertible linear operators acting on an arbitrary banach space has the quasi shadowing property we also obtain a continuous time version of this result as an application of our main result we prove that a certain class of partially dichotomic sequences of linear operators is stable up to the movement in the central direction

a semi implicit level set method for structural shape and topology optimization this paper proposes a new level set method for structural shape and topology optimization using a semi implicit scheme structural boundary is represented implicitly as the zero level set of a higher dimensional scalar function and an appropriate time marching scheme is included to enable the discrete level set processing in the present study the hamilton jacobi partial differential equation pde is solved numerically using a semi implicit additive operator splitting aos scheme rather than explicit schemes in conventional level set methods the main feature of the present method is it does not suffer from any time step size restriction as all terms relevant to stability are discretized in an implicit manner the semi implicit scheme with additive operator splitting treats all coordinate axes equally in arbitrary dimensions with good rotational invariance hence the present scheme for the level set equations is stable for any practical time steps and numerically easy to implement with high efficiency resultantly it allows enhanced relaxation on the time step size originally limited by the courant friedrichs lewy cfl condition of the explicit schemes the stability and computational efficiency can therefore be greatly improved in advancing the level set evolvements furthermore the present method avoids additional cost to globally reinitialize the level set function for regularization purpose it is noted that the periodically applied reinitializations are time consuming procedures in particular the proposed method is capable of creating new holes freely inside the design domain via boundary incorporating splitting and merging processes which makes the final design independent of initial guess and helps reduce the probability of converging to a local minimum the availability of the present method is demonstrated with two widely studied examples in the framework of the structural stiffness designs

on minima of a functional of the gradient necessary conditions in this paper we propose a new method for texture segmentation based on the use of texture feature detectors derived from a decorrelation procedure of a modified version of a pseudo wigner distribution pwd the decorrelation procedure is accomplished by a cascade recursive least squared crls principal component pc neural network the goal is to obtain a more eacient analysis of images by combining the advantages of using a high resolution joint representation given by the pwd with an eaective adaptive principal component analysis pca through the use of feedforward neural networks 1999 elsevier science b v all rights reserved

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

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

implementation of an adaptive finite element approximation of the mumford shah functional we present and detail a method for the numerical solving of the mumford shah problem based on a finite element method and on adaptive meshes we start with the formulation introduced in 13 detail its numerical implementation and then propose a variant which is proved to converge to the mumford shah problem a few experiments are illustrated

relaxed minimax control the relaxation of the optimal control problem with cost functional which is the supremum in time of some function h t x z is determined the trajectory is convexified in the usual way but the cost functional is convexified in a nonobvious manner thus if the original value function is v t x inf zeta in z left h s xi s zeta s right l infty t t cr then the relaxed value function is hat v t x mathop inf limits mu in hat z left t t right left left h s hat xi s z right l infty rm z mu s right l infty left t t right cr where the inner norm is the essential sup of h over z in rm z cr with respect to the measure mu s cr we prove that v and hat v coincide

distributional chaos for linear operators abstract we characterize distributional chaos for linear operators on frechet spaces in terms of a computable condition dcc and also as the existence of distributionally irregular vectors a sufficient condition for the existence of dense uniformly distributionally irregular manifolds is presented which is very general and can be applied to many classes of operators distributional chaos is also analyzed in connection with frequent hypercyclicity and the particular cases of weighted shifts and composition operators are given as an illustration of the previous results

optimization of dispersive coefficients in the homogenization of the wave equation in periodic structures we study dispersive effects of wave propagation in periodic media which can be modelled by adding a fourth order term in the homogenized equation the corresponding fourth order dispersive tensor is called burnett tensor and we numerically optimize its values in order to minimize or maximize dispersion more precisely we consider the case of a two phase composite medium with an eightfold symmetry assumption of the periodicity cell in two space dimensions we obtain upper and lower bound for the dispersive properties along with optimal microgeometries

a multi material topology optimization algorithm based on the topological derivative abstract we present a level set based topology optimization algorithm for design optimization problems involving an arbitrary number of different materials where the evolution of a design is solely guided by topological derivatives our method can be seen as an extension of the algorithm that was introduced in amstutz and andra 2006 for two materials to the case of an arbitrary number m of materials we represent a design that consists of multiple materials by means of a vector valued level set function which maps into r m 1 we divide the space r m 1 into m sectors each corresponding to one material and establish conditions for local optimality of a design based on certain generalized topological derivatives the optimization algorithm consists in a fixed point iteration striving to reach this optimality condition like the two material version of the algorithm also our method possesses a nucleation mechanism such that it is not necessary to start with a perforated initial design we show numerical results obtained by applying the algorithm to an academic example as well as to the compliance minimization in linearized elasticity

multi material topology optimization considering interface behavior via xfem and level set method abstract in most of the existing topology optimization studies of multi material structures the interface of different materials was assumed to be perfectly bonded optimal design based on the perfect interface assumption may introduce the risk of failure caused by interface debonding this paper presents an efficient multi material topology optimization strategy for seeking the optimal layout of structures considering the cohesive constitutive relationship of the interface based on the color level set method to describe the topology and the interface the interface behavior is simulated by combining the extended finite element method xfem and the cohesive model on fixed mesh this enables modeling of possible separation of material interfaces and thus provides a more realistic model of multi material structures furthermore this interface modeling technique avoids the difficulty of re meshing when tracking the moving cohesive interface positions during the optimization process in the topology optimization model the normal velocities defined on the level set points are considered as design variables in conjunction with the adjoint variable sensitivity analysis these design variables are updated by using the mathematical programming approach and then used to interpolate the boundary velocities these boundary velocities are extrapolated to the whole domain with the fast marching method and used to advance the structural boundary through the hamilton jacobi equation this topology optimization technique can handle multiple constraints easily in the framework of level set method and at the same time preserve the signed distance property of the level set functions two numerical examples are given to demonstrate the effectiveness of the present method it is also revealed that the optimal design considering interface behavior may exhibit tension compression non symmetric topology in which material interfaces mainly undergo compression

explicit level set based topology optimization using an exact heaviside function and consistent sensitivity analysis summary this paper presents a level set based topology optimization method based on numerically consistent sensitivity analysis the proposed method uses a direct steepest descent update of the design variables in a level set method the level set nodal values an exact heaviside formulation is used to relate the level set function to element densities the level set function is not required to be a signed distance function and reinitialization is not necessary using this approach level set based topology optimization problems can be solved consistently and multiple constraints treated simultaneously the proposed method leads to more insight in the nature of level set based topology optimization problems the level set based design parametrization can describe gray areas and numerical hinges consistency causes results to contain these numerical artifacts we demonstrate that alternative parameterizations level set based or density based regularization can be used to avoid artifacts in the final results the effectiveness of the proposed method is demonstrated using several benchmark problems the capability to treat multiple constraints shows the potential of the method furthermore due to the consistency the optimizer can run into local minima a fundamental difficulty of level set based topology optimization more advanced optimization strategies and more efficient optimizers may increase the performance in the future copyright 2012 john wiley sons ltd

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

a multi resolution method for 3d multi material topology optimization abstract this paper presents a multi resolution implementation in 3d for multi material topology optimization problem an alternating active phase algorithm where the problem at hand is divided into a series of the traditional material void phase topology optimization is employed for the multi material problem different levels of discretization are used for the displacement mesh design variable mesh and density mesh which provides higher resolution designs for the solutions a projection scheme is employed to compute the element densities from design variables and control the length scale of the material density simple block coordinate descent method similar to the gauss seidel technique is used to solve the subproblems several 3d numerical examples are presented to demonstrate the ease and the effectiveness of the proposed implementation incorporating the multi resolution method into the multi material approach robust designs with improved resolution can be achieved for real life problems with complex geometries

shape optimization and its extension to topological design based on isogeometric analysis in most of structural optimization approaches finite element method fem has been employed for structural response analysis and sensitivity calculation however the approaches generally suffer certain drawbacks in shape optimization cumbersome parameterization of design domain is required and time consuming remeshing task is also necessary in topology optimization design results are generally restricted on the initial design space and additional post processing is required for communication with cad systems these drawbacks are due to the use of different mathematical languages in design or geometric modeling and numerical analysis spline basis functions are used in design and geometric modeling whereas lagrangian and hermitian polynomials in analysis isogeometric analysis is a very attractive and promising alternative to overcome the limitations resulting from the use of the conventional fem in structural optimization in isogeometric analysis the same spline information such as control points and spline basis functions which represent geometries in cad systems are also used in numerical analysis such unification of the mathematical languages in cad analysis and design optimization can resolve the issues mentioned above in this work structural shape optimization using isogeometric analysis is studied on 2d and shell problems the proposed framework is extended to topology optimization using trimming techniques new inner fronts are introduced by trimming spline curves in topology optimization trimmed surface analysis which was recently proposed to analyze arbitrary complex topology problems is employed for topology optimization some benchmarking problems in shape and topology optimization are treated using the proposed approach

hypercyclic weighted shifts an operator t acting on a hilbert space is hypercyclic if for some vector x in the space the orbit tnx n 0 is dense in this paper we characterize hypercyclic weighted shifts in terms of their weight sequences and identify the direct sums of hypercyclic weighted shifts which are also hypercyclic as a consequence we show within the class of weighted shifts that multi hypercyclic shifts and direct sums of fixed hypercyclic shifts are both hypercyclic for general hypercyclic operators the corresponding questions were posed by d a herrero and they still remain open using a different technique we prove that i t is hypercycic whenever t is a unilateral backward weighted shift thus answering in more generality a question recently posed by k c chan and j h shapiro

design of multi layer materials using inverse homogenization and a level set method abstract this work is concerned with the micro architecture of multi layer material that globally exhibits desired mechanical properties for instance a negative apparent poisson s ratio we use inverse homogenization the level set method and the shape derivative in the sense of hadamard to identify material regions and track boundary changes within the context of the smoothed interface the level set method and the shape derivative obtained in the smoothed interface context allows to capture within the unit cell the optimal micro geometry we test the algorithm by computing several multi layer auxetic micro structures the multi layer approach has the added benefit that contact during movement of adjacent branches of the micro structure can be avoided in order to increase its capacity to withstand larger stresses

li yorke and distributionally chaotic operators abstract we study li yorke chaos and distributional chaos for operators on banach spaces more precisely we characterize li yorke chaos in terms of the existence of irregular vectors sufficient computable criteria for distributional and li yorke chaos are given together with the existence of dense scrambled sets under some additional conditions we also obtain certain spectral properties finally we show that every infinite dimensional separable banach space admits a distributionally chaotic operator which is also hypercyclic

a level set based parameterization method for structural shape and topology optimization this paper presents an effective parametric approach by extending the conventional level set method to structural shape and topology optimization using the compactly supported radial basis functions rbfs and the optimality criteria oc method the structural design boundary is first represented implicitly by embedding into a higher dimensional level set function as its zero level set and the rbfs of a favorable smoothness are then applied to interpolate the level set function the original initial value problem is thus converted to a parametric optimization with the expansion coefficients of the interplant posed as the design variables the oc method is then applied to advance the structure boundary in terms of the velocity field derived from the parametric optimization hence the structural shape and topology optimization is now transformed into a process of iteratively finding coefficients to update the level set function to achieve an optimal configuration the numerical considerations of the conventional level set method including upwind schemes velocity extension and reinitialization are eliminated the proposed scheme is capable of addressing structural shape fidelity and topology change simultaneously and of keeping the boundary smooth during the optimization process furthermore numerical convergence is expected to be improved a widely investigated example in the framework of structural stiffness designs is applied to demonstrate the efficiency and accuracy of the proposed approach copyright 2007 john wiley sons ltd

a mass constraint formulation for structural topology optimization with multiphase materials this work is focused on the topology optimization of lightweight structures consisting of multiphase materials instead of adopting the common idea of using volume constraint a new problem formulation with mass constraint is proposed meanwhile recursive multiphase materials interpolation rmmi and uniform multiphase materials interpolation ummi schemes are discussed and compared based on numerical tests and theoretical analysis it is indicated that the nonlinearity of the mass constraint introduced by rmmi brings numerical difficulties to attain the global optimum of the optimization problem on the contrary the ummi 2 scheme makes it possible to formulate the mass constraint in a linear form with separable design variables one such formulation favors very much the problem resolution by means of mathematical programming approaches especially the convex programming methods moreover numerical analysis indicates that fully uniform initial weighting is beneficial to seek the global optimum when ummi 2 scheme is used besides the relationship between the volume constraint and mass constraint is theoretically revealed the filtering technique is adapted to avoid the checkerboard pattern related to the problem with multiphase materials numerical examples show that the ummi 2 scheme with fully uniform initial weighting is reliable and efficient to deal with the structural topology optimization with multiphase materials and mass constraint meanwhile the mass constraint formulation is evidently more significant than the volume constraint formulation copyright 2011 john wiley sons ltd

a multiphase level set framework for image segmentation using the mumford and shah model we propose a new multiphase level set framework for image segmentation using the mumford and shah model for piecewise constant and piecewise smooth optimal approximations the proposed method is also a generalization of an active contour model without edges based 2 phase segmentation developed by the authors earlier in t chan and l vese 1999 in scale space 99 m nilsen et al eds lncs vol 1682 pp 141 151 and t chan and l vese 2001 ieee ip 10 2 266 277 the multiphase level set formulation is new and of interest on its own by construction it automatically avoids the problems of vacuum and overlaps it needs only log n level set functions for n phases in the piecewise constant cases it can represent boundaries with complex topologies including triple junctionss in the piecewise smooth case only two level set functions formally suffice to represent any partition based on the four color theorem finally we validate the proposed models by numerical results for signal and image denoising and segmentation implemented using the osher and sethian level set method

a tour of the theory of absolutely minimizing functions a detailed analysis of the class of absolutely minimizing functions in euclidean spaces and the relationship to the infinity laplace equation

curve evolution implementation of the mumford shah functional for image segmentation denoising interpolation and magnification we first address the problem of simultaneous image segmentation and smoothing by approaching the mumford shah 1989 paradigm from a curve evolution perspective in particular we let a set of deformable contours define the boundaries between regions in an image where we model the data via piecewise smooth functions and employ a gradient flow to evolve these contours each gradient step involves solving an optimal estimation problem for the data within each region connecting curve evolution and the mumford shah functional with the theory of boundary value stochastic processes the resulting active contour model offers a tractable implementation of the original mumford shah model i e without resorting to elliptic approximations which have traditionally been favored for greater ease in implementation to simultaneously segment and smoothly reconstruct the data within a given image in a coupled manner various implementations of this algorithm are introduced to increase its speed of convergence we also outline a hierarchical implementation of this algorithm to handle important image features such as triple points and other multiple junctions next by generalizing the data fidelity term of the original mumford shah functional to incorporate a spatially varying penalty we extend our method to problems in which data quality varies across the image and to images in which sets of pixel measurements are missing this more general model leads us to a novel pde based approach for simultaneous image magnification segmentation and smoothing thereby extending the traditional applications of the mumford shah functional which only considers simultaneous segmentation and smoothing

on lower subdifferentiable functions this paper studies the notion of lower subdifferentiability from the view point of generalized conjugation by this method the main properties of lower sub differentiable functions are analysed a related conjugation theory for holder and lipschitz functions is developed which is used to characterize those functions which can be expressed as a supremum of holder functions some applications to quasiconvex optimization and optimal control theory are examined

a multi material level set based topology and shape optimization method abstract this paper proposes a new multi material level set mm ls topology description model for topology and shape optimization of structures involving multiple materials each phase is represented by a combined formulation of different level set functions with a total number of m level set functions this approach provides a representation of m materials and one void phase totally m 1 phases the advantages of the proposed method include 1 it can guarantee that each point contains exactly one phase without overlaps between each two phases and redundant regions within the design domain 2 it possesses an explicit mathematical expression which greatly facilitates the design sensitivity analysis and 3 it retains the merits of the level set method including smooth boundary and distinct interface a parametric level set method is applied to evolve the topology and shape of multi material structures with a high computational efficiency several numerical examples are presented to demonstrate the effectiveness of the proposed method

filters in topology optimization in this article a modified filtered version of the minimum compliance topology optimization problem is studied the direct dependence of the material properties on its pointwise density is replaced by a regularization of the density field by the mean of a convolution operator in this setting it is possible to establish the existence of solutions moreover convergence of an approximation by means of finite elements can be obtained this is illustrated through some numerical experiments the filtering technique is also shown to cope with two important numerical problems in topology optimization checkerboards and mesh dependent designs copyright 2001 john wiley sons ltd

recurrence properties of hypercyclic operators we generalize the notions of hypercyclic operators mathfrak u frequently hypercyclic operators and frequently hypercyclic operators by introducing a new concept in linear dynamics namely mathcal a hypercyclicity we then state an mathcal a hypercyclicity criterion inspired by the hypercyclicity criterion and the frequent hypercyclicity criterion and we show that this criterion characterizes the mathcal a hypercyclicity for weighted shifts we also investigate which density properties can the sets n x u n in mathbb n t nx in u have for a given hypercyclic operator and we study the new notion of reiteratively hypercyclic operators

a level set based topology optimization method targeting metallic waveguide design problems in this paper we propose a level set based topology optimization method targeting metallic waveguide design problems where the skin effect must be taken into account since the metallic waveguides are generally used in the high frequency range where this effect critically affects performance one of the most reasonable approaches to represent the skin effect is to impose an electric field constraint condition on the surface of the metal to implement this approach we develop a boundary tracking scheme for the arbitrary lagrangian eulerian ale mesh pertaining to the zero iso contour of the level set function that is given in an eulerian mesh and impose dirichlet boundary conditions at the nodes on the zero iso contour in the ale mesh to compute the electric field since the ale mesh accurately tracks the zero iso contour at every optimization iteration the electric field is always appropriately computed during optimization for the sensitivity analysis we compute the nodal coordinate sensitivities in the ale mesh and smooth them by solving a helmholtz type partial differential equation the obtained smoothed sensitivities are used to compute the normal velocity in the level set equation that is solved using the eulerian mesh and the level set function is updated based on the computed normal velocity finally the utility of the proposed method is discussed through several numerical examples copyright 2011 john wiley sons ltd

a new multi material level set topology optimization method with the length scale control capability abstract multi material level set topology optimization methods conventionally rely on the overlap of multiple level set functions to realize the multi material structural representation however this representation may produce redundant material phases and the signed distance information is no longer available within the individual material regions to fix these issues a new multi material level set topology optimization method is proposed in this paper where m level set functions represent m material phases plus the void and the signed distance information is straightforwardly available in each material phase to be specific each level set function corresponds to a material phase and the overlapping areas are filled with an artificial material type the property of which is weaker than any of the involved material types this artificial material plays the role of penalizing the overlap areas to vanish which is different from the close to zero property material type for the void hence the optimization process starts with any multi level set interpolated input and the overlapping areas will gradually vanish based on this new method we have successfully realized the component length scale control on multi material structures and innovatively an approach has been proposed to realize the uniform component thickness control without need of pre specifying the thickness target

a structural optimization method based on the level set method using a new geometry based re initialization scheme structural optimization methods based on the level set method are a new type of structural optimization method where the outlines of target structures can be implicitly represented using the level set function and updated by solving the so called hamilton jacobi equation based on a eulerian coordinate system these new methods can allow topological alterations such as the number of holes during the optimization process whereas the boundaries of the target structure are clearly defined however the re initialization scheme used when updating the level set function is a critical problem when seeking to obtain appropriately updated outlines of target structures in this paper we propose a new structural optimization method based on the level set method using a new geometry based re initialization scheme where both the numerical analysis used when solving the equilibrium equations and the updating process of the level set function are performed using the finite element method the stiffness maximization eigenfrequency maximization and eigenfrequency matching problems are considered as optimization problems several design examples are presented to confirm the usefulness of the proposed method copyright 2010 john wiley sons ltd

existence of minimizers for non quasiconvex integrals keywords quasiconvexity existence of minimizers reference caa article 1995 002doi 10 1007 bf00380915 record created on 2008 11 25 modified on 2017 05 12

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

convexity criteria and uniqueness of absolutely minimizing functions we show that an absolutely minimizing function with respect to a convex hamiltonian h mathbb r n rightarrow mathbb r is uniquely determined by its boundary values under minimal assumptions on h along the way we extend the known equivalences between comparison with cones convexity criteria and absolutely minimizing properties to this generality these results perfect a long development in the uniqueness existence theory of the archetypal problem of the calculus of variations in l

invariant measures for frequently hypercyclic operators abstract we investigate frequently hypercyclic and chaotic linear operators from a measure theoretic point of view among other things we show that any frequently hypercyclic operator t acting on a reflexive banach space admits an invariant probability measure with full support which may be required to vanish on the set of all periodic vectors for t that there exist frequently hypercyclic operators on the sequence space c 0 admitting no ergodic measure with full support and that if an operator admits an ergodic measure with full support then it has a comeager set of distributionally irregular vectors we also give some necessary and sufficient conditions which are satisfied by all the known chaotic operators for an operator t to admit an invariant measure supported on the set of its hypercyclic vectors and belonging to the closed convex hull of its periodic measures finally we give a baire category proof of the fact that any operator with a perfectly spanning set of unimodular eigenvectors admits an ergodic measure with full support

the euler equation and absolute minimizers of l functionals the aronsson euler equation for the functional on w g 1 m i e w 1 with boundary data g is this equation has been derived for smooth absolute minimizers i e a function which minimizes f on every subdomain we prove in this paper that for m 1 n 1 or n 1 m 1 an absolute minimizer of f exists in w g 1 m and for m 1 n 1 any absolute minimizer of f must be a viscosity solution of the aronsson euler equation

irregular vectors of hilbert space operators abstract a vector x in a hilbert space h is called irregular for an operator t h h provided that sup n t n x and inf n t n x 0 we establish some basic properties of operators having irregular vectors and present examples that highlight the relationship or lack thereof between irregularity and hypercyclicity

topology design of structures using a dual algorithm and a constraint on the perimeter dual optimization algorithms are well suited for the topology design of continuum structures in discrete variables since in these problems the number of constraints is small in comparison to the number of design variables the raw dual algorithm which was originally proposed for the minimum compliance design problem worked well when a perimeter constraint was added in addition to the volume constraint however if the perimeter constraint was gradually relaxed by increasing the upper bound on the allowable perimeter the algorithm tended to behave erratically recently a simple strategy has been suggested which modifies the raw dual algorithm to make it more robust in the absence of the perimeter constraint in particular the problem of checkerboarding which is frequently observed with the use of lower order finite elements is eliminated in this work we show how the perimeter constraint can be incorporated in this improved algorithm so that it not only provides a designer with a control over the topology but also generates good topologies irrespective of the value of the upper bound on the perimeter

an efficient interface capturing method for a large collection of interacting bodies immersed in a fluid an efficient method to capture an arbitrary number of fluid structure interfaces in a level set framework is built following ideas introduced for contour capturing in image analysis using only three label maps and two distance functions we succeed in locating and evolving the bodies independently in the whole domain and get the distance between the closest bodies in order to apply a collision force whatever the number of cells is the method is applied to rigid solid bodies in order to compare to the results available in the literature in that case a global penalization model uses the label maps to follow the solid bodies all together without a separate computation of each body velocity numerical simulations are performed in two and three dimensions an application to immersed vesicles is also proposed and shows the capability and efficiency of the method to handle numerical contacts between elastic bodies at low resolution two dimensional simulations of vesicles under various flow conditions are presented

design of piezoelectric actuators using a multiphase level set method of piecewise constants this paper presents a multiphase level set method of piecewise constants for shape and topology optimization of multi material piezoelectric actuators with in plane motion first an indicator function which takes level sets of piecewise constants is used to implicitly represent structural boundaries of the multiple phases in the design domain compared with standard level set methods using n scalar functions to represent 2 n phases each constant value in the present method denotes one material phase and 2 n phases can be represented by 2 n pre defined constants thus only one indicator function including different constant values is required to identify all structural boundaries between different material phases by making use of its discontinuities in the context of designing smart actuators with in plane motions the optimization problem is defined mathematically as the minimization of a smooth energy functional under some specified constraints thus the design optimization of the smart actuator is transferred into a numerical process by which the constant values of the indicator function are updated via a semi implicit scheme with additive operator splitting aos algorithm in such a way the different material phases are distributed simultaneously in the design domain until both the passive compliant host structure and embedded piezoelectric actuators are optimized the compliant structure serves as a mechanical amplifier to enlarge the small strain stroke generated by piezoelectric actuators the major advantage of the present method is to remove numerical difficulties associated with the solution of the hamilton jacobi equations in most conventional level set methods such as the cfl condition the regularization procedure to retain a signed distance level set function and the non differentiability related to the heaviside and the delta functions two widely studied examples are chosen to demonstrate the effectiveness of the present method

a level set method for topology optimization of continuum structures with bounded design domains in practice a continuum structure is usually designed to carry the traction applied to the boundary of the structure subject to prescribed displacements imposed on its boundary the design domains of practical structures are often limited and significantly affect the final optimal design of the structures structural boundaries under traction and prescribed displacements should be treated as a zero level set in the level set methods firstly to overcome the limitations of current level set methods and the stopping issue of structural boundary movements this paper presents a set of new level set based optimization formulae for the optimal design of continuum structures with bounded design domains a set of new normal velocities required by level set movements is given a level set based optimization algorithm is developed and implemented with several robust and efficient numerical techniques which include the level set regularization algorithm gradient projection algorithm nonlinear velocity mapping algorithm and return mapping algorithm secondly in order to overcome the difficulty in nucleating holes in the design domain in the level set based optimization method this paper introduces a new optimization strategy with a small possibility random topology mutations and crossovers a mixed topology optimization algorithm is implemented and presented for the compliance minimization problems of continuum structures with material volume constraints the validity and effectiveness of the proposed method are demonstrated with two examples

existence of minimizers for nonlevel convex supremal functionals the paper is devoted to determining necessary and sufficient conditions for existence of solutions to the problem inf mathop rm ess sup x in omega f nabla u x u in u 0 w 1 infty 0 omega when the supremand f is not necessarily level convex these conditions are obtained through a comparison with the related level convex problem and are written in terms of a differential inclusion involving the boundary datum several conditions of convexity for the supremand f are also investigated

a level set method for shape and topology optimization of both structure and support of continuum structures abstract we present a level set method for the shape and topology optimization of both structure and support two level set functions are used to implicitly represent a structure the traction free boundary and the dirichlet boundary are represented separately and are allowed to be continuously changed during the optimization the optimization problem of minimum compliance is considered the shape derivatives of both boundaries are derived by using a lagrangian function and the adjoint method the finite element analysis is done by modifying a fixed background mesh and we do not use the artificial weak material numerical examples in two dimensions are investigated

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

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

optimal design of multiphase composites under elastodynamic loading abstract an algorithm is proposed to optimize the performance of multiphase structures composites under elastodynamic loading conditions the goal is to determine the distribution of material in the structure such that the time averaged total stored energy of structure is minimized a penalization strategy is suggested to avoid the checkerboard instability simultaneously to generate near 0 1 topologies as a result of this strategy the solutions of presented algorithm are sufficiently smooth and possess the regularity of h 1 function space a simple method for the continuum adjoint sensitivity analysis of the corresponding pde constrained optimization problem is presented it is general and can be easily applied to a wide range of alternative problems the success of the introduced algorithm is studied by numerical experiments on two dimensional model problems for different numbers of phases according to numerical results the objective functional is reduced monotonically with iterations moreover the final topologies at the optimal solutions are near 0 1 the dynamic behavior of optimal designs is compared to that of initial ones to show the impact of optimization on the performance of structures

level set based structural optimization of distributed piezoelectric modal sensors for plate structures abstract this paper presents structural optimization of piezoelectric modal sensors and actuators for plate structures unlike for beam structures the design of constant thickness distributed modal transducers for plate structures is not straightforward in this work the design problem for plate structures is tackled as a multi material structural optimization problem where the optimal distribution of the piezoelectric material as well as its polarization profile have to be found for the solution a parametric level set method is proposed this allows simultaneous shape and topology optimization in contrast to most conventional level set approaches we choose a parametric level set method that does not involve the direct solution of the hamilton jacobi equation using numerical methods such as the finite difference method the optimization problem is directly posed in terms of the parameters of the parameterization and is solved with an evolutionary algorithm

expansivity and shadowing in linear dynamics abstract in the early 1970 s eisenberg and hedlund investigated relationships between expansivity and spectrum of operators on banach spaces in this paper we establish relationships between notions of expansivity and hypercyclicity supercyclicity li yorke chaos and shadowing in the case that the banach space is c 0 or l p 1 p we give complete characterizations of weighted shifts which satisfy various notions of expansivity we also establish new relationships between notions of expansivity and spectrum moreover we study various notions of shadowing for operators on banach spaces in particular we solve a basic problem in linear dynamics by proving the existence of nonhyperbolic invertible operators with the shadowing property this contrasts with the expected results for nonlinear dynamics on compact manifolds illuminating the richness of dynamics of infinite dimensional linear operators

level set methods for optimization problems involving geometry and constraints i frequencies of a two density inhomogeneous drum many problems in engineering design involve optimizing the geometry to maximize a certain design objective geometrical constraints are often imposed in this paper we use the level set method devised in osher and sethian j comput phys 79 12 1988 the variational level set calculus presented in zhao et al j comput phys 127 179 1996 and the projected gradient method as in rudin et al physica d 60 259 1992 to construct a simple numerical approach for problems of this type we apply this technique to a model problem involving a vibrating system whose resonant frequency or whose spectral gap is to be optimized subject to constraints on geometry our numerical results are quite promising we expect to use this approach to deal with a wide class of optimal design problems in the future

maximal stiffness design of two material structures by topology optimization with nonprobabilistic reliability basedon the multi ellipsoid convex modelandthe quantified measure of the nonprobabilistic reliability topology optimization of two material structures in the presence of parameter uncertainties is investigated the task of the optimal design problem is to distribute a given amount of two candidate materials into the design domain for acquiring the maximal stiffness while satisfying the reliability requirement the extended power law interpolation scheme for material properties is employed for relaxing the two material topological design problem into a continuous valued optimization problem in addition through transforming the minimax type optimization problemintoaseriesofdeterministiconesbyusingasequentialapproximateprogrammingstrategy thispaperaims tomaketheoptimizationdesignnumericallytractable theresultingmathematicalprogrammingproblemsarethen efficientlysolvedbytheassociationofthemethodofmovingasymptoteswithaheuristiciterativemanner numerical investigations reveal that system uncertainties may have considerable effects on the optimal material layout of twomaterial structures the proposed topology optimization methodology could yield a more reasonable two material structuraldesignthan theconventional deterministic counterpart whenthe samereliabilityrequirements need tobe achieved

a level set based topology optimization method for simultaneous design of elastic structure and coupled acoustic cavity using a two phase material model abstract this papers proposes a level set based topology optimization method for the simultaneous design of acoustic and structural material distributions in this study we develop a two phase material model that is a mixture of an elastic material and acoustic medium to represent an elastic structure and an acoustic cavity by controlling a volume fraction parameter in the proposed model boundary conditions at the two phase material boundaries are satisfied naturally avoiding the need to express these boundaries explicitly we formulate a topology optimization problem to minimize the sound pressure level using this two phase material model and a level set based method that obtains topologies free from grayscales the topological derivative of the objective functional is approximately derived using a variational approach and the adjoint variable method and is utilized to update the level set function via a time evolutionary reaction diffusion equation several numerical examples present optimal acoustic and structural topologies that minimize the sound pressure generated from a vibrating elastic structure

stress related topology optimization of continuum structures involving multi phase materials considering stress constraints in structural topology optimization is very important from both theoretical and application perspectives most of the corresponding studies in literature however are only focused on the problems involving single phase homogenous materials in the present paper stress constrained topology optimization of continuum structures involving multi phase heterogeneous materials is investigated a level set based variational consistent solution framework is developed numerical examples are also presented and discussed to illustrate the effectiveness of the proposed approach

uniqueness of lipschitz extensions minimizing the sup norm of the gradient in this paper we examine the problem of minimizing the sup norm of the gradient of a function with prescribed boundary values geometrically this can be interpreted as finding a minimal lipschitz extension due to the weak convexity of the functional associated to this problem solutions are generally nonunique by adopting g aronsson s notion of absolutely minimizing we are able to prove uniqueness by characterizing minimizers as the unique solutions of an associated partial differential equation in fact we actually prove a weak maximum principle for this partial differential equation which in some sense is the euler equation for the minimization problem this is significantly difficult because the partial differential equation is both fully nonlinear and has very degenerate ellipticity to overcome this difficulty we use the weak solutions of m g crandall and p l lions also known as viscosity solutions in conjunction with some arguments using integration by parts

operators with the specification property abstract we study a version of the specification property for linear dynamics operators having the specification property are investigated and relationships with other well known dynamical notions such as mixing devaney chaos and frequent hypercyclicity are obtained

hopf lax type formula forut h u du 0 abstract an explicit formula is derived for the hamilton jacobi equation u t h u du 0 on 0 r n with u 0 x g x the hamiltonian h p is assumed to be nondecreasing in r 1 and convex and positively homogeneous of degree 1 in p r n the unique solution is given by formula where h is a quasiconvex function given as the quasiconvex dual of h that is h q inf r 1 h p p q p

a level set method for topology optimization of heat conduction problem under multiple load cases in this paper we present a numerical approach of topology optimization under multiple load cases for heat conduction problem this framework is based on the theories of topological derivative and shape derivative for elliptic system we employ level set model to implicitly represent geometric boundary of thermal conductive material introducing topological derivative will generate new topology in the design domain which suppresses the dependence of initial topology guess to some extent the shape optimization is obtained by combining shape derivative with level set method the functional of quadratic temperature gradient is taken as the objective function in our analysis which is subjected to the state equation of steady heat conduction and volume constrain the shape of material domain is treated as the design variable and the final result is achieved by updating level set function gradually we develop an effective numerical technique to implement the optimal design with multiple load cases for heat conduction problem numerical examples demonstrate that our proposed approach is effective and robust for topology optimization of heat conduction problem

topology optimization with multiple materials via moving morphable component mmc method summary with the fast development of additive manufacturing am technology topology optimization involving multiple materials has received ever increasing attention traditionally this kind of optimization problem is solved within the implicit solution framework using the simp solid isotropic material with penalization or level set method this treatment however will inevitably lead to a large number of design variables especially when many types of materials are involved and three dimensional 3d problems are considered this is due to the fact that for each type of material a corresponding density field level function defined on the entire design domain must be introduced to describe its distribution in the present paper a novel approach for topology optimization with multiple materials is established based on the moving morphable component mmc framework with use of this approach topology optimization problems with multiple materials can be solved with much less numbers of design variables and degrees of freedom numerical examples provided demonstrate the effectiveness of the proposed approach

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

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 polytree based adaptive polygonal finite element method for multi material topology optimization abstract this study presents a polytree based adaptive methodology for multi material topology optimization mmtop polytree data structure is introduced as a general recursive multi level mesh that is automatically refined in processing based on error analysis in order to resolve hanging nodes in element edges the wachspress coordinate is employed on a reference element before using a mapping scheme to obtain shape functions and their derivatives for any polygons a new definition of filter radius is also proposed to improve the efficiency of filters and optimized results the combination of polytree meshes and adaptive filters not only clarifies the interfaces between material phases including void phase but also decreases the computing time of the overall process in comparison to using the regular fine meshes several benchmark and practical problems are considered to show distinct features of the proposed method

capturing the behavior of bubbles and drops using the variational level set approach we reproduce the general behavior of complicated bubble and droplet motions using the variational level set formulation introduced by the authors earlier our approach here ignores inertial effects thus the motion is only correct as an approximation for very viscous problems however the steady states are true equilibrium solutions inertial forces will be added in future work the problems include soap bubbles colliding and merging drops falling or remaining attached to a generally irregular ceiling and liquid penetrating through a funnel in both two and three dimensions each phase is identified with a particular level set function the zero level set of this function is that particular phase boundary the level set functions all evolve in time through a constrained gradient descent procedure so as to minimize an energy functional the functions are coupled through physical constraints and through the requirements that different phases do not overlap and vacuum regions do not develop both boundary conditions and inequality constraints are cast in terms of either local or global equality constraints the gradient projection method leads to a system of perturbed by curvature if surface tension is involved hamilton jacobi equations coupled through a constraint the coupling is enforced using the lagrange multiplier associated with this constraint the numerical implementation requires much of the modern level set technology in particular we achieve a significant speed up by using the fast localization algorithm of h k zhao m kang b merriman d peng and s osher

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

piecewise constant level set method for structural topology optimization in this paper a piecewise constant level set pcls method is implemented to solve a structural shape and topology optimization problem in the classical level set method the geometrical boundary of the structure under optimization is represented by the zero level set of a continuous level set function e g the signed distance function instead in the pcls approach the boundary is described by discontinuities of pcls functions the pcls method is related to the phase field methods and the topology optimization problem is defined as a minimization problem with piecewise constant constraints without the need of solving the hamilton jacobi equation the result is not moving the boundaries during the iterative procedure thus it offers some advantages in treating geometries eliminating the reinitialization and naturally nucleating holes when needed in the paper the pcls method is implemented with the additive operator splitting numerical scheme and several numerical and procedural issues of the implementation are discussed examples of 2d structural topology optimization problem of minimum compliance design are presented illustrating the effectiveness of the proposed method

multimaterial topology optimization by volume constrained allen cahn system and regularized projected steepest descent method abstract a new computational algorithm is introduced in the present study to solve multimaterial topology optimization problems it is based on the penalization of the objective functional by the multiphase volume constrained ginzburg landau energy functional the update procedure is based on the gradient flow of the objective functional by a fractional step projected steepest descent method in the first step the new design is found based on the projected steepest descent method to ensure the reduction in the objective functional simultaneously satisfying the control constraints in the second step regularization step an h 1 regularity of the solution is ensured while keeping the feasibility of solution with respect to the set of control constraints the presented algorithm could be accounted as a constrained h 1 optimization algorithm which according to our knowledge has not been reported to solve such kind of problems yet the success and efficiency of the presented method are shown through several test problems numerical results show that the presented algorithm ends with a near 0 1 topology and its computational cost scales sub linearly by the number of phases for the sake of reader convenience and the ease of further extension the matlab implementation of the presented algorithm is included in the appendix

topology optimization on two dimensional manifolds abstract this paper presents topology optimization on general two dimensional manifolds for phenomena described by second order partial differential equations where the material interpolation is implemented by using the material distribution method when a physical field is defined on a two dimensional manifold the material interpolation is implemented on a material parameter in the partial differential equation used to describe the distribution of the physical field when the physical field is defined on a three dimensional domain with its boundary conditions defined on a two dimensional manifold corresponding a surface or an interface of this three dimensional domain the material density is used to formulate a mixed boundary condition of the partial differential equation for the physical field and implement the penalization between two different boundary types based on the homeomorphic property of two dimensional manifolds typical two dimensional manifolds e g sphere torus mobius strip and klein bottle are included in the numerical tests which are used to demonstrate this topology optimization approach for the design problems of fluidic mechanics heat transfer and electromagnetics

a topology optimization method based on the level set method incorporating a fictitious interface energy this paper proposes a new topology optimization method which can adjust the geometrical complexity of optimal configurations using the level set method and incorporating a fictitious interface energy derived from the phase field method first a topology optimization problem is formulated based on the level set method and the method of regularizing the optimization problem by introducing fictitious interface energy is explained next the reaction diffusion equation that updates the level set function is derived and an optimization algorithm is then constructed which uses the finite element method to solve the equilibrium equations and the reaction diffusion equation when updating the level set function finally several optimum design examples are shown to confirm the validity and utility of the proposed topology optimization method

optimization of structures with thin layer functional device on its surface through a level set based multiple type boundary method abstract functional structure i e native base structure equipped with thin layer functional devices on its surface is important in many engineering applications the performance of such a functional structure is seriously affected by the configuration of the base structure and the surface functional devices and the optimization of configuration is a simultaneous topology optimization of the boundaries that belong to different types in the present study load cell is taken as an example of the functional structure and strain gauge is taken as an example of the functional device the task of designing a load cell is accomplished by solving a multi objective topology optimization problem whose objective makes a compromise between stiffness and flexibility of the base structure the design variables include two types of boundaries functional boundary where a strain gauge is installed the free boundary where there does not exist any connection to the environment the optimization problem is solved by using a level set based multiple type boundary method two level set functions are used to represent a functional structure and the two types of boundaries evolution of the two boundaries is modeled by two independent hamilton jacobi equations re meshing is conducted in each iteration of optimization quadrilateral body fitted mesh is used for the finite element analysis and the artificial weak material is not used numerical examples in two dimensions are investigated

the method of moving asymptotes a new method for structural optimization a new method for non linear programming in general and structural optimization in particular is presented in each step of the iterative process a strictly convex approximating subproblem is generated and solved the generation of these subproblems is controlled by so called moving asymptotes which may both stabilize and speed up the convergence of the general process

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

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

voigt reuss topology optimization for structures with linear elastic material behaviours the desired results of variable topology material layout computations are stable and discrete material distributions that optimize the performance of structural systems to achieve such material layout designs a continuous topology design framework based on hybrid combinations of classical reuss compliant and voigt sti mixing rules is investigated to avoid checkerboarding instabilities the continuous topology optimization formulation is coupled with a novel spatial ltering procedure the issue of obtaining globally optimal discrete layout designs with the proposed formulation is investigated using a continuation method which gradually transitions from the sti voigt formulation to the compliant reuss formulation the very good performance of the proposed methods is demonstrated on four structural topology design optimization problems from the literature 1997 by john wiley sons ltd

composite material design of two dimensional structures using the homogenization design method summary composite materials of two dimensional structures are designed using the homogenization design method the composite material is made of two or three dierent material phases designing the composite material consists of nding a distribution of material phases that minimizes the mean compliance of the macrostructure subject to volume fraction constraints of the constituent phases within a unit cell of periodic microstructures at the start of the computational solution the material distribution of the microstructure is represented as a pure mixture of the constituent phases as the iteration procedure unfolds the component phases separate themselves out to form distinctive interfaces the eective material properties of the articially mixed materials are dened by the interpolation of the constituents the optimization problem is solved using the sequential linear programming method both the macrostructure and the microstructures are analysed using the nite element method in each iteration step several examples of optimal topology design of composite material are presented to demonstrate the validity of the present numerical algorithm copyright 2001 john wiley sons ltd