a level set method for shape and topology optimization of large displacement compliant mechanisms a parameterization level set method is presented for structural shape and topology optimization of compliant mechanisms involving large displacements a level set model is established mathematically as the hamilton jacobi equation to capture the motion of the free boundary of a continuum structure the structural design boundary is thus described implicitly as the zero level set of a level set scalar function of higher dimension the radial basis function with compact support is then applied to interpolate the level set function leading to a relaxation and separation of the temporal and spatial discretizations related to the original partial differential equation in doing so the more difficult shape and topology optimization problem is now fully parameterized into a relatively easier size optimization of generalized expansion coefficients as a result the optimization is changed into a numerical process of implementing a series of motions of the implicit level set function via an existing efficient convex programming method with the concept of the shape derivative the geometrical non linearity is included in the rigorous design sensitivity analysis to appropriately capture the large displacements of compliant mechanisms several numerical benchmark examples illustrate the effectiveness of the present level set method in particular its capability of generating new holes inside the material domain the proposed method not only retains the favorable features of the implicit free boundary representation but also overcomes several unfavorable numerical considerations relevant to the explicit scheme the reinitialization procedure and the velocity extension algorithm in the conventional level set method

fast sweeping algorithms for a class of hamilton jacobi equations we derive a godunov type numerical flux for the class of strictly convex homogeneous hamiltonians that includes h p q sqrt ap 2 bq 2 2cpq c 2 ab we combine our godunov numerical fluxes with simple gauss seidel type iterations for solving the corresponding hamilton jacobi hj equations the resulting algorithm is fast since it does not require a sorting strategy as found e g in the fast marching method in addition it providesa way to compute solutions to a class of hj equations for which the conventional fast marching method is not applicable our experiments indicate convergence after a few iterations even in rather difficult cases

a geometry projection method for shape optimization we present a new method for shape optimization that uses an analytical description of the varying design geometry as the control in the optimization problem a straightforward filtering technique projects the design geometry onto a fictitious analysis domain to support simplified response and sensitivity analysis however the analytical geometry model is referenced directly for all purely geometric calculations the method thus combines the advantages of direct geometry representations with the simplified analysis procedures that are possible with fictitious domain analysis methods such as the material distribution methods commonly used in topology optimization the projected geometry measure converges to the indicator function of the analytical geometry model in the limit of numerical mesh refinement consequently optimal designs obtained with the new method converge to solutions of well defined continuum optimization problems in the limit of mesh refinement this property is confirmed in example computations for minimum compliance design of an elastic structure subject to a volume constraint and for minimum volume design subject to a maximum stress constraint copyright 2004 john wiley sons ltd

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

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

active contours without edges we propose a new model for active contours to detect objects in a given image based on techniques of curve evolution mumford shah 1989 functional for segmentation and level sets our model can detect objects whose boundaries are not necessarily defined by the gradient we minimize an energy which can be seen as a particular case of the minimal partition problem in the level set formulation the problem becomes a mean curvature flow like evolving the active contour which will stop on the desired boundary however the stopping term does not depend on the gradient of the image as in the classical active contour models but is instead related to a particular segmentation of the image we give a numerical algorithm using finite differences finally we present various experimental results and in particular some examples for which the classical snakes methods based on the gradient are not applicable also the initial curve can be anywhere in the image and interior contours are automatically detected

the sharp hardy uncertainty principle for schrodinger evolutions we give a new proof of hardy s uncertainty principle up to the end point case which is only based on calculus the method allows us to ex tend hardy s uncertainty principle to schrodinger equations with non constant coefficients we also deduce optimal gaussian decay bounds for solutions to these schrodinger equations

topology optimization using hyper radial basis function network in this paper we present the application of hyper radial basis function network hrbfn as a topology description function tdf hrbfn is used to parameterize the material distribution density for topology optimization thus the topology optimization problem is to determine the parameters governing hrbfn in order to satisfy certain design criteria here we present two kinds of problems minimum compliance and maximum fundamental natural frequency design hrbfn can be used to parameterize material density during the finite element analysis stage or after analysis i e to post process the results obtained by traditional topology optimization both the solutions are presented in this paper an efficient optimization algorithm to obtain the hrbfn parameters was developed that makes use of optimality criteria examples are presented to demonstrate the proposed approach as also a comparison with the traditional topology optimization the advantage of proposed approach is that it can yield checkerboard free manufacturable topologies using coarse mesh fe analysis models as opposed to the traditional approach that requires fine meshes this results in reduction in solution time for a response using fe analysis while conserving the ability to yield smooth topology we feel the proposed approach has a great deal of application in topology optimization type of problems i introduction opology optimization 1 aims at obtaining the best possible arrangement of the given volume of structural material within a spatial domain to obtain an optimal mechanical performance of the concept design the topology optimization is a material distribution problem in which the material content of each of the points of reference domain must be determined the traditional approach to solve this problem is to mesh the reference domain and obtain the material content of each element to determine the optimal material distribution if the material content of an element is above the cut off level corresponding to the percentage of total material material density of 1 is assigned to that element whereas if the material content of an element is below the cut off level void or material density of 0 is assigned to that element also various methods have been suggested by previous researchers to model the intermediate densities eg level set method 2 to improve the performance of this traditional approach all these methods compute the optimal density of each element and hence there must be as many design variables as the number of finite elements the number of elements must be sufficient to obtain a correct representation of geometrical features as well as obtain a correct physical response the idea of using topology description function 3 4 tdf stems from the fact that the number of elements required to represent the geometrical features far exceed that required to obtain a correct physical response as such describing the geometrical features using a tdf instead of the traditional approach would lead to a drastic reduction in number of parameters required to describe the geometrical features assuming that a tdf can be represented by a small number of parameter as compared to the corresponding number of elements required this would permit the use of a coarse mesh sufficient to obtain correct physical response in solving the fe analysis problem in this work the application of hyper radial basis function hrbfn as tdf is presented the layout of the rest of the paper is as follows hrbfn is briefly described in the second section in the third section hrbfn is coupled with fe analysis to obtain compliance strain energy and the fundamental natural frequency of vibration semi analytical sensitivity to the hrbfn parameters analysis for the aforementioned two types of problems is briefly outlined the fourth section describes the algorithm to solve the topology optimization problem based on optimality criteria

crystal growth and dendritic solidification we present a numerical method which computes the motion of complex solid liquid boundaries in crystal growth the model we solve includes physical effects such as crystalline anisotropy surface tension molecular kinetics and undercooling the method is based on two ideas first the equations of motion are recast as a single history dependent boundary integral equation on the solid liquid boundary a fast algorithm is used to solve the integral equation efficiently second the boundary is moved by solving a hamiltonjacobi type equation on a fixed domain formulated by osher and sethian for a function in which the boundary is a particular level set this equation is solved by finite difference schemes borrowed from the technology of hyperbolic conservation laws the two ideas are combined by constructing a smooth extension of the normal velocity off the moving boundary in a way suggested by the physics of the problem our numerical experiments show the evolution of complex crystalline shapes development of large spikes and corners dendrite formation and side branching and pieces of solid merging and breaking off freely

a moving grid framework for geometric deformable models geometric deformable models based on the level set method have become very popular in the last decade to overcome an inherent limitation in accuracy while maintaining computational efficiency adaptive grid techniques using local grid refinement have been developed for use with these models this strategy however requires a very complex data structure yields large numbers of contour points and is inconsistent with the implementation of topology preserving geometric deformable models tgdms in this paper we investigate the use of an alternative adaptive grid technique called the moving grid method with geometric deformable models in addition to the development of a consistent moving grid geometric deformable model framework our main contributions include the introduction of a new grid nondegeneracy constraint the design of a new grid adaptation criterion and the development of novel numerical methods and an efficient implementation scheme the overall method is simpler to implement than using grid refinement requiring no large complex hierarchical data structures it also offers an extra benefit of automatically reducing the number of contour vertices in the final results after presenting the algorithm we demonstrate its performance using both simulated and real images

on level set regularization for highly ill posed distributed parameter estimation problems the recovery of a distributed parameter function with discontinuities from inverse problems with elliptic forward pdes is fraught with theoretical and practical difficulties better results are obtained for problems where the solution may take on at each point only one of two values thus yielding a shape recovery problem this article considers level set regularization for such problems however rather than explicitly integrating a time embedded pde to steady state which typically requires thousands of iterations methods based on gauss newton are applied more directly one of these can be viewed as damped gauss newton utilized to approximate the steady state equations which in turn are viewed as the necessary conditions of a tikhonov type regularization with a sharpening sub step at each iteration in practice this method is eclipsed however by a special finite time or levenberg marquardt type method which we call dynamic regularization applied to the output least squares formulation our stopping criterion for the iteration does not involve knowledge of the true solution the regularization functional is applied to the smooth level set function rather than to the discontinuous function to be recovered and the second focus of this article is on selecting this functional typical choices may lead to flat level sets that in turn cause ill conditioning we propose a new quartic non local regularization term that penalizes flatness and produces a smooth level set evolution and compare its performance to more usual choices two numerical test cases are considered a potential problem and the classical eit dc resistivity problem

level set based topology optimization for electromagnetic dipole antenna design this paper presents a level set framework for a typical electromagnetic design problem of dipole antenna in this study the geometrical configuration of an antenna is represented by the zero level contour of a higher dimensional level set function the governing equation for the induced current flow on a metal surface is the electric field integral equation efie which takes into account the electric component of the incident wave the design objective is formulated in terms of the surface current and incident electric field the normal velocity of the level set model which reflects the sensitivity of the objective function is derived from the adjoint variable method and shape derivative by optimizing the objective function the area with the highest current density to which the voltage feeding should be applied can be reshaped the advantages of adopting the level set technique for electromagnetic design lie in its capacity for capturing sophisticated topological changes and facilitation in mathematical representation of the design configuration the demonstrative examples of dipole antenna design show that the level set method results in a fairly smooth optimization process where the vacuum metal interface gradually attains its optimal configuration a series of design cases with self adjoint and non self adjoint sensitivity analyses are studied and compared to the benchmarking problems in dipole antenna

construction by isotopy ii construction by isotopy is a technique introduced by lain r aitchison for obtaining doubly slice fibered knots in any dimension we show that if k is any doubly slice fibered n 2 knot n 5 such that 7 snk z then k is constructible by isotopy we also prove that the m twist spin of any doubly slice knot is constructible by isotopy consequently there exists a double slice knot constructible by isotopy that is not the double of any disk knot we also give an example of a doubly slice fibered 6 knot that is not constructible by isotopy we continue an investigation sil of construction by isotopy a technique introduced by lain r aitchison a for obtaining doubly slice fibered knots of any dimension we introduce the problem of finding a fibered double slice for a knot and show that in higher dimensions this problem is equivalent to producing the khot by the construction as a result we are able to expand the class of doubly slice fibered knots that are known to be constructible by isotopy we show for example that any twist spin of a doubly slice knot of any dimension is constructible by isotopy using a result of levine le we then confirm that construction by isotopy can produce knots that are not doubles of disk knots this provides an affirmative answer to a question posed in a p 67 on the other hand work of cappell and ruberman car r1 r2 enables us to give an example of a doubly slice fibered 6 knot that does not arise from the construction consequently we disprove aitchison s main conjecture a sil every doubly slice fibered knot is constructible by isotopy all manifolds that appear are smooth and oriented we will smooth corners when they appear maps are smooth the symbols cl int nbd denote closure interior and tubular neighborhood respectively homology groups have integer coefficients and homomorphisms are induced by inclusion unless otherwise indicated an n 2 knot n 3 is a submanifold k of the n sphere sn diffeomorphic to sn 2 the knot is trivial if it bounds an embedded ball the knot k is fibered if x cl sn nbd k admits a locally trivial fibration f x s1 trivial on the boundary ax we will refer to f as a fibration of k in this case we can identify x with a mapping torus f xh s1 f x i x 0 h x 1 received by the editors june 24 1988 1980 mathematics subject classification 1985 revision primary 57m25 57q45 partially supported by the university of south alabama research council 1990 american mathematical society 0002 9947 90 1 00 25 per page

an eulerian method for capturing caustics a robust numerical method for the localization of caustics is proposed for general hamiltonians it is based on the direct resolution of a system of partial differential equations obtained through a local change of the time variable in the hamilton jacobi equation and complemented by a set of transport equations numerical results 1 to 3 d are presented

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

high order essentially nonsocillatory schemes for hamilton jacobi equations hamilton jacobi h j equations are frequently encountered in applications e g in control theory and differential games h j equations are closely related to hyperbolic conservation laws in one space dimension the former is simply the integrated version of the latter similarity also exists for the multidimensional case and this is helpful in the design of difference approximations in this paper high order essentially nonoscillatory eno schemes for h j equations are investigated which yield uniform high order accuracy in smooth regions and sharply resolve discontinuities in the derivatives the eno scheme construction procedure is adapted from that for hyperbolic conservation laws the schemes are numerically tested on a variety of one dimensional and two dimensional problems including a problem related to control optimization and high order accuracy in smooth regions good resolution of discontinuities in the derivatives and convergence to viscosity solutions are observed

hardy uncertainty principle and unique continuation properties of covariant schrodinger flows abstract we prove a logarithmic convexity result for exponentially weighted l 2 norms of solutions to electromagnetic schrodinger equation without needing to assume smallness of the magnetic potential as a consequence we can prove a unique continuation result in the style of the hardy uncertainty principle which generalizes the analogous theorems which have been recently proved by escauriaza kenig ponce and vega

hamilton jacobi equations on a manifold and applications to grid generation or refinement in this paper we consider hamilton jacobi equations on a manifold typically on the graph of some previously computed function z x y and we show how the corresponding level set method allows us to generate and or to refine a mesh in regions where this function z has large derivatives such as it is the method needs to be strongly improved and accelerated but the principle is awfully natural and in principle the method is fully automatic similar ideas are also useful in image processing in particular for the active contours method

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 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

convergence analysis for a class of high order semi lagrangian advection schemes the convergence properties of a class of high order semi lagrangian schemes for pure advection equations are studied here in the framework of the theory of viscosity solutions we review the general convergence results for discrete time approximation schemes belonging to that class and we prove some a priori estimates in l infty and l2 for the rate of convergence of fully discrete schemes we prove then that a careful coupling of time and space discretizations can allow large time steps in the numerical integration still preserving the accuracy of the solutions several examples of schemes and numerical tests are presented

an enhanced genetic algorithm for structural topology optimization genetic algorithms gas have become a popular optimization tool for many areas of research and topology optimization an effective design tool for obtaining efficient and lighter structures in this paper a versatile robust and enhanced ga is proposed for structural topology optimization by using problem specific knowledge the original discrete black and white 0 1 problem is directly solved by using a bit array representation method to address the related pronounced connectivity issue effectively the four neighbourhood connectivity is used to suppress the occurrence of checkerboard patterns a simpler version of the perimeter control approach is developed to obtain a well posed problem and the total number of hinges of each individual is explicitly penalized to achieve a hinge free design to handle the problem of representation degeneracy effectively a recessive gene technique is applied to viable topologies while unusable topologies are penalized in a hierarchical manner an efficient fem based function evaluation method is developed to reduce the computational cost a dynamic penalty method is presented for the ga to convert the constrained optimization problem into an unconstrained problem without the possible degeneracy with all these enhancements and appropriate choice of the ga operators the present ga can achieve significant improvements in evolving into near optimum solutions and viable topologies with checkerboard free mesh independent and hinge free characteristics numerical results show that the present ga can be more efficient and robust than the conventional gas in solving the structural topology optimization problems of minimum compliance design minimum weight design and optimal compliant mechanisms design it is suggested that the present enhanced ga using problem specific knowledge can be a powerful global search tool for structural topology optimization copyright 2005 john wiley sons ltd

a reformulation of the conservative level set reinitialization equation for accurate and robust simulation of complex multiphase flows abstract this paper presents an accurate and robust reinitialization equation for the conservative level set that does not significantly deform stationary surfaces the compression and diffusion term of the reinitialization equation are reformulated to use a distance level set directly mapped from the conservative level set the normals are calculated using a distance level set reconstructed from the interface using a fast marching method increasing robustness and allowing the use of high order non tvd transport schemes using this new reinitialization equation we present results for canonical test cases such as zalesak s disk and spurious currents which show significant improvement a simulation of a liquid gas jet with r e 5000 is also presented to demonstrate the volume conservation properties of the method in more complex flows

a level set approach for computing solutions to incompressible two phase flow a level set approach for computing solutions to incompressible two phase flow is presented the interface between the two fluids is considered to be sharp and is described as the zero level set of a smooth function a new treatment of the level set method allows us to efficiently maintain the level set function as the signed distance from the interface we never have to explicitly reconstruct or find the zero level set consequently we are able to handle arbitrarily complex topologies large density and viscosity ratios and surface tension on relatively coarse grids we use a second order projection method along with a second order upwinded procedure for advecting the momentum and level set equations we consider the motion of air bubbles and water drops we also compute flows with multiple fluids such as air oil and water

two approximations of solutions of hamilton jacobi equations abstract equations of hamilton jacobi type arise in many areas of application including the calculus of variations control theory and differential games the associated initial value problems almost never have global time classical solutions and one must deal with suitable generalized solutions the correct class of generalized solutions has only recently been established by the authors this article establishes the convergence of a class of difference approximations to these solutions by obtaining explicit error estimates analogous results are proved by similar means for the method of vanishing viscosity author

a geometrical optics based numerical method for high frequency electromagnetic fields computations near fold caustics part i this paper presents an eulerian numerical method for the computation of a bi valued solution of hamilton jacobi type equation in a particular geometric setting more precisely we consider high frequency electromagnetic fields in the vicinity of fold caustics

integrated segmentation and interpolation of sparse data we address the two inherently related problems of segmentation and interpolation of 3d and 4d sparse data and propose a new method to integrate these stages in a level set framework the interpolation process uses segmentation information rather than pixel intensities for increased robustness and accuracy the method supports any spatial configurations of sets of 2d slices having arbitrary positions and orientations we achieve this by introducing a new level set scheme based on the interpolation of the level set function by radial basis functions the proposed method is validated quantitatively and or subjectively on artificial data and mri and ct scans and is compared against the traditional sequential approach which interpolates the images first using a state of the art image interpolation method and then segments the interpolated volume in 3d or 4d in our experiments the proposed framework yielded similar segmentation results to the sequential approach but provided a more robust and accurate interpolation in particular the interpolation was more satisfactory in cases of large gaps due to the method taking into account the global shape of the object and it recovered better topologies at the extremities of the shapes where the objects disappear from the image slices as a result the complete integrated framework provided more satisfactory shape reconstructions than the sequential approach

approximations for viscosity solutions of hamilton jacobi equations with locally varying time and space grids a new monotone finite difference scheme is introduced that approximates viscosity solutions of first order nonlinear hamilton jacobi equations the main feature of the scheme is that it allows for locally varying time and space grids making it ideal for use with adaptive algorithms explicit a priori error estimates are given to establish convergence numerical examples including adaptive mesh refinement examples demonstrate the effectiveness of the proposed scheme

a simple explicit operator splitting method for effective hamiltonians understanding effective hamiltonians quantitatively is essential for the homogenization of hamilton jacobi equations we propose in this article a simple efficient operator splitting method for computing effective hamiltonians when the hamiltonian is either convex or nonconvex in the gradient variable to speed up our lie scheme based operator splitting method we further propose a cascadic initialization strategy so that the steady state of the underlying time dependent hamilton jacobi equation can be reached more rapidly extensive numerical examples demonstrate the efficiency and accuracy of the new algorithm

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

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

the casson gordon invariants in high dimensional knot theory the casson gordon invariants of knots in all dimensions are interpreted in terms of surgery theory applications are given to finding nondoubly slice knots and doubly slice knots which are not the double of a disk knot in even dimensions the property of being doubly slice is shown to be largely homotopy theoretic while in odd dimensions the surgery theoretic method shows such properties to depend on more than the homotopy type in 1975 a casson and c gordon used the equivariant signatures defined in as to show that classical knot cobordism disers radically from the high dimensional theory the author s thesis r1 developed a generalization of the casson gordon invariants to all odd dimensions to attack the problem of double null concordance see also st su gl this is the question of whether an n dimensional knot is a slice of an n 1 dimensional unknot we showed further that analogous invariants could be defined in even dimensions and applied to the same problem in this paper we give some new applications of the casson gordon invariants in high dimensional knot theory and strengthen the results of r1 and s1 it was shown by j tevine l that doubly slice knots are obtained if one doubles a disk pair along its boundary doubly slice knots have many properties in common with such doubled disk knots in fact according to l the concepts are equivalent for simple knots levine asked whether in fact all doubly slice knots were obtained as doubled disk knots and provided examples of knots in s3 and s4 which are not in his ph d thesis s1 and more recent work s2 l smolinsky gave examples in higher odd dimensions of knots which are doubly slice without being the double of a disk pair using the casson gordon invariants in a manner similar to r1 as an application of the even dimensional invariants we show that in all even dimensions there are doubly slice knots which are not doubled disk knots to construct such examples we show that an even dimensional knot which has the z z homology type of a doubly slice knot is also doubly slice this contradicts a theorem of r1 which asserts a bound on the even dimensional casson gordon invariants of a doubly slice knot to reestablish the main result of r1 namely that in all even dimensions there are algebraically doubly slice knots which are not doubly slice we extend the desnition of the even dimensional invariants to a more general context than that used in r1 in that paper q valued invariants were defined for homomorphisms to a cyclic group zd d was assumed to be a prime power and only certain homomorphisms were allowed in the approach taken in lleceived by the editors december 12 1986 1980 mathematics xubject lassification 1985 revision primary 57q45 57m25 57r67

finite difference methods for the infinity laplace and p laplace equations abstract we build convergent discretizations and semi implicit solvers for the infinity laplacian and the game theoretical p laplacian the discretizations simplify and generalize earlier ones we prove convergence of the solution of the wide stencil finite difference schemes to the unique viscosity solution of the underlying equation we build a semi implicit solver which solves the laplace equation as each step it is fast in the sense that the number of iterations is independent of the problem size this is an improvement over previous explicit solvers which are slow due to the cfl condition

level set topology optimization of fluids in stokes flow we propose the level set method of topology optimization as a viable robust and efficient alternative to density based approaches in the setting of fluid flow the proposed algorithm maintains the discrete nature of the optimization problem throughout the optimization process leading to significant advantages over density based topology optimization algorithms specifically the no slip boundary condition is implemented directly this is accurate removes the need for interpolation schemes and continuation methods and gives significant computational savings by only requiring flow to be modeled in fluid regions topological sensitivity information is utilized to give a robust algorithm in two dimensions and familiar two dimensional power dissipation minimization problems are solved successfully computational efficiency of the algorithm is also clearly demonstrated on large scale three dimensional problems

radial basis functions and level set method for structural topology optimization level set methods have become an attractive design tool in shape and topology optimization for obtaining lighter and more efficient structures in this paper the popular radial basis functions rbfs in scattered data fitting and function approximation are incorporated into the conventional level set methods to construct a more efficient approach for structural topology optimization rbf implicit modelling with multiquadric mq splines is developed to define the implicit level set function with a high level of accuracy and smoothness a rbf level set optimization method is proposed to transform the hamilton jacobi partial differential equation pde into a system of ordinary differential equations odes over the entire design domain by using a collocation formulation of the method of lines with the mathematical convenience the original time dependent initial value problem is changed to an interpolation problem for the initial values of the generalized expansion coefficients a physically meaningful and efficient extension velocity method is presented to avoid possible problems without reinitialization in the level set methods the proposed method is implemented in the framework of minimum compliance design that has been extensively studied in topology optimization and its efficiency and accuracy over the conventional level set methods are highlighted numerical examples show the success of the present rbf level set method in the accuracy convergence speed and insensitivity to initial designs in topology optimization of two dimensional 2d structures it is suggested that the introduction of the radial basis functions to the level set methods can be promising in structural topology optimization

a fast marching like algorithm for geometrical shock dynamics we develop a new algorithm for the computation of the geometrical shock dynamics gsd model the method relies on the fast marching paradigm and enables the discrete evaluation of the first arrival time of a shock wave and its local velocity on a cartesian grid the proposed algorithm is based on a first order upwind finite difference scheme and reduces to a local nonlinear system of two equations solved by an iterative procedure reference solutions are built for a smooth radial configuration and for the 2d riemann problem the link between the gsd model and p systems is given numerical experiments demonstrate the efficiency of the scheme and its ability to handle singularities

regular article a pde based fast local level set method we develop a fast method to localize the level set method of osher and sethian 1988 j comput phys 79 12 and address two important issues that are intrinsic to the level set method a how to extend a quantity that is given only on the interface to a neighborhood of the interface b how to reset the level set function to be a signed distance function to the interface efficiently without appreciably moving the interface this fast local level set method reduces the computational effort by one order of magnitude works in as much generality as the original one and is conceptually simple and easy to implement our approach differs from previous related works in that we extract all the information needed from the level set function or functions in multiphase flow and do not need to find explicitly the location of the interface in the space domain the complexity of our method to do tasks such as extension and distance reinitialization is o n where n is the number of points in space not o n log n as in works by sethian 1996 proc nat acad sci 93 1591 and helmsen and co workers 1996 spie microlithography ix p 253 this complexity estimation is also valid for quite general geometrically based front motion for our localized method

spectral viscosity approximations to hamilton jacobi solutions the spectral viscosity approximate solution of convex hamilton jacobi equations with periodic boundary conditions is studied it is proved in this paper that the approximation and its gradient remain uniformly bounded formally spectral accurate and converge to the unique viscosity solution the l1 convergence rate of the order 1 varepsilon forall varepsilon 0 is obtained

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

hybrid control synthesis for eventuality specifications using level set methods this paper is concerned with the extraction of controllers for hybrid systems with respect to eventuality specifications given a hybrid system modelled by a hybrid automaton and a target set of states the objective is to compute the maximal set of initial states together with the hybrid control policy such that all the trajectories of the controlled system reach the target in finite time due to the existence of set valued disturbance inputs the problem is studied in a game theoretic framework having shown that a least restrictive solution does not exist we propose a dynamic programming algorithm that computes the maximal initial set and a controller with the desired property to implement the algorithm reachable sets of pursuit evasion differential games need to be computed for that reason level set methods are employed where the boundary of the reachable set is characterized as the zero level set of a hamilton jacobi equation the procedure for the numerical extraction of the controller is presented in detail and examples illustrate the methodology finally to demonstrate the practical character of our results a control design problem in the benchmark system of the batch evaporator is considered as an eventuality synthesis problem and solved using the proposed methodology

the generalized finite element method this paper describes a pilot design and implementation of the generalized finite element method gfem as a direct extension of the standard finite element method sfem or fem which makes possible the accurate solution of engineering problems in complex domains which may be practically impossible to solve using the fem the development of the gfem is illustrated for the laplacian in two space dimensions in domains which may include several hundreds of voids and or cracks for which the construction of meshes used by the fem is practically impossible the two main capabilities are 1 it can construct the approximation using meshes which may overlap part or all of the domain boundary 2 it can incorporate into the approximation handbook functions which are known analytically or are generated numerically and approximate well the solution of the boundary value problem in the neighborhood of corner points voids cracks etc the main tool is a special integration algorithm which we call the fast remeshing approach which is robust and works for any domain with arbitrary complexity the incorporation of the handbook functions into the gfem is done by employing the partition of unity method pum the presented formulations and implementations can be easily extended to the multi material medium where the voids are replaced by inclusions of various shapes and sizes and to the case of the elasticity problem this work can also be understood as a pilot study for the feasibility and demonstration of the capabilities of the gfem which is needed before analogous implementations are attempted in the three dmensional and nonlinear cases which are the cases of main interest for future work

incorporating topological derivatives into level set methods the aim of this paper is to investigate the use of topological derivatives in combination with the level set method for shape reconstruction and optimization problems we propose a new approach generalizing the standard speed method which is obtained by using a source term in the level set equation that depends on the topological derivative of the objective functional the resulting approach can be interpreted as a generalized fixed point iteration for the optimality system with respect to topological and shape variations moreover we apply the new approach for a simple model problem in shape reconstruction where the topological derivative can be computed without additional effort finally we present numerical tests related to this model problem which demonstrate that the new method based on shape and topological derivative successfully reconstructs obstacles in situations where the standard level set approach fails

a manifold learning based reduced order model for springback shape characterization and optimization in sheet metal forming abstract the parameters of a stamping process include the geometry of the tools the shape of the initial sheet blank the material constitutive law and the process parameters when designing the overall process one has to also take into account the springback effect that appears when the tools are removed and additional surfaces are cut off the goal then is to obtain a final shape as close as possible to the desired shape while satisfying the admissibility constraints on the variable parameters as well as the feasibility constraints frequently expressed in the form of forming limit diagrams in the present paper we represent the post springback shape by a level set function then rather than rely on arbitrarily selected case dependent measurement locations as in the numisheet benchmark problems we build a reduced order shape space where this level set evolves by extending our recent shape manifold approach to the problem of springback assessment for 3d shapes next we propose an optimization algorithm designed to minimize the gap between the post springback and the desired final shapes the required level set functions are generated from a corresponding set of springback shapes predicted by finite element simulations using our approach we determine the minimal number of parameters needed in order to uniquely characterize the final formed shape regardless of complexity finally we demonstrate the approach using an industrial test case springback assessment of the deep drawing operation of an automotive strut tower

discrete time high order schemes for viscosity solutions of hamilton jacobi bellman equations a general method for constructing high order approximation schemes for hamilton jacobi bellman equations is given the method is based on a discrete version of the dynamic programming principle we prove a general convergence result for this class of approximation schemes also obtaining under more restrictive assumptions an estimate in l infty of the order of convergence and of the local truncation error the schemes can be applied in particular to the stationary linear first order equation in bbb r n we present several examples of schemes belonging to this class and with fast convergence to the solution

a three dimensional model based inversion algorithm using radial basis functions for microwave data we present a three dimensional 3d model based inversion algorithm for inverting electromagnetic data in our approach the models are described by points in the 3d space and the so called radial basis functions are used as the interpolation functions for connecting these points the use of the radial basis functions renders the surface of the target intrinsically smooth the l2 norm and weighted l2 norm regularization schemes are employed to constrain the curvature and to further smooth the surface of the target we tested this inversion algorithm using both synthetic and experimental data sets the inversion results demonstrate that both the shapes and material properties of the target are well reconstructed as a complementary algorithm to the pixel based tomographic inversion algorithms the model based inversion algorithm has fewer parameters to invert hence it has a higher computational efficiency the model based inversion algorithm also gives the users the freedom to construct an initial model based on their knowledge and other related information it helps to reduce the non uniqueness in the data interpretation

gaussian decay of harmonic oscillators and related models abstract we prove that the decay of the eigenfunctions of harmonic oscillators uniform electric or magnetic fields is not stable under 0 order complex perturbations even if bounded of these hamiltonians in the sense that we can produce solutions to the evolutionary schrodinger flows associated to the hamiltonians with a stronger gaussian decay at two distinct times we then characterize in a quantitative way the sharpest possible gaussian decay of solutions as a function of the oscillation frequency or the strength of the field depending on the hamiltonian which is considered this is connected to the hardy s uncertainty principle for free schrodinger evolutions

topology optimization with local stress constraint based on level set evolution via reaction diffusion abstract this work focuses the structural topology optimization problem of mass minimization subject to local stress constraints to this aim two related issues are addressed the first one is the successful strategy used to define local stress constraints by means of an augmented lagrangian approach the second and main contribution of the present paper is the use of a reaction diffusion equation to guide via evolution of a level set the design optimization sequence the advantages of this strategy are twofold firstly it allows the creation of new holes during the optimization process a significant feature for a true topological optimization method secondly reinitialization steps usually found in classical hamilton jacobi based evolution are eliminated with a significant improvement in convergence ease a set of benchmark examples in two dimensions are presented numerical results show the efficiency of the algorithm to create new holes identify stress concentrations and to provide stable optimization sequences converging to local minima defined by stress saturated designs

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

magnetic virial identities weak dispersion and strichartz inequalities we show a family of virial type identities for the schrodinger and wave equations with electromagnetic potentials as a consequence some weak dispersive inequalities in space dimension n 3 involving morawetz and smoothing estimates are proved finally we apply them to prove strichartz inequalities for the wave equation with a non trapping electromagnetic potential with almost coulomb decay

an eulerian approach for computing the finite time lyapunov exponent we propose efficient eulerian methods for approximating the finite time lyapunov exponent ftle the idea is to compute the related flow map using the level set method and the liouville equation there are several advantages of the proposed approach unlike the usual lagrangian type computations the resulting method requires the velocity field defined only at discrete locations no interpolation of the velocity field is needed also the method automatically stops a particle trajectory in the case when the ray hits the boundary of the computational domain the computational complexity of the algorithm is o dx d 1 with d the dimension of the physical space since there are the same number of mesh points in the x t space the computational complexity of the proposed eulerian approach is optimal in the sense that each grid point is visited for only o 1 time we also extend the algorithm to compute the ftle on a co dimension one manifold the resulting algorithm does not require computation on any local coordinate system and is simple to implement even for an evolving manifold

parametric structural shape topology optimization with a variational distance regularized level set method abstract the signed distance function sdf gives the shortest distance from a given point to the boundary and the sign indicates whether this point is inside or outside the closed boundary or enclosed region the sdf property is highly preferred in classical level set methods to maintain the numerical stability during the topology optimization process and provide a metric for the distance based interpolation of different material properties in conventional level set methods a common way of achieving a level set function with the signed distance property is to periodically implement the so called reinitialization scheme by solving an additional hamilton jacobi partial differential equation however such reinitialization scheme is implemented outside the optimization loop with the optimization process suspended which may shift the optimization result and bring convergence issues in this paper a double well potential functional is employed for distance regularization inside the topology optimization loop which can enforce the signed distance property of the level set function in a narrow band along the design boundaries while keeping the level set function flat in the rest area of the computational domain the radial basis function rbf based parameterization technique is combined with mathematical programming to improve the performance of the proposed distance regularized topology optimization method in handling problems with non convex objective functions and multiple constraints the flatness of the level set function in the material region also enables easy creation of new holes to the design in the topology optimization process both 2d and 3d benchmark examples are employed to demonstrate the validity of the proposed method

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

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

unique continuation for schrodinger evolutions with applications to profiles of concentration and traveling waves we prove unique continuation properties for solutions of the evolution schrodinger equation with time dependent potentials as an application of our method we also obtain results concerning the possible concentration profiles of blow up solutions and the possible profiles of the traveling waves solutions of semi linear schrodinger equations

level set methods an overview and some recent results the level set method was devised by s osher and j a sethian 1988 j comput phys 79 12 49 as a simple and versatile method for computing and analyzing the motion of an interface in two or three dimensions bounds a possibly multiply connected region the goal is to compute and analyze the subsequent motion of under a velocity field v this velocity can depend on position time the geometry of the interface and the external physics the interface is captured for later time as the zero level set of a smooth at least lipschitz continuous function x t i e t x x t 0 is positive inside negative outside and is zero on t topological merging and breaking are well defined and easily performed in this review article we discuss recent variants and extensions including the motion of curves in three dimensions the dynamic surface extension method fast methods for steady state problems diffusion generated motion and the variational level set approach we also give a user s guide to the level set dictionary and technology and couple the method to a wide variety of problems involving external physics such as compressible and incompressible possibly reacting flow stefan problems kinetic crystal growth epitaxial growth of thin films vortex dominated flows and extensions to multiphase motion we conclude with a discussion of applications to computer vision and image processing

adjoint based optimization of pdes in moving domains in this investigation we address the problem of adjoint based optimization of pde systems in moving domains as an example we consider the one dimensional heat equation with prescribed boundary temperatures and heat fluxes we discuss two methods of deriving an adjoint system necessary to obtain a gradient of a cost functional in the first approach we derive the adjoint system after mapping the problem to a fixed domain whereas in the second approach we derive the adjoint directly in the moving domain by employing methods of the noncylindrical calculus we show that the operations of transforming the system from a variable to a fixed domain and deriving the adjoint do not commute and that while the gradient information contained in both systems is the same the second approach results in an adjoint problem with a simpler structure which is therefore easier to implement numerically this approach is then used to solve a moving boundary optimization problem for our model system

an extended level set method for shape and topology optimization in this paper the conventional level set methods are extended as an effective approach for shape and topology optimization by the introduction of the radial basis functions rbfs the rbf multiquadric splines are used to construct the implicit level set function with a high level of accuracy and smoothness and to discretize the original initial value problem into an interpolation problem the motion of the dynamic interfaces is thus governed by a system of coupled ordinary differential equations odes and a relatively smooth evolution can be maintained without reinitialization a practical implementation of this method is further developed for solving a class of energy based optimization problems in which approximate solution to the original hamilton jacobi equation may be justified and nucleation of new holes inside the material domain is allowed for furthermore the severe constraints on the temporal and spatial discretizations can be relaxed leading to a rapid convergence to the final design insensitive to initial guesses the normal velocities are chosen to perform steepest gradient based optimization by using shape sensitivity analysis and a bi sectioning algorithm a physically meaningful and efficient extension velocity method is also presented the proposed method is implemented in the framework of minimum compliance design and its efficiency over the existing methods is highlighted numerical examples show its accuracy convergence speed and insensitivity to initial designs in shape and topology optimization of two dimensional 2d problems that have been extensively investigated in the literature

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

eigenvalue stability of radial basis function discretizations for time dependent problems differentiation matrices obtained with infinitely smooth radial basis function rbf collocation methods have under many conditions eigenvalues with positive real part preventing the use of such methods for time dependent problems we explore this difficulty at theoretical and practical levels theoretically we prove that differentiation matrices for conditionally positive definite rbfs are stable for periodic domains we also show that for gaussian rbfs special node distributions can achieve stability in 1 d and tensor product nonperiodic domains as a more practical approach for bounded domains we consider differentiation matrices based on least squares rbf approximations and show that such schemes can lead to stable methods on less regular nodes by separating centers and nodes least squares techniques open the possibility of the separation of accuracy and stability characteristics

factored singularities and high order lax friedrichs sweeping schemes for point source traveltimes and amplitudes in the high frequency regime the geometrical optics approximation for the helmholtz equation with a point source results in an eikonal equation for traveltime and a transport equation for amplitude because the point source traveltime field has an upwind singularity at the source point all formally high order finite difference eikonal solvers exhibit first order convergence and relatively large errors in this paper we propose to first factor out the singularities of traveltimes takeoff angles and amplitudes and then we design high order lax friedrichs sweeping schemes for point source traveltimes takeoff angles and amplitudes numerical examples are presented to demonstrate the performance of our new method

a variational level set method for the topology optimization of steady state navier stokes flow the smoothness of topological interfaces often largely affects the fluid optimization and sometimes makes the density based approaches though well established in structural designs inadequate this paper presents a level set method for topology optimization of steady state navier stokes flow subject to a specific fluid volume constraint the solid fluid interface is implicitly characterized by a zero level contour of a higher order scalar level set function and can be naturally transformed to other configurations as its host moves a variational form of the cost function is constructed based upon the adjoint variable and lagrangian multiplier techniques to satisfy the volume constraint effectively the lagrangian multiplier derived from the first order approximation of the cost function is amended by the bisection algorithm the procedure allows evolving initial design to an optimal shape and or topology by solving the hamilton jacobi equation two classes of benchmarking examples are presented in this paper 1 periodic microstructural material design for the maximum permeability and 2 topology optimization of flow channels for minimizing energy dissipation a number of 2d and 3d examples well demonstrated the feasibility and advantage of the level set method in solving fluid solid shape and topology optimization problems

binary level set methods for topology and shape optimization of a two density inhomogeneous drum abstract we optimize eigenvalues in optimal shape design using binary level set methods the interfaces of subregions are represented implicitly by the discontinuities of binary level set functions taking two values 1 or 1 at convergence a binary constraint is added to the original model problems we propose two variational algorithms to solve the constrained optimization problems one is a hybrid type by coupling the lagrange multiplier approach for the geometry constraint with the augmented lagrangian method for the binary constraint the other is devised using the lagrange multiplier method for both constraints the two iterative algorithms are both largely independent of the initial guess and can satisfy the geometry constraint very accurately during the iterations intensive numerical results are presented and compared with those obtained by level set methods which demonstrate the effectiveness and robustness of our algorithms

integrated topology optimization with embedded movable holes based on combined description by material density and level sets abstract this paper presents a novel topology description model for topology optimization problems with embedded movable holes combining the ability of the level set model for accurate geometrical description of the prescribed hole shapes and the high efficiency of the material density based method by this means any arbitrary hole shapes can be represented accurately and smoothly while the topological changes can be easily handled by the material distribution model an explicit mathematical expression is defined to obtain the actual structural layout by combing both models moreover an effective model for the non overlap constraint is proposed in a unified and systematic manner to avoid the overlaps between the holes and between each hole and the design domain boundary therein the non overlap constraint for all the embedded holes is imposed as a single explicit integral constraint over the design domain thus avoiding the difficulties in the overlap detection of multiple arbitrary shaped geometries such a non overlap constraint is accurate and differentiable facilitating an analytical design sensitivity analysis numerical examples are given to demonstrate the effectiveness and efficiency of the present method

a moving superimposed finite element method for structural topology optimization level set methods are becoming an attractive design tool in shape and topology optimization for obtaining efficient and lighter structures in this paper a dynamic implicit boundary based moving superimposed finite element method s version fem or s fem is developed for structural topology optimization using the level set methods in which the variational interior and exterior boundaries are represented by the zero level set both a global mesh and an overlaying local mesh are integrated into the moving s fem analysis model a relatively coarse fixed eulerian mesh consisting of bilinear rectangular elements is used as a global mesh the local mesh consisting of flexible linear triangular elements is constructed to match the dynamic implicit boundary captured from nodal values of the implicit level set function in numerical integration using the gauss quadrature rule the practical difficulty due to the discontinuities is overcome by the coincidence of the global and local meshes a double mapping technique is developed to perform the numerical integration for the global and coupling matrices of the overlapped elements with two different co ordinate systems an element killing strategy is presented to reduce the total number of degrees of freedom to improve the computational efficiency a simple constraint handling approach is proposed to perform minimum compliance design with a volume constraint a physically meaningful and numerically efficient velocity extension method is developed to avoid the complicated pde solving procedure the proposed moving s fem is applied to structural topology optimization using the level set methods as an effective tool for the numerical analysis of the linear elasticity topology optimization problems for the classical elasticity problems in the literature the present s fem can achieve numerical results in good agreement with those from the theoretical solutions and or numerical results from the standard fem for the minimum compliance topology optimization problems in structural optimization the present approach significantly outperforms the well recognized ersatz material approach as expected in the accuracy of the strain field numerical stability and representation fidelity at the expense of increased computational time it is also shown that the present approach is able to produce structures near the theoretical optimum it is suggested that the present s fem can be a promising tool for shape and topology optimization using the level set methods copyright 2005 john wiley sons ltd

on the reinitialization procedure in a narrow band locally refined level set method for interfacial flows a local level set algorithm for simulating interfacial flows described by the two dimensional incompressible navier stokes equations is presented the governing equations are solved using a finite difference discretization on a cartesian grid and a second order approximate projection method the level set transport and reinitialization equations are solved in a narrow band around the interface using an adaptive refined grid which is reconstructed every time step and refined using a simple uniform cell splitting operation within the band instabilities at the border of the narrow band are avoided by smoothing the level set function in the outer part of the band the influence of different pde based reinitialization strategies on the accuracy of the results is investigated the ability of the proposed method to accurately compute interfacial flows is discussed using different tests namely the advection of a circle of fluid in two different time reversed vortex flows the advection of zalesak s rotating disk the propagation of small amplitude gravity and capillary waves at the interface between two superposed viscous fluids in deep water and a classical test of rayleigh taylor instability with and without surface tension effects the interface location error and area loss for some of the results obtained are compared with those of a recent particle level set method copyright 2005 john wiley sons ltd

a multilevel level set method for optimizing eigenvalues in shape design problems in this paper we consider optimal design problems that involve shape optimization the goal is to determine the shape of a certain structure such that it is either as rigid or as soft as possible to achieve this goal we combine two new ideas for an efficient solution of the problem first we replace the eigenvalue problem with an approximation by using inverse iteration second we use a level set method but rather than propagating the front we use constrained optimization methods combined with multilevel continuation techniques combining these two ideas we obtain a robust and rapid method for the solution of the optimal design problem

unconditionally stable methods for hamilton jacobi equations we present new numerical methods for constructing approximate solutions to the cauchy problem for hamilton jacobi equations of the form ut h dxu 0 the methods are based on dimensional splitting and front tracking for solving the associated non strictly hyperbolic system of conservation laws pt dx h p 0 where p dx u in particular our methods depend heavily on a front tracking method for one dimensional scalar conservation laws with discontinuous coefficients the proposed methods are unconditionally stable in the sense that the time step is not limited by the space discretization and they can be viewed as large time step godunov type or front tracking methods we present several numerical examples illustrating the main features of the proposed methods we also compare our methods with several methods from the literature

a uniformly second order fast sweeping method for eikonal equations a uniformly second order method with a local solver based on the piecewise linear discontinuous galerkin formulation is introduced to solve the eikonal equation with dirichlet boundary conditions the method utilizes an interesting phenomenon referred as the superconvergence phenomenon that the numerical solution of monotone upwind schemes for the eikonal equation is first order accurate on both its value and gradient when the solution is smooth this phenomenon greatly simplifies the local solver based on the discontinuous galerkin formulation by reducing its local degrees of freedom from two 1 d or three 2 d or four 3 d to one with the information of the gradient frozen when considering the eikonal equation with point source conditions we further utilize a factorization approach to resolve the source singularities of the eikonal by decomposing it into two parts either multiplicatively or additively one part is known and captures the source singularities the other part serves as a correction term that is differentiable at the sources and satisfies the factored eikonal equations we extend the second order method to solve the factored eikonal equations to compute the correction term with second order accuracy then recover the eikonal with second order accuracy numerical examples are presented to demonstrate the performance of the method

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

a variational level set approach to multiphase motion a coupled level set method for the motion of multiple junctions of e g solid liquid and grain boundaries which follows the gradient flow for an energy functional consisting of surface tension proportional to length and bulk energies proportional to area is developed the approach combines the level set method of s osher and j a sethian with a theoretical variational formulation of the motion by f reitich and h m soner the resulting method uses as many level set functions as there are regions and the energy functional is evaluated entirely in terms of level set functions the gradient projection method leads to a coupled system of perturbed by curvature terms hamilton jacobi equations the coupling is enforced using a single lagrange multiplier associated with a constraint which essentially prevents a regions from overlapping and b the development of a vacuum the numerical implementation is relatively simple and the results agree with and go beyond the theory as given in 12 other applications of this methodology including the decomposition of a domain into subregions with minimal interface length are discussed finally some new techniques and results in level set methodology are presented

variational piecewise constant level set methods for shape optimization of a two density drum we apply the piecewise constant level set method to a class of eigenvalue related two phase shape optimization problems based on the augmented lagrangian method and the lagrange multiplier approach we propose three effective variational methods for the constrained optimization problem the corresponding gradient type algorithms are detailed the first uzawa type algorithm having applied to shape optimization in the literature is proven to be effective for our model but it lacks stability and accuracy in satisfying the geometry constraint during the iteration the two other novel algorithms we propose can overcome this limitation and satisfy the geometry constraint very accurately at each iteration moreover they are both highly initial independent and more robust than the first algorithm without penalty parameters the last projection lagrangian algorithm has less severe restriction on the time step than the first two algorithms numerical results for various instances are presented and compared with those obtained by level set methods the comparisons show effectiveness efficiency and robustness of our methods we expect our promising algorithms to be applied to other shape optimization and multiphase problems

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

computation of nonclassical solutions to hamilton jacobi problems this paper is devoted to the construction of numerical methods for the approximation of nonclassical solutions to multidimensional hamilton jacobi equations for both scalar and vectorial problems recent theoretical results have yielded existence of solutions in many cases for which the usual viscosity approach was ill suited or not applicable the selection criterion used here is based on a viscoelasticity capillarity approach common in solid mechanics numerical methods adapted to this framework are built consistency of the model equation with the given selection criterion is essential it is achieved here through the use of high order finite difference schemes by considering applications to potential well problems the convergence of the methods are investigated

bilateral filtering for structural topology optimization filtering has been a major approach used in the homogenization based methods for structural topology optimization to suppress the checkerboard pattern and relieve the numerical instabilities in this paper a bilateral filtering technique originally developed in image processing is presented as an efficient approach to regularizing the topology optimization problem a non linear bilateral filtering process leads to a suitable problem regularization to eliminate the checkerboard instability pronounced edge preserving smoothing characteristics to favour the 0 1 convergence of the mass distribution and computational efficiency due to its single pass and non iterative nature thus we show that the application of the bilateral filtering brings more desirable effects of checkerboard free mesh independence crisp boundary computational efficiency and conceptual simplicity the proposed bilateral technique has a close relationship with the conventional domain filtering and range filtering the proposed method is implemented in the framework of a power law approach based on the optimality criteria and illustrated with 2d examples of minimum compliance design that has been extensively studied in the recent literature of topology optimization and its efficiency and accuracy are highlighted copyright 2005 john wiley sons ltd