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

band structure optimization of two dimensional photonic crystals in h polarization in the study of photonic crystals the question arises naturally which crystals produce the largest band gaps this question is investigated by means of an optimization based evolution algorithm which given two dielectric materials seeks to produce a material distribution within the fundamental cell which produces a maximal band gap at a given point in the spectrum the case of h polarization in two dimensions is considered several numerical examples are presented

crossed products of nuclear c algebras and their traces abstract this work addresses the blackadar kirchberg question bkq as it pertains to group and reduced crossed product c algebras the bkq asks if every stably finite c algebra admits the matricial field mf property we characterize the mf property for a class of reduced crossed product c algebras and for their traces we prove that every ordered abelian group is in a sense mf and show that there is no k theoretic obstruction to an affirmative answer to the bkq using classification techniques and induced k theoretic dynamics we give an affirmative answer to the bkq for reduced crossed products of ah algebras of real rank zero by free groups combining our results with recent progress in elliott s classification program shows that the reduced crossed product of a separable simple unital nuclear uct and monotracial c algebra by the free group is always mf we also examine traces on these crossed products and show they always admit certain finite dimensional approximation properties by appealing to a result of ozawa rordam and sato we show that semidirect products of discrete amenable groups by free groups admit mf reduced group c algebras

abelian mathrm c ast subalgebras of mathrm c ast algebras of real rank zero and inductive limit mathrm c ast algebras

level set and total variation regularization for elliptic inverse problems with discontinuous coefficients we propose a level set approach for elliptic inverse problems with piecewise constant coefficients the geometry of the discontinuity of the coefficient is represented implicitly by level set functions the inverse problem is solved using a variational augmented lagrangian formulation with total variation regularization of the coefficient the corresponding euler lagrange equation gives the evolution equation for the level set functions and the constant values of the coefficients we use a multiple level set representation which allows the coefficient to have multiple constant regions knowledge of the exact number of regions is not required only an upper bound is needed numerical experiments show that the method can recover coefficients with rather complicated geometries of discontinuities under moderate amount of noise in the observation data the method is also robust with respect to the initial guess for the geometry of the coefficient discontinuities

on the synthesis of boundary control laws for distributed port hamiltonian systems this paper is concerned with the energy shaping of 1 d linear boundary controlled port hamiltonian systems the energy casimir method is first proposed to deal with power preserving systems it is shown how to use finite dimensional dynamic boundary controllers and closed loop structural invariants to partially shape the closed loop energy function and how such controller finally reduces to a state feedback when dissipative port hamiltonian systems are considered the casimir functions do not exist anymore dissipation obstacle and the immersion via a dynamic controller reduction through invariants method cannot be applied the main contribution of this paper is to show how to use the same ideas and state functions to shape the closed loop energy function of dissipative systems through direct state feedback i e without relying on a dynamic controller and a reduction step in both cases the existence of solution and the asymptotic stability by additional damping injection of the closed loop system are proven the general theory and achievable closed loop performances are illustrated with the help of a concluding example the boundary stabilization of a longitudinal beam vibrations

classifying c algebras via ordered mod p k theory we introduce an order structure on k 0 9 k0 z p this group may also be thought of as ko 7z z p we exhibit new examples of real rank zero c algebras that are inductive limits of finite dimensional and dimension drop algebras have the same ordered graded k theory with order unit and yet are not isomorphic in fact they are not even stably shape equivalent the order structure on k0 z 3 z p naturally distinguishes these algebras the same invariant is used to give an isomorphism theorem for such realrank zero inductive limits as a corollary we obtain an isomorphism theorem for all real rank zero approximately homogeneous c algebras that arise from systems of bounded dimension growth and torsion free k0 group at the 1980 kingston conference effros posed the problem of finding suitable invariants for use in studying c algebras that are limits of sequences of homogeneous c algebras these are now called almost homogeneous ah c algebras the classification of ah algebras is a rapidly developing field and we will not attempt to summarize all this activity instead we will focus on the growth of the invariants used specifically we consider an ah algebra a that is the direct limit o f a system of the form

classification ofc algebras of real rank zero and unsuspendede equivalence types abstract in this article examples are given to prove that the graded scaled ordered k group is not the complete invariant for a c algebra in the class of unital separable nuclear c algebras of real rank zero and stable rank one even for a c algebra in the subclass which consists of those real rank zero stable rank one c algebras being expressed as inductive limits of k n i 1 m n i c x n i where x n i are two dimensional finite cw complexes and n i are positive integers in the case of simple such c algebras it has been proved that the above invariant is the complete invariant by george elliott and the author these examples prove that the classification conjecture of elliott for the case of non simple real rank zero c algebras should be revised one needs extra invariants the obstruction preventing two such c algebras with the same graded scaled ordered k group from being isomorphic is that they have different unsuspended e equivalence types a refinement of kk equivalence type of c algebras due to connes and higson in this article it is proved that for the above class of inductive limit c algebras the obstruction of unsuspended e equivalence type is the only obstruction i e if two c algebras in the class are unsuspended e equivalence then they are isomorphic it is a surprise that in the case of simple such c algebras or even the case of c algebras with finitely many ideals the obstruction will disappear see section 4

simple c algebras with perforation abstract simple c algebras are constructed for which the ordered k 0 group is not weakly unperforated the algebras are approximately homogeneous and do not have slow dimension growth

level set method for the inverse elliptic problem in nonlinear electromagnetism an inverse problem of inhomogeneity identification inside a nonlinear magnetic material from the local measurements of the magnetic induction is investigated the representation of the shape of the inhomogeneity and its evolution during an iterative reconstruction process is achieved by the level set method the reconstruction is realized by the minimization of a cost function using the steepest descent method the gradient directions are evaluated using the sensitivity equation and the adjoint variable method simulations has been performed showing the robustness of the algorithm and its ability to reconstruct single inhomogeneities convex and non convex as well as multiple inhomogeneities

dirac structures and boundary control systems associated with skew symmetric differential operators associated with a skew symmetric linear operator on the spatial domain a b we define a dirac structure which includes the port variables on the boundary of this spatial domain this dirac structure is a subspace of a hilbert space naturally associated with this dirac structure is an infinite dimensional system we parameterize the boundary port variables for which the c 0 semigroup associated with this system is contractive or unitary furthermore this parameterization is used to split the boundary port variables into inputs and outputs similarly we define a linear port controlled hamiltonian system associated with the previously defined dirac structure and a symmetric positive operator defining the energy of the system we illustrate this theory on the example of the timoshenko beam

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

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

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

approximations of c algebras and the ideal property we introduce several classes of c algebras using for this local approximations by nice c algebras that generalize the ah algebras among others we initiate their study proving mainly results about the ideal property but also about the ideals generated by their projections the real rank zero the weak projection property minimal tensor products extensions quasidiagonal extensions ideal structure the largest ideal with the ideal property and short exact sequences some of the previous results of the second named author are generalized 2007 elsevier inc all rights reserved

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

exponential stability of a class of boundary control systems we study a class of partial differential equations with variable coefficients on a one dimensional spatial domain with control and observation at the boundary for this class of systems we provide simple tools to check exponential stability this class is general enough to include models of flexible structures traveling waves heat exchangers and bioreactors among others the result is based on the use of a generating function the energy for physical systems and an inequality condition at the boundary furthermore based on the port hamiltonian approach we give a constructive method to reduce this inequality to a simple matrix inequality

exponential stabilization of port hamiltonian boundary control systems via energy shaping this article is concerned with the exponential stabilization of a class of linear boundary control systems bcss in port hamiltonian form through energy shaping starting from a first feedback loop that is in charge of modifying the hamiltonian function of the plant a second control loop that guarantees exponential convergence to the equilibrium is designed in this way a major limitation of standard energy shaping plus damping injection control laws applied to linear port hamiltonian bcss namely the fact that only asymptotic convergence is assured has been removed

incorporating topological derivatives into shape derivatives based level set methods shape derivatives and topological derivatives have been incorporated into level set methods to investigate shape optimization problems the shape derivative measures the sensitivity of boundary perturbations while the topological derivative measures the sensitivity of creating a small hole in the interior domain the combination of these two derivatives yields an efficient algorithm which has more flexibility in shape changing and may escape from a local optimal examples on finding the optimal shapes for maximal band gaps in photonic crystal and acoustic drum problems are demonstrated

on accuracy of approximate boundary and distributed h1 shape gradient flows for eigenvalue optimization abstract the boundary and distributed shape gradients of elliptic eigenvalues in shape optimization are approximated by the finite element method we show a priori error estimates for the two approximate shape gradients in h 1 shape gradient flows the convergence analysis shows that the volume integral formula converges faster and offers higher accuracy when the finite element method is used for discretization numerical results verify the theory for the dirichlet case shape optimization examples solved by algorithms illustrate the more effectiveness of distributed shape gradients for the dirichlet case for optimizing a neumann eigenvalue the boundary and volume h 1 flows have the same efficiency moreover we observe that the distributed h 1 shape gradient flow is more efficient than the boundary l 2 shape gradient flow in literature

on the classification of simple inductive limit c algebras ii the isomorphism theorem in this article it is proved that the invariant consisting of the scaled ordered k group and the space of tracial states together with the natural pairing between them is a complete invariant for the class of unital simple c algebras which can be expressed as the inductive limit of a sequence a 1 to a 2 to cdots to a n to cdots with a n bigoplus i 1 t n p n i m n i c x n i p n i where xn i is a compact metrizable space and pn i is a projection in m n i c xn i for each n and i and the spaces xn i are of uniformly bounded finite dimension note that the c algebras in the present class are not assumed to be of real rank zero as they were in eg2

a binary level set model and some applications to mumford shah image segmentation in this paper we propose a pde based level set method traditionally interfaces are represented by the zero level set of continuous level set functions instead we let the interfaces be represented by discontinuities of piecewise constant level set functions each level set function can at convergence only take two values i e it can only be 1 or 1 thus our method is related to phase field methods some of the properties of standard level set methods are preserved in the proposed method while others are not using this new method for interface problems we need to minimize a smooth convex functional under a quadratic constraint the level set functions are discontinuous at convergence but the minimization functional is smooth we show numerical results using the method for segmentation of digital images

image segmentation using some piecewise constant level set methods with mbo type of projection in this work we are trying to propose fast algorithms for mumford shah image segmentation using some recently proposed piecewise constant level set methods pclsm two variants of the pclsm will be considered in this work the first variant which we call the binary level set method needs a level set function which only takes values 1 to identify the regions the second variant only needs to use one piecewise constant level set function to identify arbitrary number of regions for the mumford shah image segmentation model with these new level set methods one needs to minimize some smooth energy functionals under some constrains a penalty method will be used to deal with the constraint aos additive operator splitting and mos multiplicative operator splitting schemes will be used to solve the euler lagrange equations for the minimization problems by doing this we obtain some algorithms which are essentially applying the mbo scheme for our segmentation models advantages and disadvantages are discussed for the proposed schemes

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

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

repetitive control system a new type servo system for periodic exogenous signals a control scheme called repetitive control is proposed in which the controlled variables follow periodic reference commands a high accuracy asymptotic tracking property is achieved by implementing a model that generates the periodic signals of period l into the closed loop system sufficient conditions for the stability of repetitive control systems and modified repetitive control systems are derived by applying the small gain theorem and the stability theorem for time lag systems synthesis algorithms are presented by both the state space approach and the factorization approach in the former approach the technique of the kalman filter and perfect regulation is utilized while coprime factorization over the matrix ring of proper stable rational functions and the solution of the hankel norm approximation are used in the latter one

dissipativity based boundary control of linear distributed port hamiltonian systems the main contribution of this paper is a general synthesis methodology of exponentially stabilising control laws for a class of boundary control systems in port hamiltonian form that are dissipative with respect to a quadratic supply rate being the total energy the storage function more precisely general conditions that a linear regulator has to satisfy to have at first a well posed and secondly an exponentially stable closed loop system are presented the methodology is illustrated with reference to two specific stabilisation scenarios namely when the distributed parameter plant is in impedance or in scattering form moreover it is also shown how these techniques can be employed in the analysis of more general systems that are described by coupled partial and ordinary differential equations in particular the repetitive control scheme is studied and conditions on the finite dimensional linear plant to have asymptotic tracking of generic periodic reference signals are determined

finite element approximation to the extremal eigenvalue problem for inhomogenous materials in this paper an extremal eigenvalue problem corresponding to an inhomogeneous membrane which is composed of two different materials with different densities is investigated the convergence of the finite element discretization and the error order for the smallest eigenvalue are obtained a monotonic decreasing algorithm is presented to solve the discretized problem and numerical examples are given to demonstrate the error estimation as well as the efficiency of the method

a way of updating the density function for the design of the drum abstract designing an acoustic drum can be categorized into a class of eigenvalue optimization problems in the structural engineering area in this paper we propose an algorithm that is based on the g teaux derivative of the objective function with respect to the density functions and analyze our algorithm in detail in the algorithm we deal with the geometry constraint by exchanging the densities of two domains occupied by two kinds of different materials finally we apply this algorithm to some practical examples frequently used by the researchers and present some numerical results to show its feasibility stability and efficiency

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

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

tensor products of c algebras with the ideal property abstract combining a construction of dadarlat of a unital simple non exact c algebra c of real rank zero and stable rank one which is shape equivalent to a uhf algebra with results of kirchberg and a result obtained by dadarlat and the first named author we show that b h c contains an ideal that is not generated by its projections we also find a unital separable sub c algebra a of b h such that a is of real rank zero and a c has an ideal that is not generated by its projections

convergence analysis of mixed finite element approximations to shape gradients in the stokes equation abstract eulerian derivatives of shape functionals in shape optimization can be written in two formulations of boundary and volume integrals the former is widely used in shape gradient descent algorithms the latter holds more generally although rarely being used numerically in literature for shape functionals governed by the stokes equation we consider the mixed finite element approximations to the two types of shape gradients from corresponding eulerian derivatives the standard mini and taylor hood elements are employed to discretize the state equation its adjoint and the resulting shape gradients we present thorough convergence analysis with a priori error estimates for the two approximate shape gradients the theoretical analysis shows that the volume integral formula has superconvergence property numerical results are presented to verify the theory and show that the volume formulation is more accurate

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

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

a level set method for shape optimization in semilinear elliptic problems abstract we develop a finite element based level set method for numerically solve shape optimization problems constrained by semilinear elliptic problems by combining the shape sensitivity analysis and level set method a gradient descent algorithm is proposed to solve the model problem different from solving the nonlinear hamilton jacobi equations with finite differences in traditional level set methods we solve the linear convection equation and reinitialization equation using the characteristic galerkin finite element method the methodology can handle topology as well as shape changes in both regular and irregular design regions numerical results are presented to demonstrate the effectiveness of our algorithm as well as to verify symmetry preserving and breaking properties of optimal subdomains

a phase field method for shape optimization of incompressible flows abstract in this paper we present a phase field method applied to the fluid based shape optimization the fluid flow is governed by the incompressible navier stokes equations a phase field variable is used to represent material distributions and the optimized shape of the fluid is obtained by minimizing the certain objective functional regularized the shape sensitivity analysis is presented in terms of phase field variable which is the main contribution of this paper it saves considerable amount of computational expense when the meshes are locally refined near the interfaces compared to the case of fixed meshes numerical results on some benchmark problems are reported and it is shown that the phase field approach for fluid shape optimization is efficient and robust

a simple mesh generator in matlab creating a mesh is the first step in a wide range of applications including scientific computing and computer graphics an unstructured simplex mesh requires a choice of meshpoints vertex nodes

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

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

sequential and parallel splitting methods for bilinear control problems in hilbert spaces an augmented lagrangian method with second order update is developed and its relationship to the sequential quadratic programming method is described the rate of convergence proof depends on a second order sufficient optimality condition which is shown to be satisfied for a class of nonlinear optimal control problems of tracking type numerical examples are included which demonstrate the globalizing effect of the augmented lagrangian method

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 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

a complete invariant foradalgebras with real rank zero and bounded torsion ink1 abstract for each integerp 1 we consider an algebraic invariant forc algebras the invariant consists ofk0 k1 thek0 group with z pcoefficients the order structures these groups possess and the natural maps between the three groups we prove that this invariant is complete for the class ofadalgebras of real rank zero ifpannihilates every torsion element ofk1 theadalgebras arec algebras which are inductive limits of finite direct sums ofc algebras of continuous matrix valued functions over the circle or over the interval where in the interval case we require that the value over the endpoints is a scalar multiple of the unit matrix examples show that the condition on the torsion ink1is necessary when only one modp k0 group is considered applying a theorem of d d rlat the result also applies to a subclass of theahalgebras

classification of simple c algebras inductive limits of matrix algebras over one dimensional spaces abstract in this article we will give a complete classification of simple c algebras which can be written as inductive limits of algebras of the form a n i 1 k n m n i c x n i where x n i are arbitrary variable one dimensional compact metrizable spaces the results unify and generalize the previous results for the case x n i s 1 and for the case of x n i being trees we obtain our classification results by reducing the case of general one dimensional spaces to the case of circles the techniques in this paper play important roles in the study of the case of higher dimensional spaces

exponential stabilization of boundary controlled port hamiltonian systems with dynamic feedback it is shown that a strictly input passive linear finite dimensional controller exponentially stabilizes a large class of partial differential equations actuated at the boundary of a one dimensional spatial domain this follows since the controller imposes exponential dissipation of the total energy the result can by use for control synthesis and for the stability analysis of complex systems modeled by sets of coupled pde s and ode s the result is specialized to port hamiltonian control systems and a simplified dna manipulation process is used to illustrate the result

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

a variant of the level set method and applications to image segmentation in this paper we propose a variant of the level set formulation for identifying curves separating regions into different phases in classical level set approaches the sign of n level set functions are utilized to identify up to 2 phases the novelty in our approach is to introduce a piecewise constant level set function and use each constant value to represent a unique phase if 2 phases should be identified the level set function must approach 2 predetermined constants we just need one level set function to represent 2 unique phases and this gains in storage capacity further the reinitializing procedure requested in classical level set methods is superfluous using our approach the minimization functional for our approach is locally convex and differentiable and thus avoids some of the problems with the nondifferentiability of the delta and heaviside functions numerical examples are given and we also compare our method with related approaches

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

on inductive limits of matrix algebras over the two torus it will be shown in this paper that certain real rank zero c algebras which are inductive limits of c algebras of the form i m k i c inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i can be expressed as inductive limits of c algebras of the form i m k i c s 1 in particular if both a and b are of real rank zero and are inductive limits of c algebras of the form i m k i c s 1 then also a b is an inductive limit of c algebras of the form i m k i c s 1 hence a b can be classified by its k theory this is a key step in the general classification theory of inductive limit c algebras

a second order shape optimization approach for image segmentation the problem of segmentation of a given image using the active contour technique is considered an abstract calculus to find appropriate speed functions for active contour models in image segmentation or related problems based on variational principles is presented the speed method from shape sensitivity analysis is used to derive speed functions which correspond to gradient or newton type directions for the underlying optimization problem the newton type speed function is found by solving an elliptic problem on the current active contour in every time step numerical experiments comparing the classical gradient method with newton s method are presented

on casimir functionals for infinite dimensional port hamiltonian control systems we consider infinite dimensional port hamiltonian systems with respect to control issues in contrast to the well established representation relying on stokes dirac structures that are based on skew adjoint differential operators and the use of energy variables we employ a different port hamiltonian framework based on this system representation conditions for casimir functionals will be derived where in this context the variational derivative plays an extraordinary role furthermore the coupling of finite and infinite dimensional systems will be analyzed in the spirit of the control by interconnection problem

at structure of ah algebras with the ideal property and torsion free k theory abstract let a be an ah algebra that is a is the inductive limit c algebra of a 1 1 2 a 2 2 3 a 3 a n with a n i 1 t n p n i m n i c x n i p n i where x n i are compact metric spaces t n and n i are positive integers and p n i m n i c x n i are projections suppose that a has the ideal property each closed two sided ideal of a is generated by the projections inside the ideal as a closed two sided ideal suppose that sup n i dim x n i this condition can be relaxed to a certain condition called very slow dimension growth in this article we prove that if we further assume that k a is torsion free then a is an approximate circle algebra or an a t algebra that is a can be written as the inductive limit of b 1 b 2 b n where b n i 1 s n m n i c s 1 one of the main technical results of this article called the decomposition theorem is proved for the general case i e without the assumption that k a is torsion free this decomposition theorem will play an essential role in the proof of a general reduction theorem where the condition that k a is torsion free is dropped in the subsequent paper gong et al preprint 31 of course in that case in addition to space s 1 we will also need the spaces t ii k t iii k and s 2 as in gong 2002 29