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

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

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

a hybrid particle level set method for improved interface capturing in this paper we propose a new numerical method for improving the mass conservation properties of the level set method when the interface is passively advected in a flow field our method uses lagrangian marker particles to rebuild the level set in regions which are underresolved this is often the case for flows undergoing stretching and tearing the overall method maintains a smooth geometrical description of the interface and the implementation simplicity characteristic of the level set method our method compares favorably with volume of fluid methods in the conservation of mass and purely lagrangian schemes for interface resolution the method is presented in three spatial dimensions

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

a reduced multiscale model for nonlinear structural topology optimization abstract this paper presents a reduced multiscale model for macroscopic structural design considering microscopic material nonlinear microstructures this work introduces reduced order model rom to alleviate the heavy computational demand of nonlinear nested multiscale procedures particularly in an optimization framework which requires multiple loops involving similar computations the surrogate model constructed using proper orthogonal decomposition pod and diffuse approximation reduces the computational effort for solving the microscopic boundary value problems multiscale analysis model fe 2 is applied to link structure and microstructures in the two scales maximum stiffness design of the macroscopic structure is realized using a discrete level set topology optimization model it is shown by means of numerical tests that the reduced multiscale model provides reasonable designs as compared to those obtained by the unreduced model while with a significantly reduced computational effort

stochastic algorithms for inverse problems involving pdes and many measurements inverse problems involving systems of partial differential equations pdes can be very expensive to solve numerically this is so especially when many experiments involving different combinations of sources and receivers are employed in order to obtain reconstructions of acceptable quality the mere evaluation of a misfit function the distance between predicted and observed data often requires hundreds and thousands of pde solves this article develops and assesses dimensionality reduction methods both stochastic and deterministic to reduce this computational burden we assume that all experiments share the same set of receivers and concentrate on methods for reducing the number of combinations of experiments called simultaneous sources that are used at each stabilized gauss newton iteration algorithms for controlling the number of such combined sources are proposed and justified evaluating the misfit approximately except for the final verification for terminating the process always involves

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

a meshfree level set method for topological shape optimization of compliant multiphysics actuators abstract this paper proposes a topology optimization method for compliant multiphysics actuators of geometrically nonlinear structures using meshfree galerkin weak forms and level set methods the design boundary is implicitly represented as the zero level set of a higher dimensional level set function leading to a level set model capable of handling complex shape and topological changes with flexibilities a family of compactly supported radial basis functions csrbfs is firstly used to interpolate the level set function of lipschitz continuity and then augmented to construct the shape function for meshless approximation by satisfying basic requirements in particular the predetermined consistency and the kronecker delta function property a meshless galerkin method mgm with global weak forms is established to implement the discretization of the state equations the design of actuators is transformed into an easier size optimization from a more difficult shape and topology optimization the design boundary evolution is just a question of advancing the discrete level set function in time by updating the design variables of the size optimization compared to most conventional level set methods the proposed meshless level set method is able to implement the free moving boundary discontinuities without remeshing and unify two different numerical procedures in propagating the discrete level set function e g eulerian grid and approximating the state equation e g lagrangian mesh respectively this method can also avoid numerical difficulties in solving a series of complicate hamilton jacobi partial differential equations pdes with explicit time schemes two typical numerical examples are used to demonstrate the effectiveness of the proposed method

inversion of 3d electromagnetic data in frequency and time domain using an inexact all at once approach we present a general formulation for inverting frequencyor time domain electromagnetic data using an all at once approach in this methodology the forward modeling equations are incorporated as constraints and thus we need to solve a constrained optimization problem where the parameters are the electromagnetic fields the conductivity model and a set of lagrange multipliers this leads to a much larger problem than the traditional unconstrained formulation where only the conductivities are sought nevertheless experience shows that the constrained problem can be solved faster than the unconstrained one the primary reasons are that the forward problem does not have to be solved exactly until the very end of the optimization process and that permitting the fields to be away from their constrained values in the initial stages introduces flexibility so that a stationary point of the objective function is found more quickly in this paper we outline the all atonce approach and apply it to electromagnetic problems in both frequency and time domains this is facilitated by a unified representation for forward modeling for these two types of data the optimization problem is solved by finding a stationary point of the lagrangian numerically this leads to a nonlinear system that is solved iteratively using a gauss newton strategy at each iteration a large indefinite matrix is inverted and we discuss how this can be accomplished as a test we invert frequency domain synthetic data from a grounded electrode system that emulates a field csamt survey for the time domain we invert borehole data obtained from a current loop on the surface

on effective methods for implicit piecewise smooth surface recovery this paper considers the problem of reconstructing a piecewise smooth model function from given measured data the data are compared to a field which is given as a possibly nonlinear function of the model a regularization functional is added which incorporates the a priori knowledge that the model function is piecewise smooth and may contain jump discontinuities regularization operators related to total variation tv are therefore employed two popular methods are modified tv and huber s function both contain a parameter which must be selected the huber variant provides a more natural approach for selecting its parameter and we use this to propose a scheme for both methods our selected parameter depends both on the resolution and on the model average roughness thus it is determined adaptively its variation from one iteration to the next yields additional information about the progress of the regularization process the modified tv operator has a smoother generating function nonetheless we obtain a huber variant with comparable and occasionally better performance for large problems e g high resolution the resulting reconstruction algorithms can be tediously slow we propose two mechanisms to improve efficiency the first is a multilevel continuation approach aimed mainly at obtaining a cheap yet good estimate for the regularization parameter and the solution the second is a special multigrid preconditioner for the conjugate gradient algorithm used to solve the linearized systems encountered in the procedures for recovering the model function

analytical solution of a generalized penna model in 1995 penna introduced a simple model of biological aging a modified penna model has been demonstrated to exhibit behavior of real life systems including catastrophic senescence in salmon and a mortality plateau at advanced ages we present a general steady state analytic solution to the penna model which is able to deal with arbitrary birth and survivability functions this solution is employed to solve standard variant penna models studied by simulation different verhulst factors regulating both the birth rate and external death rate are considered

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

on the similarity classes among products of m nonsingular matrices in various orders abstract it is shown that for any k 1 2 m 1 there exist m invertible complex matrices such that among the m products a a 1 a 2 a m s m exactly k different similarity classes occur the cases in which the matrices a i are upper triangular or are 2 by 2 are considered in detail in the former case it is shown that any m 1 unispectral matrices with common eigenvalue may be found among the a and in the latter case it is shown explicitly how to achieve m 1 and m 1 2 similarity classes as well as any number from 1 to 6 when m 4 other particular results are given as well as a discussion of further natural questions

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

structural shape and topology optimization using a meshless galerkin level set method summary this paper aims to propose a meshless galerkin level set method for shape and topology optimization of continuum structures to take advantage of the implicit free boundary representation scheme the design boundary is represented as the zero level set of a scalar level set function to flexibly handle complex shape fidelity and topology changes by maintaining concise and smooth interface compactly supported radial basis functions csrbfs are used to parameterize the level set function and construct the shape functions for meshfree approximations based on a set of unstructured field nodes the meshless galerkin method with global weak form is used to implement the discretization of the state equations this provides a pathway to unify the two different numerical stages in most conventional level set methods 1 the propagation of discrete level set function on a set of eulerian grid and 2 the approximation of discrete equations on a set of lagrangian mesh the original more difficult shape and topology optimization based on the level set equation is transformed into a relatively easier size optimization to which many efficient optimization algorithms can be applied the proposed level set method can describe the moving boundaries without remeshing for discontinuities the motion of the free boundary is just a question of advancing the discrete level set function in time by solving the size optimization several benchmark examples are used to demonstrate the effectiveness of the proposed method the numerical results show that the proposed method can simplify numerical process and avoid numerical difficulties involved in most conventional level set methods it is straightforward to apply the proposed method to more advanced shape and topology optimization problems copyright 2011 john wiley sons ltd

the coc algorithm part ii topological geometrical and generalized shape optimization abstract after outlining analytical methods for layout optimization and illustrating them with examples the coc algorithm is applied to the simultaneous optimization of the topology and geometry of trusses with many thousand potential members the numerical results obtained are shown to be in close agreement up to twelve significant digits with analytical results finally the problem of generalized shape optimization finding the best boundary topology and shape is discussed

velocity extension for the level set method and multiple eigenvalues in shape optimization in the context of structural optimization by the level set method we propose an extension of the velocity of the underlying hamilton jacobi equation the gradient method is endowed with a hilbertian structure based on the h1 sobolev space numerical results for compliance minimization and mechanism design show a strong improvement of the rate of convergence of the level set method another important application is the optimization of multiple eigenvalues

analysis of regularization methods for the solution of ill posed problems involving discontinuous operators we consider a regularization concept for the solution of ill posed operator equations where the operator is composed of a continuous and a discontinuous operator a particular application is level set regularization where we develop a novel concept of minimizers the proposed level set regularization is capable of handling changing topologies a functional analytic framework explaining the splitting of topologies is given the asymptotic limit of the level set regularization method is an evolution process which is implemented numerically and the quality of the proposed algorithm is demonstrated by solving an inverse source problem

contragredient equivalence a canonical form and some applications abstract let a and c be m by n complex matrices and let b and d be n by m complex matrices the pair a b is contragrediently equivalent to the pair c d if there are square nonsingular complex matrices x and y such that xay 1 c and ybx 1 d contragredient equivalence is a common generalization of four basic equivalence relations similarity consimilarity complex orthogonal equivalence and unitary equivalence we develop a complete set of invariants and an explicit canonical form for contragredient equivalence and show that a a t is contragrediently equivalent to c c t if and only if there are complex orthogonal matrices p and q such that c paq using this result we show that the following are equivalent for a given n by n complex matrix a 1 1 a qs for some complex orthogonal q and some complex symmetric s 2 2 a t a is similar to aa t 3 3 a a t is contragrediently equivalent to a t a 4 4 a q 1 a t q 2 for some complex orthogonal q 1 q 2 5 5 a pa t p for some complex orthogonal p we then consider a linear operator on n by n complex matrices that shares the following properties with transpose operators for every pair of n by n complex matrices a and b a preserves the spectrum of a b a a and c ab b a we show that a a is contragrediently similar to b b if and only if a x 1 bx 2 for some nonsingular x 1 x 2 that satisfy x 1 1 x 1 and x 1 2 x 2 we also consider a factorization of the form a xy where x 1 x and y y we use the canonical form for the contragredient equivalence relation to give a new proof of a theorem of flanders concerning the relative sizes of the nilpotent jordan blocks of ab and ba we present a sufficient condition for the existence of square roots of ab and ba and close with a canonical form for complex orthogonal equivalence

a new algorithm for topology optimization using a level set method the level set method has been recently introduced in the field of shape optimization enabling a smooth representation of the boundaries on a fixed mesh and therefore leading to fast numerical algorithms however most of these algorithms use a hamilton jacobi equation to connect the evolution of the level set function with the deformation of the contours and consequently they can hardly create new holes in the domain at least in 2d in this work we propose an evolution equation for the level set function based on a generalization of the concept of topological gradient this results in a new algorithm allowing for all kinds of topology changes

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

shape optimization with topological changes and parametric control recent advances in shape optimization rely on free form implicit representations such as level sets to support boundary deformations and topological changes by contrast parametric shape optimization is formulated directly in terms of meaningful geometric design variables but usually does not support free form boundary and topological changes we propose a novel approach to shape optimization that combines and retains the advantages of the earlier optimization techniques the shapes in the design space are represented implicitly as level sets of a higher dimensional function that is constructed using b splines to allow free form deformations and parameterized primitives combined with r functions to support desired parametric changes our approach to shape design and optimization offers great flexibility because it provides explicit parametric control of geometry and topology within a large space of free form shapes the resulting method is also general in that it subsumes most other types of shape optimization as special cases we describe an implementation of the proposed technique with attractive numerical properties the explicit construction of an implicit representation supports straightforward sensitivity analysis that can be used with most gradient based optimization methods furthermore our implementation does not require any error prone polygonization or approximation of level sets isocurves and isosurfaces the effectiveness of the method is demonstrated by several numerical examples

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

parameter estimation in flow through partially saturated porous materials a class of numerical simulators were developed and critically evaluated to be incorporated as the solver of a forward problem in the framework of an inverse modeling strategy the strategy couples a mass lumped galerkin linear finite element solution of the mixed form richards equation with an experimental time space series and the osborne more revised form of the levenberg marquardt algorithm to retrieve hydraulic parameters of a partially saturated porous medium the numerical simulator shows excellent agreement with a reference solution obtained on a dense grid and infinitesimal time step in terms of fluid pressure head fluid content and fluid volumetric flux density and perfectly conserves the global mass an adaptive algorithm was implemented to estimate sensitivity matrix in the inverse algorithm a multi criterion stopping rule was developed and successfully implemented to end the inverse code at the solution the result of the optimization was compared with a large scale in situ soil moisture space time series measured during the course of a drainage experiment and excellent agreements were found analysis of the parameter response surfaces and hyper space plots closeness of the gradient of the penalty function at minimum to zero and positive definiteness of the approximation for the hessian at the solution eigs h 0 indicate that the obtained solution is a strong local minimum a state of the art sensitivity analysis carried out to quantify sensitivity of the state variable with respect to uncertainty and changes in different model parameters

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

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

structural topology and shape optimization using a level set method with distance suppression scheme abstract in level set methods for structural topology and shape optimization the level set function gradients at the design interface need to be controlled in order to ensure stability of the optimization process one popular way to do this is to enforce the level set function to be a signed distance function by periodically using initialization schemes which is commonly known as re initialization however such re initialization schemes are time consuming as additional partial differential equations need to be solved in every iteration step furthermore the use of re initialization brings some undesirable problems for example it may move the zero level set away from the expected position this paper presents a level set method with distance suppression scheme for structural topology and shape optimization an energy functional is introduced into the level set equation to maintain the level set function to close to a signed distance function near the structural boundaries meanwhile forcing the level set function to be a constant at locations far away from the structural boundaries as a result the present method not only can avoid the need for re initialization but also can simplify the setting of the initial level set function the validity of the proposed method is tested on the mean compliance minimization problem and the compliant mechanisms synthesis problem different aspects of the proposed method are demonstrated on a number of benchmarks from the literature of structural optimization

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

explicit jordan form for certain block triangular matrices ii abstract let a mn b mm and c be given for x mn m we seek to determine the jordan structure of m a x o b associated with in terms of the jordan structure of a and b associated with and properties of x the principal result is an explicit prescription for the jordan structure of m associated with in the case that the geometric multiplicity of in b is one

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

a multilevel method for image registration in this paper we introduce a new framework for image registration our formulation is based on consistent discretization of the optimization problem coupled with a multigrid solution of the linear system which evolves in a gauss newton iteration we show that our discretization is h elliptic independent of parameter choice and therefore a simple multigrid implementation can be used to overcome potential large nonlinearities and to further speed up computation we use a multilevel continuation technique we demonstrate the efficiency of our method on a realistic highly nonlinear registration problem

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

fast simulation of 3d electromagnetic problems using potentials we consider solving three dimensional electromagnetic problems in parameter regimes where the quasi static approximation applies the permeability is constant the conductivity may vary significantly and the range of frequencies is moderate the difficulties encountered include handling solution discontinuities across interfaces and accelerating convergence of traditional iterative methods for the solution of the linear systems of algebraic equations that arise when discretizing maxwell s equations in the frequency domain we use a potential current formulation a with a coulomb gauge the potentials a and decompose the electric field e into components in the active and null spaces of the operator we develop a finite volume discretization on a staggered grid that naturally employs harmonic averages for the conductivity at cell faces after discretization we eliminate the current and the resulting large sparse linear system of equations has a block structure that is diagonally dominant allowing an efficient solution with preconditioned krylov space methods a particularly efficient algorithm results from the combination of bicgstab and an incomplete lu decomposition we demonstrate the efficacy of our method in several numerical experiments

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

level set topology optimization for mechanical metamaterials under hybrid uncertainties abstract this paper proposes a level set based robust topology optimization rto method for computational design of metamaterials under hybrid uncertainties e g auxetics with negative poisson s ratio where the young s modulus of the solid is described as a random variable while the poisson s ratio is regarded as an interval variable firstly the robust objective function is formulated by a combination of interval mean and interval variance of the deterministic objective function secondly the interval mean and interval variance are computed by a hybrid uncertain analysis approach termed as polynomial chaos chebyshev interval pcci method thirdly the design sensitivities of the robust objective function are obtained after the implementation of the pcci method finally a powerful parametric level set method plsm in conjunction with the numerical homogenization method is applied to achieve the robust topological design for the auxetic microstructure several numerical cases are used to demonstrate the effectiveness of the proposed method for the robust topology optimization problems this method is non intrusive and general and can be easily extended to a range of design problems of micro structured metamaterials

order invariant spectral properties for several matrices abstract the collections of m n by n matrices over a field such that the products in any of the m orders share a common similarity class resp spectrum trace are studied the spectral and trace order invariant properties are characterized and the similarity invariant one is related to them in several cases a complete explicit description is given in case m 3 and n 2

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

topological sensitivity analysis the so called topological derivative concept has been seen as a powerful framework to obtain the optimal topology for several engineering problems this derivative characterizes the sensitivity of the problem when a small hole is created at each point of the domain however the greatest limitation of this methodology is that when a hole is created it is impossible to build a homeomorphic map between the domains in study because they have not the same topology therefore some specific mathematical framework should be developed in order to obtain the derivatives this work proposes an alternative way to compute the topological derivative based on the shape sensitivity analysis concepts the main feature of this methodology is that all the mathematical procedure already developed in the context of shape sensitivity analysis may be used in the calculus of the topological derivative this idea leads to a more simple and constructive formulation than the ones found in the literature further to point out the straightforward use of the proposed methodology it is applied for solving some design problems in steady state heat conduction

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

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

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

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 new level set method for systematic design of hinge free compliant mechanisms abstract this paper presents a new level set based method to realize shape and topology optimization of hinge free compliant mechanisms a quadratic energy functional used in image processing applications is introduced in the level set method to control the geometric width of structural components in the created mechanism a semi implicit scheme with an additive operator splitting aos algorithm is employed to solve the hamilton jacobi partial differential equation pde in the level set method the design of compliant mechanisms is mathematically represented as a general non linear programming with a new objective function augmented by the high order energy term the structural optimization is thus changed to a numerical process that describes the design as a sequence of motions by updating the implicit boundaries until the optimized structure is achieved under specified constraints in doing so it is expected that numerical difficulties such as the courant friedrichs lewy cfl condition and periodically applied re initialization procedures in most conventional level set methods can be eliminated in addition new holes can be created inside the design domain the final mechanism configurations consist of strip like members suitable for generating distributed compliance and solving the de facto hinge problem in the design of compliant mechanisms two widely studied numerical examples are studied to demonstrate the effectiveness of the proposed method in the context of designing distributed compliant mechanisms

an efficient interface preserving level set redistancing algorithm and its application to interfacial incompressible fluid flow in sussman smereka and osher j comp phys 94 1994 pp 146 159 a numerical scheme was presented for computing incompressible air water flows using the level set method crucial to the above method was a new iteration method for maintaining the level set function as the signed distance from the zero level set in this paper we implement a constraint along with higher order difference schemes in order to make the iteration method more accurate and efficient accuracy is measured in terms of the new computed signed distance function and the original level set function having the same zero level set we apply our redistancing scheme to incompressible flows with noticeably better resolved results at reduced cost we validate our results with experiment and theory we show that our distance level set scheme with the added constraint competes well with available interface tracking schemes for basic advection of an interface we perform basic accuracy checks and more stringent tests involving complicated interfacial structures as with all level set schemes our method is easy to implement

efficient image segmentation using partial differential equations and morphology the goal of this paper is to investigate segmentation methods that combine fast preprocessing algorithms using partial differential equations pdes with a watershed transformation with region merging we consider two well founded pde methods a nonlinear isotropic diffusion filter that permits edge enhancement and a convex nonquadratic variational image restoration method which gives good denoising for the diffusion filter an efficient algorithm is applied using an additive operator splitting aos that leads to recursive and separable filters for the variational restoration method a novel algorithm is developed that uses aos schemes within a gaussian pyramid decomposition examples demonstrate that preprocessing by these pde techniques significantly improves the watershed segmentation and that the resulting segmentation method gives better results than some traditional techniques the algorithm has linear complexity and it can be used for arbitrary dimensional data sets the typical cpu time for segmenting a 256 image on a modern pc is far below one second

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 level set method for vibration and multiple loads structural optimization abstract we extend the level set method for shape and topology optimization to new objective functions such as eigenfrequencies and multiple loads this method is based on a combination of the classical shape derivative and of the osher sethian level set algorithm for front propagation in two and three space dimensions we maximize the first eigenfrequency or we minimize a weighted sum of compliances associated to different loading configurations the shape derivative is used as an advection velocity in a hamilton jacobi equation for changing the shape this level set method is a low cost shape capturing algorithm working on a fixed eulerian mesh and it can easily handle topology changes

gompertz mortality law and scaling behavior of the penna model the penna model is a model of evolutionary ageing through mutation accumulation where traditionally time and the age of an organism are treated as discrete variables and an organism s genome is represented by a binary bit string we reformulate the asexual penna model and show that a universal scale invariance emerges as we increase the number of discrete genome bits to the limit of a continuum the continuum model introduced by almeida and thomas int j mod phys c 11 1209 2000 can be recovered from the discrete model in the limit of infinite bits coupled with a vanishing mutation rate per bit finally we show that scale invariant properties may lead to the ubiquitous gompertz law for mortality rates for early ages which is generally regarded as being empirical

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

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

flandersʼ theorem for many matrices under commutativity assumptions we analyze the relationship between the jordan canonical form of products in different orders of k square matrices a 1 a k our results extend some classical results by h flanders motivated by a generalization of fiedler matrices we study permuted products of a 1 a k under the assumption that the graph of non commutativity relations of a 1 a k is a forest under this condition we show that the jordan structure of all nonzero eigenvalues is the same for all permuted products for the eigenvalue zero we obtain an upper bound on the dierence between the sizes of jordan blocks for any two permuted products and we show that this bound is attainable for k 3 we show that moreover the bound is exhaustive

electrical impedance tomography using level set representation and total variational regularization in this paper we propose a numerical scheme for the identification of piecewise constant conductivity coefficient for a problem arising from electrical impedance tomography the key feature of the scheme is the use of level set method for the representation of interface between domains with different values of coefficients numerical tests show that our method can recover sharp interfaces and can tolerate a relatively high level of noise in the observation data results concerning the effects of number of measurements noise level in the data as well as the regularization parameters on the accuracy of the scheme are also given

numerical solution of the penna model of biological aging with age modified mutation rate in this paper we present results of numerical calculation of the penna bit string model of biological aging modified for the case of a dependent mutation rate m a where a is the parent s age the mutation rate m a is the probability per bit of an extra bad mutation introduced in offspring inherited genome we assume that m a increases with age a as compared with the reference case of the standard penna model based on a constant mutation rate m the dynamics of the population growth shows distinct changes in age distribution of the population here we concentrate on mortality q a a fraction of items eliminated from the population when we go from age a to a 1 in simulated transition from time t to next time t 1 the experimentally observed q a dependence essentially follows the gompertz exponential law for a above the minimum reproduction age deviation from the gompertz law is however observed for the very old items close to the maximal age this effect may also result from an increase in mutation rate m with age a discussed in this paper the numerical calculations are based on analytical solution of the penna model presented in a series of papers by coe et al j b coe y mao and m e cates phys rev lett 89 288103 2002 results of the numerical calculations are supported by the data obtained from computer simulation based on the solution by coe et al

a level set approach for topology optimization with local stress constraints summary the purpose of this work is to present a level set based approach for the structural topology optimization problem of mass minimization submitted to local stress constraints the main contributions are threefold first the inclusion of local stress constraints by means of an augmented lagrangian approach within the level set context second the proposition of a constraint procedure that accounts for a continuous activation deactivation of a finite number of local stress constraints during the optimization sequence finally the proposition of a logarithmic scaling of the level set normal velocity as an additional regularization technique in order to improve the minimization sequence a set of benchmark tests in two dimensions achieving successful numerical results assesses the good behavior of the proposed method in these examples it is verified that the algorithm is able to identify stress concentrations and drive the design to a feasible local minimum copyright 2014 john wiley sons ltd