computational design and additive manufacturing of periodic conformal metasurfaces by synthesizing topology optimization with conformal mapping abstract in this paper we present a computational framework for computational design and additive manufacturing of free form periodic metasurfaces the proposed scheme rests on the level set based topology approach and the conformal mapping theory a metamaterial with pre specified performance is created using a level set based topology optimization method the achieved unit cell is further mapped to the 3d quad meshes on a free form surface by applying the conformal mapping method which can preserve the local shape and angle during the mapping with embedded geometric information the proposed level set based optimization methods not only can act as a motivator for design synthesis but also can be seamlessly hooked with additive manufacturing without the need of cad reconstructions the current computational framework provides a solution to increasing applications involving innovative metamaterial designs on free form surfaces for different fields of interests the performance of the proposed scheme is illustrated through two benchmark examples where a negative poisson s ratio unit cell pattern and a stiff and light inner structure are mapped to 3d free form surfaces and fabricated through additive manufacturing

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

topology optimization with closed b splines and boolean operations abstract in this work free form curves of closed b splines cbs are introduced as basic design primitives to realize structural topology optimization in the way of shape optimization unlike the existing density and level set methods the cbs takes full advantage of its parametric form for the design deformability and flexibility with a few number of design variables and its implicit form for boolean operations of the structural topology accounting for any overlapping merging and separation of different cbs no matter how geometric complexities of the cbs and the design domain are the cbs is implemented within the computing framework of finite cell method fcm so that structural analysis and topology optimization can efficiently be carried out using the fixed mesh typical examples are tested to demonstrate the effectiveness and generality of the proposed method in dealing with engineering design problems

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

cbs based topology optimization including design dependent body loads abstract in this paper the inherent problem of the so called parasitic effect of low density region caused by design dependent loads is investigated for topology optimization a cbs closed b splines based method is developed to solve efficiently the problem in the way of shape optimization compared to the standard density method design variables are unattached to the finite element model and defined by control parameters dominating the parametric equation of the cbs as a result design dependent loads associated with the material layout of a structure are made change indirectly by the boundary variation of the structure to favour structural reanalyses and sensitivity analysis the implicit form of the cbs i e level set function lsf is used in conjunction with the fixed computing grid of finite cell method fcm a variety of design dependent body loads is considered for the mathematical formulation of the cbs based method typical examples are given to illustrate that the cbs based method is free of low density regions in the optimized topology owing to the proper definitions of design variables

explicit feature control in structural topology optimization via level set method abstract the present paper aims to address a long standing and challenging problem in structural topology optimization explicit feature control of the optimal topology the basic idea is to resort to the level set solution framework and impose constraints on the extreme values of the signed distance level set function used for describing the topology of the structure numerical examples are also presented and discussed to illustrate the effectiveness of the proposed approach

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

a shape topological control problem for nonlinear crack defect interaction the antiplane variational model we consider the shape topological control of a singularly perturbed variational inequality the geometry dependent state problem that we address in this paper concerns a heterogeneous medium with a micro object defect and a macro object crack modeled in two dimensions the corresponding nonlinear optimization problem subject to inequality constraints at the crack is considered within a general variational framework for the reason of asymptotic analysis singular perturbation theory is applied resulting in the topological sensitivity of an objective function representing the release rate of the strain energy in the vicinity of the nonlinear crack the antiplane strain energy release rate is expressed by means of the mode iii stress intensity factor that is examined with respect to small defects such as microcracks holes and inclusions of varying stiffness the result of shape topological control is useful either for arrests or rise of crack growth

optimal fine scale structures in compliance minimization for a uniaxial load we consider the optimization of the topology and geometry of an elastic structure o r2 subjected to a fixed boundary load i e we aim to minimize a weighted sum of material volume vol o structur

level set methods for optimization problems involving geometry and constraints ii optimization over a fixed surface in this work we consider an optimization problem described on a surface the approach is illustrated on the problem of finding a closed curve whose arclength is as small as possible while the area enclosed by the curve is fixed this problem exemplifies a class of optimization and inverse problems that arise in diverse applications in our approach we assume that the surface is given parametrically a level set formulation for the curve is developed in the surface parameter space we show how to obtain a formal gradient for the optimization objective and derive a gradient type algorithm which minimizes the objective while respecting the constraint the algorithm is a projection method which has a pde interpretation we demonstrate and verify the method in numerical examples

probability distribution function inspired structural optimization for frequency response problems abstract this study introduces an innovative topological optimization scheme for frequency response problems using the phase field design method in the proposed approach the frequency response function is mapped onto a probability distribution function where the mean value of the probability distribution function becomes the resonance frequency and the variance corresponds to its bandwidth using the mapping concept the frequency characteristics such as the resonance frequency and bandwidth can be directly included in the formulation of the design objective function therefore the mapping process enables to handle frequency characteristics in topological design for frequency response problems as numerical examples the resonance frequency maximization problem of a cantilever beam bandwidth maximization of a clamped end beam and the bandwidth minimization problem of a plasmonic grating coupler are presented to demonstrate the validity of the proposed method

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

an alternative truly mixed formulation to solve pressure load problems in topology optimization abstract the paper deals with an alternative formulation for the topology optimization of structures acted upon by pressure loads exploiting finite elements techniques that are able to handle incompressible materials the method firstly presented in o sigmund p m clausen topology optimization using a mixed formulation an alternative way to solve pressure load problems comput methods appl mech engrg 196 2007 1874 1889 33 consists in exploiting the modeling of fluid incompressibility within a topology optimization framework the implementation of a fluid phase enables to transfer pressure loads from the domain boundaries to the evolving edges of optimal design without relying on more complex techniques traditionally employed to recover the load application surfaces at each step of the minimization process in this context the main numerical trouble is therefore the application of finite elements techniques to solve incompressible materials analysis this topic in fact cannot be tackled using most of the approaches of the current literature that are mainly based on displacement finite elements which are well known to be affected by the locking phenomenon while sigmund and clausen 2007 uses a displacement pressure finite element discretization to solve the problem the approach herein presented consists in the adoption of a truly mixed variational formulation coupled to a discretization based on the johnson and mercier finite element that both pass the inf sup conditions of the problem even in the presence of incompressible materials the well known method of moving asymptotes mma is adopted in the numerical studies presented along with a particular density interpolation to model the presence of a fluid and solid phase within the same design the adopted scheme is especially conceived to avoid the arising of numerical instabilities that may arise within the optimization procedure when handling incompressible material moreover the accuracy and stability in stress evaluation provided by the truly mixed setting are herein exploited to introduce an alternative procedure that implements pressure constraints to avoid optimal designs that present cavities filled by fluid

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

multiscale isogeometric topology optimization for lattice materials abstract this paper presents isogeometric topology optimization ito for periodic lattice materials where non uniform rational b spline nurbs basis functions of cad models are directly used in the finite element analysis to improve computational accuracy and efficiency two to schemes that use asymptotic homogenization ah for the calculation of the mechanical properties are proposed for lattice materials with uniform and graded relative density respectively to accelerate ito for graded lattice materials the mechanical properties are expressed as a function of the relative density of the unit cell a step that avoids their iterative calculations during ito three benchmark examples are presented to validate the proposed scheme with results that show tangible advantages such as reduced computational time and faster convergence of ito over conventional to

a quadratically convergent unstructured remeshing strategy for shape optimization a novel unstructured remeshing environment for gradient based shape optimization using triangular finite elements is presented the remeshing algorithm is based on a truss structure analogy in solving for the equilibrium position of the truss system the quadratically convergent newton s method is used exact analytical sensitivity information is therefore made available to the shape optimization algorithm the overall computational efficiency in gradient based shape optimization is very high in solving the truss structure analogy we compare our quadratically convergent newton solver with a previously proposed forward euler solver this includes notes regarding mesh uniformity element quality convergence rates and efficiency we present three numerical examples it is then shown that remeshing may introduce discontinuities and local minima we demonstrate that the effects of these on gradient based algorithms are alleviated to some extent through mesh refinement and may largely be overcome with a simple multi start strategy copyright 2005 john wiley sons ltd

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

the topological asymptotic for pde systems the elasticity case the aim of the topological sensitivity analysis is to obtain an asymptotic expansion of a design functional with respect to the creation of a small hole in this paper such an expansion is obtained and analyzed in the context of linear elasticity for general functionals and arbitrary shaped holes by using an adaptation of the adjoint method and a domain truncation technique the method is general and can be easily adapted to other linear pdes and other types of boundary conditions

shape topology optimization for navier stokes problem using variational level set method we consider the shape topology optimization of the navier stokes problem a new algorithm is proposed based on the variational level set method by this algorithm a relatively smooth evolution can be maintained without re initialization and drastic topology change can be handled easily finally the promising features of the proposed method are illustrated by two benchmark examples

inverse inclusion problem a stable method to determine disks in this paper we are interested in inverse inclusion problem in the plane we derive uniqueness and holder stability results of the inclusion recovery problem using a single boundary measurement under the assumption that the inclusion has a circular shape we also propose a simple minimizing scheme for the recovery of a disk our numerical results show that the holder power in the stability estimate is close to one

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

modal basis approaches in shape and topology optimization of frequency response problems the optimal design of mechanical structures subject to periodic excitations within a large frequency interval is quite challenging in order to avoid bad performances for non discretized frequencies it is necessary to finely discretize the frequency interval leading to a very large number of state equations then if a standard adjoint based approach is used for optimization the computational cost both in terms of cpu and memory storage may be prohibitive for large problems especially in three space dimensions the goal of the present work is to introduce two new non adjoint approaches for dealing with frequency response problems in shape and topology optimization in both cases we rely on a classical modal basis approach to compute the states solutions of the direct problems in the first method we do not use any adjoint but rather directly compute the shape derivatives of the eigenmodes in the modal basis in the second method we compute the adjoints of the standard approach by using again the modal basis the numerical cost of these two new strategies are much smaller than the usual ones if the number of modes in the modal basis is much smaller than the number of discretized excitation frequencies we present numerical examples for the minimization of the dynamic compliance in two and three space dimensions

shape optimization of dirichlet boundaries based on weighted b spline finite cell method and level set function abstract this paper addresses extended shape optimization problems where structural supports i e dirichlet boundaries and free boundaries are simultaneously optimized unlike traditional fem the weighted b spline finite cell method fcm is applied as structural analysis tool and combined with the level set function lsf to take into account dirichlet boundary condition dbc automatically through penalization of the displacement field the proposed shape optimization method achieves a comprehensive integration of fixed grid b spline shape function and level set function as both the structure and dirichlet boundaries are described in the form of lsf any modification of dirichlet boundaries can be made in a straightforward way as easily as free boundaries by changing continuous design variables meanwhile the computing accuracy is ensured within the framework of fixed grid owing to the quadtree refinements of boundary cells stress related shape optimization problems are finally solved to demonstrate the merit and validity of the proposed optimization method in dealing with shape optimization of dirichlet boundaries

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

topology optimization of steady navier stokes flow with body force abstract this paper presents the topology optimization of steady navier stokes flows with body forces that influence the optimal shape and topology of fluid flows based on the implicit expression of the fluid flow with the level set method an optimization problem is formulated and analyzed using the continuous adjoint method the shape and topological sensitivities are computed based on the adjoint and asymptotic analysis of the optimization problem in the optimization procedure the level set surface is evolved based on the shape sensitivity and nucleated based on the topological sensitivity simultaneously three kinds of body forces that are commonly used in the design of fluid devices i e constant nonuniform and solution dependent body forces are considered in the two dimensional and three dimensional numerical examples numerical results demonstrate that this method can effectively achieve the topology optimization of the navier stokes flows with body forces

risk averse structural topology optimization under random fields using stochastic expansion methods abstract this work proposes a level set based approach for solving risk averse structural topology optimization problems considering random field loading and material uncertainty the use of random fields increases the dimensionality of the stochastic domain which poses several computational challenges related to the minimization of the excess probability as a measure of risk awareness this problem is addressed both from the theoretical and numerical viewpoints first an existence result under a typical geometrical constraint on the set of admissible shapes is proved second a level set continuous approach to find the numerical solution of the problem is proposed since the considered cost functional has a discontinuous integrand the numerical approximation of the functional and its sensitivity combine an adaptive anisotropic polynomial chaos pc approach with a monte carlo mc sampling method for uncertainty propagation furthermore to address the increment of dimensionality induced by the random field an anisotropic sparse grid stochastic collocation method is used for the efficient computation of the pc coefficients a key point is that the non intrusive nature of such an approach facilitates the use of high performance computing hpc to alleviate the computational burden of the problem several numerical experiments including random field loading and material uncertainty are presented to show the feasibility of the proposal

topology optimization of acoustic structure interaction problems using a mixed finite element formulation the paper presents a gradient based topology optimization formulation that allows to solve acoustic structure vibro acoustic interaction problems without explicit boundary interface representation in acoustic structure interaction problems the pressure and displacement fields are governed by helmholtz equation and the elasticity equation respectively normally the two separate fields are coupled by surface coupling integrals however such a formulation does not allow for free material re distribution in connection with topology optimization schemes since the boundaries are not explicitly given during the optimization process in this paper we circumvent the explicit boundary representation by using a mixed finite element formulation with displacements and pressure as primary variables a u p formulation the helmholtz equation is obtained as a special case of the mixed formulation for the elastic shear modulus equating to zero hence by spatial variation of the mass density shear and bulk moduli we are able to solve the coupled problem by the mixed formulation using this modelling approach the topology optimization procedure is simply implemented as a standard density approach several two dimensional acoustic structure problems are optimized in order to verify the proposed method copyright 2006 john wiley sons ltd

a finite element based level set method for structural optimization a finite element based level set method is implemented for structural optimization the streamline diffusion finite element method is used for solving both the level set equation and the reinitialization equation the lumped scheme is addressed and the accuracy is compared with the conventional finite difference based level set method a dirichlet boundary condition is enforced during the reinitialization to prevent the boundary from drifting numerical examples of minimum mean compliance design illustrate the reliability of the proposed optimization method copyright 2009 john wiley sons ltd

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

topology optimization of nonlinear heat conduction problems involving large temperature gradient abstract in this work we develop a topology optimization method for steady state nonlinear heat conduction problems involving large temperature gradient to minimize the maximum structural temperature temperature dependent material properties tdmps are taken into account to break the widely used assumption of constant material properties cmps in conventional topology optimization the kreisselmeier steinhauser function is adopted as an aggregated measure of the maximum temperature over a specific region and the adjoint method is used to derive the sensitivity expressions to effectively solve the well known unsymmetric adjoint problem caused by material nonlinearity an engineering oriented stationary iterative method esim is constructed to transform the unsymmetric adjoint problem into a symmetric one with a series of right hand sides that can be efficiently solved with mature linear system solvers both 2d and 3d numerical examples are provided to illustrate the validity and utility of the proposed method including the accuracy and convergence of the esim the results show that the assumption of cmps can result in significant inaccuracy in the analysis and design of heat conduction systems working under large temperature gradient on the contrary by considering tdmps and employing the proposed method reasonable designs are achieved and thermal performance is greatly improved

a density result in two dimensional linearized elasticity and applications we show that in a two dimensional bounded open set whose complement has a finite number of connected components the vector fields u h 1 2 are dense in the space of fields whose symmetrized gradient e u is in l 2 4 this allows us to show the continuity of some linearized elasticity problems with respect to variations of the set with applications to shape optimization or the study of crack evolution

robust shape optimization of continuous structures via the level set method abstract this work proposes a stochastic shape optimization method for continuous structures using the level set method such a method aims to minimize the expected compliance and its variance as measures of the structural robustness the behavior of continuous structures is modeled by linear elasticity equations with uncertain loading and material this uncertainty can be modeled using random variables with different probability distributions as well as random fields the proper problem formulation is ensured by the proof of the existence colorrev of solution under certain geometrical constraints on the set of admissible shapes the proposed method addresses the stochastic linear elasticity problem in its weak form obtaining the explicit expressions for the continuous shape derivatives some numerical examples are presented to show the effectiveness of the proposed approach

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

a semismooth newton method for topology optimization abstract this paper deals with elliptic optimal control problems for which the control function is constrained to assume values in 0 1 based on an appropriate formulation of the optimality system a semismooth newton method is proposed for the solution convergence results are proved and some numerical tests illustrate the efficiency of the method

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

explicit control of structural complexity in topology optimization abstract in topology optimization design specific requirements over the complexity of component geometry often need to be satisfied for practical purposes in this paper with the introduction of a novel complexity control measure built on the structural skeletons of solid and void regions an explicit structural complexity control approach for the topology optimization of continuum structures is proposed under the framework of level set representation the use of skeletons enables us to control the minimum sizes of structural holes so as to meet the manufacturing demands through numerical comparison with the cases that have been investigated implicitly we show that the proposed approach is an effective tool to control the structural complexity in an explicit way

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

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

isogeometric shape optimization of smoothed petal auxetic structures via computational periodic homogenization an important feature that drives the auxetic behaviour of the star shaped auxetic structures is the hinge functional connection at the vertex connections this feature poses a great challenge for manufacturing and may lead to significant stress concentrations to overcome these problems we introduced smoothed petal shaped auxetic structures where the hinges are replaced by smoothed connections to accommodate the curved features of the petal shaped auxetics a parametrisation modelling scheme using multiple nurbs patches is proposed next an integrated shape design frame work using isogeometric analysis is adopted to improve the structural performance to ensure a minimum thickness for each member a geometry sizing constraint is imposed via piece wise bounding polynomials this geometry sizing constraint in the context of isogeometric shape optimization is particularly interesting due to the non interpolatory nature of nurbs basis the effective poisson ratio is used directly as the objective function and an adjoint sensitivity analysis is carried out the optimized designs smoothed petal auxetic structures are shown to achieve low negative poisson s ratios while the difficulties of manufacturing the hinges are avoided for the case with six petals an in plane isotropy is achieved

interpolation scheme for fictitious domain techniques and topology optimization of finite strain elastic problems abstract the focus of this paper is on interpolation schemes for fictitious domain and topology optimization approaches with structures undergoing large displacements numerical instability in the finite element simulations can often be observed due to excessive distortion in low stiffness regions a new energy interpolation scheme is proposed in order to stabilize the numerical simulations the elastic energy density in the solid and void regions is interpolated using the elastic energy densities for large and small deformation theory respectively the performance of the proposed method is demonstrated for a challenging test geometry as well as for topology optimization of minimum compliance and compliant mechanisms the effect of combining the proposed interpolation scheme with different hyperelastic material models is investigated as well numerical results show that the proposed approach alleviates the problems in the low stiffness regions and for the simulated cases results in stable topology optimization of structures undergoing large displacements

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

design sensitivity analysis for shape optimization based on the lie derivative abstract the paper presents a theoretical framework for the shape sensitivity analysis of systems governed by partial differential equations the proposed approach based on geometrical concepts borrowed from differential geometry shows that sensitivity of a performance function i e any function of the solution of the problem with respect to a given design variable can be represented mathematically as a lie derivative i e the derivative of that performance function along a flow representing the continuous shape modification of the geometrical model induced by the variation of the considered design variable theoretical formulae to express sensitivity analytically are demonstrated in detail in the paper and applied to a nonlinear magnetostatic and a linear elastic problem following both the direct and the adjoint approaches following the analytical approach one linear system of which only the right hand side needs be evaluated the system matrix being known already has to be solved for each of the design variables in the direct approach or for each performance functions in the adjoint approach a substantial gain in computation time is obtained this way compared to a finite difference evaluation of sensitivity which requires solving a second nonlinear system for each design variable this is the main motivation of the analytical approach there is some freedom in the definition of the auxiliary flow that represents the shape modification we present a method that makes benefit of this freedom to express sensitivity locally as a volume integral over a single layer of finite elements connected to both sides of the surfaces undergoing shape modification all sensitivity calculations are checked with a finite difference in order to validate the analytic approach convergence is analyzed in 2d and 3d with first and second order finite elements

phase field method to optimize dielectric devices for electromagnetic wave propagation we discuss a phase field method for shape optimization in the context of electromagnetic wave propagation the proposed method has the same functional capabilities as the level set method for shape optimization the first advantage of the method is the simplicity of computation since extra operations such as re initialization of functions are not required the second is compatibility with the topology optimization method due to the similar domain representation and the sensitivity analysis structural shapes are represented by the phase field function defined in the design domain and this function is optimized by solving a time dependent reaction diffusion equation the artificial double well potential function used in the equation is derived from sensitivity analysis we study four types of 2d or 2 5d axisymmetric optimization problems two are the classical problems of photonic crystal design based on the bloch theory and photonic crystal wave guide design and two are the recent topics of designing dielectric left handed metamaterials and dielectric ring resonators

topology optimization of a reinforced wing box for enhanced roll maneuvers in this paper the aileron reversal speed of a wing torsion box is maximized by reinforcing its upper skin we propose a topology optimization approach using the spectral level set methodology this is based on the level set methods which represent theinterface describing thereinforcements asthe zerolevel setofafunction accordingto the proposed methodology this function is discretized using its fourier coefficients these coefficients become the design variables of the optimization problem assigned to define the reinforcing layout results show that approximately the sameoptimal reinforcement layout isobtained from topologically different initialconfigurations the two underlying ideas of the present work are to propose the spectral level set methodology as a framework for topology optimization given its potential for reducing the design space dimension and considering the possibilities of new custom made materials to readdress the reinforcement perspective in aircraft structures without the weight penalty nomenclature ai bi problem domain boundaries corresponding to componenti ofx at t smoothness parameter av v smoothness parameter a0 lift curve slope bt minimum thickness c positive constant cl derivative of the rolling moment coefficient of the rigid airplane with respect to c y n twist aty produced by a unit moment at

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

robust topology optimization minimization of expected and variance of compliance robust topology optimization has long been considered computationally intractable as it combines two highly expensive computational strategies this paper considers simultaneous minimization of expectancy and variance of compliance in the presence of uncertainties in loading magnitude using exact formulations and analytically derived sensitivities it shows that only a few additional load cases are required which scales in polynomial time with the number of uncertain load cases the approach is implemented using the level set topology optimization method shape sensitivities are derived using the adjoint method several examples are used to investigate the effect of including variance in robust compliance optimization

neural networks orientations of the hypercube and algebraic threshold functions a class of possible generalizations of current neural networks models is described using local improvement algorithms and orientations of graphs a notation of dynamical capacity is defined and by computing bounds on the number of algebraic threshold functions it is proven that for neural networks of size n and energy function of degree d this capacity is o n sup d 1 stable states are studied and it is shown that for the same networks the storage capacity is o n sup d 1 in the case of random orientations it is proven that the expected number of stable states is exponential applications to coding theory are indicated and it is shown that usual codes can be embedded in neural networks but only at high cost cycles and their storage are also examined

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

damage and fracture evolution in brittle materials by shape optimization methods this paper is devoted to a numerical implementation of the francfort marigo model of damage evolution in brittle materials this quasi static model is based at each time step on the minimization of a total energy which is the sum of an elastic energy and a griffith type dissipated energy such a minimization is carried over all geometric mixtures of the two healthy and damaged elastic phases respecting an irreversibility constraint numerically we consider a situation where two well separated phases coexist and model their interface by a level set function that is transported according to the shape derivative of the minimized total energy in the context of interface variations hadamard method and using a steepest descent algorithm we compute local minimizers of this quasi static damage model initially the damaged zone is nucleated by using the so called topological derivative we show that when the damaged phase is very weak our numerical method is able to predict crack propagation including kinking and branching several numerical examples in 2d and 3d are discussed

on structural shape optimization using an embedding domain discretization technique summary this contribution presents a novel approach to structural shape optimization that relies on an embedding domain discretization technique the evolving shape design is embedded within a uniform finite element background mesh which is then used for the solution of the physical state problem throughout the course of the optimization we consider a boundary tracking procedure based on adaptive mesh refinement to separate between interior elements exterior elements and elements intersected by the physical domain boundary a selective domain integration procedure is employed to account for the geometric mismatch between the uniform embedding domain discretization and the evolving structural component thereby we avoid the need to provide a finite element mesh that conforms to the structural component for every design iteration as it is the case for a standard lagrangian approach to structural shape optimization still we adopt an explicit shape parametrization that allows for a direct manipulation of boundary vertices for the design evolution process in order to avoid irregular and impracticable design updates we consider a geometric regularization technique to render feasible descent directions for the course of the optimization this article is protected by copyright all rights reserved

determinant structure of the rational solutions for the painleve iv equation rational solutions for the painleve iv equation are investigated by hirota bilinear formalism it is shown that the solutions in one hierarchy are expressed by 3 reduced schur functions and those in another two hierarchies by casorati determinants of the hermite polynomials or by a special case of the schur polynomials

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

explicit three dimensional topology optimization via moving morphable void mmv approach abstract three dimensional 3d topology optimization problems always involve huge numbers of degrees of freedom dofs in finite element analysis fea and design variables in numerical optimization respectively this will inevitably lead to large computational efforts in the solution process in the present paper an efficient and explicit topology optimization approach which can reduce not only the number of design variables but also the number of degrees of freedom in fea is proposed based on the moving morphable voids mmvs solution framework this is achieved by introducing a set of geometry parameters e g control points of b spline surfaces to describe the boundary of a structure explicitly and removing the unnecessary dofs from the fe model at every step of numerical optimization numerical examples demonstrate that the proposed approach does can overcome the bottleneck problems associated with a 3d topology optimization problem in a straightforward way and enhance the solution efficiency significantly

a density gradient approach to topology optimization under design dependent boundary loading abstract the paper proposes a density gradient based approach to topology optimization under design dependent boundary loading since there is no explicit boundary representation in density based topology optimization the design dependent boundary loads are implicitly imposed through a domain integration of spatial gradient of the density field the density gradient based loading is formally derived and justified based on the smoothed boundary method for numerical solution of pde the heaviside projection density filtering and local refinement are combined to efficiently control the thickness of the loading interface and the number of elements within it for problems with both loading and non loading boundary an auxiliary density field is introduced to keep track of the loading boundary both 2d and 3d heat conduction problems linear elasticity problems and coupled thermoelastic problems under design dependent boundary loading are presented to illustrate the effectiveness and efficiency of the method

an immersed boundary element method for level set based topology optimization summary in this paper we propose a new bem for level set based topology optimization in the proposed bem the nodal coordinates of the boundary element are replaced with the nodal level set function and the nodal coordinates of the eulerian mesh that maintains the level set function because this replacement causes the nodal coordinates of the boundary element to disappear the boundary element mesh appears to be immersed in the eulerian mesh therefore we call the proposed bem an immersed bem the relationship between the nodal coordinates of the boundary element and the nodal level set function of the eulerian mesh is clearly represented and therefore the sensitivities with respect to the nodal level set function are strictly derived in the immersed bem furthermore the immersed bem completely eliminates grayscale elements that are known to cause numerical difficulties in topology optimization by using the immersed bem we construct a concrete topology optimization method for solving the minimum compliance problem we provide some numerical examples and discuss the usefulness of the constructed optimization method on the basis of the obtained results copyright 2012 john wiley sons ltd

regularized perimeter for topology optimization the perimeter functional is known to oppose serious difficulties when it has to be handled within a topology optimization procedure in this paper a regularized perimeter functional mbox per varepsilon defined for two and three dimensional domains is introduced on one hand the convergence of mbox per varepsilon to the exact perimeter when varepsilon tends to zero is proved on the other hand the topological differentiability of mbox per varepsilon for varepsilon 0 is analyzed these features lead to the design of a topology optimization algorithm suitable for perimeter dependent objective functionals several numerical results illustrate the method

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

feature driven topology optimization method with signed distance function abstract in this paper a feature driven topology optimization method is developed this is the first study on layout design of multiple engineering features using level set functions lsfs and boolean operations the novelty of this work is threefold first multiple engineering features of arbitrary shape are considered as basic design primitives and topology variation is achieved via the layout and shape optimization of the features kreisselmeier steinhauser ks function constructed by means of boolean operations is adopted as the lsf which uses an implicit function to ensure a smooth description and topological changes of basic features and the whole structure second using a modified heaviside function to smooth the void solid material transition over a fixed computing mesh a narrow band domain integral scheme is developed for the efficient sensitivity analysis third the gray material distribution regions at the feature connecting portions are analyzed and the underlying reason for that is traced to the non equidistant distribution of level set contours of specific features to avoid the gray regions an approximated signed distance function is proposed to regularize the lsf and ks function the bounded normalization property of the ks function is highlighted for its construction with the signed distance functions or normalized first order approximations numerical examples are finally tested to demonstrate the validity and merits of the proposed feature driven topology optimization for complicated design problems

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

structural topology optimization with minimum distance control of multiphase embedded components by level set method abstract this paper presents a novel topology optimization method for designing structures with multiphase embedded components under minimum distance constraints in the level set framework by using the level set representation for both the component layout and the host structure topology the shapes of the components can be easily preserved and optimal structural topologies with smooth boundary material interface can be obtained with the purpose of preventing the components moving too close to each other a minimum distance constraint based on virtual boundary offset is proposed different from existing distance detection methods relying on explicit topology representation the proposed constraint is imposed as a unified integral form for which the design sensitivity can be readily obtained moreover this constraint is effective for detecting the distance between any complex shaped components several numerical examples are presented to demonstrate the validity and effectiveness of the proposed method

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

an isogeometrical approach to structural level set topology optimization abstract this paper aims to utilize the isogeometric analysis iga for the level set structural topology optimization the level set function is parametrized using the non uniform rational b spline nurbs basis functions in a higher dimension the same basis functions are employed for approximating the unknown deformations geometry modeling of the design domain and the level set function in this research three optimization problems including minimization of the mean compliance considering a certain amount of material minimization of weight with avoiding local stress concentration as well as minimization of weight and strain energy under local stress constraints are dealt with the sensitivity analyses for the optimization problems are carried out to obtain velocity functions on the boundaries in order to move the boundaries towards optimum the hamilton jacobi h j equation is solved using the forward euler scheme in order to illustrate the performance of the method to obtain reasonable results with smooth boundaries several numerical examples are presented and compared with well known problems in literature

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

integrated layout design of multi component systems using xfem and analytical sensitivity analysis this study presents the integrated layout optimization of multi component systems using a fixed mesh the optimization formulation is established under the framework of the extended finite element method xfem the level set method is used to represent components and is combined with the xfem to describe material discontinuities across elements sensitivity analysis is proposed with respect to geometric variables of components and pseudo densities of the basic structure an analytical shape sensitivity analysis method with respect to positions and shapes of components is developed both solid and void components are considered to show the efficiency and accuracy of the proposed shape sensitivity analysis method furthermore a revised finite circle method that adapts shape changes of elliptical components is proposed for the definition of non overlapping constraints finally numerical examples of maximizing the structural stiffness are tested to demonstrate the proposed method

short note shape and topology optimization for electrothermomechanical microactuators using level set methods in this short note a shape and topology optimization method is presented for multiphysics actuators including geometrically nonlinear modeling based on an implicit free boundary parameterization method a level set model is established to describe structural design boundary by embedding it into the zero level set of a higher dimensional level set function the compactly supported radial basis functions csrbf are introduced to parameterize the implicit level set surface with a high level of accuracy and smoothness the original more difficult shape and topology optimization driven by the hamilton jacobi partial differential equation pde is transferred into a relatively easier parametric size optimization to which many well founded optimization algorithms can be applied thus the structural optimization is transformed to a numerical process that describes the design as a sequence of motions of the design boundaries by updating the expansion coefficients of the size optimization two widely studied examples are chosen to demonstrate the effectiveness of the proposed method

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

reconstruction of a piecewise constant conductivity on a polygonal partition via shape optimization in eit abstract in this paper we develop a shape optimization based algorithm for the electrical impedance tomography eit problem of determining a piecewise constant conductivity on a polygonal partition from boundary measurements the key tool is to use a distributed shape derivative of a suitable cost functional with respect to movements of the partition numerical simulations showing the robustness and accuracy of the method are presented for simulated test cases in two dimensions