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

tangent cone and contingent cone to the intersection of two closed sets abstract for a nonempty closed set c in a real normed vector space x and an inequality solution set we present several sufficient conditions for the tangent and contingent cones to their intersection to contain the intersections of the corresponding cones we not only express the contingent cone to a solution set of inequalities and equalities by the directional or frechet derivatives of the active inequality constraint functions and the frechet derivatives of the equality constraint functions but also the tangent cone by the clarke or lower dini or upper dini derivatives of the active inequality constraint functions and the directional derivatives of the equality constraint functions by using a simple property of the function d c d c c we characterize these cones by the hypertangent and hypercontingent vectors to the set c furthermore these results allow us to present new constraint qualifications for the karush kuhn tucker conditions

optimality conditions for shape and topology optimization subject to a cone constraint this paper provides first order necessary optimality conditions for simultaneous shape and topology optimization subject to a cone constraint these conditions are expressed with the help of the shape and topological derivatives of the objective and constraint functionals several examples of applications are given

a stabilisation approach for topology optimisation of hyperelastic structures with the simp method abstract this paper presents a novel computational approach for simp based topology optimisation to of hyperelastic materials at large strains during the to process for structures subjected to very large deformations and especially in the presence of intermediate density regions the standard newton solver or its arc length variant have been reported not to converge refer to references wang et al 2014 lahuerta et al 2013 and liu et al 2017 in this paper the new to stabilisation technique proposed in ortigosa et al 2019 in the context of level set to initially devised to alleviate numerical instabilities inherent to level set to is extended for the to by means of the simp method the success of the methodology rests on the combination of two distinct key ingredients first the nonlinear equilibrium equations of motion for intermediate to design stages are solved in a non exact albeit consistent incrementally linearised fashion by splitting the design load into a number of load increments second the resulting linearised tangent elasticity tensor is locally stabilised regularised in order to prevent its loss of positive definiteness and thus avoid the loss of convexity of the discrete tangent operator this solution strategy is shown to be extremely robust in the context of density based to where the constitutive law of the underlying evolving solid structure is a mixture of solid and void constituents the latter classically defined by means of a fictitious strain energy the robustness and applicability of this to methodological approach are thoroughly demonstrated through an ample spectrum of challenging numerical examples ranging from benchmark two dimensional plane stress examples to larger scale three dimensional applications crucially the performance of all the final designs has been tested at a post processing stage without adding any source of artificial stiffness specifically an arc length newton raphson method has been employed in conjunction with a ratio of the material parameters for void and solid regions of 1 0 12

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

why the high frequency inverse scattering by topological sensitivity may work this study deciphers the topological sensitivity ts as a tool for the reconstruction and characterization of impenetrable anomalies in the high frequency regime it is assumed that the anomaly is simply connected and convex and that the measurements of the scattered field are of the far field type in this setting the formula for ts which quantifies the perturbation of a cost functional due to a point like impenetrable scatterer is expressed as a pair of nested surface integrals one taken over the boundary of a hidden obstacle and the other over the measurement surface using multipole expansion the latter integral is reduced to a set of antilinear forms featuring green s function and its gradient the remaining expression is distilled by evaluating the scattered field on the surface of an obstacle via kirchhoff approximation and pursuing an asymptotic expansion of the resulting fourier integral in this way the ts is found to survive upon three asymptotic lynchpins namely i the near boundary approximation for sampling points close to the exposed surface of an obstacle ii uniform expansions synthesizing the diffraction catastrophes for sampling points near caustic surfaces lines and points and iii stationary phase approximation within the framework of catastrophe theory it is shown that in the case of the full source aperture the ts is asymptotically dominated by the explicit near boundary term which explains the previously reported reconstruction capabilities of this class of indicator functionals the analysis further shows that when the dirichlet or neumann character of an anomaly is unknown beforehand the latter can be effectively exposed by assuming point like dirichlet perturbation and considering the sign of the leading term inside the reconstruction

stability and resolution analysis for a topological derivative based imaging functional the aim of this paper is to study a topological derivative based anomaly detection algorithm we compare its performance with other imaging approaches such as multiple signal classification backpropagation and kirchhoff migration we also investigate its stability with respect to medium and measurement noises as well as its resolution a simple postprocessing of the data set is introduced and shown to be essential in order to obtain an efficient topological based imaging functional both in terms of resolution and stability

the topological asymptotic for the helmholtz equation the aim of the topological sensitivity analysis is to obtain an asymptotic expansion of a functional with respect to the creation of a small hole in the domain in this paper such an expansion is obtained for the helmholtz equation with a dirichlet condition on the boundary of a circular hole some applications of this work to waveguide optimization are presented

topological sensitivity analysis in heterogeneous anisotropic elasticity problem theoretical and computational aspects the topological sensitivity analysis for the heterogeneous and anisotropic elasticity problem in two dimensions is performed in this work the main result of the paper is an analytical closed form of the topological derivative for the total potential energy of the problem this derivative displays the sensitivity of the cost functional the energy in this case when a small singular perturbation is introduced in an arbitrary point of the domain in this case we consider a small disc with a completely different elastic material full mathematical justification for the derived formula and derivations of precise estimates for the remainders of the topological asymptotic expansion are provided finally the influence of the heterogeneity and anisotropy is shown through some numerical examples of structural topology optimization

on convergence in elliptic shape optimization the present paper aims at analyzing the existence and convergence of approximate solutions in shape optimization motivated by illustrative examples an abstract setting of the underlying shape optimization problem is suggested taking into account the so called two norm discrepancy a ritz galerkin type method is applied to solve the associated necessary condition existence and convergence of approximate solutions are proved provided that the infinite dimensional shape problem admits a stable second order optimizer the rate of convergence is confirmed by numerical results

multi frequency topological derivative for approximate shape acquisition of curve like thin electromagnetic inhomogeneities abstract in this paper we investigate a non iterative imaging algorithm based on the topological derivative in order to retrieve the shape of penetrable electromagnetic inclusions when their dielectric permittivity and or magnetic permeability differ from those in the embedding homogeneous space the main objective is the imaging of crack like thin inclusions but the algorithm can be applied to arbitrarily shaped inclusions for this purpose we apply multiple time harmonic frequencies and normalize the topological derivative imaging function by its maximum value in order to verify its validity we apply it for the imaging of two dimensional crack like thin electromagnetic inhomogeneities completely hidden in a homogeneous material corresponding numerical simulations with noisy data are performed for showing the efficacy of the proposed algorithm

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

asymptotic models for optimizing the contour of multiply connected cross section of an elastic bar in torsion an asymptotic solution is obtained for the problem of maximizing the torsional rigidity of elastic multiply connected cylindrical bars for a given area of cross section the shapes of the inner contours of the multiply connected cross section are specified while the outer contour is determined as a result of the shape optimization we apply the method of matched asymptotic expansions to construct a first order asymptotic model the conditions for unique solvability of the asymptotic model have been established under some restrictions imposed on the location of the inner contours and their polarization matrices the economy achieved by optimization is estimated

energy change due to the appearance of cavities in elastic solids abstract the paper presents an overview of the problem of assessing an increment of strain energy due to the appearance of small cavities in elastic solids the following approaches are discussed the compound asymptotic method by mazja et al the eshelby like method used in the classical works on the mechanics of composites the homogenization method and the topological derivative method proposed by sokolowski and ochowski the increment of energy is expressed by a quadratic form with respect to strains referring to the virgin solid all the methods lead to the same formula for the increment of energy it is expressed by a quadratic form with respect to strains referring to the virgin solid this quadratic form turns out to be unconditionally positive definite explicit formulae are derived for an elliptical hole and for a spherical cavity the results derived determine the characteristic function of the bubble method of the optimal shape design of elastic 2d and 3d structures

optimal topological design through insertion and configuration of finite sized heterogeneities in this paper we develop a procedure for optimal topological design by sequentially inserting finite sized non spherical inclusions or holes within a homogeneous domain we propose a new criterion for topology change that results in a trade off problem to achieve the greatest least change in the objective for the least greatest change in the size of the inclusion hole respectively we derive the material derivative of the proposed objective termed as the configurational derivative that describes sensitivity of arbitrary functionals to arbitrary motions of the inclusion hole as well as the domain boundaries we specifically utilize the sensitivity to position orientation and scaling of finite sized heterogeneities to effect topological design we simplify the configurational derivative to the special case of infinitesimally small spherical inclusions or holes and show that the developed derivative is a generalization of the classical topological derivative the computational implementation relies on b spline isogeometric approximations we demonstrate through a series of examples optimal topology achieved through sequential insertion of a heterogeneity of fixed shape and optimization of its configuration location orientation and scale

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

performance analysis of multi frequency topological derivative for reconstructing perfectly conducting cracks abstract this paper concerns a fast one step iterative technique of imaging extended perfectly conducting cracks with dirichlet boundary condition in order to reconstruct the shape of cracks from scattered field data measured at the boundary we introduce a topological derivative based electromagnetic imaging functional operated at several nonzero frequencies the structure of the imaging functionals is carefully analyzed by establishing relationships with infinite series of bessel functions for the configurations of both symmetric and non symmetric incident field directions identified structure explains why the application of incident fields with symmetric direction operated at multiple frequencies guarantees a successful reconstruction various numerical simulations with noise corrupted data are conducted to assess the performance effectiveness robustness and limitations of the proposed technique

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

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

the topological asymptotic for the helmholtz equation with dirichlet condition on the boundary of an arbitrarily shaped hole 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 the domain in this paper such an expansion is obtained for the helmholtz equation in two and three space dimensions with a dirichlet condition on the boundary of an arbitrarily shaped hole in this case the main difficulty is related to the nonhomogeneous symbol of the helmholtz operator in the numerical part of this work we will show that the topological sensitivity method is very promising for solving shape inverse problems in electromagnetic applications

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

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

combination of topology optimization and optimal control method this paper presents the combination of topology optimization and optimal control method to find the optimal match between the material topology and control in the presented method the material topology is determined using the simp solid isotropic material with penalization method which has been popularly used in topology optimization in the simp method the design variable is relaxed and bounded in the interval 0 1 and the evolution of the design variable is usually implemented by the method of moving asymptotes mma which can be used to deal with optimization problem with multiple integral constraints and bound constraint of the design variable in the combination of topology optimization and optimal control method the control variable appears along with the design variable in order to evolve the control variable and design variable using mma simultaneously the control variable is regularized using a bound constraint and the corresponding bound constraint is projected onto the interval 0 1 which is the same as the bound constraint of the design variable the optimization problem is analyzed using the adjoint method to obtain the adjoint sensitivity during the optimization procedure the design and control variables are filtered by the helmholtz filters to ensure the smoothness of the distribution to ensure the minimum scale length and remove the gray area in the material topology the filtered design variable is projected by the threshold method the feasibility and robustness of the combination of these two methods are demonstrated by several test problems including heat transfer fluid flow and compliance minimization

risk averse shape optimization risk averse optimization has attracted much attention in finite dimensional stochastic programming in this paper we propose a risk averse approach in the infinite dimensional context of shape optimization we consider elastic materials under stochastic loading as measures of risk awareness we investigate the expected excess and the excess probability the developed numerical algorithm is based on a regularized gradient flow acting on an implicit description of the shapes based on level sets we incorporate topological derivatives to allow for topological changes in the shape optimization procedure numerical results in two dimensions demonstrate the impact of the risk averse modeling on the optimal shapes and on the cost distribution over the set of scenarios

a consistent grayscale free topology optimization method using the level set method and zero level boundary tracking mesh summary this paper proposes a level set based topology optimization method incorporating a boundary tracking mesh generating method and nonlinear programming because the boundary tracking mesh is always conformed to the structural boundary good approximation to the boundary is maintained during optimization therefore structural design problems are solved completely without grayscale material previously we introduced the boundary tracking mesh generating method into level set based topology optimization and updated the design variables by solving the level set equation in order to adapt our previous method to general structural optimization frameworks the incorporation of the method with nonlinear programming is investigated in this paper to successfully incorporate nonlinear programming the optimization problem is regularized using a double well potential furthermore the sensitivities with respect to the design variables are strictly derived to maintain consistency in mathematical programming we expect the investigation to open up a new class of grayscale free topology optimization the usefulness of the proposed method is demonstrated using several numerical examples targeting two dimensional compliant mechanism and metallic waveguide design problems copyright 2014 john wiley sons ltd

a goal oriented error estimator for the analysis of simplified designs abstract simplifying an engineering design by removing geometric details can significantly reduce the complexity of downstream tasks of mesh generation and field solution computation however lack of a proper analysis and estimates of the errors induced by design simplification remains a bottleneck in seamless cad cae integration to address this issue this paper develops a general framework for providing a posteriori estimates of goal oriented engineering analysis error caused by removing a negative feature or void where material is absent from an engineering design our approach is built upon the idea of representing it as a modeling error defined in the same geometry which is then estimated using the dual weighted residual method the derived error estimator still involves an uncomputable term involving dual errors which is removed by further simplification into a term on the feature s boundary via an exterior solution in linear cases or problem specific approaches in nonlinear cases utilizing classical theories of differential operators we illustrate our approach via a semilinear second order elliptic equation and explicit error estimators are also derived for the classical poisson equation and linear elasticity results of numerical tests are shown and comparisons made with results obtained with related approaches

a fem based topological sensitivity approach for fast qualitative identification of buried cavities from elastodynamic overdetermined boundary data a time domain topological sensitivity ts approach is developed for elastic wave imaging of media of arbitrary geometry the ts which quantifies the sensitivity of the misfit cost functional to the creation at a specified location of an infinitesimal hole is expressed in terms of the time convolution of the free field and a supplementary adjoint field as a function of that specified location following previous studies performed under mostly static or time harmonic conditions the ts field is here considered as a natural and computationally efficient approach for defining a defect indicator function this study emphasizes the implementation and exploitation of ts fields using standard displacement based fem approaches a straightforward task once the correct sensitivity formulation is available a comprehensive set of numerical experiments on 3d and 2d elastodynamic and acoustic configurations is reported allowing to assess and highlight many features of the proposed ts based fast qualitative identification such as its ability to identify multiple defects and its robustness against data noise

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

modelling of topological derivatives for contact problems the problem of topology optimization is considered for free boundary problems of thin obstacle types the formulae for the first term of asymptotics for energy functionals are derived the precision of obtained terms is verified numerically the topological differentiability of solutions to variational inequalities is established in particular the so called outer asymptotic expansion for solutions of contact problems in elasticity with respect to singular perturbation of geometrical domain depending on small parameter are derived by an application of nonsmooth analysis such results lead to the topological derivatives of shape functionals for contact problems the topological derivatives are used in numerical methods of simultaneous shape and topology optimization

second order topological sensitivity analysis abstract the topological derivative provides the sensitivity of a given cost function with respect to the insertion of a hole at an arbitrary point of the domain classically this derivative comes from the second term of the topological asymptotic expansion dealing only with infinitesimal holes however for practical applications we need to insert holes of finite size therefore we consider one more term in the expansion which is defined as the second order topological derivative in order to present these ideas in this work we apply the topological shape sensitivity method as a systematic approach to calculate first as well as second order topological derivative for the poisson s equations taking the total potential energy as cost function and the state equation as constraint furthermore we also study the effects of different boundary conditions on the hole neumann and dirichlet both homogeneous finally we present some numerical experiments showing the influence of the second order topological derivative in the topological asymptotic expansion which has two main features it allows us to deal with hole of finite size and provides a better descent direction in optimization process

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

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

energy change to insertion of inclusions associated with the reissner mindlin plate bending model abstract the topological derivative concept has been proved to be useful in many relevant applications such as topology optimization inverse problems image processing multi scale constitutive modeling fracture mechanics and damage evolution modeling the topological asymptotic analysis has been fully developed for a wide range of problems modeled by partial differential equations on the other hand the topological derivatives associated with coupled problems have been derived only in their abstract forms in this paper therefore we deal with the reissner mindlin plate bending model which is written in the form of a coupled system of partial differential equations in particular the topological asymptotic analysis of the associated total potential energy is developed and the topological derivative with respect to the nucleation of a circular inclusion is derived in its closed form finally we provide the estimates for the remainders of the topological asymptotic expansion and perform a complete mathematical justification for the derived formulas

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

shape and topology optimization of compliant mechanisms using a parameterization level set method in this paper a parameterization level set method is presented to simultaneously perform shape and topology optimization of compliant mechanisms the structural shape boundary is implicitly embedded into a higher dimensional scalar function as its zero level set resultantly establishing the level set model by applying the compactly supported radial basis function with favorable smoothness and accuracy to interpolate the level set function the temporal and spatial hamilton jacobi equation from the conventional level set method is then discretized into a series of algebraic equations accordingly the original shape and topology optimization is now fully transformed into a parameterization problem namely size optimization with the expansion coefficients of interpolants as a limited number of design variables design of compliant mechanisms is mathematically formulated as a general optimization problem with a nonconvex objective function and two additionally specified constraints the structural shape boundary is then advanced as a process of renewing the level set function by iteratively finding the expansion coefficients of the size optimization with a sequential convex programming method it is highlighted that the present method can not only inherit the merits of the implicit boundary representation but also avoid some unfavorable features of the conventional discrete level set method such as the cfl condition restriction the re initialization procedure and the velocity extension algorithm finally an extensively investigated example is presented to demonstrate the benefits and advantages of the present method especially its capability of creating new holes inside the design domain

undercut and overhang angle control in topology optimization a density gradient based integral approach summary we present an approach for controlling the undercut and the minimal overhang angle in density based topology optimization which are useful for reducing support structures in additive manufacturing we cast both the undercut control and the minimal overhang angle control that are inherently constraints on the boundary shape into a domain integral of heaviside projected density gradient such a heaviside projection based integral of density gradient leads to a single constraint for controlling the undercut or controlling the overhang angle in the optimization it effectively corresponds to a constraint on the projected perimeter that has undercut or has slope smaller than the prescribed overhang angle in order to prevent trivial solutions of intermediate density to satisfy the density gradient constraints a constraint on density grayness is also incorporated into the formulations numerical results on messerschmitt bolkow blohm beams cantilever beams and 2d and 3d heat conduction problems demonstrate the proposed formulations are effective in controlling the undercut and the minimal overhang angle in the optimized designs copyright 2016 john wiley sons ltd

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

strain energy change to the insertion of inclusions associated to a thermo mechanical semi coupled system abstract the topological derivative measures the sensitivity of a given shape functional with respect to an infinitesimal singular domain perturbation according to the literature the topological derivative has been fully developed for a wide range of one single physical phenomenon modeled by partial differential equations in addition the topological asymptotic analysis associated to multi physics problems has been reported in the literature only on the level of mathematical analysis of singularly perturbed geometrical domains in this work we present the topological derivative in its closed form for the total potential mechanical energy associated to a thermo mechanical semi coupled system when a small circular inclusion is introduced at an arbitrary point of the domain in particular we consider the linear elasticity system modeled by the navier equation coupled with the steady state heat conduction problem modeled by the laplace equation the mechanical coupling term comes out from the thermal stress induced by the temperature field since this term is non local we introduce a non standard adjoint state which allows to obtain a closed form for the topological derivative finally we provide a full mathematical justification for the derived formulas and develop precise estimates for the remainders of the topological asymptotic expansion

pod assisted strategies for structural topology optimization abstract we propose a new numerical tool for structural optimization design to cut down the computational burden typical of the solid isotropic material with penalization simp method we apply proper orthogonal decomposition on simp snapshots computed on a fixed grid to construct a rough structure predictor which becomes the input of a simp procedure performed on an anisotropic adapted mesh corrector the benefit of the proposed design tool is to deliver smooth and sharp layouts which require a contained computational effort before moving to the 3d printing production phase

topological derivative based topology optimization of structures subject to drucker prager stress constraints abstract an algorithm for topology optimization of elastic structures under plane stress subject to the drucker prager stress constraint is presented the algorithm is based on the use of the topological derivative of the associated objective functional in conjunction with a level set representation of the structure domain in this context a penalty functional is proposed to enforce the point wise stress constraint and a closed formula for its topological derivative is derived the resulting algorithm is of remarkably simple computational implementation it does not require post processing procedures of any kind and features only a minimal number of user defined algorithmic parameters this is in sharp contrast with current procedures for topological structural optimization with local stress constraints the effectiveness and efficiency of the algorithm presented here are demonstrated by means of numerical examples the examples show in particular that it can easily handle structural optimization problems with underlying materials featuring strong asymmetry in their tensile and compressive yield strengths

asymptotic imaging of perfectly conducting cracks in this paper we consider cracks with dirichlet boundary conditions we first derive an asymptotic expansion of the boundary perturbations that are due to the presence of a small crack based on this formula we design a noniterative approach for locating a collection of small cracks in order to do so we construct a response matrix from the boundary measurements the location and the length of the crack are estimated respectively from the projection onto the noise space and the first significant singular value of the response matrix indeed the direction of the crack is estimated from the second singular vector we then consider an extended crack with dirichlet boundary conditions we rigorously derive an asymptotic expansion for the boundary perturbations that are due to a shape deformation of the crack to reconstruct an extended crack from many boundary measurements we develop two methods for obtaining a good guess several numerical experiments show how the proposed techniques for imaging small cracks as well as those for obtaining good initial guesses toward reconstructing an extended crack behave

topological sensitivity and shape optimization for the stokes equations the topological sensitivity analysis provides an asymptotic expansion of a shape function with respect to the insertion of a small hole or obstacle inside a domain this expansion can then be used for shape optimization in this paper such an expansion is obtained for the stokes equations with general shape functions and arbitrarily shaped holes a numerical example illustrates the use of the topological sensitivity in a shape optimization problem

stability of finite difference models containing two boundaries or interfaces it is known that the stability of finite difference models of hyperbolic initial boundary value problems is connected with the propagation and reflection of parasitic waves here the waves point of view is applied to models containing two boundaries or interfaces where repeated reflection of trapped packets is a potential new source of instability our analysis accounts for various known instability phenomena in a unified way and leads to several new results three of which are as follows 1 dissipativity does not ensure stability when three or more formulas are concatenated at a boundary or internal interface 2 algebraic gks instabilities can be converted by a second boundary to exponential instabilities only when an infinite numerical reflection coefficient is present 3 gks stability and p stability can be established in certain problems by showing that the numerical reflection coefficient matrices have norm less than one

asymptotic analysis polarization matrices and topological derivatives for piezoelectric materials with small voids asymptotic formulae for the mechanical and electric fields in a piezoelectric body with a small void are derived and justified such results are new and useful for applications in the field of design of smart materials in this way the topological derivatives of shape functionals are obtained for piezoelectricity the asymptotic formulae are given in terms of the so called polarization tensors matrices which are determined by the integral characteristics of voids the distinguishing feature of the piezoelectricity boundary value problems under consideration is the absence of positive definiteness of a differential operator which is non self adjoint two specific gibbs functionals of the problem are defined by the energy and the electric enthalpy the topological derivatives are defined in different manners for each of the governing functionals actually the topological derivative of the enthalpy functional is local i e defined by the pointwise values of the mechanical and electric fields which is contrary to the energy functional and some other suitable shape functionals which admit nonlocal topological derivatives i e depending on the whole problem data an example with weak interaction between mechanical and electric fields provides the analytic asymptotic expansions and can be used in numerical procedures of optimal design for smart materials

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

qualitative identification of cracks using 3d transient elastodynamic topological derivative formulation and fe implementation a time domain topological derivative td approach is developed for transient elastic wave imaging of buried cracks the td which quantifies the sensitivity of the misfit cost functional to the creation at a specified location of an infinitesimal trial crack is expressed in terms of the time convolution of the free field and an adjoint field as a function of that specified location and of the trial crack shape following previous studies on cavity identification in similar conditions the td field is here considered as a natural and computationally efficient approach for defining a crack location indicator function this study emphasizes the implementation and exploitation of td fields using the standard displacement based fem a straightforward exploitation of the relevant sensitivity formulation established here results on several numerical experiments on 3d elastodynamic and acoustic configurations are reported and discussed allowing to assess and highlight many features of the proposed td based fast qualitative crack identification including its ability to identify multiple cracks and its robustness against data noise

sounding of finite solid bodies by way of topological derivative this paper is concerned with an application of the concept of topological derivative to elastic wave imaging of finite solid bodies containing cavities building on the approach originally proposed in the elastostatic theory of shape optimization the topological derivative which quantifies the sensitivity of a featured cost functional due to the creation of an infinitesimal hole in the cavity free reference body is used as a void indicator through an assembly of sampling points where it attains negative values the computation of topological derivative is shown to involve an elastodynamic solution to a set of supplementary boundary value problems for the reference body which are here formulated as boundary integral equations for a comprehensive treatment of the subject formulas for topological sensitivity are obtained using three alternative methodologies namely i direct differentiation approach ii adjoint field method and iii limiting form of the shape sensitivity analysis the competing techniques are further shown to lead to distinct computational procedures methodologies i and ii are implemented within a bem based platform and validated against an analytical solution a set of numerical results is included to illustrate the utility of topological derivative for 3d elastic wave sounding of solid bodies an approach that may perform best when used as a pre conditioning tool for more accurate gradient based imaging algorithms despite the fact that the formulation and results presented in this investigation are established on the basis of a boundary integral solution the proposed methodology is readily applicable to other computational platforms such as the finite element and finite difference techniques

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

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

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

control of a mechanical aeration process via topological sensitivity analysis the topological sensitivity analysis method gives the variation of a criterion with respect to the creation of a small hole in the domain in this paper we use this method to control the mechanical aeration process in eutrophic lakes a simplified model based on incompressible navier stokes equations is used only considering the liquid phase which is the dominant one the injected air is taken into account through local boundary conditions for the velocity on the injector holes a 3d numerical simulation of the aeration effects is proposed using a mixed finite element method in order to generate the best motion in the fluid for aeration purposes the optimization of the injector location is considered the main idea is to carry out topological sensitivity analysis with respect to the insertion of an injector finally a topological optimization algorithm is proposed and some numerical results showing the efficiency of our approach are presented

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

narrow escape problem for the unit sphere homogenization limit optimal arrangements of large numbers of traps and the n 2 conjecture a narrow escape problem is considered to calculate the mean first passage time mfpt needed for a brownian particle to leave a unit sphere through one of its n small boundary windows traps a procedure is established to calculate optimal arrangements of n 1 equal small boundary traps that minimize the asymptotic mfpt based on observed characteristics of such arrangements a remarkable property is discovered that is the sum of squared pairwise distances between optimally arranged n traps on a unit sphere is integer equal to n 2 it is observed for 2 n 1004 with high precision it is conjectured that this is the case for such optimal arrangements for all n a dilute trap limit of homogenization theory when n can be used to replace the strongly heterogeneous dirichlet neumann mfpt problem with a spherically symmetric robin problem for which an exact solution is readily found parameters of the robin homogenization problem are computed that capture the first four terms of the asymptotic mfpt close agreement of asymptotic and homogenization mfpt values is demonstrated the homogenization approach provides a radically faster way to estimate the mfpt since it is given by a simple formula and does not involve expensive global optimization to determine locations of n 1 boundary traps

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

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

detection of point forces location using topological algorithm in stokes flows we consider a viscous incompressible fluid under the action of a finite number of particles located inside the flow domain each particle exerts a pointwise force on the fluid the main objective is the detection of point forces locations through minimization of a given performance functional the regularity lack of the state does not allow to derive optimalities conditions and build numerical algorithms to overcome this difficulty a relaxation technique is used and the detection problem is formulated as a topological optimization one a topological sensitivity analysis is derived and efficient numerical algorithms are proposed

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

topological sensitivity for 3d elastodynamic and acoustic inverse scattering in the time domain building on previous work for 3d inverse scattering in the frequency domain this article develops the concept of topological derivative for 3d elastic and acoustic wave imaging of media of arbitrary geometry using data in the time domain the topological derivative which quantifies the sensitivity of the cost functional associated with the inverse scattering problem due to the creation at a specified location of an infinitesimal hole for the elastodynamic case or rigid inclusion for the acoustic case is found to be expressed in terms of the time convolution of the free field and a supplementary adjoint field the derivation of the topological derivative follows the generic pattern proposed in previous studies which is transposable to a variety of other physical problems a numerical example where the featured cost function is defined in terms of synthetic data arising from the scattering of plane acoustic waves by a rigid spherical inclusion illustrates the utility of the topological derivative concept for defect identification using time varying data