binary spatial partitioning of the central difference time integration scheme for explicit fast transient dynamics this paper proposes a generalization of the explicit central difference time integration scheme using a time step variable not only in time but also in space the solution at each element node is advanced in time following local rather than global stability limitations this allows substantial saving of computer time in realistic applications with non uniform meshes especially in multi field problems like fluid structure interactions a binary scheme in space is used time steps are not completely arbitrary but stay in a constant ratio of two when passing from one partition level to the next one this choice greatly facilitates implementation via an integer based logic ensures inherent synchronization and avoids any interpolations necessary in other partitioning schemes in the literature but which may reduce numerical stability the mesh partition is automatically built up and continuously updated by simple spatial adjacency considerations the resulting algorithm deals automatically with large variations in time of stability limits the paper introduces the core spatial partitioning technique in the lagrangian formulation some academic numerical examples allow a detailed comparison with the standard spatially uniform algorithm a final more realistic example shows the application of partitioning in simulations with arbitrary lagrangian eulerian formulation and fully coupled boundary conditions fluid structure interaction copyright 2008 john wiley sons ltd

generalized topological derivative for the navier equation and inverse scattering in the time domain abstract in this work the concept of generalized topological sensitivity developed recently to deal with inverse elastodynamic scattering by penetrable obstacles in the frequency domain is extended to permit non invasive defect reconstruction and material characterization by means of transient elastic waves this quantity which signifies the first order perturbation of a given cost functional due to nucleation of an infinitesimal elastic defect in the reference defect free solid is intended to be used as an obstacle indicator though an assembly of interior sampling points where it attains pronounced negative values from an asymptotic analysis of the transient scattered field emanating from a defect with vanishing size in the reference elastic solid the expression for topological sensitivity explicit in terms of i the green s function and ii the free i e incident field for the reference domain is obtained by employing the so called direct approach and a convolution based boundary integral representation of the featured elastodynamic wavefields for generality the proposed formula is recast in terms of its adjoint field counterpart that caters for general reference domains where the green s function may not be available through numerical simulations it is shown that the generalized topological sensitivity in the time domain provides an effective preliminary tool for the 3d reconstruction and material identification of discrete internal heterogeneities the results also demonstrate that the defect illumination by transient elastic waveforms results in a higher quality of reconstruction relative to its time harmonic monotonal counterpart

a semi implicit level set method for structural shape and topology optimization this paper proposes a new level set method for structural shape and topology optimization using a semi implicit scheme structural boundary is represented implicitly as the zero level set of a higher dimensional scalar function and an appropriate time marching scheme is included to enable the discrete level set processing in the present study the hamilton jacobi partial differential equation pde is solved numerically using a semi implicit additive operator splitting aos scheme rather than explicit schemes in conventional level set methods the main feature of the present method is it does not suffer from any time step size restriction as all terms relevant to stability are discretized in an implicit manner the semi implicit scheme with additive operator splitting treats all coordinate axes equally in arbitrary dimensions with good rotational invariance hence the present scheme for the level set equations is stable for any practical time steps and numerically easy to implement with high efficiency resultantly it allows enhanced relaxation on the time step size originally limited by the courant friedrichs lewy cfl condition of the explicit schemes the stability and computational efficiency can therefore be greatly improved in advancing the level set evolvements furthermore the present method avoids additional cost to globally reinitialize the level set function for regularization purpose it is noted that the periodically applied reinitializations are time consuming procedures in particular the proposed method is capable of creating new holes freely inside the design domain via boundary incorporating splitting and merging processes which makes the final design independent of initial guess and helps reduce the probability of converging to a local minimum the availability of the present method is demonstrated with two widely studied examples in the framework of the structural stiffness designs

a posteriori error estimation for unilateral contact with matching and non matching meshes in this paper we consider the unilateral contact problem between elastic bodies we propose an error estimator based on the concept of error in the constitutive relation in order to evaluate the finite element approximation involving matching and non matching meshes on the contact zone the determination of the a posteriori error estimate is linked to the building of kinematically admissible stress fields and statically admissible stress fields we then propose a finite element method for approximating the unilateral contact problem taking into account matching and non matching meshes on the contact zone then we describe the construction of admissible fields lastly we present optimized computations by using both the error estimates and a convenient mesh adaptivity procedure 2000 elsevier science s a all rights reserved

a variational principle for the formulation of partitioned structural systems a continuum based variational principle is presented for the formulation of the discrete governing equations of partitioned structural systems this application includes coupled substructures as well as subdomains obtained by mesh decomposition the present variational principle is derived by a series of modifications of a hybrid functional originally proposed by atluri for finite element development the interface is treated by a displacement frame and a localized version of the method of lagrange multipliers interior displacements are decomposed into rigid body and deformational components to handle floating subdomains both static and dynamic versions are considered an important application of the present principle is the treatment of nonmatching meshes that arise from various sources such as separate discretization of substructures independent mesh refinement and global local analysis the present principle is compared with that of a globalized version of the multiplier method copyright 2000 john wiley sons ltd

a method of finite element tearing and interconnecting and its parallel solution algorithm a novel domain decomposition approach for the parallel finite element solution of equilibrium equations is presented the spatial domain is partitioned into a set of totally disconnected subdomains each assigned to an individual processor lagrange multipliers are introduced to enforce compatibility at the interface nodes in the static case each floating subdomain induces a local singularity that is resolved in two phases first the rigid body modes are eliminated in parallel from each local problem and a direct scheme is applied concurrently to all subdomains in order to recover each partial local solution next the contributions of these modes are related to the lagrange multipliers through an orthogonality condition a parallel conjugate projected gradient algorithm is developed for the solution of the coupled system of local rigid modes components and lagrange multipliers which completes the solution of the problem when implemented on local memory multiprocessors this proposed method of tearing and interconnecting requires less interprocessor communications than the classical method of substructuring it is also suitable for parallel vector computers with shared memory moreover unlike parallel direct solvers it exhibits a degree of parallelism that is not limited by the bandwidth of the finite element system of equations

discussion of second order topological sensitivity analysis by j rocha de faria et al abstract the aim of this discussion is to expose incorrect results in a previous ijss article

the nsluc property and klee envelope a notion called norm subdifferential local uniform convexity nsluc is introduced and studied it is shown that the property holds for all subsets of a banach space whenever the norm is either locally uniformly convex or k fully convex the property is also valid for all subsets of the banach space if the norm is kadec klee and its dual norm is g teaux differentiable off zero the nsluc concept allows us to obtain new properties of the klee envelope for example a connection between attainment sets of the klee envelope of a function and its convex hull it is also proved that the klee envelope with unit power plus an appropriate distance function is equal to some constant on an open convex subset as large as we need as a consequence of obtained results the subdifferential properties of the klee envelope can be inherited from subdifferential properties of the opposite of the distance function to the complement of the bounded convex open set moreover the problem of singleton property of sets with unique farthest point is reduced to the problem of convexity of chebyshev sets

sensitivity of the macroscopic response of elastic microstructures to the insertion of inclusions this paper proposes an exact analytical formula for the topological sensitivity of the macroscopic response of elastic microstructures to the insertion of circular inclusions the macroscopic response is assumed to be predicted by a well established multi scale constitutive theory where the macroscopic strain and stress tensors are defined as volume averages of their microscopic counterpart fields over a representative volume element rve of material the proposed formula a symmetric fourth order tensor field over the rve domain is a topological derivative which measures how the macroscopic elasticity tensor changes when an infinitesimal circular elastic inclusion is introduced within the rve in the limits when the inclusion matrix phase contrast ratio tends to zero and infinity the sensitivities to the insertion of a hole and a rigid inclusion respectively are rigorously obtained the derivation relies on the topological asymptotic analysis of the predicted macroscopic elasticity and is presented in detail the derived fundamental formula is of interest to many areas of applied and computational mechanics to illustrate its potential applicability a simple finite element based example is presented where the topological derivative information is used to automatically generate a bi material microstructure to meet pre specified macroscopic properties

on chebyshev functions and klee functions abstract associated to a lower semicontinuous function one can define its proximal mapping and farthest mapping the function is called chebyshev klee if its proximal mapping farthest mapping is single valued everywhere we show that the function f is 1 hypoconvex if its proximal mapping p f is single valued when the function f is bounded below and p f is single valued for every 0 the function must be convex similarly we show that the function f is 1 strongly convex if the farthest mapping q f is single valued when the function is the indicator function of a set this recovers the well known chebyshev problem and klee problem in r n we also give an example illustrating that a continuous proximal mapping farthest mapping needs not be locally lipschitz which answers one open question by hare and poliquin

a direct proof and a generalization for a kantorovich type inequality a direct proof for a kantorovich type inequality due to bauer and householder is presented a generalization of the inequality is also established by the theory of compound matrices

on a subproblem of trust region algorithms for constrained optimization we study a subproblem that arises in some trust region algorithms for equality constrained optimization it is the minimization of a general quadratic function with two special quadratic constraints properties of such subproblems are given it is proved that the hessian of the lagrangian has at most one negative eigenvalue and an example is presented to show that the hessian may have a negative eigenvalue when one constraint is inactive at the solution

on the stress wave imaging of cavities in a semi infinite solid the problem of mapping underground cavities from surface seismic measurements is investigated within the framework of a regularized boundary integral equation bie method with the ground modeled as a uniform elastic half space the inverse analysis of elastic waves scattered by a three dimensional void is formulated as a task of minimizing the misfit between experimental observations and theoretical predictions for an assumed void geometry for an accurate treatment of the gradient search technique employed to solve the inverse problem sensitivities of the predictive bie model with respect to cavity parameters are evaluated semi analytically using an adjoint problem approach and a continuum kinematics description several key features of the formulation including the rigorous treatment of the radiation condition for semi infinite solids modeling of an illuminating seismic wave field and treatment of the prior information are highlighted a set of numerical examples with spherical and ellipsoidal cavity geometries is included to illustrate the performance of the method it is shown that the featured adjoint problem approach reduces the computational requirements by an order of magnitude relative to conventional finite difference estimates thus rendering the three dimensional elastic wave imaging of solids tractable for engineering applications

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

development of pareto topology optimization considering thermal loads abstract in this research we developed a level set based topology optimization with a topological derivative formulation considering thermal load thermo elasticity equations were utilized to obtain the sensitivity of the objective function after inserting a small hole in the domain total strain energy and the maximum stress in the design domain were taken as the objective functions after taking the thermal loading effect into account the total strain energy density function became a nonhomogeneous function of the strain in addition temperature variation changed hooke s law from a linear homogeneous to a linear nonhomogeneous expression including a zero order term we derived the sensitivity value of the selected objective functions with respect to a perturbation in the structural domain under mechanical and thermal loads while considering these changes in the governing equations we performed several numerical optimization problems to demonstrate the validity of the present level set based pareto topology optimization two types of examples compliance and stress minimization were solved based on the chosen objective functions furthermore in the stress minimization examples the derived formula was extended to consider thermal effects in the failure theories for pressure independent and dependent materials

the moreau envelope function and proximal mapping in the sense of the bregman distance abstract in this paper we explore some properties of the moreau envelope function e f x of f and the associated proximal mapping p f x in the sense of the bregman distance induced by a convex function g precisely we study the continuity differentiability and clarke regularity of the moreau envelope function and the upper semicontinuity and single valuedness of the proximal mapping as well as its relation to the convexity of f g where is a positive parameter

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

sensitivity analysis in kinematically incompatible models abstract a natural question that arises when working with the so called kinematically incompatible models is about the positioning of the internal artificial boundary that splits the domain of analysis into the sub domains for which the different incompatible models are established although the experience in modeling might provide an ad hoc solution to such matter it may also draw misleading conclusions here the sensitivity analysis furnishes a systematic and reliable framework to study the impact of the positioning of the artificial boundary over the solution of the problem in this sense the present work is concerned with the sensitivity analysis of the variational formulation corresponding to the kinematically incompatible models when the boundary over which the incompatibility between the models takes place is changed assuming the presence of a discontinuity in the fields over a given interface between two incompatible models this analysis allows to measure the sensitivity of a given cost functional when the location of such interface is changed the application for which this analysis is envisaged is to assess the correctness or incorrectness in the definition of the positioning of such coupling interface between dimensionally heterogeneous domains also some numerical results are provided to give numerical evidence of the usefulness of the present tool

higher order topological derivatives in elasticity the topological derivative provides the variation of a response functional when an infinitesimal hole of a particular shape is introduced into the domain in this work we compute higher order topological derivatives for elasticity problems so that we are able to obtain better estimates of the response when holes of finite sizes are introduced in the domain a critical element of our algorithm involves the asymptotic approximation for the stress on the hole boundary when the hole size approaches zero it consists of a composite expansion that is based on the responses of elasticity problems on the domain without the hole and on a domain consisting of a hole in an infinite space we present a simple example in which the higher order topological derivatives of the total potential energy are obtained analytically and by using the proposed asymptotic expansion we also use the finite element method to verify the topological asymptotic expansion when the analytical solution is unknown

a staggered approach to shape and topology optimization using the traction method and an evolutionary type advancing front algorithm abstract we investigate a novel approach for structural shape optimization on the basis of complementary shape and topological sensitivity analysis as in early approaches to shape optimization the domain variation is specified by modification of boundary nodal points hence leading to an updated lagrangian description for the course of optimization to overcome the formation of oscillating boundaries in the optimal design trials we employ the traction method to establish smooth descent directions for shape variation therein an auxiliary elastic problem is solved in which the shape sensitivity constitutes the external loading and the deformation of the auxiliary elastic body is used as a descent direction for shape variation rather than the shape sensitivity itself we complement this method through an evolutionary type element removal procedure that is based on the topological sensitivity such that an advancing front algorithm is gradually removing elements from the design boundary of the domain once the minimum topological sensitivity is no longer encountered at the design boundary we create a hole in the domain again using the topological sensitivity to specify its exact location and resume the element removal procedure for the newly established design boundary since this approach yields only a vague estimate of the true optimal shape of newly established holes the traction method is again used for shape variation of the extended design boundary

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

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

topological and shape gradient strategy for solving geometrical inverse problems abstract in this paper we present a technique for shape reconstruction based on the topological and shape gradients the shape in consideration is a solution of an inverse conductivity problem to solve such a problem numerically we compute the topological gradient of a kohn vogelius type cost function when the domain under consideration is perturbed by the introduction of a small inclusion instead of a hole the reconstruction is done by considering the shape as a superposition of very thin elliptic inclusions to get a first approximation then we use a gradient type algorithm to perform a good reconstruction various numerical experiments of single and multiple inclusions demonstrate the viability of the designed algorithm

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

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

topology optimization of unsteady incompressible navier stokes flows this paper discusses the topology optimization of unsteady incompressible navier stokes flows an optimization problem is formulated by adding the artificial darcy frictional force into the incompressible navier stokes equations the optimization procedure is implemented using the continuous adjoint method and the finite element method the effects of dynamic inflow reynolds number and target flux on specified boundaries for the optimal topology of unsteady navier stokes flows are presented numerical examples demonstrate the feasibility and necessity of this topology optimization method for unsteady navier stokes flows

non conforming mesh gluing in the finite elements method a methodology to connect two or more non conforming meshes generated from a geometric model is presented the problem leads to a variational problem subject to constraints the emphasis of the work is on constraints construction two methods namely discrete connection and continuous connection are developed the relation with hybrid formulations is considered and two application examples are presented

multi scale domain decomposition method for large scale structural analysis with a zooming technique application to plate assembly this article is concerned with a multi scale domain decomposition method ddm based on the feti dp solver for large scale structural elastic analysis and suited to problems that exhibit structural heterogeneities such as plate assemblies in the presence of structural details in this approach once a partition of the global fine mesh into subdomains has been performed all subdomains possess a fine mesh and to optimize the computational time the fine mesh is preserved only in the zones of interest with local phenomena due to discontinuity hole etc while the remaining subdomains are replaced by numerical homogenized coarse elements indeed the multi scale aspect is introduced by the description of subdomains with either a fine or a coarse scale mesh as a result an extension of the feti dp ddm is proposed in this article called herein feti dp micro macro that allows the simultaneous usage of different discretizations fine microscopic mesh for subdomains in zones of interest and coarse macroscopic or homogenized mesh for the complementary part of the structure using this strategy raises the problem of the determination of the stiffness of coarse subdomains and of the incompatible finite element connection between fine and coarse subdomains two approaches collocation and mortar are presented and compared the article ends with patch tests and some numerical examples in 2d and 3d the obtained numerical results exemplify the efficiency and capability of the feti dp micro macro approach and reveal that the mortar approach is more accurate at constant cost than the collocation approach copyright 2009 john wiley sons ltd

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

an approach to the connection between subdomains with non matching meshes for transient mechanical analysis this paper describes a general method for coupling non matching linear finite element meshes in transient dynamic analysis we propose a method based on schur s dual formulation whose main advantage is to provide equilibrium as well as kinematic continuity throughout the interface the essence of our work lies in the particular discretization of the space of lagrange multipliers and in the validation of the method through two and three dimensional static calculations as well as two dimensional dynamic calculations an example is also presented and the results are compared to those of the mortar method copyright 2002 john wiley sons ltd

topological sensitivity analysis for three dimensional linear elasticity problem in this work we use the topological shape sensitivity method to obtain the topological derivative for three dimensional linear elasticity problems adopting the total potential energy as cost function and the equilibrium equation as constraint this method based on classical shape sensitivity analysis leads to a systematic procedure to calculate the topological derivative in particular firstly we present the mechanical model later we perform the shape derivative of the corresponding cost function and finally we calculate the final expression for the topological derivative using the topological shape sensitivity method and results from classical asymptotic analysis around spherical cavities in order to point out the applicability of the topological derivative in the context of topology optimization problems we use this information as a descent direction to solve a three dimensional topology design problem furthermore through this example we also show that the topological derivative together with an appropriate mesh refinement strategy are able to capture high quality shapes even using a very simple topology algorithm

a monolithic energy conserving method to couple heterogeneous time integrators with incompatible time steps in structural dynamics a new hybrid multi time method for multi time scales structural dynamics simulations is described a monolithic method in a schur dual domain decomposition framework is proposed and allows to consider heterogeneous time integrators with their own time discretization and possible large ratio between the time steps for each subdomain in the proposed method zero numerical dissipation is ensured at the interface this implies that the global stability of the coupling method is governed by the stability of each time integrator without influence of the interface for that purpose velocity continuity is ensured in a weak sense at the interfaces and time integrators newmark hht simo krenk verlet are introduced in a unified framework incremental velocity formulation furthermore dynamics governing equations are introduced from a weak formulation in time in other words equilibrium equation is no more ensured in a strong sense at a given time step but rather on average on a time interval some numerical examples illustrate the efficiency and the robustness of the proposed method for ratio of time scales close to 1000 without any numerical dissipation at the interfaces

shape optimisation with multiresolution subdivision surfaces and immersed finite elements abstract we develop a new optimisation technique that combines multiresolution subdivision surfaces for boundary description with immersed finite elements for the discretisation of the primal and adjoint problems of optimisation similar to wavelets multiresolution surfaces represent the domain boundary using a coarse control mesh and a sequence of detail vectors based on the multiresolution decomposition efficient and fast algorithms are available for reconstructing control meshes of varying fineness during shape optimisation the vertex coordinates of control meshes are updated using the computed shape gradient information by virtue of the multiresolution editing semantics updating the coarse control mesh vertex coordinates leads to large scale geometry changes and conversely updating the fine control mesh coordinates leads to small scale geometry changes in our computations we start by optimising the coarsest control mesh and refine it each time the cost function reaches a minimum this approach effectively prevents the appearance of non physical boundary geometry oscillations and control mesh pathologies like inverted elements independent of the fineness of the control mesh used for optimisation on the immersed finite element grid the domain boundary is always represented with a relatively fine control mesh of fixed resolution with the immersed finite element method there is no need to maintain an analysis suitable domain mesh in some of the presented two and three dimensional elasticity examples the topology derivative is used for introducing new holes inside the domain the merging or removing of holes is not considered

numerical implementation of two nonconforming finite element methods for unilateral contact we consider the finite element approximation of the unilateral contact problem between elastic bodies we are interested in a practical problem which often occurs in finite element computations concerning two independently discretized bodies in unilateral contact it follows that the nodes of both bodies located on the contact surface do not fit together we present two different approaches in order to define unilateral contact on nonmatching meshes the first is an extension of the mortar finite element method to variational inequalities that defines the contact in a global way on the contrary the second one expresses local node on segment contact conditions in both cases the theoretical approximation properties are given then we implement and compare the two methods

the topological asymptotic expansion for the dirichlet problem the topological sensitivity analysis provides an asymptotic expansion of a shape function when creating a small hole inside a domain this expansion yields a descent direction which can be used for shape optimization if one wishes to keep a classical domain throughout the optimization process in this paper such an expansion is obtained for the poisson equation for a large class of cost functions and arbitrarily shaped holes in the three dimensional case this expansion depends on the shape of the hole but not on its orientation if the cost function involves only the solution u to the underlying partial differential equation whereas it may also depend on its orientation if the cost function involves the gradient nabla u in contrast the asymptotic expansion is independent of the shape in the two dimensional case a numerical example illustrates the use of the asymptotic expansion which yields a minimizing sequence of classical domains in a case where no classical solution exists

using mesh adaption for the identification of a spatial field of material properties it is well known that the solution of an inverse problem is ill posed and not unique to avoid difficulties caused by this when solving such a problem tikhonov s regularization terms are usually added to the norm quantifying the discrepancy between the model s predictions and experimental data this regularization term however is often inadequate to perform the identification of a field of material properties that varies spatially this is all the more difficult when dealing with the numerical solution of this inverse problem for the sought field is spatially discretized and this discretization can influence the result of the identification we will here examine an overall strategy using classical adaptive meshing methods used to circumvent these drawbacks the first step consists of using two distinct meshes one associated with the discretization of the sought spatial field and the other associated with the solution of the mechanical problems forward and adjoint states in the second step we will introduce local error estimators that allow an oriented refinement of the mesh associated with the sought parameters this general strategy is applied to a practical case study the detection of underground cavities using experimental data obtained by an interferometric device on a satellite we will then address the question of how the regularization terms and the error estimator driving the mesh refinement were selected copyright 2011 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

connectivity of quadratic hypersurfaces and its applications in optimization part i general theory in this paper we establish a basic theory on the connectivity of quadratic hypersurfaces in addition we examine how both the connectivity and the hidden convexity of quadratic functions are related to each other and prove several connectivity theorems on quadratic functions that combine and generalize dines s and brickman s results on the hidden convexity of quadratic mappings in real field many classical results in quadratic optimization are reproved by the unified approach and more properties involving the matrix pencils are given

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

stabilized lagrange multiplier methods for bilateral elastic contact with friction in most finite element fe codes contact is checked only at the nodes corresponding to the use of pointwise constraints however this approach might not be stable in case the bodies coming into contact have non matching grids at the contact interface to alleviate this problem we propose a stabilized lagrange multiplier method based on a global polynomial multiplier for the finite element solution of non linear elastic contact problems with non matching grids in particular our approach allows us to avoid integrating products of different finite element basis functions on the surface meshes at the contact zone

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

isogeometric topology optimization using trimmed spline surfaces in the present work a new spline based topology optimization using trimmed spline surfaces and the isogeometric analysis is proposed in the proposed approach the trimmed surface analysis which can treat topologically complex spline surfaces using trimming information provided by cad systems is employed for structural response analysis and sensitivity calculation in the topology optimization the outer and inner boundaries of design models are represented by a spline surface and trimming curves design variables used in this approach are the coordinates of control points of a spline surface and those of trimming curves new sensitivity formulations for the control points in the trimmed surface analysis are proposed and their efficiency and accuracy are verified the creation of new inner fronts during optimization is allowed for the topological flexibility an inner front merging algorithm is also presented the proposed spline based topology optimization is used to solve some benchmarking problems design space dependency which is one of serious shortcomings in conventional topology optimization approaches is naturally eliminated by the proposed spline based optimization design dependent load problems which are difficult to treat with conventional grid based topology optimization methods are easily dealt with by the proposed one it is also shown that post processing effort for converting to cad model is eliminated by using the same spline information in numerical analysis and design optimization

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

multi time step explicit implicit method for non linear structural dynamics we present a method with domain decomposition to solve time dependent non linear problems this method enables arbitrary numeric schemes of the newmark family to be coupled with different time steps in each subdomain this coupling is achieved by prescribing continuity of velocities at the interface we are more specifically interested in the coupling of implicit explicit numeric schemes taking into account material and geometric non linearities the interfaces are modelled using a dual schur formulation where the lagrange multipliers represent the interfacial forces unlike the continuous formulation the discretized formulation of the dynamic problem is unable to verify simultaneously the continuity of displacements velocities and accelerations at the interfaces we show that within the framework of the newmark family of numeric schemes continuity of velocities at the interfaces enables the definition of an algorithm which is stable for all cases envisaged to prove this stability we use an energy method i e a global method over the whole time interval in order to verify the algorithms properties then we propose to extend this to non linear situations in the following cases implicit linear explicit non linear explicit non linear explicit non linear and implicit non linear explicit non linear finally we present some examples showing the feasibility of the method copyright 2001 john wiley sons ltd

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

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

shape sensitivity analysis and the energy momentum tensor for the kinematic and static models of torsion abstract the aim of this paper is to bridge shape sensitivity analysis and configurational mechanics by means of a widespread use of the shape derivative concept this technique will be applied as a systematic procedure to obtain the eshelby s energy momentum tensor associated to the problem under consideration in order to highlight special features of this procedure and without loss of generality we focus our attention in the application of shape sensitivity analysis to the problem of twisted straight bars within the framework of linear elasticity kinematic and static variational formulations as well as the direct method of sensitivity analysis are used to perform shape derivatives of both models integral expressions of first and second order shape derivatives of the total potential energy and the complementary potential energy with respect to an arbitrary transverse cross section shape change are achieved these integral expressions put in evidence the relationship between shape sensitivity analysis and the first and second order eshelby s energy momentum tensors also the null divergence property of these tensors is easily proved by comparing in each case the domain and boundary integral shape derivative arrived at finally an example with a known exact solution corresponding to an elastic bar with elliptical transverse cross section submitted to twist is presented in order to illustrate the usefulness of these tensors to compute the corresponding shape derivatives

a level set based topology optimization method for simultaneous design of elastic structure and coupled acoustic cavity using a two phase material model abstract this papers proposes a level set based topology optimization method for the simultaneous design of acoustic and structural material distributions in this study we develop a two phase material model that is a mixture of an elastic material and acoustic medium to represent an elastic structure and an acoustic cavity by controlling a volume fraction parameter in the proposed model boundary conditions at the two phase material boundaries are satisfied naturally avoiding the need to express these boundaries explicitly we formulate a topology optimization problem to minimize the sound pressure level using this two phase material model and a level set based method that obtains topologies free from grayscales the topological derivative of the objective functional is approximately derived using a variational approach and the adjoint variable method and is utilized to update the level set function via a time evolutionary reaction diffusion equation several numerical examples present optimal acoustic and structural topologies that minimize the sound pressure generated from a vibrating elastic structure

time domain transient elastodynamic analysis of 3 d solids by bem the numerical implementation of the direct boundary element formulation for time domain transient analysis of three dimensional solids is presented in a most general and complete manner the present formulation employs the space and time dependent fundamental solution stokes solution and graffi s dynamic reciprocal theorem to derive the boundary integral equations in the time domain a time stepping scheme is then used to solve the boundary initial value problem by marching forward in time higher order shape functions are used to approximate the field quantities in space as well as in time and a combination of analytical time integration and numerical spatial integration integration is carried out to form a system of linear equations at the end of each time step these equations are solved to obtain the unknown field quantities at that time finally the accuracy and reliability of this algorithm is demonstrated by solving a number of example problems and comparing the results against the available analytical and numerical solution

second derivatives of a convex function and of its legendre fenchel transformate in 1977 crouzeix established a simple relationship between the second order differentials of a convex function f mathbb r n to mathbb r cup infty and its legendre fenchel transformate f mathbb r n to mathbb r cup infty in the first part of this paper the importance of crouzeix s formula is enhanced by illustrating how it can be applied to a large number of classical and modern mathematical problems in the second part the result of crouzeix is extended to the case in which the functions f and f are not necessarily smooth this generalization is based on the works of hiriart urruty and seeger concerning the so called second order subdifferential of a convex function

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

a perturbed lagrangian formulation for the finite element solution of contact problems making use of a perturbed lagrangian formulation a finite element procedure for contact problems is developed for the general case in which node to node contact no longer holds the proposed procedure leads naturally to a discretization of the contact interface into contact segments within the context of a bilinear interpolation for the displacement field a mixed finite element approximation is introduced by assuming discont naous contact pressure constant on the contact segment because of this piece wise constant approximation the gap function enters into the formulation in an average sense instead of through a point wise definition numerical examples are presented that illustrate the performance of the proposed procedure current finite element formulations for contact problems based on either the classical lagrange parameter procedure l 3 12 203 or the penalty function method 4 6 11 are characterized by a point wise enforcement of the contact constraint condition in the sense that penetration of the bodies is established on a nodal basis moreover in this methods the recovery of the contact pressure over the element from the contact nodal forces generally requires an additional procedure within the framework of classical lagrange multiplier methods the contact condition is exactly satisfied by transforming the constrained problem into an unconstrained one with the introduction of additional variables lagrange multipliers these extra variables add computational effort to the solution process which often requires special procedures to handle the presence of zero diagonal terms penalty methods on the other hand enable one to transform the constrained problem into an unconstrained one without introducing additional variables the constraint condition is now satisfied only approximately for finite values of the penalty parameter the main difhculty associated with these methods however lies in the poor conditioning of the problem as the penalty is increased to more accurately enforce the constraint condition this is a well understood phenomenon particularly in the context of the incompressible and nearly incompressible problem in solid and fluid mechanics e g see l 22 25 for a review recently augmented lagrangian procedures have been proposed as a promising way to partially overcome these difficulties and regularize the penalty formulation e g see the survey in 7 8 within the framework of zinearited e u c it is possible to restrict the finite element formulation of contact problems by assuming that node to node contact occurs this is in fact the case often considered in the literature l 2 4 6 12 20 211 in the general context of fully

a new method for inverse electromagnetic casting problems based on the topological derivative the inverse electromagnetic casting problem consists in looking for a suitable set of electric wires such that the electromagnetic field induced by an alternating current passing through them makes a given mass of liquid metal acquire a predefined shape in this paper we propose a new method for the topology design of such inductors the inverse electromagnetic casting problem is formulated as an optimization problem and topological derivatives are considered in order to locate new wires in the right position several numerical examples are presented showing that the proposed technique is effective to design suitable inductors

differential properties of the moreau envelope in a vector space endowed with a uniformly g teaux differentiable norm it is proved that the moreau envelope enjoys many remarkable differential properties and that its subdifferential can be completely described through a certain approximate proximal mapping this description shows in particular that the moreau envelope is essentially directionally smooth new differential properties are derived for the distance function associated with a closed set moreover the analysis when applied to the investigation of the convexity of tchebyshev sets allows us to recover several known results in the literature and to provide some new ones

the shape and topological optimizations connection abstract in 1973 jean cea introduced a topological shape optimization algorithm based on a fixed point method the recent topological gradient concept provides a mathematical justification of cea s powerful method the aim of the topological gradient is to compute the sensitivity of a cost function when we drill a little hole in the domain some optimality conditions are derived when classical shape gradient and topological gradient are used simultaneously and numerical results are shown both in mechanics and electromagnetics

higher order topological sensitivity for 2 d potential problems application to fast identification of inclusions abstract this article concerns an extension of the topological derivative concept for 2 d potential problems involving penetrable inclusions whereby a cost function j is expanded in powers of the characteristic size e of a small inclusion the o e4 approximation of j is established for a small inclusion of given location shape and conductivity embedded in a 2 d region of arbitrary shape and conductivity and then generalized to several such inclusions simpler and more explicit versions of this result are obtained for a centrally symmetric inclusion and a circular inclusion numerical tests are performed on a sample configuration for i the o e4 expansion of potential energy and ii the identification of a hidden inclusion for the latter problem a simple approximate global search procedure based on minimizing the o e4 approximation of j over a dense search grid is proposed and demonstrated

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

potpourri of conjectures and open questions in nonlinear analysis and optimization we present a collection of fourteen conjectures and open problems in the fields of nonlinear analysis and optimization these problems can be classified into three groups problems of pure mathematical interest problems motivated by scientific computing and applications and problems whose solutions are known but for which we would like to know better proofs for each problem we provide a succinct presentation a list of appropriate references and a view of the state of the art of the subject

defeaturing a posteriori error analysis via feature sensitivity it is well known that small geometric features within a cad model can significantly impact the computational cost and often undermine the reliability of finite element analysis engineers therefore resort to defeaturing or detail removal wherein the offending features are suppressed prior to computational analysis however this results in a defeaturing induced analysis error in this paper we estimate this error in an a posteriori sense through the novel concept of feature sensitivity the latter determines the first order change in quantities of interest when an arbitrary cluster of small geometric features is deleted from a model a formal theory and a set of associated algorithms are provided to compute the feature sensitivity associated with a scalar elliptic partial differential equation the theory is supported through numerical experiments in 2 d involving both internal and boundary features copyright 2008 john wiley sons ltd

on the discrete variant of the traction method in parameter free shape optimization abstract one of the major challenges in the parameter free approach to computational shape optimization is the avoidance of oscillating i e non smooth boundaries in the optimal design trials in order to achieve this we investigate a method for regularization that corresponds to the discrete variant of the so called traction method in this approach the design updates are generated in terms of a displacement field which is obtained as the solution to an auxiliary boundary value problem that is defined on the actual design domain the main idea herein is to apply fictitious nodal forces corresponding to the discrete sensitivity of the objective function we propose an algorithm in which constraint functions will be taken into account by using an augmented lagrangian formulation and a step length control ensures a sufficient decrease condition in terms of the objective function within each iteration we examine the benefits of the proposed regularization method on the basis of some numerical examples in comparison to an unregularized steepest descent algorithm

a topology optimization method based on the level set method incorporating a fictitious interface energy this paper proposes a new topology optimization method which can adjust the geometrical complexity of optimal configurations using the level set method and incorporating a fictitious interface energy derived from the phase field method first a topology optimization problem is formulated based on the level set method and the method of regularizing the optimization problem by introducing fictitious interface energy is explained next the reaction diffusion equation that updates the level set function is derived and an optimization algorithm is then constructed which uses the finite element method to solve the equilibrium equations and the reaction diffusion equation when updating the level set function finally several optimum design examples are shown to confirm the validity and utility of the proposed topology optimization method

a level set based topology optimisation for acoustic elastic coupled problems with a fast bem fem solver abstract this paper presents a structural optimisation method in three dimensional acoustic elastic coupled problems the proposed optimisation method finds an optimal allocation of elastic materials which reduces the sound level on some fixed observation points in the process of the optimisation configuration of the elastic materials is expressed with a level set function and the distribution of the level set function is iteratively updated with the help of the topological derivative the topological derivative is associated with state and adjoint variables which are the solutions of the acoustic elastic coupled problems in this paper the acoustic elastic coupled problems are solved by a bem fem coupled solver in which the fast multipole method fmm and a multi frontal solver for sparse matrices are efficiently combined along with the detailed formulations for the topological derivative and the bem fem coupled solver we present some numerical examples of optimal designs of elastic sound scatterer to manipulate sound waves from which we confirm the effectiveness of the present method

a time and space mortar method for coupling linear modal subdomains and non linear subdomains in explicit structural dynamics this paper deals with an extension of dynamic subdomain methods it relies on time multi scale algorithms in the particular case of explicit dynamics it introduces new opportunities in terms of proper modeling on each subdomain adapted to its effective loading in order to reduce the calculation time of the simulations treatment of non matching meshes and local modal projection methods are thus especially discussed the efficiency of the method is then illustrated with two large scale examples

minimizing within convex bodies using a convex hull method we present numerical methods to solve optimization problems on the space of convex functions or among convex bodies hence convexity is a constraint on the admissible objects whereas the functionals are not required to be convex to deal with this our method mixes geometrical and numerical algorithms we give several applications arising from classical problems in geometry and analysis alexandrov s problem of finding a convex body of prescribed surface function cheeger s problem of a subdomain minimizing the ratio surface area on volume newton s problem of the body of minimal resistance in particular for the latter application the minimizers are still unknown except in some particular classes we give approximate solutions better than the theoretical known ones hence demonstrating that the minimizers do not belong to these classes

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