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This article examines the accuracy for large times of asymptotic expansions from periodic homogenization of wave equations. As usual, $\epsilon$ denotes the small period of the coefficients in the wave equation. We first prove that the standard two scale asymptotic expansion provides an accurate approximation of the exact solution for times $t$ of order $\epsilon^{-2+\delta}$ for any $\delta>0$. Second, for longer times, we show that a different algorithm, that is called criminal because it mixes different powers of $\epsilon$, yields an approximation of the exact solution with error $O(\epsilon^N)$ for times $\epsilon^{-N}$ with $N$ as large as one likes. The criminal algorithm involves high order homogenized equations that, in the context of the wave equation, were first proposed by Santosa and Symes and analyzed by Lamacz. The high order homogenized equations yield dispersive corrections for moderate wave numbers. We give a systematic analysis for all time scales and all high order corrective terms.

We study topology optimization in quasi-static plasticity with linear kinematic and linear isotropic hardening using a level-set method. We consider the primal variational formulation for the plasticity problem. This formulation is subjected to penalization and regularization, resulting in an approximate problem that is shape-dierentiable. The shape derivative for the approximate problem is computed using the adjoint method. Thanks to the proposed penalization and regularization, the time discretization of the adjoint problem is proved to be well-posed. For comparison purposes, the shape derivative for the original problem is computed in a formal manner. Finally, shape and topology optimization is performed numerically using the level-set method, and 2D and 3D case studies are presented. Shapes are captured exactly using a body-tted mesh at every iteration of the optimization algorithm.

In the context of a diffusion equation, this work is devoted to a two-phase optimal design problem where the interface, separating the phases, is imperfect, meaning that the solution is discontinuous while the normal flux is continuous and proportional to the jump of the solution. The shape derivative of an objective function with respect to the interface position is computed by the adjoint method. Numerical experiments are performed with the level set method and an exact remeshing algorithm so that the interface is captured by the mesh at each optimization iteration. Comparisons with a perfect interface are discussed in the setting of optimal design or inverse problems.

This paper is concerned with shape and topology optimization of parts and their supports, taking into account constraints coming from the metal powder bed additive manufacturing process. Despite the high complexity of this process, it is represented by the simple inherent strain model, which has the advantage of being computationally cheap. Three optimization criteria, evaluated with this model, are proposed to minimize defects caused by additive manufacturing: vertical displacements, residual stresses and deflection of the part after baseplate separation. Combining these criteria with a constraint on the compliance for the final use of the part leads to optimization problems which deliver optimized manufacturable shapes with only a slight loss on the final use performance. The numerical results are assessed by manufacturing some optimized and reference geometries. These experimental results are also used to calibrate the inherent strain model by an inverse analysis. The same type of optimization is applied to supports in the case of a fixed non-optimizable part. For our 3-d numerical tests we rely on the level set method, the notion of shape derivatives and an augmented Lagrangian algorithm for optimization.

Supports are an important ingredient of the building process of structures by additive manufacturing technologies. They are used to reinforce overhanging regions of the desired structure and/or to facilitate the mitigation of residual thermal stresses due to the extreme heat flux produced by the source term (laser beam). Very often, supports are, on purpose, weakly connected to the built structure for easing their removal. In this work, we consider an imperfect interface model for which the interaction between supports and the built structure is not ideal, meaning that thedisplacement is discontinuous at the interface while the normal stress is continuous and proportional to the jump of the displacement. The optimization process is based on the level set method, body-fitted meshes and the notion of shape derivative using the adjoint method. We provide 2-d and 3-d numerical examples, as well as a comparison with the usual perfect interface model. Completely different designs of supports are obtained with perfect or imperfect interfaces.

This paper studies the partial differential equation describing the charge distribution of an electrolyte in a porous medium.Realistic non-ideal effects are incorporated through the mean spherical approximation(MSA) model which takes into account finite size ions and screening effects. The main novelty is the consideration of a non-constant surface charge density on the pore walls. Indeed, a chemical equilibrium reaction is considered on the boundary to represent the dissociation of ionizable sites on the solid walls. The surface charge density is thus given as a non-linear function of the electrostatic potential. Even in the ideal case, the resulting system is a new variant of the famous Poisson-Boltzmann equation, which still has a monotone structure under quantitative assumptions on the physical parameters. In the non-ideal case, the MSA model brings in additional non-linearities which break down the monotone structure of the system. We prove existence, and sometimes uniqueness, of the solution. Some numerical experiments are performed in 2-d to compare this model with that for a constant surface charge.

This paper investigates path planning strategies for additive manufacturing processes such as powder bed fusion. The state of the art mainly studies trajectories based on existing patterns. Parametric optimization on these patterns and allocating them to the object areas are the main strategies. We propose in this work a more systematic optimization approach without any a priori restriction on the trajectories. The typical optimization problem is to melt the desired structure, without over-heating (to avoid thermally induced residual stresses) and possibly with a minimal path length. The state equation is the heat equation with a source term depending on the scanning path. First, in a steady-state context, shape optimization tools are applied to trajectories. Second, for time-dependent problems, an optimal control method is considered instead. In both cases, gradient-type algorithms are deduced and tested on 2-D examples. Numerical results are discussed, leading to a better understanding of the problem and thus to short- and long-term perspectives.

One of the issues for the automotive industry is weight reduction. For this purpose, topology optimization is used for mechanical parts and usually involves a single part. Its connections to other parts are assumed to be fixed. This paper deals with a coupled topology optimization of both the structure of a part and its connections (location and number) to other parts. The present work focuses on two models of connections, namely rigid support and spring that prepares work for bolt connection. Rigid supports are modeled by Dirichlet boundary conditions while bolt-like connections are idealized and simplified as a non-local interaction to be representative enough at a low computational cost. On the other hand, the structure is modeled by the linearized elasticity system and its topology is represented by a level set function. A coupled optimization of the structure and the location of rigid supports is performed to minimize the volume of an engine accessories bracket under a compliance constraint. This coupled topology optimization (shape and connections) provides more satisfactory performance of a part than the one given by classical shape optimization alone. The approach presented in this work is therefore one step closer to the optimization of assembled mechanical systems. Thereafter, the concept of topological derivative is adapted to create an idealized bolt. The main idea is to add a small idealized bolt at the best location and to test the optimality of the solution with this new connection. The topological derivative is tested with a 3d academic test case for a problem of compliance minimization.

In the formulation of shape optimization problems, multiple geometric constraint functionals involve the signed distance function to the optimized shape Ω. The numerical evaluation of their shape derivatives requires to integrate some quantities along the normal rays to Ω, a challenging operation to implement, which is usually achieved thanks to the method of characteristics. The goal of the present paper is to propose an alternative, variational approach for this purpose. Our method amounts, in full generality, to compute integral quantities along the characteristic curves of a given velocity field without requiring the explicit knowledge of these curves on the spatial discretization; it rather relies on a variational problem which can be solved conveniently by the finite element method. The well-posedness of this problem is established thanks to a detailed analysis of weighted graph spaces of the advection operator β ⋅ ∇ associated to a C1 velocity field β. One novelty of our approach is the ability to handle velocity fields with possibly unbounded divergence: we do not assume div(β) ∈ L∞. Our working assumptions are fulfilled in the context of shape optimization of C2 domains Ω, where the velocity field β = ∇dΩ is an extension of the unit outward normal vector to the optimized shape. The efficiency of our variational method with respect to the direct integration of numerical quantities along rays is evaluated on several numerical examples. Classical albeit important implementation issues such as the calculation of a shape’s curvature and the detection of its skeleton are discussed. Finally, we demonstrate the convenience and potential of our method when it comes to enforcing maximum and minimum thickness constraints in structural shape optimization.

In the framework of the level-set method we propose a topology optimization algorithm for linear elastic structures which can exhibit fractures. In the spirit of Grith theory, brittle fracture is modeled by the Francfort-Marigo energy model, with its Ambrosio-Tortorelli regularization, which can also be viewed as a gradient damage model. This quasi-static and irreversible gradient damage model is approximated using penalization to make it amenable to shape-dierentiation. The shape derivative is determined using the adjoint method. The shape optimization is implemented numerically using a level-set method with body-tted remeshing, which captures shapes exactly while allowing for topology changes. The eciency of the proposed method is demonstrated numerically on 2D and 3D test cases. The method is shown to be ecient in conceiving crack-free structures.

This work introduces a new coupled topology optimization approach for a structural assembly. Considering several parts connected by bolts, the shape and topology of potentially each part, as well as the position and number of bolts are simultaneously optimized. The main ingredients of our optimization approach are the level-set method for structural optimization, a new notion of topological derivative of an idealized model of bolt in order to decide where it is advantageous to add a new bolt, coupled with a parametric gradient-based algorithm for its position optimization. Both idealized bolt and its topological derivative handle prestressed state complexity. Several 3d numerical test cases are performed to demonstrate the efficiency of the proposed strategy for mass minimization, considering Von Mises and fatigue constraints for the bolts and compliance constraint for the structure. In particular, a simplified but industrially representative example of an accessories bracket for car engines demonstrates significant benefits. Optimizing both the structure and its connections reduces the mass by 24% compared to classical "structure-only" optimization.

Lattice structures are periodic porous bodies which are becoming popular since they are a good compromise between rigidity and weight and can be built by additive manufacturing techniques. Their optimization has recently attracted some attention, based on the homogenization method, mostly for compliance minimization [1], [2], [3]. The goal of the present two-part work is to extend this lattice optimization to an objective function involving stress minimization. As is well known in structural optimization, stress optimization is a very dicult problem. While the second part of our work will be devoted to the macroscopic optimization process itself, the present rst part is devoted to the choice of a parametrized periodicity cell that will be optimally selected in the second part of this work. Designing the right periodicity cell is of paramount importance for the success of the optimization process. For manufacturability reasons it is crucial that this cell is parametrized by just a few parameters. According to homogenization theory, one has to compute the eective elasticity tensor, as well as the corrector terms accounting for possible stress concentrations at the cell or microscopic scale. For compliance minimization a standard choice in 2-d is a square cell with a rectangular hole, or a rank-2 laminate. However, since these microstructures feature corners, they are not optimal for stress minimization. Therefore we propose a square cell with a super-ellipsoidal hole which exhibits no corners. This type of cell is parametrized in 2-d by one orientation angle, two semi-axis and an exponent in its dening equation which can be interpreted as a corner smoothing parameter. We rst analyse the inuence of these parameters on the stress norm by performing some numerical experiments. Second, the optimal corner smoothing parameter is found for each possible micro-structure and macroscopic stress. In order to obtain an optimal micro-structure that depends only on geometrical parameters and not on the stress value, we further average (with specic weights) the optimal smoothing exponent with respect to the macroscopic stress. For simplicity, the optimal values of the corner smoothing parameter are interpolated by an analytical approximated formula. Finally, to validate the results, we compare our optimal super-ellipsoidal hole with the Vigdergauz micro-structure which is known to be optimal for stress minimization in some special cases.

We introduce a model and several constraints for shape and topology optimization of structures, built by additive manufacturing techniques. The goal of these constraints is to take into account the thermal residual stresses or the thermal deformations, generated by processes like Selective Laser Melting, right from the beginning of the structural design optimization. In other words, the structure is optimized concurrently for its final use and for its behavior during the layer-by-layer production process. It is well known that metallic additive manufacturing generates very high temperatures and heat fluxes, which in turn yield thermal deformations that may prevent the coating of a new powder layer, or thermal residual stresses that may hinder the mechanical properties of the final design. Our proposed constraints are targeted to avoid these undesired effects. Shape derivatives are computed by an adjoint method and are incorporated into a level set numerical optimization algorithm. Several 2D and 3D numerical examples demonstrate the interest and effectiveness of our approach.

PurposeThis paper is devoted to the theoretical and numerical study of a new topological sensitivity concerning the insertion of a small bolt connecting two parts in a mechanical structure. First, an idealized model of bolt is proposed which relies on a non-local interaction between the two ends of the bolt (head and threads) and possibly featuring a pre-stressed state. Second, a formula for the topological sensitivity of such an idealized bolt is rigorously derived for a large class of objective functions. Third, numerical tests are performed in 2D and 3D to assess the efficiency of the bolt topological sensitivity in the case of no pre-stress. In particular, the placement of bolts (acting then as springs) is coupled to the further optimization of their location and to the shape and topology of the structure for volume minimization under compliance constraint.Design/methodology/approachThe methodology relies on the adjoint method and the variational formulation of the linearized elasticity equations in order to establish the topological sensitivity.FindingsThe numerical results prove the influence of the number and locations of the bolts which strongly influence the final optimized design of the structure.Originality/valueThis paper is the first one to study the topology optimization of bolted systems without a fixed prescribed number of bolts.

In this paper, scanning paths optimization for the powder bed fusion additive manufacturing process is investigated. The path design is a key factor of the manufacturing time and for the control of residual stresses arising during the building, since it directly impacts the temperature distribution. In the literature, the scanning paths proposed are mainly based on existing patterns, the relevance of which is not related to the part to build. In this work, we propose an optimization algorithm to determine the scanning path without a priori restrictions. Taking into account the time dependence of the source, the manufacturing time is minimized under two constraints: melting the required structure and avoiding any over-heating causing thermally induced residual stresses. The results illustrate how crucial the part’s shape and topology is in the path quality and point out promising leads to define path and part design constraints.

International audience; We present a topology optimization approach for the design of fluid-to-fluid heat exchangers which rests on an explicit meshed discretization of the phases at stake, at every iteration of the optimization process. The considered physical situations involve a weak coupling between the Navier-Stokes equations for the velocity and the pressure in the fluid, and the convection-diffusion equation for the temperature field. The proposed framework combines several recent techniques from the field of shape and topology optimization, and notably a level-set based mesh evolution algorithm for tracking shapes and their deformations , an efficient optimization algorithm for constrained shape optimization problems, and a numerical method to handle a wide variety of geometric constraints such as thickness constraints and non-penetration constraints. Our strategy is applied to the optimization of various types of heat exchangers. At first, we consider a simplified 2D cross-flow model where the optimized boundary is the section of the hot fluid phase flowing in the transverse direction, which is naturally composed of multiple holes. A minimum thickness constraint is imposed on the cross-section so as to account for manufacturing and maximum pressure drop constraints. In a second part, we optimize the design of 2D and 3D heat exchangers composed of two types of fluid channels (hot and cold), which are separated by a solid body. A non-mixing constraint between the fluid components containing the hot and cold phases is enforced by prescribing a minimum distance between them. Numerical results are presented on a variety of test cases, demonstrating the efficiency of our approach in generating new, realistic, and unconventional heat exchanger designs.

This article is concerned with the coupled optimization of the external boundary of a structure and its infill made of some graded lattice material. The lattice material is made of a periodic cell, macroscopically modulated and oriented. The external boundary may be coated by a layer of pure material with a fixed prescribed thickness. The infill is optimized by the homogenization method while the macroscopic shape is geometrically optimized by the Hadamard method of shape sensitivity. A first original feature of the proposed approach is that the infill material follows the displacement on the exterior boundary during the geometric optimization step. A second key feature is the dehomogenization or projection step which build a smoothly varying lattice infill from the optimal homogenized properties. Several numerical examples illustrate the effectiveness of our approach in 2-d, which is especially convenient when considering design-dependent loads. Agence Nationale de la Recherche

The purpose of this article is to introduce a gradient-flow algorithm for solving generic equality or inequality constrained optimization problems, which is suited for shape optimization applications. We rely on a variant of the Ordinary Differential Equation (ODE) approach proposed by Yamashita for equality constrained problems: the search direction is a combination of a null space step and a range space step, which are aimed to reduce the value of the minimized objective function and the violation of the constraints, respectively. Our first contribution is to propose an extension of this ODE approach to optimization problems featuring both equality and inequality constraints. In the literature, a common practice consists in reducing inequality constraints to equality constraints by the introduction of additional slack variables. Here, we rather solve their local combinatorial character by computing the projection of the gradient of the objective function onto the cone of feasible directions. This is achieved by solving a dual quadratic programming subproblem whose size equals the number of active or violated constraints, and which allows to identify the inequality constraints which should remain tangent to the optimization trajectory. Our second contribution is a formulation of our gradient flow in the context of-infinite-dimensional-Hilbert space settings. This allows to extend the method to quite general optimization sets equipped with a suitable manifold structure, and notably to sets of shapes as it occurs in shape optimization with the framework of Hadamard's boundary variation method. The cornerstone of this latter setting is the classical operation of extension and regularization of shape derivatives. Some numerical comparisons on simple academic examples are performed to illustrate the behavior of our algorithm. Its numerical efficiency and ease of implementation are finally demonstrated on more realistic shape optimization problems.

International audience; Supports are often required to safely complete the building of complicated structures by additive manufacturing technologies. In particular, supports are used as scaffoldings to reinforce overhanging regions of the structure and/or are necessary to mitigate the thermal deformations and residual stresses created by the intense heat flux produced by the source term (typically a laser beam). However, including supports increase the fabrication cost and their removal is not an easy matter. Therefore, it is crucial to minimize their volume while maintaining their efficiency. Based on earlier works, we propose here some new optimization criteria. First, simple geometric criteria are considered like the projected area and the volume of supports required for overhangs: they are minimized by varying the structure orientation with respect to the baseplate. In addition, an accessibility criterion is suggested for the removal of supports, which can be used to forbid some parts of the structure to be supported. Second, shape and topology optimization of supports for compliance minimization is performed. The novelty comes from the applied surface loads which are coming either from pseudo gravity loads on overhanging parts or from equivalent thermal loads arising from the layer by layer building process. Here, only the supports are optimized, with a given non-optimizable structure, but of course many generalizations are possible, including optimizing both the structure and its supports. Our optimization algorithm relies on the level set method and shape derivatives computed by the Hadamard method. Numerical examples are given in 2-d and 3-d.

International audience; We study dispersive effects of wave propagation in periodic media, which can be modelled by adding a fourth-order term in the homogenized equation. The corresponding fourth-order dispersive tensor is called Burnett tensor and we numerically optimize its values in order to minimize or maximize dispersion. More precisely, we consider the case of a two-phase composite medium with an 8-fold symmetry assumption of the periodicity cell in two space dimensions. We obtain upper and lower bound for the dispersive properties, along with optimal microgeometries.

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

International audience; We consider the optimal design of composite laminates by allowing a variable stacking sequence and in-plane shape of each ply. In order to optimize both variables we rely on a decomposition technique which aggregates the constraints into one unique constraint margin function. Thanks to this approach, a rigorous equivalent bi-level optimization problem is established. This problem is made up of an inner level represented by the combinatorial optimization of the stacking sequence and an outer level represented by the topology and geometry optimization of each ply. We propose for the stacking sequence optimization an outer approximation method which iteratively solves a set of mixed integer linear problems associated to the evaluation of the constraint margin function. For the topology optimization of each ply, we lean on the level set method for the description of the interfaces and the Hadamard method for boundary variations by means of the computation of the shape gradient. Numerical experiments are performed on an aeronautic test case where the weight is minimized subject to different mechanical constraints, namely compliance, reserve factor and buckling load.

International audience; Hadamard's method of shape differentiation is applied to topology optimization of a weakly coupled three physics problem. The coupling is weak because the equations involved are solved consecutively, namely the steady state Navier-Stokes equations for the fluid domain, first, the convection diffusion equation for the whole domain, second, and the linear thermo-elasticity system in the solid domain, third. Shape sensitivities are derived in a fully Lagrangian setting which allows us to obtain shape derivatives of general objective functions. An emphasis is given on the derivation of the adjoint interface condition dual to the one of equality of the normal stresses at the fluid solid interface. The arguments allowing to obtain this surprising condition are specifically detailed on a simplified scalar problem. Numerical test cases are presented using the level set mesh evolution framework of [4]. It is demonstrated how the implementation enables to treat a variety of shape optimization problems. keywords. Topology and shape optimization, adjoint methods, fluid structure interaction, convective heat transfer, adaptive remeshing.

International audience; Porosity is a well-known phenomenon occurring during various manufacturing processes (casting, welding, additive manufacturing) of solid structures, which undermines their reliability and mechanical performance. The main purpose of this article is to introduce a new constraint functional of the domain which controls the negative impact of porosity on elastic structures in the framework of shape and topology optimization. The main ingredient of our modelling is the notion of topological derivative, which is used in a slightly unusual way: instead of being an indicator of where to nucleate holes in the course of the optimization process, it is a component of a new constraint functional which assesses the influence of pores on the mechanical performance of structures. The shape derivative of this constraint is calculated and incorporated into a level set based shape optimization algorithm. Our approach is illustrated by several two-and three-dimensional numerical experiments of topology optimization problems constrained by a control on the porosity effect.