Anca-Maria Toader

A level-set method for shape optimization

We study a level-set method for numerical shape optimization of elastic structures. Our approach combines the level-set algorithm of Osher and Sethian with the classical shape gradient. Although this method is not specifically designed for topology optimization, it can easily handle topology changes for a very large class of objective functions. Its cost is moderate since the shape is captured on a fixed Eulerian mesh. To cite this article: G. Allaire et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 1125–1130.

Scale Convergence in Homogenization

In order to treat non-periodic oscillations we extend the concept of two-scale convergence, using Young measures. We present examples and applications.

Optimization of bodies with locally periodic microstructure

This article describes a numerical study of the optimization of elastic bodies featuring a locally periodic microscopic pattern. The authors’ approach makes the link between the microscopic level and the macroscopic one. Two-dimensional linearly elastic bodies are considered; the same techniques can be applied to three-dimensional bodies. Homogenization theory is used to describe the macroscopic (effective) elastic properties of the body. The macroscopic domain is divided in (rectangular) finite elements and in each of them the microstructure is supposed to be periodic; the periodic pattern is allowed to vary from element to element. Each periodic microstructure is discretized using a finite element mesh on the periodicity cell, by identifying the opposite sides of the cell in order to handle the periodicity conditions in the cellular problem. Shape optimization and topology optimization are used at the microscopic level, following an alternate directions algorithm. Numerical examples are presented, in which a cantilever is optimized for different load cases, one of them being multi-load. The problem is numerically heavy, since the optimization of the macroscopic problem is performed by optimizing in simultaneous hundreds or even thousands of periodic structures, each one using its own finite element mesh on the periodicity cell. Parallel computation is used in order to alleviate the computational burden.

Structural Optimization by the Level-Set Method

In the context of structural optimization, we describe a new numerical method based on a combination of the classical shape derivative and of the level-set method for front propagation. We have implemented this method in two and three space dimensions for models of linear or non-linear elasticity, with various objective functions and constraints on the volume or on the perimeter. The shape derivative is computed by an adjoint method. The cost of our numerical algorithm is moderate since the shape is captured on a fixed Eulerian mesh. Although this method is not specifically designed for topology optimization, it can easily handle topology changes.

The Topological Derivative of Homogenized Elastic Coefficients of Periodic Microstructures

In the present paper we compute in full mathematical rigor the topological derivative of the elastic homogenized coefficients of periodic microstructures. The expression, here proven for the topological derivative, was successfully used for optimizing homogenized coefficients in an alternate directions optimization algorithm (jointly with the shape derivative) in [C. Barbarosie and A.-M. Toader, Struct. Multidiscip. Optim., 40 (2010), pp. 393–408]. This optimization problem can be viewed as a control problem: the homogenized coefficients are controlled by the shape/topology of the subdomain occupied by material in the cellular problem. The main ingredients for proving the formula of the topological derivative are a generalized adjoint method and a Dirichlet-to-Neumann operator. The techniques employed are general and may be adapted to different functionals depending on elliptic PDEs under periodicity conditions.

Combining topological and shape derivatives in structural optimization

Two recent methods in shape and topology optimization of structures are combined in order to obtain an efficient optimization algorithm that benefits of advantages from both methods. The level set method, based on the classical shape derivative, is known to easily handle boundary propagation with topological changes. However, in practice it does not allow for the nucleation of new holes (at least in 2-d). The bubble or topological gradient method of Schumacher, Masmoudi, Sokolowski and their co-workers, is precisely designed for introducing new holes in the optimization process. Therefore, the coupling of these two methods yields a robust algorithm which can escape from local minima in a given topological class of shapes. The method we propose is a logical sequel of our previous work [1], [2] where we proposed a numerical method of shape optimization based on the level set method and on shape differentiation. The novelty is in the coupling and in the robustness of the proposed numerical implementation. Our basic algorithm is to iteratively use the shape gradient or the topological gradient in a gradient-based descent algorithm. The tricks are to carefully monitor the decrease of the objective function (to avoid large changes in shape and topology) and to choose the right ratio of successive iterations in each method.We provide several 2-d and 3-d numerical examples for compliance minimization and mechanism design. The main advantage of our coupled algorithm is to make the resulting optimal design largely independent of the initial guess, although local minima may still exist (even in the class of shapes sharing the same topology). Similar numerical results where discussed in [3].

Convergence of an algorithm in optimal design

The convergence of an algorithm in optimal design for problems in which the material properties are described by a second-order tensor is proved in this paper. The heat conductance context has been chosen for the presentation. Numerical results by using this kind of algorithm have already been obtained by Allaireet al. (1996) in elasticity.

An Algorithm for Constrained Optimization with Applications to the Design of Mechanical Structures

We propose an algorithm for minimizing a functional under constraints. It uses first order derivatives of both the objective function and the constraints. The step is computed as a sum between a steepest descent step (which minimizes the objective functional) and a correction step related to the Newton method (which aims to solve the equality constraints). The linear combination between these two steps involves coefficients similar to Lagrange multipliers which are computed in a natural way based on the Newton method. The algorithm uses no projection and thus the iterates are not feasible; the constraints are only satisfied in the limit (after convergence). Although the algorithm can be used as a general-purpose optimization tool, it is designed specifically for problems where first order derivatives of both objective and constraint functionals are available but not second order derivatives (as is often the case in structural optimization).

Detection of holes in an elastic body based on eigenvalues and traces of eigenmodes

Abstract We consider the numerical solution of an inverse problem of finding the shape and location of holes in an elastic body. The problem is solved by minimizing a functional depending on the eigenvalues and traces of corresponding eigenmodes. We use the adjoint method to calculate the shape derivative of this functional. The optimization is performed by BFGS, using a genetic algorithm as a preprocessor and the Method of Fundamental Solutions as a solver for the direct problem. We address several numerical simulations that illustrate the good performance of the method.

Optimization of bodies with locally periodic microstructure by varying the periodicity pattern

This paper describes a numerical method to optimize elastic bodies featuring a locally periodic microscopic pattern. A new idea, of optimizing the periodicity cell itself, is considered. In previously published works, the authors have found that optimizing the shape and topology of the model hole gives a limited flexibility to the microstructure for adapting to the macroscopic loads. In the present study the periodicity cell varies during the optimization process, thus allowing the microstructure to adapt freely to the given loads. Our approach makes the link between the microscopic level and the macroscopic one. Two-dimensional linearly elastic bodies are considered, however the same techniques can be applied to three-dimensional bodies. Homogenization theory is used to describe the macroscopic (effective) elastic properties of the body. Numerical examples are presented, in which a cantilever is optimized for different load cases, one of them being multi-load. The problem is numerically heavy, since the optimization of the macroscopic problem is performed by optimizing in simultaneous hundreds or even thousands of periodic structures, each one using its own finite element mesh on the periodicity cell. Parallel computation is used in order to alleviate the computational burden.