Cristian Barbarosie

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.

Study of the Cost Functional for Free Material Design Problems

We study properties of the cost functional arising in free material optimization problems, with special emphasis on semicontinuity and its relation to convexity.

Reducing the wrapping effect

When solving ODEs by interval methods, the main difficulty is reducing the wrapping effect. Various solutions have been put forward, all of which are applicable for narrow initial intervals or to particular classes of equations only. This paper describes an algorithm which, instead of intervals, uses a larger family of sets. The algorithm exhibits a very small wrapping effect and applies to any type of equation and initial region. For the time being it handles only two-dimensional equations. Wenn Systeme gewöhnlicher Differentialgleichungen mit Intervallmethoden gelöst werden, besteht die Hauptschwierigkeit in der Reduktion des Wrappingeffekts. Die verschiedenen bis jetzt vorgeschlagenen Lösungen sind nur bei engen Anfangsintervallen oder speziellen Gleichungsklassen anwendbar. Diese Arbeit beschreibt einen Algorithmus, der statt Intervallen eine größere Familie von Mengen verwendet. Der Algorithmus führt zu einem sehr geringen Wrappingeffekt und ist bei beliebigem Gleichungstyp und weiten Anfangsintervallen anwendbar. Zum gegenwärtigen Zeitpunkt können nur 2-dimensionale Probleme behandelt werden.

Representation of divergence-free vector fields

This paper focuses on a representation result for divergence-free vector fields. Known results are recalled, namely the representation of divergence-free vector fields as curls in two and three dimensions. The representation proposed in the present paper expresses the vector field as an exterior product of gradients and remains valid in arbitrary dimensions. Links to computer graphics and to partial differential equations are discussed.

Optimization of perforated domains through homogenization

We begin by explaining briefly why some shape/topology optimization problems need to be relaxed through homogenization. In Section 2 we present, from a mechanical viewpoint, the formula for the homogenized coefficients for a periodic infinitesimal perforation, and then briefly discuss the locally periodic ones (Section 3). Sections 4–6 describe a program which minimizes a certain functional over the set of model holes, and then its integration into a larger program, intended to treat topology and shape optimization problems. Numerical results are presented.

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.