Claude Fleury

CONLIN: An efficient dual optimizer based on convex approximation concepts

The Convex Linearization method (CONLIN) exhibits many interesting features and it is applicable to a broad class of structural optimization problems. The method employs mixed design variables (either direct or reciprocal) in order to get first order, conservative approximations to the objective function and to the constraints. The primary optimization problem is therefore replaced with a sequence of explicit approximate problems having a simple algebraic structure. The explicit subproblems are convex and separable, and they can be solved efficiently by using a dual method approach.In this paper, a special purpose dual optimizer is proposed to solve the explicit subproblem generated by the CONLIN strategy. The maximum of the dual function is sought in a sequence of dual subspaces of variable dimensionality. The primary dual problem is itself replaced with a sequence of approximate quadratic subproblems with non-negativity constraints on the dual variables. Because each quadratic subproblem is restricted to the current subspace of non zero dual variables, its dimensionality is usually reasonably small. Clearly, the Hessian matrix does not need to be inverted (it can in fact be singular), and no line search process is necessary.An important advantage of the proposed maximization method lies in the fact that most of the computational effort in the iterative process is performed with reduced sets of primal variables and dual variables. Furthermore, an appropriate active set strategy has been devised, that yields a highly reliable dual optimizer.

First and second order convex approximation strategies in structural optimization

In this paper, various methods based on convex approximation schemes are discussed, that have demonstrated strong potential for efficient solution of structural optimization problems. First, theconvex linearization method (CONLIN) is briefly described, as well as one of its recent generalizations, themethod of moving asymptotes (MMA). Both CONLIN and MMA can be interpreted as first order convex approximation methods, that attempt to estimate the curvature of the problem functions on the basis of semi-empirical rules. Attention is next directed toward methods that use diagonal second derivatives in order to provide a sound basis for building up high quality explicit approximations of the behaviour constraints. In particular, it is shown how second order information can be effectively used without demanding a prohibitive computational cost. Various first and second order approaches are compared by applying them to simple problems that have a closed form solution.

A UNIFIED APPROACH TO STRUCTURAL WEIGHT MINIMIZATION

Abstract It is shown that the two classical approaches to structural optimization have now reached a stage where they employ the same basic principles. Indeed, the well-known optimality criteria approach can be viewed as transforming the initial problem in a sequence of simple explicit problems in which the constraints are approximated from virtual work considerations. On the other hand, the mathematical programming approaches have progressively evoluated to a linearization method using the reciprocals of the design variables — this powerful method is proven here to be identical to a generalized optimality criteria approach. Finally, new efficient methods are proposed: (a) a hybrid optimality criterion based on first-order approximations of the most critical stress constraints and zeroth-order approximations of the others and (b) a mixed method which lies between a strict primal mathematical programming method and a pure optimality criteria (or linearization) approach. Simple numerical problems illustrate the concepts established in the paper.

A primal-dual approach in truss topology optimization

Abstract The objective of truss topology optimization is to find, for a given weight, the stiffest truss, defined as a subset of an initially choosen set of bars called the ground structure. The restrictions are bounds on bar volumes and the satisfaction of equilibrium equations. For a single load case and without bounds on bar volumes, the problem can be written as a linear programming problem with nodal displacements as variables. The addition to the objective function of a quadratic term vanishing at the optimum allows us to use a dual solution scheme. A conjugate gradient method is well suited to maximize the dual function. Several ground structures are introduced. For large networks, those where each node is connected to those situated in a certain vicinity give a good compromise between generating a reasonable number of bars, and obtaining a sufficient number of possible directions. Some examples taken from the literature are treated to illustrate the quality of the solution and the influence of initial topology.

Shape Optimal Design by the Convex Linearization Method

The main goal of the present paper is to discuss the choice of suitable numerical methods for shape optimal design of elastic structures discretized in finite elements. After a short description of the approach we followed to create an appropriate geometric model involving a relatively small number of design variables, attention is mainly directed toward the selection of an adequate optimization algorithm. To this aim the paper will briefly present the various attempts that we have successively undertaken before adopting the convex linearization method as the basic optimizer, not only for shape optimal design problems but also for all our other structural synthesis capabilities. Various examples of applications to optimum shape problems are offered to demonstrate the efficiency of this new algorithm. Finally some comments are made about future developments needed to effectively implement shape optimization concepts into the real design cycle.

A MATHEMATICAL CONVERGENCE ANALYSIS OF THE CONVEX LINEARIZATION METHOD FOR ENGINEERING DESIGN OPTIMIZATION

Abstract This paper is concerned with the convex linearization method recently proposed by Fleury and Braibant for structural optimization. We give here a mathematical convergence analysis or this method. We also discuss some modifications of it.

Sensitivity and optimization of composite structures in MSC/NASTRAN

Abstract The main purpose of this paper is to present the considerations and the resultant approach used to implement design sensitivity capability for composites into a large-scale, general-purpose finite element system ( msc/nastran ). The design variables for composites can be lamina thicknesses, orientation angles, material properties, or a combination of all three. As a secondary goal, the sensitivity analysis has been coupled with a general-purpose optimizer to validate sensitivity analysis results and also demonstrate how easy it is couple to an optimizer once a general senssitivity capability is available. This preliminary version of the optimizer is capable of dealing with minimum weight structural design with a rather general design variable linking capability at the element level or the system level. Only sizing type of design variables (i.e., lamina thicknesses) can be handled by the optimizer. Test cases have been run and validated by comparison witj independent finite element packages. The linking of a design sensitivity capability for composites in msc / nastran with an optimizer would give designsrs a powerful, automated tool to carry out practical optimization design of real-life, complicated composite structures.

MIXED VARIABLE STRUCTURAL OPTIMIZATION USING CONVEX LINEARIZATION TECHNIQUES

Abstract In this paper, the use of convex approximation techniques is investigated for optimizing the configuration of planar trusses. It has been previously shown that simultaneous treatment of sizing and configuration variables is feasible. This work demonstrates that not only do the two types of variables not require separation but that they can be handled efficiently together. Problems with both types of variables can be solved within a small number of analyses (typically ten) This work also indicates that approximation techniques can be used to great advantage on problems with both configuration and sizing variables. The addition of the second type of design variable causes little increase in the computational difficulty of the problem. The use of approximation techniques improves the convergence properties and greatly reduces the number of analyses required Various convex linearization schemes are investigated on problems of shape optimization of truss structures. In particular, the Convex Lineariza...

Convex Approximation Strategies in Structural Optimization

The design optimization problem considered in this paper consists of minimizing some ob-jective function subject to constraints insuring the feasibility of the structural design.

An interactive capability for shape optimal design

Abstract Shape optimal design of two-dimensional structures discretized in finite elements is investigated with emphasis on creating an interactive Computer-Aided Design capability. The paper will first provide a brief description of the approach followed to create an appropriate geometric model, involving a relatively small number of design variables. Next a general procedure to perform the sensitivity analysis is proposed, that takes advantage of the parametric modeling concepts used to create the design model. This sensitivity analysis method is based on the so-called semi-analytical approach, i.e. a finite difference approximation of the derivative of the stiffness matrix is employed. The numerical solution to the shape optimal design problem is obtained by resorting to an efficient optimizer based on the convex linearization method. A two-dimensional pre- and post-processing module is described, which is aimed at performing interactive shape optimal design on an engineering workstation. This module exhibits some innovative visualization techniques that should highly facilitate the task of a designer. Examples of application to shape optimal design problems are offered to illustrate the various functions of the new interactive optimization system. Although the described capability is restricted to simple two-dimensional structures, it is based upon very general concepts that could be readily implemented in general-purpose finite element systems.