Optimal material design in composites: An iterative ... - Science Direct

Computer methods in applied mechanics and engineering ELSEVIER

Comput. Methods Appl. Mech. Engrg. 169 (1999) 31-42

Optimal material design in composites: An iterative approach based on homogenized cells •

RS.

Theocarts

a ~

" , G.E.

Stavroulakis

h

~National Academy o1:Athens, P.O. Box 77230. GR-175 10 Athens. Greece hTechnical Universi~ of Braunschweig, Germany

Received 2 September 1997

Abstract

An iterative method is proposed for the optimal material design of structures. The case of optimal material design with composite materials, with materials of variable microstructures and some classes of optimal topology design problems are discussed. Homogenized elastic properties can be used based either on analytic results or on numerical homogenization techniques. Several numerical examples demonstrate the theory. © 1999 Elsevier Science S.A. All fights reserved.

1. I n t r o d u c t i o n

The problem of optimal material design for structures made of composite materials is treated in this paper. Starting from a given initial condition, this design is gradually adjusted towards a stiffer structure by means of an iterative method, which is based on multilevel optimization and structural analysis techniques. For the finite-element discretized structure, the relation between the design parameters and the elasticity constants is either known from existing analytic solutions of the material homogenization problem, or is produced by means of a separate detailed analysis of a unit-periodicity cell, which takes into account all material and geometrical details of the composite studied. Numerical examples illustrate the applicability of the method and its potentialities for the design of sophisticated structures. The problem of optimal material design, which includes the topology optimization method has been studied in [ 1 - 4 ] among others. This formulation is used here in connection with an implicit description of the homogenized parameters of the composite m a t e r i a l The rigorous formulation of the optimal design problem leads to two-level optimization problems. Here, the one level concerns with the optimal structural design problem, while the second level is a modified structural analysis problem, parametrized with the design parameters adopted. Various mathematical aspects of this two-level optimization problem are considered [5-16]. The more effective available techniques for the numerical solution of problems of this type are the optimality criteria methods [17]. Only simple optimization problems concerning the determination of the stiffest structure by means of variation of the material within the structure from a set of predetermined composite materials are treated here. The stiffest structure is obtained by maximizing the potential energy of the structure at equilibrium or, equivalently, by minimizing the complementary energy at equilibrium. The choice of the material at each point o f the structure, or at each finite element in a discretized problem, is governed by the design variables of the optimal design problem. These variables define either the type of microstructure o f the material, or the material

* Corresponding author. 0045-7825/99/$ - see front matter © 1999 Elsevier Science S.A. All fights reserved PlI: S0045-7825(98)00174- 1

P.S. Theocaris. G.E. Stavroulakis / Comput. Methods Appl, Mech. Engrg. 169 (1999) 3 1 - 4 2

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characteristics of the composite (e.g. the fiber volume, fraction, the shape and the arrangement of the fibers, eventually, the characteristics of the mesophase developed between phases, etc.). Obviously, these variables change the overall (homogenized) elastic properties of the composite material as well. Thus, they give a possibility to modify the structural response of the whole structure. For the material selection problem, either classical fibre-reinforced composites and porous cellular materials with given analytically-obtained effective elastic properties [18,19] are considered, or numerical overall material properties are produced by using the numerical homogenization technique [20,21]. For the topology optimization problem a material with elasticity modulus depending on a power law of a single design variable is used [1,22-24]. The aim of this paper is to outline this approach and 1:o discuss several numerical experiments, which concern two-dimensional plane-stress structures with various loadings and boundary conditions. The potentialities of this method to produce either practically realizable microstructural arrangements, or optimal material designs, and to lead to considerable enhancement of the mechanical performance of an elastic structure are shown by means of these academic examples. In the preliminary numerical examples which are presented in this paper emphasis is posed on the comparison between the power elasticity law which is suitable for topology optimization problems and for models laws which are suitable for optimal composite material design. The latter case includes analytic fiber-reinforced composite formulae and numerical homogenization techniques (including cellular microstructures with even negative Poisson's ratio). Future work includes the consideration of concrete applications and the elaboration of optimal design problems with var:ious goals (not only stiffness maximization problems, as it is the case with this paper) and problems of multiple; loading cases.

2. Formulation of the optimal design problem A distributed parameter, minimum compliance (or equivalently maximum stiffness) problem is assumed. Let the design vector be denoted by [p(x), c~(x)]~, where p(x) is the density of each point x of the structure and a(x) is the set of design variables that describe the microstructure of the material. For instance, in a composite material structure the fibre-volume fraction, the elasticity constants of the matrix and the fibre, the mesophase variables, etc. may be used as design variables in o~(x). Analogously, for a cellular (or foamy) structure a(x) describes the design parameters of the unit cell, as it will be discussed in details later on with respect to concrete applications. Note also that in a discretized structure [p(x), a(x)] "r is defined element WiSe.

In a displacements based formulation, the minimum compliance problem is written in the following form (cf. [1]): max

min ( 2 ~ E~lk~(x,p(x), a(x))eij(u)ek,(u) d~(2-1(u) } ,

[p(x),~(r)]GAad tt E L"ad

(1)

Here, a small displacements linear elastic problem is assumed, where E~jk~(X,p(x), a(x)) is the fourth-order elastic stiffness tensor. Nevertheless, note that the previous relation is a linearized one, in the sense that the components of the elasticity tensor depend, possibly on a nonlinear way, on the design variables. Moreover A,j is the admissible set of the design variables which, on the assumption that a structure with maximum mass V is sought, reads:

A ,d ={ p(x), c~(x), x ~ ~2 such that fs~ p d~(2