Composite Materials Research Progress

74
Michaël Bruyneel
It results that robust approximation schemes must be available to efficiently optimize
laminated structures. The characteristics of such a reliable approximation are explained in the
following, and tests are carried out to show the efficiency and the applicability of the method.
7. Optimization Algorithm for Industrial Applications
7.1. The Approximation Concepts Approach
In the approximation concepts approach, the solution of the primary optimization problem
(2.1) is replaced with a sequence of explicit approximated problems generated through first
order Taylor series expansion of the structural functions in terms of specific intermediate
variables (e.g. direct xi or inverse 1/xi variables). The generated structural approximations
built from the information known at least at the current design point (via a finite element
analysis), are convex and separable. As will be explained latter a dual formulation can then be
used in a very efficient way for solving each explicit approximated problem.
According to section 2, it is apparent that the approximation concepts approach is well
adapted to structural optimization including sizing, shape and topology optimization
problems. However, the use of the existing schemes (section 7.2) can sometimes lead to bad
approximations of the structural responses and slow convergence (or no convergence at all)
can occur (Figure 7.1).
x2
* global
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x2
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* global
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* local
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a. A too conservative approximation b. A too few conservative approximation and unfeasible intermediate
solutions c. An approximation not adapted to the problem, leading to zigzagging
* local
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Figure 7.1. Difficulties appearing in the approximation of highly non linear structural responses.
x1
(k ) *
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* local
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