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Mathematically, a structural optimization problem P is generally stated in the following general form: (P )        min x g0(x) s.t.: gj(x) ≤ gj j = 1 . . . m x_{i ≤ x}i ≤ xi i= 1 . . . n (5.1)

where x denotes the vector of variables xi which are the design parameters of the problem. In

shape optimization, the design variables are related to geometrical quantities such as the radius of a circle, the position of NURBS curve control point whereas in topology optimization or in sizing, the design variables are respectively the elementary densities and an element thickness for instance. The n variables xi are subject to side constraints xi and xi corresponding to

the admissible design domain for the different variables coming from physical limitations or manufacturability considerations. The objective function, noted g₀, is generally a non linear implicit function that can represent the stresses, the mass, the compliance or the displacement of the structure... The inequalities defined with the gj(x) functions represent the m design

restrictions on the structural behavior of the problem. Practical constraints are usually related to the stresses of the structure, the mass or specific eigenfrequency range... The bound is noted g_j. These functions are generally non-linear and non explicit with respect to the design variables. They limit the region of the design domain to be feasible design points as shown in Figure 5.1.

x

x

g

g

g

x

x

x

x

g

=a

g

=b

Figure 5.1: Representation of a general optimization problem

In shape optimization, the number of design variables n remains usually small whereas the num- ber of constraints m can be quite large when treating problems with stress constraints, multiple load cases or multiple eigenfrequencies. In topology optimization the situation is identical to sizing but the number of variables n can be very high as it generally corresponds to the number of elements.

In the most general case, all the functions involved in (5.1) are non linear and implicit with respect to the design variables. Moreover, the evaluation of these functions and their derivatives can be very ’expensive’ as they require the solution of the structural finite element problem. Following the idea proposed by Fleury [62], the solution of the implicit optimization problem (5.1) can be replaced by a sequence of simpler sub-problems having a simple explicit algebraic structure in the design variables. The original optimization problem P is then solved iteratively by solving the successive explicit approximated sub-problems ˜P . The idea being to generate a sequence of sub-problems for which the successive solutions xk∗ _{obtained forms a sequence of optimal}

design points that steadily improves the solution of the real problem P and converges to the real solution. Mathematically, for each iteration k of the optimization problem, the expression of the approximated sub-problem is stated as:

( ˜Pk)        min xk ˜ g0k(xk) s.t.: g˜jk(x) ≤ gj j= 1 . . . m x_{i− α}k_{i ≤ x}i ≤ xi+ βik j= 1 . . . n

whereg˜jk(j = 0...m) are the explicit approximations of the gjk(j = 0...m) functions around the

current design point xk_{. The parameters α}k

i and βik are move limits introduced to control the

variations of the current design pointxk _{to a certain neighborhood.}

This approach has been generalized with the notion of sequential convex programming (SCP) [67] where each function is replaced with a convex approximation. The problem ˜Pk is schematically represented in Figure 5.2 where the current approximations of constraints and the objective function are built around the current design point xk_{. Given the value of the responses at the}

current design point, the first order derivatives and sometimes the second order derivatives, the approximated functionsg are constructed using a kind of Taylor series expansion. Then the serie˜ of sub-problems are solved iteratively to find the next design point until convergence. Hopefully, this sequence of sub-problems yield to an optimal solution xk∗ _{towards the real optimal point}

x∗_.

g

(a) Convex approximation of problem P at itera- tion k x2 x1 x0 x x x g (x) = 01 g (x) = 03 g (x) = 02 1 * f(x) 2

(b) Convergence of the successive sub- problem solutions

Figure 5.2: Convex approximation The optimization problem can be summarized with this following steps:

2. Compute the sensitivity analysis for the objective functions and the constraints;

3. Form the approximate optimization sub-problem given the information obtained at step 2 and 3;

4. Solve the explicit (convex and separable) approximated model with efficient mathematical programming methods;

5. Update the model with the new parameters obtained from the solution of the approximated sub-problem;

6. Verify if the stopping criterion is satisfied. If not, the procedure is restarted from the step 2 given the new design variables at step k+ 1.

To succeed with such techniques, it is very important to obtain a high quality approximation ˜g without being to expensive to compute. Over the years, several approaches have been proposed to solve efficiently the SCP problem. The most important step to achieve an efficient method has been suggested by Fleury [63] when he proposed to resort to a dual approach. The dual method consists in replacing the constrained primal problem with a high number of variables by its dual problem of lower dimension. In the dual space of the Lagrange multipliers, the problem is transformed into a quasi-unconstrained maximization of the dual function. The power and effectiveness of the method is directly related to the fact the maximization of the Lagrangian is realized in a sub space, which dimension is limited to the set of active constraints. Hence, this approach is very effective when the number of active constraints is much lower than the number of design variables. Moreover, if we suppose that the primal problem is convex and separable, the dual approach appears really attractive and powerful as the computation of the dual function is very effective. When these conditions hold, the n-dimensional Lagrangian problem can be broken into n one-dimensional problems. Over the years, this approach have been the subject of a continuous improvement to obtain high quality approximations for the functions combined to effective SQP and dual approach [66] for solving the sub-problems.