SOLUTION OF THE LARGE MINIMUM COMPLIANCE PROBLEM USING BILEVEL OPTIMIZATION

The topology optimization problem have a great industrial interest. Using the subdomains method, we have formulated the decomposed topology optimization problem as a bilevel one. In this paper, we reformulate our bilevel problem as a single level optimization problem by replacing the lower level optimization problem with its KKT optimality conditions, we give also a new algorithm and numerical results.


INTRODUCTION
The problem of topology optimization is typically regarded as material distribution problem for the optimal shape design [8]. The distribution of the material is limited to the design domain, which forms part of a larger domain which can include areas prescribed to be solid or void. The problem considered in this paper is the minimal compliance problem. Minimizing compliance turned out to be a standard problem in topology optimization [3]. This problem aims at the design of the stiffest (or least compliant) structure under a given fixed load, possible support conditions and restriction on the volume of the used material.
We specify the problem mathematically in the SIMP approach as follows [8]: with the following constraints on ρ : U is the set of admissible displacements, V is a limit on the amount of material at our disposal, represents the material properties of a given isotropic material, ρ which is interpreted as a density of material is the design variable and p is the penalty factor which penalizes intermediate densities in f is the vector of volume forces acting on the body, Topology optimization has for a considerable time been applied successfully in the automotive industry, but still has not become a mainstream technology for the design of large structures and complex materials like aircraft components. The explanation for this is partly to be sought in the larger problem sizes.
The Domain Decomposition Method (DDM) is one of the most effective parallel methods for large scale problems due to the fact that it provides a high level of concurrency and is simple to implement on most modern parallel computers. The principle of the method is to split the original domain of computation in smaller simpler subdomains, which may or may not overlap. Next, the original problem can be reformulated upon each subdomain, yielding a family of subproblems of reduced size that are coupled one to another through the values of the unknown solution at subdomain interfaces.

PROBLEM STATEMENT
Suppose that Ω is partitioned into two nonoverlapping subdomains Ω1 and Ω2 with interface Γ0 i.e. . Let and . Applying domain decomposition method in minimum compliance problem (1), we have obtained the following optimization problem: [1] with the constraints on ρ1, ρ2 : where and "g * =g opt " is the unique choice of g for which i.e.
(see [1] Remark 3.7) In [2] we have proved that our problem (2), which involve an hierarchical making process, is a bilevel programming problem. In fact, on one hand, the functions l(u1; u2; g) and can't be minimized simultaneously, so, bicriteria optimization is no more suitable, and the hierarchical nature of the two levels is a natural justification for this choice.
On the other hand, structural optimization problems have an inherent bilevel form. The upper level objective function measures some performance of the structure, such as the stiffness in our example. The lower level problem describes the behavior of the structure given the choices of the design variables and the external forces acting on it.
The theory of bilevel programming focuses on forms of optimality conditions and complexity results, the complexity of the problem has been addressed by a number of authors, it has been proved that even the linear programming problem, where all the involved functions are affine, is a strongly NP-hard problem, in addition, due to the hierarchical structure the BLPP is not convex and not differentiable.
The formulation of optimality conditions for bilevel programming problems usually starts with a suitable reformulation of the problem as a one-level one by replacing the lower level optimization problem with its KKT optimality conditions, which are necessary and sufficient for defining the optimum of the inner level problem only under convexity condition and a first order constraint qualification. When the inner problem constraints are non convex, the KKT conditions are only necessary. A further difficulty arises in locating the global optimum of the resulting single level problem after the KKT transformation, the bilinear nature of complementarity condition introduce nonconvexities even if the original problem is linear.
A topology optimization problem is an optimization problem where a generalized equation(modeling an equilibrium) arises among the constraints. This new class of optimization problems which play an important role in mechanics, is known as The MPEC problems coming from the topology optimization of structures, solids and fluids violates standard non linear programming constraint qualifications, and even the novel qualification, such as the strict complementarity conditions, or strong regularity assumptions. Therefore, Evgrafov [9] proposed approximation techniques for the numerical solution of such problems. As mentioned above, one tool often used to reformulate the bilevel programming problem as an one level problem are the Karush-Kuhn-Tucker conditions. If a regularity condition is satisfied for the lower level problem, then the KKT conditions are necessary optimality conditions. They are also sufficient in the case when the lower level problem is convex, this is the approach used in this paper.
The paper is organized in the following way: in section 3 the mathematical analysis of the bilevel formulation is given, and a new single level optimization problem is deduced, and an algorithm is proposed for numerical results in section 4.

REFORMULATION OF THE DECOMPOSED TOPOLOGY OPTIMIZATION PROBLEM USING THE KUHN-TUCKER APPROACH
Consider the lower level problem: The conventional solution approach to the BLPP is, as mentioned earlier, to transform the original two level problem into a single level one by replacing the lower optimization problem with the set of equations that define its KKT optimality conditions.
Noting that, the KKT optimality conditions are necessary and sufficient for definig the optimum of the inner level problem only under convexity condition ans a first order constraint qualifications. When the inner problem constraints are nonconvex, the KKT conditions are only necessary. Without inequality constraints in the lower level problem, the resulting single level problem after the KKT transformation lacks complementarity conditions which introduce nonconvexities even if the original problem is linear.
In our case, the convex nature of the inner problem (3) enable us to apply the Kuhn-Tucker approach to transform the bilevel problem (2) into a single one and derive the KKT optimality conditions.
For the lower level problem (3) define the Lagrangian: where Then, a necessary and sufficient condition for g* to be an optimal solution to the inner level problem is that: gives the optimality condition gives the adjoint, or co-state equations gives the state equations in the two subdomains that is respectively: Given that is convex when f and g are convex and f not decreasing, then the function is convex because and are convex and increases for , therefore the objective function is convex in u1, u2 and g.
The convexity of the Lagrangian in u1, u2 and g then follows from the fact that is convex and the state equations are linear.
It follows that a necessary condition for to be an optimal solution of the bilevel problem (2), ( must satisfy the above conditions at fixed , thus the bilevel programming problem (2) is transformed into a single level problem of the form: with the constraints on ρ1, ρ2 :

The optimality system
Define the Lagrange function for the single level problem (5) as follows: where, are the Lagrange multipliers for the equilibrium constraints, and are also Lagrange multipliers corresponding to the co-state equations and optimality condition for the lower level problem, the Lagrange multipliers are related to the constraints in .
The KKT optimality condition for the problem(5) can be formally interpreted as stationary points of .

The proposed algorithm
The algorithm described in [8, p.14] to solve the compliance topology design problem consists of the following modules: 1. Finite element analysis 2. Sensitivity analysis 3. Mesh independency filter

Update design variables
In our algorithm, only FEA and sensitivity analysis are parallelized, because on one hand, the main part of the computational time is devoted to the solution of the equilibrium equations. However, the effort required to update design variables at each optimization iteration is small compared to the corresponding finite element solve, hence, the FEA is the most important part of the entire solution algorithm to parallelize.
On the other hand, mesh independency is not parallelized, since filtering is not local by nature and requires the average of the sensitivities over several elements. Also, calculations related to sensitivity filtering are negligible compared to those required for the solution of the equilibrium equation and calculation of the element sensitivities.
Because the Lagrange multiplier used to calculate in (16) is a global parameter, and when filtering (which is not parallelized) is applied, the optimization update (15) won't be parallelized. The equations (11) (12) (13) are coupled, that's why a gradient method is used to uncouple the search of local displacements , and a loop on the control variable satisfies the continuity of the displacement on the interface on .
To update g let: The simple gradient method is defined as follows: where is a step size. Combining with (21) yields a formula to update : Hence following the discussion above the proposed algorithm to solve (5)

Numerical results
The numerical test of the proposed algorithm is the beam (see figure 1).

Fig 1: Design domain with boundary conditions
The beam is decomposed in two equal beams. Figure 2 gives the result of the direct topology optimization of this structure using the Matlab code proposed in [8].  The algorithm converges in three iterations. One can show that if the two substructures are assembled, we don't find the result of the figure 3 form the initial structure, because the uniqueness of the solution of the topology optimization is not shown in any reference. So, the finding result stay reasonable and good.

CONCLUSION
Using a bilevel analysis of the decomposed topology optimization problem, we have developed a new method to solve this complicated problem and the numerical results show her efficiency. In the future work, we want to do numerical implementation of this new algorithm for complex structures issue from the industry to show her efficiency.