Section: New Results

Shape and topology optimization

Incorporating manufacturing constraints in topology optimization

Participant : Grégoire Allaire.

With G. Michailidis and F. Jouve we study how to incorporate manufacturing constraints in topology optimization of structures using the level set method. The goal is to obtain a structure with optimal mechanical behaviour, which at the same time respects some predefined constraints imposed by the fabrication process. In this way, the final optimal shape is manufacturable and thus the method of shape and topology optimization turns to be industrially applicable.

The first constraints we have tackled are related to the limits of thickness a structure is forced to respect. We need to avoid optimal shape that contain very thin or thick members or even members that are very close between them. To achieve this, we have adopted two different approaches, a geometrical and a mechanical one. In the geometrical one, we have made extensive use of the notion of the signed-distance function to a domain. We have formulated a global constraint which guarantees that, at the end of the optimization process, the optimal structure respects the thickness limits. In the mechanical approach, we have tried to simulate the solidification process of a structure constructed via casting. We have set a time contraint, i.e. we have required that the structure cools earlier than some predefined time limit. We have started working on a more complicated thermal equation, a non-linear model with phase change, in order to describe more accurately the solidification process.

Optimization of composite materials draping

Participant : Grégoire Allaire.

With G. Delgado we work on the optimization of composite materials draping. These composite structures are constructed by lamination of a sequence of unidirectional reinforced layers or plies. Each ply is typically a thin sheet of carbon fibers impregnated with polymer matrix material. The optimization variables are the geometries of these layers and they are parameterized by a level set function. In a first instance, we treat the problem of mass minimization (with a constraint on the maximal compliance) for a cantilever type composite structure, laminated with four layers of a given orthotropic material at different angles. The elasticity analysis is performed with the software Freefem++, coupled with a C++ routine to solve, by a finite difference scheme, the evolution of the level sets.

A hybrid optimization method

Participant : Grégoire Allaire.

With Ch. Dapogny and P. Frey we develop a new method of geometric optimization for structures that relies on two alternative descriptions of shapes : on the one hand, they are exactly meshed so that mechanical evaluations by finite elements are accurate ; on the other hand, we resort to a level-set characterization to describe their deformation along the shape gradient. The key ingredient is a meshing algorithm for building a mesh, suitable for numerical computations, out of a piecewise linear level-set function on an unstructured mesh. Therefore, our approach is at the same time a geometric optimization method (since shapes are exactly meshed) and a topology optimization method (since the topology of successive shapes can change thanks to the power of the level-set method). Our first results in 2-d have been announced. We continue to work on the 3-d case.

DeHomogenization

Participant : Olivier Pantz.

In most shape optimization problems, the optimal solution does not belong to the set of genuine shapes but is a composite structure. The homogenization method consists in relaxing the original problem thereby extending the set of admissible structures to composite shapes. From the numerical viewpoint, an important asset of the homogenization method with respect to traditional geometrical optimization is that the computed optimal shape is quite independent from the initial guess (even if only a partial relaxation is performed). Nevertheless, the optimal shape being a composite, a post-treatment is needed in order to produce an almost optimal non-composite (i.e. workable) shape. The classical approach consists in penalizing the intermediate densities of material, but the obtained result deeply depends on the underlying mesh used and the details level is not controllable. We proposed in [40] a new post-treatment method for the compliance minimization problem of an elastic structure. The main idea is to approximate the optimal composite shape with a locally periodic composite and to build a sequence of genuine shapes converging toward this composite structure. This method allows us to balance the level of details of the final shape and its optimality. Nevertheless, it was restricted to particular optimal shapes, depending on the topological structure of the lattice describing the arrangement of the holes of the composite. We lifted this restriction in order to extend our method to any optimal composite structure for the compliance minimization problem in [39] . Since, the method has been improved and a new article presenting the last results is in preparation. Moreover, we intend to extend this approach to other kinds of cost functions. A first attempt, based on a gradient method, has been made. Unfortunately, it was leading to local minima. Thus a new strategy has to be worked out. It will be mainly based on the same ideas than the one developed for the compliance minimization problem, but some difficulties are still to be overcome.

Level-Set Method

Participant : Olivier Pantz.

We have begin to work, with Gabriel Delagado, on a new level-set optimization method, based on a gradient method. The key idea consists in computing directly the derivative of the discretized cost functions. The main advantage is that it is usually more simple to implement than the standard approach (consisting in using a discretized version of the gradient of the cost function). Moreover, the results obtained are as good or even better than the one obtained in previous works. Nevertheless, this method has its drawbacks, since the cost function is only derivable almost everywhere (the zero level-set has to be transverse to the triangulation of the mesh). It follows that convergence toward the minimum by the gradient method is not granted. To overcome this problem, we intend to use a mix-formulation for the state function. An article is in preparation on this subject.

Robust Optimization

Participant : Olivier Pantz.

One of the main problem in shape optimization problems is due to the fact that the gradient is never computed exactly. When the current solution is far from a local optimum, this is not a problem: even a rough approximation of the gradient enable us to exhibit a descent direction. On the contrary, when close to a local optimal, a very precise computation of the gradient is needed. We intend, with G. Delgado, to use a-posteriori error estimates evaluate the errors made on the computation of the gradient and to ensure that at each step, a genuine descent direction is used in the gradient method.

Optimization of a sodium fast reactor core

Participants : Grégoire Allaire, Olivier Pantz.

In collaboration with D. Schmidt, G. Allaire and E. Dombre, we apply the geometrical shape optimization method for the design of a SFR (Sodium Fast reactor) core in order to minimize a thermal counter-reaction known as the sodium void effect. In this kind of reactor, by increasing the temperature, the core may become liable to a strong increase of reactivity $\rho$, a key-parameter governing the chain-reaction at quasi-static states. We first use the 1 group energy diffusion model and give the generalization to the 2 groups energy equation. We then give some numerical results in the case of the 1 group energy equation. Note that the application of our method leads to some designs whose interfaces can be parametrized by very smooth curves which can stand very far from realistic designs. We don't explain here the method that it would be possible to use for recovering an operational design but there exists several penalization methods that could be employed to this end. This work was partially sponsored by EDF. Our results will be published in the proceedings of the CEMRACS'11, during which part of the results have been obtained.