Frédéric Messine

Shape and topology optimization is currently enjoying renewed interest, largely thanks to the rise of 3D printing techniques that make it possible to manufacture increasingly complex designs. In nonlinear 3D magnetostatics, however, significant mathematical and numerical challenges remain such as obtaining designs with well-defined boundaries which is essential to ensure manufacturability. To address this issue, we propose a sequential optimization strategy that combines topology and shape optimization to overcome the limitations of each method when they are used independently. First, a SIMP density-based topology optimization is performed to explore a wide design space. Next, an isosurface of the resulting material distribution is extracted and used as the initial geometry for a shape optimization phase, based on Hadamard boundary variation method and a level-set representation. This hybrid approach enables the generation of designs with well-defined boundaries, avoiding both the intermediate densities typical of SIMP and the sensitivity to initialization inherent in pure shape optimization. The effectiveness of the methodology is demonstrated through two threedimensional examples: the design of a C-core actuator and a preliminary design of a magnetic switch exploiting nonlinear material effects. These results highlight the potential of our approach to produce innovative, high-performance magnetic devices that are physically consistent and provide a solid basis for manufacturing.

The present work deals with the numerical resolution of the telegrapher equation. We present various numerical explicit and implicit schemes for problems defined in one-dimensional domains. In this case, we consider, on one hand, Dirichlet boundary value conditions and in the other hand, the mixed ones. In both cases, we analyze the stability by using the classical Von Neumann method or the matricial one. Moreover, the truncation error of the presented schemes are computed. The cases of two- and three-dimensional cases, is also briefly considered with only the Dirichlet boundary value conditions and in simple geometries. Concerning the implicit schemes, we briefly survey the main classical numerical linear algebra algorithms and we show that they can be applied successfully to the numerical solution of the target problem. We briefly consider the cases where, the coefficients of the problem are not constant and, also, the case where the linear problem is perturbed by a singlevalued or a multivalued diagonal operator. This study is completed by numerical simulations in the case of one-dimensional problems.

The present work deals with the numerical resolution of the damped wave equation. Thus, we present various numerical explicit and implicit schemes for problems defined in one-, two-and three-dimensional domain. In the case of one-dimensional problem, we consider, on one hand, Dirichlet boundary value conditions and in the other hand, mixed boundary value conditions. In the two-and three-dimensional cases, we consider only the case of Dirichlet boundary value conditions. However, in all cases, we analyze the stability and the truncation error of the presented schemes. Concerning the implicit schemes, we survey the main classical numerical linear algebra algorithms and show that they can be applied successfully to the numerical solution of the target problem, in the two-and three-dimensional cases on sequential or multiprocessor computers. We briefly consider also the cases where, on the one hand, the coefficients of the problem are not constant and, on the other hand, the case where the linear problem is perturbed by a singlevalued or a multivalued diagonal operator and we explain how to solve implicitly this type of problem on a multiprocessor. This study is completed by numerical simulations in the case of one-dimensional problems. Finally, to study the stability of the schemes, we can either use the classical Von Neumann method or the matricial one when the discretization matrix in space is normal. In this case, it is necessary to have the eigenvalues of the space discretization matrices. Thus, an appendix presents the determinination of these eigenelements in academic situations.

Spatial Global Optimization branch and bound (B&B) methods aim at enclosing global minimum points in a guaranteed way with a certain accuracy. We extend simplicial B&B (sBB) concepts to polytopal B&B (pBB), with polytope subsets. The main challenges are: polytope division and extension of monotonicity tests theoretically and algorithmically. We compare the performance of interval B&B with linear constraints (iBBLC), sBB and pBB algorithms, to determine the most efficient B&B algorithm for different types of instances.

In deterministic global optimization, techniques for linear relaxation of a non-convex program are used in the lower bound calculation phase. To achieve this phase, most deterministic global optimization codes use reformulation-linearization techniques. However, there exist also two interval-based polyhedral relaxation techniques which produce reliable bounds without adding new auxiliary variables, and which can take into account mathematical operations and most transcendental functions: (i) the affine relaxation technique, used in the IBBA code, based on affine forms and affine arithmetic, and (ii) the extremal Taylor technique, used in the Ibex-Opt code, which is based on a specific interval-based Taylor form. In this paper, we describe how these two interval-based linear relaxation techniques can be hybridized. These two approaches appear to be complementary, and such a hybrid method performs well on a representative sample of constrained global optimization instances.

We consider a simplicial branch and bound Global Optimization algorithm, where the search region is a simplex. Apart from using longest edge bisection, a simplicial partition set can be reduced due to monotonicity of the objective function. If there is a direction in which the objective function is monotone over a simplex, depending on whether the facets that may contain the minimum are at the border of the search region, we can remove the simplex completely, or reduce it to some of its border facets. Our research question deals with finding monotone directions and labeling facets of a simplex as border after longest edge bisection and reduction due to monotonicity. Experimental results are shown over a set of global optimization problems where the feasible set is defined as a simplex, and a global minimum point is located at a face of the simplicial feasible area.

Topology optimization is currently enjoying renewed interest thanks to the recent development of 3D printing techniques, which offer the possibility of producing these new complex designs. One of the difficulties encountered in manufacturing topologically optimized magnetostatic structures is that they are not necessarily mechanically stable. In order to take this mechanical constraint into account, we have developed a SIMP-based topology optimization algorithm which relies on numerical simulations of both the mechanical deformation and the magnetostatic behavior of the structure. Two variants are described in this paper, respectively taking into account the compliance or the von Mises constraint. By comparing the designs obtained with those from magnetostatic optimization alone, our approach proves effective in obtaining efficient and robust designs.

Electrohydraulic servovalves used in various aircraft systems (flight control system, auxiliary power unit, etc.) are a key element in fluid flow and pressure control. This paper presents a 3D analytical modelling method of the electromagnetic performance of servovalves torque motor. This analytical model, based on a reluctance network, allows to quickly evaluate the actuator performance taking into account the complex 3D geometry and the high flux leakage rate. This model enables to reduce the performance analysis time by about 89% compared to the conventional 3D finite element calculation time. This is valuable for the design and optimization of the actuator geometry.