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The presented study aims to minimize the energy consumed by a Hall effect thruster (HET) under a constraint which makes it possible to generate a specified magnetic field in a target region of the thruster.

Herein topology optimization (TO) is used to reduce the energy consumption of an HET while keeping its performance unchanged. The design variables are the current densities in the coils and the distribution of materials in the polar pieces of the thruster. Intermediate values of material distribution are penalized using the solid isotropic material with penalization method to favor binary solutions. By means of the adjoint method, this paper provides the derivatives of the objective and constraint functions with respect to material distribution and current density variables.

The TO-based design methodology is developed and validated on a design example involving 2,051 variables. The approach shows its interest and its effectiveness of on a large scale two-criteria problem.

In this paper, TO is presented as a tool that has allowed to explore new and innovative designs. However, although the design presented is original, its fabrication is not feasible. Despite this, the designs found give a good idea of the starting points for shape and parametric optimization tools.

Through the HET design problem, TO shows the ability to explore more original design possibilities of a complex magnetostatic design problem and to discover designs that make a HET more efficient with respect to several criteria at the same time.

A new way to reduce the energy consumption of a HET is presented. To achieve this, an adjoint-based TO method is developed and then implemented in a simple way. This approach shows that, for efficiency purposes, TO is a key tool for extending the state of the art of HET designs.

In this work, a method to design a slotless permanent magnet machine (SPMM) based on the joint use of an analytical model and deterministic global optimization algorithms is addressed. The purpose of this study is to propose to include torque ripples as an extra constraint in the optimization phase involving de facto the study of a semi-infinite optimization problem.

Based on the use of a well-known analytical model describing the electromagnetic behavior of an SPMM, this analytical model has been supplemented by the calculus of the dynamic torque and its ripples to carry out a more accurate optimized sizing method of such an electromechanical converter. As a consequence, the calculated torque depends on a continuous variable, namely, the rotor angular position, resulting in the definition of a semi-infinite optimization problem. The way to solve this kind of semi-infinite problem by discretizing the rotor angular position by using a deterministic global optimization solver, that is to say COUENNE, via the AMPL modeling language is addressed.

In this study, the proposed approach is validated on some numerical tests based on the minimization of the magnet volume. Efficient global optimal solutions with torque ripples about 5% (instead of 30%) can be so obtained.

The analytical model does not use results from the solution of two-dimensional field equations. A strong assumption is put forward to approximate the distribution of the magnetic flux density in the air gap of the SPMM.

The problem to design an SPMM can be efficiently formulated as a semi-infinite global optimization problem. This kind of optimization problems are hard to solve because they involve an infinity of constraints (coming from a constraint on the torque ripple). The authors show in this paper that by using analytical models, a discretization method and a deterministic global optimization code COUENNE, this problem is efficiently tackled. Some numerical results show that the deterministic global solution of the design can be reached even if the step of discretization is small.

In this work, the authors deal with topology optimization in electromagnetism using solid isotropic material with penalization (SIMP) method associated with a gradient-based algorithm. The purpose of this study is to propose and investigate the impact of new generalized material interpolation scheme (MIS) used in SIMP approaches.

The variable domains of this kind of electromagnetism design problem are decomposed into small squares which represent a material point (iron here) or void. A least square function where the magnetic field in a target zone has to be as close as possible to a fixed one is minimized. Then the binary optimization problem is relaxed to a continuous one. By using the adjoint variable method, the gradient is provided. By penalizing the objective function using MIS, gradient-based algorithms can then be directly applied to provide efficient solutions close to the binary ones.

In this work, new general MISs are proposed. It is shown on numerous numerical instances that the so-obtained design solutions are more precise to define the zones with or without materials.

Only the linearity of the materials is addressed because the associated adjoint method needs this assumption. However, the new penalization approaches are not dependent directly on this assumption.

The new MISs are efficiently applied to design of a hall effect thruster (HET) magnetic circuits. Furthermore, these schemes are generic and can then be applied to other topology optimization applications in electromagnetism as well as and in mechanism.

The purpose of this sudy is to provide a complete optimization-based methodology to design waveguides with metamaterial walls. The present methodology is based on optimization. Indeed, the inverse problems of design are formulated as nonlinear black-box optimization problems with constraints. Two inequality black-box constraints are taken into account as penalized terms that are added to the objective function when the constraints are not satisfied. The numerical steps are done by using a finite element method solver (GetDP). Thus, different optimization software are tested to solve the nonlinear black-box optimization problems such as IPOPT, NLOPT and NOMAD from the Opti ToolBox in MatLab.

In this work, a methodology to design waveguides with metamaterial walls is proposed. The aim is to solve an inverse problem to find the best design where the electric field cartography is the closest to an imposed one.

The present methodology is applied to solve inverse problems of design and satisfactory results were provided by the three solvers IPOPT, NLOPT and NOMAD. Those numerical experiments show that NOMAD is the most efficient method to optimize the design of those cylindrical waveguide structures with metamaterial walls.

The model is set to find solutions using a specific pattern of metamaterials. This is promising to take those geometries as variables of the optimization problems. Moreover, in this exploratory work, no constraint on the fabrication limits has been taken into account.

The originality is to formulate design problems of waveguide with metamaterial walls into optimization problems. These optimization problems are difficult to solve because the objective function and two inequality constraints are computed via a numerical simulation code based on finite element methods. Thus, an original approach based on penalization is implemented and three optimization software are used. Hence, the authors propose an optimization-based methodology and apply to solve two inverse problems of design.

In magnetostatics, topology optimization (TO) addresses the problem of finding the distributions of both current densities and ferromagnetic materials to comply with fixed magnetic specifications. The purpose of this paper is to develop TO in order to design Hall-effect Thrusters (HETs).

In fact, TO problems are known to be large-scale optimization problems. The authors therefore adopt the adjoint method to reduce the computation time required to obtain the gradient information. In this paper, they illustrate the continuous variant of the adjoint method in the context of magnetostatics TO. Herein, the authors propose an implementation of the adjoint method then use it within a gradient-based optimization solver fmincon-MATLAB to solve a HET TO design problem.

By comparison with finite difference method, the authors validate the accuracy of the suggested implementation of the adjoint method. Then, they solve a large-scale HET TO design problem. The resultant design of TO is distinctly original and not intuitive.

In this paper, the authors introduce TO as a tool that has allowed them to explore new and innovative design of a HET. However, although the design presented is original, its manufacture is not feasible. Thus, a discussion section has been included at the end of paper to suggest a possible way to concretize topological solutions.

TO helps to explore more original design possibilities. In this paper, the authors present an implementation of the adjoint method that makes it possible to solve efficiently and in less central processing unit time large-scale TO design problem.

An easy implementation of the adjoint method is presented in magnetostatics TO. This implementation was first validated by comparison with the finite difference method and then used to solve a large-scale design problem. The result of the TO design problem is distinctly original and non-intuitive.

The purpose of this paper is to present two optimization methodologies based on interval branch‐and‐bound algorithm.

These techniques decrease the total time of computation, even in spite of discrete nature of some of the design variables. Computational experiments performed on multivariable optimization problem reveal great accuracy and technical validity of developed approaches. As an example, the optimal design of the induction machine (IM) was treated, where the aim was to find the set of the most efficient and, at the same time, cheapest in the manufacturing configurations.

In this paper, two approaches were developed for resolving the problem of optimal design of IM with discrete variables. The strategy of constructing the meta‐models is utilized and put in practice. The methods show relatively high efficiency and robustness of obtained results.

These approaches are the core technics of the developed industrial application, which help identify the set of optimal configurations of IM with the criteria of optimality such as total cost of manufacturing and the efficiency of IM.

The purpose of this paper is to propose a methodology to seek the optimal topology of electromagnetic devices using the density method while taking into account the non-linear behaviour of ferromagnetic materials. The tools and methods used are detailed and applied to a three-dimensional (3D) electromagnet for analysis and validation. Resulting topologies with and without the non-linear behaviour are investigated.

The polynomial mapping is used with the density method for material distribution in the optimisation domain. To consider the non-linear behaviour of the materials, an analytical approximation based on the Marrocco equation is used and combined with the polynomial mapping to solve the problem. Furthermore, to prevent the occurrence of intermediate materials, a weighted sum of objectives is used in the optimisation problem to eliminate these undesired materials.

Taking into account the non-linear materials behaviour and 3D model during topology optimisation (TO) is important, as it produces more physically feasible and coherent results. Moreover, the use of a weighted sum of objectives to eliminate intermediate materials increases the number of evaluations to reach the final solution, but it is efficient.

Considering non-linear materials behaviour yields results closer to reality, and physical feasibility of structures is more obvious in absence of intermediate materials.

This work tackles an obstacle of TO in electromagnetism which is often overlooked in literature, that is, non-linear behaviour of ferromagnetic materials by proposing a methodology.

Discrete highly constrained optimization of induction machine taking into consideration two objective functions: efficiency and total costs of production. The paper aims to discuss these issues.

Interactive and semi-interactive interval-based optimization methods were used. Two concepts of multi-objective discrete optimization were proposed.

Proposed methodology and algorithms allow decision maker (DM) participate in the process of optimal design and therefore decrease the total time of optimization process. The search procedure is straightforward and it does not require special skills of DM. Presented methods were successfully versified for the problem of optimal design with discrete variables.

Three interval algorithms suitable for inverse problems are researched and verified. It generally can be used for multi-objective problems. The dominance principles for interval boxes are showed in the paper. Proposed algorithms are based on the idea of hybridization of exact and evolutionary methods.

Proposed approaches were successfully implemented within computer-aided application which is used by manufacturer of high power induction machine.

The concept of pareto-domination using the interval boxes can be treated as original. The paper researched several elimination rules and discusses the difference between different approaches.