Frederic Messine

University of Toulouse, ENSEEIHT-INPT

This paper proposes a formulation to minimize the torque ripple of switched reluctance machines by means of topology optimization. The novelty lies in targeting critical harmonics which allows for more control, both in the design of the rotor and the harmonic content of the torque, during the ripple minimization. An efficient gradient-based algorit...

This study focuses on exhaustive global optimization algorithms over a simplicial feasible set with simplicial partition sets. Bounds on the objective function value and its partial derivative are based on interval automatic differentiation over the interval hull of a simplex. A monotonicity test may be used to decide to either reject a simplicial...

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 q...

We consider a simplicial branch and bound (BnB) Global Optimization (GO) algorithm, where the search region is a simplex. Apart from using longest edge bisection, a simplicial partition set can be re-moved/reduced due to monotonicity of the objective function. Here we use Automatic Differentiation over the interval hull of a simplex to get bounds o...

The solution to a biobjective optimization problem is composed of a collection of trade-off solution called the Pareto set. Based on a computer assisted proof methodology, the present work studies the question of certifying numerically that a conjectured set is close to the Pareto set. Two situations are considered. First, we analyze the case where...

The concept of exploiting proven monotonicity for dimension reduction and elimination of partition sets is well known in the field of Interval Arithmetic Branch and Bound (B &B). Part of the concepts can be applied in simplicial B &B over a box. The focus of our research is here on minimizing a function over a lower simplicial dimension feasible se...

We address the problem of covering a rectangle with six identical circles, whose radius is to be minimized. We focus on open cases from Melissen and Schuur (Discrete Appl Math 99:149–156, 2000). Depending on the rectangle side lengths, different configurations of the circles, corresponding to the different ways they are placed, yield the optimal co...

Purpose 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 i...

In this paper, a complete methodology to design a modular brushless wound rotor synchronous machine is proposed. From a schedule of conditions and a chosen structure (with 7 phases, 7 slots and 6 poles), a non-linear and non-convex optimization problem is defined and solved using NOMAD (a derivative free local optimization code): the external volum...

Use of derivative bounds in simplical branch and bound based on interval arithmetic bounding. The presentation can be found in https://player.vimeo.com/video/601710760?h=9b3aecdc86

Branch and Bound (B&B) algorithms in Global Optimization are used to perform an exhaustive search over the feasible area. One choice is to use simplicial partition sets. Obtaining sharp and cheap bounds of the objective function over a simplex is very important in the construction of efficient Global Optimization B&B algorithms. Although enclosing...

In this paper, we design a Branch and Bound algorithm based on interval arithmetic to address nonconvex robust optimization problems. This algorithm provides the exact global solution of such difficult problems arising in many real life applications. A code was developed in MatLab and was used to solve some robust nonconvex problems with few variab...

Simplicial based Global Optimization branch and bound methods require tight bounds on the objective function value. Recently, a renewed interest appears on bound calculation based on Interval Arithmetic by Karhbet and Kearfott (2017) and on exploiting second derivative bounds by Mohand (2021). The investigated question here is how partial derivativ...

Purpose 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-inf...

Purpose 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. Design/methodology/approach Herein topology optimization (TO) is used to reduce the energy consumption of an HET while keeping its per...

In magnetics, topology optimizationTopology optimization (TO) is a tool helping to find a suitable ferromagnetic material space distribution in order to meet magnetic specifications. TO is a tool that becomes very interesting when the designer looks for new and original structures. Herein, TO is used to design a Hall-effect thruster. But, the topol...

This study presents a comparative study of different modular brushless wound rotor synchronous machines (MB‐WRSM) using non‐overlapping fractional slot double‐layer concentrated windings. The goal of the study is to highlight the structure which offers the best fault‐tolerance capability and the highest output performance. The fundamental winding f...

Purpose 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). Design/methodology/approach In fact, TO problems a...

In order to obtain reliable deterministic global optima, all the computed bounds have to be certified in a way that no numerical error due to floating-point operations can discard a feasible solution. Interval arithmetic Branch and Bound algorithms which are developed since the 1980th possess this property of reliability. However, some new accelera...

We consider the problem of covering a square with exactly 6 identical circles of minimal radius. In the literature, a covering is presented by Melissen and Schuur, and conjectured to be optimal. We adress the problem proposing a mathematical programming formulation and solving it to global optimality. We prove that the conjectured optimal covering...

Purpose 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. Design/me...

Our aim is to solve a problem of optimal control with free final time using the Pontryagin's maximum principle. As an illustration, we consider a navigation problem which is solved analytically and numerically by the shooting method in the case without constraint. The two approaches are compared. In the second case, we solve numerically the same pr...

In hybrid electric vehicles, the electrical powertrain system has multiple energy sources that it can gather power from to satisfy the propulsion power requested by the vehicle at each instant. This paper focusses on the minimization of the fuel consumption of such a vehicle, taking advantage of the different energy sources. Based on global optimiz...

This paper deals with optimal control applied to one of the most crucial and challenging problems in Air Traffic Management that of aircraft conflict avoidance. We propose an optimal control model where aircraft separation is achieved by changing the speeds of aircraft, and the integral over a time window of their squared accelerations is minimized...

Electrical propulsion in space is currently experiencing significant improvements. Among the electrical space propulsion technologies available, Hall effect thrusters work on the generation of plasma inside a cylindrical channel. These thrusters incorporate a magnetic circuit that has to generate a very specific magnetic mapping inside the plasma c...

In this paper, some improvements of spatial Branch and Bound (sBB) algorithms are discussed to solve aircraft conflict avoidance problems formulated as MINLP. We propose a new quadratic convex relaxation technique based on affine arithmetic. Moreover, a branching strategy is also proposed for the considered problem. Preliminary numerical results va...

In this paper, an exact Branch and Bound Algorithm has been developed to solve a difficult global optimization problem concerning the design of space thrusters. This optimization problem is hard to solve mainly because the objective function to be minimized is implicit and must be computed by using a Finite Element method code. In a previous paper,...

We propose a simple, but efficient method for design problems. Instead of using directly, a heavy numerical code in an optimization process, we successively define some approximated subproblems because of an appropriate analytical model. These subproblems can then be solved exactly by a deterministic global optimization solver and the procedure is...

The studied problem concerns the optimal regulation with a fixed horizon of two vertical ovens having respectively three or twelve heating areas. The model of the thermal process is described thanks to a linear state equation with a quadratic criterion to minimize. Furthermore, the terminal state is fixed and the state is subjected to some constrai...

This work deals with the optimal regulation of a large thermal process when the final state is fixed and the control is subject to some constraints, for which we propose a relaxation method coupled with the shooting one. We study the behavior of this method. The studied example concerns the optimal control law for two ovens with three and twelve he...

In this paper, we present a method to solve inverse problems of electromagnetic circuit design, which are formulated as a topology optimization problem. Indeed, by imposing the magnetic field inside a region, we search the best material distribution into variable domains. To perform this, we minimize the quadratic error between the prescribed magne...

In this paper, we show how optimization methods can be used efficiently to determine the parameters of an oscillatory model of handwriting. Because these methods have to be used in real-time applications, this involves that the optimization problems must be rapidely solved. Hence, we developed an original heuristic algorithm, named FHA. This code w...

An automatic method for constructing linear relaxations of constrained global optimization problems is proposed. Such a construction is based on affine and interval arithmetics and uses operator overloading. These linear programs have exactly the same numbers of variables and inequality constraints as the given problems. Each equality constraint is...

We present a control problem for an electrical vehicle. Its motor can be operated in two discrete modes, leading either to acceleration and energy consumption, or to a recharging of the battery. Mathematically, this leads to a mixed-integer optimal control problem (MIOCP) with a discrete feasible set for the controls taking into account the electri...

In satellite communication, Spatial Division Multiple Access (SDMA) has become one of the most promising techniques that can accommodate continuing increase in the number of users and traffic demands. The technology is based on radio resource sharing that separates communication channels in space. It relies on adaptive and dynamic beam-forming tech...

The problem we address in this paper, is the minimization of the energy consumption of an electrical vehicle achievable on a given driving cycle. This can be formulated as an optimal control problem with a discrete switch control. In this paper, we present a new formulation of this problem by taking into account the electrical and mechanical parts...

This paper focuses on finding an optimal control for a doubly fed induction machine (DFIM) in motor mode. Our purpose is not to improve the quality of the DFIM functioning but to find a method to improve it. Thus we will start with a fixed system made by a DFIM and two ideal 3 phases voltage inverter with an open loop static V/f control. A method w...

This work deals with simulation on an Inverted Pendulum (IP). The control strategy of an IP is split into two main control phases: (i) swing-up control to bring back the pendulum from the downward position to the upward one, and (ii) upright stabilization control to maintain the pendulum to the upright vertical position. In the case (ii), a feedbac...

This article focusses on the well-known problem of energy management for hybrid-electric vehicles. Although researches on this problem have recently intensified. Dynamic programming (DP) is still considered as the reference method because it obtains the best solutions of the literature so far, even though it requires a significant computational tim...

One of the decisive tasks within the air traffic management is the resolution of aircraft conflict avoidance problems. To avoid conflict, aircraft have to preserve a minimal safety distance between them. In this paper, we present optimal control models and approaches based on speed regulation to perform aircraft conflict avoidance. We consider some...

Purpose – The purpose of this paper is to investigate the impact of different mathematical formulations of the problem of optimal design of electrical machines on the results obtained using a local optimization solver. The aim is to investigate the efficiency and reliability of standard local solvers when handling different mathematical formulation...

The paper answers an open problem introduced by Bezdek and Fodor in 2000. The width of any unit-diameter octagon is shown to be less than or equal to $\frac14\sqrt{10 + 2\sqrt7}$ and there are infinitely many small octagons having this optimal width. The proof combines geometric and analytical reasoning as well as the use of a recent version of the...

Pour prévenir le risque de collision, le rôle du contrôle aérien est dassurer une distance minimale de séparation entre tout couple d'avions. Cette dernière correspond a 1000 ft verticalement et 5 NM horizontalement ; on dit qu'il y a conflit si elle n'est pas respectée. Face à la densité croissante du trafic, différentes approches de détection et...

Aircraft conflict avoidance is a crucial issue arising in air traffic management. The problem is to keep a given separation distance for aircraft along their trajectories. We focus on an optimal control model based on speed regulation to achieve aircraft separation. We propose a solution strategy based on the decomposition of the problem and on the...

Interval Branch-and-Bound (B&B) algorithms are powerful methods which aim for guaranteed solutions of Global Optimisation problems. Lower bounds for a function in a given interval can be obtained directly with Interval Arithmetic. The use of lower bounds based on Taylor forms show a faster convergence to the minimum with decreasing size of the sear...

Reduced RLT constraints are a special class of Reformulation-Linearization Technique (RLT) constraints. They apply to nonconvex (both continuous and mixed-integer) quadratic programming problems subject to systems of linear equality constraints. We present an extension to the general case of polynomial programming problems and discuss the derived c...

Spatial Division Multiple Access (SDMA) is a principle of radio resource sharing that separates communication channels in space. It relies on adaptive and dynamic beam-forming technology and well-designed algorithms for resource allocation. As satellite communication systems move towards greater capacity in both the number of users and throughput,...

Dans le contexte du trafic aérien, pour éviter le risque de collision, des distances de séparation doivent être respectées. On dit que deux avions sont en conflit si les distances qui les séparent sont inférieures à 5 NM horizontalement ou à 1000 ft 1 verticalement. Motivées par des raisons de sécurité et d?efficacité pour des trafics en constante...

Interval Branch-and-Bound (B&B) algorithms are powerful methods which aim for guaranteed solutions of Global Optimization problems. Lower bounds for a function in a given interval can be obtained directly with Interval Arithmetic. The use of lower bounds based on Taylor forms show a faster convergence to the minimum with decreasing size of the sear...

Branch and Bound Algorithms based on Interval Arithmetic permit to solve exactly continuous (as well as mixed) non-linear and non-convex global optimization problems. However, their intrinsic exponential time-complexities do not make it possible to solve some quite large problems. The idea proposed in this paper is to limit the memory available dur...

A small polygon is a convex polygon of unit diameter. We are interested in small polygons which have the largest area for a given number of vertices $n$. Many instances are already solved in the literature, namely for all odd $n$, and for $n=4, 6$ and 8. Thus, for even $n\geq 10$, instances of this problem remain open. Finding those largest small p...

The hexagon and heptagon with unit diameter and maximum sum of Euclidean distances between vertices are determined by enumerating diameter configurations, and by using a branch and cut algorithm for nonconvex quadratic programming. Lower bounds on the value on this sum are presented for polygon with a larger number of vertices.

Relationships between the diameter of a set of n points in the plane at mutual distance at least one, the diameter of an equilateral n-gon and the radius of a circle including n unit disks are explored. Upper bounds on the minimal diameter of a point set at mutual distance at least one are presented for up to 30 points.

Interval branch-and-bound (B&B) algorithms are powerful methods which look for guaranteed solutions of global optimisation problems. The computational effort needed to reach this aim, increases exponentially with the problem dimension in the worst case. For separable functions this effort is less, as lower dimensional sub-problems can be solved ind...

The value is shown to be an upper bound on the width of any n-sided polygon with unit perimeter. This bound is reached when n is not a power of 2, and the corresponding optimal solutions are the regular polygons when n is odd and clipped regular Reuleaux polygons when n is even but not a power of 2. Using a global optimization algorithm, we show...

Let a polygon Vn be given with n sides, perimeter Pn, diameter Dn, area An, sum of distances between vertices Sn and width Wn. In the paper, the authors present a survey of recent (since 2005) publications related to optimization problems, consisting in minimizing or maximizing one of the above quantities while the other ones are fixed. Isometric a...

The purpose of this paper is to use a new rational approach for the design of some magnetic couplings. Our method is based on the association of analytical models and an exact global optimization algorithm named IBBA and developed by the third author. The analytical model presented in this paper is more sophisticated than those previously dealt usi...

A polygon is said to be simple if the only points of the plane belonging to two of its edges are its vertices. We answer the question of finding, for a given integer n, a simple n-sided polygon contained in a disk of radius 1 that has the longest perimeter. When n is even, the optimal solution is arbitrarily close to a line segment of length 2n. Wh...

From the pentagon onwards, the area of the regular convex polygon with n sides and unit diameter is greater for each odd number n than for the next even number n + 1. Moreover, from the heptagon onwards, the difference in areas decreases as n increases. Similar properties hold for the perimeter. A new proof of a result by K. Reinhardt follows.

Abstract The value , 2n � is shown to be an upper bound on the width of any n-sided polygon with unit perimeter. This bound is reached when n is not a power of 2, and the corresponding optimal solutions are the regular polygons when n is odd, and

This paper presents a new methodology of design of electrical rotating machines. The methodology is an extension of previous works of the second author. Indeed, associating combinatorial analytical models with exact global optimization algorithms leads to rational solutions of predesign. These solutions need to be validated by a numerical tool (usi...

This paper deals with a deterministic and rational way to design piezoelectric transformers in radial mode. The proposed approach is based on the study of the inverse problem of design and on its reformulation as a mixed constrained global optimization problem. The methodology relies on the association of the analytical models for describing the co...

Consider a convex polygon V n with n sides, perimeter P n , diameter D n , area A n , sum of distances between vertices S n and width W n . Minimizing or maximizing any of these quantities while fixing another defines 10 pairs of extremal polygon problems (one of which usually has a trivial solution or no solution at all). We survey research...

Two ways for bounding n-variables functions over a box, based on interval evaluations of first order derivatives, are compared. The optima