3.3 L’algorithme par essaim particulaire (Particle Swarm Optimisation)
3.3.2 La disposition du voisinage
La topologie de voisinage illustrée dans la figure 3.6 permet de définir la com- munication entre chaque particule. Il existe bien sûr de nombreuses combinaisons. Les plus connues sont :
1. topologie en anneau : chaque rapport de particules avec n autres (générale- ment n = 2), la topologie la plus courante
2. topologie du rayon : chaque rapport de particules avec uniquement la parti- cule centrale
3. topologie en étoile : chaque particule se rapporte à toutes les autres. L’opti- mum de voisinage est l’optimum global
3.3. L’ALGORITHME PAR ESSAIM PARTICULAIRE (PARTICLE SWARM
OPTIMISATION) CHAPITRE 3. CHOIX DE L’ALGORITHME
Figure 3.6 – Combinaison commune de communication entre les particules : (a)- anneau (n=2) ; (b)-rayon ; (c)-étoile, (tiré de Bouzid A. [2])
3.3.3
La distribution aléatoire
Les variables de distribution pèsent la tendance des particules à suivre leur ins- tinct de conservation ou leur panurgisme. Ces variables aléatoires suivent des lois uniformes et sont toujours positives. Cela permet "d’explorer" l’espace défini de manière plus aléatoire.
3.3.4
La vitesse
Afin d’empêcher les particules d’approcher la valeur optimale "trop vite" et de ne pas la remarquer, il peut être nécessaire de régler la vitesse maximale pour améliorer la convergence de l’algorithme. Une autre méthode de limitation de la vitesse consiste à définir un facteur limite pour permettre de réduire l’espace de recherche. Les recherches d’Eberhart et Shi [11] nous permettent d’affirmer que le coefficient limite apportera une meilleure vitesse de convergence.
3.3.5
Le coefficient d’inertie
Le coefficient d’inertie w introduit par Eberhart et Shi [11] peut définir la capacité de détection de chaque particule pour améliorer la convergence de la méthode. Par conséquent, une grande inertie est égale à une grande amplitude de mouvement, elle est donc équivalente à la détection globale des particules. Une faible inertie aura tendance à explorer localement les particules. Ce coefficient peut trouver un compromis entre deux explorations possibles.
Ce facteur affecte grandement l’exploration de l’espace, mais ne peut garantir la convergence vers la solution optimale. Eberhart et Shi, [11], ont déterminé la meilleure plage d’utilisation de ce facteur. Cependant, la caractéristique la plus importante de cette étude est qu’une fois l’espace de recherche pleinement exploré, il est possible de changer le coefficient au fil du temps pour passer de l’exploration globale à l’exploration locale.
3.3.6
L’initialisation de l’essaim
La position des particules ainsi que leur vitesse initiale doivent être initialisées aléatoirement selon une loi uniforme sur [0 : 1].
3.4. CHOIX CHAPITRE 3. CHOIX DE L’ALGORITHME
3.3.7
Le critère d’arrêt
Même si cette méthode est très efficace, rien ne garantit qu’elle convergera vers la solution optimale globale dans tous les cas. Par conséquent, l’algorithme doit avoir une porte de sortie : le nombre maximal d’itérations. Par conséquent, tant que l’un des critères de convergence n’est pas respecté, l’algorithme est exécuté :
1. le nombre maximum d’itérations est atteint ; 2. le changement de vitesse est proche de zéro ; 3. la solution trouvée est intéressante.
Selon Bratton et Kennedy [19], malgré sa simplicité, le grand nombre de chan- gements subtils dans l’algorithme PSO conduit à un large éventail de choix, et il est souvent difficile de déterminer quelle version peut fournir les meilleures performances ou les plus pertinentes pour une question de recherche donnée.
3.4
Choix
Au regard des définitions des algorithmes définis précédemment, voici un tableau récapitulatif en fonction du besoin de l’étude : temps de calcul court, compréhension de l’algorithme et facilité d’utilisation.
Table 3.1 – Avantages et inconvénients des algorithmes précédemment définis
Algorithme Avantages Inconvénients
Recuit simulé Résolution ra-
pide, grande
adaptabilité et
taux de réussite élevé
Choix des para- mètres difficiles et habituel- lement faits de manière empiriques, impossible ici Algorithme génétique Très bon lorsque le temps de cal- cul est élevé
La fonction coût est répété indéfiniment, il faut définir entièrement l’algorithme Essaim particu- laire Utilisation simple, s’adapte à tout calcul de solution et temps de calcul court Paramètres d’optimisations à déterminer
Le choix se porte donc sur l’algorithme d’optimisation par essaim particulaire, qui, malgré le fait de devoir déterminer une partie de son comportement, est simple à utiliser et s’adapte aux changements de fonctions cout sans avoir à changer sa nature.
Chapitre 4
Conception et optimisation d’un
panneau structurel en aluminium à
âmes corruguées
4.1
Avant-propos
Ce mémoire est constitué d’un article scientifique soumis à la revue scientifique Journal of Sandwiches Structures and Materials le 1er juillet 2020. Une copie du courriel de confirmation de soumission se trouve en annexe D.
Titre en français : Conception et optimisation d’un panneau structurel en alu- minium à âmes corruguées
Revue : Journal of Sandwiches Structures and Materials Date de soumission 01 juillet 2020
Auteurs et affiliation :
Florian Domenge, étudiant à la maitrise, Université de Sherbrooke, Faculté de Génie, Département de Génie Mécanique.
Alain Desrochers, ing. Dr ing., Professeur à l’Université de Sherbrooke, Faculté de Génie, Département de Génie Mécanique.
Elijah Van Houten, ing. Dr. ing., Professeur à l’Université de Sherbrooke, Faculté de Génie, Département de Génie Mécanique.
Résumé en Français Afin de répondre à des spécifications géométriques ou topologiques, il a été nécessaire de trouver une structure d’optimisation fondée sur de nombreux paramètres. De nombreux modes d’optimisation ont donc été recherchés et testés afin de trouver le programme d’optimisation le plus rapide et le plus fiable. Lorsqu’un niveau acceptable eut été trouvé, les objectifs étaient de mettre en œuvre tous les calculs analytiques pour trouver les meilleurs paramètres en fonction de la demande. Cet article contribue à trouver un moyen d’optimiser les panneaux ondulés structurels en aluminium selon la demande.
4.2. MANUSCRIT
CHAPITRE 4. CONCEPTION ET OPTIMISATION D’UN PANNEAU STRUCTUREL EN ALUMINIUM À ÂMES CORRUGUÉES
4.2
Manuscrit
Design and optimisation of aluminium
structural corrugated panels
Journal Title XX(X):1–8
c
The Author(s) 2016 Reprints and permission:
sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/ToBeAssigned www.sagepub.com/
SAGE
Alain Desrochers, Elijah Van Houten and Florian Domenge
Abstract
A numerical optimization approach was developed in order to achieve geometric or topological specifications for aluminum corrugated structural panels. After consideration of a variety of different optimization methods, a particle swarm method was chosen, and the controlling parameters were adjusted to achieve rapid and robust optimal structural specifications. The optimization method was based on an analytic model for the effective rigidity of corrugated structural panels. This article illustrates the application of recent stochastic optimization methods to structural element optimization.
Keywords
Sandwich Structure, Corrugated, Aluminium, Optimisation, Stochastic Algorithm, Particle Swarm
What is already known ?
It is known that sandwich structures can be used to improve the stiffness-to-mass ratio of a panel of a given thickness. Foam core and honeycomb panels are used commonly in industrial and transport applications. Honeycomb panels are the generally preferred choice, due to their light weight, moisture resistance and improved rigidity.
What is the contribution of the study ?
The corrugated panels proposed in this study can achieve stiffness-to-mass ratios similar to those observed in honeycomb panels, with the added advantage of allowing the passage of services such as electric wiring, hydraulic lines or plumbing. The possibility of integrated service passage is of particular interest as it allows a concomitant reduction in floor thickness, and therefore ride height. However, given the wide range of parameters controlling corrugated panel design, and the important influence these parameters can have on the overall rigidity of the panel, specialized optimization methods are needed for efficient design.
Introduction
Urban public transport concepts (bus, light rail, etc.) are confronted with numerous challenges as they are required to address an increasingly varied range of criteria; from energy efficiency and durability concerns to issues of safety, flexibility and capacity to accept a wide range of passenger needs. Importantly, public transport energy efficiency is a key factor for continued development in the current ecological context. Transport design must therefore include substantial optimizations in terms of weight and part-count reduction to achieve these objectives. Moreover, there are significant advantages to modular designs which allow the rapid adaption of a design concept to particular configurations depending on length, motorization, and intended purpose.
The goal of the current research is to develop a corrugated structural panel design process capable of achieving
rigidities that are competitive with honeycomb panels while offering design flexibility, modularity and service passage. The focus will be on the design of floor panels for public transport buses, as they present a challenging range of design specifications.
The structural panels using in engineering generally consist of two plates separated by a core. The choice of core material defines the properties of the panel (foam, honeycomb, corrugated). The face plates are designed to carry bending moments while the core is designed to carry shear loads. The objective in general is to have a low- density core, providing high flexural resistance through the separation of the face plates while contributing very little weight, leading to a high stiffness-to-mass ratio. Figure 1, from Binetruy (1), shows several different sandwich panel designs.
Figure 1. Different core typology by Binetruy (1)
Corresponding author:
Florian Domenge University of Sherbrooke, Department of mecanical engineering
Email: florian.domenge@usherbrooke.ca Prepared usingsagej.cls[Version: 2017/01/17 v1.20]
2 Journal Title XX(X)
Inspired by nature, honeycomb panels are lightweight and provide more structural resistance than foam core panels. Despite certain production challenges, they are one of the most common structural panel concepts in current use, particularly in the aircraft industry. Although they have a good stiffness-to-mass ratio, honeycomb panels do not offer sufficient structural resistance to support heavy loads, which can lead to thick panel designs. In addition, the honeycomb core structure does not allow service passage.
Figure 2. Example of structural cores by Difs and Ro (2)
Corrugated Panel Design
There are several different structural panel core designs currently used in the transport industry, particularly in naval, aircraft or spacecraft applications. Examples of these structures are shown in figure 2, from (2). Difs et al. provide an excellent overview of the different properties of these various core designs. For the target application of floor panels for public transport buses, the corrugated core panel was chosen (3). Indeed, by thinking about the final geometry of the panel, only the corrugated core solution can, by tressing strips in both directions, be able to let services passage in each pattern. In addition, the other core, by increasing their height can have a effective stiffness but there is a weakness in the material resistance between each strips. Zhang et al. (4) as well as Boorle et al. (5) have shown that corrugated structural panels are most efficient in terms of response to flexural loads. In addition, a corrugated core allows a variety of configurations to permit service passage and allow modular, flexible design.
One disadvantage of a traditional corrugated core design is that service passage is only possible in one direction. Additionally, the effective panel rigidity is often anisotropic. Sun et al. (6) have shown that tressed core structures made of corrugated strips provide service passage in two directions as well as allowing more variability in effective panel rigidity, including the ability to generate isotropic rigidities.
Figure 3. Tressed core structures by Sun (6)
Rigidity Calculation
Given the critical role of the stiffness-to-mass ratio in structural panel design, efficient methods for the calculation of the rigidity of a particular panel design were needed. Given the extreme computational costs of finite-element based methods for this calculation, an analytic model for corrugated panel rigidity was developed and verified, see (7), based on the work of Libove et al. (8).
For the present study, two different panel configurations were considered: (a) one-way corrugated panels; (b) one-way corrugated strips. These two configurations are illustrated in figures 4 and 5, respectively.
The One-Way Corrugated Panel (OWCP) Model
Figure 4. The one-way corrugated panel (OWCP) configuration
The model for the effective rigidity of a one-way corrugated panel is given by the NACA paper (8) :
Dx = EIx = Ec∗ IC + (E1 ∗ t1 ∗ kEIx2+ Ec
∗ AC ∗ (kc − kEIx)2
+ E2∗ t2 ∗ (1 − kEIx)2)∗ (h + t1/2 + t2/2)2 (1) Where :
1. Ec, E1 and E2 represent the Young’s modulus of the core, the bottom plate and the top plate.
2. IC, I1 and I2 represent the moment of Inertia per unit width of the core, the bottom plate and the top plate. 3. EIx represents the bending stiffness, per unit width. 4. kEIx represents the distance between middle surface
of lower face ans centroidal axis associated with EIx 5. kc represents the distance between middle surface of
lower face and centroidal axis of corrugation
Figure 5. The NACA’s modelisation used to determine rigidities
(8)
Florian Domenge 3
EIy = (E1∗ t1 ∗ kEIy2+ E2∗ t2 ∗ (1 − kEIy)2)∗ h2 (2)
Dy = EIy/(1− mu22∗ (1 − EIy/EIx) − (µ12− µ22)∗ (kEIx− kEIx ∗ EIy/EIx) − (µ1 − µ2)2∗
((1− kEIx) ∗ kEIx ∗ EIy/EIx − EIy/(EAx ∗ h2)))
(3) Where :
1. E1 and E2 represent the Young‘s modulus of the core, the bottom plate and the top plate.
2. µ1 and µ2 represent the Poisson’s ratios of the plates. 3. EIx and EIy represent the bending stiffness, per unit
width, in both patterns.
4. kEIx and kEIy represent the distance between middle surface of the lower face and the centroidal axis associated with EIx and EIy
5. EAx represents the extensional stiffness of the corrugated core sandwich plate in the x-direction
The One-Way Corrugated Strip (OWCS) Model
Figure 6. The one-way corrugated strip (OWCS) configuration
The effective stiffness of the one-way corrugated strip model was developed using the fill ratio of the core strips applied to the effective core rigidity of the equivalent one- way corrugated panel model.
Local Rigidity Calculation
The 2013 American Public Transportation Association (APTA) norms for urban buses (9) provide specifications for the floor structure, including the frame. The majority of these norms concern the overall mechanical response of the frame to twisting and bending loads. However, norms are also provided for the minimal resistance to local passenger loads. APTA requires that the floor is able to resist the so- called heel test, a load of 150 lbs. applied over a 0.5 inch diameter area. This local rigidity test is critically important for the corrugated panel designs considered here.
The points most susceptible to local deflections in the corrugated panel design are regions where the panel surface consists of only one sheet of material. A simple model based on beam theory and the Bernoulli principle was developed to determine the response of the panel to the heel test in these locations. The beam was modeled as having simple supports
on both ends, with a local distributed load, equivalent to the 0.5 inch diameter, applied symmetrically about the center as shown in Figure 7.
Figure 7. Beam configuration and loading
The thickness of the beam is determined by the thickness of the panel sheet material, while the width of the beam was fixed at a small, finite value 0.5 inches, to represent the line in the center of a circular deflection, where transverse bending approaches zero. The heel test includes consideration of the maximum deflection and the maximum stress observed in the material. The maximum deflection is given by Eqs. 4 (for the case of span lengths longer than 0.5 inches), Eqs. 5 (for spans shorter than 0.5 inches):
f max = q∗ (8 ∗ b ∗ l 3 − 4 ∗ l ∗ b3+ b4) 384∗ E ∗ I (4) f max = 5∗ q ∗ l 4 384∗ E ∗ I (5) Where l represents the span length, q represents the linear loading (pressure of 5.29 N/mm2 on a 0.5 inches line : 67.183 N/mm), and b represents the the loading diameter (0.5 inches).
The maximum stress in the sheet material is estimated by Eq 6:
σmax = M f ∗ 0.5 ∗ t
I (6)
Where Mf is the maximum bending moment in the span estimated by Eq 7, t is the sheet thickness, and I is the moment of inertia, given by Eq 8.
M f = R∗ l/2 − 0.5 ∗ q ∗ (b/2).2; (7) Where R represents the support reaction of the beam, l represents the length of the beam, q represents the linear loading and b represents the loading diameter (0.5 inches).
I = w∗ t.3/12 (8) Where w represents the fixes width of the beam and t is the sheet thickness.
Based on these estimates, optimal panel geometries will be generated that meet the heel test requirements for a maximum local deflection of 15 mm and a maximum sheet stress of 193 MPa, (Tensile Yield Strength of 5052 H32 sheets of aluminium).
4 Journal Title XX(X)
Important parameters
As seen above, the properties of the corrugated structural panels depend on multiple parameters. The design process involves determining the overall dimensions of the panel, as well determining the corrugation geometry. The parameters defining the corrugation geometry are shown in figure 5.
Figure 8. Corrugated strip geometry
The final optimization parameters for the two panel configurations are thus:
Parameter Configuration Min Value Max Value Plate thickness (t) OWCP and OWCS 0.9 mm 1.1 mm Iteration length (L) OWCP and OWCS 50 mm 500 mm
Folding radius (r) OWCP and OWCS 0 rad π 2
Length in contact (cl) OWCP and OWCS 20 mm 100 mm Strip height (h) OWCP and OWCS 10 mm 50 mm Strip width (w) OWCS 20 mm 500 mm
Glue Length OWCP and OWCS 20 mm Table 1. Parameters of the corrugated strip geometry
The particle swarm definition
The particle swarm optimization method
The implementation of the particle swarm optimization (PSO) method used here was is part of the the Matlab Optimization Toolbox (Mathworks, Natick, Massachusetts, USA). Here we present a study of the effect of different optimization parameters on the structural optimization problem presented above. In general, PSO is a stochastic optimization approach where a set of particles, each representing a particular structural configuration, are introduced at random into the parameter space determined by the limits defined in Table 1. For each particle, the structural configuration is used to calculate the value of a Cost Function, F, which is used to rank the particles in terms of how well the meet the specified design objectives. After initiation, the position of each particle within the parameter space, xi
k, is adjusted iteratively, to xik+1, with
the goal of reducing the cost function value for each particle and thus, over time, discovering the lowest cost function values possible within the specified parameter space. At each iteration the particle positions are adjusted according to Eq. 9. xik+1= xik+ wk∗ vik+ R1∗ (pi− xik) + R2∗ (p g k− xik) (9) Where:
1. Wkrepresents the inertia of particle i.
2. Vi
k represents the motion of particle i at iteration k.
3. R1 represents the coefficient of influence memory on the particle i.
4. Pirepresents the best known position of the particle i.
5. R2 represents the coefficient of swarm influence on the particle i.
6. pg
krepresents the best known position of the swarm.
The different components of the particle iteration step from xi
kto xik+1are shown in Figure 9.
Figure 9. Change from iteration k to k + 1 The cost function
The definition of the cost function is the most important aspect of any optimization problem. The cost function must consider all the criteria important for the design specification, as well as the characteristics of the material and the loading case used to determine the panel rigidity. The key design criteria in this case where panel weight and rigidity, particularly in comparison with honeycomb panels, which are the current industry standard. In PSO, the cost function can also enforce limits on the design parameters (shown in Table 1).
The complete formulation of the cost function for the structural optimization problem considered here is given by Eq 10 :
F = W/Whoneycomb (10)
Where W represents the weight of the optimized panel, Whoneycombis the weight of the honeycomb panel with the
same dimensions as the optimized panel.
Before returning this result, the function cost make sure that criteria are respected : heel test, minimum rigidity value, no plasticity, glue length and geometry possibilities.
PSO control parameters
PSO is governed by a number of different algorithmic control parameters, which are studied here for this particular case of structural panel optimization.
Number of particles The number of particles used in the PSO algorithm is one of the more important control