Application of imperialist competitive algorithm for optimizing a thin resistant interphase

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1 Available olie at Procedia Techology (0 ) INSODE 0 Applicatio of imperialist competitive algorithm for optimizig a thi resistat iterphase Hamid Mozafari a*, Behzad Abdi a, Amra Ayob a a Faculty of Mechaical Egieerig, Uiversiti Tekologi Malaysia, 830 UTM Skudai, Johor, Malaysia Abstract The Imperialist Competitive Algorithm (ICA) is used i optimal desig of a thi resistat iterphase layer for several temperature distributios o a specime. I applicatios of iterphase layers, the compoets of thickess ad heat source are very importat ad they are assiged as multiobjective optimizatio desig fuctios. The results of ICA are the compared with Fiite Elemet Method (FEM) ad aother optimizatio method, the Geetic Algorithm (GA). The ICA results show good performace i optimal desig of thi adhesive layers. Keywords: Iterphase layer, Imperialist competitive algorithm (ICA), Numerical simulatio, Geetic algorithm (GA).. Itroductio Thi iterphases have bee icreasigly used i importat parts of scietific processes ad compoets []. A ihomogeeous structure cosistig of iterphase layers may exhibit a wide variety of thermal ad mechaical properties. As a example, adhesive layers allow for high quality joiig of materials with essetially differet properties. Differet areas about thi iterphase layers were ivestigated by [, 3, 4, 5] Global optimizatio methods have bee used to solve may problems i sciece ad egieerig, such as data aalysis [6-8], material desig [9-0], chemistry [] ad may others. Evolutioary algorithms [- 3] have bee widely proposed for solvig global optimizatio problems. Im et al [4] described the optimizatio procedure usig the fiite elemet aalysis ad the sequetial ucostraied miimizatio techique. The ICA is a ew evolutioary algorithm for optimizig a wide variety of problems. ICA was first proposed by [5] who used it to solve a cotiuous optimizatio problem. I the preset paper, a optimal desig is ivestigated o a thi resistat iterphase layer, by usig ICA, with liear temperature source. The iterphase, with several thermal distributios, forms a layer i a hybrid model structure (see Fig. ). The results are the compared with fiite elemet method ad aother optimal method based o Geetic Algorithm (GA).. Problem Formulatio Cosider a plae problem domai with a thi adhesive layer betwee two differet materials (Fig. ). The adhesive layer is assumed to be very thi ad two materials are applied at the top ad the bottom of the adhesive * Hamid Mozafari. address: Mozafari.h@gmail.com Published by Elsevier Ltd. doi:0.06/j.protcy Ope access uder CC BY-NC-ND licese.

2 88 Hamid Mozafaria et al. / Procedia Techology ( 0 ) layer i the same elemet size. The top ad the bottom layers temperature ca be expected to be relatively idepedet of the boded thickess. This paper ivestigates three cases of boudary coditios. The cases are meat to set temperature distributio at the top (y = +H/), at bottom (y = -H/) ad part of the surface alog the x- axis. The first case is a costat temperature distributio (CTD) alog the x-axis. The secod case is a liear temperature distributio (LTD) ad the third case is a parabolic temperature distributio (PTD). Figure : Specime i heat coductio problem The differetial form of Fourier's Law of thermal coductio shows that the heat flux, q, is equal to the product of thermal coductivity, k, ad the egative temperature gradiet, T. The heat flux is the amout of eergy that flows through a particular surface per uit area per uit time [7], q k T () The total thermal resistace from the bottom ad the top layers to ambiet air ca be expressed as the resistace across the thickess of the adhesive layers: x x x R () ka ka ka BL TL adh where k is the thermal coductivity, A is the area of the surface. The adhesive layer, the bottom ad top layers are show with adh, BL ad TL idexes, respectively. The temperature rise at the top ad bottom surface ca be ormalized with a maximum allowable temperature differece T T T T TL (3) f RQ T T f T, as: where T f is defied as the temperature at which failure occurs. The bottom layer temperature ca be rewritte i the followig form: TBL TBL T Q( RBL Radh) (4) 3. Fiite Elemet Modelig I this sectio the FEM is egaged to validate the optimizatio results. The commercial fiite elemet code MSC-Marc is used for the simulatio of thermal behaviour of a thi iterphase layer located betwee two similar adherets. The boded fibre-reiforced adherets have mechaical properties, Youg s module E MPa ad Poisso s ratio The thermal properties of the adherets are, coductivity k 37 w/(m.k) at 300 K,

3 Hamid Mozafaria et al. / Procedia Techology ( 0 ) mass desity kg/m 3 ad specific heat c 898. J/(kg.K ). The thi iterphase layer is made of a ~ epoxy resi with properties k 0. w/(m.k), 00 kg/m 3 ad c 790 J/(kg.K ) ad exhibits differet liear temperature depedece of the source. The geometrical dimesios are L = 0 mm, H = mm ad h = 0.0 mm. so that the value of the small parameter ca be estimated as h / H = 0.0. Figure : Two-dimesioal fiite elemet mesh The D fiite elemet mesh cosists of four-odes (see Fig. ), isoparametric elemets with biliear iterpolatio fuctios. The source or sik formulatio is implemeted by meas of a special user subroutie writte i FORTRAN. The applicatio of this program requires a trasiet solutio i order to icorporate the source expressio. 4. Imperialist Competitive Algorithm Imperialist Competitive Algorithm (ICA) is a ew evolutioary algorithm based o the huma socio-political evolutio. The algorithm starts with a iitial radom populatio called coutries. Some of the best coutries i the populatio are selected to be the imperialists ad the rest form the coloies of these imperialists. I a N dimesioal optimizatio problem, a coutry is a array. This array is defied as, =,,, (5) The cost of a coutry is foud by evaluatig the cost fuctio f at the variables (,,, ). The, = () = (,,, ) (6) The algorithm starts with N iitial coutries ad the best of them (coutries with miimum cost) are chose as the imperialists. The remaiig coutries are coloies that each belogs to a empire. The iitial coloies belog to the imperialists i coveiece with their powers. To distribute the coloies amog imperialists proportioally, the ormalized cost of a imperialist is defied as, ={ } (7) where c is the cost of the th imperialist ad C is its ormalized cost. Havig ormalized the cost of all imperialists, the ormalized power of each imperialist is defied by: p C N i C i (8) The iitial coloies are divided amog empires based o their powers. The the iitial umber of coloies of the th empire will be: N. C = roud {p. N c } (9) where N. C is the iitial umber of coloies of the th empire ad N c is the umber of iitial coloies. To divide the coloies, N. C of the coloies are radomly chose ad give to the th imperialist. These coloies alog with the th imperialist form the th empire.

4 90 Hamid Mozafaria et al. / Procedia Techology ( 0 ) The imperialist coutries absorb the coloies usig the absorptio policy. The absorptio policy (show i Fig. ) makes up the mai core of this algorithm ad causes the coutries to move towards their miimum optima. The imperialists absorb these coloies i lie with their power, as described by Eq. (0). The total power of each imperialist is determied by the power of both of its compoets, the empire power ad percetage of its average coloies power. C.C =cost fuctio (imperialist ) + λ mea {cost (coloies of empires )} (0) where C.C is the total cost of the th empire ad λ is a positive umber which is cosidered to be less tha oe. ~ (0, ) () I the absorptio policy, the coloy moves towards the imperialist by a amout x uit. The directio of movemet is the vector from coloy to imperialist, as show i Fig.. I this figure, the distace betwee the imperialist ad coloy, show by d ad x, is a radom variable with uiform distributio. is a umber, greater tha ad earer to. So, a proper choice is =. I the preset implemetatio, is /4 (r). The deflected directio of coloy movemet is, ~(,) () I ICA algorithm, to search for differet poits aroud the imperialist, a radom amout of deviatio is added to the directio of coloy movemet towards the imperialist. I Fig., this deflectio agle is show as, which is chose radomly ad with a uiform distributio. While movig towards the imperialist coutries, a coloy may reach a better positio, so the coloy positio chages accordig to the positio of the imperialist. Figure 3: Illustratio of coloies movig toward their imperialist. After a while all the empires except the most powerful oe will collapse ad all the coloies will be uder the cotrol of this uique empire. 5. Optimizatio Problem Referece [6] shows the details of the heat-coductio problem. I this sectio, the same objective fuctio as [6] is chose. The ability of ICA is to be compared with fiite elemet method (FEM) ad geetic algorithm (GA) i the optimizatio solutio. The flowchart of this algorithm is show i Figure 3.

5 Hamid Mozafaria et al. / Procedia Techology ( 0 ) Figure 4: Illustratio of imperialist competitive algorithm (ICA) [5] To obtai a optimal desig which cosiders both heat source ad thickess of adhesive layer, the objective fuctio is defied as follows [6], J x,..., x. x,..., x ( x,..., x) where, are costat ad x x,..., are desig variables. Whe efficiecy is more importat tha power factor,, 0 are selected. Whe source factor is more importat 0, are selected. By cosiderig both efficiecy ad source factor will be optimize simultaeously. I this optimizatio problem the goal fuctio is the iverse of Equatio (3). The optimizatio variables are the thickess of the layers. After fidig the optimal thickess, the upper ad lower compoets of heat flux o the iterface q, q ad the upper ad lower temperatures o the iterface T, T ca be evaluated. 6. Results ad Discussio This Figures 5 show the temperature distributio alog the y-axis, i the iterphase (x = 0) where there are liear temperature distributios while at the same time a heat source (Q 0 > 0) is set alog the iterphase. Fig. 6 shows the results for parabolic temperature distributio alog y-axis i the iterphase (x = 0) Q = +000 T Q = -000 T (3) Q=+000 T Q=-000 T y-coordiate y-coordiate Figure 5: Temperature distributio alog the y-axis i the iterphase, for liear temperature distributio

6 9 Hamid Mozafaria et al. / Procedia Techology ( 0 ) Q = +000 T Iterphase Q=+000 T 50 Q=-000 T 50 Q = -000 T y-coordiate y-coordiate Figure 6: Temperature distributio alog the y-axis i the iterphase, for parabolic temperature y-compoet of heat flux, W/m Q=0 iterphase Q= 000T Q= 000T Opt. ICA Opt. GA y-coordiate, m Q=0 Q= 000T Q= 000T iterphase BCs: T(y = 0.5) = 360 K T(y=-0.5)=90K y-coordiate, m Figure 7: Compariso of ICA with GA ad FE aalysis heat flux i iterphase layer for costat temperature distributio Figures 7 illustrates the compariso betwee fiite elemet aalysis with optimal desig based o ICA ad GA, for o-mootoic heat source to the iterphase layer with costat temperature distributio applied to the upper ad lower iterface surfaces. 7. Coclusio The preset work demostrates that the ew optimizatio procedure Imperialist competitive algorithm (ICA) is able to reproduce the same results at the classical approach which is based o the fiite elemet method. The compariso shows that the excellet predictio by ICA makes it a viable tool for optimizig heat-coductig problems o adhesive layers. Future works would iclude aalysis ad simulatio of thi reactive iterphase layers ad optimizatio based o geetic algorithm ad coloial competitive algorithm. Refereces. F. Rosseli, P. Carbutt: Sampe J. Vol. 37 (00), p. 7. K. Salomosso, T. Adersso. Mechaics of Materials 40 (008) K. Leffler, KS. Alfredsso, U. Stigh. It. J. Solids ad Structures. Vol. 44(007). pp Chirtoc, M. Chirtoc, I. Pittois, S. Glorieux, C. Thoe, J. Rev. Sci. Is. Vol. 74-: (003). P J.Y. Cogard, P. Davies, L. Sohier, R. Creac. Composite Structures. Vol. 76 (006), pp S. A. Maier ad H. A. Atwater, J. Appl. Phys. 98, 00 (005).

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