Evolutionary Algorithm in Identification of Stochastic Parameters of Laminates

Transcription

1 Evolutonary Algorthm n Identfcaton of Stochastc Parameters of Lamnates Potr Orantek 1, Wtold Beluch 1 and Tadeusz Burczyńsk 1,2 1 Department for Strength of Materals and Computatonal Mechancs, Slesan Unversty of Technology, Konarskego 18a, Glwce, e-mal: potr.orantek@polsl.pl, wtold.beluch@polsl.pl 2 Insttute of Computer Modellng, Artfcal Intellgence Department, Cracow Unversty of Technology, Warszawska 24, Kraków, e-mal: tadeusz.burczynsk@polsl.pl Abstract. The paper deals wth the dentfcaton of materal constants n mult-layered compostes. Smple and hybrd (wth lamnas made of dfferent materals) lamnates are consdered. Materal constants are presented n a stochastc form to model ther uncertanty. The Evolutonary Algorthm based on such representaton of the data s used as the global optmzaton method. Chromosomes are represented by multdmensonal random vectors consstng of random genes n the form of ndependent random varables wth the Gaussan densty probablty functons. The stochastc optmzaton problem s replaced by a determnstc one by evolutonary computng of chromosomes havng genes consstng of mean values and standard devatons. Modal analyss methods are employed to collect measurement data necessary for the dentfcaton. The Fnte Element Method n stochastc verson s used to solve the boundary-value problem for lamnates. Numercal examples showng effcency of the method are presented. 1 Introducton Compostes are materals constructed by jonng (at least) two materals together on the macroscopc level. They usually consst of the matrx and the renforcement. The resultant propertes of compostes depend on: ) the propertes of phases; ) the volume fracton of the renforcement; ) the dstrbuton of the renforcement n the matrx; v) the geometry of the renforcement. The most commonly used group of compostes are lamnates fbre-renforced compostes made of many layers (ples). The fbres are usually stuated drectonally n each ply of the lamnate, but have dfferent drectons n partcular ples. There are two man reasons of the popularty of lamnates: ) the hgh strength/weght (or stffness/weght) rato, especally n comparson wth the conventonal, usually sotropc materals; ) the possblty of desgnng the materal propertes accordng to the requrements by manpulatng the components materals, stackng sequence, fbers orentaton and layer thcknesses. The cost of lamnates quckly ncreases wth ther strength. To fnd the balance between cost and requred propertes, the ples n lamnates may be composed of more than one materal [1]. The lamnates obtaned n such a way are called the hybrd ones. The nterply hybrd lamnates wth ples composed of two dfferent materals are consdered n the present paper. The nternal

2 layers are bult of a weaker and cheaper materal whle the external layers are made of a hghstffness but more expensve materal. The am of the paper s to dentfy materal constants n multlayered, fbre-renforced compostes (lamnates). Smple and hybrd lamnates are taken nto account. An dentfcaton problem can be formulated as the mnmzaton of some objectve functons dependng on measured and calculated state felds. The dentfed materal constants of lamnates are assumed to be nondetermnstc ones due to the manufacturng process. The uncertantes are ntroduced to decrease dscrepances between real structures and ther mathematcal model. The uncertantes can be ntroduced by usng dfferent granularty models, lke nterval numbers, fuzzy numbers or n the stochastc way [3]. The dentfcaton of the materal constants n lamnates wth nterval and fuzzy representaton of the desgn varables was presented n [4, 5]. In the present paper the stochastc approach to dentfcaton s taken nto account. The parameters are modelled by means of random varables characterzed by probablty densty functons. The classcal approach to the soluton of such problems s based on stochastc programmng [6]. In the proposed the evolutonary algorthm wth stochastc representaton of genes s used as the global optmzaton method. 2 The Formulaton of the Stochastc Identfcaton Problem A general non-lnear stochastc programmng problem can be descrbed as searchng for a random vector [7]: X ( γ ) = [ X ( γ), X ( γ),..., X ( γ),..., X ( γ)] (1) 1 2 whch mnmzes the objectve functon F( γ ) = F( X ( γ )) and satsfes the constrants: P g j( X ) 0 pj, j = 1, 2,..., m. (2) The probablty space (Γ, F, P) plays the basc role n the theoretcal model of random phenomena [10]. The set Γ s called the space of elementary events and t represents all the possble smplest outcomes of a tral assocated wth a gven random phenomenon. F s a σ- algebra of subset of the set Γ and the elements of the F are called random events. P denotes the probablty defned on F. If the problem s solved by the evolutonary algorthm, the vector X ( γ ) s called a chromosome, where X ( γ ), =1,2,,n, are random genes. Each gene s represented by a random varable, whch s a real functon X = X( γ ), γ Γ, defned on a sample space Γ and measurable wth respect to P:.e., for every real number x, the set { γ : X ( γ ) < x} s an event n F. The chromosome X ( γ ) s a functon (measurable wth respect to P) whch takes every element γ Γ nto a pont x Rn. The mean value of the chromosome X ( γ ) s gven as: m = E[ X ( γ )] = [ m1, m2,..., m,..., mn] (3) where: + [ ( γ) ] ( γ) ( γ) ( ) Γ m = E X = X dp = x p x dx (4) n

3 s the mean value of the gene X ( γ ) and p( x ) s the probablty densty functon (PDF) of ths gene. The matrx of covarance s gven as: The covarance between ( ) where (, j) T ( ) ( ) K = [ kj ] = E X( γ) m X( γ ) m (5) X γ and X ( γ ) s defned as: j + + ( ( γ) )( ( γ) ) ( )( ) (, ) k = E X m X m = x m x m p x x dx x (6) j j j j j j j p x x s the jont PDF of X ( γ ) and X j ( γ ). If = j, the covarance s represented by a varance. In the present paper the random chromosome X ( γ) = [ X1( γ), X2( γ),..., X( γ),..., Xn( γ)] consstng of n genes has a n-dmensonal Gaussan dstrbuton functon. It s assumed that random genes are ndependent random varables. The jont probablty densty functon s expressed by the probablty densty functons of sngle random genes: where: px (, x,..., x,..., x) = p( x) p( x)... p( x)... p( x) (7) 1 2 n n n ( x m ) 2 1 p( x) = N( m, σ ) = exp 2 σ 2 2σ π s the probablty densty functon of the random gene X ( γ ),, σ X γ. (8) denotes the standard devaton of ( ) If the random genes ( ) functon can by fully descrbed by means of two parameters: the mean value X γ, =1,2,,n are ndependent random Gaussan varables, the PDF m and the standard devaton σ. Compostes are ansotropc materals. Multlayered lamnates can be usually treated as thn plates made of orthotropc materals. If the ples are dstrbuted symmetrcally to the mdplane, the lamnate s called symmetrcal. Man advantage of such lamnates s that there does not exst a couplng between sheld and bendng states [8]. The sngle ply of the lamnate, beng n the plane-stress state, has 4 ndependent materal constants: axal and transverse Young's module (E 1, E 2 ), axal-transverse shear modulus (G 12 ) and axal-transverse Posson rato (ν 12 ). It s assumed, that materal constants are random varables wth Gaussan PDF. The dentfcaton of the materal constants can be treated as the mnmzaton of the objectve functon F wth respect to the vector of the desgn varables x: ( F ) F = ( q ˆ ) 2 j qj mn, where: x N j= 1. (9) The functonal F depends on N measured ˆ j q and N calculated q j values of the state felds. It s assumed, that measured and calculated values of the state felds have stochastc character. The dentfcaton results strongly depend on the number of measurement data. To reduce the number of sensors, the dynamc propertes of lamnates can be taken nto account and the modal analyss methods can be employed. The egenfrequences are used as the measurement data [11]. The numbers of ples, ther thcknesses, fbres orentaton and the number of layers made of each materal are assumed to be known.

4 3 Evolutonary Algorthm for Stochastc Problems The applcaton of evolutonary algorthms to solve stochastc optmzaton problems requres some modfcatons of the classcal evolutonary approaches. The man reason s that chromosomes consst of random genes X ( γ ), =1, 2,, n. As the results the genes are descrbed by the moments, e.g. by the mean value m and the standard devaton σ (n the case of Gaussan ndependent random genes). The mean dea of presented Evolutonary Algorthm s analogous to the determnstc evolutonary algorthm [2], but all steps of the algorthm requre some modfcatons due to the stochastc character of the data. Each ndvdual (chromosome) n employed EA expresses a stochastc soluton of the optmzaton problem. The ftness functon value for each ndvdual s evaluated and a stochastc value of the ftness functon s obtaned as the result of calculatons. To calculate the stochastc ftness functon value the Stochastc Fnte Element Method (SFEM) s employed [9]. The orgnal stochastc problem can be reduced to the determnstc one. Random chromosome X ( γ ) can be replaced by a determnstc chromosome ch j (x) n whch each gene x s a stochastc varable represented by 2 values: mean value and standard devaton: x=( m, σ ), =1,2,,n correspondng to the random varable X ( γ ): [ ] [ σ σ σ σ ] j ch ( x ) = x ; x ;... x ;...; x = ( m, );( m, );...;( m, );...;( m, ) (10) 1 2 n n n Two knds of constrants are mposed for each chromosome x=( m, σ ), =1,2,,n,: m m m (11) mn max σ σ σ (12) mn max where: mn and max ndces denote the maxmum and mnmum values of m and σ. The selecton method s based on the classcal tournament selecton. The mnmzaton problem s taken nto account. Consder the ftness functons for two dfferent random chromosomes: F1( γ ) = F( X 1( γ )) and F2( γ ) = F( X 2( γ )). The random values F ( γ ) and 1 F ( ) 2 γ are descrbed by the moments: F1( γ ) ( m F 1, σ F 1) and F2( γ ) ( m F 2, σ F 2), respectvely. The parameters β 1 and β 2, whch decde about the probablty of the survval of chromosomes: X ( γ ) and X ( γ ), 1 2 respectvely, are ntroduced. At the begnnng the parameters β 1 and β 2 are equal to β 0 (n consdered case β 0 =0.1). In the next step, the condtons: m < m (13) σ F1 F 2 F1 F 2 < σ (14) are checked. If the condtons (12) and (13) are fulflled, the probablty of the survval of the frst chromosome s ncreased by Δ βm and Δ βσ, respectvely (n present paper Δ βm =0.7, Δ βσ =0.3). In the contrary cases the probablty of the survval of the second chromosome s ncreased by Δ β m and Δ βσ, respectvely. If the both stochastc values of the ftness functons are dentcal, the probabltes of the survval of both ndvduals are the same. Fnally, the survved ndvdual s sampled wth respect to the survval parameters β 1 and β 2. The dedcated stochastc versons of mutaton and crossover operators are also appled. Two types of the Gaussan mutaton are ntroduced. In the frst type of mutaton the mean value m of the random gene X ( γ ) s modfed. In the second type of the mutaton the standard devaton σ of the random gene X ( γ ) s modfed. A dedcated arthmetc crossover operator creatng two offsprng chromosomes from two parents s proposed for the stochastc evolutonary algo-

5 rthm. The evolutonary operators appled n presented EA are more comprehensvely descrbed n [7]. In consdered case chromosomes have one of the followng forms (the denstes of materals also have to be dentfed n the case of hybrd lamnates): 1. For smple lamnates: ch j (x) = (E 1, E 2, G 12, ν 12 ) (15) 2. For hybrd lamnates (superscrpts specfy the materal number): where ρ - the densty of the -th materal. ch j (x) = (E 1 1, E 1 2, G 1 12, ν 1 12, ρ 1, E 2 1, E 2 2, G 2 12, ν 2 12, ρ 2 ) (16) 4 Numercal Examples 4.1 Identfcaton of the smple lamnate A rectangular smple lamnate plate made of the glass-epoxy s consdered (Fgure 2a). Each ply of the symmetrcal lamnate has the same thckness h =0.002m. The stackng sequence of the lamnate s: (0/45/90/-45/0/90/0/90)s. To solve the drect problem the plate s dvded nto node plane fnte elements. The frst 10 egenfrequences ω of the plate are taken nto account as the measurement data. It s assumed, that measurement are random varables wth the Gaussan dstrbuton. The measurements were repeated 200 tmes to collect data. Each chromosome chj(x) n populaton conssts of 4 genes. Each gene x n chromosome chj(x) s a random number represented by 2 moments: mean value m and standard devaton σ. y 0.2 x M { 1 M 2 symmetry 0.5 a) b) Fgure 2. a) The lamnate plate a) shape and dmensons; b) materals locaton n hybrd lamnate. The parameters of the EA (expermentally selected) are: - the number of chromosomes pop sze = 200; - the number of generatons gen_num = 400; - the arthmetc crossover probablty p ac = 0.2; - the gaussan mutaton probablty: p gm = 0.4. The varable ranges, actual values and dentfcaton results are collected n Table 1.

6 Table 1. Identfcaton results for the smple lamnate. E 1 [Pa] E 2 [Pa] ν 12 G 12 [Pa] m σ m σ m σ m σ Mn 2.00E E9 4.00E9 0.00E E9 0.10E8 Max 6.00E E9 9.00E9 0.30E E9 0.70E8 Actual 3.86E E9 8.28E9 0.20E E9 0.50E8 Found 3.92E E9 8.14E9 0.17E E9 0.22E8 All materal parameters, represented by mean value and standard devaton, have been found wth satsfactory precson. 4.2 Identfcaton of the hybrd lamnate A rectangular hybrd lamnate plate of shape and dmensons presented n Fgure 2a) s consdered. Each ply of the lamnate has thckness h = 0.002m. The symmetrcal lamnate plate conssts of 10 ples of the stackng sequence: (0/15/-15/45/-45)s. The external ples of the lamnate are made of materal M 1, the nternal lamnas are made of the materal M 2 (Fgure 2b). The plate FEM model conssts of node plane fnte elements. The frst 10 egenfrequences of the plate are consdered. It s assumed, that measurement data have stochastc nature and they are descrbed by the normal dstrbuton. The measurements were repeated 200 tmes to collect data. Each of 10 genes x n chromosome chj(x) s a random number represented by 2 moments: mean value m and standard devaton σ. The parameters of the EA (expermentally selected) are: - the number of chromosomes pop sze = 400; - the number of generatons gen_num = 1200; - the arthmetc crossover probablty p ac = 0.2; - the gaussan mutaton probablty: p gm = 0.4. The varable ranges, actual values and dentfcaton results are collected n Table 2 for materal M 1 and n Table 3 for materal M 2. Table 2. Identfcaton results for the hybrd lamnate materal M 1. E 1 [Pa] E 2 [Pa] ν 12 G 12 [Pa] ρ [kg/m 3 ] m σ m σ m σ m σ m σ Mn 2.00E E9 4.00E9 0.00E E9 0.10E8 1.00E3 0.00E2 Max 6.00E E9 9.00E9 0.30E E9 0.70E8 3.00E3 0.50E2 Actual 3.86E E9 8.28E9 0.20E E9 0.50E8 1.80E3 0.20E2 Found 3.75E E9 8.12E9 0.28E E9 0.61E8 1.81E3 0.17E2

7 Table 3. Identfcaton results for the hybrd lamnate materal M 2. E 1 [Pa] E 2 [Pa] ν 12 G 12 [Pa] ρ [kg/m 3 ] m σ m σ m σ m σ m σ Mn 1.20E E E E E9 0.10E8 1.00E3 0.00E2 Max 2.50E E E E E9 0.70E8 3.00E3 0.50E2 Actual 1.80E E E E E9 0.50E8 1.60E3 0.20E2 Found 2.01E E E E E9 0.52E8 1.63E3 0.19E2 All materal parameters for both materals (M 1 and M 2 ) of hybrd lamnate have been found wth acceptable precson. 5 Fnal Conclusons An effcent dentfcaton method based on the stochastc representaton of the dentfed parameters has been presented. A global optmzaton method n the form of the Evolutonary Algorthm has been employed to solve the dentfcaton task for smple and hybrd lamnates. Presented EA works on the stochastc genes converted to determnstc ones by representng stochastc varables by two moments: the mean value and the standard devaton. Postve dentfcaton results have been obtaned for smple as well as for hybrd lamnates. The future work s to combne global optmzaton methods wth local ones. To reduce nconvenences connected wth the necessty of the ftness functon gradent calculatons, t s possble to employ Artfcal Neural Network (ANN). Ths atttude was successfully tested on the fuzzy and nterval representaton of the dentfed parameters [5]. The EA s used n the frst step; afterwards the local optmzaton method supported by ANN s employed to fnsh the computatons. The ANN s used to approxmate the ftness functon and the ftness functon gradent. Acknowledgements The research s partally fnanced from the Polsh scence budget resources as the research project and the Foundaton for Polsh Scence ( ). Bblography [1] Adal, S. et al. Optmal desgn of symmetrc hybrd lamnates wth dscrete ply angles for maxmum bucklng load and mnmum cost, Composte Structures, 32: , [2] Arabas, J. Lectures on Evolutonary Algorthms. WNT, 2001 (n Polsh). [3] Bargela, A., and Pedrycz, W. Granular Computng: An ntroducton. Kluwer Academc Publshers Boston/Dordrecht/London, [4] Beluch, W., Burczyńsk, T., and Orantek, P. Evolutonary Identfcaton of Fuzzy Materal Constants n Lamnates. Evolutonary Computaton and Global Optmzaton 2007, Ofcyna Wyd. Pol. Warszawskej, Prace Naukowe, Elektronka, z.160, (Ed. J. Arabas), Warszawa, pages 17-24, [5] Beluch, W., Burczyńsk, T., and Orantek, P. The fuzzy strategy n dentfcaton of lamnates' elastc constants. Full papers. Computer Methods n Mechancs CMM-2007, Spała, 2007, CD-ROM.

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