EVOLUTIONARY COMPUTING FOR METALS PROPERTIES MODELLING

Transcription

1 EVOLUTIONARY COMPUTING FOR METALS PROPERTIES MODELLING M.F. Abbod*, M. Mahfouf, D.A. Linkens and Sellas, C.M. IMMPETUS Institute fo Micostuctue and Mechanical Popeties Engineeing, The Univesity of Sheffield Depatment of Automatic Contol and Systems Engineeing, The Univesity of Sheffield, Sheffield, UK * Abstact: Duing the last decade Genetic Pogamming (GP) has emeged as an efficient methodology fo teaching computes how to pogam themselves. This pape pesents eseach wok which utilises GP fo developing mathematical equations fo the esponse sufaces that have been geneated using hybid modelling techniques fo pedicting the popeties of hot defomation of mateials. Collected data fom the liteatue and expeimental wok on aluminium ae utilised as the initial taining data fo the GP to develop the mathematical models unde diffeent defomation conditions and compositions. Keywods: Alloy mateials, genetic pogamming, mateial popeties, modelling, stain, ste.. INTRODUCTION One of the cental challenges of compute science is to get a compute to pefom cetain tasks without instucting it how to do it. Genetic pogamming addees this challenge by poviding a method fo automatically ceating a woking compute pogam fom a high-level statement of the poblem. Genetic Pogamming (GP), invented by Came in 985 and futhe developed by Koza (99), solved the poblem of fixed length solutions by ceating non-linea entities with diffeent sizes and shapes. The epetoie used to ceate these entities was also moe vaied, ceating a iche, moe vesatile system of epesentation. GP achieves this goal of automatic pogamming (also sometimes called pogam synthesis o pogam induction) by genetically beeding a population of compute pogams using the pinciples of Dawinian natual selection and biologically inspied opeations. The opeations include epoduction, coove (sexual ecombination), mutation, and achitectue-alteing opeations pattened afte gene duplication and gene deletion in natue. This pape is based on developing mathematical models fo hybid models that have been developed fo modelling metals popeties unde hot defomation conditions. Collected data fom the liteatue and expeimental wok fo aluminium alloys ae utilised as the databases fo the models. Unde diffeent defomation conditions and compositions, data points ae available fo the flow ste and ecystallisation.. GENETIC PROGRAMMING Genetic pogamming stats with a pimodial lage set of andomly-geneated compute pogams. The set of functions that may appea at the intenal points of a pogam tee may include odinay aithmetic functions and conditional opeatos. The set of teminals appeaing at the extenal points typically include the pogam's extenal inputs (such as the independent vaiables X and Y) and andom constants (such as. and.4). The andomly ceated pogams typically have diffeent sizes and shapes Conside, fo example, the following algebaic expeion:

2 a b c It can also be epesented as a diagam which is shown in Figue. d e () + / Q * c - a b d e Fig.. Genetic pogamming tee. whee Q epesents the squae oot function. This kind of diagam epesentation is in fact the phenotype of GP chomosomes, being the genotype easily infeed fom the phenotype as follows: /Q*c-abde () which is the staightfowad eading of the equation fom left to ight and fom top to bottom (exactly as ones ead a page of text). The expeion is an Open Reading Fame, stating at + (position ) and teminating at e (position 9). The subtaction and division ae functions of two aguments, and theefoe in the next line ae placed fou moe nodes, in this case, the symbols at positions, 4, 5, and 6. Now, we have two diffeent functions in the thid line: one is a function of one agument (Q), and anothe a function of two aguments (+). Theefoe, thee moe nodes ae equied in the next line. In this case, they ae filled with the elements at positions 7, 8, and 9... Solving Poblems with Genetic Pogamming The aim of this study is to utilise GP fo finding mathematical equations as a solution fo modelling the behaviou of metals based on specified input conditions. In symbolic egeion o function-finding the goal is to find an expeion that satisfactoily explains the dependent vaiable. The input into the system is a set of fitne cases in the fom (a(i,), a(i,),..., a(i,n-), yi) whee a(i,) - a(i,n-) ae the independent vaiables and yi is the dependent vaiable. The set of fitne cases consists of the adaptation envionment whee solutions adapt, thus discoveing, solutions to poblems. Thee ae five majo steps in pepaing to use GP. The fist is to choose the fitne function. Fo instance, we could measue the fitne fi of an individual pogam i by the following expeion: f i c t j ( M C T () ( i, j) j whee M is the ange of selection, C(i,j) the value etuned by the individual chomosome i fo fitne case j (out of Ct fitne cases), and Tj is the taget value fo fitne case j. If C(i,j) - Tj (the pecision) is le than o equal to., then the pecision is equal to zeo, and fi = fmax = Ct.M. (Feeia ). The second majo step consists of choosing a set of teminals T and a set of functions F to ceate the chomosomes. The thid majo step is to choose the chomosomal achitectue, i.e., the length of the individual and the numbe of genes. The fouth majo step in pepaing to use GP is to choose the linking function. Finally, the fifth majo step is to choose the set of genetic opeatos that cause vaiation and thei ates.

3 . MATERIAL PROPERTIES MODELLING One method fo modelling of metals popeties is the hybid modelling technique which povides a good modelling tool which combines diffeent modelling appoaches in one dynamic model (Abbod et al, ). Combinations of physical models and neuo-fuzzy models wok vey efficiently. While the physical model ensues that the esults ae physically sensible, the neuo-fuzzy model can be utilised to incease the accuacy of the esults. One of the poible hybid modelling techniques is the semi-paametic model which utilises seial and paallel paametic and non-paametic models. This gives advantages fo compensating fo spase data, extapolation impovements, uncetainty and the bias of the default model. The ultimate aim of modelling the metals behaviou, howeve, is to be able to povide tools that can be extended to all conditions and fo othe mateial. Cuently, physically-based modelling is difficult to apply to extended conditions and mateials. To extend the egion of application equies the caying out of a huge amount of expeimental eseach, which is a vey time-consuming poce. Theefoe, it is desiable to develop a modelling tool capable of being used fo pediction unde extended defomation conditions and fo othe kinds of metals. The hybid modelling is caied out by combining the developed neuo-fuzzy models to pedict the intenal states, while the physically-based models all used to pedict the final ecystallisation and flow ste conditions (Zhu et al, ). In the hybid model, the inputs ae the defomation conditions such as stain ate, tempeatue T and defomation level, i.e. stain fo a given mateial. The fiction ste is at pesent difficult to calculate theoetically and hence is calculated using a neuo-fuzzy model based on expeimental esults. The intenal state vaiables such as intenal dislocation density, subgain size and misoientation between subgains ae calculated by neuo-fuzzy models. The neuo-fuzzy models wee tained based on data extacted fom fitted smoothed cuves of the oiginal expeimental measuements. Then, using the physical equations the final flow ste and the ecystallisation behaviou (N v, d ex, t 5 ) ae detemined. Since the model coves a wide ange of tempeatue and stain ate fo each individual composition, tuning the model was impotant to adjust some of the paametes fo scaling the intenal state vaiables and calculating the individual intenal stees. 4. FLOW STRESS MODELLING The genealisation of the model to a wide ange of defomation conditions and compositions stated with collecting data fom the liteatue fo diffeent authos and diffeent defomation conditions. The validation poce of the hybid model is based on the input-output elationships, in othe wods the input defomation conditions, composition, and the output flow ste and the ecystallisation kinetics. The intemediate vaiables (intenal states) ae validated based on the output of the model using the physical equations. 4..Model Developments Model pediction has been pefomed fo diffeent alloys based on the expeimental esults fom diffeent eseaches. It was obseved that the most impotant elements which have a significant effect on the flow ste ae, Mn and Fe. Theefoe, in ode to develop a global model, the effects of the alloying elements have to be consideed as the majo inputs to the composition block. In designing expeimental conditions fo developing such a model, the input paametes to be consideed ae the stain, stain ate, tempeatue, and composition (, and othe alloying elements). The collected data fom the liteatue wee based on the input defomation conditions, the composition of the mateial, and defomation conditions, and any othe elated infomation which might be of use in the futue in ode to explain some of the unexpected behaviou that might occu duing the modelling stage. Infomation of paticula impotance is the composition of the mateial which has been ecoded fo most of the collected data. One of the impotant factos that affect the final popeties of the metals is the level of impuities. Duing the validation poce it was noted that fo commecial alloys, the impuities might be vey high such that they exceed the solubility limits. Theefoe, coase second phase paticles ae fomed which do not affect the final flow ste of the mateial. Theefoe, it was impotant to test the limits of the solubility of the impuities because it is consideed to be an impotant facto fo the model pediction. The developed hybid model is based on commecial puity Al-% alloy which has some impuities, theefoe it is expected that fo high puity aluminium the pedicted flow ste should be highe than the actual values. The same statement is valid fo commecial puity aluminium, as the inclusion of moe impuities should incease the flow ste while the hybid model will unde-pedict the flow ste. Fo commecial alloys, the

4 model unde-pedicts the flow ste due to the effect of the magnesium whose pesence inceases the flow ste. Using the calculated slopes fo diffeent compositions, the final effect of the impuities on the slope of the pedicted flow ste behaviou is divided into two steps; fo pue aluminium, and fo commecial aluminium. The two elationships fo the pue aluminium and aluminium alloys can be combined togethe to geneate a D suface that epesents the effects of the impuities and content on the final pedicted flow ste of the mateial. The dynamic evolution of the flow ste should be based on some paametes, which ae the initial flow ste, the final flow ste and the stain when steady-state conditions ae eached. The dynamic behaviou of the flow ste is govened by the following equations: a b (4) whee a, b and c should be composition-dependant only. c (5) The final flow ste can be calculated using the following equation: The stat of steady-state s is defined as follows: / ( )[( exp( ))] (6) ( ).95 ; then. (7) A gaphical epesentation of the flow ste cuve behaviou is shown in figue. Table shows the data collected fom the liteatue via diffeent authos. All the elated infomation is listed in this table. The initial values fo the flow ste ae not easy to quantify. This is because the figues ae digitally quantised fom diffeent souces, theefoe,it was not always clea what the initial values wee. Since these locations ae not easy to distinguish and quantify, it has been decided to select the steady-state flow ste ( ), the elaxation ste ( ) and the elaxation stain ( ). Using the collected and quatitised data, thee -D sufaces wee geneated which descibe the elationship between the contents, the impuities and the steady-state flow ste ( ), elaxation ste ( ) and elaxation stain ( ) as shown in Figue -5(a) espectively. The sufaces wee geneated using a neuo-fuzzy modelling technique. 5. MODELLING RESULTS One impotant featue of GP is that the initial stuctue of the equations can be pedefined and constained fo both the paametes limits and the algebaic fomulas. With the initial metallugical knowledge of such fomulation, the equations can be pedefined and fomulated to evolve to semi-physical equations. Diffeent input vaiables can be used such as the defomation conditions and the composition of the mateial. The developed -D sufaces which descibe the elationship between the impuities, contents and the pedicted slope ae based on neuo-fuzzy models which ae data-diven. Such models ae linguistically intepetable in tems of the input and output vaiables. Nevethele, a mathematical equation can be used to model the sufaces based on the data geneated by the neuo-fuzzy models. The modelling of such sufaces equies knowledge of the basic components of a polynomial equation. The stuctue of the polynomial equation can be made flexible to select the best fit that minimises the eo between the pedicted and modelled sufaces. GP is one method which allows such flexibility in the stuctue.

5 Typically, an initial function stuctue can be based on the basic mathematical opeatos (+-/*) and the input vaiables, with some knowledge being available about the sot of suface. Some functions ae supplied to the algoithm to stat as an initial pogamming seed. Such functions ae exp(impuities),,, and. The pedicted equations fo the steady state flow ste and wee found to be espectively: ste 64( G( exp( 6 9) im im T ) A B C D (8) ste D G( exp( im T )) A B C (9) stain D G ( R exp( im T )) A B C () whee im denotes the impuities. The equations paametes ae as shown in Table. Figues -5(b) show the pediction accuacy fo the flow ste, elaxation ste and elaxation stain espectively. paametes ste_ ste_ stain_ G R.669 T A B C D eo Table : GP paametes fitting. 6. CONCLUSIONS The developed hybid models have been extensively validated against diffeent expeimental esults fom diffeent authos and diffeent defomation conditions. The developed model can be used by both the eseach community and industy as it povides the intenal states as well as the final metals popeties. Futhemoe, GP was utilised to geneate mathematical models fo a wide ange of compositions that can be used to pedict the mateial popeties without the need fo expeimental testing which may pove to be an expensive and time consuming execise. ACKNOWLEDGMENT The authos gatefully acknowledge the UK EPSRC (Engineeing and Physical Sciences Reseach Council) fo thei financial suppot unde gant numbe GR/R754/.

6 REFERENCES Abbod, M.F., D.A. Linkens and Q. Zhu (). Semi-physical modelling of intenal states fo aluminium alloys. Mateials Science and Technology, vol A, pp Came, N.L. (985). A epesentation fo the adaptive geneation of simple sequential pogams. In J. J. Gefenstette, ed., Poceedings of the Fist Intenational Confeence on Genetic Algoithms and Thei Applications, Elbaum. Feeia, C. (). Gene expeion pogamming: a new adaptive algoithm fo solving poblems. Complex Systems. Koza, J.R. (99). Genetic Pogamming: On the Pogamming of Computes by Means of Natual Selection, Cambidge, MA: MIT Pe. Zhu, Q., M.F. Abbod, J. Talamantes-Silva, C.M. Sellas, D.A. Linkens and J.H. Beynon (). Hybid modelling of aluminium-magnesium alloys duing themomechanical poceing in tems of physicallybased, neuo-fuzzy and finite elements models, Acta Mat, (in pe). mateial (%) Impuities (MPa) (MPa) CP Al AL Al Al-Mn Al Al Al Al Al Al--Mn AA Table : Calculated paametes fo the developed coection sufaces. 95% 6% Fig.. Gaphical pesentation of the dynamic behaviou of the flow ste.

7 GP ste (MPa) GP ste (MPa).5.5 slope.5.5 impuities.5.5 NF ste (MPa) (a) NF esponse suface (b) GP equation pediction Fig.. Steady-state flow ste modelling..5.5 slope impuities NF ste (MPa) (a) NF esponse suface (b) GP equation pediction Fig. 4. Relaxation ste modelling.

8 GP stain slope impuities NF stain (a) NF esponse suface (b) GP equation pediction Fig. 5. Relaxation stain modelling.