SURFACE PROFILE EVALUATION BY FRACTAL DIMENSION AND STATISTIC TOOLS USING MATLAB V. Hotař, A. Hotař Techncal Unversty of Lberec, Department of Glass Producng Machnes and Robotcs, Department of Materal Scence Abstract A lot of natural structures n ndustry applcatons can be hardly descrbed by conventonal methods mostly statstc tools, because they are complex and rregular. A relatvely new approach s the applcaton of the fractal geometry that s successfully used n scence, but an applcaton n ndustry s sporadc and expermental only. However, the fractal geometry can be used as a useful tool for an explct, objectve and automatc descrpton of producton process data (laboratory, off-lne and potental on-lne). The artcle s ntended on applcaton of the fractal geometry wth combnaton of statstc tool for a quantfcaton of metal surface changes of relatvely new materals: ron alumndes n comparson wth currently used chrome-nckel steels n contact wth a glass melt. The software Matlab was used for the generaton and classfcaton of dvdng lnes (a surface profle or a surface roughness evaluaton) and for evaluaton of results. Basc statstc and roughness parameters were used. The obtaned data n a dgtal form can be descrbed by the fractal geometry, whch expressng the complexty degree of structured data (deally) by means of a sngle number, the fractal dmenson. The dmenson can be estmated by many dfferent methods. A compass method s one of them and the method s based on the measurement of the dvdng lne by dfferent sze of a ruler. 1 Introducton Although contnuously growng a compettve press to ncreasng qualty of products actvates a requrement of an objectve measurement and control methods for materals, processes and productons, many structures (e.g. defects, surface, crack, tme seres from dynamc processes) can be hardly descrbed by conventonal methods, because they are complex and rregular. The fractal geometry [1-3] s a useful tool for analyss of complextes and rregulartes that use fractal dmenson (sngle number) for quantfcaton; however an applcaton n ndustry s sporadc and expermental only [4]. Currently, there are tools to montor tree basc data format types: dgtalzed photos (applcaton for the corrugaton test of wndows glass shelds), tme seres (analyss for control systems) and topologcal one dmensonal dvdng lnes (especally surface roughness). On account of ths, we are developng three off-lne programs and some are converted nto montorng systems. The analyss of topologcal one dmensonal dvdng lnes was developed for the quantfcaton of metal surface changes of relatvely new materals: ron alumndes n comparson wth currently used chrome-nckel steels n contact wth the glass melt a for laboratory use. The methodology was also used for an objectfcaton of the corrugaton test of glass sheet [5, n Czech]. The analyss was prepared n software Matlab that has sutable functon for ths reason. 2 Classfcaton of Dvdng Lnes Surface Profle Evaluaton Analyses were performed on samples of the ron alumnde Fe28Al4Cr0,1Ce and the chromenckel steel X15CrNS25 21 - EN 10095 (AISI 310) that were exposed to statc and dynamc glass melt effects n dfferent temperatures. The methodology of the surface profle evaluaton s shown n Fg. 1. Frstly, a dgtal camera takes a photo of a surface layer profle from a mcroscoped metallographc sample, Fg. 1, A.
Secondly, a dvdng lne s generated from the dgtal photography, Fg. 1, B, by the software tool developed n Matlab (usng the command contourc ) that exactly defned the curve between materal alloys and a surroundng - the dvdng s obtaned. The wdth of mages s 2272 that matches 57,7 µm. Statstc tools and (or) the fractal dmenson can descrbe the curve Fg.1, C, D. Photo of surface layer (chrome-nckel steel after dynamc test) Dvdng lne generaton (lne between metal alloy and surroundng - profle) r 1 =250 N 1 =18,5 L 1 = 4625 r 2 =500 N 2 =4,5 L 2 = 2250 Computng of compass dmenson: measurement of profle length by dfferent ruler r : L ( r ) = N ( r ) r Central slope Computng of compass dmenson: generaton of Rchardson Mandedlbrot plot, choosng central slope and a compass dmenson compute D R from central slope by relaton: D R log L( r) = 1 log r Fgure 1: Analyss of surface layer, dvdng lne generaton from photography, evaluaton by statstc and compass dmenson 3 Fractal Dmenson The fractal dmenson (also named the Hausdorff-Bescovtch dmenson) s closely connected to fractals that were defned by Benot Mandelbrot, though scentsts found some geometrc problems wth specfc objects before hm (e.g. the measurement of coast lnes per dfferent length of rulers by Rchardson) [1]. A potentally powerful property of the fractal dmenson (FD) s descrbng complexty by usng sngle number that defnes and quantfes structures. The number s mostly a no nteger value and FD s hgher then the topologcal dmenson. For example, the Koch curve (one of the most famous mathematcal determnstc fractal, Fg. 2) has the topologcal dmenson D T = 1, but the FD D F = 1.2619. A smooth curve as a lne has the topologcal dmenson D T = 1 and the FD D F =
1. The FD can be computed for set of ponts, curves, surfaces, topologcal 3D objects, etc. and f the FD s hgher than the topologcal dmenson, we name the objects fractals [1-3]. Fgure 2: Koch curve Analyses of data from producton processes, qualty controls, producton tools, etc. should correspond wth ther characters and the computng the FD s sutable for hghly structured data sets. Furthermore, the computaton of the FD s explct, objectve and fast and t enables the applcaton of the results for producton control and qualty montorng. The FD can be estmated by many dfferent methods [1-3]. A compass method [2] s one of them and the method s based on the measurement of the dvdng lne (roughness profle) by dfferent sze of a ruler (Fg. 1, C) va the equaton: L ( r ) = N ( r ) r (1) L s a length n -step of the measurement, r s a ruler sze and N s a number of steps needed for the measurement that s gven by a power law: D R ) = const. r N( r (2) If the lne s fractal and hence the FD s larger than the topologcal dmenson, the measured length ncreases as the ruler sze s reduced (Fg. 1, C). Usng equatons (1) and (2): DR 1 D R r = const. r L ( r ) = N ( r ) r = const. r (3) D R s the compass dmenson. Logarthmc dependence between log 2 N(r ) and log 2 r s called the Rchardson-Mandelbrot plot (Fg. 1, D). The compass dmenson s then determned from slope s of the regresson lne (Fg. 1, D): D log2 L( r) = 1 s = 1. (4) R log r 2 Although the typcal dependence conssts of three-parts slope, only central part (the central slope) s mportant for the compass dmenson computng. The compass dmenson D R s multpled by 1000 for better confrontaton, D R 1000. 4 Statstc Tools Basc statstc and roughness parameters were used such as: the Standard Devaton, the Range, the Average Surface Roughness (R a ), the Maxmum Roughness (R m or R max ), the Average Maxmum Heght of the Profle (R z ), the Peak Count (Peak Densty, P c ), the Mean Spacng (Sm),... (Fg. 1, C).
5 Fractal Dmenson versus Statstcs A comparson of statstcal tools and the FD s possble, but should be done wth care. The FD gves added nformaton about the character of descrbng data sets and to say that the FD s better than statstcs and vce versa s mpossble. Furthermore, the FD should not be used separately because the dmenson does not gve all the nformaton about data set captures. Usng added parameters (statstcs, topology, spectral analyss, etc.) together wth the dmenson brngs benefts and s recommended. A decsve number (a testng number) for producton control or qualty montorng (for example) can be computed from obtaned parameters (ncludng FD) by weght coeffcents. In hs thess [6] the author shows some other methods to use the cooperaton between the FD and statstcs and gves examples of ther applcaton. 6 Examples of Results The examples are shown n Fg. 3. Sx dgtal photos of every metal sample profle n dfferent poston were made and analysed. The presented results R a, R m and D R 1000 are an average of sx measurements on a tested sample and Fg. 3 shows only one example of the sx dvdng lnes. 7 Concluson Although fractal and statstc results correlate n these examples of results (a profle wth hgher R a and R m has hgher D R 1000 ), the estmated fractal dmenson (n ths artcle the compass dmenson) s nformaton about structure, but R a and R m are statstc nformaton. Ths nformaton can correlate, but t s not rule. The compass dmenson ndcates complexty of profle, whch can be used as added nformaton to statstc or as a sngle profle specfcaton. The estmated fractal dmenson can also be used for others dvdng lnes types such as a surface roughness classfcaton. The results of our research (from applcaton to dgtalzed pctures, tme seres or a dvdng lne) show that the fractal dmenson s potentally a powerful tool for explct, objectve and automatc descrpton and quantfcaton of complex data [5-8]. The possbltes of successful applcatons n ndustry are beleved to be large. Ths work was supported by the Czech Scence Foundaton (GA CR 106/05/P167).
Chrome-nckel steel EN 10095 (AISI 310) FA - ron alumnde on base Fe 3 Al Dvdng lne - roughness profle (sze n pxels) R m [µm] R a [µm] R m [µm] R a [µm] Dvdng lne - roughness profle (sze n pxels) D o 1000 [-] D o 1000 [-] 3.07 1.56 Ground state 0.79 1029.2 0.29 1055.8 4.28 1.54 1100 O C, 24 hour 1.01 1056.4 0.47 1012.0 28.11 5.52 1250 O C, 96 hour 8.59 1121.6 1.39 1100.3 24.08 22.64 1350 O C, 96 hour 6.09 1185.4 8.09 1124.0 Fgure 3: Examples of dvdng lnes chrome-nckel steel materal and ron alumnde, after statc glass melt effects n dfferent temperatures and results of analyses
References [1] MANDELBROT. B. B. 1982. The fractal geometry of nature. New York: W. H. Freeman and Co.,1982. [2] PEITGEN, H.O., JUERGENS, H. and SAUPE, D. 1992. Chaos and Fractals: New Fronters of Scence. New York; Berln; Hedelberg: Sprnger-Verlag, 1992. [3] BUNDE, A. and HAVLIN, S. Fractals n scence. Berln: Sprnger, 1994. [4] LEVY VEHEL, J., LUTTON, E., TRICOT, C. 1997. Fractals n Engneerng. New York; Berln; Hedelberg: Sprnger-Verlag, 1997. [5] HOTAŘ, V. Systém hodnocení vlntost plochého skla vyvnutý v prostředí Matlab R 14. In Techncal Computng Prague 2005 [CD + onlne, n Czech language]. Praha: Humusoft, 2005 [ct. 2007-10-16]. 5. s. ISBN 80-7080-577-3, avalable from: http://www.humusoft.cz/akce/matlab05/sbor05.htm [6] HOTAŘ, V. Evaluaton of ndustral data usng fractal geometry. [n Czech language] In: Thess Lberec: Techncal Unversty of Lberc, 2005. 123 s. [7] HOTAŘ, V., NOVOTNÝ, F.: Surface Profle Evaluaton by Fractal Dmenson and Statstc Tools. In: Proceedngs of 11th Internatonal Conference on Fracture. Turn (Italy): CCI Centro Congress Internazonale s.r.l., 2005. p. 588. [8] HOTAŘ, V., NOVOTNÝ, F. Some Advanced Analyse for Qualty Montorng. In proceedngs: 21st Internatonal Congress on Glass [on CD]. Strasbourg: Internatonal Commsson on Glass, 2007. Vlastml Hotar Department of Glass Producng Machnes and Robotcs, Techncal Unversty of Lberec, Halkova 6, 461 17 Lberec, Czech Republc, tel. +420 485 354 129, fax: +420 485 354 157, e-mal: vlastm.hotar@tul.cz Adam Hotar Department of Materal Scence, Techncal Unversty of Lberec, Halkova 6, 461 17 Lberec, Czech Republc, tel. +420 485 353 136, fax: +420 485 353 631, e-mal: adam.hotar@tul.cz