The international journal of advanced manufacturing technology, tập 60, số 9 12, 2012

Int J Adv Manuf Technol (2012) 60:841–851 DOI 10.1007/s00170-011-3647-1 ORIGINAL ARTICLE An image-based methodology to establish correlations between porosity and cutting force in micromilling of porous titanium foams M Abolghasemi Fakhri & E V Bordatchev & O R Tutunea-Fatan Received: August 2010 / Accepted: 14 September 2011 / Published online: 12 October 2011 # Her Majesty the Queen in Right of Canada 2011 Abstract Porous titanium foam is now a standard material for various dental and orthopedic applications due to its light weight, high strength, and full biocompatibility properties In practical biomedical applications, outer surface geometry and porosity topology significantly influence the adherence between implant and neighboring bone New microfabrication technologies, such as micromilling and laser micromachining opened new technological possibilities for shape generation of this class of products Besides typical geometric alterations, these manufacturing techniques enable a better control of the surface roughness that in turn affects to a large extent the friction between implant and surrounding bone tissue This paper proposes an image analysis approach for optical investigation of the porosity that is tailored to the specifics of micromilling process, with emphasis on cutting force monitoring According to this method, the area of porous material removed during micromilling operation is estimated from optical images of the micromachined surface, and then the percentage of solid material cut is calculated for each tool revolution The employment of the aforementioned methodology in micromilling of the porous titanium foams revealed reasonable statistical correlations between porosity and cutting forces, especially when they were characterized by low-frequency M Abolghasemi Fakhri : E V Bordatchev : O R Tutunea-Fatan (*) Department of Mechanical and Materials Engineering, The University of Western Ontario, London, ON N6A 5B9, Canada e-mail: rtutunea@eng.uwo.ca M Abolghasemi Fakhri : E V Bordatchev Centre for Automotive Materials and Manufacturing, Industrial Materials Institute, National Research Council of Canada, 800 Collip Circle, London, ON N6G 4X8, Canada variations The developed procedure unlocks new opportunities in optimization of the implant surface micro-geometry, to be characterized by an increased roughness with minimal porosity closures in an attempt to maximize implant fixation through an appropriate level of bone ingrowth Keywords Porous titanium foam Optical imaging Image processing Porosity Micromilling Cutting force Introduction Porous sintered metals are typically produced by powder metallurgy and represent a class of relatively new materials with wide variety of industrial applications, especially biomedical applications, e.g., dental and orthopedic applications [1–3], due to their light weight, high strength, and full biocompatibility properties However, outer surface geometry and porosity topology for fabricated components, e.g., implants and prostheses, significantly influence the adherence of bone cells to implant material Therefore, most components fabricated from porous foam metals are produced in near-net shapes, and still require secondary machining operations [4] that can provide the desired roundness, smearing, and other surface quality parameters which cannot be obtained during sintering [5] Also, there are many 3D shapes and geometries that are difficult or almost impossible to produce by conventional forming technologies [2] without secondary machining, e.g., slots, bevels, blind holes, threads, cross-holes, and re-entrants normal to the pressing directions At present, about half of the components produced by powder metallurgy parts require secondary machining operations [6] New microfabrication technologies, such as micromilling and laser micromachining [7], opened new technological possibilities 842 Int J Adv Manuf Technol (2012) 60:841–851 options for characterization and visualization of 3D foam structures [11] The micro-CT technique performs segmentation of 3D porosity into a set of 2D CT slice images providing acceptable image quality, precise density profile, high spatial resolutions (i.e., below μm), good pore contrast, and reliable pore shape anisotropy Micro-CT is still under development especially in terms of 3D structure reconstruction and extraction of reliable information on structural parameters [18, 19] As mentioned above, metal-foam-based functional components (e.g., biomedical implants and prostheses) require a rough outer surface with fully open pores in order to attain high levels of bonding between implant and bone Among the options available to generate implant surfaces, material removal operations have always been well regarded, especially when moving into the micro-scale domain In this sense, the use of small diameter cutting tools (around 25 μm) in micromilling operations is believed to be capable of significantly enhancing the accuracy of the generated surfaces However, when the size of the cutting tools becomes comparable with pore size, this induces significant fluctuations in the cutting force [20] leading to excessive tool wear and/or breakage Cross-correlation between micromachining parameters and porosity has been relatively little investigated in the literature and therefore represents one of the objectives of the current work Intuitively, it is easy to understand that each micromilled slot cut in a porous sample will be characterized by a unique process signature that is strongly dependent on porosity distribution This difference was outlined in the past [20] by comparing resultant cutting forces during micromilling of solid and foam Ti As shown in Fig 1, the resultant cutting force amplitude FR is increased (0.52 N vs 1.05 N) and peak-to-valley size of the cutting force is reduced (0.4 N vs 0.1 N) when micromilling porous compared to solid titanium The cutting forces were in porous material geometry modification by allowing a superior control of surface roughness and even porosity closure amount These two surface characteristics influence to a large extent the friction coefficient between implant and surrounding bone tissue However, two main interrelated technical challenges are associated with obtaining the desired surface geometry and roughness on outer porous surfaces: accurate characterization of real 2D/3D porosity values and optimization of micromachining process parameters with respect to the physical porosity amounts From this perspective, the present study focuses on optical image analysis of porosity with respect to material removal process through micromilling operations Physical–mechanical properties of porous metals and their adherence with bone cells significantly depend on 2D/ 3D porosity characterized through parameters like: quantity of pores (i.e., the fractional porosity), interconnectivity, size, morphology, permeability, and spatial distribution [8– 11] Classical non-destructive optical image analysis methodologies have been successfully applied in the past to analysis of cell morphology and microstructure of porous metals [8] In most cases, pore size distribution and shape analysis was performed by means of commercial image analysis programs The main drawback of this approach resides in destructive techniques involved in preparation of sample surface because image analysis is significantly dependent on appropriate differentiation between solid material and internal cavities Optical image analysis is limited to 2D spatial analysis only Nevertheless, optical micrographs were used before in the context of porous Ti materials used in load-bearing implants [12] Within machining environment, the most common application of vision-based methods is represented by tool wear analysis and monitoring [13–17] New developments in X-ray-based microcomputer tomography (micro-CT) have opened advanced non-destructive Fig Comparison of resultant cutting forces measured during micromilling of solid and porous Ti [20] 1.2 solid Ti 1.0 FR [N] 0.8 porous Ti 0.6 0.4 0.2 0.0 -0 0.0 0.5 1.0 1.5 2.0 l [mm] 2.5 3.0 3.5 4.0 Int J Adv Manuf Technol (2012) 60:841–851 recorded in space domain as a function of cutting tool center position l The present research was primarily focused on the investigation of the correlations between optically determined porosity and cutting forces recorded during micromilling of porous titanium foams For this purpose, an original imagebased methodology was developed to estimate the percentage of solid material removed during each revolution of the cutter along a linear tool path Once the porosity profile along the tool path became available, standard statistical measures were used to determine the amount of correlation between porosity and cutting force measured dynamically along the intended tool path All details of this analysis are presented in the following sections Optical assessment of porosity in context of micromilling operations Functional parts and components made from Ti–6Al–4V foams are characterized by a complex 3D porosity with significant variations in pore size and morphology as shown in Fig For these materials, micromilling with cutting tools having a diameter smaller than 0.4 mm can be used as a secondary shaping procedure that is capable of achieving high surface precision and complex geometries Since these procedures are accounted for as finish operations, the axial cutting depth is typically small, rarely exceeding 20 μm This allows treatment of the porosity as a function dependent on the presence of solid material along the microtool path trajectory This study considers only X–Y (planar/linear) motions, however, the developed method can be extended also to X–Y–Z (tridimensional) microtool path trajectories During microslot end-micromilling, each cutting tooth moves along the trochoidal trajectory formed as a superposiFig Typical sample of a porous Ti foam (Ti–6Al–4V) characterized by a complex 3D porosity structure 843 tion of tooth rotational motion around the cutter axis and linear translation of tool center along the intended tool path, as shown in Fig It is important to emphasize that a clear distinction has to be made between the volumetric porosity defined as the fraction of the voids spread throughout the volume of the porous material (a common physical property of porous materials) and cutting-force-related porosity defined as the fraction of the voids encountered by the cutting edge of the tool along its trochoidal trajectory Obviously, when analyzing the correlations between porosity and cutting force, only the distribution of the latter is relevant As a result, the remainder of this section focuses on the assessment of cutting-force-related porosity, essentially derived in a particular manner from generic volumetric porosity In this regard, the area swept per ith tool revolution Ai, represents the geometric difference between two consecutive, ith and (i−1)-th, cutting tooth trajectories This area can be approximated as the difference of two consecutive circles separated by a feed per tooth value, f (Fig 3) When machining solid material, the area Si and the volume of material removed per tooth revolution Vi, will be always constant: Si, Vi =const Moreover, the swept area Ai, and area of material removed, Si, per tooth revolution are absolutely identical By contrast, the presence of randomly distributed porosity makes the estimation of the amount of solid material removed more difficult In this situation, the area and the corresponding volume of material removed per tooth revolution exhibit large variations, Si, Vi =var, although the swept area Ai, remains unchanged (Fig 4) Proportion of the material within the swept area per tooth revolution pi can vary anywhere between and according to the following relationship: pi ¼ Si Ai ð1Þ pores solid Ti 0.2 mm 1x1x1mm 844 Fig Area removed during one tooth revolution Int J Adv Manuf Technol (2012) 60:841–851 trochoidal tooth cutting trajectory Y f i-1 i X v machined slot cutting tool workpiece In Eq 1, pi =0 corresponds to the case when swept area Ai is fully covered by a pore (void), and pi =1 corresponds to the case when only solid material is removed Because cutting force is directly proportional to the volume (area) of material removed, it is important to recognize that cutting forces will vary with respect to porosity, specifically the fraction of solid material contained within the area removed during one complete revolution of the cutter This fraction will be quantified herein based on optical considerations Area swept per tooth revolution Ai The absolute novelty of the methodology proposed for estimation of the porosity fraction per tool revolution resides in correlation of the optical images of the micromilled surface with micromachining parameters recorded in real time during cutting (cutting forces and axial positions) According to the proposed technique, the swept area is determined by overlapping cutting tooth path trajectories with optical images acquired for the top surface of the sample, followed by selection of the “on” pixels located between two consecutive tooth revolutions The accuracy of Fig Visual appearance of porosity enclosed within the area swept in one cutter revolution f i i-2 porosity tool diameter i-1 v Si-1, Vi-1 Si, Vi Si-1 ≠ Si Vi-1 ≠ Vi Int J Adv Manuf Technol (2012) 60:841–851 845 the method is primarily influenced by optical image resolution Δh, and spatial sampling period of linear tool motions Δl Figure depicts a graphical representation of the area covered in one tooth revolution Each tooth path is offset with a distance equal to feed per tooth f After swept area and corresponding pixels for each tooth revolution are determined, the porosity and the material presence contained within swept areas of single tool revolutions can be calculated along the machining direction (tool path trajectory) If f is comparable to optical image resolution Δh, calculation of the percentage of solid material contained within the area swept in one cutter revolution (2D case) can be reduced to an 1D problem by estimating the proportion of solid material positioned along the circumference of circular tooth trajectories This can be achieved by counting the “on” pixels that are intersected by these circular paths By doing this, the original “static” image of porosity will be converted into a “dynamic” image of porosity as observed by the cutting edge of the tool The key element of this image conversion resides in tooth path “linearization”, which essentially means that the semicircular tooth path is unwrapped along a straight line This unwrapping is accompanied by an adequate mapping of pixels encountered along the semicircular tooth trajectory, as shown in Fig To preserve the aspect ratio of “dynamic” image pixels, the semicircular tooth trajectories have to be sampled at intervals equal to pixel size Δh In order to perform the aforementioned conversion, the image has to be initially pre-processed to a binary format such that it contains only white (“on”) and black (“off”) pixels that correspond to solid material and pores, respectively As a result, the surface is converted to an array [Xi, Yi, Zi] defined only through zeros (Zi =0 for solid material) and ones (Zi =1 for voids) After that, tooth trajectories are calculated based on the position in real time of the bottom tool center and tool size The newly obtained tooth trajectories are then overlapped with the optical image, such that each tooth trajectory becomes a vector with {xi, yi, zi} components The zi values are initially unknown as they correspond to material presence for a particular (xi, yi) location It is also necessary to note that the sampling period of the tooth trajectory,$l ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðxj À xjÀ1 Þ2 þ ðyj À yjÀ1 Þ2 , is set to be equal to the pixel resolution of acquired optical image Δh In the next iteration step, all zi within the swept area are determined as a subset of the surface matrix [Xi, Yi, Zi] as following: zi ¼fZi jfxi ; yi g \ ½Xi ; Yi Šg Fig Estimation of material presence within the area swept in one cutter revolution through tooth path linearization f porosity material “on” machining direction Ai-2 Ai-1 Ai “off” ð2Þ 846 Int J Adv Manuf Technol (2012) 60:841–851 Based on these definitions, the ratio between the number ni of “on” pixels (zi =0) and total number of pixels intersected by particular tooth trajectories Ni will be fractionally identical to the amount of solid material distributed within the area swept in one cutter rotation: pi ¼ ni Ni ð3Þ An example of the pixel quantification procedure described above is provided in Fig As it can be noticed, the circular tooth path has been sampled at distances equal to pixel size (Δh) Then, the “on” or “off” attribute of each pixel corresponding to the sampled locations—identifiable through an adjacent star—has been recorded and eventually transcribed into a linear column array as part of the “linearization” procedure involved in cutting-force-related porosity signature analysis Experimental verification Applicability of the developed approach for vision-based estimation of porosity was verified experimentally in the context of porous Ti foam micromilling Real-time measurements of cutting forces and motions were performed during cutting trials Then, the actual porosity per single tooth revolution was estimated and subsequently correlated with per revolution-averaged cutting force amplitudes 3.1 Material description The Ti–6Al–4V foam used in micromilling trials was manufactured by Industrial Material Institute of the National Research Council of Canada (Boucherville, Quebec) by h= l sampled tooth path location porosity i solid material means of a patented powder metallurgy procedure [21] Structural and mechanical properties of the Ti foam are presented in Table The samples had a cubic shape with an approximate side length of 13 mm (Fig 2) 3.2 Experimental setup and methodology for micromilling Micromilling experiments were carried out on a custom-built five-axis CNC micromilling system equipped with an airbearing spindle capable of rotational speeds between 5,000 and 100,000 rpm range The system is capable of providing a static positional accuracy of μm over a 300-mm maximum travel range Cutting experiments were performed as micromilling of linear slots along X-axes using a two-flute, uncoated, tungsten carbide flat-end micromill having a diameter of 800 μm diameter and helix and clearance angles of 25° and 6°, respectively Linear slots having a width of 800 μm, depth of 40 μm, and length of 12 mm were machined on the top face of the foam Ti specimen with a feed rate of 120 mm/min and spindle rotational speed of 10,020 rpm An in-house developed real-time data acquisition system written in Labview was used to measure the X, Y, and Z components of the cutting force with a Kistler dynamometer (type 9256C2) and a Kistler dual mode charge amplifier (type 5010B) The X, Y, and Z cutting motions were tracked based on a signal from position encoders and three-phase current consumed by spindle drive (Fig 7) Spindle current and cutting forces F(t)={FX(t), FY(t), FZ(t)}, were recorded with a sampling frequency of 25 kHz with a minimum threshold of 0.002 N and were filtered out of the resonance frequency of the dynamometer (around 4.8 kHz) Simultaneously, the cutting position l(t)= {lX(t), lY(t), lZ(t)}, along the tool path trajectory was recorded with a sampling frequency of 500 Hz Further, a mean value of the cutting force in each direction was calculated for each tooth revolution using spindle current signal as point of synchronization in space domain After that, cutting forces were rearranged as a spatial function of the cutting tool path trajectory F(l)={FX(l), FY(l), FZ(l)}, using synchronization in time–space domain between cutting forces and their corresponding axial positions Table Structural and mechanical properties of Ti foams [3] h analyzed pixels Fig Analyzing pixels and their associated material/void attributes along circular tooth paths Property Value Density Porosity Pore size Surface area Compressive yield strength Compressive elastic modulus 1.6–2.25 g/cm3 50–65% 50–400 μm 0.05 m2/g 25–125 MPa 5–20 GPa Int J Adv Manuf Technol (2012) 60:841–851 847 I(t) Tool outline Tool path t FR(t) Microslot profile FRi = FR(ti) l li Tooth engagement area ti t FR(l) FRi l(t) li = l(ti) li l space domain ti-1 ti ti+1 t time domain Fig Achieving space domain synchronization of porosity and cutting force through concurrent time domain representations 3.3 Experimental setup and methodology for image analysis of porosity Optical image acquisition of micrographs was performed after micromachining by using a CANON EPS 450D camera with a 12.2-megapixel image sensor equipped with an EF 100 mm macro lens (f/2.8 Macro USM) that provides a spatial image resolution Δh of approximately μm/pixel A custom indirect LED light was used to reduce light reflections and enhance picture contrast for better porosity assessments In this study, the length of the machined slot was 12.56 mm and the feed per tooth f was 0.012 mm Due to the comparable values of Δh and f, estimations of the solid material fraction located along the circular tooth trajectory can be performed accurately through the proposed 1D approach as described in Section 2, above It will be assumed here that the porosity has a minimal cross sectional variation over the investigated axial depth of cut (20–40 μm) that is characteristic to finish operations As a result, the porosity amount measured on top of the sample surface before and after the microslot cutting operation will be assumed to be approximately identical The general methodology for vision-based estimation of porosity involves the following steps: a Acquire and convert the optical image of the micromilled surface to the binary format b Synchronize tooth position with respect to space domain and superimpose its trajectory on the optical image of the surface c Perform circular to linear tooth path conversions (e.g., “linearizations”) d Calculate the total number of optical pixels intersected by a singular tooth path Ni e Calculate the total number of “on” pixels ni (zi =0) along the investigated tooth path f Establish the fraction of solid material removed pi during each revolution of the tool (Eq 3) g Track the amount of solid material removed with respect of axial position of the tool p(l) In step (a), gray-scale micrographs of the sample surface captured by a high-resolution camera are converted to binary format by using imaging threshold set by Otsu’s method [22] to clearly delimit the porosity (black) and solid material (white) This value of threshold ensures accurate conversions of voids into black, especially when considering pore boundaries (Fig 8) Any dark spot covering less than 10 pixels was considered noise (light reflected on nonplanar foam inclusions) and therefore was filtered out The accuracy of thresholding procedure used was validated by comparing void dimensions against those obtained with a Wyko interferometric microscope Figure illustrates the steps (b) to (f) involved in visionbased estimation of the porosity First, circular trajectories of the cutting tooth rotation along tool path motion are superimposed on the black and white micrographs according to step (b) Then, in step (c), the circular tooth trajectory is “linearized” After performing the quantitative assessments involved in steps (d) through (f), the outcome of step (g) is a 848 Int J Adv Manuf Technol (2012) 60:841–851 Fig Converting gray porosity micrographs (top) into binary format (bottom) through thresholding solid material porosity Fig Optical image processing and porosity assessment Circular tooth trajectory Binary micrograph of micromilled porous surface 2r A machining direction A’ Tooth path conversion A Porosity cross-section πr “Linearization” of tooth trajectory Instantaneous amount of solid material removed along tool path Percentage of material p(l), % A’ Calculation of Ni, ni, pi, 100 p(l) 50 p(A) A Position of cutting tool center l, mm Int J Adv Manuf Technol (2012) 60:841–851 100 849 2.5 p(l ) 90 80 2.0 60 1.5 50 40 FR [N] p(l) [%] 70 1.0 FR(l) 30 20 0.5 Stable cutting 10 -1 l [mm] 0.0 10 11 12 13 Fig 10 Comparison between optically determined porosity and cutting force in space domain function p(l) expressing the fraction of solid material removed for any position of the tool along the intended tool path As noticeable here, the cutting-force-related porosity (Fig 9b) whose spatial variation along the tool path is plotted in Fig 9c represents nothing else but a transformed (mapped) form of the real surface porosity (Fig 9a) as observed at the top of the surface to be micromilled Evidently, the transformation function between the two types of porosities depends primarily on the size and cutting regime of the micromilling operation and it relies on the “linearization” process described above The real surface porosity constitutes in fact a discrete value of the volumetric porosity as measured at a certain depth within the sample Conversely, the volumetric porosity averages all real surface porosities measured throughout the height of the sample However, other than on the originally exposed exterior surfaces of the raw sample, noninvasive accurate determinations of the real surface porosity are relatively difficult at the current development level of technology 3.4 Preliminary experimental analysis of correlation between porosity and cutting forces To analyze the interplay between porosity and cutting forces from a statistical standpoint, they should be both dependent on the same variable For this application, the most rational choice for the common variable would be the space domain, specifically the axial position of the micromill along its trajectory l(t)={lX(t), lY(t), lZ(t)} As a result, both porosity and cutting forces were expressed as a function of l, respectively Figure 10 depicts a space domain comparison between the amount of solid material removed and the corresponding signature of the resultant cutting force A simple visual inspection of the graph reveals that cutting force variations follow closely the percentage of solid material removed in each tool revolution Coherence value, dimensionless Fig 11 Coherence function between proportion of material and resultant cutting force While correlations between cutting forces and real top surface porosity can be established in an absolutely similar manner as those between cutting forces and linearized/ transformed porosity, they will be always lower and thereby less convincing and/or useful Cutting-force-related porosity reflects in a more accurate manner material discontinuities as perceived by the cutting edge of the tool However, as Fig itself shows, when material voids are large enough, then significant drops will occur on both transformed and actual porosities plots, and they will propagate further on cutting force signature To summarize these observations, while linearized porosity represents a more accurate instrument for prediction of the cutting force variation, the real top surface porosity enables quick visual estimations of the correlations between material discontinuities and cutting forces pattern These visual correlations tend to be more obvious in case of larger material discontinuities characterized by more pronounced effects on cutting force signature 0.8 0.6 0.4 0.2 1.6 mm-1 0 Spatial frequency, mm -1 10 1230 Int J Adv Manuf Technol (2012) 60:1223–1235 Fcc is called swarm attractant cost θg is the position of the global optimum bacterium and m represents the mth parameter of bacterium location dattract is the depth of the attractant released by the cell, ωattract measures the width of the attractant signal, hrepellent is the depth of the repellent effect, and ωrepellant is the measure of width of the repellent signal Since it is not possible for two bacterium to have same location, it is assumed that hrepellent =dattract (c) Reproduction: The original set of bacteria, after getting evolved through several chemotactic stages, reaches the reproduction stage Here, the best set of bacteria (chosen out of all the chemotactic stages) gets divided into two groups The healthier half replaces the other half of bacteria, which gets eliminated, owing to their poorer foraging abilities This makes the population of bacteria constant during the evolution process Mathematically, for reproduction, the population is sorted in terms of accumulated cost (Fsw) which is sum of cost function value (F) added with the swarm attractant cost (Fcc) À Á Fsw ði; j; k; l Þ ¼ F ði; j; k; l Þ þ Fcc qg ðj; k; l Þ; q ðj; k; l Þ ð14Þ Half of the bacteria population having better cost will survive and remaining half are replaced by randomly generated new population (d) Elimination and dispersal: In the evolution process, a sudden unforeseen event can occur, which may drastically alter the smooth process of evolution and cause the elimination of the set of bacteria and/or disperse them to a new environment From a broad perspective, elimination and dispersal are parts of the population level long distance motile behavior In its application to optimization, it helps in reducing the behavior of stagnation, i.e., being trapped in a premature solution point or local optima Suppose that θ is the position of a bacterium and F (θ) represent the objective function with the basic goal to find È É the minimum Let Pðj; k; l Þ ¼ qi ðj; k; l Þji ¼ 1; ::; S represent the position of each bacterium in the population of s bacteria at the jth chemotactic step, kth reproduction step, and lth elimination–dispersal event Let F (i, j, k, l) denote the cost at the location of the ith bacterium θi (j, k, l) The steps of the algorithm are presented below: Step 1: Initialization: Number of parameters (p) to be optimized Number of bacteria (s) to be used for searching the total region Maximum swimming length (SLmax) after which tumbling of bacteria will be undertaken in a chemotaxis step Number of iteration (Nc) to be undertaken in a chemotaxis loop Maximum number of reproduction (NR) cycles Maximum number of elimination–dispersal (Ne) events imposed on bacteria Probability (Ped) with which elimination–dispersal will continue Location P(p, s, 1) of initial set of bacteria Random swim direction (Ø(j)) and step length (C(i)) 10 Swarming coefficients (dattract, wattract, hrepelent, and wrepelent) Step 2: Elimination–dispersal loop With some probability (Ped), the existing set of bacteria gets eliminated and dispersed in a new random direction Increment l=l+1 Continue step if l[...]... 0.400 0.247 0.372 7 8 9 22.5 25 5 74.676 106.68 42.672 24.20 24.20 73 .9 0 .99 90 0 .99 90 0 .93 09 0.283 0.258 0.242 0. 296 0.273 0.225 0. 297 0.260 0.213 0.301 0.258 0.227 10 11 12 7.5 10 15 74.676 106.68 42.672 73 .9 73 .9 73 .9 0 .93 09 0 .93 09 0 .93 09 0.206 0.220 0.312 0.220 0.2 19 0.284 0.210 0.210 0.302 0.2 19 0.211 0.304 13 14 15 17.5 20 25 74.676 106.68 42.672 73 .9 73 .9 73 .9 0 .93 09 0 .93 09 0 .93 09 0.244 0.222 0.411... 0.202 0. 394 16 17 18 19 20 21 22 2.5 5 10 12.5 15 20 22.5 74.676 106.68 42.672 74.676 106.68 42.672 74.676 1 39. 44 1 39. 44 1 39. 44 1 39. 44 1 39. 44 1 39. 44 1 39. 44 0 .90 28 0 .90 28 0 .90 28 0 .90 28 0 .90 28 0 .90 28 0 .90 28 0.208 0.200 0.2 79 0.218 0.204 0.347 0.255 0.202 0.204 0.242 0.2 29 0.223 0. 290 0.256 0.200 0. 195 0.264 0.223 0.206 0.322 0.253 0.205 0. 199 0.252 0.221 0. 196 0.343 0.241 23 24 25 26 27 28 29 30 RMS... Heat of vaporization, Qv (J/m3) Absorptivity (a) Deptha experimental, D (mm) Deptha model (mm) Deptha MGGP (mm) Deptha ANN (mm) 1 2 3 2.5 5 10 74.676 106.68 42.672 24.20 24.20 24.20 0 .99 90 0 .99 90 0 .99 90 0.208 0.210 0.325 0.206 0.211 0.273 0.207 0.208 0. 297 0.215 0. 195 0.306 4 12.5 74.676 24.20 0 .99 90 0.226 0.251 0.247 0.228 5 6 15 20 106.68 42.672 24.20 24.20 0 .99 90 0 .99 90 0.232 0.404 0.242 0.352 0.2 29. .. subsequent runs The analysis of influence of the individual terminal gene on accuracy of the prediction of the depth of cut gave interesting results The laser power and the cutting speed variables always remained in the model This implies that the laser power and, particularly, the cutting speed are the most influential parameters on which the depth of cut depends to the greatest extent The laser beam... model The probabilistic value, Prob>F, and the lack of fit of each model were calculated and the linear model was selected for having the least probabilistic value and the most insignificant lack of fit The analysis of variance was then performed to test the significance of the selected regression model and its coefficients (Table 4) As before, the maximum probabilis- tic values of 5% were set for the. .. now 0 .98 This very close to unity R2 indicates that the model closely approximates the tool life data The model’s adequate precision ratio, which compares the range of the predicted values at the design points to the average prediction error, is well beyond the minimum adequacy limit of 4 The obtained final equation of the model could then be presented in terms of coded factors as: ln T ¼ 1:63 À 0 :92 x1... Performance of ANN model on training set 0.8 0 .9 Int J Adv Manuf Technol (2012) 60:865–882 877 F5 ¼ 3:667203  P þ 7:636680  V À 4:7 498 05 F9 ¼ 1:236501  P þ 6:577310  V À 9: 087621  Qv þ 1:7603 29  aþ1:286081  Qv þ 2 :91 1266  aþ7:656582 F6 ¼ 1:2 097 49  P À 1: 899 521  V À 2:8 393 01 F10 ¼ À8:476 491  P À 1:420375  V À 2:006352  Qv þ 3:715367  aþ6: 693 3262  Qv þ 1:335370  aÀ8:775026 F7 ¼ 9: 6 698 08 Â... to meet the costumer’s demand of geometrical accuracy of the machined component On the other hand, machine shops will be more efficient when the cutting tool lasts longer Therefore, a compromised solution seems appropriate to select the cutting parameters Being able to quantify the effect of each input factor to the machining responses, the models offer an optimizing option to select the range of cutting... performance The study carried on CO2 laser cutting of mild steel focused on the effects of high-pressure assistant gas flow on cutting quality [8] The study on the laser cutting of ceramic plates emphasizes the importance of cutting power, the feed-rate of the specimens and the material properties, namely specific heat, conductivity and thermal expansion coefficient in controlling the formation of cracks... diameter So, on the small area, the depth of cut is obtain from Fig 5 Point cloud transferred to Microsoft Excel software ti ¼ D V ð4Þ where Imax is the peak intensity at the centre and D is the laser beam diameter Clearly, in the case of a spatial distribution of power intensity of axially symmetric (Gaussian) type, the relationship between output power of P which is focused on the work piece and the peak