- Email: [email protected]
Scanning schemes in white light photoelasticity – Part II: Novel fringe resolution guided scanning scheme
Optics and Lasers in Engineering ∎ (∎∎∎∎) ∎∎∎–∎∎∎
Contents lists available at ScienceDirect
Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Scanning schemes in white light photoelasticity – Part II: Novel fringe resolution guided scanning scheme Vivek Ramakrishnan, K. Ramesh n Dept. of Applied Mechanics, Indian Institute of Technology Madras, India
art ic l e i nf o
a b s t r a c t
Article history: Received 15 December 2015 Received in revised form 16 April 2016 Accepted 17 May 2016
Varied spatial resolution of isochromatic fringes over the domain influences the accuracy of fringe order estimation using TFP/RGB photoelasticity. This has been brought out in the first part of the work. The existing scanning schemes do not take this into account, which leads to the propagation of noise from the low spatial resolution zones. In this paper, a method is proposed for creating a whole field map which represents the spatial resolution of the isochromatic fringe pattern. A novel scanning scheme is then proposed whose progression is guided by the spatial resolution of the fringes in the isochromatic image. The efficacy of the scanning scheme is demonstrated using three problems – an inclined crack under biaxial loading, a thick ring subjected to internal pressure and a stress frozen specimen of an aerospace component. The proposed scheme has use in a range of applications. The scanning scheme is effective even if the model has random zones of noise which is demonstrated using a plate subjected to concentrated load. This aspect is well utilised to extract fringe data from thin slices cut from a stereolithographic model that has characteristic random noise due to layered manufacturing. & 2016 Elsevier Ltd. All rights reserved.
Keywords: Digital photoelasticity Image processing RGB photoelasticity Twelve fringe photoelasticity Stereo-lithography Isochromatics
1. Introduction Recent advances in TFP/RGB Photoelasticity have made it possible to obtain isochromatic data from a single colour image of the photoelastic fringe patterns with a high level of accuracy [1,2]. Colour image based fringe order estimation has been applied to a variety of problems ranging from stress analysis in glass [3,4] to analysis of dry masonry walls [5]. Since photoelastic techniques are increasingly being used for inter-disciplinary research [6,7], it is essential that even personnel with limited background in photoelastic analysis are able to perform the post-processing of isochromatic images with ease. Use of the technique to solve innovative problems requires it to be robust to accommodate complex model geometries and handle fringe patterns of varied shapes and gradients. The existing scanning schemes reported in the literature [8–11] for refining the fringe order data have been critically evaluated in the first part of this work [12]. The accuracy of isochromatic results obtained depends on the spatial resolution of the fringe pattern and the location of the seed point. Key factor is the spatial resolution as insufficient resolution leads to the origin and propagation of noise thus, making the seed point location important. The existing scanning schemes either do not take this into n
Corresponding author. E-mail address: [email protected] (K. Ramesh).
consideration or involve complex interactive methods to handle this issue. Among the various scanning schemes, the advancing front scanning scheme in conjunction with multiple seed points can minimise this noise propagation. However, it requires careful selection of multiple seed points by analysing the fringe order results as well as progression of the results interactively. In this paper, a novel scanning scheme which accounts for the spatial resolution of the fringe pattern in the isochromatic image is proposed. Initially, a map resembling the gradient of the fringe pattern is generated for the given problem and this map is then used to guide the scanning scheme for refining the fringe order data. The proposed scanning scheme has use in a range of applications and these are demonstrated using problems having appropriate example problems.
2. Fringe resolution guided scanning scheme 2.1. Creation of resolution map The first step is to create a whole field map of the spatial resolution or the gradient of the isochromatic fringe patterns. Hitherto, there is no standard method to quantify the whole field spatial resolution of photoelastic fringe patterns. In this paper, a novel method based on intensity gradients of the photoelastic fringe pattern is used to create a map that represents the fringe gradients in the model. Initially, the method is demonstrated using
http://dx.doi.org/10.1016/j.optlaseng.2016.05.010 0143-8166/& 2016 Elsevier Ltd. All rights reserved.
Please cite this article as: Ramakrishnan V, Ramesh K. Scanning schemes in white light photoelasticity – Part II: Novel fringe resolution guided scanning scheme. Opt Laser Eng (2016), http://dx.doi.org/10.1016/j.optlaseng.2016.05.010i
V. Ramakrishnan, K. Ramesh / Optics and Lasers in Engineering ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
Fig. 1. (a) Theoretically simulated fringe pattern in a circular disc under diametral compression (b) The intensity gradient map G (c) Magnified view of a portion in Fig. 1(b) (d) Whole field map resembling the fringe gradient (FR_map) (e) Magnified view of a portion in Fig. 1(d). (For interpretation of the references to colour in this figure, the reader is referred to the web version of this article.)
analytically obtained fringe pattern in a circular disc under compression. Later this method is applied to experimental isochromatic fringe patterns recorded using white light. 2.1.1. Circular disc under compression Theoretically simulated photoelastic fringe pattern in a circular disc under diametral compression (Load¼492 N, Dia¼ 60 mm) is used to illustrate the proposed method. Fig. 1(a) shows the dark field isochromatics in the disc and it can be observed that the fringe gradient is very high near the loading points. The objective here is to create a whole field map whose values are representative of the fringe gradient in the model. Initially, the intensity gradients are computed in the horizontal as well as the vertical directions. The directional gradients are calculated by the central difference formula given by,
Gx =
I (i, j + 1) − I (i, j − 1) I (i + 1, j ) − I (i − 1, j ) ; Gy = 2 2
(1)
where, Gx and Gy are the intensity gradients in the horizontal and the vertical directions respectively and (i, j) represent the pixel coordinates of the point under consideration. Next, the absolute gradient is calculated for each pixel from the directional gradients as,
G=
Gx2 + Gy2
(2)
Fig. 1(b) shows the absolute gradient G for the circular disc. It has high values (shown as red) in most of the zones near the loading points, which are also zones of stress concentration where fringe gradient is high. However, it can be seen from the magnified image (Fig. 1(c)) that there are still pixels in which the values are low (appear as blue). These pixels correspond to points near the centre of the fringes where the intensity variation is low. Ideally, all the pixels in the high fringe gradient zones should have comparatively higher values, which is not the case now. Hence, the quantity G is not adequate to capture the fringe gradient. The quantity that captures the fringe gradient is obtained by taking the sum of the absolute intensity gradient over a k k pixel window centred around the pixel under consideration. This is given by, m= i + p n= j + p
FR(i, j ) =
∑ ∑ m= i − p n= j − p
G(m, n) where p =
1 (k − 1) 2
(3)
where (i, j) represents the global coordinates of the pixel under consideration and (m, n) represents their local coordinates in the k k neighbourhood. A kernel size of 11 is found to be sufficient for the problems studied. The quantity FR is calculated for every pixel in the model domain which is plotted as an image and the resulting image is labelled as FR_map. Fig. 1(d) shows the FR_map obtained for the circular disc and the magnified view of the map near the loading point is shown in Fig. 1(e). It can be seen that the values in the map increase towards the high fringe gradient zones near the loading point where the spatial resolution of fringes is low. A low value of FR indicates higher spatial resolution of the fringes and vice-versa. 2.1.2. Experimental isochromatic colour images Fig. 2(a) shows the dark field isochromatics in a bi-axially loaded cruciform specimen having an inclined crack (bi-axial load ratio (Fx/Fy)¼ 0.5, thickness¼6 mm) recorded using white light [13]. The gradient of the fringe pattern increases towards the crack-tip and the spatial resolution of the fringe pattern is very less in this zone (o10 pixels per fringe order). The method described in the previous section is extended for the isochromatics obtained under white light (as in TFP/RGB photoelasticity). The absolute intensity gradients are computed using Eq. (2) for each colour planes (red, green and blue) separately. Fig. 2(b–d) show the contours of G obtained using the red, green and blue planes respectively. Next, the maps obtained using the individual colour components are added to obtain the total intensity gradient (Gt) map (Fig. 2(e)). The FR_map is then created from the total intensity gradient map by using Gt instead of G in Eq. (3), which is shown in Fig. 2(f). It can be seen that values in the map increase towards the crack-tip and hence, capturing the fringe gradient in the model. Further, it can also be noted that the map identifies even minute marks in the model domain. Though, this is not of any advantage for the current problem, it is helpful in analysing models having random noise which is illustrated in Section 5. Fig. 3(a) shows the dark field isochromatics in a thick ring subjected to internal pressure (load¼ 3.93 MPa, thickness¼6 mm). The fringe order in the model increases from 1.5 in the outer diameter to nearly 6 at the inner boundary. The fringe gradient increases towards the inner diameter of the ring whereas their spatial resolution decreases. The FR_map obtained for the problem using Eqs. (1–3) is shown in Fig. 3(b). It can be seen that the values of FR increases towards the inner diameter of the ring which is in accordance with the variation of the fringe gradient in the model.
Please cite this article as: Ramakrishnan V, Ramesh K. Scanning schemes in white light photoelasticity – Part II: Novel fringe resolution guided scanning scheme. Opt Laser Eng (2016), http://dx.doi.org/10.1016/j.optlaseng.2016.05.010i
V. Ramakrishnan, K. Ramesh / Optics and Lasers in Engineering ∎ (∎∎∎∎) ∎∎∎–∎∎∎
3
Fig. 2. (a) Dark field isochromatics in the bi-axially loaded cruciform specimen having an inclined crack. Absolute gradient for different colour planes of the image (b) Red (c) Green (d) Blue. (e) Sum of the absolute gradients (Gt). (f) Contour plot of the FR map. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 3. (a) Dark field isochromatics in the thick ring subjected to internal pressure. (b) FR_map obtained using Eqs. (1–3). (For interpretation of the references to colour in this figure, the reader is referred to the web version of this article.)
The FR_map thus obtained is used for guiding the proposed scanning scheme. 2.2. Scanning scheme The new scanning scheme envisaged is designed to resolve the low fringe gradient zones having higher spatial resolution first. Fig. 4 shows the flowchart of the proposed Fringe Resolution Guided Scanning in TFP (FRSTFP). Initially, the fringe orders are estimated by the least squares method using the colour difference formula. The FR_map is obtained for the problem at hand using Eqs. (1–3) as discussed in the previous section by processing the colour image. To start the refining process, a point that is correctly resolved by the least squares method is selected. At each step, all the unresolved adjacent pixels corresponding to the resolved pixels are considered for deciding the progression of the scanning scheme. Initially, the eight neighbouring pixels corresponding to the seed point are stored in a list. Subsequently, this list of adjacent unresolved pixels is updated. The algorithm searches for the least value of FR in the list of adjacent unresolved pixels at every step.
The method used for refining may be Refined TFP (RTFP) [9] or window search method [10,11]. If the pixel under consideration has more than one neighbouring resolved pixel, an average value of the fringe order is taken as Np to maximise the use of neighbouring resolved pixel data. Hence, the proposed scanning scheme retains the advantages of advancing front scanning scheme while addressing the issue of varied spatial resolution of the isochromatic fringes. The generation of FR_map and the proposed scanning scheme are implemented using visual C þ þ. The values in the FR_map depend on the intensity of the experimentally recorded isochromatic image and generally range from 50–2000. The scanning scheme is independent of the absolute values in the FR_map as it works in a comparative manner by identifying the pixel having the least value of FR in the neighbourhood.
3. Fringe order refinement using FRSTFP In this section, the efficacy of the proposed fringe resolution guided scanning scheme is demonstrated by applying it two problems – bi-axially loaded cruciform specimen with an inclined
Please cite this article as: Ramakrishnan V, Ramesh K. Scanning schemes in white light photoelasticity – Part II: Novel fringe resolution guided scanning scheme. Opt Laser Eng (2016), http://dx.doi.org/10.1016/j.optlaseng.2016.05.010i
4
V. Ramakrishnan, K. Ramesh / Optics and Lasers in Engineering ∎ (∎∎∎∎) ∎∎∎–∎∎∎
Fig. 4. Flowchart of the fringe resolution guided scanning scheme in TFP.
crack (Fig. 2) and a thick ring subjected to internal pressure (Fig. 3). The limitations of the existing scanning schemes in refining these two problems are clearly demonstrated in the first part of this work [12]. Later, the proposed scanning scheme is applied on a stress frozen model of an aerospace component and is compared with the advancing front scanning scheme (AFSTFP). Initially, the fringe orders for the inclined crack and the thick ring problems are obtained by the least squares method. Refining is carried out by FRSTFP in conjunction with the modified window search method [9]. The FR_maps corresponding to the isochromatics fringe patterns are shown in Figs. 2(f) and 3(b) respectively. Fig. 5(a–e) show the progression of the scanning scheme for the inclined crack problem at an interval of 100,000 pixels. It can be seen that the area near the crack tip which has the least spatial resolution is refined only at the end of the scan. Hence, the noise in the results due to inadequate resolution is encapsulated within this region and its propagation is restricted. Fig. 5(f–j) show the progression of the scanning scheme for the thick ring problem at an interval of 40,000 refined pixels. Unlike the existing scanning schemes [12], this scanning scheme proceeds in a concentric fashion following the increase in the fringe gradient towards the inner diameter. Further, the proposed scanning scheme is able to scan the entire model domain with a single seed point (shown as yellow dot in Fig. 5) even for multiply connected models. The proposed scheme needs only minimal user interaction, which will enable the use of photoelastic analysis for a wide range of interdisciplinary problems [6,7]. Smoothing of the isochromatic data is usually necessary to remove minor jumps in fringe order values that may arise due to localised colour variations in the models. Fig. 6(a) and (b) show the smoothed fringe order results corresponding to the isochromatic data shown in Fig. 5(e) and (j) obtained for the two problems.
Multi-directional smoothing scheme involving local regression using weighted linear least squares proposed in Ref. [14] is used to smooth the isochromatic data. Fig. 6(c) and (d) show the fringe order variation along lines AB (Fig. 6(a)) and CD (Fig. 6(b)) respectively. The fringe order results are compared with the integer fringe order values obtained by skeletonisation of the fringe pattern in the green plane image. It can be seen that the results obtained are consistent with the integer fringe orders. The mean errors in the fringe order values at the fringe skeleton points are only 0.05 and 0.06 for the inclined crack problem and the thick ring problem respectively. 3.1. Immunity to seed point location To study the influence of seed point location on the fringe order results, the inclined crack problem is refined using seed points selected at different locations in the model domain other than the point shown in Fig. 5(e). The contour plots of the smoothed fringe order results and the corresponding seed points are shown in Fig. 7(a) and (b). A whole field map of the difference in the fringe order values showed that they are the same though the location of the seed points used are different. Similarly, seed points selected at different locations in the problem of the thick ring subjected to internal pressure (Fig. 7(c) and (d)) also yielded similar results. Hence, unlike the existing scanning schemes [12], the fringe order values obtained by FRSTFP is independent of the location of the seed point selected.
4. Comparison of FRSTFP and advancing front scanning In this section, the proposed FRSTFP is used to extract the
Please cite this article as: Ramakrishnan V, Ramesh K. Scanning schemes in white light photoelasticity – Part II: Novel fringe resolution guided scanning scheme. Opt Laser Eng (2016), http://dx.doi.org/10.1016/j.optlaseng.2016.05.010i
V. Ramakrishnan, K. Ramesh / Optics and Lasers in Engineering ∎ (∎∎∎∎) ∎∎∎–∎∎∎
5
Fig. 5. Progression of the fringe resolution guided scanning scheme applied to (a)–(e) Bi-axial fracture problem (interval ¼ 100,000 pixels) (f)–(j) Thick vessel subjected to internal pressure (interval ¼40,000 pixels). The seed points are shown as yellow dots. Progression images are shrunk for optimal spacing.
Fig. 6. Contour plot of the smoothed fringe order results obtained by refining using the fringe resolution guided scanning scheme for (a) Inclined crack in the bi-axially loaded cruciform specimen (b) Thick vessel subjected to internal pressure. (c), (d) Fringe order variation along lines AB and CD respectively. (For interpretation of the references to colour in this figure, the reader is referred to the web version of this article.)
isochromatic data in a stress-frozen slice of an industrial problem. Fig. 8(a) shows the dark field isochromatics of the problem. Initially, the fringe orders are estimated using the least squares method and is shown as greyscale in Fig. 8(b). The fringe orders thus obtained are then refined using only a single seed point (shown as yellow dot). The refined results obtained using FRSTFP are shown in Fig. 8(c). Fig. 8(d) shows the contour plot of the fringe order values after multi-directional smoothing. The smooth
variation of the colours indicates that the fringe orders are correctly resolved in the entire model domain. For comparison, the advancing front scanning scheme, which is the most sophisticated among the existing schemes [12] is also used. Fig. 8(e) shows the contour plot of the results obtained using the advancing front scanning scheme using the same seed point. It can be observed that noise has propagated to the zone opposite to the location of the seed point. Low resolution zones near the hole
Please cite this article as: Ramakrishnan V, Ramesh K. Scanning schemes in white light photoelasticity – Part II: Novel fringe resolution guided scanning scheme. Opt Laser Eng (2016), http://dx.doi.org/10.1016/j.optlaseng.2016.05.010i
6
V. Ramakrishnan, K. Ramesh / Optics and Lasers in Engineering ∎ (∎∎∎∎) ∎∎∎–∎∎∎
Fig. 7. (a), (b) Contour plot of the smoothed fringe order results obtained for the inclined crack problem using different seed points. (c), (d) Contour plot of the smoothed fringe order results obtained for the thick ring using different seed points. Seed points are shown as black dots. (For interpretation of the references to colour in this figure, the reader is referred to the web version of this article.)
Fig. 8. (a) Dark field isochromatics in the stress frozen slice of an aerospace component (b) Greyscale representation of the fringe order obtained by least squares method. Yellow dot shows the seed point used for refining. (c) Refined results obtained using the fringe resolution guided scanning scheme using a single seed point (d) Fringe order values after smoothing (e) Refined results obtained using advancing front scanning scheme using the same seed point. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
have triggered this noise propagation, which is totally absent in FRSTFP as it is designed to resolve these zones only at the end. In both the scanning schemes, the maximum fringe order that could be resolved in the model is 4 as the spatial resolution near the hole is low.
5. Problems with random zones of noise In many situations, one has to analyse models from industries having certain marks or even having foreign particles at random locations which may have crept in as part of their studies/specimen preparation. However, these marks/particles act as source of noise from digital photoelastic point of view. In such cases, the technique adopted should provide accurate results in all the other zones of the model by encapsulating noise within the zones containing the marks/particles. The flexibility in creating multiple masks proposed in Ref. [14] is useful if the noisy zones are few and are clearly identifiable. In the subsequent sections, the use of the proposed scanning scheme in handling such noisy zones is explored.
5.1. Plate under concentrated load with simulated noise To demonstrate the efficacy of the scanning scheme, initially a problem is considered in which the noisy zones are simulated artificially. The problem of a plate subjected to a concentrated load is considered for this. On the experimental isochromatics recorded, spots of noise are introduced by modifying the intensity values of these zones. Many problems in industry seek fringe order values in the rest of the model domain even in the presence of such noise. Fig. 9(a) and (b) show the original dark field isochromatics in the plate and the one with simulated noise randomly scattered across the model domain. The seed point selected for refining the fringe order data is shown as yellow dot. The FR_map for the problem (Fig. 9(c)) shows that the values are higher at the boundary of the noisy zones due to higher intensity gradient at the interface. Hence, the scanning scheme first scans all the pixels surrounding the dark spots before proceeding to the noisy pixels inside it. This is also clear from the progression of the scanning scheme shown in Fig. 9(d)–(g). This eliminates any chance of noise propagation from the dark spots. The contour plot of the fringe
Please cite this article as: Ramakrishnan V, Ramesh K. Scanning schemes in white light photoelasticity – Part II: Novel fringe resolution guided scanning scheme. Opt Laser Eng (2016), http://dx.doi.org/10.1016/j.optlaseng.2016.05.010i
V. Ramakrishnan, K. Ramesh / Optics and Lasers in Engineering ∎ (∎∎∎∎) ∎∎∎–∎∎∎
7
Fig. 9. Dark field isochromatics in a plate under concentrated load (a) Original image (b) Isochormatics with simulated noise. (c) FR_map of the isochromatics (d)– (g) Greyscale representation of the progression of the scanning scheme at an interval of 60,000 steps (h) Contour plot of the results obtained using fringe resolution guided scanning scheme (i) Whole field plot of the difference in the fringe order values obtained with and without the simulated noise. (For interpretation of the references to colour in this figure, the reader is referred to the web version of this article.)
order obtained after refining is shown in Fig. 9(h). The isochromatic data is also extracted from the original image without noise shown in Fig. 9(a) using the proposed scanning scheme. The whole field map of the differences in fringe orders are shown in Fig. 9(i). It is found that the proposed scanning scheme gives the same results as obtained from the original image throughout the model (blue denotes zero difference) leaving the noise points (shown as brown). This demonstrates the efficacy of the model in handling problems with random noise spread across the model domain. 5.2. Stereo-lithographic model The recent developments in stereo-lithography have enabled one to make photoelastic models of desired shapes through layer by layer manufacturing. Stress analysis in such 3-D models is performed by stress-freezing followed by slicing. However, these models have dark spots due to porosity randomly spread across the domain which are potential sources of noise for photoelastic analysis. It is shown that more noise shows up in the isochromatic image as the thickness of the slice decreases [15]. However, to capture the variation of stresses better, use of thin slices is recommended. A circular disc (diameter¼60 mm and thickness ¼6 mm) made using stereo-lithography is subjected to a diametral load of 64 N during stress freezing. Due to the layered manufacturing process, some levels of micro-porosity exist in each layer of the model. A slice of 2 mm thickness is then cut from the central portion of the disk carefully. Fig. 10(a) shows the dark field isochromatics in a quarter of the disc recorded using an incandescent lamp. It can be seen that there are intermittent spots spread across the model domain which makes the creation of appropriate boundary mask very difficult. The FR_map generated for the problem using Eqs. (1– 3) is shown in Fig. 10(b). Apart from the high fringe gradient zones near the loading point, the map is able to capture the small spots
owing to the sudden intensity gradients. The fringe order is initially estimated by the colour difference equation involving the combined use of RGB and HSV followed by refining using window search method. The progression of the scan after the first 210,000 pixels shows that the scheme refines the areas surrounding the noise spots first and then gradually proceeds into the pores (Fig. 10 (c)). The white spots in Fig. 10(c) correspond to the noisy spots in Fig. 10(a) and hence, removing the chance of noise propagation from these spots. The fringe order results obtained after refining using the proposed scanning scheme is shown in Fig. 10(d). Isochromatic demodulation is carried out accurately only till four fringe orders as the intensity modulation is weak beyond this owing to the use of incandescent lamp. This fringe order data is then smoothed by the multi-directional progressive smoothing scheme and the results obtained are shown in Fig. 10(e). The variation of fringe order values along line XY (Fig. 10(e)) is plotted in Fig. 10(f). It can be seen that the fringe order values obtained closely follow the corresponding theoretical values. Thus, the proposed scanning scheme is also capable of handling stereo-lithographic models with random noise of varied sizes. This would allow users to cut thinner slices from the stressfrozen 3-D photoelastic model to study their stress variation better.
6. Conclusions A novel scanning scheme for TFP (Twelve Fringe Photoelasticity) is proposed that takes into account the spatial resolution of the isochromatic fringe pattern. The proposed fringe resolution guided scanning scheme is capable of solving a variety of problems without the propagation of noise from the zones where spatial resolution of the fringes is low. Initially, a method based on the intensity gradients is proposed to generate a map that represents the fringe gradients in the model. The procedure for creating the map is discussed using
Please cite this article as: Ramakrishnan V, Ramesh K. Scanning schemes in white light photoelasticity – Part II: Novel fringe resolution guided scanning scheme. Opt Laser Eng (2016), http://dx.doi.org/10.1016/j.optlaseng.2016.05.010i
V. Ramakrishnan, K. Ramesh / Optics and Lasers in Engineering ∎ (∎∎∎∎) ∎∎∎–∎∎∎
8
Fig. 10. (a) Dark field isochromatics in the circular disc under compression made by stereo-lithography (b) FR map (c) Progression of the scanning scheme after 210,000 steps. Contour plot of the fringe order results obtained after (d) refining (e) refining and smoothing (f) Fringe order variation along line XY compared with the corresponding values from theory. (For interpretation of the references to colour in this figure, the reader is referred to the web version of this article.)
theoretically simulated fringe pattern in a circular disc and later extended to experimentally recorded isochromatics using white light. The proposed scanning scheme is successfully applied to two problems – bi-axially loaded cruciform specimen with an inclined crack and thick ring subjected to internal pressure. The existing scanning schemes have limitations in solving both these problems as they do not consider the spatial resolution of the fringes. The proposed scheme eliminates the propagation of noise and restricts them within the low resolution zones since the algorithm postpones the scanning of noisy zones towards the end. The performance of the scanning scheme is independent of location of the seed point selected and hence, the user interaction is minimal. The efficacy of the method is also demonstrated by applying it to an industrial stress frozen slice and the results are compared with the advancing front scanning scheme which is the most robust among the existing schemes. Further, it is found that the proposed scanning method is also capable of handling problems with random noise in the model. This is first demonstrated using a plate subjected to concentrated load with simulated noise. Finally, a problem of a disc under diametral compression made using stereo-lithographic model making approach having random noise due to layered manufacturing is also solved.
Acknowledgements Partial financial support from the Indian National Academy of Engineering (INAE) (Ref No. INAE/201/WLBS) through their short term research proposal is gratefully acknowledged. The authors acknowledge Dr. S.A. Annamala Pillai and Mr. Jeby Philip of VSSC
Trivandrum, India for providing the stress frozen slice of the aerospace component.
References [1] Ajovalasit A, Petrucci G, Scafidi M. Review of RGB photoelasticity. Opt Lasers Eng 2015;68:58–73. [2] Ramesh K, Kasimayan T, Neethi Simon B. Digital photoelasticity – a comprehensive review. J Strain Anal Eng Des 2011;46:245–66. [3] Scafidi M, Toscano A, Petrucci G, Alessi S, Ajovalasit A. Review of photoelastic image analysis applied to structural birefringent materials: glass and polymers. Opt Eng 2015;54(8):081206. [4] Ajovalasit A, Petrucci G, Scafidi M. RGB photoelasticity applied to the analysis of membrane residual stress in glass. Meas Sci Technol 2012;23:025601. [5] Iqbal Baig, Ramesh K, Hariprasad MP. Analysis of stress distribution in dry masonry walls using three fringe photoelasticity. In: Proceedings of the SPIE 9302, international conference on experimental mechanics, 93022P; 2014. http://dx.doi.org/10.1117/12.2081235. [6] Ramesh K, Hariprasad MP, Bhuvaneshwari S. Digital photoelasticity applied to implant dentistry. Opt Lasers Eng 2016. http://dx.doi.org/10.1016/j.optlaseng.2016.03.022 (article in press). [7] Antony SJ. Imaging shear stress distribution and evaluating the stress concentration factor of the human eye. Sci Rep 2015;5:8899. [8] Swain D, Philip J, Pillai SA. A modified regularized scheme for isochromatic demodulation in RGB photoelasticity. Opt Lasers Eng 2014;61:39–51. [9] Madhu KR, Ramesh K. Noise removal in three fringe photoelasticity by adaptive colour difference estimation. Opt Lasers Eng 2007;45:175–82. [10] Ajovalasit A, Petrucci G, Scafidi M. RGB photoelasticity: review and improvements. Strain 2010;46:137–47. [11] Kale S, Ramesh K. Advancing front scanning approach for three-fringe photoelasticity. Opt Lasers Eng 2013;51:592–9. [12] Vivek R, Ramesh K. Scanning schemes in white light photoelasticity, part i – critical assessment of the existing scanning schemes. Opt Lasers Eng 2015 (under review). [13] Shreesudha B. Study of digital photoelastic evaluation of stress field parameters for cracks – examples from bi-axially loaded cracked cruciform specimen (MS Thesis).Madras, India: Dept. of applied mechanics, Indian Institute
Please cite this article as: Ramakrishnan V, Ramesh K. Scanning schemes in white light photoelasticity – Part II: Novel fringe resolution guided scanning scheme. Opt Laser Eng (2016), http://dx.doi.org/10.1016/j.optlaseng.2016.05.010i
V. Ramakrishnan, K. Ramesh / Optics and Lasers in Engineering ∎ (∎∎∎∎) ∎∎∎–∎∎∎ of Technology (IIT); 2016. [14] Ramesh K, Vivek R, Ramya C. New initiatives in single colour image based fringe order estimation in digital photoelasticity. J Strain Anal 2015;50(7):488– 504. [15] Ashokan K, Prasath RGR, Ramesh K. Noise-free determination of isochromatic parameter of stereolithography-built models. Exp Tech 2012;36:70–5.
Vivek Ramakrishnan received his B.Tech degree in Mechanical Engineering from the University of Kerala, India in 2011. He is currently a Ph.D student at the Department of Applied Mechanics, Indian Institute of Technology Madras, India. His research interests include optical methods in mechanics, digital photoelasticity, residual stress analysis of glass and non-destructive evaluation.
9
K. Ramesh received his undergraduate degree from the Regional Engineering College, Trichy (now NIT, Trichy), Postgraduate degree from the Indian Institute of Science, Bangalore and the Doctoral Degree from the Indian Institute of Technology Madras. He is currently a Professor at the Department of Applied Mechanics, IIT Madras; as its Chairman during (2005–2009) and formerly a Professor at the Department of Mechanical Engineering, IIT Kanpur. In recognition of his significant contributions in photoelastic coatings, he was awarded the Zandman award by SEM in 2012. He has received several recognitions: Fellow of the Indian National Academy of Engineering (2006), Distinguished Alumnus Award of NIT, Trichy (2008), President of India Cash Prize (1984), Member of the Editorial Boards of the International Journals: Strain (since 2001), Journal of Strain Analysis for Engineering Design (2009–10), Optics and Lasers in Engineering, and Steering committee member of ASEM.
Please cite this article as: Ramakrishnan V, Ramesh K. Scanning schemes in white light photoelasticity – Part II: Novel fringe resolution guided scanning scheme. Opt Laser Eng (2016), http://dx.doi.org/10.1016/j.optlaseng.2016.05.010i
Contents lists available at ScienceDirect
Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Scanning schemes in white light photoelasticity – Part II: Novel fringe resolution guided scanning scheme Vivek Ramakrishnan, K. Ramesh n Dept. of Applied Mechanics, Indian Institute of Technology Madras, India
art ic l e i nf o
a b s t r a c t
Article history: Received 15 December 2015 Received in revised form 16 April 2016 Accepted 17 May 2016
Varied spatial resolution of isochromatic fringes over the domain influences the accuracy of fringe order estimation using TFP/RGB photoelasticity. This has been brought out in the first part of the work. The existing scanning schemes do not take this into account, which leads to the propagation of noise from the low spatial resolution zones. In this paper, a method is proposed for creating a whole field map which represents the spatial resolution of the isochromatic fringe pattern. A novel scanning scheme is then proposed whose progression is guided by the spatial resolution of the fringes in the isochromatic image. The efficacy of the scanning scheme is demonstrated using three problems – an inclined crack under biaxial loading, a thick ring subjected to internal pressure and a stress frozen specimen of an aerospace component. The proposed scheme has use in a range of applications. The scanning scheme is effective even if the model has random zones of noise which is demonstrated using a plate subjected to concentrated load. This aspect is well utilised to extract fringe data from thin slices cut from a stereolithographic model that has characteristic random noise due to layered manufacturing. & 2016 Elsevier Ltd. All rights reserved.
Keywords: Digital photoelasticity Image processing RGB photoelasticity Twelve fringe photoelasticity Stereo-lithography Isochromatics
1. Introduction Recent advances in TFP/RGB Photoelasticity have made it possible to obtain isochromatic data from a single colour image of the photoelastic fringe patterns with a high level of accuracy [1,2]. Colour image based fringe order estimation has been applied to a variety of problems ranging from stress analysis in glass [3,4] to analysis of dry masonry walls [5]. Since photoelastic techniques are increasingly being used for inter-disciplinary research [6,7], it is essential that even personnel with limited background in photoelastic analysis are able to perform the post-processing of isochromatic images with ease. Use of the technique to solve innovative problems requires it to be robust to accommodate complex model geometries and handle fringe patterns of varied shapes and gradients. The existing scanning schemes reported in the literature [8–11] for refining the fringe order data have been critically evaluated in the first part of this work [12]. The accuracy of isochromatic results obtained depends on the spatial resolution of the fringe pattern and the location of the seed point. Key factor is the spatial resolution as insufficient resolution leads to the origin and propagation of noise thus, making the seed point location important. The existing scanning schemes either do not take this into n
Corresponding author. E-mail address: [email protected] (K. Ramesh).
consideration or involve complex interactive methods to handle this issue. Among the various scanning schemes, the advancing front scanning scheme in conjunction with multiple seed points can minimise this noise propagation. However, it requires careful selection of multiple seed points by analysing the fringe order results as well as progression of the results interactively. In this paper, a novel scanning scheme which accounts for the spatial resolution of the fringe pattern in the isochromatic image is proposed. Initially, a map resembling the gradient of the fringe pattern is generated for the given problem and this map is then used to guide the scanning scheme for refining the fringe order data. The proposed scanning scheme has use in a range of applications and these are demonstrated using problems having appropriate example problems.
2. Fringe resolution guided scanning scheme 2.1. Creation of resolution map The first step is to create a whole field map of the spatial resolution or the gradient of the isochromatic fringe patterns. Hitherto, there is no standard method to quantify the whole field spatial resolution of photoelastic fringe patterns. In this paper, a novel method based on intensity gradients of the photoelastic fringe pattern is used to create a map that represents the fringe gradients in the model. Initially, the method is demonstrated using
http://dx.doi.org/10.1016/j.optlaseng.2016.05.010 0143-8166/& 2016 Elsevier Ltd. All rights reserved.
Please cite this article as: Ramakrishnan V, Ramesh K. Scanning schemes in white light photoelasticity – Part II: Novel fringe resolution guided scanning scheme. Opt Laser Eng (2016), http://dx.doi.org/10.1016/j.optlaseng.2016.05.010i
V. Ramakrishnan, K. Ramesh / Optics and Lasers in Engineering ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
Fig. 1. (a) Theoretically simulated fringe pattern in a circular disc under diametral compression (b) The intensity gradient map G (c) Magnified view of a portion in Fig. 1(b) (d) Whole field map resembling the fringe gradient (FR_map) (e) Magnified view of a portion in Fig. 1(d). (For interpretation of the references to colour in this figure, the reader is referred to the web version of this article.)
analytically obtained fringe pattern in a circular disc under compression. Later this method is applied to experimental isochromatic fringe patterns recorded using white light. 2.1.1. Circular disc under compression Theoretically simulated photoelastic fringe pattern in a circular disc under diametral compression (Load¼492 N, Dia¼ 60 mm) is used to illustrate the proposed method. Fig. 1(a) shows the dark field isochromatics in the disc and it can be observed that the fringe gradient is very high near the loading points. The objective here is to create a whole field map whose values are representative of the fringe gradient in the model. Initially, the intensity gradients are computed in the horizontal as well as the vertical directions. The directional gradients are calculated by the central difference formula given by,
Gx =
I (i, j + 1) − I (i, j − 1) I (i + 1, j ) − I (i − 1, j ) ; Gy = 2 2
(1)
where, Gx and Gy are the intensity gradients in the horizontal and the vertical directions respectively and (i, j) represent the pixel coordinates of the point under consideration. Next, the absolute gradient is calculated for each pixel from the directional gradients as,
G=
Gx2 + Gy2
(2)
Fig. 1(b) shows the absolute gradient G for the circular disc. It has high values (shown as red) in most of the zones near the loading points, which are also zones of stress concentration where fringe gradient is high. However, it can be seen from the magnified image (Fig. 1(c)) that there are still pixels in which the values are low (appear as blue). These pixels correspond to points near the centre of the fringes where the intensity variation is low. Ideally, all the pixels in the high fringe gradient zones should have comparatively higher values, which is not the case now. Hence, the quantity G is not adequate to capture the fringe gradient. The quantity that captures the fringe gradient is obtained by taking the sum of the absolute intensity gradient over a k k pixel window centred around the pixel under consideration. This is given by, m= i + p n= j + p
FR(i, j ) =
∑ ∑ m= i − p n= j − p
G(m, n) where p =
1 (k − 1) 2
(3)
where (i, j) represents the global coordinates of the pixel under consideration and (m, n) represents their local coordinates in the k k neighbourhood. A kernel size of 11 is found to be sufficient for the problems studied. The quantity FR is calculated for every pixel in the model domain which is plotted as an image and the resulting image is labelled as FR_map. Fig. 1(d) shows the FR_map obtained for the circular disc and the magnified view of the map near the loading point is shown in Fig. 1(e). It can be seen that the values in the map increase towards the high fringe gradient zones near the loading point where the spatial resolution of fringes is low. A low value of FR indicates higher spatial resolution of the fringes and vice-versa. 2.1.2. Experimental isochromatic colour images Fig. 2(a) shows the dark field isochromatics in a bi-axially loaded cruciform specimen having an inclined crack (bi-axial load ratio (Fx/Fy)¼ 0.5, thickness¼6 mm) recorded using white light [13]. The gradient of the fringe pattern increases towards the crack-tip and the spatial resolution of the fringe pattern is very less in this zone (o10 pixels per fringe order). The method described in the previous section is extended for the isochromatics obtained under white light (as in TFP/RGB photoelasticity). The absolute intensity gradients are computed using Eq. (2) for each colour planes (red, green and blue) separately. Fig. 2(b–d) show the contours of G obtained using the red, green and blue planes respectively. Next, the maps obtained using the individual colour components are added to obtain the total intensity gradient (Gt) map (Fig. 2(e)). The FR_map is then created from the total intensity gradient map by using Gt instead of G in Eq. (3), which is shown in Fig. 2(f). It can be seen that values in the map increase towards the crack-tip and hence, capturing the fringe gradient in the model. Further, it can also be noted that the map identifies even minute marks in the model domain. Though, this is not of any advantage for the current problem, it is helpful in analysing models having random noise which is illustrated in Section 5. Fig. 3(a) shows the dark field isochromatics in a thick ring subjected to internal pressure (load¼ 3.93 MPa, thickness¼6 mm). The fringe order in the model increases from 1.5 in the outer diameter to nearly 6 at the inner boundary. The fringe gradient increases towards the inner diameter of the ring whereas their spatial resolution decreases. The FR_map obtained for the problem using Eqs. (1–3) is shown in Fig. 3(b). It can be seen that the values of FR increases towards the inner diameter of the ring which is in accordance with the variation of the fringe gradient in the model.
Please cite this article as: Ramakrishnan V, Ramesh K. Scanning schemes in white light photoelasticity – Part II: Novel fringe resolution guided scanning scheme. Opt Laser Eng (2016), http://dx.doi.org/10.1016/j.optlaseng.2016.05.010i
V. Ramakrishnan, K. Ramesh / Optics and Lasers in Engineering ∎ (∎∎∎∎) ∎∎∎–∎∎∎
3
Fig. 2. (a) Dark field isochromatics in the bi-axially loaded cruciform specimen having an inclined crack. Absolute gradient for different colour planes of the image (b) Red (c) Green (d) Blue. (e) Sum of the absolute gradients (Gt). (f) Contour plot of the FR map. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 3. (a) Dark field isochromatics in the thick ring subjected to internal pressure. (b) FR_map obtained using Eqs. (1–3). (For interpretation of the references to colour in this figure, the reader is referred to the web version of this article.)
The FR_map thus obtained is used for guiding the proposed scanning scheme. 2.2. Scanning scheme The new scanning scheme envisaged is designed to resolve the low fringe gradient zones having higher spatial resolution first. Fig. 4 shows the flowchart of the proposed Fringe Resolution Guided Scanning in TFP (FRSTFP). Initially, the fringe orders are estimated by the least squares method using the colour difference formula. The FR_map is obtained for the problem at hand using Eqs. (1–3) as discussed in the previous section by processing the colour image. To start the refining process, a point that is correctly resolved by the least squares method is selected. At each step, all the unresolved adjacent pixels corresponding to the resolved pixels are considered for deciding the progression of the scanning scheme. Initially, the eight neighbouring pixels corresponding to the seed point are stored in a list. Subsequently, this list of adjacent unresolved pixels is updated. The algorithm searches for the least value of FR in the list of adjacent unresolved pixels at every step.
The method used for refining may be Refined TFP (RTFP) [9] or window search method [10,11]. If the pixel under consideration has more than one neighbouring resolved pixel, an average value of the fringe order is taken as Np to maximise the use of neighbouring resolved pixel data. Hence, the proposed scanning scheme retains the advantages of advancing front scanning scheme while addressing the issue of varied spatial resolution of the isochromatic fringes. The generation of FR_map and the proposed scanning scheme are implemented using visual C þ þ. The values in the FR_map depend on the intensity of the experimentally recorded isochromatic image and generally range from 50–2000. The scanning scheme is independent of the absolute values in the FR_map as it works in a comparative manner by identifying the pixel having the least value of FR in the neighbourhood.
3. Fringe order refinement using FRSTFP In this section, the efficacy of the proposed fringe resolution guided scanning scheme is demonstrated by applying it two problems – bi-axially loaded cruciform specimen with an inclined
Please cite this article as: Ramakrishnan V, Ramesh K. Scanning schemes in white light photoelasticity – Part II: Novel fringe resolution guided scanning scheme. Opt Laser Eng (2016), http://dx.doi.org/10.1016/j.optlaseng.2016.05.010i
4
V. Ramakrishnan, K. Ramesh / Optics and Lasers in Engineering ∎ (∎∎∎∎) ∎∎∎–∎∎∎
Fig. 4. Flowchart of the fringe resolution guided scanning scheme in TFP.
crack (Fig. 2) and a thick ring subjected to internal pressure (Fig. 3). The limitations of the existing scanning schemes in refining these two problems are clearly demonstrated in the first part of this work [12]. Later, the proposed scanning scheme is applied on a stress frozen model of an aerospace component and is compared with the advancing front scanning scheme (AFSTFP). Initially, the fringe orders for the inclined crack and the thick ring problems are obtained by the least squares method. Refining is carried out by FRSTFP in conjunction with the modified window search method [9]. The FR_maps corresponding to the isochromatics fringe patterns are shown in Figs. 2(f) and 3(b) respectively. Fig. 5(a–e) show the progression of the scanning scheme for the inclined crack problem at an interval of 100,000 pixels. It can be seen that the area near the crack tip which has the least spatial resolution is refined only at the end of the scan. Hence, the noise in the results due to inadequate resolution is encapsulated within this region and its propagation is restricted. Fig. 5(f–j) show the progression of the scanning scheme for the thick ring problem at an interval of 40,000 refined pixels. Unlike the existing scanning schemes [12], this scanning scheme proceeds in a concentric fashion following the increase in the fringe gradient towards the inner diameter. Further, the proposed scanning scheme is able to scan the entire model domain with a single seed point (shown as yellow dot in Fig. 5) even for multiply connected models. The proposed scheme needs only minimal user interaction, which will enable the use of photoelastic analysis for a wide range of interdisciplinary problems [6,7]. Smoothing of the isochromatic data is usually necessary to remove minor jumps in fringe order values that may arise due to localised colour variations in the models. Fig. 6(a) and (b) show the smoothed fringe order results corresponding to the isochromatic data shown in Fig. 5(e) and (j) obtained for the two problems.
Multi-directional smoothing scheme involving local regression using weighted linear least squares proposed in Ref. [14] is used to smooth the isochromatic data. Fig. 6(c) and (d) show the fringe order variation along lines AB (Fig. 6(a)) and CD (Fig. 6(b)) respectively. The fringe order results are compared with the integer fringe order values obtained by skeletonisation of the fringe pattern in the green plane image. It can be seen that the results obtained are consistent with the integer fringe orders. The mean errors in the fringe order values at the fringe skeleton points are only 0.05 and 0.06 for the inclined crack problem and the thick ring problem respectively. 3.1. Immunity to seed point location To study the influence of seed point location on the fringe order results, the inclined crack problem is refined using seed points selected at different locations in the model domain other than the point shown in Fig. 5(e). The contour plots of the smoothed fringe order results and the corresponding seed points are shown in Fig. 7(a) and (b). A whole field map of the difference in the fringe order values showed that they are the same though the location of the seed points used are different. Similarly, seed points selected at different locations in the problem of the thick ring subjected to internal pressure (Fig. 7(c) and (d)) also yielded similar results. Hence, unlike the existing scanning schemes [12], the fringe order values obtained by FRSTFP is independent of the location of the seed point selected.
4. Comparison of FRSTFP and advancing front scanning In this section, the proposed FRSTFP is used to extract the
Please cite this article as: Ramakrishnan V, Ramesh K. Scanning schemes in white light photoelasticity – Part II: Novel fringe resolution guided scanning scheme. Opt Laser Eng (2016), http://dx.doi.org/10.1016/j.optlaseng.2016.05.010i
V. Ramakrishnan, K. Ramesh / Optics and Lasers in Engineering ∎ (∎∎∎∎) ∎∎∎–∎∎∎
5
Fig. 5. Progression of the fringe resolution guided scanning scheme applied to (a)–(e) Bi-axial fracture problem (interval ¼ 100,000 pixels) (f)–(j) Thick vessel subjected to internal pressure (interval ¼40,000 pixels). The seed points are shown as yellow dots. Progression images are shrunk for optimal spacing.
Fig. 6. Contour plot of the smoothed fringe order results obtained by refining using the fringe resolution guided scanning scheme for (a) Inclined crack in the bi-axially loaded cruciform specimen (b) Thick vessel subjected to internal pressure. (c), (d) Fringe order variation along lines AB and CD respectively. (For interpretation of the references to colour in this figure, the reader is referred to the web version of this article.)
isochromatic data in a stress-frozen slice of an industrial problem. Fig. 8(a) shows the dark field isochromatics of the problem. Initially, the fringe orders are estimated using the least squares method and is shown as greyscale in Fig. 8(b). The fringe orders thus obtained are then refined using only a single seed point (shown as yellow dot). The refined results obtained using FRSTFP are shown in Fig. 8(c). Fig. 8(d) shows the contour plot of the fringe order values after multi-directional smoothing. The smooth
variation of the colours indicates that the fringe orders are correctly resolved in the entire model domain. For comparison, the advancing front scanning scheme, which is the most sophisticated among the existing schemes [12] is also used. Fig. 8(e) shows the contour plot of the results obtained using the advancing front scanning scheme using the same seed point. It can be observed that noise has propagated to the zone opposite to the location of the seed point. Low resolution zones near the hole
Please cite this article as: Ramakrishnan V, Ramesh K. Scanning schemes in white light photoelasticity – Part II: Novel fringe resolution guided scanning scheme. Opt Laser Eng (2016), http://dx.doi.org/10.1016/j.optlaseng.2016.05.010i
6
V. Ramakrishnan, K. Ramesh / Optics and Lasers in Engineering ∎ (∎∎∎∎) ∎∎∎–∎∎∎
Fig. 7. (a), (b) Contour plot of the smoothed fringe order results obtained for the inclined crack problem using different seed points. (c), (d) Contour plot of the smoothed fringe order results obtained for the thick ring using different seed points. Seed points are shown as black dots. (For interpretation of the references to colour in this figure, the reader is referred to the web version of this article.)
Fig. 8. (a) Dark field isochromatics in the stress frozen slice of an aerospace component (b) Greyscale representation of the fringe order obtained by least squares method. Yellow dot shows the seed point used for refining. (c) Refined results obtained using the fringe resolution guided scanning scheme using a single seed point (d) Fringe order values after smoothing (e) Refined results obtained using advancing front scanning scheme using the same seed point. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
have triggered this noise propagation, which is totally absent in FRSTFP as it is designed to resolve these zones only at the end. In both the scanning schemes, the maximum fringe order that could be resolved in the model is 4 as the spatial resolution near the hole is low.
5. Problems with random zones of noise In many situations, one has to analyse models from industries having certain marks or even having foreign particles at random locations which may have crept in as part of their studies/specimen preparation. However, these marks/particles act as source of noise from digital photoelastic point of view. In such cases, the technique adopted should provide accurate results in all the other zones of the model by encapsulating noise within the zones containing the marks/particles. The flexibility in creating multiple masks proposed in Ref. [14] is useful if the noisy zones are few and are clearly identifiable. In the subsequent sections, the use of the proposed scanning scheme in handling such noisy zones is explored.
5.1. Plate under concentrated load with simulated noise To demonstrate the efficacy of the scanning scheme, initially a problem is considered in which the noisy zones are simulated artificially. The problem of a plate subjected to a concentrated load is considered for this. On the experimental isochromatics recorded, spots of noise are introduced by modifying the intensity values of these zones. Many problems in industry seek fringe order values in the rest of the model domain even in the presence of such noise. Fig. 9(a) and (b) show the original dark field isochromatics in the plate and the one with simulated noise randomly scattered across the model domain. The seed point selected for refining the fringe order data is shown as yellow dot. The FR_map for the problem (Fig. 9(c)) shows that the values are higher at the boundary of the noisy zones due to higher intensity gradient at the interface. Hence, the scanning scheme first scans all the pixels surrounding the dark spots before proceeding to the noisy pixels inside it. This is also clear from the progression of the scanning scheme shown in Fig. 9(d)–(g). This eliminates any chance of noise propagation from the dark spots. The contour plot of the fringe
Please cite this article as: Ramakrishnan V, Ramesh K. Scanning schemes in white light photoelasticity – Part II: Novel fringe resolution guided scanning scheme. Opt Laser Eng (2016), http://dx.doi.org/10.1016/j.optlaseng.2016.05.010i
V. Ramakrishnan, K. Ramesh / Optics and Lasers in Engineering ∎ (∎∎∎∎) ∎∎∎–∎∎∎
7
Fig. 9. Dark field isochromatics in a plate under concentrated load (a) Original image (b) Isochormatics with simulated noise. (c) FR_map of the isochromatics (d)– (g) Greyscale representation of the progression of the scanning scheme at an interval of 60,000 steps (h) Contour plot of the results obtained using fringe resolution guided scanning scheme (i) Whole field plot of the difference in the fringe order values obtained with and without the simulated noise. (For interpretation of the references to colour in this figure, the reader is referred to the web version of this article.)
order obtained after refining is shown in Fig. 9(h). The isochromatic data is also extracted from the original image without noise shown in Fig. 9(a) using the proposed scanning scheme. The whole field map of the differences in fringe orders are shown in Fig. 9(i). It is found that the proposed scanning scheme gives the same results as obtained from the original image throughout the model (blue denotes zero difference) leaving the noise points (shown as brown). This demonstrates the efficacy of the model in handling problems with random noise spread across the model domain. 5.2. Stereo-lithographic model The recent developments in stereo-lithography have enabled one to make photoelastic models of desired shapes through layer by layer manufacturing. Stress analysis in such 3-D models is performed by stress-freezing followed by slicing. However, these models have dark spots due to porosity randomly spread across the domain which are potential sources of noise for photoelastic analysis. It is shown that more noise shows up in the isochromatic image as the thickness of the slice decreases [15]. However, to capture the variation of stresses better, use of thin slices is recommended. A circular disc (diameter¼60 mm and thickness ¼6 mm) made using stereo-lithography is subjected to a diametral load of 64 N during stress freezing. Due to the layered manufacturing process, some levels of micro-porosity exist in each layer of the model. A slice of 2 mm thickness is then cut from the central portion of the disk carefully. Fig. 10(a) shows the dark field isochromatics in a quarter of the disc recorded using an incandescent lamp. It can be seen that there are intermittent spots spread across the model domain which makes the creation of appropriate boundary mask very difficult. The FR_map generated for the problem using Eqs. (1– 3) is shown in Fig. 10(b). Apart from the high fringe gradient zones near the loading point, the map is able to capture the small spots
owing to the sudden intensity gradients. The fringe order is initially estimated by the colour difference equation involving the combined use of RGB and HSV followed by refining using window search method. The progression of the scan after the first 210,000 pixels shows that the scheme refines the areas surrounding the noise spots first and then gradually proceeds into the pores (Fig. 10 (c)). The white spots in Fig. 10(c) correspond to the noisy spots in Fig. 10(a) and hence, removing the chance of noise propagation from these spots. The fringe order results obtained after refining using the proposed scanning scheme is shown in Fig. 10(d). Isochromatic demodulation is carried out accurately only till four fringe orders as the intensity modulation is weak beyond this owing to the use of incandescent lamp. This fringe order data is then smoothed by the multi-directional progressive smoothing scheme and the results obtained are shown in Fig. 10(e). The variation of fringe order values along line XY (Fig. 10(e)) is plotted in Fig. 10(f). It can be seen that the fringe order values obtained closely follow the corresponding theoretical values. Thus, the proposed scanning scheme is also capable of handling stereo-lithographic models with random noise of varied sizes. This would allow users to cut thinner slices from the stressfrozen 3-D photoelastic model to study their stress variation better.
6. Conclusions A novel scanning scheme for TFP (Twelve Fringe Photoelasticity) is proposed that takes into account the spatial resolution of the isochromatic fringe pattern. The proposed fringe resolution guided scanning scheme is capable of solving a variety of problems without the propagation of noise from the zones where spatial resolution of the fringes is low. Initially, a method based on the intensity gradients is proposed to generate a map that represents the fringe gradients in the model. The procedure for creating the map is discussed using
Please cite this article as: Ramakrishnan V, Ramesh K. Scanning schemes in white light photoelasticity – Part II: Novel fringe resolution guided scanning scheme. Opt Laser Eng (2016), http://dx.doi.org/10.1016/j.optlaseng.2016.05.010i
V. Ramakrishnan, K. Ramesh / Optics and Lasers in Engineering ∎ (∎∎∎∎) ∎∎∎–∎∎∎
8
Fig. 10. (a) Dark field isochromatics in the circular disc under compression made by stereo-lithography (b) FR map (c) Progression of the scanning scheme after 210,000 steps. Contour plot of the fringe order results obtained after (d) refining (e) refining and smoothing (f) Fringe order variation along line XY compared with the corresponding values from theory. (For interpretation of the references to colour in this figure, the reader is referred to the web version of this article.)
theoretically simulated fringe pattern in a circular disc and later extended to experimentally recorded isochromatics using white light. The proposed scanning scheme is successfully applied to two problems – bi-axially loaded cruciform specimen with an inclined crack and thick ring subjected to internal pressure. The existing scanning schemes have limitations in solving both these problems as they do not consider the spatial resolution of the fringes. The proposed scheme eliminates the propagation of noise and restricts them within the low resolution zones since the algorithm postpones the scanning of noisy zones towards the end. The performance of the scanning scheme is independent of location of the seed point selected and hence, the user interaction is minimal. The efficacy of the method is also demonstrated by applying it to an industrial stress frozen slice and the results are compared with the advancing front scanning scheme which is the most robust among the existing schemes. Further, it is found that the proposed scanning method is also capable of handling problems with random noise in the model. This is first demonstrated using a plate subjected to concentrated load with simulated noise. Finally, a problem of a disc under diametral compression made using stereo-lithographic model making approach having random noise due to layered manufacturing is also solved.
Acknowledgements Partial financial support from the Indian National Academy of Engineering (INAE) (Ref No. INAE/201/WLBS) through their short term research proposal is gratefully acknowledged. The authors acknowledge Dr. S.A. Annamala Pillai and Mr. Jeby Philip of VSSC
Trivandrum, India for providing the stress frozen slice of the aerospace component.
References [1] Ajovalasit A, Petrucci G, Scafidi M. Review of RGB photoelasticity. Opt Lasers Eng 2015;68:58–73. [2] Ramesh K, Kasimayan T, Neethi Simon B. Digital photoelasticity – a comprehensive review. J Strain Anal Eng Des 2011;46:245–66. [3] Scafidi M, Toscano A, Petrucci G, Alessi S, Ajovalasit A. Review of photoelastic image analysis applied to structural birefringent materials: glass and polymers. Opt Eng 2015;54(8):081206. [4] Ajovalasit A, Petrucci G, Scafidi M. RGB photoelasticity applied to the analysis of membrane residual stress in glass. Meas Sci Technol 2012;23:025601. [5] Iqbal Baig, Ramesh K, Hariprasad MP. Analysis of stress distribution in dry masonry walls using three fringe photoelasticity. In: Proceedings of the SPIE 9302, international conference on experimental mechanics, 93022P; 2014. http://dx.doi.org/10.1117/12.2081235. [6] Ramesh K, Hariprasad MP, Bhuvaneshwari S. Digital photoelasticity applied to implant dentistry. Opt Lasers Eng 2016. http://dx.doi.org/10.1016/j.optlaseng.2016.03.022 (article in press). [7] Antony SJ. Imaging shear stress distribution and evaluating the stress concentration factor of the human eye. Sci Rep 2015;5:8899. [8] Swain D, Philip J, Pillai SA. A modified regularized scheme for isochromatic demodulation in RGB photoelasticity. Opt Lasers Eng 2014;61:39–51. [9] Madhu KR, Ramesh K. Noise removal in three fringe photoelasticity by adaptive colour difference estimation. Opt Lasers Eng 2007;45:175–82. [10] Ajovalasit A, Petrucci G, Scafidi M. RGB photoelasticity: review and improvements. Strain 2010;46:137–47. [11] Kale S, Ramesh K. Advancing front scanning approach for three-fringe photoelasticity. Opt Lasers Eng 2013;51:592–9. [12] Vivek R, Ramesh K. Scanning schemes in white light photoelasticity, part i – critical assessment of the existing scanning schemes. Opt Lasers Eng 2015 (under review). [13] Shreesudha B. Study of digital photoelastic evaluation of stress field parameters for cracks – examples from bi-axially loaded cracked cruciform specimen (MS Thesis).Madras, India: Dept. of applied mechanics, Indian Institute
Please cite this article as: Ramakrishnan V, Ramesh K. Scanning schemes in white light photoelasticity – Part II: Novel fringe resolution guided scanning scheme. Opt Laser Eng (2016), http://dx.doi.org/10.1016/j.optlaseng.2016.05.010i
V. Ramakrishnan, K. Ramesh / Optics and Lasers in Engineering ∎ (∎∎∎∎) ∎∎∎–∎∎∎ of Technology (IIT); 2016. [14] Ramesh K, Vivek R, Ramya C. New initiatives in single colour image based fringe order estimation in digital photoelasticity. J Strain Anal 2015;50(7):488– 504. [15] Ashokan K, Prasath RGR, Ramesh K. Noise-free determination of isochromatic parameter of stereolithography-built models. Exp Tech 2012;36:70–5.
Vivek Ramakrishnan received his B.Tech degree in Mechanical Engineering from the University of Kerala, India in 2011. He is currently a Ph.D student at the Department of Applied Mechanics, Indian Institute of Technology Madras, India. His research interests include optical methods in mechanics, digital photoelasticity, residual stress analysis of glass and non-destructive evaluation.
9
K. Ramesh received his undergraduate degree from the Regional Engineering College, Trichy (now NIT, Trichy), Postgraduate degree from the Indian Institute of Science, Bangalore and the Doctoral Degree from the Indian Institute of Technology Madras. He is currently a Professor at the Department of Applied Mechanics, IIT Madras; as its Chairman during (2005–2009) and formerly a Professor at the Department of Mechanical Engineering, IIT Kanpur. In recognition of his significant contributions in photoelastic coatings, he was awarded the Zandman award by SEM in 2012. He has received several recognitions: Fellow of the Indian National Academy of Engineering (2006), Distinguished Alumnus Award of NIT, Trichy (2008), President of India Cash Prize (1984), Member of the Editorial Boards of the International Journals: Strain (since 2001), Journal of Strain Analysis for Engineering Design (2009–10), Optics and Lasers in Engineering, and Steering committee member of ASEM.
Please cite this article as: Ramakrishnan V, Ramesh K. Scanning schemes in white light photoelasticity – Part II: Novel fringe resolution guided scanning scheme. Opt Laser Eng (2016), http://dx.doi.org/10.1016/j.optlaseng.2016.05.010i