This section presents the iterative IPM process as mentioned in the introduction to this chapter. The main idea behind the iterative IPM is to perform the springback prediction and the application process repeatedly until a “best”Ccorr cloud is obtained. Best in this
context refers to aCcorr cloud that is expected to minimise the springback phenomena.
The Iterative IPM process is presented in block diagram form in Figure 9.2. The iterative element of the process can clearly be seen from the figure. From the figure it can be observed that the process incorporates three types of clouds: (i)Cin inputcloud
(ii) the predicted cloudCpredand (iii) the corrected cloudCcorr. TheCincloud (coloured
in red) is the description of the desired shape (the target shape). The predicted cloud Cpred(coloured inpurple) is used to see whether the currentinputis close enough to the
desired shape Cin. The corrected cloudCcorr (coloured inblue) is generated to provide
the Iterative IPM with a newinputfor the next iteration (where required). The iterative IPM operates as follows.
Figure 9.2: Iterative IPM.
At start up the process is similar to that for the Single Pass IPM process (see above). The Cin cloud is preprocessed using one of the proposed 3D surface representations to
predicted springback values E={e1, e2, . . . , en}one value per grid centre point in Gin.
Recall that errors values are given a negative sign if the springback is upwards with respect to Gout and a positive sign otherwise (as described previously in Chapter 3).
Recall that the springback values are already inverse, so to get the original value of the error the “inverse of the inverse”E are considered and then applied toGin to produce
a predicted shape Cpred as shown in Equation 9.1.
Cpred=Cin−E (9.1)
If, on this first iteration,Cpred ≈Cin(to a certain level of tolerance) then the process
finishes and the originalCin is passed on ready for forming. However, this is an unlikely
event as this will mean that the predicted springback errors all equated to approximately 0 (or below the tolerance value). In practice the predicted shape will not be the same as the desired shape on the first iteration. As presented previously in Chapter 6, a tolerance of 0.08 mm has been suggested by BS EN ISO 1101:2005 [29] and used with respect to the PS representation to identify the similarity between two curves; this tolerance value can be also used in the context of the iterative IPM in order to identify the maximum difference of error (springback) betweenCpred andCin that can be acceptable. Thus, as
in the case of the Single Pass IPM, the predicted springback errors are applied to the Cin cloud to produce a corrected cloudCcorr as shown in Equation 9.2.
Ccorr =Cin+E (9.2)
In the case of the Single Pass IPM theCcorr shape was then passed on for manufacture.
In the case of the Iterative IPM the process is repeated. TheCcorr cloud is preprocessed
in the same way as before and fed back into the classifier and a new Cpred produced.
If this new Cpred is now near enough to Cin the process stops and Ccorr passed on for
manufacture. If it is not the set of differencesdiff between the newCpredand the original
Cinclouds is determined and this set of difference applied to theinputcloud to produce
Ccorr which is a newinput for the iterative IPM:
diff =Cin−Cpred
Ccorr =input+diff
The diff values are described in terms of a magnitude and a direction and calculated in the same manner as springback as described previously in Chapter 3 (Section 3.4). The magnitude is thedistance along the normal from the point located on Cin obtained
from the Cpred cloud while the direction is determined according the position of the
Cpred points with respect to theCin points. If thediff value is upwards with respect to
previously in Chapter 3). Note that the diff values will not necessarily be the same across the shape.
The process continues until either: (i) Cpred approximates toCin where the average
difference between both clouds satisfy a prescribed tolerance or (ii) a maximum number of iterations has been reached. Algorithm 9.2 presents the iterative IPM process. For experimental purposes n = 12 was used. The inputs to the algorithm are the desired (target) cloud Cin, a classifier and the max number of iterationsn.
The algorithm starts by generating Gin using one of the proposed surface represen-
tation techniques (line 1). The classifier is then applied toGin so that each grid centre
point will be associated with one predicted error value. The set of all the predicted error values define the set E (line 2). Cin is assigned to the input cloud (line 3). The pre-
dicted cloudCpred is then generated using FunctionGenerateP redictedCloud (line 4).
IfCpredis approximately the same as the desired (target) shapeCin, then theCcorr is the
inputand the algorithm stops. Otherwise, theCcorr0 is generated (line 9) using Function GenerateCorrectedCloud1 (described earlier for the Single Pass IPM) and used as an input for the next iteration (line 9) where Algorithm 9.3 is applied to obtain the best Ccorr (line 10).
Algorithm 9.2:Main Iterative IPM process Input: Cin, classifier, number of iterationsn
Output: Ccorr cloud
1 Gin ←Process Cin using appropriate 3D surface representation ;
2 E ← Set of springback prediction values obtained from applying classifier toGin,
each value in E is correlated to a pointp inGin;
3 input←Cin ;
4 Cpred← GeneratePredictedCloud (E,Gin,input);
5 if Cpred ≈Cin then
6 Ccorr ←input ;
7 end
8 else
9 Ccorr0 ← GenerateCorrectedCloud1(E,Gin,input) ;
10 Ccorr ← IterativeIPM(Ccorr0 ,Cin,n) ; // Algorithm 9.3
11 end
12 return Ccorr
The Cpred of a given C cloud is generated using Function GenerateP redictedCloud
where the set of predicted error values E, Gin and the C are the inputs to the func-
tion. Recall that each centre grid point in Gin is associated with an e value, thus e0
is the inverse of the e (opposite sign). The Cpred of the given C cloud is obtained by
apply the inverse of the predicted error values e0 to the points of C cloud (Function GeneratePredictedCloud line 6).
Algorithm 9.3 describes the iterative part of the Iterative IPM starting from itera- tion 2 (as shown in line 1). The inputs are: (i) the originalCin cloud, (ii) the maximum
FunctionGeneratePredictedCloud(E,Gin,C)
Input: E,Gin,C
Output: Cpred cloud
1 Cpred←C;
2 for allp∈C and ppred∈Cpred do
3 p← Grid centre point in Gin corresponding to (x, y) ; 4 e← Value inE corresponding top;
5 e0 ← Value ofewith reversed sign ;
6 ppred←p+e0 ;
7 end
8 return Cpred
initially to the Ccorr0 resulting from the first iteration with respect to the main itera- tive process described in Algorithm 9.2, then it assigned to the Ccorr cloud (Algorithm
9.3 line 2). An iteration counter is used to compare with n as the process proceeds. Again a Cpred is generated for each iteration using Function GenerateP redictedCloud
(line 6) in such a way that if Cpred is close enough to the Cin, then the Ccorr is the
output. Otherwise, the Ccorr for the current input is generated using Function Gen-
erateCorrectedCloud2 and used as the new input for the next iteration (lines 8 and 9 respectively). Thus, the algorithm will be terminated when the iterative IPM is applied for n times (line 3). Finally, the output from Algorithm 9.3 is the final Ccorr cloud.
FunctionGenerateCorrectedCloud2 is used to generate theCcorr cloud for the Iterative
IPM from the second iteration onwards. TheCcorris assigned initially to theinputcloud
(line 1). The differences (diffs) between the points located inCin and the points located
in Cpred are obtained in line 3. The intuition is that if the diffs are repeatedly added
to the input then the Cpred for the next input will gradually converge to the desired
(target) shape Cin.
Algorithm 9.3:Iterative IPM
Input: input,Cin, number of iterations n
Output: Ccorr cloud
1 counter ←2; 2 Ccorr ←input;
3 while counter < n do
4 Gin ← ProcessCin using appropriate 3D surface representation;
5 E← Set of springback prediction values obtained from applying classifier to
Gin, each value inE is correlated to a pointp inGin;
6 Cpred← GeneratePredictedCloud (E,Gin,input);
7 if Cpred 6=Cin then
8 Ccorr ← GenerateCorrectedCloud2(Cin,input,Cpred);
9 input←Ccorr;
10 end
11 end
FunctionGenerateCorrectedCloud2(Cin,input,Cpred)
Input: Cin,input,Cpred
Output: Ccorr cloud
1 Ccorr ←input;
2 for all (pin∈Cin), (ppred∈Cpred), (pcorr ∈Ccorr) do
3 diff =pin−ppred;
4 pcorr =input+ diff;
5 end
6 return Ccorr
Table 9.1 presents a detailed example for the Iterative IPM. The iteration ID, the average predicted error (springback)eand the average differencediff between theinput and the Cpred for a given shape were recoded for 6 iterations (n= 6). From the table
it can be seen that the average error equalled the average absolute diff on the first iteration where theCincloud double as theinput. From the example it can be seen that
the difference diff gradually decreases and that the minimum diff was obtained on the fifth iteration with a difference of 0.016 (bold font). Thus, the input cloud arrived at on iteration 5 produce the“best”Ccorr that could be used to form the given shape with
minimum potential effect of springback.
Table 9.1: An example on the iterative IPM process for a given shape where the
average predicted error (e) and the average absolute difference between theCpred and
theCin (diff) are recorded for six iterationsn= 6.
Iteration ID e diff 1 0.731 0.731 2 0.664 0.067 3 0.735 0.071 4 0.763 0.029 5 0.747 0.016 6 0.646 0.101
9.4
Experiments and Evaluation
A sequence of experiments were conducted to evaluate the iterative IPM process. The Iterative IPM settings were as follows.
1. To be consistent with the evaluation of the Single Pass IPM: (i) |L|= |LE| = 5
using equal width discretisation and (ii) the LGM (level one) was used to generate a classifier.
2. Two grid sizes d= 1 mm and d= 10 mm. The first was used because this is the most appropriate resolution for manufacturing purposes and the second because earlier results indicated that this was best suited to the level one LGM technique.
3. The eight Gonzalo and Modified data sets were used: GSV1, GSV2, GTV1, GTV2, MSV1, MSV2, MTV1 and MTV2.
4. It was considered that n = 12 was sufficient to obtain a deep insight in to the operation of the Iterative IPM in order to: (i) provide a sufficient analysis on the generated Cpred and Ccorr clouds obtained by each iteration of the iterative IPM
and (ii) to facilitate the identification of the “best” Ccorr.
The results of each iteration of the Iterative IPM were recorded in terms of the average of absolute diff values between the Cpred and Cin, and the average springback
distribution of the Cpred cloud. It should be noted that only the magnitude of thediff
values was considered as the intuition is to see if theCpred and Cin converge from each
other and the absolutediff values converge to zeros as well.
The average absolutediff values and the springback distribution of the Cpred clouds
for the GSV1 data set, using d = 10 mm and d= 1 mm are presented in Figures 9.3 and 9.4 respectively. Similarly, for the GSV2 data set, the average absolutediff values and the springback distribution of theCpredclouds usingd= 10 mm and d= 1 mm are
presented in Figures 9.5 and 9.6 respectively. In the same manner the results obtained using GTV1, GTV2, MSV1, MSV2, MTV1 and MTV2 data sets are presented in Figures 9.7, 9.8, 9.9, 9.10, 9.11, 9.12, 9.13, 9.14, 9.15, 9.16, 9.17, 9.18 respectively. From the figures, it can be seen that:
• For the first iteration, the average absolute diff values are similar to the average springback distributions since theinputfor the Iterative IPM in the first iteration is the desired shape itself Cin so the predicted error (springback) would be the
difference between the Cin and Cpred as shown in the figures.
• The convergence of Cpred towards Cin can be clearly observed in all the datasets
as the average absolute diff tends to be zero. Note that the actual springback distribution will not be zero as the error labels are associated with mean values.
• The overall behaviour of the springback distribution is totally related to the opera- tion of the classifier and the nature of the discretisation (equal width discretisation in this case). Recall that the predicted error labels are characterised by their un- even distribution and this may have some implications on the mean values used to replace the labels. For instance, suppose one of the records was labelled with a labelL1 that had a mean value ofv1 in iterationi1 and this label is changed in the
next iteration i2 to be L2 which had a mean value of v2 and if this happened for nrecords then the springback distribution would be affected by this change either in an increasing or decreasing manner.
• The fluctuations in the average absolute diff values (as in the case, for example of Figure 9.6) are explained by: (i) the uneven distribution of the labels, (ii) the dominant error (springback) distributed over the shape, (iii) the differences between the mean values of the error labels and (iv) the changing from one label to another during IPM iterations with respect to a single record which probably will not only affect the average springback distribution but also the overall average
absolutediff values. Despite the fluctuating behaviour of the average absolutediff values, the overall behaviour was tending to decrease throughout the IPM process.
In practice, it would be of interest to supply the manufacturer with the the “best” Ccorrcloud rather than the one finally arrived at (there may be local minima). Therefore,
Table 9.2 presents a summary related to the “best” iteration ID where the “best” Ccorr
cloud was obtained (from Figures 9.3 to 9.18) with respect to the eight datasets using d= 10 and d= 1. From the table it can be noted that:
• The maximum number of iterations required to obtain the “best” input using d= 10 and d= 1 was 9 iterations (GTV2) and 8 iterations (MTV2) respectively.
• The minimum number of iterations required to obtain the “best” input using d= 10 and d= 1 was 2 iterations (MSV2, MTV2 and GTV2).
• It was found thatd= 1 mm required more iterations to obtain the “best” input (Ccorr cloud) than d= 10 in five of the eight cases (GSV1, GSV2, GTV1, MSV2
and MTV2). The average number of iterations required to obtain the best input for d= 10 and d= 1 was 4.25 and 4.88 respectively and thus we can argue that there was no significant difference between using different grid sizes to obtain the bestinputwith respect to the number of iterations required.
• The final average absolute diff obtained when d = 1 is always 0. It is therefore suggestedd= 1 will produce slightly a more accurate end result (by small margin) than whend= 10 is used although d= 10 required fewer iterations and was more efficient in terms of run time (as will be demonstrated in Section 9.5).
• The material of the shape (steel or titanium) had no significant impact with respect to the iterative IPM process.
Finally, these results show that the iterative IPM would be able to supply manufac- turers with a most suitableinputcloud for the desired shape so as to serve to limit the effect of springback.
9.5
Run Time Analysis
This section presents the run time analysis for the IPM process using the two different grid sizes, d = 10 and d = 1, for the eight data sets GSV1, GSV2, GTV1, GTV2, MSV1, MSV2, MTV1, MTV2. The experiments were carried out using a 2.7 GHz Intel Core i5 PC with 4 GB 1333 MHz DDR3 memory, running OS X 10.8.1 (12B19). The implementation of the IPM was conducted using the Java programming language. The run time for each iteration of the IPM includes the following:
1. All the preprocessing steps required to generate: (i) grids, (ii) the LGM representa- tion and (iii) equal width discretisation using|L|=|LE|= 5 and error calculation
0.07 0.07 0.03 0.02 0.10 0.06 0.12 0.02 0.02 0.05 0.02 0.73 0.66 0.73 0.76 0.75 0.65 0.59 0.71 0.73 0.75 0.70 0.73 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 3 4 5 6 7 8 9 10 11 12 Th e av er ag e va lu e of th e sp rin gb ac k distr ibu6 on and the diffe re nc e be tw ee n Cp red a nd Ci n Iteration ID Difference(diff) Springback dist.
Figure 9.3: A average absolute diff
values and the average springback dis-
tribution of Cpred for the GSV1 using
d= 10 mm. 0.14 0.02 0.01 0.02 0.00 0.00 0.02 0.01 0.02 0.00 0.00 1.24 1.39 1.36 1.37 1.40 1.39 1.40 1.38 1.37 1.39 1.39 1.39 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1 2 3 4 5 6 7 8 9 10 11 12 Th e av er ag e va lu e of th e sp rin gb ac k distr ibu6 on and the diffe re nc e be tw ee n Cpr ed and C in Iteration ID Difference(diff) Springback dist.
Figure 9.4: The average absolutediff
values and the average springback dis-
tribution ofCpredfor GSV1 usingd= 1
mm. 0.68 0.04 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.72 0.71 0.72 0.71 0.72 0.71 0.72 0.71 0.72 0.71 0.72 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1 2 3 4 5 6 7 8 9 10 11 12 Th e av er ag e va lu e of th e sp rin gb ac k distr ibu6 on and the diffe re nc e be tw ee n Cp red a nd Ci n Iteration ID Difference (diff) Springback dist.
Figure 9.5: The average absolutediff
values and the average springback dis-
tribution ofCpred for GSV2 using d=
10 mm. 0.39 0.01 0.62 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.55 0.16 0.15 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0.47 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 1 2 3 4 5 6 7 8 9 10 11 12 Th e av er ag e va lu e of th e sp rin gb ac k distr ibu6 on and the diffe re nc e be tw ee n Cp red a nd Ci n Iteration ID Difference (diff) Springback dist.
Figure 9.6: The average absolutediff
values and the average springback dis-
tribution ofCpredfor GSV2 usingd= 1
mm.
Table 9.2: The best iteration ID, the average absolutediff and the springback distri-
bution for the GSV1, GSV2, GTV1, GTV2, MSV1, MSV2, MTV1 and MTV1 datasets
ford= 10 mm andd= 1 mm.
Datasets d best iteration ID The average absolutediff
GSV1 d=10 5 0.02 d=1 6 0.00 GSV2 d=10 3 0.01 d=1 5 0.00 GTV1 d=10 3 0.00