acuracy of ssm processes

Materials & Design. Vol. 17, No. 3, pp. 159-166, 1996 0 1997 Elsevier Science Ltd

Printed in Great Britain. All rights reserved 0261-3069/96 $15.00 + 0.00

PII: SO261-3066(96100060-7

Experimental design and analysis of in-plane processing accuracy for SSM process

Wanlong Wang,‘,” Wei Feng,b Yongnian Yan,b and Jerry Y.H. Fuh,’

aDepattment of Mechanical and Production Engineering, National University of Singapore, IO Kent Ridge Crescent, 119260 Singapore bDepattment of Mechanical Engineering, Tsinghua University, Beijing 100084, P.R. China

Received 20 August 1996; accepted 24 August 1996

Processing accuracy is an important area studied in rapid prototyping (RP) research. It is mainly dependent on the processing parameters, material characteristics and many other factors. Studies on processing accuracy can be divided into two categories: in-plane processing accuracy and vertical processing accuracy (that determines the staircase on the surfaces of prototypes). This work focuses on the in-plane processing accuracy. Similar to laminated object manufacturing (LOM) process, slicing solid manufacturing (SSM) uses paper and CO2 laser as material and energy source, respectively. This paper introduces an integrated method that combines orthogonal experimental design and analysis, and neural network analysis to determine the optimal processing conditions. The key processing parameters and their degree of influences on the processing accuracy, and the quantitative relations between input parameters and output accuracy will be investigated. This method of experimental design and analysis is not only effective for the SSM process, but also applicable to other RP processes that use the principle of 3D layered manufacture. 0 1997 Elsevier Science Ltd. All rights reserved

Keywords: rapid prototyping; slicing solid manufacturing (SSM); processing accuracy; orthogonal experimental design; back propagation network

Introduction

As more and more global competition arises, manufacturing industries are facing new challenges of bringing concept design to manufacture very quickly. This puts demands on enterprises not only to produce high quality products with low cost, but also to be content with a short lead time into market. Rapid prototyping techniques are new born manufacturing technologies, that will bring revolutionary changes in new products development. But the present status of part accuracy cannot meet the industry requirements. Many approaches have been taken to improve the techniques, such as looking for new processes, enhancing the existing processes and studying the relations between processing accuracy and processing parameters.

Owing to the nature of layer-by-layer fabrication, the studies on processing accuracy can be divided into two categories: in-plane processing accuracy and vertical processing accuracy (that determines the staircase on the surface of prototypes). Rapid prototyping (RP) processes produce the parts layer-by-layer on a certain plane, e.g. on the X-Y plane. The in-plane accuracy, that contributes to the overall accuracy is, thereby, an important area studied. Whether in-plane or vertical accuracy, a good experimental design and analysis can save time and cost, and also produce valuable information on improving the process. In the RP processes, there are many factors affecting their

*Correspondence to W. Wang.

accuracy. For example, in the SLA process, laser spot size (diameter), laser power, scanning speed, slicing pitch, resin, recoating, etc. will all affect the final part accuracy.

The sliding solid manufacturing (SSM) process (as shown in Figure I) is developed by the Centre for the Laser Rapid Prototyping in the Tsinghua University, China. The process uses paper and COZ laser as materials and energy source, respectively. In this process, there are many factors affecting the processing accuracy. In such a case, the design of the experiments for studying accuracy will be very important.

Generally speaking, a good design scheme should possess the following characteristics:

1. requires a lower cost, shorter time and higher reliability; 2. can produce more relevant information; 3. the results of experiment are easily to be interpreted.

In this paper, the method of orthogonal experimental design will be used to study the relations between the processing parameters and the resulting part accuracy.

Overview of present status on accuracy research Although many researches are devoted to the experimental design and analysis for studying the RP accuracy, Jacobs’ is the first one to systematically investigate the processing accuracy in terms of distortion and warpage in the SLA process. Bugeda et ~2.’ have studied distortion in the SLA

Materials & Design Volume 17 Number 3 1996 159

Processing accuracy for SSM process: Wanlong Wang et

(a) Cross-sectionalProfile

elevator

/:/li: 3#,

n

(b) Layered manufacturing

Figure 1 The SSM process

curing process by means of the finite element method. Wiedemann et CZL.~ studied the factors causing processing inaccuracy due to part distortion in the SLA process. Lee et a1.4 investigated the layer position accuracy in powder- based rapid prototyping, analyzed the main parameters affecting position accuracy and suggested appropriate selection of materials and techniques to achieve higher packing density for accuracy improvement. Lu et al.’ examined the depth and width of the laser curing lines for the stereolithography (SL) process, and relationships between the result and process parameters: the laser power, scan pitch and scanning pattern. Chen et aL6 studied the parameters affecting the Z-height inaccuracy of the SL process. All the above-mentioned researches are mainly devoted to the resulting accuracy or experimental studies, the relations between the processing parameters and processing accuracy being not much investigated. In our work, the emphasis is placed on studying the factors that determine part accuracy through a systematic approach.

Experimental design

Experimental objectives The objective7 of the processing accuracy experiments is to systematically study the main factors affecting processing accuracy in the SSM process, so as to realise the following three aspects through a good experimental design and analysis:

1. to identify the relationships between in-plane processing accuracy and processing parameters;

2.

3.

al.

to study the overall processing accuracy of prototypes; to study the mechanical properties, glue quality and carbonisation rate. According to experimental observation and analysis, the

factors affecting processing accuracy are: laser spot diameter, laser power and focus distance, laser cutting speed, paper and its thickness, adhesive material, temperature, pressure, and rolling speed. For a certain situation, this can be further simplified. For example, when the equipment decided upon, the laser spot size and focus distance are also decided. If the paper and adhesive material are known, the other five will be the only influencing factors. Those factors are the laser power, laser cutting speed, temperature, pressure and rolling speed. Assuming processing accuracy has a correlation with the above-mentioned five parameters, the following function exists:

d = f(W, v, T, f’, u) where

(1)

d is the laser cutting width, W is the laser power, Y is the laser cutting speed, T is the temperature, P is the pressure, and u is the rolling speed.

Measurement of the laser cutting width A rectangular profile (Figure 2) is used to measure the laser cutting width. After cutting, the distances between the inner and the outer borders will be determined using the following relation:

d = (u - b)/2 (2)

where a is the outer distance and b is the inner distance.

Orthogonal experimental design Orthogonal experimental designsV9 method, also known as the Latin square method, is widely used in the engineering fields. It is mainly applied in determining the key influencing factors. It has many standard tables that can be used to decide experimental conditions and arrange experimental schemes.

The advantages of the orthogonal tables are that they can be used to decide the optimal experimental conditions, and thus reduce the number of iterations required in extensive experiments. Besides, this method also enables further

Figure 2 Measurement of the laser cutting width

160 Materials & Design Volume 17 Number 3 1996

analysis on each factor and obtaining more useful information from the experimental results.

Orthogonal table and orthogonal experiment Orthogonal table is a kind of table which is constructed according to certain laws based on the combinatorial theory. It has found wide applications in experimental design. An experiment that employs orthogonal tables to arrange experimental schemes and conducts result analysis is called an orthogonal experiment. It is more appropriate to apply this to experiments involving multi-factor, multi- target (i.e. results examined in an experiment), interactions between multi-factors (i.e. correlation among factors), and experiments with random errors. Through orthogonal experiment, the influences of the factors and their interactions on the experimental results can be analyzed, the sequence of the key factors can be ranked, and the optimal process conditions can be determined. In an orthogonal experiment, each factor considered must be controllable. The value of each factor is called the level of that factor.

The symbol of the orthogonal table is L,(b’): where L is orthogonal table; the subscript a is the row number representing the number of experiments; c is the column number representing the maximum number of factors arranged in the experiment; b is the number of the different values, meaning the number of values for each factor. For example, in table L&27) (shown in Table I), 8 means that there are 8 rows, i.e. the number of experiments is 8; 7 means that there are 7 columns, and the maximum number of experimental factors (including the interactional columns) is 7; 2 means that each factor has only two levels. This type of orthogonal table is also called a two-level orthogonal table.

According to the former analysis on the main factors affecting processing accuracy, there are five key parameters used in the orthogonal experiment design. Considering the experimental quantities and conditions, four levels are used in the orthogonal table. In such a case, table L16(45) (shown in Table 2) is obtained. This table does not take into account the interactions among the factors. The head designs of the orthogonal table, L16(45), are shown in Table 3.

Interactional columns in orthogonal tables If factors are not independent in the experiments, the interactional effect must be considered. Once any two columns are determined, the interaction between the two columns can be represented by other columns. This column is called the interactional column. There is one interactional column in a two-level orthogonal table, but two in a three- level one. For example, in the interactional table of table

Table 1 The orthogonal table of Ls(2’)

Factors 1 2 3 4 5 6 7

1 1 1 1 1 1 1 1 2 1 1 1 2 2 2 2 3 1 2 2 1 1 2 2 4 1 2 2 2 2 1 1 5 2 1 2 1 2 1 2 6 2 1 2 2 1 2 1 7 2 2 1 1 2 2 1 8 2 2 1 2 1 1 2

‘able 2 The orthogonal table of &(4’)

Factors l(A) 2(B) 3(C) 4(D) 503

1 1 1 1 1 1 2 1 2 2 2 2 3 1 3 3 3 3 4 1 4 4 4 4 5 2 1 2 3 4 6 2 2 1 4 3 I 2 3 3 1 2 8 2 4 4 2 1 9 3 1 3 4 2 10 3 2 4 3 1 11 3 3 1 2 4 12 3 4 2 1 3 13 4 1 4 2 3 14 4 2 3 1 4 15 4 3 2 4 1 16 4 4 1 3 2

Table 3 The header design for 5 factors, 4 levels not considering interactions 46(45)

Column number 1 2 3 4 5

Factors A B C D E

Note: A denotes the laser power, B is the cutting speed, C is the temperahue, D is the pressure, and E is the rolling speed.

Table 4 The interactional table for Z&2’)

1 2 3 4 5 6 7 Column number

(1) 3 2 5 4 7 6 (2) 1 6 7 4 5 :

(3) 7 6 5 4 3 (4) 1 2 3 4

(5) 3 2 5 (6) 1 6

(7) 7

&(27) (shown in Table 4), the interactional column of column 1 and column 2 is column 3, the interactional column of column 3 and column 5 is column 6, etc.

In this paper, for comparison, another orthogonal table, L16(215), (shown in Table 5), considering interactions between two factors, is adopted. Its interactional table can be obtained (shown in Table 6). The head design of the orthogonal table, Li6(215), is shown in Table 7. Thus, two types of orthogonal tables - one considering and the other not considering the interactions - are used in this work.

The orthogonality of the orthogonal tables The orthogonal tables possess the following orthogonality:

l In any column, all iterative numbers for each level are equal. For example, in table Ls(27), each level of any column repeats itself four times.

l In any two of the columns, the number of pairs composed at one level embody all possible number pairs (under that level), and the iterative numbers for each number matched are equal. For example, in table Z&34), the number of pairs composed of any two of the columns embody all possible number pairs under three levels: (1, 11, (1,2), (1,3), (2, 0, C&2), C&3), (3, l), (3, 2), (3, 3), and all iterative numbers equal 1.

Processing accuracy for SSM process: Wanlong Wang et al.



Table 5 The orthogonal table of L16(215)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 4 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1

5 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 6 1 2 2 1 1 2 2 2 2 1 1 2 2 1 1 I 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1 8 1 2 2 2 2 1 1 2 2 1 1 1 1 2 2

9 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 10 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 11 2 1 2 2 1 2 1 1 2 1 2 2 1 2 1 12 2 1 2 2 1 2 1 2 1 2 1 1 2 1 2

13 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 14 2 2 1 1 2 2 1 2 1 1 2 2 1 1 2 15 2 2 1 2 1 1 2 1 2 2 1 2 1 1 2 16 2 2 1 2 1 1 2 2 1 1 2 1 2 2 1

Table 6 The interactional table for L1~(2~‘)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

1 (1) :2,

2 5 4 7 6 9 8 11 10 13 12 15 14 2 :3) 6 I 4 5 10 11 8 9 14 15 12 13 3 7 6 5 4 11 10 9 8 15 14 13 12 4 (4) :5) 2 3 12 13 14 15 8 9 10 11 5 :6) 2 13 12 15 14 9 8 11 10 6

f7) 14 15 12 13 10 11 8 9

I ta: 14 13 12 11 10 9 8 8 :9, 2 3 4 5 6 I 9 :w 2 5 4 I 6 10 fw 6 I 4 5 11 12

712) 6 5 4 :13) 2 3

13 :14)

2 14 1

Table 7 The header design for 5 factors, 2 levels while considering interactions &(215)

Column Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

A B AB c AC BC DE D AD BD CE CD BE Al? E

Note: A denotes the laser power, B is the cutting speed, C is the temperature, D is the pressure, and E is the rolling speed. AxC is simplified as AC.

Because of this orthogonality, the orthogonal experiment can be arranged in equilibrium and uniformity.

Steps and methods of experimental scheme The steps of arranging an orthogonal experiment are as follows:

Determine the number of varying factors and levels of each factor. Preliminarily analyze the interactions of the factors, and determine the one that must be considered by experience. Determine the approximate number of experiments according to the experimental conditions, such as manpower, equipment, time and cost. Choose appropriate orthogonal tables and arrange experimental schemes.

The methods of arranging an orthogonal experiment are as follows:


In the case of non-consideration of the interactions, arrange each factor into any column of the orthogonal table. Then the experimental conditions (level adopted for each factor) of each experiment (corresponding to the row of the orthogonal table) are determined by the levels of each of the factors arranged. For example, in the in-plane processing accuracy experiment, five factors and four levels are adopted to arrange the experiment. The experimental target is the cutting width of the laser. Table 8 gives this arrangement. In the case of consideration of the interactions, the factors cannot be randomly arranged. They must be arranged using the corresponding table title design. Note that different factors (including interactions) cannot be placed in the same column. This is because, it is difficult to analyze the actions of different factors in the same column. If this requirement is not met, then a large table should be adopted. For example, in the m-plane processing accuracy experiment, five factors and two levels are adopted to arrange the experiment. The


Table 8 Factors and four levels while not considering interaction

Laser power Cutting speed Temperature Pressure Rolling speed

25 W 40 mm/s 120°C -0.2 40 mm/s 27 w 48 mm/s 130°C -0.1 49 mm/s 30 w 60 mm/s 140°C 0.1 63 mm/s 32.5 W 80 mm/s 150°C 0.2 88 mm/s

Table 9 Factors and two levels while considering interaction

Laser power Cutting speed Temperature Pressure Rolling speed

1 25 w 48 mm/s 120°C -0.1 49 mm/s 2 30 w 80 mm/s 140°C 0.1 88 mm/s

experimental target, here also, is the cutting width of the laser. Table 9 gives this arrangement.

Analysis of the orthogonal tables

Calculation of the lever! sum and mean of the level ‘i’ For example, using table L16(45), the analysis is shown in Table 10. In Table 10, the variables are calculated as follows:MV = the sum of y values of level i in column j;

mu = r Mu (the average value of i levels in j column : n

n is the experimental number, r is the level number);

(3)

(the divergence square sum of each column); (4)

(the maximum difference of each column); (5)

Table 10 The calculation of orthogonal table

1 (4 2 @I 3 (C) 4 (D) 5 (E) Result (y)

1 Al Bl Cl Dl El Yl 2 Al B2 c2 D2 E2 Y2 3 Al B3 c3 D3 E3 Y3 4 Al B4 c4 D4 E4 Y4 5 A2 Bl c2 D3 E4 Y5 6 A2 B2 Cl D4 E3 Y6 I A2 B3 c4 Dl E2 Y7 8 A2 B4 c3 D2 El YS 9 A3 Bl c3 D4 E2 Y9 10 A3 B2 c4 D3 El YlO 11 A3 B3 Cl D2 E4 Yll 12 A3 B4 c2 Dl E3 Yl2 13 A4 Bl c4 D2 E3 Y13 14 A4 B2 c3 Dl E4 Y14 15 A4 B3 c2 D4 El Yl5 16 A4 B4 Cl D3 E2 Y1b Mlj MII M12 Ml3 M14 M15

4 M21 M22 M23 M24 M25

M3j M31 M32 M33 M34 M35

M4j M41 M42 M43 M44 M45

mlj ml1 ml2 ml3 ml4 ml5

mzj m21 m22 m23 m24 m25

m3j m31 m32 m33 m34 m35

m4j m41 m42 m43 m44 m45

Ri Rl R2 R1 R4 R5

si s4 s5

cq = mu - 7 = i [rMg - T]

(the effect of leveliin column j); (6)

Ranking of key factors According to the maximum difference of each column, the important ranking of the key factors can be determined. The bigger the difference is, the more important the factor will be.

The optimal process conditions In the case of non-consideration of the interactions, find the best point for each factor on the experimental target, combine all corresponding experimental conditions and form the optimal process conditions. This can also be achieved by choosing a highest or lowest point on the curve figured out from the analysis results.

In the case of consideration of the interactions, if the analysis results show that the interaction of two of the factors has a big influence on the experimental target, compare all the experimental results of all possible combinations of the two factors (if there are several experimental results, the mean value should be used), and choose the best one. Finally, consider the other factors synthetically and determine the optimal process conditions.

Experimental results and analysis The following are the experimental results and analysis for a certain kind of paper with one adhesive material.

Five factors, two levels experimental results and analysis considering interactions

The values of laser cutting width measured in the experiments are as follows:

y1 = 0.244, y2 = 0.205, y3 = 0.205, y4 = 0.192, y5 = 0.123, y6 = 0.093, yl = 0.096, y8 = 0.127, y9 = 0.170, ylo = 0.019, yll = 0.202, y12 = 0.203,

~13 = 0.087, ~14 = 0.085, ~15 = 0.113, y16 = 0.098.

Hence,

7 = eyi/ 16 = 0.152 i=l

where yi is the value measured, and y is the mean value. Following the orthogonal analysis method, Table 11 can

be obtained. Obviously, the divergence square sum of each


Table 11 Calculation results of five factors and two levels while considering interaction

Factors MI M2 ml m2 Ri s,

A 1.283 1.153 0.160 0.144 0.130 0.0011 B 1.614 0.822 0.202 0.103 0.792 0.0392 AB 1.227 1.209 0.153 0.151 0.018 0.0000 C 1.175 1.236 0.147 0.155 0.061 0.0002 AC 1.279 1.157 0.160 0.145 0.122 0.0009 BC 1.246 1.190 0.156 0.149 0.056 00002 DE 1.241 1.189 0.156 0.149 0.058 0.0002 D 1.238 1.198 0.155 0.150 0.040 0.0001 AD 1.247 1.189 0.156 0.149 0.058 0.0002 BD 1.222 1.214 0.153 0.152 0.008 0.0000 CE 1.265 1.171 0.158 0.146 0.094 0.0006 CD 1.242 1.194 0.155 0.149 0.048 0.0001 BE 1.279 1.157 0.160 0.145 0.122 0.0009 AE 1.194 1.242 0.149 0.155 0.018 0.0001 E 1.205 1.231 0.151 0.154 0.026 o.oooo

factor is ranked as:

SB > SA > SAxC = SBxE > SCxE

That means that B, A, AxC, BxE, CxE are the key factors. In other words, the laser cutting speed, laser power, interactions between laser power and temperature, cutting speed and rolling speed, temperature, and rolling speed are the key factors affecting cutting width. Especially, the divergence square sum of laser cutting speed is 36 times larger than that of the laser power (see Table 11). Hence, the cutting speed is the most significant factor.

The optimal experimental conditions can be determined from Table 11. For B, A, AxC, BxE and CxE, the cutting widths are narrower in the high level than that in the low level. But for C and D, the cases are reverse. Taking into account that the interactions of AxC, B xE and C xE are larger than that of C and D, the following conditions are the optimal experimental conditions:

Bz A2 (A x C>, (B x E)2 (C x El,

It could be simplified as:

B2 A2 C2 E2

Because of the little influence of the factor D, a lower conditions is chosen; the final optimal experimental conditions are:

A2 B2 C2 Dl E2

The mean value under the optimal experimental conditions is:

p=y+&p- y + a2 + bz + cz + ez = 0.098 kcj

This means that if the optimal experimental conditions are used, the laser cutting width will be 0.098 mm. It is great to get such an accuracy. In such a case, the optimal experimental conditions are: the laser power 30 W, the cutting speed 80 mm/s, the temperature 14O”C, the thermal pressure -0.1 mm (compression distance in the z direction), and the rolling speed 88 mm/s.

Five factors, four levels experimental results and analysis not considering interactions

The values of laser cutting width measured in the experiments are as follows: 164 Materials & Design Volume 17 Number 3 1996


yl = 0.230, y2 = 0.163, y3 = 0.138, y4 = 0.087, y5 = 0.273, y6 = 0.223, y7 = 0.142, ys = 0.108, Yg = 0.293, ylo = 0.222, yll = 0.190, y12 = 0.098,

~13 = 0.232, y14 = 0.263, yls = 0.130, y16 = 0.152.

Thus,

3 = gyi / 16 = 0.184 i=l

where yi is the value measured, and F is the mean value.The values of Mii, mu, R and S are shown in Table 12. Obviously, the ranking of S can be obtained from the above table is

s2 > Sl > s3 > s5 > s4

that is, the sequence of the important factors is B, A, C, E, D. This result is consistent with the case considering the interaction effect.

From Table 12, the values of cutting width of Ai, Bq. C2, Dz and El are the smallest under four levels. So, the optimal processing conditions are A1B4CZDZE1, i.e. the laser power 25 W, the cutting speed 80 mm/s, the temperature 13o”C, the thermal pressure 0.1 mm (compression distance in the z direction), and the rolling speed 40 mm/S.

By comparing the two kinds of analysis results, it is found that the sequences of important factors are the same. This means that the interaction effects should be taken into account. The results of considering interaction effects are to be used in the following calculation. The results of the orthogonal experimental analysis show that the possibility of process improvements is quite high.

Principle and analysis of the back-propagation neural network

Artificial neural network (ANN) is a kind of non-linear system that is composed of a large number of processing neurons. It possesses abilities of learning, memory, calculation and other intelligent processing functions, and can also simulate the information processing, storing and indexing abilities of a human brain at different degrees and levels. It is widely used in information processing, pattern recognition and function approximation.

At present, there are three kinds of methods in the ANN studies”. The fust is a network composed using physical models, which includes Hopfield network based on non- possibility neurons and Boltzmann machine (BM) model based on possibility neurons. The second is developed from

Table 12 Calculation results of five factors and four levels without considering interaction

Factors

Ml M2 M3 M4 ml

A B C D E

m2 m3

0.618 1.208 0.795 0.733 0.690 0.746 0.871 0.664 0.693 0.750 0.803 0.600 0.802 0.785 0.691 0.777 0.445 0.683 0.733 0.813 0.1545 0.2570 0.1988 0.1833 0.1725 0.1865 0.2178 0.1660 0.1733 0.1875 0.2008 0.1508 0.2005 0.1963 0.1728 0.1943 0.1113 0.1708 0.1833 0.2033 0.185 0.588 0.138 0.092 0.123 0.00505 0.05167 0.00396 0.00107 0.00257


the theory of auto-adaptive signal processing, and is best known as forward multi-layer neural network and back propagation algorithm (abbreviated as BP). The third is composed using self-organisation methods, which includes auto-adaptive resonance theory (ART) and self-organisation feature mirror network. From the system integrity point of view, the first and second are better than the third, but for simulating human recognition procedure and intelligent processing, the third one is better than the remaining two. From the maturity and application point of view, the second one is better than the remaining two. In this work, the second method, which is the BP network, is adopted.

Basic algorithm of BP network The learning algorithm” of a BP network is divided into two stages. In the first stage (forward propagation), the data is inputted, then the hidden layers process these data and calculate the actual values of the outputs. If the outputs of the final layer are not the expected ones, the second stage (back propagation) calculates the differences (errors) between the actual outputs and expected outputs for each layer, then adjust the weight coefficients until the outputs are satisfactory.

For a general case, assume that the network has L layers, an N-dimension input vector, and an M-dimension output vector, given P learning samples (+, yp),p = 1, 2, . . . , P. The S function is adopted to demonstrate the characteristics of the neurons. While the pth sample is the input, there is:

ne& = c w! of-’ I’ ‘P

Ojp = f(ietjp)

Ep = i C(Yip - jjp)’ I

(7)

(8)

(9)

where ~jp denotes the actual output of neuronj, and the total error is:

E=&kEp p=l

(10)

Define Sip = 3, then IP

aEP _ aEp anetjp _ aEp %--‘--

-. &etjp d W,:

o!-’ = 6” d-1 &etjp lp ip @ (11)

(1) If the neuron j is the output unit, then

Of = jp

(y! _ aEP *P .- Jp &. &zetjp

(12)

= -(rp - jpP) fYne$pL (: 1: 1 M-l (13) > > . . . 7

(2) If the neuron j is the hidden layer neuron, then

where qp is theapput for the next layer (1+1). If it is desirable to get &-, it is calculated back from the (1+1) layer. Generally sp&king,

i

= (16)

In the above equations, k is used to substitute for j so as to avoid confusion.

In short, the BP network algorithm is summarised as follows:

(1) Choose the initial values of weights. These could be generated by random or by given.

(2) Repeat the following procedures until the results are convergent.

a. From p = 1 to P

l For every neuron of each layer, calculate ($-‘and ne$p,jp,l = 2,3, . . . ,L. (the forward propagation procedure);

l For every neuron of each layer, calculate 6jp, 1 = L- 1, . . . ,2 (the back propagation procedure).

b. adjust weights by

wji = Wji - Pq

Input Data

Input Layer

Hidden Layer One or Multi-layer

output Layer

Output Data

Figure 3 Figure 3

Materials & Design Volume 17 Number 3 1996 Materials & Design Volume 17 Number 3 1996 166 165


Table 13 The weight coefficients of the BP network, Wi,

-0.362 -0.109 -0.589 -0.561

0.261 -0.343 0.166 0.207 0.016 -0.569

-0.203 -0.386 -0.361 0.511 -0.472 -0.286 -0.261 0.110 -0.115 0.556

0.265 -0.354 0.418 -0.452

-0.050 -0.645 0.026 -0.819 0.908 0.685 0.179 -0.643 0.320 0.349 0.449 -0.182

-0.582 0.122 -1.692 -0.670

-0.474 0.473 0.018 0.409 -0.454 -0.073 0.437 0.428 -0.054

-0.242 0.065 -0.423 0.266 0.176 0.079

-0.417 0.295 -0.115 0.115 -0.220 0.860 0.414 -0.438 -0.400

-0.137 0.091 0.651 0.468 -0.273 0.567

-0.039 -0.291 -0.761 0.375 -0.298 -1.260

-0.157 -0.123 -0.359 0.610 -0.126 -0.817

-0.292 -0.045 -0.211 0.135 0.648 0.549 0.090 -0.644 -0.129 0.252 0.398 -0.647 0.582 0.508 0.225

-0.727 -0.911 0.136

0.056 0.062

-0.039 -0.224 -0.429 -0.242 -0.176

0.013 0.316

-0.486 -0.342

0.259 0.073 0.810

-0.05 1 -0.008 -0.043

0.099 -0.069

dE 1 c paEp

l?Wji = iF p=l aWji

l Synthesis of two analytical methods to determine the sequence of key parameters affecting the accuracy and build qualitative and quantitative relationships between the accuracy and process parameters.

The orthogonal experimental design is an effective experimental design method, which could provide more experimental information with limited number of experiments. The analysis gives the ranking of the key factors, the optimal processing conditions and the mean value. But it cannot build quantitative relationships between the processing accuracy and processing parameters.

The advantage of the BP neural network is that it can build the quantitative relationships within a complex system. In the SSM process, it can build the quantitative relationships between the processing accuracy and processing parameters. The two methods can be complementary to each other. The integration of these two is, thereby, used for this study.

This method has been proved useful in the design and analysis of the in-plane processing accuracy for the SSM process. It is also applicable to other RP processes for experimental design and analysis.

where p is the adjusting coefficient in the above equation. References

The BP network model for analysis of in-plane processing accuracy For the in-plane processing accuracy experiment, the input vector has five elements: laser power, laser cutting speed, temperature, thermal pressure and rolling speed. The output vector has just one element, that is the cutting width. Following the former equations, a program has been written to solve this problem. As shown in Figure 3, the control variables and results are: 5 input variables; 1 output variable; 16 learning samples; the learning step n = 0.3; the weight adjusting coefficient CY = 0.2; the total error is 0.0001; the maximum individual error is 0.00005; the maximum repeating number is 3000; three hidden layers; and the number of each layer are 5, 8 and 4.

The result shows that when the cycle number reaches 223 1, the algorithm is convergent. The weight coefficients of the BP networks are shown in Table 13. There are a total of 119 weight coefficients. It is noted that in each hidden layer, one constant input is added so as to increase the convergent speed.

Conclusions A new integrated method for studying processing accuracy has been presented. This method has the combined advantages of the orthogonal experiment design and artificial neural network, and possesses the following characteristics:

l Minimum number of experiments (i.e. the lowest cost and shortest time spent) to obtain the necessary information for the experimental analysis.

l Limited experimental data to determine the optimal process conditions and provide the theoretical and experimental bases for process improvement.

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