Modeling of CVI process in fabrication of carbon/carbon composites by an artificial neural network

Vol. 46 No.2SCIENCE IN CHINA (Series E)April 2003Modeling of CVI process in fabrication of carbon/carboncomposites by an a rtificial neural networkLI Aijun (李爱军), LI Hejun (李贺军), LI Kezhi (李克智)& GU Zhengbing (顾正彬)Superhigh Temperature Composites Key Laboratory , Northwestern Polytechnical University, Xi an 710072, ChinaCorrespondence should be addressed to Li Hejun (email: lihejun@nwpu.edu.cn)Received May 8, 2002Abstract The chemical vapor infiltration(CVI) process in fabrication of carbon-carbon compositesis very complex and highly inefficient, which adds considerably to the cost of fabrication and limitsthe application of the material. This paper tries to use a supervised artificial neural network (ANN) tomodel the nonlinear relationship between parameters of isothermal CVI(ICVI) processes andphysical properties of C/C composites. A model for preprocessing dataset and selecting itstopology is developed using the Levenberg-Marquardt training algorithm and trained withcomprehensive dataset of tubal C/C components collected from experimental data and abundantsimulated data obtained by the finite element method. A basic repository on the domain knowledgeof CVI processes is established via sufficient data mining by the network. With the help of therepository stored in the trained network, not only the time-dependent effects of parameters in CVIprocesses but also their coupling effects can be analyzed and predicted. The results show that theANN system is effective and successful for optimizing CVI processes in fabrication of C/Ccomposites.Keywords: CIC composites, ICVI process, artificial neural network, Levenberg-Marquard algorithm, finite elementmethod.C/C composites maintain unique intensity features at temperatures as high as 2600C. Becau-se of their superior thermal and mechanical properties, which can persist at ultra-high temper a-tures, CIC composites are used in many areas including national defense, aviation aerospace, etc.At present, high performance C/C composites are mainly fabricated by using CVI processes 1.2.But it should be pointed out that the CVI process is controlled by many factors such as the infil-tration temperature, the pressure in furnace and the volume ratio between propylene and nitrogen.The process has only been studied empirically by trial and-error method so far. For this reason, itis important and indispensable to simulate the procedures of CVI processes by numeric methodsin order to optimize them'3]. Most classical numeric methods, finite element method (FEM) andfinite difference method (FDM) have already been employed in this fieldl4.51. Unfortunately nowa-days it is still difficult to develop an accurate mathema中国煤化工or deposition ofpyrocarbon.THCNMHGAs a kind of data mining and artificial intelligence Tecnniques, neural networks are developedto model the way the human brain processes information. A neural network is a massively paral-SCIENCE IN CHINA (Series E)Vol.46lelly distributed processor that has a neural propensity for storing experiential knowledge andmaking it available for future use. Unlike conventional, explicitly programmed computer pro-grams, neural networks are trained by using previous example data and then iteratively adjustingthe weights of the neurons until the output for a specific network is close to the desired output.Furthermore, neural networks possess many excellent properties such as outstanding nonlinearapproximation, auto-adaptation and association capability. As a complex nonlinear system, NNmodels have been widely employed to map the indeterminate relationship between cause and f-fect variables in many fields(6.7]. In the present work a universal ANN program is designed on thebasis of improved BP training algorithms. Using this program, a three-hidden-layer network isconstructed to simulate ICVI processes in fabrication of C/C composites. It is trained with com-prehensive data including simulated data by FEM. The trained network can acquire useful know -edge and rules implicated in the dataset successfully, which helps to further comprehend CVIprocesses and then to optimize them. Moreover the knowledge is beneficial for deriving moreaccurate mathematical and physical models in this field.1 Design of BP training algorithm and network architecture1.1 Improvement on BP training algorithmsIn this paper, an error back- propagation (BP) network is selected because of its greater capa-bility of association and generalization. Basically, the error BP process consists of two passesthrough the different layers of the network: a forward pass for training data and a backward passfor error data. The weights of the neurons are iteratively adjusted in accordance with the errorcorrection rule until the output for a specific network is close to the desired output. The classicalerror correction rule is the steepest descent algorithm, but the method suffers from the drawbackof rapidly reducing rate of convergence near the extremum points of the objective function; whileto the second order algorithms such as Gauss-Newton algorithm, the rate of convergence is re-duced rapidly far away from the extremum points of the objective functionl8]. The method used inthis study is the Levenberg-Marquardt algorithm (fig. 1) which is a kind of quasi-Newton methods.The weights of the neurons are iteratively adjusted bywW(n+1)=w(n)- (J"J+ 21)-'8n(1)where 8n三g/2, g is the gradient of the error function E with respect to the weight and bias vari-ables w; JT is the transposed matrix of J; I is the identity matrix which has the same dimensionswith JI;is a adjustable constant multiplier and when it is down to zero, formula (1) is just ap-proximate Newton' s method; when is large, it becomes the steepest descent algorithm with asmall step size (fig. 2). Newton' s method is faster and more accurate near an error minimum, sothe aim is to shift towards Newton' s method as quickYH中国煤化工decresed aftereach successful step (reduction in performance functiocNMHGwhenatentative.step increases the performance function!9!.No.2MODELING OF CVI PROCESS IN FABRICATION OF CARBONCARBON COMPOSITES175+@“团" +O*团长\ w"+②-+<+0团，> +@"0爷+Q-国NAw二-(J TJ+λI"'J'esw=J[J+1'J}e sw=-(JJ+A,Ir'J{e=些Low」[图]Fig. 1. The model of the Levenberg - Marquardt algorithm.The training error is usually reduced without a cor-Contour lineresponding increase in the generalization performanceof the networks during the whole training phase. Some-times the validating error is increased abruptly whenthe training error is reduced gradually. In order to over-come this drawback and save training time, effort isThg direction ofGauss-Newtonmade to improve the generalization by means of“earlystopping with validation" in this paper.1.2 Training samples and input/output parametersFig.2. The descent directions of three algorithms.The knowledge of a specific field is implicated inthe existing training samples, so an appropriate dataset with good distribution is significant forreliable training and performance of neural networks. To ensure reasonable distribution andenough information of the dataset, different parameters of ICVI processes are covered with differ-ent levels and sampled using an uneven orthogonal method for reference. As we know, mostparameters of ICVI processes have time-dependent effects. So the parameter--the infiltrationtime-- should be sampled at different intervals and more levels than other parameters: () at thebeginning of CVI process intervals are short; (i) subsequently they become long lttle by lttle.The selection of input/output variables is a very important aspect of neural network modeling,and is usually based on the physical background of a process and the existing dataset. Generallymore dimensions of the input variables require more samples for reliable performance of neuralnetworks, making it much more difficult to train the n中国煤化工fore it is benefi-cial to preprocess training samples by ignoring some:THCN M H Gables and elimi-nating those redundant data. In the present work the inpul parameters Incluae ne infiltration tem-perature Te, the pressure in furnace P, the volume ratio of propylene to nitrogen R、and the infil-176SCIENCE IN CHINA (Series E)Vol.46tration time T, andRatio =VcH。P=PcH6 +PR, + Pesidua,(2)(Vc,H6 +Vx)’where V is the volume of gases; Presidual is the partial pressure of the residual gases in the furnace.Because nitrogen functions as a carrier gas in pyrolysis reactions and p keeps a very small value inthe CVI processes, we havePcH6 =愿a，RH。 =k2.P. Rati,(3)where k| and k are constants; P感H。 represents the approximate partial pressure of propylenewhich proves to be of importance in CVI processes. After preprocessing the input variables inaccordance with formulae (2) and (3), the dimensions of the input variables, as well as necessarysamples are considerably reduced..3 Design of hidden layers and neuronsHidden layers perform abstract functions; that is, they can extract characteristic knowledgeimplicated in input data. So it is the hidden layers that enable neural networks to deal robustlywith nonlinear and complex problems. In 1988 Cybenko pointed out that on the one hand, two-hidden-layer neural networks are accurate enough to approximate any output function of any inputfigure when transfer functions of neurons are sigmoid; on the other hand, the generalization andthe capability of dealing with problems of the networks would be enhanced if the hidden layersare increased in number properly. Certainly, the training procedure turns out to be much morecomplicated and the training time is also increased. In 1989 Robert Hecht-Nielson proved that asingle-hidden-layer neural network could approximate any continuous function in a closed interval.In other words, a three-layer neural network can perfectly map the relationship between n-dimen-sion input variables and m-dimension output variables.However different algorithms of BP networks have different limitations in practice. For in-stance, it is difficult for a single-hidden-layer network to improve its closeness-of-fit if it has toofew hidden nodes; while too many hidden nodes enable it to memorize (over-fit) the training da-taset, which produces poor generalization performance. At present there is not any valid analysisformula for designing hidden layers and“it is an art to decide the quantity of nodes per hiddenlayer", so a trade-off exists between generalization performance and the complexity of trainingprocedure when designing the topology of an ANN.In this paper a lot of computational instances show that: denoting the dimension of in-put/output layers and the quantity of samples by N (not too great), M and K respectively, we haveN=N.中国煤化工(4)According to the conditions under which specific inco.MHC N M H Ghe least squaressolution, a formula for two-hidden-layer networks can be given as follows:N.N; +N, +N:.N2 +N2+N: M+M≤KNo.2MODELING OF CVI PROCESS IN FABRICATION OF CARBONCARBON COMPOSITES177→N2≤(K-M- N- N2)/(M +N),(5)where N; and N2 are the quantities of nodes in the first and the second hidden layer respectively.Usually suitable N2 ensures that both the generalization performance and the rate of the conver-gence are satisfactory. Sometimes it cannot be made to work perfectly by only adjusting N2. Con-sequently it is considered to add another hidden layer N3 that should also accord with some condi-tions like formula (5).2 Modeling of the CVI processThree main aspects, i.e. the transportation of gaseous hydrocarbon, the course of pyrolysisreactions and the exhaust of residual gases have impact on the densification course in CVI proc-esses. As is well known, the infiltration temperature is so important that it has direct influence onnot only the competitive relation between the first two aspects but also the matrix structure ofpyrolytic carbon. The quantity of propylene and residual gases can be adjusted by controlling Pand R, to change the competitive relationship between the two latter aspects. In the present workfour parameters一- the infiltration temperature, the pressures in furnace, the volume ratio of pro-pylene and nitrogen and the infiltration time一are treated in terms of formulae (2) and (3). Lev-els of these input variables are designed as shown in tables1 and 2.Table 1 Factors and levels before preprocessingT,/h050100200300400T。C80090011001200PY 10-1 atm .0.20.30.40.5R0.).50.6Table 2 Factors and levels after preprocessingT/h206(TJT1000甩H6 /10-atm1.21.62.02.5.0In this study we assume that the mean density ρ and mean porosity ε of CIC workpieces arethe output variables (i.e. M = 2). Samples of tubal CIC workpieces are collected corresponding totable 2 (i.e. N = 3) with the help of the FEM program for ICVI processes written by Lil4]. As weknow, the infiltration course of gas phases will stop once the pores on the surface of C/C areclosed unless those closed pores are opened in time by mechanical treatment methods. In order toensure better representativeness of these samples, the data XYend besides the grid data sampledaccording to table 2 are also collected as soon as the surface pores are closed. Sequentially thequantity of training samples is up to 259 (i.e. K = 259). Let there be three hidden layers in the NN中国煤化工model of the CVI process. Then, from (4) and (5) we haN=3,YHCNMHG(6)4N2+ (N 2+ 3)Nz≤245.Set the sum squared error (SSE) criterion at 1.8e-3. A perfect topology ({3,3,15,2,2}) of the178SCIENCE IN CHINA (Series E)Vol.46model is found (i.e. N2= 15 and Nz = 2) after many times of trial-and-error computation by theANN program. The network with 259 training samples is trained successfully on a computer withtwo PII 866 CPUs after 939 epochs. The training procedure is shown in fig. 3. To test the gener-alization performance of the trained network, 46 samples different from the training ones are alsocollected in a similar way. The test result is shown in fig. 4.0.9.10Al 939 epuths.8-Performance 0.001 79984Goal is 0.0018).7 t1goal. validationtraining1.40~210~30!1002003004005006007008009000.2102030s0EpochSampleFig. 3. The training procedure of the NN model.Fig.4. The result of validating the generalization. 1, Desiredρ;2, desiredε ;3, predictedp ;4, predicated P .As shown in fig.3, the training SSE is always reduced in the training procedure, but the vali-dating SSE is inconsistent; the training error and the validating error are hardly changed after theepoch is up to 500 times. So the training process can be terminated at 500 epochs, thus savingtraining time at very lttle cost of increasing the system error. A very good agreement between thepredicted values from the trained neural network and the validating data is achieved (fig. 4), whichindicates that the trained network has optimal generalization performance. This also demonstratesthat, as a typical data mining technique, neural networks can find the basic pattern informationcontained in a great number of experimental data, extract useful rules and then use these rules toobtain reasonable predicted results.3 The repository of tubal CIC workpiecesAfter neural networks are trained successfully, all domain knowledge extracted out from theexisting samples is stored in digital forms in weights associated with each connection betweenneurons. Fig. 5 is a Hinton figure一- an easy method to describe the weights. The area of asquare in Hinton figures is used to scale relative size of a weight; a white square represents a plusweight, while a black one represents a negative weight: a square in the same column with“0”represents a bias of a neuron in the former layer. MakirH中国煤化Inowledge storedin the trained network, three- dimensional graphs are draCNMH Gns (fgs.6-10).Obviously those graphs exhibit much more professional knowledge than Hinton figures.No.2MODELING OF CVI PROCESS IN FABRICATION OF CARBONCARBON COMPOSITES1791.8-1.6.121.0 10.8. ,0.6.400”12003000010000200 100T0800.Neurons of the third hidden layerFig. 5. The Hinton figure of the weights between the thirdFig. 6. The curved surface of P (g/cm) with regard toT。(C)hidden layer and the output layer.andT; (h)(电H。=0.0144 atmn).3.1 The time-dependent effects of parameters on CVI processesCVI processes are greatly affected by the infiltration temperature as shown in the fig. 6. .Whether the temperature is high or low, all CVI processes have a similarity: the rate of densific a-tion is great and uniform in all processes at the beginning of processes and then the rate is reduceddue to the pyrogenation of propylene; the mean density of workpieces is up to around 1.0 g/cm'after 18- -20 h deposition . The difference is that the rate is reduced rapidly and then keeps stableafter early rapid deposition if the temperature is low; at high temperatures the rate decreases astime goes on but the mean rate is greater while the valid infiltration time is reduced because mostsurface pores of preforms are rapidly closed earlier.As shown in figs. 7(T。= 980C) and8 (T。= 840C), the higher the approxi-mate partial pressure of propylene, the1.8.greater the rate of the densification andthe shorter the infiltration time. It can be1.4.explained as follows: as P and R、are“1.2-raised (see formulae (2) and (3)), a much1.0more propylene can infiltrate into the0.8.inside of carbon-fibred performs, which0250.030speeds up the pyrolysis reaction of unit2001000 0.005 "00010.0volume and makes most pores on the sur-face closed earlier.Fig. 7. The curved surface of D(g/cm') with regard to臣H6 (atm)中国煤化工3.2 Coupling effects of parameters orand I, (h)(T。fYHCNMHGCVI processesFigs. 6 and 7 help further understand time-dependent effects of the partial pressure of propyl-180SCIENCE IN CHINA (Series E)Vol.46ene and the infiltration temperature 0CVI processes. Using the knowledgestored in the trained network, figs. 9 (T =0.5-380 h) and 10 T;= 160 h) are drawn in0.4。order to analyze their coupling effects1.3-intuitively.0.2.As is evident in the two figures, the).1.CVI processes with higher temperature1000m0.0are more sensitive to the partial pressure200020200.015'of propylene than those with lower tem-400 0.030perature; the CVI processes with higherpressure are more sensitive to the changein the infiltration temperature than theFig. 8. The curved surface of cwith regard to琶H6 (atm) and T (b)ones with lower pressure. In the CVI pro-(T。= 840C).cesses, the density of C/C workpieces is almost increased linearly from 1.4 to around 1.6 g/cm asthe partial pressure of propylene is changed from 0.01 to 0.03 atm when infiltration temperature is800C, but when it is 1000C the density is increased in sigmoid (fig. 9). In the CVI processes,the mean porosity is almost reduced linearly from 0.3 to 0.1 or so, as the infiltration temperature ischanged from 800 to 1000C when partial pressure of propylene is 0.0latm, but when it is 0.03atm, the mean porosity is also reduced in sigmoid (fig. 10). It is clear that the infiltration tem-perature has the same effect on CVI processes as the partial pressure of propylene. So how to ad-just the two factors is the key to optimization of the processes.210.05 0.10 0.150.20 025 0.301.9-The grcy scaleof c0.030-1.8-1.0.025-1.6.Cuntour linesof 01.40.015-12000.011101000 9008000.030.020.010-0.00580090010001100Fig.9. The curved surface ofo(g/cm') with regard toT，民出6 (atm) andT。(C) Ti =380h).Fig.中国煤化工rd 10电s。(a)4 ConclusionsandT。(MYHCNM HGA neural network model of CVI processes has been built for the first time in this paper. Highprecision of the model and good generalization performance are demonstrated.No.2MODELING OF CVI PROCESS IN FABRICATION OF CARBONCARBON COMPOSITES181(1) A new token variable感H is obtained by preprocessing P and R v(2) The basic database of the ICVI process for tubal CIC workpieces is established.(3) The repository on ICVI processes comes into being by knowledge acquirement of the NNmodel; it is demonstrated that T。has similar effect on CVI processes with比H。 , which points outthe optimum direction to optimize CVI processes.Acknowledgements The work was supported by the National Natural Science Foundation of China (Grant No.50072019) and the Aeronautical Foundation of China under Grant No.99G53092.References1. Meyer, R. 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