Applied Soft Computing 1 (2001) 119–125

Prediction of mechanical properties of cast irons

Jan Voracek∗Department of Information Processing, Lappeenranta University of Technology,

P.O. Box 20, FIN 53851, Lappeenranta, Finland

Accepted 26 June 2001

Abstract

The adaptive hierarchical model reflecting behavior of different types of cast irons was designed and verified. It enables toevaluate an industrial metallurgical process with respect to its parameters, such as many possible combinations of chemicalcomposition or various approaches to heat treatment and, consequently to estimate mechanical properties of the final product.Because of the multivariate nature of the presented problem, it is impossible to make technological conclusions based onlyon formulae or diagrams. That is why an intelligent computational technique can help, for example, in the development ofnew metal materials or act as a powerful tool in total quality management systems.

In the proposed method, inductive learning and classification principle was selected and justified as suitable prediction tool.Thanks to the built-in adaptivity the suggested technique is more flexible than commonly used statistical methods and, incontrast to the numerical simulation of material characteristics, it can easily incorporate also historical or technology-specificvalues.

After the series of computational experiments the presented intelligent approach to prediction was found as an applicablealternative to the traditional ways of laboratory investigations and processing of experimental data. Its responses were verifiedon real samples and compared with experts’ opinion. © 2001 Elsevier Science B.V. All rights reserved.

Keywords: Prediction; Classification; Tree structure; Mechanical properties; Cast irons

1. Introduction

Mechanical properties of a material reveal the elas-tic and inelastic reaction when force is applied orinvolve the relationship between stress and strain,such as elasticity, tensile strength or fatigue limit. Inthe metal industry, they are typically derived from mi-crostructures of investigated materials [1–3]. It meansthat the precise metallographic sample must be cutand analyzed visually. If the image corresponds withthe expected template, then the behavior of the metal

∗ Tel.: +358-5-621-2811; fax: +358-5-621-2899.E-mail address: jan.voracek@lut.fi (J. Voracek).

is predictable. Although, a set of separate mechanicaltests is always realized on top of the microstructuralstudies, the microscopic observation is still consideredas the primary and the most informative method.

That is why, for example, a steel without a heattreatment is improper for manufacturing of rollsbecause its dendritic microstructure, coarse grain,internal non-homogeneities and stresses lead to un-satisfactory hardness. Another common property isdocumented in Fig. 1. For spheroidal graphite castirons the size, shape and number of black roundnodules of the graphite phase can affect the finalapplicability significantly. Typical microstructures ofbasic types of cast irons with different magnifications

1568-4946/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved.PII: S1568 -4946 (01 )00012 -6

120 J. Voracek / Applied Soft Computing 1 (2001) 119–125

Fig. 1. Microstructure of spheroidal graphite cast iron.

are in Figs. 1–3 and the structure-dependent behaviorcan be summarized as follows [4,5]:

• Grey iron◦ low toughness;◦ low tensile strength;◦ cheap;◦ easily castable.

• Spheroidal graphite cast iron◦ generally good mechanical properties;◦ expensive;◦ less castable;◦ less machinable.

Fig. 2. Microstructure of grey cast iron.

Fig. 3. Microstructure of malleable cast iron.

• Malleable iron◦ good mechanical properties;◦ difficult to cast;◦ expensive to heat treat.

Despite of its routine utilization, the outlined opto-analytic technique is rather slow and asynchronouswith the manufacturing process. Moreover, the finalmicrostructure does not tell all the details concerningits origination. Varying chemical composition, lateinoculation with a minor alloying element or thermaltreatment applied in different stages of solidificationprocess can lead to the similar structural expression.Modern prediction techniques are striving to excludedemanding and subjectively oriented microscopicmethods from the evaluation process and left themonly an auxiliary control function.

Because, the data describing the behavior of mate-rials is primarily obtained from a set of laboratorymeasurements, also statistical tools used for the de-sign and evaluation of experiments [6] are widelyused in the metallurgy. Results from hypothesis tests,ANOVA or regression analysis are either general-ized and used for inter- or extrapolation or takenas a ground for design of empirical formulae, dia-grams and coefficients. In such way, the natural

J. Voracek / Applied Soft Computing 1 (2001) 119–125 121

non-determinism of internal materials originating pro-cesses is incorporated and the ability of generalizationis preserved. On the other hand, traditional statisticallimitations such as the cardinality of evaluated dataor set of a priori presumptions about its distributionmust be still taken into account. Moreover, a commonincompatibility of experiments among single researchlaboratories world-wide (different types of analyzedsamples or various testing methodologies) disables toshare the raw data directly.

On the other hand, dynamic models of solidifi-cation can be used for enumeration of the desiredquantity and completed with experimental results[7–9]. Similarly, with other disciplines dealing withthe visualization of dynamic systems, various modifi-cations of the finite elements method (FEM) [10,11]are employed also in metallurgy [12]. Beyond thecomputational complexity of FEM, there are alsoother limitations related, such as the primary need fora vector drawing of the analyzed casting, the absenceof a feedback from the real metallurgical process and,consequently, the impossibility to learn from historicaldata.

Employment of soft computing techniques rep-resents a qualitatively different possibility. Insteadof a set of formulae, an appropriate algorithm witha suitable user interface could replace successfullydemanding mathematical methods. Archive data canbe used for creation of a proper internal structure andthis model can be verified with additional experimen-tal measurements. There are, however, only few suchapplications in the presented area [13–15].

Considering the prediction problem from compu-tational point of view, it represents interpolation taskin non-linear multidimensional space. Using the tra-ditional terminology from the theory of systems, thenthe quantities of interest can be interpreted in thefollowing way:

• Input variables: basic chemical composition andextra additives, various technological temperaturesand delays.

• State variables: metal structure, shape, size andlocation of graphite nodules.

• Output variables: selected mechanical characteri-stic(s).

If we consider, in accordance with the above givenpresumption, the set of state variables stable but

irrelevant, then the system behavior can be found asa direct input–output transformation.

2. Method

An inductively learning classifier [16] is a func-tion that maps an unlabelled instance to a label usinginternal data structures and a given data set. Suchfunctionality is in accordance with the acceptable taskformalization, suggested in Section 1. The only differ-ence is that the real-world domain is continuous andclassification outputs are discrete. Such disadvantagecan be, however, solved either with an appropriatequantification of the full range of output or incor-porating a regression into the concrete classificationalgorithm.

In the present study, industrial data with cardinal-ity, N was transformed into a set of feature vectors(patterns), xi and assigned to K corresponding classesdi such that every relation

xi → di

is unambiguous for distinct xi , i ∈ 〈1, N〉 and di ∈〈1, K〉. Mainly, the complete chemical compositionof a cast iron, transformation temperatures and delayswere used as components of feature vectors xi andthe categorized value of searched mechanical propertyrepresented the output class, di .

The two-stage algorithm was used to train theclassifier with available data. During the first, pre-learning stage, hierarchical clustering [17] of the sin-gle original pattern was realized. Depending on theexternal parameters near feature vectors were clus-tered into so called working classes (different fromreal classes, di). These classes were a priori declaredas linearly separable and ordered into a binary treestructure.

In the second learning stage, the linear separabilityfor all internal nodes was verified with the original setof patterns by adjusting the weights of single-layeredneural networks, located in these nodes. If both theworking classes for a particular tree node were reallylinearly separable, the weights of neural inputs wereadjusted properly and found as coefficients of the dis-criminating hyperplane.

The following algorithm gives a detailed descriptionof the above outlined process.

122 J. Voracek / Applied Soft Computing 1 (2001) 119–125

Beginning of algorithm

Stage 1: Partition training data into clusters.Step 1: Normalize each feature vector, xi based on

limit values from the training set.Step 2: Find the reference feature vector, xref

belonging to the real output class, dref and locatednearest to the origin of coordinates. Search for suchneighboring vectors, xn for which holds:

||xn − xref || ≤ SL ∧ dn = dref 0

where SL ∈ 〈0, 1〉 is the external variable expressingthe similarity level between patterns and dn the realoutput class for xn.

Step 3: Join xref with all acceptable xn and computethe centroid of this cluster. Then, every such cluster Cn

represents one working class and the correspondingcentroid its etalon. It is evident that for the number ofworking classes, kp holds:

kp = K for SL = 1, kp = N for SL = 0

Step 4: Remove all processed feature vectors, xi

from the original training set and repeat from Step 2until at least one feature vectors, xi is available forclustering.

Step 5: Create the distance table from etalons ofworking classes.

Step 6: Separate linearly two farthest etalons (e.g.with hyperplane perpendicular to and halving thejoin of the both etalons) and remove them from thetable.

Step 7: According to their belonging into one ofnewly originated half-hyperspaces (signs of a discrim-inating function) divide the set of remaining etalonsinto two parts. Such distribution corresponds witha particular estimated node of binary classificationtree.

Step 8: Apply recursively Step 7 to the both newlyoriginated hyperspaces until at least two unprocessedetalons are available. This results to the binary tree ofetalons for the given SL value.

Stage 2: Verify the validity of binary tree for singlepatterns xi

Step 9: Find the exact position of the linear discrimi-nation function starting from the root node using back-propagation algorithm, applied on single layer neuralnetwork (perceptrone) [18].

Step 10: Follow the error function:

E(x) =N∑

i=1

(di − f (vi, xi))2

where vi are weight vectors and f (vi, xi) the actualoutput of perceptrone. If it does not fall smoothly,decrease SL value and continue from Step 2, otherwiseselect the next node in tree hierarchy and continuefrom Step 9.

After the minimization of E(x) weights vi assignedto the corresponding inputs xi represent coefficients ofsearched hyperplane. Note that its validity is limitedwith a particular position in tree hierarchy and globalboundaries from Step 1 as well.

Step 11: Replace working classes with real classesof clustered patterns, i.e. join single hyperplanesand realize piecewise linear discrimination function(boundary).

Step 12: Prune the final tree in such a way that neigh-boring terminal nodes (different working classes) withcorresponding real output class, dk are removed fromthe tree and their common superior node is replacedwith the new terminal, belonging to the same class, dk .

Step 13: Repeat Step 12, until at least one pair ofactual terminals suitable for pruning exists.

End of algorithm

In the working stage, tree structure with knownweights of internal nodes was loaded, every testingsample was let down through and according to thesign of discrimination function (output of a particularperceptrone) turned in every internal node to an ap-propriate direction, until the leaf node, expressing thedesired real class, di was reached.

Advantageous properties of the above algorithm canbe summarized as follows:

1. Robust behavior is achieved by the combiningunsupervised and supervised training.

2. Absolutely convergent learning with no risk of alocal minimum deadlock is guaranteed with theadaptive adjustment of SL parameter. From Step 3it is evident that for SL = 0 every becomes a singleworking class and must be necessarily separatedfrom the others.

3. There are no a priori presumptions concerningprocessed data because of deterministic characterof inductive learners.

J. Voracek / Applied Soft Computing 1 (2001) 119–125 123

4. Backtracking of suspicious results is possible sincethe readable binary tree structure and particularequations of separating hyperplanes are available.Intersecting parts of feature space can be easily vi-sualized. Also, the detailed interactive analysis ofthe problem and possible alternative solutions areoffered by the user interface.

5. System is capable to process data sets with arbitrarycardinality and dimensionality.

3. Experiments

Data sets for single tasks were divided into thetraining set and artificial testing set with the ratioabout 80:20 in order to preliminarily verify the suit-ability of an investigated classification model. Apreliminary classifier was built based on the trainingset in accordance with the above proposed algorithm.Moreover, the following rules were applied duringthe learning stage:

• Minimal amount of training patterns for one expe-riment was 1000.

• Amount of training patterns in single classes canvary ±5% at worst.

• Outlying and redundant values were not considered.

If the artificial testing set was classified with theaccuracy lower than 65%, then the particular modelwas rejected from the next experiments. For all theremaining, properly formulated problems, hitherto un-known laboratory or industrial data was used as a realtesting set (for details, see Tables 2 and 4). Althoughsuch conditions represent an extensive loss of infor-mation, they give at least a realistic estimation of thesystem’s behavior. Two tasks from the set of successfulexperiments are described in the following sections.

3.1. Task 1. bainitic spheroidal graphite cast iron

This modern material [19–21] has higher strength,hardness and wear resistance than standard spheroidalgraphite cast iron. There are several ways how toproduce bainitic microstructure based mainly on acontrolled heat treatment and cooling down to roomtemperature.

Presented data was taken from laboratory experi-ments described in [22], where the isotropic austenite

Table 1Complete set of input variables used for the prediction of A5 ofbainitic spheroidal graphite cast iron

Input variables for ductility (A5) prediction

Name Range

Minimum Maximum

Transformation delay tt (s) 180 2.104

Transformation temperature Tt (◦C) 300 400Carbon (%) 3.25 3.50Silicon (%) 2.43 2.64Manganese (%) 0.25 0.28Nickel (%) 0.00 3.82Phosphorus (%) 0.03 0.06Sulphur (%) 0.01 0.10Magnesium (%) 0.05 0.62

→ bainite transformation was realized. Cast iron spec-imen was warmed up to the austeniting temperature,Ta cooled down to the transformation temperature, Ttkept there for a certain transformation time, tt and,finally, cooled down to room temperature. Then, theductility, A5 of the final material was analyzed as afunction of transformation temperature, transforma-tion delay and chemical composition with constantaustenitizing temperature, i.e.

A5 = A5(Tt, tt, C)Ta

Output variable A5 was discretized into four classes.The ranges of single inputs are shown in Table 1,cardinalities of related data sets and the overall accu-racy can be found in Table 2.

3.2. Task 2: grey cast iron

Different types of commercial castings realizedwith the same technology in the same foundry wereanalyzed. Hardness (HB) was specified as the mostimportant customer’s parameter and measured on pre-defined parts of castings using Brinell method. In this

Table 2Cardinalities of available data sets and overall classification accu-racy.

Cardinalities of processed data sets Classificationaccuracy for A5 (%)Training set Testing set

2358 712 87.5

124 J. Voracek / Applied Soft Computing 1 (2001) 119–125

Table 3Complete set of input variables used for prediction of HB of theindustrially manufactured grey cast iron

Input variables for HB prediction

Name Range

Minimum Maximum

Casting type 1 6Carbon (%) 3.20 3.62Silicon (%) 1.51 2.43Manganese (%) 0.33 0.70Phosphorus (%) 0.10 0.42Sulfur (%) 0.07 0.11Chromium (%) 0.04 0.18Copper (%) 0.02 0.11Tin (%) 0.00 0.08Eutecticity 0.90 1.02Carbon equivalent (%) 3.91 4.38Antimony (inoculation) (%) 0 1

Table 4Cardinalities of available data sets and overall classification accu-racy.

Cardinalities of processed data sets Classificationaccuracy for HB (%)Training set Testing set

6732 2013 92.3

case, hardness as a function of basic and extendedchemical composition, i.e.

HB = HB(C)technology

Output variable HB was discretized into 11 classes.Data similar with the previous experiment are listedin Tables 3 and 4.

4. Discussion and conclusions

The main computational problem in Task 1consisted in the wide range of the transformationdelay, which resulted to the long learning time andlower classification accuracy. Such an extensive delaysare, however, well founded only in research-orientedexperiments. In the practical applications consider-ably shorter time periods are typical and so a higherprediction precision can be expected there.

Very promising results were obtained for Task 2.Detailed inspection of the classification structure

eliminated the possible risk concerning a misleadingrole of the both discrete features (categorical type ofcasting and binary inoculation with antimony) andconcluded that the hierarchy of the model expressedthe task in a natural way. Because all measurementswere realized in a stable technological process it alsomeans that the proposed method is directly applicableto the industrial environment, where larger series ofcastings are manufactured and an adequate amount ofarchive data is available.

Our method for prediction of mechanical propertiesof cast irons directly from input parameters insteadof traditional numerical simulation or statistical gen-eralization was formulated and realized. Althoughavailable data did not cover the whole feature spaceequally, obtained results are applicable and docu-ment that the analyzed prediction problem is suitablefor deterministic interpolation using soft computingtechniques. From the both described experiments itis evident that the overall accuracy obtained for theunknown data is high. This conclusion fully supportsthe initial hypothesis concerning the prospective ex-clusion of the metallographic stage from the routineprediction of mechanical properties of cast irons.

In the future research, functional regression in leafnodes of the decision tree shell replace current dis-cretization. Next important task is to find a suitablemethod for self-tuning of classifier parameters. Also,fully automated data acquisition is sill a subject of ourintensive interest.

Acknowledgements

This research was realized as a part of INTASProject No. 397, Data Mining Technologies andImage Processing: Theory and Applications.

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