Application of the immune algorithm IRM for solving the inverse problem of metal alloy solidification including the shrinkage phenomenon

Application of the immune algorithm IRM for solving the inverse problem of metal alloy solidification including the shrinkage phenomenon

Adam Zielonka, Edyta Hetmaniok, Damian Słota

Institute of Mathematics, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland.



In the paper the mathematical model of the inverse one-dimensional problem of binary alloy solidification, with the material shrinkage phenomenon taken into account, is defined. The process is described by using the model of solidification in the temperature interval, whereas the shrinkage of material is modeled basing on the mass balance equation. The inverse problem consists in reconstruction of the heat transfer coefficient on the boundary of the casting mould separating the cast from the environment. Lack of this data is compensated by the measurements of temperature in the control point located inside the mould. The method of solving the investigated problem is based on two procedures: the implicit scheme of finite difference method supplemented by the procedure of correcting the field of temperature in the vicinity of liquidus and solidus curves and the immune optimization algorithm IRM.

Cite as:

Zielonka, A., Hetmaniok, E., Słota, D. (2018). Application of the immune algorithm IRM for solving the inverse problem of metal alloy solidification including the shrinkage phenomenon. Computer Methods in Materials Science, 18(1), 1 – 10.

Article (PDF):


Solidification, Binary Alloy, Material shrinkage, Immune algorithm


Beck, J.V., Blackwell, B., 1988, Inverse Problems.Handbook of Numerical Heat Transfer, WileyIntersc., New York.

Bersini, H., Varela, F., 1991, The Immune RecruitmentMechanism: a selective evolutionary strategy, R.Belew, L. Booker, eds., Proceedings of the 4thInternational Conference on Genetic Algorithms,Morgan Kaufman, San Mateo, 520-526.

Cheung, N., Santos, N.S., Quaresma, J.M.V., G.S.Dulikravich, G.S., Garcia, A., 2009, Interfacial heattransfer coefficients and solidification of an aluminumalloy in a rotary continuous caster, Int. J. Heat MassTransfer, 52, 1-2, 451-459.

Hetmaniok, E., Nowak, I., Słota, D., Zielonka, A., 2012,Determination of optimal parameters for the immunealgorithm used for solving inverse heat conductionproblems with and without a phase change,Numerical Heat Transfer B, 62, 462-478.

Hetmaniok, E., Słota, D., Zielonka, A., 2017, Solution ofthe direct alloy solidification problem including thephenomenon of material shrinkage, Thermal Science,21, 1A, 105-115.

Hojny, M., Głowacki, M., 2009, The methodology ofstrain-stress curves determination for steel in semisolidstate, Arch. Metall. Mater., 54, 475-483.

Majchrzak, E., Mochnacki, B., 1995, Application of theBEM in the thermal theory of foundry, Eng. Anal.Bound. Elem., 16, 2, 99-121.

Mochnacki, B., Suchy, J.S., 1995, Numerical Methods inComputations of Foundry Processes, PFTA, Cracow,Poland.

Nawrat, A., Skorek, J., Sachajdak, A., 2009, Identificationof the heat fluxes and thermal resistance on the ingotmouldsurface in continuous casting of metals,Inverse Probl. Sci. Eng., 17, 3, 399-409.

Matlak, J., Słota, D., 2015, Solution of the pure metalssolidification problem by involving the materialshrinkage and the air-gap between material and mold,Arch. Foundry Eng., 15, 47-52.

O’Mahoney, D., Browne, D.J., 2000, Use of experimentand an inverse method to study interface heat transferduring solidification in the investment castingprocess, Exp. Therm. Fluid Sci., 22, 3-4, 111-122.

Piekarska, W., Kubiak, M., Bokota, A., 2011, Numericalsimulation of thermal phenomena and phasetransformations in laser-arc hybrid welded joint,Arch. Metall. ater., 56, 409-421.

Ryfa, A., Bialecki, R., 2011, Retrieving the heat transfercoefficient for jet impingement from transienttemperature meaurements, Int. J. Heat Fluid Flow, 32,1024-1035.

Sczygiol, A., Dyja, R., 2007, Evaluating the influence ofselectedparameters on sensitivity of a numericalmodel of solidification, Archives of FoundryEngineering, 7(4), 159-164.

Shestakov, N.I., Lukanin, Y.U.V., Kostin, Y.U.P., 1994,Heat exchange regularities in a crystallizer, Izv.V.U.Z. Chernaya Metall., 1, 22-23.

Sowa, L., Bokota, A., 2007, Numerical modeling ofthermal and fluid flow phenomena in the mouldchannel, Archives of Foundry Engineering, 7(4), 165-168.

Szeliga, D., Gaweda, J., Pietrzyk, M., 2004, Parametersidentification of material models based on the inverseanalysis, Int. J. Appl. Math. Comput. Sci., 14, 549-556.

Talar, J., Szeliga, D., Pietrzyk, M., 2002, Application ofgenetic algorithm for identification of rheological andfriction parameters in copper deformation process,Arch. Metallurgy, 47, 27-41.

Telejko, T., Malinowski, Z., 2004, Application of aninverse solution to the thermal conductivityidentification using the finite element method, J. Mater. Process. Technol., 146 (2), 145-155.

Zielonka, A., Hetmaniok, E., Słota, D., 2017, Inverse alloy solidification problem including the material shrinkage phenomenon solved by using the bee algorithm, Int. Comm. Heat Mass Transf., 87, 295-301.