Geodesic
Control of substance solidification in a complex-geometry mold
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 52 (2012) no. 12, pp. 2149-2162
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The control of metal solidification in a mold of complex geometry is studied. The underlying mathematical model is based on a three-dimensional two-phase initial-boundary value problem of the Stefan type. The mathematical formulation of the optimal control problem for the solidification process is presented. This problem was solved numerically using gradient optimization methods. The gradient of the cost function was computed by applying the fast automatic differentiation technique, which yields the exact value of the cost function gradient for the chosen discrete version of the optimal control problem. The results of the study are described and analyzed. Some of the results are illustrated as plots. 
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A. V. Albu; A. F. Albu; V. I. Zubov. Control of substance solidification in a complex-geometry mold. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 52 (2012) no. 12, pp. 2149-2162. http://geodesic.mathdoc.fr/item/ZVMMF_2012_52_12_a3/

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[3] Albu A. V., Albu A. F., Zubov V. I., “Vychislenie gradienta funktsionala v odnoi zadache optimalnogo upravleniya slozhnoi dinamicheskoi sistemoi”, Zh. vychisl. matem. i matem. fiz., 51:5 (2011), 814–833 | MR | Zbl

[4] Evtushenko Y. G., “Computation of Exact Gradients in Distributed Dynamic Systems”, Optimizat. Meth. and Software, 1998, no. 9, 45–75 | DOI | MR | Zbl

[5] Albu A. F., Zubov V. I., “Vychislenie gradienta funktsionala v odnoi zadache optimalnogo upravleniya, svyazannoi s kristallizatsiei metalla”, Zh. vychisl. matem. i matem. fiz., 49:1 (2009), 51–75 | MR | Zbl