The aim of this work is to present a computationally efficient algorithm to simulate the deformations suffered by a viscoplastic body in a solidification process. This type of problems involves a nonlinearity due to the considered thermo-elastic-viscoplastic law. In our previous papers, this difficulty has been solved by means of a duality method, known as Bermúdez-Moreno algorithm, involving a multiplier which was computed with a fixed point algorithm or a Newton method. In this paper, we will improve the former algorithms by means of a generalized duality method with variable parameters and we will present numerical results showing the applicability of the resultant algorithm to solidification processes. Furthermore, we will describe a numerical procedure to choose a constant parameter for the Bermúdez-Moreno algorithm which gives good results when it is applied to solidification processes.

Classification: 74C10, 74D10, 65N30

Keywords: viscoplastic materials, duality methods, solidification process

@article{M2AN_2014__48_1_87_0, author = {Barral, P. and Quintela, P. and S\'anchez, M. T.}, title = {A Berm\'udez-Moreno algorithm adapted to solve a viscoplastic problem in alloy solidification processes}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique}, publisher = {EDP-Sciences}, volume = {48}, number = {1}, year = {2014}, pages = {87-106}, doi = {10.1051/m2an/2013095}, zbl = {1286.74025}, mrnumber = {3177838}, language = {en}, url = {http://www.numdam.org/item/M2AN_2014__48_1_87_0} }

Barral, P.; Quintela, P.; Sánchez, M. T. A Bermúdez-Moreno algorithm adapted to solve a viscoplastic problem in alloy solidification processes. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 1, pp. 87-106. doi : 10.1051/m2an/2013095. http://www.numdam.org/item/M2AN_2014__48_1_87_0/

[1] Numerical solution of a 1-d elastohydrodynamic problem in magnetic storage devices. ESAIM: M2AN 42 (2008) 645-665. | Numdam | MR 2437777 | Zbl 1203.76052

, , and ,[2] Numerical simulation of some problems related to aluminium casting. J. Mater. Process. Technol. 142 (2003) 383-399.

, , , , and ,[3] A numerical algorithm for a Signorini problem associated with Maxwell-Norton materials by using generalized Newton's methods. Comput. Methods Appl. Mech. Engrg. 195 (2006) 880-904. | MR 2195293 | Zbl 1115.74046

, , and ,[4] A numerical algorithm for prediction of thermomechanical deformation during the casting of aluminium alloy ingots. Finite Elem. Anal. Des. 34 (2000) 125-143. | Zbl 1023.74042

and ,[5] Asymptotic justification of the treatment of a metallostatic pressure type boundary condition in an aluminium casting. Math. Models Methods Appl. Sci. 11 (2001) 951-977. | MR 1850558 | Zbl 1205.74040

and ,[6] Duality methods for solving variational inequalities. Comput. Math. Appl. 7 (1981) 43-58. | MR 593554 | Zbl 0456.65036

and ,[7] Numerical solution of a three-dimensional solidification problem in aluminium casting. Finite Elem. Anal. Des. 40 (2004) 1885-1906. | MR 2075434

and ,[8] Thermomechanical effects during direct chill and electromagnetic casting of aluminum alloys. Part I: Experimental investigation. Light Metals (1995) 931-940.

and ,[9] Thermomechanical effects during direct chill and electromagnetic casting of aluminum alloys. Part II: numerical simulation. Light Metals (1995) 941-950.

, and ,[10] Modelling of the transient and steady state periods during aluminium dc casting. Light Models (1995) 925-929.

, , and ,[11] Solidification processing. In McGraw-Hill Series in Materials Science and Engineering. McGraw-Hill, New York (1974).

,[12] Le matériau de Norton-Hoff généralisé et ses applications en analyse limite. C. R. Acad. Sci. Paris Sér. A-B 286 (1978) A953-A956. | MR 498880 | Zbl 0374.73041

,[13] A generalized duality method for solving variational inequalities. Applications to some nonlinear Dirichlet problems. Numer. Math. 100 (2005) 259-291. | MR 2135784 | Zbl 1085.65052

, and ,[14] Mécanique des matériaux solides. Dunod, Paris (1988).

and ,[15] Modelling of Thermomechanical Effects During the Start-Up Phase of the Electromagnetic Casting Process. Advances in Production and Fabrication of Light Metals and Metal Matrix Composites (1992) 175-187.

, , and ,[16] Modelling of materials with long memory. Int. J. Solids Struct. 45 (2008) 6133-6156. | MR 2477703 | Zbl 1168.74325

and ,[17] On the convergence of the Bermúdez-Moreno algorithm with constant parameters. Numer. Math. 92 (2002) 113-128. | Zbl 1003.65069

, and ,[18] Development of a new starting block shape for the dc casting of sheet ingots. Part I: Experimental results. Light Metals (1995) 961-967.

, and ,[19] Aluminum extrusion as a thermally activated process. Trans. Metall. Soc. AIME 242 (1968) 2271-2280. | MR 221011

and ,