Numerical solution of two phase solidification problem using dynamic substructuring based on adaptive error estimation

Year 2015, , 127 - 144, 30.12.2015

Ozgur Uyar Ata Mugan

https://doi.org/10.17350/HJSE19030000017

Abstract

N umerical solution of solidification of metals with a sharp front, in particular solidification of lead, is investigated. Considering the fact that the associated CPU time and memory requirement may be costly for large domains, alternatives are searched. It is observed that using a substructuring technique with a local mesh refinement is promising. Following, by the use of an adaptive error estimation algorithm to find the location of solidification front and mushy zone, dynamic substructuring technique is developed to decrease the computational cost and to increase the accuracy of results. Superconvergent patch recovery technique is used to obtain the heat fluxes to evaluate the error energy norm of elements at each analysis step. Solidification front, mushy zone and elements having errors above a threshold value are captured with the error estimator. Then, elements having errors above the threshold value are refined by creating a substructure which is independent from the original global mesh. Equations of the global coarse mesh are augmented with the equations of the substructure. Employing the equations of the original coarse mesh help reduce the computational cost. Numerical solutions are presented and it is shown that the proposed approach has advantages over the alternative methods and, by the virtue of the adaptive error estimation algorithm, significantly decreases the CPU time of numerical solutions while it increases the accuracy of solutions and locates precisely the solidification front and mushy zone

Keywords

Solidification, Finite Element Methods, Substructuring, Stefan Problem, Error Estimation

References

• Franca AS and Haghigi K. Adaptive finite element analysis of transient thermal problems. Numerical Heat Transfer Part B. 26 (1994) 273-292.
• Juric D and Tryggvason G. A front-tracking method for dendritic solidification. Journal of Computational Physics. 123 (1996) 127-148.
• Chen Y, Im Y-T, Yoo J. Finite element analysis of solidification of aluminum with natural convection. Journal of Materials Processing Technology 52 (1995) 592-609.
• Provatas N, Goldenfeld N and Dantzig J. Adaptive mesh refinement computation of solidification microstructures using dynamic data structures. Journal of Computational Physics. 148 (1999) 265-290.
• Lewis RW and Ravindran K. Finite element simulation of metal casting. International Journal for Numerical Methods in Engineering. 47 (2000) 29-59.
• Merle R and Dolbow J. Solving thermal and phase change problems with the extended finite element method. Computational Mechanics 28 (2002) 339-350.
• Chessa J, Smolinski P and Belytschko T. The extended finite element method (xfem) for solidification problems. International Journal for Numerical Methods in Engineering. 53 (2002) 1959-1977.
• Ji H, Chopp D and Dolbow JE. A hybrid extended finite element/level set method for modeling phase transformations. International Journal For Numerical Methods in Engineering. 54 (2002) 1209-1233.
• Zhao P, Venere M, Heinrich JC, Poirier DR. Modeling dendritic growth of a binary alloy. Journal of Computational Physics. 188 (2003) 434-461.
• Zhang XR and Xu X. Finite element analysis of pulsed laser bending: the effect of melting and solidification. Journal of Applied Mechanics 71 (2004) 321-326.
• Takaki T, Fukuoka T, Tomita Y. Directional solidification of a binary alloy using adaptive finite element method. Journal of Crystal Growth. 283 (2005) 263-278.
• Zabaras N, Ganapathysubramanian B, Tan L. Modelling dendritic solidification with melt convection using the extended finite element method. Journal of Computational Physics. 218 (2006) 200-227.
• Zhang L, Shen H-F, Rong Y, Huang T-Y. Numerical simulation on solidification and thermal stress of continuous casting billet in mold based on meshless methods. Materials and Science Engineering. A466 (2007) 71-78.
• Wang H, Li R, Tang T. Efficient computation of dendritic growth with r-adaptive finite element methods. Journal of Computational Physics. 227 (2008) 5984-6000.
• Hu X, Li R and Tang T. A multi-mesh adaptive finite element approximation to phase field models. Communications in Computational Physics. 5 (2009) 1012-1029.
• Lee S and Sundararaghavan V. Multi-scale homogenization of moving ınterface problems with flux jumps: Application to solidification. Comput Mech. 44 (2009) 297-307.
• Bo Li and John Shopple. An Interface-Fitted Finite Element Level Set Method With Application to Solidification and Solvation. Commun. Comput. Phys. 10 (2011) 32-56.
• P. O’Hara CA and Eason DT. Transient analysis of sharp thermal gradients using coarse finite element meshes. Comput. Methods Appl. Mech. Engrg. 200 (2011) 812-829.
• Chen M, Hu X-D, Ju D-Y, Zhao H-Y. The microstructure prediction of magnesium alloy crystal growth in directional solidification. Computational Materials Science. 79 (2013) 684-690.
• Chen S, Guillemot G and Gandin C-A. 3D coupled cellular automaton (ca) – finite element (fe) modeling for solidification grain structures in gas tungsten arc welding (GTAW). ISIJ International. 54 (2014) 401-407.
• Ghoneim A. A meshfree ınterface-finite element method for modelling ısothermal solutal melting and solidification in binary systems. Finite Elements in Analysis and Design. 95 (2015) 20-41.
• Bratu V, Mortici C, Oros C, Ghiban N. Mathematical model of solidification process in steel continuous casting into account the convective heat transfer at liquid-solid ınterface. Computational Materials Science. 94 (2014) 2-7.
• Skrzypczak T, Wegrzyn-Skrzypczak E. Mathematical and numarical model of solidification process of pure metals. International Journal of Heat and Mass Transfer. 55 (2012) 4276-4284.
• Ho CY, Powell RW andLiley PE. Thermal conductivity of the elements. J. Phys. Chem. Ref. Data 1 No. 2 (1972) 279-421.
• Abu-Eishah SI, Haddad Y, Solieman A and Bajbouj A. A new correlation for the specific heat of metals, metal oxides and metal fluorides as a function of temperature. Latin American Applied Research. 34 (2004) 257-265.
• Sobolev V. Database of thermophysical properties of liquid metal coolants for GEN-IV. Scientific Report of the Belgian Nuclear Research Centre. (2011).
• Nikishkov G. Programming finite elements in java. Springer- Verlag, London. (2010).
• Hughes TJR. The finite element method linear static and dynamic finite element analysis. Prentice-Hall. (1987).
• Liu GR and Quek SS. The finite element method a practical course. Butterworth-Heinemann. (2003).
• Cook RD. Finite element modeling for stress analysis, John Wiley and Sons. (1995). Rudolph Szilard. Theories and Applications of Plate Analysis. John Wiley and Sons. (2004).
• Zienkiewicz OC and Zhu JZ. A simple error estimator and adaptive procedure for practical engineering analysis. International Journal For Numerical Methods in Engineering. 24 (1987) 337-357.Thomas J. R. Hughes. The Finite Element Method Linear Static and Dynamic Finite Element analysis, Prentice-Hall, 1987.
• Zienkiewicz OC, Taylor RL and Zhu JZ. The finite element method: its basis and fundamentals, seventh ed. Butterworth-Heinemann is an imprint of Elsevier. (2013).
• Rabizadeh E, Bagherzadeh AS, Rabczuk T. Adaptive thermo-mechanical finite element formulation based on goal-oriented error estimation. Computational Materials Science. 102 (2015) 27-44.
• Khoei AR, Gharehbaghi SA. Three-dimensional data transfer operators in large plasticity deformations using modified-spr technique. Applied Mathematical Modelling. 33 (2009) 3269-3285.
• Gonzales-Estrada OA, Rodenas JJ, Bordas SPA, Nadal E, Kerfriden P, Fuenmayor FJ. Locally equilibrated stress recovery for goal oriented error estimation in the extended finite element method. Computers and Structures. 152 (2015) 1-10.