# Efficiency of A Hybrid Parallel Algorithm For Phase-Field Simulation of Polycrystalline Solidification in 3D

## Main Article Content

## Abstract

We revisit our previously developed algorithm for phase-field simulation of solidification of an arbitrary number of crystals with random crystallographic orientations and a fully resolved 3D dendritic geometry. In this contribution, its hybrid parallel implementation based on the combination of MPI and OpenMP standards undergoes parallel efficiency tests. The results reveal the settings for optimal performance and their dependence on the number of CPUs used. Next, the performance benefits of using the algorithm for single crystal growth are explained. Finally, a very high resolution simulation is demonstrated together with its computational costs.

## Article Details

How to Cite

Strachota, P., Wodecki, A., & Beneš, M.
(2020).
Efficiency of A Hybrid Parallel Algorithm For Phase-Field Simulation of Polycrystalline Solidification in 3D.

*Proceedings Of The Conference Algoritmy,*, 131 - 140. Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/algoritmy/article/view/1567/826
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Articles

## References

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[12] R. Kobayashi. Modeling and numerical simulations of dendritic crystal growth. Physica D, 63:410–423, 1993.

[13] R. E. Napolitano and S. Liu. Three-dimensional crystal-melt Wulff-shape and interfacial stiffness in the Al-Sn binary system. Phys. Rev. B, 70(21):214103, 2004.

[14] M. PunKay. Modeling of anisotropic surface energies for quantum dot formation and morphological evolution. In NNIN REU Research Accomplishments, pages 116–117. University of Michigan, 2005.

[15] P. Stenström, T. Joe, and A. Gupta. Comparative performance evaluation of cache-coherent NUMA and COMA architectures. In ISCA '92 Proceedings of the 19th annual international symposium on Computer architecture, volume 20, pages 80–91. ACM New York, NY, USA, May 1992.

[16] P. Strachota. Analysis and Application of Numerical Methods for Solving Nonlinear Reaction-Diffusion Equations. PhD thesis, Czech Technical University in Prague, 2012.

[17] P. Strachota and M. Beneš. Design and verification of the MPFA scheme for three-dimensional phase field model of dendritic crystal growth. In A. Cangiani, R. L. Davidchack, E. Georgoulis, A. N. Gorban, J. Levesley, and M. V. Tretyakov, editors, Numerical Mathematics and Advanced Applications 2011: Proceedings of ENUMATH 2011, the 9th European Conference on Numerical Mathematics and Advanced Applications, Leicester, September 2011, pages 459–467. Springer Berlin Heidelberg, 2013.

[18] P. Strachota and M. Beneš. A hybrid parallel numerical algorithm for three-dimensional phase field modeling of crystal growth. In A. Handlovičová and D. Ševčovič, editors, ALGORITMY 2016, 20th Conference on Scientific Computing, Vysoké Tatry - Podbanské, Slovakia, March 14 - 18, 2016. Proceedings of contributed papers and posters, pages 23–32. Comenius University, Bratislava, 2016.

[19] P. Strachota and A. Wodecki. High resolution 3D phase field simulations of single crystal and polycrystalline solidification. Acta Phys. Pol. A, 134(3):653–657, September 2018.

[20] Y. Suwa. Phase-field simulation of grain growth. Technical Report 102, Nippon Steel, January 2013.

[21] Y. Suwa and Y. Saito. Computer simulation of grain growth by the phase field model. effect of interfacial energy on kinetics of grain growth. Mater. Trans., 44:2245–2251, 2003.

[2] M. Beneš. Phase Field Model or Microstructure Growth in Solidification of Pure Substances. PhD thesis, Czech Technical University in Prague, 1997.

[3] M. Beneš. Anisotropic phase-field model with focused latent-heat release. In FREE BOUNDARY PROBLEMS: Theory and Applications II, volume 14 of GAKUTO International Series in Mathematical Sciences and Applications, pages 18–30, 2000.

[4] M. Beneš. Mathematical and computational aspects of solidification of pure substances. Acta Math. Univ. Comenianae, 70(1):123–151, 2001.

[5] M. Beneš. Diffuse-interface treatment of the anisotropic mean-curvature flow. Appl. Math-Czech., 48(6):437–453, 2003.

[6] M. Beneš. Computational studies of anisotropic diffuse interface model of microstructure formation in solidification. Acta Math. Univ. Comenianae, 76:39–59, 2007.

[7] W. J. Boettinger, S. Coriell, A. L. Greer, A. Karma, W. Kurz, M. Rappaz, and R. Trivedi. Solidification microstructures: Recent developments, future directions. Acta Mater., 48:43–70, 2000.

[8] J. C. Butcher. Numerical Methods for Ordinary Differential Equations. Wiley, Chichester, 2nd edition, 2008.

[9] Z. Guo and S. M. Xiong. Study of dendritic growth and coarsening using a 3-D phase field model: Implementation of the Para-AMR algorithm. IOP Conf. Ser.: Mater. Sci. Eng., 84:012067, 2015.

[10] M. E. Gurtin. Thermomechanics of Evolving Phase Boundaries in the Plane. Oxford Mathematical Monographs. Oxford University Press, 1993.

[11] A. Karma and W.-J. Rappel. Quantitative phase-field modeling of dendritic growth in two and three dimensions. Phys. Rev. E, 57(4):4, 1998.

[12] R. Kobayashi. Modeling and numerical simulations of dendritic crystal growth. Physica D, 63:410–423, 1993.

[13] R. E. Napolitano and S. Liu. Three-dimensional crystal-melt Wulff-shape and interfacial stiffness in the Al-Sn binary system. Phys. Rev. B, 70(21):214103, 2004.

[14] M. PunKay. Modeling of anisotropic surface energies for quantum dot formation and morphological evolution. In NNIN REU Research Accomplishments, pages 116–117. University of Michigan, 2005.

[15] P. Stenström, T. Joe, and A. Gupta. Comparative performance evaluation of cache-coherent NUMA and COMA architectures. In ISCA '92 Proceedings of the 19th annual international symposium on Computer architecture, volume 20, pages 80–91. ACM New York, NY, USA, May 1992.

[16] P. Strachota. Analysis and Application of Numerical Methods for Solving Nonlinear Reaction-Diffusion Equations. PhD thesis, Czech Technical University in Prague, 2012.

[17] P. Strachota and M. Beneš. Design and verification of the MPFA scheme for three-dimensional phase field model of dendritic crystal growth. In A. Cangiani, R. L. Davidchack, E. Georgoulis, A. N. Gorban, J. Levesley, and M. V. Tretyakov, editors, Numerical Mathematics and Advanced Applications 2011: Proceedings of ENUMATH 2011, the 9th European Conference on Numerical Mathematics and Advanced Applications, Leicester, September 2011, pages 459–467. Springer Berlin Heidelberg, 2013.

[18] P. Strachota and M. Beneš. A hybrid parallel numerical algorithm for three-dimensional phase field modeling of crystal growth. In A. Handlovičová and D. Ševčovič, editors, ALGORITMY 2016, 20th Conference on Scientific Computing, Vysoké Tatry - Podbanské, Slovakia, March 14 - 18, 2016. Proceedings of contributed papers and posters, pages 23–32. Comenius University, Bratislava, 2016.

[19] P. Strachota and A. Wodecki. High resolution 3D phase field simulations of single crystal and polycrystalline solidification. Acta Phys. Pol. A, 134(3):653–657, September 2018.

[20] Y. Suwa. Phase-field simulation of grain growth. Technical Report 102, Nippon Steel, January 2013.

[21] Y. Suwa and Y. Saito. Computer simulation of grain growth by the phase field model. effect of interfacial energy on kinetics of grain growth. Mater. Trans., 44:2245–2251, 2003.