Hostname: page-component-76fb5796d-x4r87 Total loading time: 0 Render date: 2024-04-26T05:52:33.628Z Has data issue: false hasContentIssue false
  • English
  • Français

A Parallel Adaptive Treecode Algorithm for Evolution of Elastically Stressed Solids

Published online by Cambridge University Press:  03 June 2015

Hualong Feng ,
Amlan Barua ,
Shuwang Li  and
Xiaofan Li
Hualong Feng*
Affiliation:
Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA School of Science, Nanjing University of Science and Technology, Nanjing, Jiangsu 210094, P.R. China
Amlan Barua*
Affiliation:
Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA
Shuwang Li*
Affiliation:
Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA
Xiaofan Li*
Affiliation:
Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA
*
Email:hfeng8@iit.edu
Email:abarua@hawk.iit.edu
Email:sli@math.iit.edu
Corresponding author.Email:lix@iit.edu
Get access

Abstract

The evolution of precipitates in stressed solids is modeled by coupling a quasi-steady diffusion equation and a linear elasticity equation with dynamic boundary conditions. The governing equations are solved numerically using a boundary integral method (BIM). A critical step in applying BIM is to develop fast algorithms to reduce the arithmetic operation count of matrix-vector multiplications. In this paper, we develop a fast adaptive treecode algorithm for the diffusion and elasticity problems in two dimensions (2D). We present a novel source dividing strategy to parallelize the treecode. Numerical results show that the speedup factor is nearly perfect up to a moderate number of processors. This approach of parallelization can be readily implemented in other treecodes using either uniform or non-uniform point distribution. We demonstrate the effectiveness of the treecode by computing the long-time evolution of a complicated microstructure in elastic media, which would be extremely difficult with a direct summation method due to CPU time constraint. The treecode speeds up computations dramatically while fulfilling the stringent precision requirement dictated by the spectrally accurate BIM.

Keywords

65D3065N3574B05Treecodeparallelelasticityboundary integral method
Type
Research Article
Information
Communications in Computational Physics , Volume 15 , Issue 2 , February 2014 , pp. 365 - 387
Copyright
Copyright © Global Science Press Limited 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Akaiwa, N. and Meiron, D.I.Numerical-simulation of 2-dimensional late-stage coarsening for nucleation and growth. Physical Review E, 51:5408, 1995.Google Scholar
[2]Barnes, J. and Hut, P.A hierarchical O(NlogN) force-calculation algorithm. Nature, 324(6096):446–449, 1986.Google Scholar
[3]Barua, A., Li, S., Feng, H., Li, X., and Lowengrub, J.An efficient rescaling algorithm for simulating the evolution of multiple elastic precipitates. Commun. Comput. Phys., 14(4):940–959.Google Scholar
[4]Barua, A., Li, S., Li, X., and Lowengrub, J.Self-similar evolution of a precipitate in inhomoge-neous elastic media. J. Cryst. Growth, 351(1):62–71, 2012.Google Scholar
[5]Belytschko, T., Gracie, R., and Ventura, G.A review of extended/generalized finite element methods for material modeling. Mod. Simu. in Mat. Sci. Engi., 17(4), 2009.Google Scholar
[6]Boisse, J., Lecoq, N., Patte, R., and Zapolsky, H.Phase-field simulation of coarsening of γ precipitates in an ordered γ ′ matrix. Acta Materialia, 55(18):6151–6158, 2007.Google Scholar
[7]Carrier, G.F., Krook, M., and Pearson, C.E. Function of a Complex Variable. McGraw-Hill, 1966.Google Scholar
[8]Cheng, H., Greengard, L., and Rokhlin, V.A fast adaptive multipole algorithm in three dimensions. J. Comput. Phys., 155(2):468–498, 1999.Google Scholar
[9]Draghicescu, C.I. and Draghicescu, M.A fast algorithm for vortex blob interactions. Comput, J.1. Phys., 116(1):69–78, 1995.Google Scholar
[10]Greenbaum, A., Greengard, L., and McFadden, G. B.Laplace equation and Dirichlet-Neumann map in multiply connected domains. J. Comput. Phys., 105:267, 1993.Google Scholar
[11]Greengard, L. and Gropp, W.A parallel version of the fast algorithm method. Comp. Math. Appl., 20(7):63–71, 1990.Google Scholar
[12]Greengard, L. and Rokhlin, V.A fast algorithm for particle simulations. J. Comput. Phys., 73(2):325–348, 1987.Google Scholar
[13]Hou, T., Lowengrub, J.S., and Shelley, M.Removing the stiffness from interfacial flows with surface tension. J. Comput. Phys., 114:312, 1994.Google Scholar
[14]Hou, T., Lowengrub, J.S., and Shelley, M.Boundary integral methods for multicomponent fluids and multiphase materials. J. Comput. Phys., 169:302, 2001.Google Scholar
[15]Jou, H.-J., Leo, P.H., and Lowengrub, J.S.Microsructural evolution in inhomogeneous elastic media. J. Comput. Phys., 131:109, 1997.Google Scholar
[16]Kabadshow, I., Dachsel, H., and Hammond, J.Poster: Passing the three trillion particle limit with an error-controlled fast multipole method. In SC Companion, pages 73–74, 2011.Google Scholar
[17]Krasny, R.A study of singularity formation in a vortex sheet by the point-vortex approximation. Fluid, J.Mech., 167:65–93, 1986.Google Scholar
[18]Kwon, Y., Thornton, K., and Voorhees, P. W.Morphology and topology in coarsening of domains via non-conserved and conserved dynamics. Philosophical Magazine, 90(1-4):317–335, 2010.Google Scholar
[19]Li, B., Lowengrub, J., Raetz, A., and Voigt, A.Geometric evolution laws for thin crystalline films: modeling and numerics. Commun. Comput. Phys., 6(3):433–482, 2009.Google Scholar
[20]Li, S., Lowengrub, J., and Leo, P. H.A rescaling scheme with application to the long-time simulation of viscous fingering in a hele-shaw cell. J. Comput. Phys., 225(1):554–567, 2007.Google Scholar
[21]Lindsay, K. and Krasny, R.A particle method and adaptive treecode for vortex sheet motion in three-dimensional flow. J. Comput. Phys., 172(2):879–907, 2001.Google Scholar
[22]Liu, D., Duan, Z.H., Krasny, R., and Zhu, J.Parallel implementation of the treecode Ewald method. In Proceedings of the 18th International Parallel and Distributed Processing Symposium, Sante Fe, New Mexico, 2004.Google Scholar
[23]Mikhlin, S.G.Integral equation and their applications to certian problems of mechanics. Pergammon, New York, 1957.Google Scholar
[24]Qin, R. S. and Bhadeshia, H. K.Phase field method. Mat Sci and Tech, 26(7):803–811, 2010.Google Scholar
[25]Rahimian, A., Lashuk, I., Veerapaneni, S., Aparna, C., Malhotra, D., Moon, I., Sampath, R., Shringarpure, A., Vetter, J., Vuduc, R., Zorin, D., and Biros, G.Petascale direct numerical simulation of blood flow on 200k cores and heterogeneous architectures. In ACM/IEEE SCxy conference series, pages 1–11, 2010.Google Scholar
[26]Saad, Y. and Schultz, M.A generalized minimum residual method for solving non symmetric linear systems. SIAM J. Sci. Stat. Comput., 7:856, 1986.CrossRefGoogle Scholar
[27]Samet, H.The Design and Analysis of Spatial Data Structures. Addison-Wesley, New York, 1989.Google Scholar
[28]Sidi, A. and Israel, M.Quadrature methods for periodic singular and weakly singular Fredholm integral equations. J. Sci. Comput., 3:201, 1988.Google Scholar
[29]Singh, J.P., Holt, C., Totsuka, T., Gupta, A., and Hennessy, J.Load balancing and data locality in adaptive hierarchical N-body methods – Barnes-Hut, fast multipole, and radiosity. Parallel, J.Distrib. Comput., 27(2):118–141, 1995.Google Scholar
[30]Steinbach, I.Phase-field models in materials science. Mod. Simu. in Mat. Sci. Engi., 17(7), 2009.Google Scholar
[31]Thornton, K., Akaiwa, N., and Voorhees, P.W.Large-scale simulations of Ostwald ripening in elastically stressed solids: I. Development of microstructure. Physical Review E, 52:1353, 2004.Google Scholar
[32]Thornton, K., Argen, J., and Voorhees, P. W.Modeling the evolution of phase boundaries in solids at the meso- and nano-scales. Acta Materilia, 51(19):5675–5710, 2003.Google Scholar
[33]Warren, M.S. and Salmon, J.K.A portable parallel particle program. Comput. Phys. Comm., 87:266–290, 1995.Google Scholar
[34]Winkel, M., Speck, R., Hubner, H., Arnold, L., Krause, R., and Gibbon, P.A massively parallel, multi-disciplinary Barnes-Hut tree code for exterme-scale N-body simulations. Comput. Phys. Comm., 183:880–889, 2012.Google Scholar
[35]Yokota, R., Barba, L., and Knepley, M.PetRBF – A parallel O(N) algorithm for radial basis function interpolation. Comput. Meth. Appl. Mech. Eng., 199(25-28):1793–1804, 2010.Google Scholar
[36]Zhang, B., Huang, J., Pitsianis, N., and Sun, X.Dynamic prioritization for parallel traversal of irregularly structured spatio-temporal graphs. In Proceedings of the 3rd USENIX Workshop on Hot Topics in Parallelism, Berkeley, CA, 2011.Google Scholar