Commun. Comput. Phys., 14 (2013), pp. 940-959.

An Efficient Rescaling Algorithm for Simulating the Evolution of Multiple Elastically Stressed Precipitates

Amlan K. Barua 1, Shuwang Li 1*, Hualong Feng 1, Xiaofan Li 1, John Lowengrub 2

1 Department of Applied Mathematics, Illinois Institute of Technology, Chicago 60616, USA.
2 Department of Mathematics, University of California at Irvine, Irvine, CA 92697, USA.

Received 25 October 2012; Accepted (in revised version) 27 December 2012
Available online 19 March 2013


In this paper, we propose a space-time rescaling scheme for computing the long time evolution of multiple precipitates in an elastically stressed medium. The algorithm is second order accurate in time, spectrally accurate in space and enables one to simulate the evolution of precipitates in a fraction of the time normally used by fixed-frame algorithms. In particular, we extend the algorithm recently developed for single particle by Li et al. (Li, Lowengrub and Leo, J. Comput. Phys., 335 (2007), 554) to the multiple particle case, which involves key differences in the method. Our results show that without elasticity there are successive tip splitting phenomena accompanied by the formation of narrow channels between the precipitates. In presence of applied elastic field, the precipitates form dendrite-like structures with the primary arms aligned in the principal directions of the elastic field. We demonstrate that when the far-field flux decreases with the effective radius of the system, tip-splitting and dendrite formation can be suppressed, as in the one particle case. Depending on the initial position of the precipitates, we further observe that some precipitates grow while others may shrink, even when a positive far field flux is applied.

AMS subject classifications: 74N25, 74B15

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Key words: Diffusion, elasticity, dendrite, side branching, boundary integral method.

*Corresponding author.
Email: (A. K. Barua), (S. Li), (H. Feng), (X. Li), (J. Lowengrub)

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