## Sparse matrix algorithms for phase field models of grain growth

### April 23, 2007

Phase field models are used, among other things, for a study of grain growth [1,2]. In one type of phase field models used to study grain growth, each grain orientation is represented by an order parameter , where the index runs from 1 to , where is the total number of orientations present in the system [3,4]. Though this method is simple to implement and has been used extensively, it is computationally intensive. So, recently there had been at least two attempts to come up with an efficient numerical implementation:

- Efficient numerical algorithm for multiphase field simulations. Srikanth Vedantam and B S V Patnaik, Phys Rev E
**73**, 016703, 2006.

Phase-field models have emerged as a successful class of models in a wide variety of applications in computational materials science. Multiphase field theories, as a subclass of phase-field theories, have been especially useful for studying nucleation and growth in polycrystalline materials. In theory, an infinite number of phase-field variables are required to represent grain orientations in a rotationally invariant free energy. However, limitations on available computational time and memory have restricted the number of phase-field variables used in the simulations. We present an approach by which the time and memory requirements are drastically reduced relative to standard algorithms. The proposed algorithm allows us the use of an unlimited number of phase-field variables to perform simulations without the associated burden on computational time or memory. We present the algorithm in the context of coalescence free grain growth.

- Sparse data structure and algorithm for the phase field method. J Gruber, N Ma, Y Wang, A D Rollett, and G S Rohrer, Modelling and Simulation in Materials Science and Engineering,
**14**, 1189, 2006.

The concepts of sparse data structures and related algorithms for phase field simulations are discussed. Simulations of polycrystalline grain growth with a conventional phase field method and with sparse data structures are compared. It is shown that memory usage and simulation time scale with the number of nodes but are independent of the number of order parameters when a sparse data structure is used.

The source code for a C++ implementation of the method described in the paper of Gruber et al is available for download here (for non-profit scientific research purposes).

The idea behind these implementations is rather simple. Consider an arbitrary mesh point in a simulation cell. The mesh point either lies in the bulk of a given grain, or it lies in the interface. If it lies in the bulk, all the order parameters except the one corresponding to the grain orientation are zero, and there is nothing much to be done about the calculation at that point. On the other hand, if it lies in the interface, the total number of order parameters which have non-zero values are still a very small number as compared to the total number of orientations present in the entire system. Hence, if there is a database such that for any point (and its neighbours) we have the information of non-zero order parameters, then the calculation can be made more efficient.

Take a look and have fun!

**References**:

[1] Phase field methods for microstructure evolution. Long-Qing Chen, Annu. Rev. Mater. Res., **32**, 113, 2002.

[2] Modeling grain boundaries using a phase-field technique. J A Warren, R Kobayashi, and W Craig Carter, J Cryst Growth, **211**, 1, 18, 2000.

[3] A novel computer simulation technique for modeling grain growth. Long-Qing Chen, Scripta Metallurgica et Materialia, **32**, 1, 115, 1995.

[4] A phase field concept for multiphase systems. I Steinbach, F Pezzolla, B Nestler, M Seeszelberg, R Prieler, G J Schmitz, and J L L Rezende, Physica D, **94**, 3, 135, 1996.

December 3, 2008 at 1:21 pm

Here are two other sparse algorithms for the simulation of grain growth with phase field modelling:

– Computer simulations of two-dimensional and three-dimensional ideal grain growth. Seong Gyoon Kim, Dong Ik Kim, Won Tae Kim and Yong Bum Park, Phys. Rev. E 74, 061605, 2006: this work is based on the model of Steinbach et al.

– Bounding box algorithm for three-dimensional phase-field simulations of microstructural evolution in polycrystalline materials. Liesbeth Vanherpe, Nele Moelans, Bart Blanpain and Stefan Vandewalle, Phys. Rev. E 76, 056702, 2007: the article is based on the work of Long-Qing Chen et al. The data structure is designed for a semi-implicit discretization.

December 4, 2008 at 4:55 am

Dear Vanherpe,

Thanks a lot for the pointers!

Guru