Adaptive Mesh Refinement Computation of Solidification Microstructures using Dynamic Data Structures

TL;DR

This paper introduces an adaptive mesh refinement algorithm with dynamic data structures for simulating solidification microstructures, significantly reducing computational costs and enabling large-scale, realistic simulations.

Contribution

The authors develop a novel adaptive finite element method with dynamic data structures for phase-field models, improving efficiency and scalability for solidification microstructure simulations.

Findings

Computational complexity scales with interface arclength using adaptive meshes.

Simulations match microscopic solvability theory at high undercoolings.

Higher velocities are observed at low undercoolings, aligning with a new heuristic criterion.

Abstract

We study the evolution of solidification microstructures using a phase-field model computed on an adaptive, finite element grid. We discuss the details of our algorithm and show that it greatly reduces the computational cost of solving the phase-field model at low undercooling. In particular we show that the computational complexity of solving any phase-boundary problem scales with the interface arclength when using an adapting mesh. Moreover, the use of dynamic data structures allows us to simulate system sizes corresponding to experimental conditions, which would otherwise require lattices greater that $2^{17}\times 2^{17}$ elements. We examine the convergence properties of our algorithm. We also present two dimensional, time-dependent calculations of dendritic evolution, with and without surface tension anisotropy. We benchmark our results for dendritic growth with microscopic solvability theory, finding them to be in good agreement with theory for high undercoolings. At low undercooling, however, we obtain higher values of velocity than solvability theory at low undercooling, where transients dominate, in accord with a heuristic criterion which we derive.