Computational challenges in the large-scale simulations of fracture in disordered media

Abstract

Computational modeling of fracture in disordered materials using discrete lattice models is often limited to small system sizes due to high computational cost associated with re-solving the governing system of equations every time a new lattice bond is broken. Previously, we proposed an efficient algorithm based on multiple-rank sparse Cholesky downdating scheme for 2D simulations, and an iterative scheme using block-circulant preconditioners for 3D simulations. Based on these algorithms, we were able to simulate large 2D lattice systems (e.g., $L = 1024$). However, despite these algorithmic advances, the largest 3D lattice system that we were able to solve was limited to a size of $L = 64$. In this paper, we present three alternate approaches, namely, the efficient preconditioners, {it krylov subspace recycling}, and massive parallelization of the algorithms, the combination of which promise to significantly reduce the computational cost associated with simulating large 3D lattice systems of sizes $L = 200$. The main idea associated with krylov subspace recycling is to retain a subspace determined while solving the current system and reuse it to reduce the cost of solving the subsequent system obtained after removing the new broken bond. Preliminary numerical simulation of fracture using 3D random fuse networks of sizes $L = 64$ substantiates the efficiency of the present algorithms.