An Efficient Block Circulant Preconditioner For Simulating Fracture Using Large Fuse Networks

TL;DR

This paper introduces a block circulant preconditioner that significantly accelerates iterative solvers for large-scale fracture simulations in lattice networks, reducing critical slowing down near critical points.

Contribution

The paper proposes a novel block circulant preconditioner that improves the efficiency of iterative solvers for fracture simulations in large disordered lattice networks.

Findings

Faster convergence than Fourier accelerated PCG.

Reduces critical slowing down near critical points.

Validated with numerical results on resistor networks.

Abstract

{\it Critical slowing down} associated with the iterative solvers close to the critical point often hinders large-scale numerical simulation of fracture using discrete lattice networks. This paper presents a block circlant preconditioner for iterative solvers for the simulation of progressive fracture in disordered, quasi-brittle materials using large discrete lattice networks. The average computational cost of the present alorithm per iteration is $O(rs log s) + delops$, where the stiffness matrix ${\bf A}$ is partioned into $r$-by-$r$ blocks such that each block is an $s$-by-$s$ matrix, and $delops$ represents the operational count associated with solving a block-diagonal matrix with $r$-by-$r$ dense matrix blocks. This algorithm using the block circulant preconditioner is faster than the Fourier accelerated preconditioned conjugate gradient (PCG) algorithm, and alleviates the {\it critical slowing down} that is especially severe close to the critical point. Numerical results using random resistor networks substantiate the efficiency of the present algorithm.