An Efficient Algorithm For Simulating Fracture Using Large Fuse Networks

TL;DR

This paper introduces an efficient algorithm that significantly reduces the computational cost of simulating fracture in large lattice networks by updating solutions incrementally, outperforming iterative methods near critical points.

Contribution

The paper presents a novel combination of sparse Cholesky downdating and Sherman-Morrison-Woodbury updates to efficiently simulate progressive fracture in large disordered lattice networks.

Findings

Algorithm reduces computational complexity to O(nnz(L))

Faster than Fourier accelerated PCG solvers

Eliminates critical slowing down near fracture points

Abstract

The high computational cost involved in modeling of the progressive fracture simulations using large discrete lattice networks stems from the requirement to solve {\it a new large set of linear equations} every time a new lattice bond is broken. To address this problem, we propose an algorithm that combines the multiple-rank sparse Cholesky downdating algorithm with the rank-p inverse updating algorithm based on the Sherman-Morrison-Woodbury formula for the simulation of progressive fracture in disordered quasi-brittle materials using discrete lattice networks. Using the present algorithm, the computational complexity of solving the new set of linear equations after breaking a bond reduces to the same order as that of a simple {\it backsolve} (forward elimination and backward substitution) {\it using the already LU factored matrix}. That is, the computational cost is $O(nnz({\bf L}))$, where $nnz({\bf L})$ denotes the number of non-zeros of the Cholesky factorization ${\bf L}$ of the stiffness matrix ${\bf A}$. This algorithm using the direct sparse solver is faster than the Fourier accelerated preconditioned conjugate gradient (PCG) iterative solvers, and eliminates the {\it critical slowing down} associated with the iterative solvers that is especially severe close to the critical points. Numerical results using random resistor networks substantiate the efficiency of the present algorithm.