An efficient block-circulant preconditioner for simulating fracture using large fuse networks*

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-circulant 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 algorithm per iteration is O(rslog s) + delops, where the stiffness matrix A is partitioned into r × r blocks such that each block is an s × s matrix, and delops represents the operational count associated with solving a block-diagonal matrix with r × r dense matrix blocks. This algorithm using the block-circulant preconditioner is faster than the Fourier accelerated preconditioned conjugate gradient algorithm, and alleviates the 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.