  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.
