An Efficient Algorithm For Simulating Fracture Using Large Fuse Networks

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 $O(nnz(\mathbf{L}))$, where $nnz(\mathbf{L})$ denotes the number of non-zeros of the Cholesky factorization $\mathbf{L}$ of the stiffness matrix $\mathbf{A}$. 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.

15 pages including 1 figure. On page pp11407 of the original paper (J. Phys. A: Math. Gen. 36 (2003) 11403-11412), Eqs. 11 and 12 were misprinted that went unnoticed during the proof reading stage

References in corpus (1)

• Fracture in Three-Dimensional Fuse Networks

Cited by in corpus (14)

• Statistical Models of Fracture
• From damage percolation to crack nucleation through finite size criticality
• Crack roughness and avalanche precursors in the random fuse model
• Crack avalanches in the three dimensional random fuse model
• Dielectric breakdown and avalanches at non-equilibrium metal-insulator transitions
• Percolation and localization in the random fuse model
• Crack Roughness in the 2D Random Threshold Beam Model
• Optimization and plasticity in disordered media
• An Efficient Block Circulant Preconditioner For Simulating Fracture Using Large Fuse Networks
• Scaling of Fracture Strength in Disordered Quasi-Brittle Materials
• Energy dissipation statistics in the random fuse model
• Role of the sample thickness in planar crack propagation
• Experimental Realization of the Fuse Model of Crack Formation
• Effect of Disorder and Notches on Crack Roughness