An Efficient Algorithm For Simulating Fracture Using Large Fuse Networks

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.Comment: 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 stag

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

{\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.Comment: 16 pages including 2 figure

Electron scattering states at solid surfaces calculated with realistic potentials

Scattering states with LEED asymptotics are calculated for a general non-muffin tin potential, as e.g. for a pseudopotential with a suitable barrier and image potential part. The latter applies especially to the case of low lying conduction bands. The wave function is described with a reciprocal lattice representation parallel to the surface and a discretization of the real space perpendicular to the surface. The Schroedinger equation leads to a system of linear one-dimensional equations. The asymptotic boundary value problem is confined via the quantum transmitting boundary method to a finite interval. The solutions are obtained basing on a multigrid technique which yields a fast and reliable algorithm. The influence of the boundary conditions, the accuracy and the rate of convergence with several solvers are discussed. The resulting charge densities are investigated.Comment: 5 pages, 4 figures, copyright and acknowledgment added, typos etc. correcte

Hypercube matrix computation task

A major objective of the Hypercube Matrix Computation effort at the Jet Propulsion Laboratory (JPL) is to investigate the applicability of a parallel computing architecture to the solution of large-scale electromagnetic scattering problems. Three scattering analysis codes are being implemented and assessed on a JPL/California Institute of Technology (Caltech) Mark 3 Hypercube. The codes, which utilize different underlying algorithms, give a means of evaluating the general applicability of this parallel architecture. The three analysis codes being implemented are a frequency domain method of moments code, a time domain finite difference code, and a frequency domain finite elements code. These analysis capabilities are being integrated into an electromagnetics interactive analysis workstation which can serve as a design tool for the construction of antennas and other radiating or scattering structures. The first two years of work on the Hypercube Matrix Computation effort is summarized. It includes both new developments and results as well as work previously reported in the Hypercube Matrix Computation Task: Final Report for 1986 to 1987 (JPL Publication 87-18)

Hypercube matrix computation task

The Hypercube Matrix Computation (Year 1986-1987) task investigated the applicability of a parallel computing architecture to the solution of large scale electromagnetic scattering problems. Two existing electromagnetic scattering codes were selected for conversion to the Mark III Hypercube concurrent computing environment. They were selected so that the underlying numerical algorithms utilized would be different thereby providing a more thorough evaluation of the appropriateness of the parallel environment for these types of problems. The first code was a frequency domain method of moments solution, NEC-2, developed at Lawrence Livermore National Laboratory. The second code was a time domain finite difference solution of Maxwell's equations to solve for the scattered fields. Once the codes were implemented on the hypercube and verified to obtain correct solutions by comparing the results with those from sequential runs, several measures were used to evaluate the performance of the two codes. First, a comparison was provided of the problem size possible on the hypercube with 128 megabytes of memory for a 32-node configuration with that available in a typical sequential user environment of 4 to 8 megabytes. Then, the performance of the codes was anlyzed for the computational speedup attained by the parallel architecture