Sharp Stability Estimates for the Force-based Quasicontinuum Method
arXiv:0907.3861
Abstract
A sharp stability analysis of atomistic-to-continuum coupling methods is essential for evaluating their capabilities for predicting the formation and motion of lattice defects. We formulate a simple one-dimensional model problem and give a detailed analysis of the stability of the force-based quasicontinuum (QCF) method. The focus of the analysis is the question whether the QCF method is able to predict a critical load at which fracture occurs. Numerical experiments show that the spectrum of a linearized QCF operator is identical to the spectrum of a linearized energy-based quasi-nonlocal quasicontinuum operator (QNL), which we know from our previous analyses to be positive below the critical load. However, the QCF operator is non-normal and it turns out that it is not generally positive definite, even when all of its eigenvalues are positive. Using a combination of rigorous analysis and numerical experiments, we investigate in detail for which choices of "function spaces" the QCF operator is stable, uniformly in the size of the atomistic system. Force-based multi-physics coupling methods are popular techniques to circumvent the difficulties faced in formulating consistent energy-based coupling pproaches. Even though the QCF method is possibly the simplest coupling method of this kind, we anticipate that many of our observations apply more generally.
22 pages, 1 figure
References in corpus (1)
Cited by in corpus (6)
- A Priori and A Posteriori Analysis of the Quasi-Nonlocal Quasicontinuum Method in 1D
- Iterative Methods for the Force-based Quasicontinuum Approximation
- A Computational and Theoretical Investigation of the Accuracy of Quasicontinuum Methods
- Analysis of Energy-Based Blended Quasicontinuum Approximations
- The Spectrum of the Force-Based Quasicontinuum Operator for a Homogeneous Periodic Chain
- Linear Stationary Iterative Methods for the Force-based Quasicontinuum Approximation