13th International Conference on Fracture June 16–21, 2013, Beijing, China -6- In addition to allowing correct description of key multiple crack configurations, the A-FEM incorporates a new iteration algorithm for searching for convergence in nonlinear problems, in which the global stiffness matrix is re-written in a piecewise linear form, allowing local convergence to be achieved in one or at most a few steps. The combination of A-FEM elements that use advanced quadrature algorithms and can be made somewhat larger than the length of the fracture process zone and the new iteration algorithm leads to gains in computational speed for typical nonlinear fracture problems of 2 – 3 orders of magnitude compared to standard methods such as X-FEM embedded in the ABAQUS commercial software [35]. As discussed further below, computational speed is essential in a virtual test strategy that seeks to address stochastic variability. 45 N 85 N 135 N 120* N RT: intra-tow delaminations (a) 10 N 90 N 105 N 130* N HT: interstitial matrix damage (b) Figure 5. Internal damage in a C-SiC composite with textile-based carbon fiber reinforcements under tensile load at (a) 25°C and (b) 1600°C. 6. Monte Carlo methods vs. probabilistic theories The Monte Carlo method using ensembles of stochastic virtual specimens provides the closest analogue of a real test matrix: a statistically significant number of virtual specimens are subjected to the same virtual test (or matrix of tests) and engineering predictions are deduced from the mean and scatter in the outcomes [36, 37]. Each specimen in the tested ensemble is one instance of a random microstructure that has been constructed by feeding pseudo-random numbers into calibrated distribution functions (a Monte Carlo procedure). The variance of the microstructure in the ensemble of virtual specimens is a major source of variance in predictions. With trivial modification, the load can also be made random, e.g., to simulate random overloads in a duty cycle. Once a stochastic virtual specimen generator has been developed and constitutive laws have been calibrated, executing a Monte Carlo analysis is straightforward. Simulations are executed in sequence and predicted metrics (strength, strain to failure, etc.) are analyzed using the same statistical methods used to analyze real tests.
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