Computational issues in stochastic finite element analysis.

摘要

Stochastic finite element analysis is a method for solving structural analysis problems in the presence of uncertainties. The conventional method of structural analysis fails to account for randomness in input data such as material properties, geometry, loads, and boundary conditions. The first order second moment method can be used to compute the mean and variance of the structural response (such as displacements and stresses) and Monte Carlo simulation can be used to compute the probability distribution function of the structural response. This thesis addresses three important computational issues in stochastic finite element analysis: geometric uncertainties, discretization errors and parallelism.; Geometrical uncertainties in finite element models are studied by using the method of automatic differentiation for computing shape sensitivities. The method of automatic differentiation is easy to use and is also efficient and accurate. This method is used in the analysis of screw holes or holes due to tumor in bones, where the shape and size of the hole is not deterministic.; Discretization errors in stochastic finite element analysis are estimated by extending the deterministic error estimate to the probabilistic case. By computing the first two moments of the error estimate, one can compute the probability of the error exceeding a given threshold. The error indicator can be used to refine selectively regions of the mesh where the errors are high.; Stochastic finite element analysis of large 3-D structural models is a computationally intensive task. It is therefore desirable to exploit parallelism to solve large problems. Stochastic finite element analysis using the first order second moment method is composed of several well-defined computational sub-tasks, each of which can be performed in parallel. It is shown that stochastic fem can be performed efficiently in parallel on distributed memory, message-passing computers. Moreover, with proper data distribution, application related parallel programs are almost as simple as their sequential counterparts.; Finally, some applications of stochastic fem to problems in structural analysis of biomechanical systems are presented. Experimental data from CT scans of the proximal femur are used to compute the correlation length for the Young's modulus field in the proximal femur. Models of the proximal femur with random Young's modulus and random loads are analyzed and the relative significance of randomness in loads and Young's modulus is compared.