Sparse Collocation based Stochastic Finite Element Method for Reliability Analysis of Pile Settlement

Abstract eng:
Uncertainty and risk are essential features of geotechnical engineering. A complex geotechnical problem would only admit numerical solutions and the material parameters show random variability in spatial distribution as well as in intensity, and may be modeled as random fields rather than random variables. A standard form of a stochastic partial differential equation (SPDE) can model the propagation of input uncertainties. Collocation based stochastic finite element method (CSFEM) uncouples the finite element analysis with stochastic analysis, which means the finite element code can be treated as a black box (Huang et al., 2007). It outperforms the Monte Carlo method for problems with relative lower random dimensions. However, when CSFEM is dealing with problems with high dimension, the number of collocation points will grow very fast, sometimes even more than Monte Carlo's. Sparse grids have been applied in many fields, such as high-dimensional integration and interpolation and solution of PDEs. The sparse grid method constructs multi-dimentional interpretation functions with much fewer points. Combining CSFEM with the sparse grid method can significantly reduce the number of collocation points for high dimension problems. The settlement of an axially loaded pile with linear soil springs is used to exam the performance of CSFEM with sparse grids, where the shear modulus of soil will be modeled as a 1-D a translation lognormal field. The numerical example demonstrated that a third Hermite expansion with level 2 sparse grid is adequate to produce accurate estimates of the output CDF while requiring much fewer response evaluations as compared to direct Monte Carlo simulation. Results show that the sparse grid method outperforms the tensor product method for problems with high dimensions.

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