We investigate three-dimensional problems in solid mechanics with stochastic loading or material properties. To solve these problems, we use a spectral expansion of the solution and random inputs based on Askey-type orthogonal polynomials in terms of independent, identically distributed (i.i.d) random variables. A Galerkin procedure using these types of expansions, the generalized Polynomial Chaos (gPC) method, is employed to solve linear elasticity problems. An analagous spectral collocation formulation is used to study problems in nonlinear elasticity. These methods both cast the stochastic problem as a coupled or decoupled high-dimensional system of deterministic PDEs, which is then solved numerically using a deterministic p-finite element solver. We present algorithms for solving certain coupled systems arising from the stochastic Galerkin projection without modifying the original deterministic solver. Three-dimensional riser-sections undergoing elastic deformations due to random pressure loads are considered. We also model a riser-section with stochastic Young's modulus undergoing deterministic loads. It is demonstrated that the gPC method provides accurate and efficient results at a speed-up factor of two and three orders of magnitude compared to traditional Monte-Carlo simulations. For nonlinear problems, the stochastic collocation method is also shown to be much faster than Monte-Carlo simulation, while still rivaling this method in simplicity of implementation.
|Original language||English (US)|
|Number of pages||22|
|Journal||Computer Methods in Applied Mechanics and Engineering|
|State||Published - Sep 1 2007|
Bibliographical noteFunding Information:
The first author would like to acknowledge the support of the US Department of Energy Computational Science Graduate Fellowship under grant number DE-FG02-97ER25308 and the Krell Institute.
The authors gratefully acknowledge the support of this work by the Office of Naval Research (Ocean Engineering and Marine Systems), and the United States-Israel Binational Science Foundation. The computations were performed on DoD’s supercomputing center (NAVO, ARSC and ARL) under a HPCMP grant.
Copyright 2008 Elsevier B.V., All rights reserved.
- Karhunen-Loève (K-L) expansion
- Polynomial chaos
- Random fluid loads
- Stochastic collocation
- Uncertainty quantification