Abstract

Large deformation processes are inherently complex considering the non-linear phenomena that need to be accounted for. Stochastic analysis of these processes is a formidable task due to the numerous sources of uncertainty and the various random input parameters. As a result, uncertainty propagation using intrusive techniques requires tortuous analysis and overhaul of the internal structure of existing deterministic analysis codes. In this paper, we present an approach called non-intrusive stochastic Galerkin (NISG) method, which can be directly applied to presently available deterministic legacy software for modeling deformation processes with minimal effort for computing the complete probability distribution of the underlying stochastic processes. The method involves finite element discretization of the random support space and piecewise continuous interpolation of the probability distribution function over the support space with deterministic function evaluations at the element integration points. For the hyperelastic–viscoplastic large deformation problems considered here with varying levels of randomness in the input and boundary conditions, the NISG method provides highly accurate estimates of the statistical quantities of interest within a fraction of the time required using existing Monte Carlo methods.

Keywords: Uncertainty; Deformation processes; Stochastic Galerkin methods; Stochastic modeling

Article Outline

1. Introduction

2. Review of large deformation processes

3. Polynomial representation of random processes

3.1. Karhunen–Loève (KL) expansion

3.2. Generalized polynomial chaos expansion (GPCE)

3.3. Locally supported piecewise continuous representation

4. Non-intrusive and intrusive approaches

4.1. Intrusive coupled system

4.2. Decoupled system – NISG formulation

4.3. Extension to full-order reliability analysis

5. Examples

5.1. Problem 1 – open die forging under random preform shape and random die–workpiece friction and reliability based design of forging press

5.2. Problem 2 – effect of material heterogeneity on the response of a tension specimen

5.3. Problem 3 – stochastic estimation of die underfill caused by material porosity

5.4. Problem 4 – random material state in an extruded specimen driven by randomness in die geometry

6. Conclusions

Acknowledgements

References