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Center for Nonlinear Studies |
The Finite Element Method (FEM) has become one of the most popular numerical techniques to solve Partial Differential Equations (PDE) across various fields of engineering, since the mid twentieth century. However, it’s only been three decades since the general research community has started to extend the potential of the FEM to tackle stochastic PDEs with ran- dom input data. Input uncertainties can exist in several forms, like for example, random field representation of material properties, random forcing conditions along the boundary with white noise, uncertain domain over which the stochastic PDE has to be solved, to name a few. With the very mention of solving a stochastic PDE, one has to address two levels of discretization: discretization of the physical domain and the stochastic space. Up until now, most of the focus in solving stochastic PDE has been fixated upon constructing global/local orthogonal polyno- mials in the stochastic space, while remaining content with the discretization in the physical space using the conventional FEM. Besides, a unified approach to treat multiple independent input uncertainties using global/local polynomial expansion, depending on the smoothness of the solution in the stochastic space, has not been addressed properly in the literature. Finally, the very purpose of projecting the solution onto orthogonal polynomial basis in the stochastic space is to construct a surrogate that could approximate the stochastic response. However, in the presence of multi-dimensional input uncertainties, one has to unearth other novel methods to approximate the stochastic response other than relying on Polynomial Chaos Expansion (PCE) of random response in the stochastic space, like for e.g. Artificial Neural Network.Only little attention was given to application of novel discretization techniques in the physical space. For example, the smoothed FEM that was developed to "soften" its conventional FEM counterpart, was at the most extended to stochastic formulation using perturbation technique, where the later is limited in the allowance of input random variance to less than 10-15%. The scaled boundary finite element method (SBFEM), a semi-analytical method that overcomes the limitations, unlike the FEM on the element topology, had been recently extended to solve stochastic elliptic PDEs with random input processes. However, upon further perusal of this work, it was observed that the work was acutely limited in capturing the stochastic response across the domain. Stochastic eXtended FEM was proposed for coherent integration of stochastic level set method with stochastic Galerkin formulation, so as to take into account geometric uncertainties. Though, several offsprings of its extention to treat both stochastic scalar and vector field problems were proposed, there was a clear lack in the literature that addresses a systematic formulation in case of hybrid uncertainties. Finally, the usage of surrogate models like Neural Networks for reliability estimation of complex material structures have been habituated within the reliability community since the last decade, but at the cost of negligence in attention towards efficient sampling for training those surrogates.The objective of the research work was to advance the frontiers of stochastic FEM within linear context, by adopting novel discretisation technique in both the physical and the stochastic domain, and to develop better surrogates for stochastic response approximation in case of high- dimensional input space for reliability and sensitivity predictions. The means of achieving this objective is what forms the main essence of this work, as summarised below:
- Formulation of stochastic Galerkin cell-based smoothed FEM wherein the mapping from physical to parent element domain for numerical integration is avoided, thereby resulting in results with same accuracy but with lower computational mesh burden.
- A technique to propagate input random field process using polynomial functions over a sub-domain within a Scaled Boundary FEM context, resulting in accurate capture of internal statistical moments of the solution.
- A novel idea to seamlessly integrate multi-element stochastic integration with stochastic level set method, so as to better address stochastic PDEs with hybrid input uncertainties.
- An original idea of adaptive importance sampling to efficiently train a Neural Network based surrogate for reliability and sensitivity estimates of complex composite structures.
The advantages offered by the proposed stochastic formulation will be elaborated with numerical examples during the seminar.
Host: Svetlana Tokareva