Sensitivity analysis plays an important role in finding an optimal design of a structure under uncertainty. Quantifying relative importance of random parameters, which leads to a rank ordering, helps in developing a systematic and efficient way to reach the optimal design. In this work, lift prediction and sensitivity analysis of a potential flow around a submerged body is considered. Such flow is often used in the initial design stage of structures. The flow computation is carried out using a vortex-panel method. A few parameters of the submerged body and flow are considered as random variables. To improve the accuracy in lift prediction in a computationally efficient way, a new semi-intrusive stochastic perturbation method is proposed. Accordingly, a perturbation is applied at the linear system solving level involving the inuence coefficient matrix, as opposed to using perturbation in the lift quantity itself. This proposed method, which is partially analogous to the intrusive or Galerkin projection methods in spectral stochastic finite element methods, is found to be more accurate than using perturbation directly on the lift and faster than a direct simulation. The proposed semi-intrusive stochastic perturbation method is found to yield faster estimates of the Sobol’ indices, which are used for global sensitivity analysis. From global sensitivity analysis, the flow parameters are found to be more important than the parameters of the submerged body.

A stochastic approach to conditional simulation of flow in randomly heterogeneous media is proposed with the combination of the Karhunen-Loeve expansion and the probabilistic collocation method (PCM). The conditional log hydraulic conductivity field is represented with the Karhunen-Loeve expansion, in terms of some deterministic functions and a set of independent Gaussian random variables. The propagation of uncertainty in the flow simulations is carried out through the PCM, which relies on the efficient polynomial chaos expansion used to represent the flow responses such as the hydraulic head. With the PCM, existing flow simulators can be employed for uncertainty quantification of flow in heterogeneous porous media when direct measurements of hydraulic conductivity are taken into consideration. With illustration of several numerical examples of groundwater flow, this study reveals that the proposed approach is able to accurately quantify uncertainty of the flow responses conditioning on hydraulic conductivity data, while the computational efforts are significantly reduced in comparison to the Monte Carlo simulations.

Robust optimization is an approach for the design of a mechanical structure which takes into account the uncertainties of the design variables. It requires at each iteration the evaluation of some robust measures of the objective function and the constraints. In a previous work, the authors have proposed a method which efficiently generates a design of experiments with respect to the design variable uncertainties to compute the robust measures using the polynomial chaos expansion. This paper extends the proposed method to the case of the robust optimization. The generated design of experiments is used to build a surrogate model for the robust measures over a certain trust region. This leads to a trust region optimization method which only requires one evaluation of the design of experiments per iteration (single loop method). Unlike other single loop methods which are only based on a first order approximation of robust measure of the constraints and which does not handle a robust measure for the objective function, the proposed method can handle any approximation order and any choice for the robust measures. Some numerical experiments based on finite element functions are performed to show the efficiency of the method.

The polynomial chaos expansion (PCE) is an efficient numerical method for performing a reliability analysis. It relates the output of a nonlinear system with the uncertainty in its input parameters using a multidimensional polynomial approximation (the so-called PCE). Numerically, such an approximation can be obtained by using a regression method with a suitable design of experiments. The cost of this approximation depends on the size of the design of experiments. If the design of experiments is large and the system is modeled with a computationally expensive FEA (Finite Element Analysis) model, the PCE approximation becomes unfeasible. The aim of this work is to propose an algorithm that generates efficiently a design of experiments of a size defined by the user, in order to make the PCE approximation computationally feasible. It is an optimization algorithm that seeks to find the best design of experiments in the D-optimal sense for the PCE. This algorithm is a coupling between genetic algorithms and the Fedorov exchange algorithm. The efficiency of our approach in terms of accuracy and computational time reduction is compared with other existing methods in the case of analytical functions and finite element based functions.

A non-intrusive stochastic finite-element method is proposed for uncertainty propagation through mechanical systems with uncertain input described by random variables. A polynomial chaos expansion (PCE) of the random response is used. Each PCE coefficient is cast as a multi-dimensional integral when using a projection scheme. Common simulation schemes, e.g. Monte Carlo Sampling (MCS) or Latin Hypercube Sampling (LHS), may be used to estimate these integrals, at a low convergence rate though. As an alternative, quasi-Monte Carlo (QMC) methods, which make use of quasi-random sequences, are proposed to provide rapidly converging estimates. The Sobol' sequence is more specifically used in this paper. The accuracy of the QMC approach is illustrated by the case study of a truss structure with random member properties (Young's modulus and cross section) and random loading. It is shown that QMC outperforms MCS and LHS techniques for moment, sensitivity and reliability analyses.