Section: New Results

Formulation of stochastic methods for CFD

Participants : Gianluca Geraci, Kunkun Tang, Francesca Fusi, Pietro Marco Congedo [Corresponding member] .

A novel adaptive strategy for stochastic problems has been developed, inspired from the classical Harten’s framework. The proposed algorithm allows building, in a very general manner, stochastic numerical schemes starting from a whatever type of deterministic schemes and handling a large class of problems, from unsteady to discontinuous solutions. Its formulations permits to recover the same results concerning the interpolation theory of the classical multiresolution approach, but with an extension to uncertainty quantification problems. The present strategy permits to build numerical scheme with a higher accuracy with respect to other classical uncertainty quantification techniques, but with a strong reduction of the numerical cost and memory requirements. Moreover, the flexibility of the proposed approach allows to employ any kind of probability density function, even discontinuous and time varying, without introducing further complications in the algorithm. The advantages of the present strategy are demonstrated by performing several numerical problems where different forms of uncertainty distributions are taken into account, such as discontinuous and unsteady custom-defined probability density functions. In addition to algebraic and ordinary differential equations, numerical results for the challenging 1D Kraichnan–Orszag are reported in terms of accuracy and convergence. Finally, a two degree-of-freedom aeroelastic model for a subsonic case is presented. Though quite simple, the model allows recovering some physical key aspect, on the fluid/structure interaction, thanks to the quasi-steady aerodynamic approximation employed. The injection of an uncertainty is chosen in order to obtain a complete parameterization of the mass matrix. All the numerical results are compared with respect to classical Monte Carlo solution and with a non-intrusive Polynomial Chaos method [3] .

Moreover, in [15] , an anchored ANOVA method is proposed to decompose statistical moments. Compared to standard ANOVA with mutually orthogonal components, anchored ANOVA, with arbitrary anchor point, loses orthogonality if employing the same measure. However, an advantage consists in the considerably reduced number of deterministic solver’s computations, which renders uncertainty quantification of real engineering problems much easier. Different from existing methods, covariance decomposition of output variance is used in this paper to take account of interactions between non-orthogonal components, yielding an exact variance expansion, and thus, with a suitable numerical integration method, provides a strategy that converges. This convergence is verified by studying academic tests. In particular, sensitivity problem of existing method to anchor point is analyzed via Ishigami case, and we point out covariance decomposition survives from it. Covariance-based sensitivity indices (SI) are also used, compared to variance-based SI. Furthermore, we emphasize covariance decomposition can be generalized in a straightforward way to decompose high order moments. For academic problems, results show the method converges to exact solution regarding both skewness and kurtosis. Finally, the proposed method is applied on a realistic case, i.e. estimating chemical reactions uncertainties in a hypersonic flow around a space vehicle during an atmospheric reentry.

External contributors. This work has benefitted from the collaboration with the University of Zurich, and in particular with R. Abgrall.