Goal-oriented model adaptivity in stochastic elastodynamics: simultaneous control of discretisation, surrogate model and sampling errors

Pedro Bonilla-Villalba (1, 2) , Susanne Claus (3) , Abhishek Kundu (1, 2) , Pierre Kerfriden (4, 1)

1 School of Engineering [Cardiff]
2 UCL Department of Mathematics
3 DTIS, ONERA, Université Paris Saclay (COmUE) [Palaiseau]
4 Mines Paris - PSL (École nationale supérieure des mines de Paris)

The presented adaptive modelling approach aims to jointly control the level of refinement for each of the building-blocks employed in a typical chain of finite element approximations for stochas-tically parametrized systems, namely: (i) finite error approximation of the spatial fields (ii) surro-gate modelling to interpolate quantities of interest(s) in the parameter domain and (iii) Monte-Carlo sampling of associated probability distribution(s). The control strategy seeks accurate calculation of any statistical measure of the distributions at minimum cost, given an acceptable margin of error as only tunable parameter. At each stage of the greedy-based algorithm for spatial discreti-sation, the mesh is selectively refined in the subdomains with highest contribution to the error in the desired measure. The strictly incremental complexity of the surrogate model is controlled by enforcing preponderant discretisation error integrated across the parameter domain. Finally, the number of Monte-Carlo samples is chosen such that either (a) the overall precision of the chain of approximations can be ascertained with sufficient confidence, or (b) the fact that the computational model requires further mesh refinement is statistically established. The efficiency of the proposed approach is discussed for a frequency-domain vibration structural dynamics problem with forward uncertainty propagation. Results show that locally adapted finite element solutions converge faster than those obtained using uniformly refined grids.

Mots clés: Adaptivity; Stochastic Control and Estimation; Stochastic finite element method; Solid mechanics; Meta-modelling; Monte-Carlo Algorithm; Uncertainty Propagation

Domaines: Mécanique des structures [physics.class-ph]

Pedro Bonilla-Villalba, Susanne Claus, Abhishek Kundu, Pierre Kerfriden. Goal-oriented model adaptivity in stochastic elastodynamics: simultaneous control of discretisation, surrogate model and sampling errors. International Journal for Uncertainty Quantification, In press, ⟨10.1615/Int.J.UncertaintyQuantification.2020031735⟩. ⟨hal-02539789⟩