Predictive Reduced Order Modeling of Chaotic Multi-scale Problems Using Adaptively Sampled Projections

Abstract

An adaptive projection-based reduced-order model (ROM) formulation is presented for model-order reduction of problems featuring chaotic and convection-dominant physics. An efficient method is formulated to adapt the basis at every time-step of the on-line execution to account for the unresolved dynamics. The adaptive ROM is formulated in a Least-Squares setting using a variable transformation to promote stability and robustness. An efficient strategy is developed to incorporate non-local information in the basis adaptation, significantly enhancing the predictive capabilities of the resulting ROMs. A detailed analysis of the computational complexity is presented, and validated. The adaptive ROM formulation is shown to require negligible offline training and naturally enables both future-state and parametric predictions. The formulation is evaluated on representative reacting flow benchmark problems, demonstrating that the ROMs are capable of providing efficient and accurate predictions including those involving significant changes in dynamics due to parametric variations, and transient phenomena. A key contribution of this work is the development and demonstration of a comprehensive ROM formulation that targets predictive capability in chaotic, multi-scale, and transport-dominated problems.

A new method to interpolate POD reduced bases–Application to the parametric model order reduction of a gas bearings supported rotor

“…It is worth mentioning that other frameworks proceed differently by updating the reduced basis from new observed/computed snapshots. [35][36][37] In this work, ideally, given a new parameter 𝝁 and a set of POD bases of rank p, say X(𝝁 0 ), … , X(𝝁 q ), obtained from snapshots computed at distinct parameter values 𝝁 0 , … , 𝝁 q , respectively, we would like to compute a new basis X(𝝁) of rank p leading to a lower snapshots reconstruction error than any of the individual bases X(𝝁 0 ), … , X(𝝁 q ). If the new snapshots matrix was known, we recall that the optimal reduced basis of rank p would be obtained with a POD on the snapshots matrix.…”

“…As in Reference 45 we consider the change of variable 𝜓 = Pf v (h) in the NGT Equation ( 32) and we adapt the boundary conditions (35)(36)(37) accordingly. A Finite Difference (FD) scheme is employed to discretize in space the modified NGT equation (expressed in 𝜓).…”

“…The partial derivative operator 𝜕 z (.) in boundary condition (37) at the mid-plane interface is discretized with a backward finite difference scheme, leading to an implicit relation under the form:…”

“…Affine parameter dependence. The parameter of interest 𝜔 appears in the non-dimensional coefficients m r , g in Equations ( 29) and ( 30) and Λ, 𝜎, c s in Equations ( 32)- (37). Consequently, the FOM (44) exhibits an affine dependence in 𝜔, that is all the terms have the form 𝛾(𝜔)Ax, with 𝛾(𝜔) a known function of 𝜔 and Ax a matrix-vector product independent of 𝜔.…”

An adaptive, training‐free reduced‐order model for convection‐dominated problems based on hybrid snapshots

“…Adaptive reduced-order models (AROMs) 22,[25][26][27][28][29][30] provide a different approach by continuously combining HDM and ROM operations. Predictive capabilities can be improved by alternating between HDM and ROM generated snapshots.…”