A novel hyper-reduction framework featuring direct projection without an approximation process

Abstract

Existing methodologies for the hyper projection-based reduced order model (HPROM) fall into two categories: the approximate-then-project and the project-then-approximate approaches. Both involves approximation and projection procedures. This paper introduces a novel hyper-reduction framework, the direct hyper projection-based reduced order model (DHPROM), which exclusively relies on the projection process without the need for the online approximation of Jacobian matrices, and nonlinear residual vectors. During the offline phase, DHPROM avoids the need for gathering the Jacobian matrices and nonlinear residual vectors associated with solving discrete partial differential equations (PDEs) at each iteration, nor for applying dimensionality reduction preprocessing techniques such as proper orthogonal decomposition, the discrete empirical interpolation method, and energy-conserving sampling and weighting to the collected dataset. These characteristics of the offline and online phases contribute to DHPROM's superior speed and accuracy compared to the HPROM. In terms of model applicability, various types of projection-based reduced order models arising from different choices of the left reduced order basis (ROB) Ψ can be derived, which are correspondingly expressed in the DHPROM with the left ROB Ψ̂. The generalizability of DHPROM is demonstrated through tests on the classic turbulent flow over periodic hills with moderately extrapolated parameters. The relative L2 norm error remains at the order of 10−3, indicating good performance. Finally, it is noteworthy that the DHPROM is applicable to any physical problems necessitating the numerical solution of PDEs.