An efficient hyper reduced-order model for segregated solvers for geometrical parametrization problems

TL;DR

The results show that the hyper-reduced models closely match full-order solutions while achieving substantial reductions in computational time, demonstrating that hyper-reduction can be effectively combined with segregated solvers and geometric parametrization to enable fast and accurate CFD simulations.

Abstract

We propose an efficient hyper-reduced order model (HROM) designed for segregated finite-volume solvers in geometrically parametrized problems. The method follows a discretize-then-project strategy: the full-order operators are first assembled using finite volume or finite element discretizations and then projected onto low-dimensional spaces using a small set of spatial sampling points, selected through hyper-reduction techniques such as DEIM. This approach removes the dependence of the online computational cost on the full mesh size. The method is assessed on three benchmark problems: a linear transport equation, a nonlinear Burgers equation, and the incompressible Navier--Stokes equations. The results show that the hyper-reduced models closely match full-order solutions while achieving substantial reductions in computational time. Since only a sparse subset of mesh cells is evaluated during the online phase, the method is naturally parallelizable and scalable to very large meshes. These findings demonstrate that hyper-reduction can be effectively combined with segregated solvers and geometric parametrization to enable fast and accurate CFD simulations.

Figures (9)

• Figure 1: The geometry of the domain
• Figure 2: The mesh domain
• Figure 3: The 100 DEIM points are represented in red, the stencils of the cells and associated degrees of freedom needed for the evaluation of the discrete differential operators are in light-blue. The discarded nodes in the evolution of the dynamics are in blue. The stencil is made of 1 layer of cells
• Figure 4: Comparison of the temperature field. First row original solutions, second row hyper-reduced solutions using 7 optimal points, and third row the error associated. Snapshots are taken at $t=0.002\ $s, $t=0.01\ $s, and $t=0.052\ $s.
• Figure 5: The first 100 DEIM points for the Burgers equation are represented in red, the stencils of the cells and associated degrees of freedom needed for the evaluation of the discrete differential operators are in light-green. The discarded nodes in the evolution of the dynamics are in blue. The stencil is made of 1 layer of cells