Hyper-reduction methods for accelerating nonlinear finite element simulations: open source implementation and reproducible benchmarks

TL;DR

The findings reveal that the comparative performance between hyper-reduction methods depends on both the problem and the choice of time integration method, and underscore the need for problem specific method selection to balance accuracy and efficiency.

Abstract

Hyper-reduction methods have gained increasing attention for their potential to accelerate reduced order models for nonlinear systems, yet their comparative accuracy and computational efficiency are not well understood. Motivated by this gap, we evaluate a range of hyper-reduction techniques for nonlinear finite element models across benchmark problems of varying complexity, assessing the inevitable tradeoff between accuracy and speedup. More specifically, we consider interpolation methods based on the gappy proper orthogonal decomposition as well as the empirical quadrature procedure (EQP), and apply them to the hyper-reduction of problems in nonlinear diffusion, nonlinear elasticity and Lagrangian hydrodynamics. Our numerical results are generated using the open source libROM, Laghos and MFEM numerical libraries. Our findings reveal that the comparative performance between hyper-reduction methods depends on both the problem and the choice of time integration method. The EQP method generally achieves lower relative errors than interpolation methods and is more efficient in terms of quadrature point usage, resulting in a lower wall time for the nonlinear diffusion and elasticity problems. However, its online computational cost is observed to be relatively high for Lagrangian hydrodynamics problems. Conversely, interpolation methods exhibit greater variability, especially with respect to the use of different time integration methods in the Lagrangian hydrodynamics problems. The presented results underscore the need for problem specific method selection to balance accuracy and efficiency, while also offering useful guidance for future comparisons and refinements of hyper-reduction techniques.

Figures (15)

• Figure 1: Visualization of final-time fields from FOM simulations. The first column shows the nonlinear diffusion problem (top, temperature) and the nonlinear elasticity problem (bottom, velocity). The remaining columns correspond to Lagrangian benchmark problems, showing energy (top) and velocity (bottom) for the Sedov blast (second column), the Taylor-Green vortex (third column), and the triple-point problem (fourth column).
• Figure 2: Sample mesh comparison for the triple point problem. Left: S-OPT sample mesh with 22 sampled elements and $22\times64 = 1408$ sampled quadrature points (in dark blue). Right: EQP sample mesh with 178 sampled elements and 150 sampled quadrature points each for the energy field (in green) and the velocity field (in red).
• Figure 3: Reduced basis dimensions with respect to the residual energy fraction for the nonlinear diffusion simulations.
• Figure 4: Performance comparison for the nonlinear diffusion ROM simulations with varying number of sampled interpolation/quadrature points. The left column shows reproductive simulation results, while the right one predictive results. The top row corresponds to residual energy fraction $E_r=2$, the middle row to $E_r=4$ and the bottom one to $E_r=6$.
• Figure 5: Overall Pareto fronts across all hyper-reduction methods for the nonlinear diffusion problem, illustrating the tradeoff between relative $L^2$ error and online phase wall time.

Theorems & Definitions (1)

• Definition 5.1: Pareto front