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.

翻译：超约化方法因其在加速非线性系统降阶模型方面的潜力而日益受到关注，然而其相对精度与计算效率尚未得到充分理解。基于这一研究空白，我们评估了一系列超约化技术在不同复杂度基准问题中的表现，针对非线性有限元模型，评估了精度与加速之间不可避免的权衡关系。具体而言，我们研究了基于间隙本征正交分解的插值方法以及经验求积程序（EQP），并将它们应用于非线性扩散、非线性弹性和拉格朗日流体动力学问题的超约化。我们的数值结果通过开源软件库libROM、Laghos和MFEM生成。研究发现，超约化方法之间的相对性能既取决于具体问题，也取决于时间积分方法的选择。EQP方法通常比插值方法获得更低的相对误差，并且在求积点使用上更为高效，从而在非线性扩散和弹性问题中实现了更短的运行时间。然而，在拉格朗日流体动力学问题中，其在线计算成本相对较高。相反，插值方法表现出更大的变异性，特别是在拉格朗日流体动力学问题中采用不同时间积分方法时。本研究结果强调了针对具体问题进行方法选择以平衡精度与效率的必要性，同时也为未来超约化技术的比较与改进提供了有益指导。