Model Consistency for Mechanical Design: Bridging Lumped and Distributed Parameter Models with A Priori Guarantees
Randi Wang, Vadim Shapiro, Morad Behandish
TL;DR
The paper tackles the problem of ensuring consistency between system-level lumped-parameter models (LPMs) and geometry-level distributed-parameter models (DPMs) in mechanical design. It introduces a simulation-free model-consistency scheme that leverages SPARK+CURE-based MOR to obtain a priori $H_2$ error guarantees between the LPM and a surrogate DPM, enabling efficient, rigorous comparisons without solving large-scale PDEs directly; the key bound relates time-domain BoIs via $ rm{y_d - y_l}_{\infty} \le \nrm{G_d - G_l}_{H_2} \sqrt{\int_0^{\infty} \nrm{h_d(t)}_2^2 dt}$ and a surrogate transfer function $G_r(s)$ with $\nrm{G_d - G_l}_{H_2} \le \bar{\varepsilon}$. The approach defines three consistency conditions (mass, ICs/BCs, and BoI matching) and demonstrates on bracket and frame designs that the method yields tight a priori bounds and meaningful time savings for large DPMs. The results show that, beyond a crossover DPM order, the simulation-free scheme outperforms direct simulation while maintaining accuracy, offering a practical path for reliable system-to-geometry design translation. Limitations include neglecting spatial discretization errors in the BoI matching and applicability primarily to linear time-invariant models, with proposed nonlinear extensions discussed for future work.
Abstract
Engineering design often involves representation in at least two levels of abstraction: the system-level, represented by lumped parameter models (LPMs), and the geometric-level, represented by distributed parameter models (DPMs). Functional design innovation commonly occurs at the system-level, followed by a geometric-level realization of functional LPM components. However, comparing these two levels in terms of behavioral outcomes can be challenging and time-consuming, leading to delays in design translations between system and mechanical engineers. In this paper, we propose a simulation-free scheme that compares LPMs and spatially-discretized DPMs based on their model specifications and behaviors of interest, regardless of modeling languages and numerical methods. We adopt a model order reduction (MOR) technique that a priori guarantees accuracy, stability, and convergence to improve the computational efficiency of large-scale models. Our approach is demonstrated through the model consistency analysis of several mechanical designs, showing its validity, efficiency, and generality. Our method provides a systematic way to compare system-level and geometric-level designs, improving reliability and facilitating design translation.