Parametric Model Order Reduction by Box Clustering with Applications in Mechatronic Systems

TL;DR

The paper tackles the high computational cost of high-fidelity simulations in mechatronic design by introducing two intrusive parametric ROM approaches that avoid parameter-space normalization. The Parametric Box Interpolation (pBI) uses a Tensor Product Weight Function (TPWF) and Box Clustering to interpolate reduced matrices with a subset of training points, while Parametric Box Reduction (pBR) extends this idea to the basis-change step to suppress non-physical influence from distant points. Across a 3D cantilever beam and a transient power-module heat transfer problem, pBI and especially pBR deliver high accuracy over wide parameter ranges with competitive offline costs and substantial online speed-ups, mitigating issues observed with traditional normalization-based pROMs. The methods are particularly valuable for design optimization in thermo-structural mechatronics, enabling robust, rapid exploration of large material and geometry parameter spaces. The study also identifies limitations related to the curse of dimensionality and highlights future work on unstructured training grids and mesh changes.

Abstract

High temperatures and structural deformations can compromise the functionality and reliability of new components for mechatronic systems. Therefore, high-fidelity simulations (HFS) are employed during the design process, as they enable a detailed analysis of the thermal and structural behavior of the system. However, such simulations are both computationally expensive and tedious, particularly during iterative optimization procedures. Establishing a parametric reduced order model (pROM) can accelerate the design's optimization if the model can accurately predict the behavior over a wide range of material and geometric properties. However, many existing methods exhibit limitations when applied to wide design ranges. In this work, we introduce the parametric Box Reduction (pBR) method, a matrix interpolation technique that minimizes the non-physical influence of training points due to the large parameter ranges. For this purpose, we define a new interpolation function that computes a local weight for each design variable and integrates them into the global function. Furthermore, we develop an intuitive clustering technique to select the training points for the model, avoiding numerical artifacts from distant points. Additionally, these two strategies do not require normalizing the parameter space and handle every property equally. The effectiveness of the pBR method is validated through two physical applications: structural deformation of a cantilever Timoshenko beam and heat transfer of a power module of a power converter. The results demonstrate that the pBR approach can accurately capture the behavior of mechatronic components across large parameter ranges without sacrificing computational efficiency.

Figures (19)

• Figure 1: Schematic representation of the main reduction steps of the classical approach Panzer2010 (green), parametric box interpolation (orange), and parametric box reduction (blue). pBI only changes the matrix interpolation step, while pBR modifies the basis change step as well.
• Figure 2: Graphical depiction of the procedure to compute $\omega$ with tensor product weight function for a two-parameter model. $N_x$ and $N_y$ are the groupings of the local weight functions.
• Figure 3: Workflow schematic of the parametric point selection methodology with box clustering. Blue circles are the training points, and green stars are the selected points.
• Figure 4: Example illustration depicting nearest neighbor challenges in normalized and non-normalized parameter space. Identification of the kNN with the dashed circle. Box clustering points are not affected by the normalization of the parameter space.
• Figure 5: Parameter space for two parameter points with low and high-value design parameters. Four and nine training points are illustrated for each case.