Sparse approximations for contact mechanics

TL;DR

The paper tackles the challenge of reduced-order modeling for frictionless, non-adhesive contact by addressing the linear inseparability of contact pressure through dictionary-based sparse approximations. It introduces a greedy active-set procedure that selects a small number of entries from a large offline dictionary to reconstruct online contact pressures while enforcing non-penetration constraints. The approach is demonstrated on Hertz and ironing problems, showing that sparse dictionary selection can achieve accurate reconstructions with substantially reduced online cost, though performance depends on the extent of contact zone variation. The work highlights practical benefits for real-time simulations and outlines bottlenecks related to nonlinear operator evaluation and manifold interpolation, suggesting avenues for further research in nonlinear ROM techniques and initialization strategies.

Abstract

Low-rank model order reduction strategies for contact mechanics show limited dimensionality reduction due to linear inseparability of contact pressure field. Therefore, a dictionary based strategy is explored for creating efficient models for frictionless non-adhesive contact. A large dictionary of contact pressure trajectories is generated using a high-fidelity finite element model, while approximating the online query with a small number of dictionary entries. This is achieved by inducing sparsity in the approximation. Accuracy, computational effort and limitations of such methods are demonstrated on few numerical examples.

Figures (19)

• Figure 1: Illustration of low-rank, localized low-rank and dictionary-based approximation. Snapshots (green circles) in an $n$-dimensional system lie on a manifold (green-curve) of dimension lower than $n$. Low-rank subspaces have a dimension $\ll n$. The approximation to the queried solution ($\oplus$) and its closeness to the manifold can be visualized for the three methods.
• Figure 2: Kinematic description of two body contact problems with possibility of contact
• Figure 3: Hertz problem: two half cylinders loaded against each other and the resulting contact pressure snapshots. Displacement $d$, imposed on the top cylinder, is treated as a parameter in the reduced model.
• Figure 4: A spy pattern of dual dictionary with 30 elements for the Hertz problem. The $79$ points on the $y$-axis are the dofs on the potential contact surface. The dictionary columns are arranged in the increasing order of loading parameter $d$.
• Figure 5: Greedy active-set reconstruction of contact pressure for certain parametric instances in the validation set of the Hertz problem. Elements of the dictionary chosen by the algorithm are shown and the reconstruction errors are given. Dictionary of size 12 is used in these examples.

Theorems & Definitions (2)

• Remark 1
• Remark 2