Design optimization in unilateral contact using pressure constraints and Bayesian optimization

TL;DR

This work tackles design optimization problems involving frictionless unilateral contact under pressure constraints, where nonsmoothness complicates optimization and sensitivity calculations. It develops a gradient-based approach using direct differentiation of the KKT system to obtain state sensitivities and solves the resulting problem with Ipopt, alongside a gradient-free constrained Bayesian optimization method using Gaussian Process surrogates and EI_C acquisition. The authors derive and implement the necessary sensitivities for pressure-constrained designs and compare both methods on two engineering-inspired examples (wedge joint and Marman clamp), showing that both can achieve meaningful improvements, with the gradient-based method delivering higher accuracy and constrained BO offering a viable alternative when sensitivities are unavailable. The results highlight the complementary roles of gradient-based and gradient-free approaches for robust design optimization in unilateral contact, and point to future work on addressing nonsmoothness more directly for improved convergence and reliability.

Abstract

Design optimization problems, e.g., shape optimization, that involve deformable bodies in unilateral contact are challenging as they require robust contact solvers, complex optimization methods that are typically gradient-based, and sensitivity derivations. Notably, the problems are nonsmooth, adding significant difficulty to the optimization process. We study design optimization problems in frictionless unilateral contact subject to pressure constraints, using both gradient-based and gradient-free optimization methods, namely Bayesian optimization. The contact simulation problem is solved via the mortar contact and finite element methods. For the gradient-based method, we use the direct differentiation method to compute the sensitivities of the cost and constraint function with respect to the design variables. Then, we use Ipopt to solve the optimization problems. For the gradient-free approach, we use a constrained Bayesian optimization algorithm based on the standard Gaussian Process surrogate model. We present numerical examples that control the contact pressure, inspired by real-life engineering applications, to demonstrate the effectiveness, strengths and shortcomings of both methods. Our results suggest that both optimization methods perform reasonably well for these nonsmooth problems.

Figures (17)

• Figure 1: Reference and deformed configurations of bodies in potential contact.
• Figure 2: Wedge problem in the $y-z$ plane. $P_1(t)$ and $P_2(t)$ are two time-dependent loads.
• Figure 3: The pressure loads $P_1(t)$ and $P_2(t)$ as they change over time. The scalars $P_1$ and $P_2$ are the stabilized values of the loads.
• Figure 4: The reference configurations of the wedges in the initial design.
• Figure 5: The deformed configurations of the optimal design at time 5$s$ (top) and 10$s$ (bottom).