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.

翻译：涉及单向接触可变形体的设计优化问题（如形状优化）具有挑战性，因其需要稳健的接触求解器、通常基于梯度的复杂优化方法以及灵敏度推导。值得注意的是，这类问题具有非光滑性，显著增加了优化过程的难度。本文研究了无摩擦单向接触中受压力约束的设计优化问题，采用基于梯度与无梯度（即贝叶斯优化）两种优化方法。接触仿真问题通过砂浆接触法和有限元法求解。对于基于梯度的方法，我们采用直接微分法计算成本函数与约束函数关于设计变量的灵敏度，随后使用Ipopt求解优化问题。对于无梯度方法，我们采用基于标准高斯过程代理模型的约束贝叶斯优化算法。受实际工程应用启发，我们通过控制接触压力的数值算例，展示了两种方法的有效性、优势与不足。结果表明，两种优化方法在这些非光滑问题中均表现出良好的性能。