  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.
