Recent mathematical tools and numerical schemes for contact problems of elastic or viscoelastic (Kelvin-Voigt type) bodies have been quite developed. Two major classical mathematical tools have been used to study the contact problems; variational inequalities and penalty methods. As a result, contact forces N are removed from the original formulations and their behavior remains mysterious. Unlike those techniques, we keep the contact forces N as a part of the complementarity conditions (CCs). Then, we can prove the existence of not only solutions u but also the contact forces satisfying CCs. In addition, CCs allow us to propose numerical schemes more efficiently.

In this talk, we sketch recent trends for dynamic contact problems with linear beams and nonlinear beams. Furthermore, we can extend those contact models with additional effects, e.g., friction, adhesion, damage, thermal heat exchange, and so on.

Numerical schemes proposed are to use Finite Difference Methods (FDMs) on the time interval and Finite Element Methods (FEMs) in the spacial domain, and compute numerical approximations satisfying the complementarity conditions (CCs). Convergence theories are investigated, as well.

We present some numerical results (simulations) and future directions for the one-dimensional problems.