Abstract: Motivated by inequality constraints appearing in optimization context, we study non-smooth variational problems and methods appropriate for their accurate solution. The use of Lagrange multipliers and merit functions leads to semi-smooth Newton methods and equivalent primal-dual active-set iterative algorithms. For application in dynamic contact mechanics, we investigate the collision of a rigid obstacle by a one-dimensional elastic bar. The problem is described by the wave equation subjected to complementarity conditions, which weak solution is characterized by a discontinuous velocity. The collision problem may have a non-unique solution for a high initial speed exceeding the propagation speed of elastic waves. Multiple solutions are constructed analytically. For the unique solution that restores the energy of the bar after rebound, the primal-dual active-set method is implemented within space-time finite elements.

Cite: Victor A. Kovtunenko
Non-unique solution with discontinuous velocity for collision of elastic bar with high initial speed by ST-PDAS restoring energy
Journal of Mathematical Sciences (United States). 2026. P.1-16. DOI: 10.1007/s10958-026-08266-w Scopus OpenAlex

Dates:

Submitted:	Dec 1, 2025
Accepted:	Feb 25, 2026
Published online:	Mar 6, 2026

Identifiers:

≡ Scopus:	2-s2.0-105033789206
≡ OpenAlex:	W7134037090

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