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<p>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.</p>

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NON-UNIQUE SOLUTION WITH DISCONTINUOUS VELOCITY FOR COLLISION OF ELASTIC BAR WITH HIGH INITIAL SPEED BY ST-PDAS RESTORING ENERGY

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  • 出版模式：

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  • 发表日期：

    2026年3月6日

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  • Victor A. Kovtunenko

摘要

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.