Fourier series and ST-PDAS solutions for non-smooth 1D collision problem with gravity

Fourier series and ST-PDAS solutions for non-smooth 1D collision problem with gravity Full article

Journal

Computational Mathematics and Modeling
ISSN: 1046-283X

Output data

Year: 2026, DOI: 10.1007/s10598-026-09695-7

Tags

Contact and impact dynamics, Discontinuous velocity, Variational inequality, Primal-dual active set, Space-time finite element

Authors

Kovtunenko Victor A. ^1,2

Affiliations

1	Department of Mathematics and Scientific Computing, Karl-Franzens University of Graz, NAWI Graz, Graz, Austria
2	Lavrentyev Institute of Hydrodynamics, Siberian Division of the Russian Academy of Sciences, Novosibirsk, Russian Federation

The one-dimensional dynamic contact problem describing a rigid obstacle collided by an elastic bar is considered in gravitational field. The collision problem is non-smooth with respect to velocity and axial strain, and formulated as a variational inequality. For the corresponding wave equation subject to complementary conditions which are imposed at the contact boundary, its nonlinear solution is expressed with the help of Fourier Series representing longitudinal vibrations. Moreover, before the bar rebound, an analytical solution comprising piece-wise quadratic function is constructed on a partition of the rectangular space-time domain along characteristics. The analytical benchmark is implemented for low initial speeds in numerical experiments solving the variational inequality over uniform space-time triangulation with the Space-Time Primal-Dual Active Set method of semi-smooth Newton type.

Kovtunenko V.A.
Fourier series and ST-PDAS solutions for non-smooth 1D collision problem with gravity
Computational Mathematics and Modeling. 2026. DOI: 10.1007/s10598-026-09695-7

Submitted:	Jan 27, 2026
Accepted:	Feb 17, 2026
Published online:	May 5, 2026

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