Semi-analytical implicit direct time integration scheme on example of 1-D wave propagation problem

Authors

DOI:

https://doi.org/10.20535/2521-1943.2022.6.2.262110

Keywords:

1D wave equation, implicit, direct time integration, suddenly applied force, numerical dissipation

Abstract

The most common approach in dynamic analysis of engineering structures and physical phenomenas consists in finite element discretization and mathematical formulation with subsequent application of direct time integration schemes. The space interpolation functions are usually the same as in static analysis. Here on example of 1-D wave propagation problem the original implicit scheme is proposed, which contains the time interval value explicitly in space interpolation function as results of analytical solution of differential equation for considered moment of time. The displacements (solution) at two previous moments of time are approximated as polynomial functions of position and accounted for as particular solutions of the differential equation. The scheme demonstrates the perfect predictable properties as to dispersion and dissipation. The crucial scheme parameter is the time interval – the lesser the interval the more correct results are obtained. Two other parameters of the scheme – space interval and the degree of polynomial approximation have minimal impact on the general behavior of solution and have influence on small zone near the front of the wave.

Author Biographies

Igor Orynyak, Igor Sikorsky Kyiv Polytechnic Institute

Department of Applied Mathematics at National Technical University Kiev Polytechnic Institute, Peremohy str, 37, Kyiv 03056, Ukraie.

Roman Mazuryk, Igor Sikorsky Kyiv Polytechnic Institute

Department of Applied Mathematics at National Technical University Kiev Polytechnic Institute, Peremohy str, 37, Kyiv 03056

Volodymyr Tsybulskyi, CEO

 Resumeget inc., 3855 Holcomb Bridge Rd., Suite 300 Norcross, GA30092, USA

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Published

2022-10-01

How to Cite

[1]
I. . Orynyak, R. Mazuryk, and V. Tsybulskyi, “Semi-analytical implicit direct time integration scheme on example of 1-D wave propagation problem”, Mech. Adv. Technol., vol. 6, no. 2, pp. 115–123, Oct. 2022.

Issue

Vol. 6 No. 2 (2022)

Section

Mechanics

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Copyright (c) 2022 Ігор Ориняк; Роман Мазурик, Володимир Цибульський

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