Mathematical modeling of dynamic related processes based on the finite element techniques

DOI: 10.31673/2412-9070.2021.033538

Abstract

In the process of investigation the functioning of information systems with the help of mathematical modeling, problems can arise that are solved using finite element techniques. For example, In the present work we solved the axially symmetric dynamic problem of related phenomena under microscale thermal loading. The statement of the problem includes: Cauchy relations, equations of motion, heat conduction equation, initial conditions, thermal and mechanical boundary conditions. The nonlinear behavior of a material is described by the unified model of flow. The problem is solved numerically by the time step integration method, iterative method and finite element method. Equations of the evolution for the inelastic flow model are integrated by the second-order Euler implicit method with the use of the rule of «middle point». The system of nonlinear transcendental equations obtained in each time step is solved by the method of simple iteration. To accelerate convergence, we apply the Stephensen – Aitken procedure. The equations of motion are integrated by the Newmark method, whereas the heat-conduction equation is integrated by the first-order implicit method. The problem of heat-conduction is linearized by finding the temperature-dependent thermal characteristics according to the temperature distribution in the previous time step or previous iteration. The main results obtained in the work are the following: quantitative estimations of temperature effects of thermostructural-mechanical coupling, which are caused by volumetric thermoelastic effect, dissipation of mechanical energy and hidden heat are obtained.

Keywords: mathematical modeling; finite element method; iterative method; pulse loading.

References
1. Чернихівський Є. М. Математичне моделювання телекомунікаційних систем та мереж: навч. посіб. Львів: вид-во Львівської політехніки, 2011. 280 с.
2. Sternberg E., Chakravorty J. G. On inertial effects in a transient termoelastic problem // ASME. J. Appl. Mech. 1959. V. 26, N 4. P. 503–509.
3. Яковенко Н. Д., Бондарчук А. П., Ковальчук О. П. Исследование динамических эффектов при микромасштабном импульсном нагружении // Проблемы управления и информатики. 2021. №4. С. 61–68.
4. Senchenkov I. K., Oksenchuk N. D. Modeling of a nonisothermal flow with regard for the dependence of plastic properties on the microstructure of a material // J. Math. Sci: електрон. версія журн. 2013. V. 190, N 6. P. 796–803. URL: https://doi.org/10.1007/s10958-013-1289-6
5. Smith I. M., Griffiths D. V. Programming the Finite Element Method / Chichester, 4th edition. New York: John Wiley and Sons, 2004. 542 p.