### October 25, 2022

A.V. Lukin and L.V. Shtukin (Peter the Great St.Petersburg Polytechnic University)

Nonlinear dynamics of N/MEMS under laser-induced opto-thermal excitations

The work is devoted to the study of the nonlinear dynamics of parametrically excited bending oscillations of a microbeam pinched at both ends - the basic sensitive element of a promising class of microsensors of various physical quantities - under laser thermooptical action in the form of periodically generated pulses acting on a certain part of the surface of the beam element. An analytical solution of the heat conduction problem is found for a steady-state harmonic temperature distribution in the resonator volume. The static and dynamic components of force factors of temperature nature - temperature axial force and bending moment - are determined. Using the Galerkin method, the discretization of nonlinear coupled partial differential equations describing the longitudinal-flexural oscillations of the resonator is performed. Using the asymptotic method of many scales, an approximate analytical solution is obtained for the problem of nonlinear system dynamics under the conditions of the main parametric resonance. The fundamental technical feasibility of laser generation of parametric oscillations of high-Q micromechanical resonators without the implementation of scenarios for the loss of elastic stability of the sensitive element or its unacceptable heating is shown.

In addition, the results of a study of the dynamics of parametrically excited bending vibrations of two weakly coupled beam microcavities under laser thermo-optical excitation are reported. The zones of possible buildup of parametric oscillations and the amplitudes of steady-state modes for both resonators are analytically obtained. It is shown that a small difference in the mass-inertia characteristics of the resonators leads to a significant change in the amplitudes of the steady-state modes for each resonator, which can be used to detect the mass of a particle deposited on one of the sensitive elements.

**
Alexei V. Lukin ** –
Associate Professor, Ph.D. High School of Mechanics and Control Processes, Physics and Mechanics Institute, Peter the Great St. Petersburg Polytechnic University. Research interests: computational mechanics, waves in continuous media, nonlinear theory of elasticity, theory of nonlinear oscillations, gyroscopy and navigation, nonlinear dynamics and control N/MEMS.

**
Lev V. Shtukin ** –
Associate Professor, Ph.D. High School of Mechanics and Control Processes, Physics and Mechanics Institute, Peter the Great St. Petersburg Polytechnic University. Research interests: computational mechanics, waves in continuous media, nonlinear theory of elasticity, theory of nonlinear oscillations, gyroscopy and navigation, nonlinear dynamics and control N/MEMS.

### October 18, 2022

### ONLINE

Khrapov S. S. (Volgograd State University, VolSU)

Numerical modeling of hydrodynamic accidents: erosion of dams and flooding of territories

A mathematical and numerical model of the joint dynamics of shallow waters and traction sediments is constructed, which takes into account the nonlinear dynamics of the fluid and bottom deformations. Shallow water dynamics is described by the Saint-Venant equations, taking into account the spatially inhomogeneous distribution of the terrain. The transport of entrained sediments is described by the Exner equation, generalized to the case of a spatially inhomogeneous distribution of the parameters of the underlying surface. The numerical model includes a digital terrain model (DTM) and a numerical method for integrating a system of equations describing the joint dynamics of shallow water and sediment. For the numerical integration of the Saint-Venant and Exner equations, a stable and well-tested CSPH-TVD method of the second order of accuracy is used, the parallel CUDA algorithm of which is implemented as the EcoGIS-Simulation software package for high-performance computing on supercomputers with graphics coprocessors (GPUs). Hydrodynamic modeling of the processes of erosion of the enclosing dam of a real hydrotechnical facility and flooding of adjacent territories was carried out. The parameters of the opening of the enclosing dam and flood zones, formed as a result of the development of a hydrotechnical accident at the tailing dump, were determined. Based on the obtained results, it was concluded that the method proposed in the work for numerical modeling of the joint dynamics of shallow water and traction sediments can be more versatile and efficient (has significantly better accuracy and performance) compared to existing methods for calculating the parameters of the opening and flood zones, especially for hydrodynamic currents with complex geometry on an inhomogeneous bottom topography.

**
Khrapov Sergey S. ** –
Associate Professor, Department of Information Systems and Computer Modeling, Volgograd State University, Ph.D.
Research interests: mathematical modeling of hydrodynamic flows, numerical methods for integrating the Saint-Venant and Exner equations, self-consistent dynamics of surface water and sediment, calculation of flood zones in case of accidents at hydraulic facilities, breakthrough and erosion of dams/dams, parallel computing on computers with graphics processors (GPUs), parallel CUDA algorithms.

### September 6, 2022

Smirnov A. S. (Peter the Great SPbPU, IPME RAS)

Optimization issues in ballistics problems
(based on the master's thesis by D. A. Penchikov
and with the participation of B. A. Smolnikov)

А series of ballistic trajectory optimization problems is considered in the report. A presentation of the classical optimization problem of hitting a given point with a minimum initial energy is given. A two-factor criterion is constructed that combines in its structure the energy and time characteristics of the flight and allows to find the best compromise between them. An optimization criterion related to the accuracy of hitting the target is considered. The problems of flight to a maximum range with a given initial speed and of flight with a minimum initial speed to a given range are discussed, taking into account the viscous resistance from the environment. For the problem of the optimal throwing of a point in a medium with a quadratic resistance according to the criterion of the maximum flight range, the known approximate solution is compared with the results of a numerical analysis based on exact formulas. In addition, the problem of maximum range flight from a cycloidal springboard is considered, which is important for ski jumping. The obtained results are presented in a visual graphical form, and they are interesting not only from a theoretical point of view, but may also be useful in solving various applied problems.