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In the science development an important role the scientific schools are played. This schools are the associations of researchers connected by the common problem, the ideas and the methods used for problems solution. Usually Scientific schools are formed around the leader and the uniting idea.

The several sciences schools were created around academician A. S. Kholodov during his scientific and pedagogical activity.

This review tries to present the main scientific directions in which the bright science collectives with the common frames of reference and approaches to researches were created. In the review this common base is marked out. First, this is development of the group of numerical methods for hyperbolic type systems of partial derivatives differential equations solution — grid and characteristic methods. Secondly, the description of different numerical methods in the undetermined coefficients spaces. This approach developed for all types of partial equations and for ordinary differential equations.

On the basis of A. S. Kholodov’s numerical approaches the research teams working in different subject domains are formed. The fields of interests are including mathematical modeling of the plasma dynamics, deformable solid body dynamics, some problems of biology, biophysics, medical physics and biomechanics. The new field of interest includes solving problem on graphs (such as processes of the electric power transportation, modeling of the traffic flows on a road network etc).

There is the attempt in the present review analyzed the activity of scientific schools from the moment of their origin so far, to trace the connection of A. S. Kholodov’s works with his colleagues and followers works. The complete overview of all the scientific schools created around A. S. Kholodov is impossible due to the huge amount and a variety of the scientific results.

The attempt to connect scientific schools activity with the advent of scientific and educational school in Moscow Institute of Physics and Technology also becomes.

Currently, different nonlinear numerical schemes of the spatial approximation are used in numerical simulation of boundary value problems for hyperbolic systems of partial differential equations (e. g. gas dynamics equations, MHD, deformable rigid body, etc.). This is due to the need to improve the order of accuracy and perform simulation of discontinuous solutions that are often occurring in such systems. The need for non-linear schemes is followed from the barrier theorem of S. K. Godunov that states the impossibility of constructing a linear scheme for monotone approximation of such equations with approximation order two or greater. One of the most accurate non-linear type schemes are ENO (essentially non oscillating) and their modifications, including WENO (weighted, essentially non oscillating) scemes. The last received the most widespread, since the same stencil width has a higher order of approximation than the ENO scheme. The benefit of ENO and WENO schemes is the ability to maintain a high-order approximation to the areas of non-monotonic solutions. The main difficulty of the analysis of such schemes comes from the fact that they themselves are nonlinear and are used to approximate the nonlinear equations. In particular, the linear stability condition was obtained earlier only for WENO5 scheme (fifth-order approximation on smooth solutions) and it is a numerical one. In this paper we consider the problem of construction and stability for WENO5, WENO7, WENO9, WENO11, and WENO13 finite volume schemes for the Hopf equation. In the first part of this article we discuss WENO methods in general, and give the explicit expressions for the coefficients of the polynomial weights and linear combinations required to build these schemes. We prove a series of assertions that can make conclusions about the order of approximation depending on the type of local solutions. Stability analysis is carried out on the basis of the principle of frozen coefficients. The cases of a smooth and discontinuous behavior of solutions in the field of linearization with frozen coefficients on the faces of the final volume and spectra of the schemes are analyzed for these cases. We prove the linear stability conditions for a variety of Runge-Kutta methods applied to WENO schemes. As a result, our research provides guidance on choosing the best possible stability parameter, which has the smallest effect on the nonlinear properties of the schemes. The convergence of the schemes is followed from the analysis.

In this paper, we propose a new software for simulation of fluid-structure interaction. The software is designed for solving coupled problems and provides an interface for synchronization synchronisation and data exchange between existing fluid and structural solvers. Algorithms of coupling solvers and mesh deformation are discussed. The software can be used on hybrid CPU/GPU platforms.

The article presents a method for linearization the effective function of material instantaneous deformation in order to generalize the torsional vibration equation to the case of nonlinearly deformable rheologically active shafts. It is considered layered and structurally heterogeneous, on average isotropic shafts made of nonlinearly viscoelastic components. The technique consists in determining the approximate shear modulus by minimizing the root-mean-square deviation in approximation of the effective diagram of instantaneous deformation.

The method allows to estimate analytically values of natural frequencies of layered and structurally heterogeneous nonlinearly viscoelastic shaft. This makes it possible to significantly reduce resources in vibration analysis, as well as to track changes in values of natural frequencies with changing geometric, physico-mechanical and structural parameters of shafts, which is especially important at the initial stages of modeling and design. In addition, the paper shows that only a pronounced nonlinearity of the effective state equation has an effect on the natural frequencies, and in some cases the nonlinearity in determining the natural frequencies can be neglected.

As equations of state of the composite material components, the article considers the equations of nonlinear heredity with instantaneous deformation functions in the form of the Prandtl’s bilinear diagrams. To homogenize the state equations of layered shafts, it is applied the Voigt’s hypothesis on the homogeneity of deformations and the Reuss’ hypothesis on the homogeneity of stresses in the volume of a composite body. Using these assumptions, effective secant and tangential shear moduli, proportionality limits, as well as creep and relaxation kernels of longitudinal, axial and transversely layered shafts are obtained. In addition, it is obtained the indicated effective characteristics of a structurally heterogeneous, on average isotropic shaft using the homogenization method previously proposed by the authors, based on the determination of the material deformation parameters by the rule of a mixture for the Voigt’s and the Reuss’ state equations.

The paper is devoted to the analysis of the proximity of the population system to dangerous boundaries. An intersection of these boundaries results in the collapse of the stable coexistence of interacting populations. As a reason of such destruction one can consider random perturbations inevitably presented in any living system. This study is carried out on the example of the well-known model of interaction between predator and prey populations, taking into account both a stabilizing factor of the competition of predators for another than prey resources, and also a destabilizing saturation factor for predators. To describe the saturation of predators, we use the second type Holling trophic function. The dynamics of the system is studied as a function of the predator saturation, and the coefficient of predator competition for resources other than prey. The paper presents a parametric description of the possible dynamic regimes of the deterministic model. Here, local and global bifurcations are studied, and areas of sustainable coexistence of populations in equilibrium and the oscillation modes are described. An interesting feature of this mathematical model, firstly considered by Bazykin, is a global bifurcation of the birth of limit cycle from the separatrix loop. We study the effects of noise on the equilibrium and oscillatory regimes of coexistence of predator and prey populations. It is shown that an increase of the intensity of random disturbances can lead to significant deformations of these regimes right up to their destruction. The aim of this work is to develop a constructive probabilistic criterion for the proximity of the population stochastic system to the dangerous boundaries. The proposed approach is based on the mathematical technique of stochastic sensitivity functions, and the method of confidence domains. In the case of a stable equilibrium, this confidence domain is an ellipse. For the stable cycle, this domain is a confidence band. The size of the confidence domain is proportional to the intensity of the noise and stochastic sensitivity of the initial deterministic attractor. A geometric criterion of the exit of the population system from sustainable coexistence mode is the intersection of the confidence domain and the corresponding separatrix of the unforced deterministic model. An effectiveness of this analytical approach is confirmed by the good agreement of theoretical estimates and results of direct numerical simulations.

A variant of import into ANSYS Mechanical APDL system of the model of behavior of flexible woven composite materials with reinforcing weaving cloth of linen at static stretching along the reinforcement yarns is offered. The import was carried out using an integration module based on the use of an analytical model of deformation of the material under study. The model is presented in the articles published earlier and takes into account the changes in the geometric structure occurring in the reinforcing layer of the material during the deformation process, the formation of irreversible deformations and the interaction of cross-lying reinforcing fabric threads. In the introduction input characteristics of the plain weave of the reinforcing fabric and the analytical model imported into ANSYS are briefly described. The input parameters of the module are the mechanical characteristics of the materials that make up the composite (binder and material of reinforcement yarns), the geometric characteristics of the interlacing of the reinforcing fabric. The algorithm for importing the model is based on the calculation and transfer in ANSYS of the calculated points of the material stress-strain diagram for uniaxial stretching along the reinforcement direction and using the Multilinear Kinematich Hardening model material embedded in the ANSYS. The analytical model imported with the help of the presented module allows to model a composite material with reinforcing fabric without a detailed description of the geometry of the interlacing of threads during modeling of the material as a whole. The imported model was verified. For verification full-scale experimental studies and numerical simulation of the stretching of samples from flexible woven composites were carried out. The analysis of the obtained results showed good qualitative and quantitative agreement of calculations.

In the work modeling method of anthropomorphic platforms is presented. An elastostatic stiffness model is used to determine positioning errors in the robot’s lower limbs. One of the main problems in achieving a fast and stable gait are deflections caused by the flexibility in the elements of the robot. This problem was solved using virtual joint modeling to predict stiffness and deformation caused by the robot weight and external forces.

To simulate a robot in the single-support phase, the robot is represented as a serial kinematic chain with a base at the supporting leg point of contact and an end effector in the swing leg foot. In the double support phase robot modeled as a parallel manipulator with an end effector in the pelvis. In this work, two cases of stiffness modeling are used: taking into account the compliance of the links and joints and taking into account only the compliance of joints. In the last case, joint compliances also include part of the link compliances. The joint stiffness parameters have been identified for two anthropomorphic robots: a small platform and a full-sized AR-601M.

Deflections maps were calculated using identified stiffness parameters and showing errors depending on the position of the robot end effector in the workspace. The errors in Z directions have maximum amplitude, due to the influence of the robot mass on its structure.