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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.

The paper reviews the existing approaches to calculating the destruction of solids. The main attention is paid to algorithms using a unified approach to the calculation of deformation both for nondestructive and for the destroyed states of the material. The thermodynamic derivation of the unified rheological relationships taking into account the elastic, viscous and plastic properties of materials and describing the loss of the deformation resistance ability with the accumulation of microdamages is presented. It is shown that the mathematical model under consideration provides a continuous dependence of the solution on input parameters (parameters of the material medium, initial and boundary conditions, discretization parameters) with softening of the material.

Explicit and implicit non-matrix algorithms for calculating the evolution of deformation and fracture development are presented. Non-explicit schemes are implemented using iterations of the conjugate gradient method, with the calculation of each iteration exactly coinciding with the calculation of the time step for two-layer explicit schemes. So, the solution algorithms are very simple.

The results of solving typical problems of destruction of solid deformable bodies for slow (quasistatic) and fast (dynamic) deformation processes are presented. Based on the experience of calculations, recommendations are given for modeling the processes of destruction and ensuring the reliability of numerical solutions.

In this paper, we proposed a two-dimensional chemo-mechanical model of the growth of invasive carcinoma in epithelial tissue. Each cell is modeled by an elastic polygon, changing its shape and size under the influence of pressure forces acting from the tissue. The average size and shape of the cells have been calibrated on the basis of experimental data. The model allows to describe the dynamic deformations in epithelial tissue as a collective evolution of cells interacting through the exchange of mechanical and chemical signals. The general direction of tumor growth is controlled by a pre-established linear gradient of nutrient concentration. Growth and deformation of the tissue occurs due to the mechanisms of cell division and intercalation. We assume that carcinoma has a heterogeneous structure made up of cells of different phenotypes that perform various functions in the tumor. The main parameter that determines the phenotype of a cell is the degree of its adhesion to the adjacent cells. Three main phenotypes of cancer cells are distinguished: the epithelial (E) phenotype is represented by internal tumor cells, the mesenchymal (M) phenotype is represented by single cells and the intermediate phenotype is represented by the frontal tumor cells. We assume also that the phenotype of each cell under certain conditions can change dynamically due to epithelial-mesenchymal (EM) and inverse (ME) transitions. As for normal cells, we define the main E-phenotype, which is represented by ordinary cells with strong adhesion to each other. In addition, the normal cells that are adjacent to the tumor undergo a forced EM-transition and form an M-phenotype of healthy cells. Numerical simulations have shown that, depending on the values of the control parameters as well as a combination of possible phenotypes of healthy and cancer cells, the evolution of the tumor can result in a variety of cancer structures reflecting the self-organization of tumor cells of different phenotypes. We compare the structures obtained numerically with the morphological structures revealed in clinical studies of breast carcinoma: trabecular, solid, tubular, alveolar and discrete tumor structures with ameboid migration. The possible scenario of morphogenesis for each structure is discussed. We describe also the metastatic process during which a single cancer cell of ameboid phenotype moves due to intercalation in healthy epithelial tissue, then divides and undergoes a ME transition with the appearance of a secondary tumor.