Mathematical modeling and optimization of multilayer systems

Abstract

In this work, the problem of calculation of temperature fields for multilayer microbiological material, which is exposed to electromagnetic effect of laser beam. The main task is to optimize technical parameters of laser emitters taking in consideration limitations on distribution of temperature fields in microbiological material and to decrease losses of expensive microbiological material. Such a formulation of the main optimization problem is typical for many biotechnological processes, particularly for laser division of an embryo or implementation of embryo cells transplantation. It should be noticed that the research conducted by the author of this work has universal character. It means that changing the set of methods during numerical implementation of applied optimization mathematical model of the process of laser effect on microbiological material, the research data can be applied for calculation of temperature fields for other multilayer technical systems, for example, for modeling of biological material welding; using in practice of microsurgery, ophthalmology, cosmetology and other industries.

                In the article, the object of study is an embryo exposed to laser effect in order to divide it. At the same time, setting and implementation of a range of applied optimization mathematical models of the process of laser effect on an embryo is needed for decreasing the expenditures of thermal and power resources of laser emitters and also for decreasing uncontrollable losses of embryo cells during the process of laser effect. It should be noticed that there is a particularity of optimization of a complicated system “embryo – laser” multiple solution of a range of boundary-value problems in the process of modeling is necessary for optimization of technical parameters of laser emitters. The novelty of the research conducted by the author is that during setting and numerical implementation of calculated mathematical model (boundary-value problem), which describes process of laser effect on an embryo, spherical form and three-layer, nonlinear, heterogeneous  internal structure of an embryo is taken in consideration.

The mathematical model of process of laser effect on an embryo was constructed by the author and its numerical implementation was conducted. As it was shown by the author, for taking in consideration spherical, three-layer, nonlinear, heterogeneous structure of an embryo, the boundary-value problem of four multidimensional, unsteady, heterogeneous differential heat equations in spherical coordinate system is fundamental for calculated mathematical model. During setting conditions of thermal contact in an embryo for calculation of time and space coordinates of control points of distribution of temperature fields in an embryo the author has used the formulae of non-uniform and uniform grid respectively. In this work the temperatures of laser heating of embryo layers were found with the help of the method of separation of variables.

     Having the information about temperature field distribution in the layers of an embryo it becomes possible to begin implementation of applied optimization mathematical models. At the stage of setting and implementation of applied optimization mathematical models accounting of complicated geometry and multilayer (three-layer) internal structure of an embryo will allow to decrease the expenditures of thermal and power resources of laser emitters and also to increase embryo cells survival during transplantation.