and Computational Methods

doi: 10.18698/2309-3684-2022-1-109128

Modelling and analysis methods for economic characteristics variation in the innovation process have become a common technique, via employing diffusion equations for a medium with given parameters. The analysis results in this case significantly depend on the measurement accuracy of the industrial environment parameters, which is hard to achieve in practice. It seems, therefore, reasonable to make a transition from the diffusion paradigm to the innovation implementation paradigm, i.e., sequential modelling of the innovation states with variables and characteristics that correspond to the practical measurement and control techniques. Applying the described approach, the economic state dynamics of the innovation development work, manufacturing and implementation can be described by systems of ordinary differential equations, where the initial conditions and coefficients depend on the parameters of the industry’s internal andexternal environments. Two discrete mathematical models developed in this work enable control of the industrial environment parameters, via application of practical measurement methods. The first discrete model is in the form of a functional (mapping), which enables conversion of the actual internal industrial environment parameters in the beginning of the innovation scaling into the coefficients of the differential equations and initial conditions that reflect the results of manufacturing process preparation. The initial data is available from the EPR data base of the industry. The second discrete model is realized as a cellular automaton. An autonomous model of the external industrial environment uses the data that can be measured by the well-developed marketing methods. The results of the computational experiments support the hypothesis of transition from the diffusion model paradigm to the paradigm of the sequential modelling of the innovation economic states.

Белов В.Ф., Гаврюшин С.С., Маркова Ю.Н., Занкин А.И. Моделирование среды предприятия с использованием дискретных вычислительных алгоритмов.Математическое моделирование и численные методы, 2022, № 1, с. 109–128

doi: 10.18698/2309-3684-2021-1-110131

Development of an analytical method for economic characteristics variation in an innovation process is a challenging task, starting from invention or entrepreneur idea and finishing with market implementation. The practical purpose here is both to minimize the risks and shorten design and implementation time. Theoretical and experimental results presented in the paper show the possibility to solve the mentioned task via applying a non-autonomous differential equation system along with the Lyapunov's First Method for stability analysis. A mathematical model of flow of funds related to the manufacturing and market implementation of the engineering innovation has been studied. It is presented in the form of the system of differential balance equations with unit impulse function in the right-hand side. An algorithm has been developed to analyze the stability of the equilibrium states of the manufacturing process, considering the influence of the external environment “from the right” (preparation of manufacturing) as well as “from the left” (market state). The requirements are determined regarding both the discrete model of manufacturing preparation to formulate initial conditions for the non-autonomous system and the discrete model of the market to calculate the market state dependent coefficients for the system of differential equations. The results of analysis of the economic characteristics variation in the stage of product manufacturing have been presented as 3D phase images.

Белов В.Ф., Гаврюшин С.С., Маркова Ю.Н. Неавтономная система как модель процесса производства технической инновации. Математическое моделирование и численные методы, 2021, № 1, с. 110–131.

doi: 10.18698/2309-3684-2014-1-115130

The article shows theoretical bases of the methods and algorithms developed to analyze the stability and supercritical behavior of thin elastic shells. The author deals with the problem of numerical analysis of nonlinear deformation of the spherical dome loaded with uniform external pressure. An algorithm for the numerical analysis method based on the parameter continuation method combined with the method of the subspace change of control parameters. The author illustrates the effectiveness of the proposed algorithm by sample calculations.

Gavryushin S. Numerical simulation of nonlinear deformation of thin elastic shells. Маthematical Modeling and Coтputational Methods, 2014, №1 (1), pp. 115-130

doi: 10.18698/2309-3684-2016-4-103121

The article centers on developing the models of complex engineering structures for monitoring their operational status. In order to organize effective monitoring of the structures state, we often need to restore the state parameters values according to the measuremen tresults and use monitoring object models. The study suggests using a model built by calculation and reduced to small size. Among the state parameters there may be either load values and their combinations on which direct restrictions are imposed, or limit states defined by their values. The work proposes both criteria for assessing the quality of the reduced model, and an efficient algorithm for searching them.

Meschikhin I., Gavryushin S. Quality criteria and algorithm for selecting reduced finite element models for technical design monitoring. Маthematical Modeling and Coтputational Methods, 2016, №4 (12), pp. 103-121