Scientists at Novosibirsk State Technical University (NSTU) are developing computational schemes for mathematical modeling of the heat transfer process in heterogeneous media based on modern finite element methods. The practical value of the development lies in solving many complex applied problems in the fields of geophysics, biology, construction, and materials science. The work is carried out within the framework of a grant from the Russian Science Foundation.
According to the project leader, Associate Professor Natalia Itkina, Head of the Department of Computing Technologies at NSTU-NETI, many physical processes in nature and technology are associated with the transitions of substances from one phase state to another, such as: melting of ice, crystallization of melts, decomposition of gas hydrates during heating, welding and many others. Mathematical modeling of physical processes, which consists of the following stages, plays an important role in the study of these phenomena.: The problem statement is the choice of a mathematical model; the choice of a method for solving a system of differential equations describing this problem.
Mathematically, the problem of determining the temperature field of a phase-changing substance (a substance that changes its state of aggregation) belongs to the class of problems with moving boundaries. The solution area consists of subdomains with matter in various states of aggregation, for example, solid and liquid (ice melting), liquid and gaseous (liquid evaporation). In some tasks, all three states are present, and over time, the boundary between the subdomains changes its location. When developing software and algorithmic support, two problems arise: it requires high accuracy in determining both the moving boundary at a given time and the value of the temperature field in the vicinity of this boundary, explains the scientist.
The most effective methods for solving the problems of modeling the heat transfer process in phase—changing media are finite element methods, especially those with extensive capabilities for constructing complex grids. In recent decades, such a finite element method as the discontinuous Galerkin method has been actively developing. Its advantage lies in the flexibility and simplicity of constructing a finite element grid, that is, in the possibility of locally changing the size and shape of the final element. This method is one of the most effective for problems with changing boundaries, the solution of which can be discontinuous. The second innovative numerical method to be implemented in the project is the virtual element method, which makes it possible to determine the solution to a problem with star-shaped inclusions (for example, composite materials).
"The goal of the project is to develop software and algorithmic support based on new schemes for numerical modeling of physical processes in multiscale heterogeneous media when solving applied problems related to heat transfer and phase transformations," Natalia Itkina said.
The scientific novelty of the project lies in the development and practical implementation of computational schemes of the modified discontinuous Galerkin method and the virtual finite element method for solving problems of modeling heat transfer processes with explicit tracking of the phase transition front in a three-dimensional formulation. From a practical point of view, the relevance of solving these problems is related to the need to study processes such as permafrost melting, dissociation of gas hydrates, conservation of hazardous chemical compounds, creation of phase-changing materials, etc.
Thus, solving the problem of mathematical modeling of processes in gas hydrates will make it possible to predict how they will behave when conditions change, which is important to prevent their uncontrolled destruction (for example, under the influence of rising temperatures). Such destruction can have serious consequences for the oil and gas industry (technological accidents) and for the environment. Hydrates store huge amounts of hydrocarbons, and their dissociation under the influence of warming can lead to greenhouse gas emissions, and modeling helps to assess these risks.
Computational schemes allow us to determine the influence of anthropogenic factors on changes in soil structure in the permafrost zone. This is important when constructing structures, laying pipelines, etc.
The theoretical value of the project lies in the development of modern numerical analysis methods focused on high-performance computing systems, as well as in the development of domestic software, which contributes to the strategic independence and scientific and technological development of Russia.
The project "Multiscale numerical modeling of the heat transfer process with phase transformations based on nonconformal and virtual finite element methods" was included in the list of supported projects following the results of the 2025 competition for grants from the Russian Science Foundation. Grants are allocated for the implementation of fundamental and exploratory scientific research.