Calculation of Three-Dimensional Stationary Potential Thermal Fields by the Finite Element Method

The article describes a clear algorithm for forming a system of nonlinear algebraic equations from the condition of the minimum of the functional to calculate the temperature distribution inside the three-dimensional domain filled with hysteresis-free nonlinear anisotropic environments. The boundary value problem of calculating the three-dimensional stationary potential temperature field is formulated. The finite element model for calculating the temperature distribution inside a three-dimensional domain filled with hysteresis-free nonlinear anisotropic environments is built. Basic formulas of the finite element method for the boundary value problem of calculating three-dimensional stationary potential thermal fields in domains filled with hysteresis-free nonlinear anisotropic environments with using Lagrangian tetrahedrons 1– 4 orders as finite elements and cubature formulas of numerical integration over the volume of the Lagrangian tetrahedron are derived. The algorithm determining the contribution of each finite element to the vector of residuals and matrix Jacobi nonlinear system of equations solved by Newton’s method using the elements of the tensor of the differential thermal conductivity of the environment was considered. The cubature formula of numerical integration over the volume of the Lagrangian tetrahedron based on the interpolation complete polynomial for the Lagrangian finite element of the first order was applied. This algorithm is suitable when using Lagrangian finite elements of the second, third, and fourth orders with the use of the corresponding cubature formulas for numerical integration over the volume of the Lagrangian tetrahedron. Formulas for calculating tensors of differential thermal conductivity for non-linear isotropic and linear environments are given. The boundary conditions of Dirichlet (first kind), Neumann (second kind), third and fourth kind and their consideration are described.