Modeling of the three-dimensional thermal and stress-strain state of elastic bodies using mixed variational formulations of the finite-element method. Report 1

The basic solving relationships of the finite-element method for three-dimensional nonstationary problems of heat conduction using local mixed variational formulations for an octagonal isoparametric element are obtained. To reduce the order of the system of algebraic equations, the components of the thermal-flux vector (CTFV) are expressed in terms of nodal temperature values from stationary conditions relative to the variation in the CTFV, which are written for a single element. The CTFV are approximated by linear Legendre polynomials. Consideration of the orthogonality of the approximating functions within the limits of an element makes it possible to avoid the formulation and inversion of the corresponding matrices relating the CTFV and temperature. The accuracy and stability of the various schemes of discretization are analyzed as an example of the determination of the thermal state in a cube with mixed boundary conditions.