# Energetic approach to direct and inverse heat conduction problems with Trefftz functions used in FEM

### Abstract

In the paper the stationary 2D inverse heat conduction problems are considered. To obtain an approximate solution of the problems three variants of the FEM with harmonic polynomials (Trefftz functions for Laplace equation) as base functions were used: the continuous FEMT, the non-continuous FEMT and the nodeless FEMT. In order to ensure physical sense of the approximate solution, one of the aforementioned physical aspects is taken into account as a penalty term in the functional, which is to be minimized in order to solve the problem. Three kinds of physical aspects that can smooth the solution were used in the work. The first is the minimization of heat flux jump between the elements, the second is the minimization of the defect of energy dissipation and third is the minimization of the intensity of numerical entropy production. The quality of the approximate solutions was verified on two test examples. The method was applied to solve inverse problem of stationary heat transfer in a rib.

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### References

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**Computer Assisted Methods in Engineering and Science**, [S.l.], v. 15, n. 3-4, p. 171-182, july 2022. ISSN 2956-5839. Available at: <https://cames.ippt.pan.pl/index.php/cames/article/view/726>. Date accessed: 14 sep. 2024.