# Comparison of Computational Complexity of Boundary Element Method and Finite Element Method for Thermal Modeling of Pin-Fin Heat Sink Rod

### Аuthors

Dobryakov V. A., Engalichev A. N.*, Nazarov A. V.**

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

*e-mail: aengalychev@inbox.ru
**e-mail: rat-rut@yandex.ru

### Abstract

The purpose of the current work is tо compare the computational complexity of the boundary element method and finite element method for the numerical example of thermal modeling of the technical object.
The rod with insulated lateral surface was selected as the object of investigation. For the correct comparison of the methods mentioned above for thermal modeling, there was derived concrete numerical expressions allowing to calculate the exact number of instructions that are implementing both methods for the typical computer system.
It is shown that one needs to solve the only linear equation with one unknown for the steady-state calculation of the temperature field provided by boundary element method at any given point of the inner rod. And such procedure typically requires only a few machine instructions. Overcoming the same problem using finite element method is minimally required to solve a system of three linear equations with three unknowns; this can take tens of machine operations.
Unlike similar works on this topic, the advantage of boundary element method over the finite element method is shown in this article with a simple example which allows one to understand clearly the cause of this advantage. The equivalency of results obtained with both methods confirms the linearity of the change in the thermal field of the object.

### Keywords:

boundary element method, finite element method, pin-fin heat sink, state variable, boundary conditions, the influence function, fundamental solution, the simplex-element, interpolation polynomial

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