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ABSTRACT
A new radial integration boundary element method (RIBEM) for solving transient heat conduction problems with heat sources and variable thermal conductivity is presented in this article. The Green’s function for the Laplace equation is served as the fundamental solution to derive the boundary-domain integral equation. The transient terms are first discretized before applying the weighted residual technique that is different from the previous RIBEM for solving a transient heat conduction problem. Due to the strategy for dealing with the transient terms, temperature, rather than transient terms, is approximated by the radial basis function; this leads to similar mathematical formulations as those in RIBEM for steady heat conduction problems. Therefore, the present method is very easy to code and be implemented, and the strategy enables the assembling process of system equations to be very simple. Another advantage of the new RIBEM is that only 1D boundary line integrals are involved in both 2D and 3D problems. To the best of the authors’ knowledge, it is the first time to completely transform domain integrals to boundary line integrals for a 3D problem. Several 2D and 3D numerical examples are provided to show the effectiveness, accuracy, and potential of the present RIBEM.
Nomenclature
| A | = | coefficients matrix |
| cp | = | mass specific heat, J/(kg.°C) |
| G | = | Green function |
| H | = | heat convective coefficient, W/(m2.°C) |
| L | = | boundary line |
| K | = | thermal conductivity, W/(m.°C) |
| N | = | total number of nodes |
| N | = | unit outward normal |
| Q | = | heat source, W/m3 |
| Q | = | heat flux, W/m2 |
| R | = | distance, m |
| R | = | distance, m |
| T | = | temperature,°C |
| T | = | time, s |
| X | = | vector of coordinates |
| Y | = | source point |
| Z | = | point on boundary |
| z′ | = | point on boundary |
| Greek symbols | ||
| A | = | vector |
| Γ | = | boundary |
| Δ | = | change in variable |
| P | = | density, W/m3 |
| Φ | = | matrix |
| Ω | = | domain |
| Subscripts | ||
| 0 | = | Initial |
| B | = | boundary |
| Back | = | back |
| Bottom | = | bottom |
| F | = | ambient surrounding |
| Front | = | front |
| I | = | internal |
| Left | = | left |
| Right | = | right |
| Upper | = | upper |
Nomenclature
| A | = | coefficients matrix |
| cp | = | mass specific heat, J/(kg.°C) |
| G | = | Green function |
| H | = | heat convective coefficient, W/(m2.°C) |
| L | = | boundary line |
| K | = | thermal conductivity, W/(m.°C) |
| N | = | total number of nodes |
| N | = | unit outward normal |
| Q | = | heat source, W/m3 |
| Q | = | heat flux, W/m2 |
| R | = | distance, m |
| R | = | distance, m |
| T | = | temperature,°C |
| T | = | time, s |
| X | = | vector of coordinates |
| Y | = | source point |
| Z | = | point on boundary |
| z′ | = | point on boundary |
| Greek symbols | ||
| A | = | vector |
| Γ | = | boundary |
| Δ | = | change in variable |
| P | = | density, W/m3 |
| Φ | = | matrix |
| Ω | = | domain |
| Subscripts | ||
| 0 | = | Initial |
| B | = | boundary |
| Back | = | back |
| Bottom | = | bottom |
| F | = | ambient surrounding |
| Front | = | front |
| I | = | internal |
| Left | = | left |
| Right | = | right |
| Upper | = | upper |