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ABSTRACT
A new boundary domain integral equation with convective heat transfer boundary is presented to solve variable coefficient heat conduction problems. Green’s function for the Laplace equation is used to derive the basic integral equation with varying heat conductivities, and as a result, domain integrals are included in the derived integral equations. The existing domain integral is converted into an equivalent boundary integral using the radial integration method by expressing the normalized temperature as a series of radial basis functions. This treatment results in a pure boundary element analysis algorithm and requires no internal cells to evaluate the domain integral. Numerical examples are presented to demonstrate the accuracy and efficiency of the present method.
Nomenclature
| aik | = | coefficients to be determined |
| ai0 | = | coefficient to be determined |
| αiA | = | coefficient to be determined |
| dA | = | support size for the application |
| Fi1 | = | coefficients to be determined |
| Fi0 | = | coefficient to be determined |
| FiA | = | coefficient to be determined |
| G | = | Green’s function |
| h | = | convection heat transfer coefficient |
| k | = | thermal conductivity (W/m K) |
| = | normalized thermal conductivity | |
| ni | = | component of the outward normal vector |
| Q | = | heat generation rate |
| q | = | heat flux (W/m2) |
| R | = | distance (m) |
| r | = | distance (m) |
| T | = | temperature (K) |
| = | normalized temperature | |
| Tf | = | external environment temperature |
| x | = | space variable (m) |
| y | = | space variable (m) |
| Ω | = | domain of the problem of interest |
| Γ | = | boundary of the domain |
| ϕ | = | radial basis function |
| Subscripts | = | |
| i | = | ith component |
| k | = | iteration number |
| Superscripts | = | |
| α | = | power of the distance |
| c | = | convection heat transfer |
Nomenclature
| aik | = | coefficients to be determined |
| ai0 | = | coefficient to be determined |
| αiA | = | coefficient to be determined |
| dA | = | support size for the application |
| Fi1 | = | coefficients to be determined |
| Fi0 | = | coefficient to be determined |
| FiA | = | coefficient to be determined |
| G | = | Green’s function |
| h | = | convection heat transfer coefficient |
| k | = | thermal conductivity (W/m K) |
| = | normalized thermal conductivity | |
| ni | = | component of the outward normal vector |
| Q | = | heat generation rate |
| q | = | heat flux (W/m2) |
| R | = | distance (m) |
| r | = | distance (m) |
| T | = | temperature (K) |
| = | normalized temperature | |
| Tf | = | external environment temperature |
| x | = | space variable (m) |
| y | = | space variable (m) |
| Ω | = | domain of the problem of interest |
| Γ | = | boundary of the domain |
| ϕ | = | radial basis function |
| Subscripts | = | |
| i | = | ith component |
| k | = | iteration number |
| Superscripts | = | |
| α | = | power of the distance |
| c | = | convection heat transfer |