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ABSTRACT
The aim of this work is the development of a space–time diffuse approximation meshless method (DAM) to solve heat equations containing discontinuous sources. This work is devoted to transient heat transfer problems with static and moving heat sources applied on a metallic plate and whose power presents temporal discontinuities. The space–time DAM using classical weight function is convenient for continuous transient heat transfer. Nevertheless, for problems including discontinuities, some spurious oscillations for the temperature field occur. A new weight function, respecting the principle of causality, is used to eradicate the physically unexpected oscillations.
Nomenclature
| A | = | calculation matrix |
| B | = | matrix system vector |
| Bi | = | Biot number |
| cx, cy, ct | = | distance coefficients |
| cp | = | specific heat capacity |
| h | = | convection exchange coefficient |
| k | = | number of neighbors |
| Lref | = | characteristic distance |
| N | = | number of node |
| P | = | polynomial basis function |
| Pref | = | reference power per unit volume |
| Rc | = | radius of influence using corrected distance. |
| r | = | dimensionless distance |
| S | = | dimensionless heat source |
| t | = | real time |
| T | = | temperature in°C |
| Tref | = | reference temperature |
| x,y,τ | = | dimensionless space–time coordinates |
| xi,yi,τi | = | dimensionless distance between calculation node and neighboring node |
| X | = | node location vector |
| Greek symbols | = | |
| α | = | vector of coefficients |
| α | = | thermal diffusivity |
| Δx | = | dimensionless x space step |
| Δy | = | dimensionless y space step |
| Δτ | = | dimensionless time step |
| θ | = | dimensionless temperature |
| λ | = | thermal conductivity |
| ρ | = | density |
| σ | = | weight function support radius |
| Φ | = | scalar field |
| ω | = | weight function |
| Ω | = | domain |
| Subscripts | = | |
| c | = | corrected |
| i, j | = | node index |
| ext | = | external |
| end | = | end of time calculation |
| init | = | initial |
| M | = | point |
| x, y, τ | = | x or y space, time |
Nomenclature
| A | = | calculation matrix |
| B | = | matrix system vector |
| Bi | = | Biot number |
| cx, cy, ct | = | distance coefficients |
| cp | = | specific heat capacity |
| h | = | convection exchange coefficient |
| k | = | number of neighbors |
| Lref | = | characteristic distance |
| N | = | number of node |
| P | = | polynomial basis function |
| Pref | = | reference power per unit volume |
| Rc | = | radius of influence using corrected distance. |
| r | = | dimensionless distance |
| S | = | dimensionless heat source |
| t | = | real time |
| T | = | temperature in°C |
| Tref | = | reference temperature |
| x,y,τ | = | dimensionless space–time coordinates |
| xi,yi,τi | = | dimensionless distance between calculation node and neighboring node |
| X | = | node location vector |
| Greek symbols | = | |
| α | = | vector of coefficients |
| α | = | thermal diffusivity |
| Δx | = | dimensionless x space step |
| Δy | = | dimensionless y space step |
| Δτ | = | dimensionless time step |
| θ | = | dimensionless temperature |
| λ | = | thermal conductivity |
| ρ | = | density |
| σ | = | weight function support radius |
| Φ | = | scalar field |
| ω | = | weight function |
| Ω | = | domain |
| Subscripts | = | |
| c | = | corrected |
| i, j | = | node index |
| ext | = | external |
| end | = | end of time calculation |
| init | = | initial |
| M | = | point |
| x, y, τ | = | x or y space, time |