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ABSTRACT
This article proposes a numerical formulation for handling mixed-dimensional elements embedded in a standard three-dimensional (3d) mesh, avoiding thus the volume meshing of filaments and strips. The method is then applied to predict the temperature heating and cooling profile of gradient coils in magnetic resonance imaging. These coils are typically constructed from copper wires or tracks and embedded in an epoxy layer. It was found that the new method significantly reduces the computational time of steady-state and transient simulations, with speedups in the range of 3.5-5. The method proved to be accurate, with relative errors below 0.5% for steady-state simulations and 1.5% with respect to a complete 3d simulation.
Nomenclature
| c | = | specific heat (J kg−1 K−1) |
| = | (i, k) entries of the discrete divergence operator | |
| = | discrete divergence matrix operator on the dual grid | |
| = | temperature gradient (K m−1) | |
| gik | = | (i, k) entries of the discrete gradient operator |
| G | = | discrete gradient matrix operator on the primal grid |
| 𝒦 | = | primal cell complex |
| = | dual cell complex | |
| Mλ | = | thermal conductance constitutive matrix (W K−1) |
| Mρc | = | thermal mass constitutive matrix (J K−1) |
| p | = | thermal power production (W) |
| = | heat flux density (W m−2) | |
| T | = | temperature (K) |
| U | = | internal energy density (J m−3) |
| u | = | internal energy (J) |
| γ | = | temperature difference (K) |
| λ | = | thermal conductivity (W m−1 K−1) |
| Φ | = | heat flux (W) |
| ρ | = | volumetric mass density (kg m−3) |
| σ | = | heat source (W m−3) |
Nomenclature
| c | = | specific heat (J kg−1 K−1) |
| = | (i, k) entries of the discrete divergence operator | |
| = | discrete divergence matrix operator on the dual grid | |
| = | temperature gradient (K m−1) | |
| gik | = | (i, k) entries of the discrete gradient operator |
| G | = | discrete gradient matrix operator on the primal grid |
| 𝒦 | = | primal cell complex |
| = | dual cell complex | |
| Mλ | = | thermal conductance constitutive matrix (W K−1) |
| Mρc | = | thermal mass constitutive matrix (J K−1) |
| p | = | thermal power production (W) |
| = | heat flux density (W m−2) | |
| T | = | temperature (K) |
| U | = | internal energy density (J m−3) |
| u | = | internal energy (J) |
| γ | = | temperature difference (K) |
| λ | = | thermal conductivity (W m−1 K−1) |
| Φ | = | heat flux (W) |
| ρ | = | volumetric mass density (kg m−3) |
| σ | = | heat source (W m−3) |
Acknowledgements
The authors acknowledge funding support from MedTeQ (Qld), the Australian Research Council, and The National Health and Medical Research Council of Australia.
Notes
Strictly speaking, global variables are associated to space and time entities. This representation, although mathematically elegant, is more difficult to understand and thus it is not used in this work.