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ABSTRACT
This work describes the fundamentals of the element-based finite volume method for anisotropic heat conduction within the framework of the finite element space. Patch tests indicate no element inconsistencies or deficiencies when facing mesh distortion and poor aspect ratio. Convergence and accuracy assessments show that the method presents asymptomatic rate of convergence with discretization errors approaching a second-order scheme. Anisotropic heat conduction in a periodical solid lattice illustrates the application of the method. Application of an optimization technique demonstrates that the choice of a proper material orientation when manufacturing the solid lattice can increase the global heat transfer coefficient.
Nomenclature
| Ac | = | heat convection area |
| c | = | specific heat |
| h | = | element/mesh size |
| = | average mesh size | |
| hc | = | convective heat transfer coefficient |
| Hl | = | volume associated with node Nl |
| Jij | = | Jacobian |
| kij | = | anisotropic conductivity tensor |
| k | = | isotropic conductivity |
| ne | = | number of elements |
| ni | = | outward normal unit vector |
| nl | = | number of local nodes |
| np | = | number of control volumes |
| ns | = | number of control surfaces of an element that compounds control volume Hl |
| nf | = | number of elements that shares a node |
| Nl | = | global node |
| p | = | polynomial order of the interpolation function |
| ph | = | error order |
| q | = | local node number |
| qi | = | heat flux |
| r | = | mesh refinement ratio |
| = | average mesh refinement ratio | |
| rp | = | position radius for patch tests |
| R | = | rotation tensor (Rij) |
| = | heat source/sink | |
| Sh | = | finite element space |
| T | = | temperature |
| u | = | internal energy |
| U | = | global heat transfer coefficient |
| xi | = | cartesian coordinates |
| Xk | = | element |
| wkl | = | weight function |
| Greek letters | = | |
| β | = | hot surface coordinate angle |
| εmnp | = | permutation tensor |
| = | exact error | |
| = | Richardson error | |
| θ | = | rotation angle of a tensor |
| θp | = | rotation angle for patch tests |
| ξi | = | local coordinates of the element |
| Ξkl | = | discretization matrix of a finite volume |
| ρ | = | specific mass |
| ϕ | = | interpolation function |
| Ω | = | continuum domain |
| ΩD | = | discrete domain |
| = | discrete boundary |
Nomenclature
| Ac | = | heat convection area |
| c | = | specific heat |
| h | = | element/mesh size |
| = | average mesh size | |
| hc | = | convective heat transfer coefficient |
| Hl | = | volume associated with node Nl |
| Jij | = | Jacobian |
| kij | = | anisotropic conductivity tensor |
| k | = | isotropic conductivity |
| ne | = | number of elements |
| ni | = | outward normal unit vector |
| nl | = | number of local nodes |
| np | = | number of control volumes |
| ns | = | number of control surfaces of an element that compounds control volume Hl |
| nf | = | number of elements that shares a node |
| Nl | = | global node |
| p | = | polynomial order of the interpolation function |
| ph | = | error order |
| q | = | local node number |
| qi | = | heat flux |
| r | = | mesh refinement ratio |
| = | average mesh refinement ratio | |
| rp | = | position radius for patch tests |
| R | = | rotation tensor (Rij) |
| = | heat source/sink | |
| Sh | = | finite element space |
| T | = | temperature |
| u | = | internal energy |
| U | = | global heat transfer coefficient |
| xi | = | cartesian coordinates |
| Xk | = | element |
| wkl | = | weight function |
| Greek letters | = | |
| β | = | hot surface coordinate angle |
| εmnp | = | permutation tensor |
| = | exact error | |
| = | Richardson error | |
| θ | = | rotation angle of a tensor |
| θp | = | rotation angle for patch tests |
| ξi | = | local coordinates of the element |
| Ξkl | = | discretization matrix of a finite volume |
| ρ | = | specific mass |
| ϕ | = | interpolation function |
| Ω | = | continuum domain |
| ΩD | = | discrete domain |
| = | discrete boundary |