![Sample our Engineering & Technology journals, sign in here to start your access, latest two full volumes FREE to you for 14 days]({"type":"image" , "src" : "/sda/5933/TOC-Subject-Sample-2021-Engineering-Tech.jpg"})
ABSTRACT
In this article, the generalized mass formulation is developed in an explicit analysis of transient transport problems. It has been well known that the time step is typically smaller in explicit analysis than in implicit analysis when the same size mesh is used. Further, the over-stiffness of conventional finite-element model may result in poor accuracy with linear triangular or tetrahedral elements. In order to improve the computational efficiency and numerical accuracy, this article proposes a generalized mass formulation by matching the mass matrix to the smoothed stiffness matrix using linear triangular elements in 2-D problems. The proposed mass matrix can be obtained by simply shifting the integration points from the conventional locations. Without loss of generality, several 2-D examples, including conduction, convection, and radiation heat transfer problems, are presented to demonstrate that the generalized mass formulation allows a larger time step in explicit analysis compared with the lumped and consistent mass matrices. In addition, it is found that the maximum allowable time step is proportional to the softened effect of the discretized model in an explicit analysis.
Nomenclature
| T | = | Temperature, K |
| X | = | Cartesian coordinate |
| kx, ky, ky | = | Thermal conductivity along x, y and z, W/(m.K) |
| Ta | = | Ambient temperature, K |
| q | = | Prescribed heat flux, W/m2 |
| Q | = | Internal heat source, W/m3 |
| h | = | Convective heat transfer coefficient, W/(m2.K) |
| c | = | Specific heat capacity of medium, J/(kg.K) |
| w | = | Test function |
| N | = | Shape function |
| t | = | Time, s |
| Greek symbols | = | |
| σ | = | Stefan Boltzmann constant, Wm−2K−4 |
| ϵ | = | The emissivity |
| ρ | = | Density kg/m3 |
| Ω | = | Integration or problem domain |
| Γ | = | Global or local boundary |
| κ | = | Thermal conductivity, W/(m.K) |
| Subscripts and superscripts | = | |
| k | = | Smoothing domain for node k |
| T | = | Transpose symbol |
| i, j, I, J | = | Node indices |
Nomenclature
| T | = | Temperature, K |
| X | = | Cartesian coordinate |
| kx, ky, ky | = | Thermal conductivity along x, y and z, W/(m.K) |
| Ta | = | Ambient temperature, K |
| q | = | Prescribed heat flux, W/m2 |
| Q | = | Internal heat source, W/m3 |
| h | = | Convective heat transfer coefficient, W/(m2.K) |
| c | = | Specific heat capacity of medium, J/(kg.K) |
| w | = | Test function |
| N | = | Shape function |
| t | = | Time, s |
| Greek symbols | = | |
| σ | = | Stefan Boltzmann constant, Wm−2K−4 |
| ϵ | = | The emissivity |
| ρ | = | Density kg/m3 |
| Ω | = | Integration or problem domain |
| Γ | = | Global or local boundary |
| κ | = | Thermal conductivity, W/(m.K) |
| Subscripts and superscripts | = | |
| k | = | Smoothing domain for node k |
| T | = | Transpose symbol |
| i, j, I, J | = | Node indices |