T2: One-dimensional heat transfer with radiation

This problem provides evidence that Abaqus can reproduce the result from the benchmark defined by NAFEMS and cited as the reference solution.

This page discusses:

Elements tested
Problem description
Reference solution
Results and discussion
Input files

ProductsAbaqus/StandardAbaqus/Explicit

Elements tested

DC1D2
DC1D3

DC2D3
DC2D4
DC2D6
DC2D8

DCAX3
DCAX4
DCAX6
DCAX8

CAX3T
CAX4RHT
CAX4RT
CAX4T

CPE3T
CPE4RHT
CPE4RT
CPE4T
CPE6MHT
CPE6MT

CPS3T
CPS4RT
CPS4T
CPS6MT

C3D4T
C3D6PHT
C3D6PT
C3D6T
C3D8RHT
C3D8RT
C3D8T
SC6RT
SC8RT

EC3D8RT

Problem description

Model:

The geometry is shown above. A uniform mesh with 10 elements along the length of the bar is used. The thickness and width of the bar are each 0.01 m. In Abaqus/Standard a steady-state simulation is performed, while in Abaqus/Explicit a transient simulation is performed. The total simulation time in the latter case is 2500 seconds. This provides enough time for the transient solution to reach steady-state conditions in this problem.

Material:

Conductivity = 55.6 W/m°C, specific heat = 460.0 J/kg°C, density = 7850 kg/m³. At end B emissivity = 0.98, Stefan-Boltzmann constant = 5.67 × 10⁻⁸ W m²/ K⁴.

For the coupled temperature-displacement elements dummy mechanical properties are used to complete the material definition.

Boundary conditions:

Prescribed temperature of 1000 K at end A. Radiation to ambient temperature of 300 K at end B. No heat flux perpendicular to AB.

Loading:

Zero internal heat generation.

Reference solution

This is a test recommended by the National Agency for Finite Element Methods and Standards (U.K.): Test T2 from NAFEMS publication TNSB, Rev. 3, “The Standard NAFEMS Benchmarks,” October 1990.

Target solution: Temperature at B=927 K (653.85°C).

Results and discussion

All elements yield the exact solution.

Input files

Abaqus/Standard input files

nt2xx12x.inp: DC1D2 elements.
nt2xx13x.inp: DC1D3 elements.
nt2xx23x.inp: DC2D3 elements.
nt2xx24x.inp: DC2D4 elements.
nt2xx26x.inp: DC2D6 elements.
nt2xx28x.inp: DC2D8 elements.
nt2xxa3x.inp: DCAX3 elements.
nt2xxa4x.inp: DCAX4 elements.
nt2xxa6x.inp: DCAX6 elements.
nt2xxa8x.inp: DCAX8 elements.
onedheattransrad_std_cpe3t.inp: CPE3T elements.
onedheattransrad_std_cpe4rht.inp: CPE4RHT elements.
onedheattransrad_std_cpe4rt.inp: CPE4RT elements.
onedheattransrad_std_cpe4t.inp: CPE4T elements.
onedheattransrad_std_cpe6mht.inp: CPE6MHT elements.
onedheattransrad_std_cpe6mt.inp: CPE6MT elements.
onedheattransrad_std_cps3t.inp: CPS3T elements.
onedheattransrad_std_cps4rt.inp: CPS4RT elements.
onedheattransrad_std_cps4t.inp: CPS4T elements.
onedheattransrad_std_cps6mt.inp: CPS6MT elements.
onedheattransrad_std_cax4rht.inp: CAX4RHT elements.
onedheattransrad_std_cax3t.inp: CAX3T elements.
onedheattransrad_std_cax4rt.inp: CAX4RT elements.
onedheattransrad_std_cax4t.inp: CAX4T elements.
onedheattransrad_std_cax6mht.inp: CAX6MHT elements.
onedheattransrad_std_cax6mt.inp: CAX6MT elements.
onedheattransrad_std_c3d4t.inp: C3D4T elements.
onedheattransrad_std_c3d6pht.inp: C3D6PHT elements.
onedheattransrad_std_c3d6pt.inp: C3D6PT elements.
onedheattransrad_std_c3d6t.inp: C3D6T elements.
onedheattransrad_std_c3d8rht.inp: C3D8RHT elements.
onedheattransrad_std_c3d8rt.inp: C3D8RT elements.
onedheattransrad_std_c3d8t.inp: C3D8T elements.

Abaqus/Explicit input files

onedheattransrad_xpl_cax3t.inp: CAX3T elements.
onedheattransrad_xpl_cax4rt.inp: CAX4RT elements.
onedheattransrad_xpl_cpe3t.inp: CPE3T elements.
onedheattransrad_xpl_cpe4rt.inp: CPE4RT elements.
onedheattransrad_xpl_cps3t.inp: CPS3T elements.
onedheattransrad_xpl_cps4rt.inp: CPS4RT elements.
onedheattransrad_xpl_c3d4t.inp: C3D4T elements.
onedheattransrad_xpl_c3d6t.inp: C3D6T elements.
onedheattransrad_xpl_c3d8rt.inp: C3D8RT elements.
onedheattransrad_xpl_c3d8t.inp: C3D8T elements.
onedheattransrad_xpl_ec3d8rt.inp: EC3D8RT elements.
onedheattransrad_xpl_sc6rt.inp: SC6RT elements.
onedheattransrad_xpl_sc8rt.inp: SC8RT elements.