T4: Two-dimensional heat transfer with convection

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

DC2D3
DC2D4
DC2D6
DC2D8

DC3D4
DC3D6
DC3D8
DC3D10
DC3D15
DC3D20

CPE3T
CPE4RHT
CPE4RT
CPE6MHT
CPE6MT
CPS3T
CPS4RT
CPS6MT
C3D4T
C3D6PHT
C3D6PT
C3D6T
C3D8RHT
C3D8RT
C3D8T
C3D10MHT
C3D10MT

EC3D8RT

SC6RT
SC8RT

Problem description

Model:

The two-dimensional geometry is shown above. Three-dimensional elements are tested with a thickness of 1.0 in the z-direction. 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 20000 seconds, which provides enough time for the transient solution to reach steady-state conditions in this problem.

Material:

Conductivity = 52 W/m°C, surface convective coefficient = 750 W/m²/°C.

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

Boundary conditions:

Temperature = 100°C along edge AB. Zero heat flux along edge DA. Convection to ambient temperature of 0°C along edges BC and CD.

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 T4 from NAFEMS publication TNSB, Rev. 3, “The Standard NAFEMS Benchmarks,” October 1990.

Target solution: Temperature of 18.3°C at point E.

Results and discussion

The results for the Abaqus/Standard analysis are shown in the following table. The values enclosed in parentheses are percentage differences with respect to the reference solution.


Element	T, Coarse Mesh	T, Fine Mesh
DC2D3	22.40°C (22.4%)	19.22°C (5.0%)
DC2D4	19.34°C (5.7%)	18.91°C (3.3%)
DC2D6	17.99°C (−1.7%)
DC2D8	17.90°C (−2.2%)
DC3D4	18.81°C (2.8%)
DC3D8	19.34°C (5.7%)	18.91°C (3.3%)
DC3D6	22.40°C (22.4%)	18.94°C (3.5%)
DC3D10	19.00°C (3.8%)
DC3D15	17.99°C (−1.7%)
DC3D20	17.89°C (−2.2%)
CPE3T	22.40°C (22.4%)	18.94°C (3.5%)
CPE4RT	19.34°C (5.7%)	18.91°C (3.3%)
CPE4RHT	19.34°C (5.7%)	18.91°C (3.3%)
CPE6MT	17.22°C (5.8%)
CPE6MHT	17.22°C (5.8%)
CPS3T	22.40°C (22.4%)	18.94°C (3.5%)
CPS4RT	19.34°C (5.7%)	18.91°C (3.3%)
CPS6MT	17.22°C (5.8%)
C3D4T	19.26°C (5.2%)
C3D6T	22.40°C (22.4%)	18.94°C (3.5%)
C3D8T	19.34°C (5.7%)	18.91°C (3.3%)
C3D8RT	19.34°C (5.7%)	18.91°C (3.3%)
C3D8RHT	19.34°C (5.7%)	18.91°C (3.3%)
C3D10MT	19.79°C (8.1%)
C3D10MHT	19.79°C (8.1%)

The results for the Abaqus/Explicit analysis are shown in the following table. The values enclosed in parentheses are percentage differences with respect to the reference solution.


Element	T, Coarse Mesh	T, Fine Mesh
CPE3T	22.26°C (21.6%)	18.90°C (3.3%)
CPE4RT	19.29°C (5.4%)	18.87°C (3.1%)
CPS3T	22.36°C (22.2%)	18.90°C (3.3%)
CPS4RT	19.29°C (5.4%)	18.87°C (3.1%)
C3D4T	19.21°C (5.0%)
C3D6T	22.36°C (22.2%)	18.90°C (3.3%)
C3D8RT	19.29°C (5.4%)	18.87°C (3.1%)
C3D8T	19.29°C (5.4%)	18.87°C (3.1%)
SC6RT	22.36°C (22.2%)	18.90°C (3.3%)
SC8RT	19.29°C (5.4%)	18.87°C (3.1%)
EC3D8RT	19.29°C (5.4%)	18.87°C (3.1%)

Input files

Abaqus/Standard input files

Coarse mesh tests:

nt4xx23c.inp: DC2D3 elements.
nt4xx24c.inp: DC2D4 elements.
nt4xx26c.inp: DC2D6 elements.
nt4xx28c.inp: DC2D8 elements.
nt4xx34c.inp: DC3D4 elements.
nt4xx36c.inp: DC3D6 elements.
nt4xx38c.inp: DC3D8 elements.
nt4xx3ac.inp: DC3D10 elements.
nt4xx3fc.inp: DC3D15 elements.
nt4xx3kc.inp: DC3D20 elements.
twodheattrconvecc_std_cpe3t.inp: CPE3T elements.
twodheattrconvecc_std_cpe4rt.inp: CPE4RT elements.
twodheattrconvecc_std_cpe4rht.inp: CPE4RHT elements.
twodheattrconvecc_std_cpe6mt.inp: CPE6MT elements.
twodheattrconvecc_std_cpe6mht.inp: CPE6MHT elements.
twodheattrconvecc_std_cps3t.inp: CPS3T elements.
twodheattrconvecc_std_cps4rt.inp: CPS4RT elements.
twodheattrconvecc_std_cps6mt.inp: CPS6MT elements.
twodheattrconvecc_std_c3d4t.inp: C3D4T elements.
twodheattrconvecc_std_c3d6pht.inp: C3D6PHT elements.
twodheattrconvecc_std_c3d6pt.inp: C3D6PT elements.
twodheattrconvecc_std_c3d6t.inp: C3D6T elements.
twodheattrconvecc_std_c3d8t.inp: C3D8T elements.
twodheattrconvecc_std_c3d8rt.inp: C3D8RT elements.
twodheattrconvecc_std_c3d8rht.inp: C3D8RHT elements.
twodheattrconvecc_std_c3d10mt.inp: C3D10MT elements.
twodheattrconvecc_std_c3d10mht.inp: C3D10MHT elements.

Fine mesh tests:

nt4xx23f.inp: DC2D3 elements.
nt4xx24f.inp: DC2D4 elements.
nt4xx36f.inp: DC3D6 elements.
nt4xx38f.inp: DC3D8 elements.
twodheattrconvecf_std_cpe3t.inp: CPE3T elements.
twodheattrconvecf_std_cpe4rt.inp: CPE4RT elements.
twodheattrconvecf_std_cpe4rht.inp: CPE4RHT elements.
twodheattrconvecf_std_cps3t.inp: CPS3T elements.
twodheattrconvecf_std_cps4rt.inp: CPS4RT elements.
twodheattrconvecf_std_c3d6t.inp: C3D6T elements.
twodheattrconvecf_std_c3d8t.inp: C3D8T elements.
twodheattrconvecf_std_c3d8rt.inp: C3D8RT elements.
twodheattrconvecf_std_c3d8rht.inp: C3D8RHT elements.

Abaqus/Explicit input files

Coarse mesh tests:

twodheattrconvecc_xpl_cpe3t.inp: CPE3T elements.
twodheattrconvecc_xpl_cpe4rt.inp: CPE4RT elements.
twodheattrconvecc_xpl_cps3t.inp: CPS3T elements.
twodheattrconvecc_xpl_cps4rt.inp: CPS4RT elements.
twodheattrconvecc_xpl_c3d4t.inp: C3D4T elements.
twodheattrconvecc_xpl_c3d6t.inp: C3D6T elements.
twodheattrconvecc_xpl_c3d8rt.inp: C3D8RT elements.
twodheattrconvecc_xpl_ec3d8rt.inp: EC3D8RT elements.
twodheattrconvecc_xpl_c3d8t.inp: C3D8T elements.
twodheattrconvecc_xpl_sc6rt.inp: SC6RT elements.
twodheattrconvecc_xpl_sc8rt.inp: SC8RT elements.

Fine mesh tests:

twodheattrconvecf_xpl_cpe3t.inp: CPE3T elements.
twodheattrconvecf_xpl_cpe4rt.inp: CPE4RT elements.
twodheattrconvecf_xpl_cps3t.inp: CPS3T elements.
twodheattrconvecf_xpl_cps4rt.inp: CPS4RT elements.
twodheattrconvecf_xpl_c3d6t.inp: C3D6T elements.
twodheattrconvecf_xpl_c3d8rt.inp: C3D8RT elements.
twodheattrconvecf_xpl_c3d8t.inp: C3D8T elements.
twodheattrconvecf_xpl_ec3d8rt.inp: EC3D8RT elements.
twodheattrconvecf_xpl_sc6rt.inp: SC6RT elements.
twodheattrconvecf_xpl_sc8rt.inp: SC8RT elements.