Radiation symmetry is verified in this test suite by comparing results obtained from models using the different symmetry options to the results obtained from the full model without symmetries. A few different configurations are used to allow the testing of all the symmetry options in two-dimensional, three-dimensional, and axisymmetric cases. Some of the configurations are also used to test radiation blocking.
Since the primary interest of this verification suite is the calculation of view factors in nontrivial geometries, all the problems consist of only a single increment in a single step of steady-state heat transfer analysis. No analytical solutions exist for the nontrivial configurations selected; therefore, verification of the results is limited to a comparison of variations of this problem, run with different types and levels of symmetry. All the results documented can be reproduced by running the input files provided with the Abaqus release.
Infinitely long square section tube
Two-dimensional models
Elements tested
DC2D4
Problem description
Four different two-dimensional models of the cross-section of the square tube are used: the full model, a half model with one reflection symmetry, a quarter model with two reflection symmetries, and a quarter model with cyclic symmetry. The full, half, and quarter models are shown in Figure 1. The two-dimensional models imply that the tube extends infinitely in the direction normal to the cross-section.
Three different models of the square section tube are used. In all cases the complete cross-section is modeled, and the infinite extent of the tube is simulated by using periodic symmetry in the direction normal to the cross-section of the tube. The three models differ in the number of repetitions used for the periodic symmetry.
Full cross-section model, DC3D8 elements, periodic symmetry (NR=20).
Infinitely long square section tube with blocking
Two-dimensional models
Elements tested
DC2D4
Problem description
Four different two-dimensional models of the cross-section of the square tube and the blocking object are used: the full model, a half model with one reflection symmetry, a quarter model with two reflection symmetries, and a quarter model with cyclic symmetry. The full, half, and quarter models are shown in Figure 3. The two-dimensional models imply that the tube and the blocking object extend infinitely in the direction normal to the cross-section.
Figure 3. Two-dimensional square tubes with blocking.
Six different models of the square section tube and the blocking object are used. These models involve different combinations of the cross-sectional model and the number of periodic symmetry repetitions used to simulate the infinite extent of the tube and the blocking object. Three cross-section models are used: the full model, a quarter model with two reflection symmetries, and a quarter model with cyclic symmetry. Figure 4 shows the cross-section models used.
Figure 4. Three-dimensional square tubes with blocking.
Quarter cross-section model with cyclic symmetry (NC=4), DC3D8 elements, periodic symmetry (NR=50).
Finite length square section tube
Three-dimensional models without blocking
Elements tested
DC3D8
Problem description
A unit-length tube with a square cross-section is analyzed. Four different models of the square section are used: the full model, a half model with one reflection symmetry, a quarter model with two reflection symmetries, and a quarter model with cyclic symmetry. Figure 5 shows the cross-section models used.
Figure 5. Three-dimensional finite square tubes with blocking.
A unit-length square cross-section tube and a blocking object are analyzed. Three cross-section models are used: the full model, a quarter model with two reflection symmetries, and a quarter model with cyclic symmetry. Figure 6 shows the cross-section models used.
Figure 6. Three-dimensional finite square tubes with blocking.
A tubular ring with a square cross-section is analyzed. Two different models of the square section are used: the full model and a half model with one reflection symmetry. Figure 7 shows the cross-section models used.
Half cross-section model, DCAX4 elements, one reflection symmetry.
Axisymmetric models with blocking
Elements tested
DCAX4
Problem description
A square cross-section tubular ring with a blocking object inside it is analyzed. Two different models of the square section are used: the full model and a half model with one reflection symmetry. Figure 8 shows the cross-section models used.
Half cross-section model, DCAX4 elements, one reflection symmetry.
Infinitely extending three-dimensional array of cubic objects
Two-dimensional models
Elements tested
DC2D4
Problem description
An infinite array of cubic objects is simulated. The two-dimensional models imply that the array extends to infinity in the third direction. Three different models are used: an array of nine by eleven objects, an array of nine objects with periodic symmetry in the direction perpendicular to the array, and a single object with periodic symmetry in two directions. The number of repetitions in the models using periodic symmetry makes these models equivalent to the nine by eleven array model. The models are shown in Figure 9 where the black square represents the model with two periodic symmetries and the gray squares represent the model with one periodic symmetry.
Single object array with two periodic symmetries (NR1=4, NR2=5), DC2D4 elements.
Three-dimensional models
Elements tested
DC3D8
Problem description
An infinite array of cubic objects is simulated. The three-dimensional models consist of a single cubic element with periodic symmetry in three directions. Two models are used where the number of periodic symmetry repetitions is varied. The single element on which the models are based is shown in Figure 10.
Figure 10. Single element used for three-dimensional cubic array.
Single object array with three periodic symmetries (NR1=8, NR2=8, NR3=10), DC3D8 elements.
Infinitely long finned tube inside another infinitely long tube
Axisymmetric models
Elements tested
DCAX4
Problem description
Radiation between an infinitely long, finned tube inside another infinitely long simple tube is simulated. The axisymmetric mesh used is shown in Figure 11. The infinite extent of the tubes is modeled with periodic symmetry in the direction of the length of the tubes. Three models with a varying number of repetitions for the periodic symmetry are used.
Figure 11. Axisymmetric mesh for finned tube models.
Axisymmetric model with periodic symmetry (NR=10), DCAX4 elements.
Three-dimensional models
Elements tested
DC3D8
Problem description
Radiation between an infinitely long finned tube inside another infinitely long simple tube is simulated. The two three-dimensional meshes used are shown in Figure 12: one is a full 360° mesh, and the other is a slice of this mesh that is used in conjunction with cyclic symmetry. The number of cycles used in the cyclic symmetry is varied. The infinite extent of the tubes is modeled with periodic symmetry in the direction of the length of the tubes.
Figure 12. Three-dimensional meshes for finned tube models.
