ABAQUS Benchmarks Manual  


1.6.7 Three-dimensional elemental cavity radiation viewfactor calculations

Product: ABAQUS/Standard  

Relatively simple configurations were selected for these verification problems to ensure that analytical solutions or tabulated results could be found. In some cases certain parameters such as the distance between two surfaces or the number of elements on a surface were varied to illustrate the effects of these parameters on viewfactor calculations within ABAQUS. To duplicate the tabulated results for the cases where parameters were varied, the user can modify the input files provided with the ABAQUS release.

Identical, directly opposed parallel rectangles

Problem description

Analytical solution

where and .

Results and discussion

  • One element per area (xrvd38n1.inp, xrvd38m1.inp, xrvds4n1.inp and xrvds8n1.inp); c can be varied to obtain the following results:


    cF
    ABAQUSAnalytical
    10.78780.7374
    30.47220.4237
    60.22880.2090
    100.10580.1001
    150.05180.0502
    250.01970.0195
    350.01020.0102
    400.00790.0078

  • Two elements per area (xrvd38n2.inp); c can be varied to obtain the following results:


    cF
    ABAQUSAnalytical
    10.73710.7374
    30.42500.4237
    60.21270.2090
    100.10100.1001
    150.05080.0502
    250.01960.0195
    350.01020.0102
    400.00780.0078

  • The ABAQUS results for c = 15 are 0.0519 (xrvds3n1.inp and xrvds6n1.inp).

Input files

xrvd38n1.inp

One DC3D8 element is used to discretize each surface of the cavity; c = 15.

xrvd38m1.inp

One DC3D8 element is used to discretize each surface of the cavity; the *MOTION option is used to vary the distance between the rectangles.

xrvd38m1.f

User subroutine UMOTION used in xrvd38m1.inp.

xrvd38n2.inp

Two DC3D8 elements are used to discretize each surface of the cavity; c = 6.

xrvds3n1.inp

Two DS3 elements are used to discretize each surface of the cavity; c = 15.

xrvds4n1.inp

One DS4 element is used to discretize each surface of the cavity; c = 15.

xrvds6n1.inp

Two DS6 elements are used to discretize each surface of the cavity; c = 15.

xrvds8n1.inp

One DS8 element is used to discretize each surface of the cavity; c = 15.

Reference

  • Howell,  J. R., A Catalog of Radiation Configuration Factors, McGraw-Hill Book Company, New York, 1982.

    • Two infinitely long, directly opposed parallel plates of the same finite width

    Problem description

    Analytical solution

    where .

    Results and discussion

    F
    ABAQUSAnalytical
    0.23150.2361

    Input file

    xrvd38p3.inp

    One DC3D8 element is used to discretize each surface of the cavity. The infinite extent of the cavity is modeled with three-dimensional periodic symmetry (NR = 15).

    Reference

  • Howell,  J. R., A Catalog of Radiation Configuration Factors, McGraw-Hill Book Company, New York, 1982.

    • Coaxial parallel squares of different sizes

    Problem description

    Analytical solution

    where , , , and . Reference solution: F = 0.5701.

    Results and discussion

    ABAQUS results for F: 0.5734 (xrvd38n4.inp); 0.5701 (xrvds3n4.inp).

    Input files

    xrvd38n4.inp

    One DC3D8 element is used to discretize each surface of the cavity.

    xrvds3n4.inp

    Two DS3 elements are used to discretize each surface of the cavity.

    Reference

  • Howell,  J. R., A Catalog of Radiation Configuration Factors, McGraw-Hill Book Company, New York, 1982.

    • Two infinitely long parallel plates of different widths; the centerlines of each plate are connected by the perpendicular between the plates

    Problem description

    Analytical solution

    where and .

    Results and discussion

    F
    ABAQUSAnalytical
    0.40700.4337

    Input file

    xrvd38p5.inp

    One DC3D8 element is used to discretize each surface of the cavity. The infinite extent of the cavity is modeled with three-dimensional periodic symmetry (NR = 15).

    Reference

  • Howell,  J. R., A Catalog of Radiation Configuration Factors, McGraw-Hill Book Company, New York, 1982.

    • Two finite rectangles of the same length, having one common edge and at an angle of 90° to each other

    Problem description

    Analytical solution

    where and . Reference solution: F = 0.1746.

    Results and discussion

    ABAQUS results for F: 0.1970 (xrvd38n6.inp); 0.1966 (xrvds6n6.inp).

    Input files

    xrvd38n6.inp

    One DC3D8 element is used to discretize each surface of the cavity.

    xrvds6n6.inp

    Two DS6 elements are used to discretize each surface of the cavity.

    Reference

  • Siegel,  R., and J. R. Howell, Thermal Radiation Heat Transfer, Hemisphere Publishing Corporation, Washington, 3rd, 1992.

    • Two infinitely long plates of unequal widths h and w, having one common edge and at an angle of 90° to each other

    Problem description

    Analytical solution

    where .

    Results and discussion

    F
    ABAQUSAnalytical
    0.22020.2229

    Input file

    xrvd38p7.inp

    One DC3D8 element is used to discretize each surface of the cavity. The infinite extent of the cavity is modeled with three-dimensional periodic symmetry (NR = 15).

    Reference

  • Siegel,  R., and J. R. Howell, Thermal Radiation Heat Transfer, Hemisphere Publishing Corporation, Washington, 3rd, 1992.

    • Two rectangles with one common edge and an included angle of Φ

    Problem description

    Analytical solution

    Definitions: ; .


    CF
    A = 0.6A = 1.0A = 2.0
    0.10.8947530.8980030.899505
    0.20.8593400.8682010.871800
    0.40.7776100.8121100.822722
    0.60.6657340.7547030.778772
    1.00.4528220.6190280.700100
    2.00.2336320.3500500.521308
    4.00.1173840.1774610.286713
    6.00.0783110.1184990.192535
    10.00.0470020.0711480.115803
    20.00.0235040.0355830.057945

    Results and discussion

    F
    ABAQUSAnalytical
    0.66110.6190

    Input files

    xrvd38n8.inp

    One DC3D8 element is used to discretize each surface of the cavity; 8.0.

    xrvds4n8.inp

    One DS4 element is used to discretize each surface of the cavity; 8.0.

    Reference

  • Howell,  J. R., A Catalog of Radiation Configuration Factors, McGraw-Hill Book Company, New York, 1982.

    • Rectangles having a common edge and forming an arbitrary angle; one rectangle is infinitely long

    Problem description

    Analytical solution

    Definition: .


    AF
    30°45°60°
    0.10.9000220.8048380.690483
    0.20.8729180.7677400.648105
    0.40.8253600.7062950.581494
    0.60.7834990.6553510.529168
    1.00.7117170.5739510.450407
    2.00.5795710.4410040.332686
    4.00.4265920.3078750.225049
    6.00.3416120.2406430.173501
    10.00.2492190.1714500.121970
    20.00.1549760.1041890.073151

    Results and discussion

    F
    ABAQUSAnalytical
    0.56580.5740

    Input file

    xrvd38n9.inp

    DC3D8 elements are used to discretize the surfaces of the cavity; one element for the finite surface and nine elements with an edge length of eight units for the infinite surface; 10.0; 45°.

    Reference

  • Howell,  J. R., A Catalog of Radiation Configuration Factors, McGraw-Hill Book Company, New York, 1982.

    • Two infinitely long plates of equal finite width w, having one common edge and having an included angle of α to each other

    Problem description

    Analytical solution

    Results and discussion

    For this test three parameters can be varied: the angle, the number of reflections, and the number of elements used to model the bottom plate. All of the variations shown in the following tables can be verified by modifying input file xrvd38p0.inp.

    • One element per plate, 60°:


      NRF
      ABAQUSAnalytical
      20.47480.5000
      40.48030.5000
      80.48060.5000
      120.48070.5000
      160.48070.5000
      200.48070.5000

    • One element per plate, NR 12:


      F
      ABAQUSAnalytical
      10°0.88370.9128
      20°0.78440.8264
      30°0.69830.7412
      40°0.62100.6580
      50°0.54920.5774
      60°0.48070.5000
      70°0.41460.4264
      80°0.35110.3572
      90°0.29070.2929

    • NR 2, 60°:


      # of elements on bottom plateF
      ABAQUSAnalytical
      10.47480.5000
      30.43280.5000
      60.45430.5000
      90.46650.5000
      120.47360.5000
      150.47830.5000

    Input file

    xrvd38p0.inp

    One DC3D8 element is used to discretize each surface of the cavity. The infinite extent of the cavity is modeled with three-dimensional periodic symmetry (NR = 12); 60°.

    Reference

  • Siegel,  R., and J. R. Howell, Thermal Radiation Heat Transfer, Hemisphere Publishing Corporation, Washington, 3rd, 1992.

    •