Cavity Radiation in Abaqus/Standard

When you define the model for a cavity radiation problem, you must:

1. define all of the surfaces in the cavity (see Defining Surfaces);

2. define the radiation properties of each surface (i.e., the emissivity) and the physical constants (see Defining Surface Radiation Properties); and

3. construct cavities from the surfaces (see Constructing a Cavity).

In the first step of a cavity radiation analysis you must associate with each cavity a radiation view factor definition, which controls the calculation of view factors for the cavity. You then may:

1. define cavity symmetries, if any (see Defining Cavity Symmetries);

2. prescribe the motion of surfaces (see Prescribing Motion during a Cavity Radiation Analysis);

3. define boundary conditions such as temperature and forced convection (see Boundary Conditions);

4. control the cavity radiation and view factor calculations in each step (the specifications from the previous step are used if they are not redefined in a step; see Controlling View Factor Calculation during the Analysis);

5. request output of heat transfer variables to the data and results files (see Requesting Surface Variable Output); and

6. request output of the radiation view factor matrices (see Writing the View Factor Matrices to the Results File).

If any of the above are included in your analysis, they must be defined within a heat transfer or coupled thermal-electrical step definition.

Surfaces that are associated with cavity radiation are subject to the following restrictions in addition to the general surface definition restrictions outlined in Element-Based Surface Definition:

• Surfaces cannot overlap because of the ambiguity that would result in the associated property definitions and in the blocking specification.

• A surface can be used only in one cavity definition (the same surface cannot appear in two different cavities).

In addition, the three-dimensional quadrilateral facets should be as close to planar as possible; otherwise, the quality of the view factor calculations will be compromised.

The radiation flux for each facet is calculated based on the average of the nodal temperatures on that facet (see Cavity radiation). This value of radiation flux is then distributed to each node in proportion to its area. Consequently, the mesh must be sufficiently fine that temperature differences across elements are small. Otherwise, computed fluxes at nodes with temperatures above the facet average will be excessively low, and the fluxes at nodes with below-average temperatures will be too high. This tends to induce a spatially oscillatory solution. This effect can be eliminated by reducing the element size in the vicinity of high temperature gradients.

Emissivity is a dimensionless quantity with a value that is greater than or equal to zero and less than or equal to one. A value of $ϵ=0$ corresponds to all radiation being reflected by the surface. A value of $ϵ=1$ corresponds to black body radiation, where all radiation is absorbed by the surface. You can define the emissivity, $ϵ$, of a surface as a function of temperature and other predefined field variables.

You must assign a name to the surface property that defines the emissivity.

You must define the Stefan-Boltzmann constant, $\sigma$, and the value of absolute zero, ${\theta }^Z$; there are no default values for these constants.

By default, a cavity is assumed to consist of surfaces for which surface properties have already been defined. Instead, you may define surface properties as part of the cavity definition.

By default, a cavity is assumed to be closed.

You can specify an open cavity by defining the reference temperature of the external medium. This ambient temperature value is converted to an absolute temperature scale based on the definition of absolute zero. You can verify the degree of opening in the cavity by specifying a tolerance for the accuracy of the view factor calculations; radiation to the external medium will take place only if the deviation of the sum of the view factors from unity is more than this tolerance. See Controlling the Accuracy of View Factor Calculations below for details.

Abaqus/Standard uses an iterative solution technique for obtaining the radiative heat fluxes when cavity parallel decomposition is enabled. This technique is based on Krylov methods, employs a preconditioner, and uses only MPI-based parallelization (see Parallel Execution in Abaqus/Standard for details). This iterative technique is used only to solve the cavity radiation equations and does not require user intervention. You may still opt to use the either the iterative or direct sparse solvers for the solution of the heat transfer finite element equations.

The exact cavity radiation equations are solved whether parallel decomposition is allowed or not; however, when parallel decomposition is active, Abaqus/Standard may require more iterations to obtain a solution. This slower rate of convergence comes from an approximation to the Jacobian (the linearization of the radiation fluxes) that is based on small changes of the irradiation (any part not due to emission from the surface). Models involving surfaces with low emissivities and steady-state analyses might be especially affected. If you encounter convergence problems with parallel decomposed cavities, you may consider

• changing the analysis from steady-state to transient (Uncoupled Heat Transfer Analysis); or

• allowing more solver iterations per time increment (Convergence Criteria for Nonlinear Problems).

Kinematic constraints (for example, coupling constraints, linear constraint equations, multi-point constraints, or surface-based tie constraints) can be applied to any node or surface belonging to a cavity where parallel decomposition is allowed. However, the nodes or surfaces must be the independent (main) nodes or surfaces in the constraint definition.

You define reflection symmetry to create a cavity that is composed of the user-defined cavity surface plus its reflected image through a line or plane. You must identify the dimensionality of the cavity when you define reflection symmetry.

Reflection of Two-Dimensional Cavities

You can define the cavity symmetry by reflecting the cavity surface through a line, as shown in Figure 1.

Reflection symmetry through a line.

This type of reflection can be used only with two-dimensional cavities.

Reflection of Three-Dimensional Cavities

You can define the cavity symmetry by reflecting the cavity surface through a plane, as shown in Figure 2.

Reflection symmetry through a plane.

This type of reflection can be used only with three-dimensional cavities.

Reflection of Axisymmetric Cavities

You can define the cavity symmetry by reflecting the cavity surface through a line of constant z-coordinate, as shown in Figure 3.

Reflection symmetry through a line of constant z-coordinate.

This type of reflection can be used only with axisymmetric cavities.

You can define cavity symmetry by periodic repetition in a given direction. Physically, periodic symmetry is understood as an infinite number of repetitions of the same image at a periodic interval. Numerically, periodic symmetry has to be represented by a finite number of repetitions of the periodic image. You can define the number of repetitions used in the numerical calculation, n.

The periodic symmetry will result in a cavity composed of the user-defined cavity plus twice n similar images, since the periodic symmetry is assumed to apply in both the positive and negative directions. By default, n=2.

Although symmetries do not increase the size of the view factor matrix, they do make its calculation more expensive. Therefore, the number of repetitions should be minimized, but the value of n should be large enough that the view factor matrix is calculated accurately. Output variable VFTOT can be used to check the amount of closure implied by the symmetry. (See Controlling the Accuracy of View Factor Calculations below.) Periodic symmetry for defining the cavity radiation view factor matrix does not impose symmetry conditions automatically in the heat transfer analysis. It may be necessary to impose appropriate constraints on the temperature and loading conditions at the nodes on the periodic symmetry planes to obtain a meaningful solution from the underlying heat transfer analysis.

You must identify the dimensionality of the cavity when you define periodic symmetry.

Periodic Symmetry of Two-Dimensional Cavities

You can create a cavity that is composed of a series of similar images generated by repetition along a two-dimensional distance vector, as shown in Figure 4.

Two-dimensional periodic symmetry.

The repeated images are bounded by lines parallel to line ab. The distance vector must be defined so that it points away from line ab and into the domain of the model. This type of periodic symmetry can be used only with two-dimensional cavities.

Periodic Symmetry of Three-Dimensional Cavities

You can create a cavity that is composed of a series of similar images generated by repetition along a three-dimensional distance vector, as shown in Figure 5. The repeated images are bounded by planes that are parallel to plane abc. The distance vector must be defined so that it points away from plane abc and into the domain of the model. This type of periodic symmetry can be used only with three-dimensional cavities.

Three-dimensional periodic symmetry.

Periodic Symmetry of Axisymmetric Cavities

You can create a cavity that is composed of a series of similar images generated by repetition in the z-direction, as shown in Figure 6.

Axisymmetric periodic symmetry.

The repeated images are bounded by lines of constant z-coordinate. The z-distance vector must be defined so that it points away from the z-constant periodic symmetry reference line and into the domain of the model. This type of periodic symmetry can be used only with axisymmetric cavities.

You can define cavity symmetry by cyclic repetition of the user-defined cavity surface about a point or an axis. The cavity defined by cyclic repetition must cover 360°.

You must define the number of cyclically similar images that compose the cavity, n. The angle of rotation about a point or axis used to create cyclically similar images is equal to 360°/n.

You must identify the dimensionality of the cavity when you define cyclic symmetry.

Cyclic Symmetry of Two-Dimensional Cavities

You can define the cavity symmetry by rotating the cavity about a point, l, as shown in Figure 7.

Cyclic symmetry about a point.

The cavity surface defined in the model must be bounded by the line lk and a line passing through l at an angle, measured counterclockwise when looking into the plane of the model, of 360°/n to lk. This type of cyclic symmetry can be used only for two-dimensional cavities.

Cyclic Symmetry of Three-Dimensional Cavities

You can define the cavity symmetry by rotating the cavity about an axis, lm, as shown in Figure 8. The cavity surface defined in the model must be bounded by the plane lmk and a plane passing through the line lm at an angle, measured clockwise when looking from l to m, of 360°/n to lmk. Line lk must be normal to line lm. This type of cyclic symmetry can be used only for three-dimensional cavities.

Cyclic symmetry about an axis.

Reflection, periodic, and cyclic symmetries can be combined as shown in Table 1. Figure 9 through Figure 12 illustrate some possible symmetry combinations.

$\mathbf{n}$, ${\mathbf{n}}_1$, ${\mathbf{n}}_2$, ${\mathbf{n}}_3$ are normals to lines or planes of reflection symmetry.

$\mathbf{d}$, ${\mathbf{d}}_1$, ${\mathbf{d}}_2$ are distance vectors used to define periodic symmetry.

$\mathbf{l}-\mathbf{m}$ is the direction of the axis of cyclic symmetry in three-dimensional cases.

Combination of two reflection symmetries in two dimensions.

Combination of two periodic symmetries in two dimensions.

Combination of one reflection symmetry and one periodic symmetry in two dimensions.

Combination of one cyclic symmetry and one periodic symmetry in three dimensions.

Translations, ${\mathbf{u}}^t$, are specified in terms of global x-, y-, and z-components unless a local coordinate system is defined at the nodes for which motion is specified; then translations are specified in terms of local x-, y-, and z-components (see Transformed Coordinate Systems).

Translational displacements are always specified as total values of translational motion. This treatment of translations is consistent with that used for displacement boundary conditions (Boundary Conditions) in stress/displacement analyses. The default is to apply translational motion.

Translational velocities can also be specified. Translational velocities always refer to the current step; therefore, the rate of translational motion specified as a velocity is in effect only during the step for which it is defined. This behavior is different from velocity boundary conditions, where velocities stay in effect in subsequent steps if they are not redefined.

Displacements due to a rigid body rotation, ${\mathbf{u}}^r$, can be defined by specifying the magnitude of the rotation and the rotation axis. In three dimensions the rotation axis is defined by specifying two points, $\mathbf{a}$ and $\mathbf{b}$, on the axis of rotation. In two dimensions the rotation axis is assumed to be normal to the plane of the model and is defined by specifying one point, $\mathbf{a}$.

The coordinates of the points defining the axis of rotation must be defined in the configuration at the beginning of the step for which rigid body rotation is being defined.

Motion due to rigid body rotation during a step is specified as the amount of rotation that takes place during that step only. Therefore, the rigid body rotation specified during a step is local to that step; if no rigid body rotation is specified in the following step, no further rotation occurs.

The treatment of rigid body rotations is different from that of translations: rigid body rotations are specified incrementally from step to step while translations are specified as total values.

MOTION, TYPE=VELOCITY, ROTATION
 node (node set), 18.84955592, 0., 0., 0., 0., 0., 1.

MOTION, TYPE=VELOCITY, ROTATION
 node (node set), 1.570796327, 0., 0., 0., 1., 0., 0.

Whenever simultaneous translational and rotational motion is specified, the total motion of a node during step k is defined as

where $\mathbf{x}\left(t\right)$ is the current location of the node due to the specified motion history, $\mathbf{X}$ is the original location of the node, ${\mathbf{u}}^t\left(t\right)$ is the displacement of the node due to the translational motion specified in the step, and ${\mathbf{u}}_i^r$ is the displacement of the node due to rigid body rotation during step i.

In these cases the translation is applied first and the rotation is then assumed to be about the translated (material) axis. In other words, the displacement ${\mathbf{u}}_i^r$ due to rigid body rotation during step i is computed as the rotation about an axis defined by points ${\mathbf{a}}_i^t$ and ${\mathbf{b}}_i^t$ where

In the preceding equations $\mathbf{a}$ and $\mathbf{b}$ are the locations of the points used to define the axis of rotation for the prescribed rotational motion (they refer to the configuration at the beginning of step i) and ${\mathbf{u}}_i^t\left(t\right)$ is the displacement due to translational motion during the step (${\mathbf{u}}_i^t\left(t\right)={\mathbf{u}}^t\left(t\right)-{\mathbf{u}}_{i-1}^t\left(T_{i-1}\right)$, where $T_{i-1}$ is the time at the end of step $i-1$).

Example

As an example, consider a three-dimensional problem with x–y planar motion as shown in Figure 13.

Planar motion example.

The centroid of the object of interest is initially located at $x=0,y=3,z=0$. In the first step the object is translated 4 length units in the x-direction while at the same time it rotates clockwise 180° ($\pi$ radians) about the z-axis at constant angular velocity. This motion moves the object from position A to position C in Figure 13. Halfway through this motion, at position B, the displacements due to the rigid body rotation are calculated by applying the translation to the z-axis (the axis of rotation) and then applying a 90° rotation about this translated axis.

In the second step the object is translated −3 length units in the y-direction only. This motion places the object at position D with no additional rotation. Finally, in the third step the object is simultaneously translated 5 length units at an angle of 53.13° to the y-direction and rotated clockwise, again at constant angular velocity, through 180° about the z-axis. This motion returns the object to its original position.

Assuming that each step time is 1.0, the input required for the above motion sequence is as follows:

First step:

MOTION
 node set, 1, 1, 4.
MOTION, ROTATION, TYPE=VELOCITY
 node set, 3.14159265, 0., 3., 0., 0., 3., -1.

Second step:

MOTION
 node set, 2, 2, -3.

Third step:

MOTION
 node set, 1, 2, 0.
MOTION, ROTATION, TYPE=VELOCITY
 node set, 3.14159265, 4., 0., 0., 4., 0., -1.

For any prescribed motion you can refer to an amplitude curve that gives the time variation of the motion throughout a step (see Amplitude Curves).

You can control how view factors are recalculated during a step as a result of prescribed motion by specifying a value for the maximum allowable motion, max, for a particular node set. View factor recalculation is triggered if a displacement component at any node in the specified node set exceeds the specified value for max.

You must respecify the value of max and the node set in every step where recalculation is required; the values do not remain in effect for subsequent steps.

View factor recalculation can be expensive; use discretion when choosing a value for max.

There are practical situations in which it may be useful to switch cavity radiation effects on and off during the analysis. For example, radiation may be taking place in a cavity that is then filled with a fluid so that radiation is no longer significant; later in the analysis, radiation may resume when the fluid is drained from the cavity. In such cases you can use a radiation view factor definition to switch the radiation on and off in any particular cavity during one or more steps of the analysis.

When cavity radiation is switched back on after having been switched off, Abaqus/Standard will use the last view factors calculated in the last step in which cavity radiation was active. However, if motion is prescribed during the time that the cavity radiation is switched off and one of the displacement components of a node in the specified node set exceeds the value for the maximum allowable motion, max, specified in the step during which cavity radiation is switched off, the view factors will be recalculated at the beginning of the step in which the cavity radiation is switched back on.

Abaqus/Standard uses a progressive integration scheme for view factor calculation. When facets are sufficiently far from each other, a lumped area approximation is used. If the facets are close to each other but one of the facets is much larger than the other, an infinitesimal-to-finite approximation is used. For all other cases a contour integral is numerically calculated to compute the view factor. See View factor calculation for details.

Two nondimensional parameters are calculated for each facet pair to determine which integration scheme is used:

where $A_{min}$ is the area of the smaller facet, $A_{max}$ is the area of the larger facet, and d is the distance between their centroids. The lumped area approximation is used whenever the nondimensional distance square parameter $r_1>P_1$, where $P_1$ has a default value of 5.0. If $r_1\le P_1$, an infinitesimal-to-finite area approximation is used if the facet area ratio $r_2>P_2$, where $P_2$ has a default value of 64.0. Otherwise, a more precise calculation is performed, involving the numerical integration of a contour integral.

You can customize the accuracy and speed of the view factor calculation by specifying the parameters $P_1$ and $P_2$ and the number of integration points per edge. For example, Abaqus/Standard will used lumped area approximations throughout the whole model if $P_1$ is set to zero. Likewise, the more precise, albeit more expensive, numerical integration method will always be used if $P_1$ and $P_2$ are set to very large numbers.

Heat is transferred between surfaces that have unobstructed direct views of each other (see Figure 14); “blocking” may occur in geometrically complex cavities.

Illustrations of blocking.

Surface blocking checks may be computationally expensive in cavities with many surfaces; therefore, significant computational time may be saved by specifying which surfaces are potential blocking surfaces, as described below.

View factor calculations with blocking surfaces are especially sensitive to mesh refinement. If a mesh is too coarse, the view factors may not add up to one (in a closed cavity). To obtain accurate results, the mesh should be refined until the view factors can be summed accurately.

In cases where there are many surfaces in the cavity, surfaces separated by more than a certain distance may not be able to “see” each other for the purposes of radiation because of blocking by other surfaces. You can specify the distance beyond which view factors need not be calculated, which reduces the computational effort required for the view factor calculations.

The cavity radiation heat transfer between facets of a surface in Abaqus is modeled using a full, unsymmetric matrix defining interactions between each node and all others in the cavity. For surfaces with large numbers of nodes this matrix may be large, resulting in memory requirements that are significantly larger than those for the finite element portion of the analysis without the cavity radiation interaction.

To minimize memory requirements and computational cost for cavity radiation heat transfer analysis, the cavity can be defined using a coarser mesh of heat transfer shell elements having a single degree of freedom per node. The overlaid element should have minimal heat capacity and conduction, and it should be used for the definition of the cavity in place of the physical, multiple-degree-of-freedom shell. The overlaid element should be used to define the main surface in a tied coupling constraint (Mesh Tie Constraints); the multiple-degree-of-freedom, physical, heat transfer shell element forms the secondary surface.