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Finally, the external flux terms contribute
\int_ {A} N ^ {N} \int_ {- h / 2} ^ {h / 2} M ^ {P} r d s _ {3} d A + \int_ {A} N ^ {N} \left[ I ^ {A} q ^ {A} + I ^ {B} q ^ {B} \right] d A
to the residual and
\int_ {A} N ^ {N} \left[ I ^ {A} \frac {d q}{d \theta} \Big | _ {t + \Delta t, A} + I ^ {B} \frac {d q}{d \theta} \Big | _ {t + \Delta t, B} \right] N ^ {M} d A
to the Jacobian, where
I ^ {A} = 1
at point A through the thickness
I ^ {A} = 0
at all other points through the thickness,
and points A and B are on the top and bottom surfaces of the shell.
2.11.3 Convection/diffusion
The formulation in this section describes a capability for modeling heat transfer with convection in ABAQUS/Standard. The resulting elements can be used in any general heat transfer mesh. These elements have a nonsymmetric Jacobian matrix: the nonsymmetric capability is invoked automatically if elements of this type are included in the model. Both steady-state and transient capabilities are provided. The transient capability introduces a limit on the time increment (the limit is defined below): the time increment is adjusted to satisfy this limit if necessary. The steady-state versions of the elements can be used in a transient analysis, which means that transient effects in the fluid are not included in the model. The formulation is based on the work of Yu and Heinrich ( 1986, 1987).
Thermal equilibrium equation
The thermal equilibrium equation for a continuum in which a fluid is flowing with velocity \mathbf { v } , is
\int \delta \theta \left[ \rho c \left\{\frac {\partial \theta}{\partial t} + \mathbf {v} \cdot \frac {\partial \theta}{\partial \mathbf {x}} \right\} - \frac {\partial}{\partial \mathbf {x}} \cdot \left(\mathbf {k} \cdot \frac {\partial \theta}{\partial \mathbf {x}}\right) - q \right] d V + \int_ {S _ {q}} \delta \theta \left[ \mathbf {n} \cdot \mathbf {k} \cdot \frac {\partial \theta}{\partial \mathbf {x}} - q _ {s} \right] d S = 0,
where \theta ( \mathbf { x } , t ) is the temperature at a point, \delta \theta ( \mathbf { x } , t ) is an arbitrary variational field, \rho ( \theta ) is the fluid density, c ( \theta ) is the specific heat of the fluid, k(µ) is the conductivity of the fluid, q is the heat added per unit volume from external sources, q _ { s } is the heat flowing into the volume across the surface on which temperature is not prescribed ( S _ { q } ) , n is the outward normal to the surface, x is spatial position, and t is time. Although most fluids will have isotropic conductivity, so that { \bf k } = k { \bf I } (where k ( \theta ) is a scalar and I is a unit matrix), we provide for anisotropic conductivity to cover such cases as that of fluid flowing through a set of baffle plates whose conductivity is smeared into that of the fluid.
Procedures
The boundary conditions are that \theta ( \mathbf { x } ) is prescribed over some part of the surface, S _ { \theta } , and that the heat flux per unit area entering the domain across the rest of the surface, q _ { s } ( \mathbf { x } ) , is prescribed or is defined by convection and/or radiation conditions. For example, the boundary layer between fluid convection elements and solid elements might be modeled by DINTERx-type elements. The boundary term in the thermal equilibrium equation defines
q _ {s} = - \mathbf {n} \cdot \mathbf {k} \cdot \frac {\partial \theta}{\partial \mathbf {x}}.
This implies that q _ { s } is the flux associated with conduction across the surface only--any convection of energy across the surface is not included in q _ { s } . This makes no difference if the surface is part of a solid body (where q _ { s } would be defined by heat transfer into the adjacent body), since then the normal velocity into that body, \mathbf { v } \cdot \mathbf { n } , is zero. But it does make a difference when there is fluid crossing the surface, as--for example--on the upstream and downstream boundaries of the mesh. In this case the choice of q _ { s } for the natural boundary condition (instead of using the total flux crossing the surface) is desirable because it avoids spurious reflections of energy back into the mesh as the fluid flows through the surface.
These equations are discretized with respect to position by using first-order isoparametric elements. The fluid velocity, v, is assumed to be known. (ABAQUS actually requires that the mass flow rate of the fluid per unit area be defined, because this is generally more convenient for the user. The velocity is computed from the mass flow rate and the density of the fluid.)
The time discretization generates the solution at time t + \Delta t from the known solution at time t .
The interpolation for the temperature, \theta , is defined over an element and over a time increment as
\theta (\mathbf {x}, t) = N ^ {N} (\mathbf {x}) A ^ {n} (t) \theta^ {(N, n)}, \quad \mathrm{for} \quad N = 1, 2, \ldots , \quad n = t, t + \Delta t,
where the N ^ { N } are standard isoparametric functions and the time interpolation, A ^ { n } , is linear:
A ^ {t} = 1 - \frac {\tau}{\Delta t}, \quad A ^ {t + \Delta t} = \frac {\tau}{\Delta t},
where \Delta t is the time increment and 0 \leq \tau \leq \Delta t .
The Petrov-Galerkin discretization proposed by Yu and Heinrich couples this linear interpolation with the weighting functions
\begin{array}{l} \delta \theta = W ^ {N} \delta \theta^ {N} \\ = \left[ N ^ {N} \bar {A} + \frac {h}{2} \left(\alpha \bar {A} + \beta \frac {\Delta t}{2} \frac {d \bar {A}}{d t}\right) \frac {\mathbf {v}}{| \mathbf {v} |} \cdot \frac {\partial N ^ {N}}{\partial \mathbf {x}} \right] \delta \theta^ {N}, \\ \end{array}
where
\bar {A} = 6 \frac {\tau}{\Delta t ^ {2}} \left(1 - \frac {\tau}{\Delta t}\right);
Procedures
v is the average fluid velocity over the element; | \mathbf { v } | is its magnitude; and h is a characteristic element length measure, defined below. ® and \beta are control parameters. The ® term in the weighting is introduced to eliminate artificial diffusion of the solution, while the \beta term is introduced to avoid numerical dispersion. Yu and Heinrich show that the optimal choices are
\alpha = \coth \frac {\gamma}{2} - \frac {2}{\gamma} \quad \mathrm{and} \quad \beta = \frac {C}{3} - \frac {2}{\gamma} \frac {\alpha}{C},
where \gamma is the local Péclet number in an element and C is the local Courant number, defined as
\gamma = | \mathbf {v} | h \frac {\rho c}{k} \quad \mathrm{and} \quad C = | \mathbf {v} | \frac {\Delta t}{h}.
The above expression for \beta yields negative values for very small fluid velocities, which may destabilize the solution; hence, for low velocities dispersion control is switched off.
The characteristic element length measure, h, is defined by Yu and Heinrich as follows.
Let \mathbf { h } _ { \alpha } be the ® isoparametric line across the element passing through its centroid. The projection of \mathbf { h } _ { \alpha } in the direction of the fluid velocity vector at the element's centroid is
h _ {\alpha} = \mathbf {h} _ {\alpha} \cdot \frac {\mathbf {v}}{| \mathbf {v} |}.
Then we define h as
h = \sum_ {\alpha} | h _ {\alpha} |.
When \beta is nonzero, these elements require that C \leq 1 for numerical stability.
Since the weighting functions are biased ("upwinding"), they are discontinuous from one element to the next. Some care is, therefore, required in manipulating the weak form of the thermal equilibrium equation (see Hughes and Brooks, 1982). In particular, the usual integration by parts of the conduction term
\delta \theta \frac {\partial}{\partial \mathbf {x}} \cdot \left(\mathbf {k} \cdot \frac {\partial \theta}{\partial \mathbf {x}}\right)
can be performed only for the continuous part of the weighting functions used to discretize \delta \theta \mathbf { \cdot } otherwise, continuity of heat flux between elements is not assured. For convenience we write the discontinuous part of the weighting as
P ^ {N} = \frac {h}{2} \left(\alpha \bar {A} + \beta \frac {\Delta t}{2} \frac {d \bar {A}}{d t}\right) \frac {\mathbf {v}}{| \mathbf {v} |} \cdot \frac {\partial N ^ {N}}{\partial \mathbf {x}}.
Procedures
The weak form of thermal equilibrium is
\begin{array}{l} \int \left[ W ^ {N} \rho c \left\{N ^ {M} \frac {d A ^ {n}}{d t} + \mathbf {v} \cdot \frac {\partial N ^ {M}}{\partial \mathbf {x}} A ^ {n} \right\} + \bar {A} \frac {\partial N ^ {N}}{\partial \mathbf {x}} \cdot \mathbf {k} \cdot \frac {\partial N ^ {M}}{\partial \mathbf {x}} A ^ {n} - P ^ {N} \mathbf {k}: \frac {\partial^ {2} N ^ {M}}{\partial \mathbf {x} \partial \mathbf {x}} A ^ {n} \right] d V \\ - \int W ^ {N} q d V - \bar {A} \int_ {S} N ^ {N} q _ {s} d S = 0. \\ \end{array}
This can be rewritten as
\begin{array}{l} \bar {A} \frac {d A ^ {n}}{d t} \int \rho c \left(N ^ {N} + \alpha \frac {h}{2} \frac {\mathbf {v}}{| \mathbf {v} |} \cdot \frac {\partial N ^ {N}}{\partial \mathbf {x}}\right) N ^ {M} d V \\ + \bar {A} A ^ {n} \int \rho c \left(N ^ {N} + \alpha \frac {h}{2} \frac {\mathbf {v}}{| \mathbf {v} |} \cdot \frac {\partial N ^ {N}}{\partial \mathbf {x}}\right) \mathbf {v} \cdot \frac {\partial N ^ {M}}{\partial \mathbf {x}} d V \\ + \beta \frac {h \Delta t}{4} \frac {d \bar {A}}{d t} \left(\frac {d A ^ {n}}{d t} \int \rho c \frac {\mathbf {v}}{| \mathbf {v} |} \cdot \frac {\partial N ^ {N}}{\partial \mathbf {x}} N ^ {M} d V + A ^ {n} \int \rho c \frac {\mathbf {v}}{| \mathbf {v} |} \cdot \frac {\partial N ^ {N}}{\partial \mathbf {x}} \mathbf {v} \cdot \frac {\partial N ^ {M}}{\partial \mathbf {x}} d V\right) \\ + \bar {A} A ^ {n} \int \left(\frac {\partial N ^ {N}}{\partial \mathbf {x}} \cdot \mathbf {k} \cdot \frac {\partial N ^ {M}}{\partial \mathbf {x}} - \alpha \frac {h}{2} \frac {\mathbf {v}}{| \mathbf {v} |} \cdot \frac {\partial N ^ {N}}{\partial \mathbf {x}} \mathbf {k}: \frac {\partial^ {2} N ^ {M}}{\partial \mathbf {x} \partial \mathbf {x}}\right) d V \\ - \beta \frac {h \Delta t}{4} \frac {d \bar {A}}{d t} A ^ {n} \int \frac {\mathbf {v}}{| \mathbf {v} |} \cdot \frac {\partial N ^ {N}}{\partial \mathbf {x}} \mathbf {k}: \frac {\partial^ {2} N ^ {M}}{\partial \mathbf {x} \partial \mathbf {x}} d V \\ - \bar {A} \int_ {V} \left[ N ^ {N} + \alpha \frac {h}{2} \frac {\mathbf {v}}{| \mathbf {v} |} \cdot \frac {\partial N ^ {N}}{\partial \mathbf {x}} \right] q d V - \frac {d \bar {A}}{d t} \int_ {V} \beta \frac {\Delta t}{2} \frac {\mathbf {v}}{| \mathbf {v} |} \cdot \frac {\partial N ^ {N}}{\partial \mathbf {x}} q d V - \bar {A} \int_ {S} N ^ {N} q _ {s} d S = 0. \\ \end{array}
We now integrate this equation from time t to t + ¢t to provide an average equilibrium statement for the increment. We use the results
\int_ {\Delta t} \bar {A} d t = 1, \quad \int_ {\Delta t} \frac {d \bar {A}}{d t} d t = 0,
\int_ {\Delta t} \bar {A} \frac {d A ^ {t + \Delta t}}{d t} d t = - \int_ {\Delta t} \bar {A} \frac {d A ^ {t}}{d t} d t = \frac {1}{\Delta t}, \quad \int_ {\Delta t} \bar {A} A ^ {t + \Delta t} d t = \int_ {\Delta t} \bar {A} A ^ {t} d t = \frac {1}{2},
and
\int_ {\Delta t} \frac {d \bar {A}}{d t} \frac {d A ^ {t + \Delta t}}{d t} d t = \int_ {\Delta t} \frac {d \bar {A}}{d t} \frac {d A ^ {t}}{d t} d t = 0, \quad \int_ {\Delta t} \frac {d \bar {A}}{d t} A ^ {t + \Delta t} d t = - \int_ {\Delta t} \frac {d \bar {A}}{d t} A ^ {t} d t = - \frac {1}{\Delta t},
to give
Procedures
\begin{array}{l} \frac {1}{\Delta t} \int \rho c \left(N ^ {N} + \alpha \frac {h}{2} \frac {\mathbf {v}}{| \mathbf {v} |} \cdot \frac {\partial N ^ {N}}{\partial \mathbf {x}}\right) N ^ {M} d V \left(\theta^ {M, t + \Delta t} - \theta^ {M, t}\right) \\ + \frac {1}{2} \int \rho c \left(N ^ {N} + \alpha \frac {h}{2} \frac {\mathbf {v}}{| \mathbf {v} |} \cdot \frac {\partial N ^ {N}}{\partial \mathbf {x}}\right) \mathbf {v} \cdot \frac {\partial N ^ {M}}{\partial \mathbf {x}} d V \left(\theta^ {M, t + \Delta t} + \theta^ {M, t}\right) \\ - \beta \frac {h}{4} \int \rho c \frac {\mathbf {v}}{| \mathbf {v} |} \cdot \frac {\partial N ^ {N}}{\partial \mathbf {x}} \mathbf {v} \cdot \frac {\partial N ^ {M}}{\partial \mathbf {x}} d V \left(\theta^ {M, t + \Delta t} - \theta^ {M, t}\right) \\ + \left(\frac {1}{2} \int \frac {\partial N ^ {N}}{\partial \mathbf {x}} \cdot \mathbf {k} \cdot \frac {\partial N ^ {M}}{\partial \mathbf {x}} d V - \alpha \frac {h}{4} \int \frac {\mathbf {v}}{| \mathbf {v} |} \cdot \frac {\partial N ^ {N}}{\partial \mathbf {x}} \mathbf {k}: \frac {\partial^ {2} N ^ {M}}{\partial \mathbf {x} \partial \mathbf {x}} d V\right) \left(\theta^ {M, t + \Delta t} + \theta^ {M, t}\right) \\ + \beta \frac {h}{4} \int \frac {\mathbf {v}}{| \mathbf {v} |} \cdot \frac {\partial N ^ {N}}{\partial \mathbf {x}} \mathbf {k}: \frac {\partial^ {2} N ^ {M}}{\partial \mathbf {x} \partial \mathbf {x}} d V \left(\theta^ {M, t + \Delta t} - \theta^ {M, t}\right) \\ - \int \left(N ^ {N} + \alpha \frac {h}{2} \frac {\mathbf {v}}{| \mathbf {v} |} \cdot \frac {\partial N ^ {N}}{\partial \mathbf {x}}\right) q d V - \int_ {S} N ^ {N} q _ {s} d S = 0. \\ \end{array}
For the steady-state case the third term in this equation is omitted. In both transient and steady-state forms the contribution of such a convective element to the system of equations for the heat transfer model is not symmetric, requiring the use of the nonsymmetric matrix storage and solution scheme.
2.11.4 Cavity radiation
The formulation described in this section provides a capability for modeling heat transfer with cavity thermal radiation (in addition to the radiation boundary conditions described in ``Uncoupled heat transfer analysis,'' Section 2.11.1). Cavities are defined in ABAQUS/Standard as collections of surfaces that are composed of facets. In axisymmetric and two-dimensional cases a facet is a side of an element; in three-dimensional cases a facet can be a face of a solid element or a surface of a shell element. For the purposes of the cavity radiation calculations, each facet is assumed to be isothermal and to have a uniform emissivity.
Based on the cavity definition, cavity radiation elements are created internally by ABAQUS. These elements can generate large matrices since they couple the temperature degree of freedom of every node on the cavity surface. Their Jacobian matrix is nonsymmetric: the nonsymmetric solution capability is automatically invoked if cavity radiation calculations are requested in the analysis. Both steady-state and transient capabilities are provided.
The theory on which this cavity radiation formulation is based is well-known and can be found in Holman (1990) and Siegel and Howell (1980). This section describes the formulation of the cavity radiation flux contributions and respective Jacobian for the Newton method used for the solution of the nonlinear radiation problem. The geometrical issues associated with the calculation of radiation viewfactors necessary in the formulation are addressed in ``Viewfactor calculation,'' Section 2.11.5.
Thermal radiation
Our formulation is based on gray body radiation theory that means that the monochromatic emissivity of the body is independent of the wavelength of propagation of the radiation. Only diffuse (nondirectional) reflection is considered. Attenuation of the radiation in the cavity medium is not considered. Using these assumptions together with the assumption of isothermal and isoemissive
Procedures
cavity facets, we can write the radiation flux per unit area into a cavity facet as
Equation 2.11.4-1
q _ {i} ^ {c} = \frac {\sigma \epsilon_ {i}}{A _ {i}} \sum_ {j} \epsilon_ {j} \sum_ {k} F _ {i k} C _ {k j} ^ {- 1} \bigg ((\theta_ {j} - \theta^ {Z}) ^ {4} - (\theta_ {i} - \theta^ {Z}) ^ {4} \bigg),
where
C _ {i j} = \delta_ {i j} - \frac {(1 - \epsilon_ {i})}{A _ {i}} F _ {i j}, \qquad \mathrm{(nosummation)}
and A _ { i } is the area of facet i (seeing all cavity facets j = 1 , n ) ; \epsilon _ { i } , \epsilon _ { j } are the emissivities of facets i , j ; \sigma is the Stefan-Boltzmann constant; F _ { i j } is the geometrical viewfactor matrix; \theta _ { i } , \theta _ { j } are the temperatures of facets i , j ; \theta ^ { Z } is the absolute zero on the temperature scale used; and \delta _ { i j } is the Kronecker delta.
In the special case of blackbody radiation, where no reflection takes place (emissivity equal to one), Equation 2.11.4-1 reduces to
Equation 2.11.4-2
q _ {i} ^ {c} = \frac {\sigma \epsilon_ {i}}{A _ {i}} \sum_ {j} \epsilon_ {j} F _ {i j} \left((\theta_ {j} - \theta^ {Z}) ^ {4} - (\theta_ {i} - \theta^ {Z}) ^ {4}\right).
Spatial interpolation
The variables used to solve the discrete approximation of the heat transfer problem with cavity radiation are the temperatures of the nodes on the cavity surface. Since we assume that for cavity radiation purposes each facet is isothermal, it is necessary to calculate an average facet temperature radiation power. To do so, we first define temperature radiation power as
\eta_ {i} = \left(\theta_ {i} - \theta^ {Z}\right) ^ {4}, \qquad \eta^ {N} = \left(\theta^ {N} - \theta^ {Z}\right) ^ {4},
where the subscript i refers to facet quantities and the superscript N refers to nodal quantities.
Then, we interpolate the average facet temperature radiation power from the facet nodal temperatures as
Equation 2.11.4-3
\eta_ {i} = \sum_ {N} P _ {i} ^ {N} \eta^ {N},
where N is the number on nodes forming the facet and P _ { i } ^ { N } are nodal contribution factors calculated from area integration as
P _ {i} ^ {N} = \frac {1}{A _ {i}} \int_ {A _ {i}} N _ {i} ^ {N} d A _ {i},
Procedures
where N _ { i } ^ { N } are the interpolation functions for facet i.
The radiation flux into facet i can now be written as
Q _ {i} = q _ {i} ^ {c} A _ {i} = \sum_ {j} R _ {i j} (\eta_ {j} - \eta_ {i}),
where
R _ {i j} = \sigma \epsilon_ {i} \epsilon_ {j} D _ {i j},
and
D _ {i j} = \sum_ {k} F _ {i k} C _ {k j} ^ {- 1}.
This can be rewritten as
Equation 2.11.4-4
Q _ {i} = \sum_ {j} \overline {{R}} _ {i j} \eta_ {j},
where
\overline {{R}} _ {i j} = R _ {i j} - \left(\sum_ {k} R _ {i k}\right) \delta_ {i j}.
Cavity radiation flux and Jacobian contributions
The nodal contributions from the radiation flux on each facet can now be written as
Q _ {i} ^ {N} = \int_ {A _ {i}} q _ {i} ^ {c} N _ {i} ^ {N} d A _ {i} = P _ {i} ^ {N} Q _ {i},
and the total radiation flux at node N is
Q ^ {N} = \sum_ {i} Q _ {i} ^ {N} = \sum_ {i} P _ {i} ^ {N} Q _ {i}.
Substituting Equation 2.11.4-3 and Equation 2.11.4-4 in the above equation:
Equation 2.11.4-5
Q ^ {N} = \sum_ {M} \overline {{R}} ^ {N M} \eta^ {M},
where
\overline {{R}} ^ {N M} = \sum_ {i} \sum_ {j} P _ {i} ^ {N} \overline {{R}} _ {i j} P _ {j} ^ {M}.
The radiation flux q _ { i } ^ { c } is evaluated based on temperatures at the end of the increment, coordinates at the end of the increment, and emissivities at the beginning of the increment. Any time variation of the coordinates during the heat transfer analysis is predefined by the *MOTION option and, therefore, provides no contribution to the Jacobian. Any variation of the emissivities as a function of temperature and predefined field variable changes with time is treated explicitly (values at the beginning of the increment are used) and, therefore, also provides no contribution to the Jacobian. The MXDEM parameter in the *HEAT TRANSFER option is used to control the accuracy of the time integration of the emissivity. Thus, the only Jacobian contribution is provided by temperature variations.
The Jacobian contribution arising from the cavity radiation flux is then written trivially as
Equation 2.11.4-6
J ^ {N M} = \frac {\partial Q ^ {N}}{\partial \theta^ {M}} = 4 \overline {{R}} ^ {N M} \left(\theta^ {M} - \theta^ {Z}\right) ^ {3} \qquad \mathrm{(nosummation)}.
In all practical cases the Jacobian is unsymmetric. This exact unsymmetric Jacobian is always used when cavity radiation analysis is performed.
2.11.5 Viewfactor calculation
Cavity radiation occurs when surfaces of the model can see each other and, thus, exchange heat with each other by radiation (Figure 2.11.5-1). Such exchange depends on viewfactors that measure the relative interaction between the surfaces composing the cavity. Viewfactor calculation is rather complicated for anything but the most trivial geometries. ABAQUS offers an automatic viewfactor calculation capability for two- and three-dimensional cases as well as for axisymmetric situations. This capability can take into account general surface blocking (or shadowing) as well as the most common forms of radiation symmetry. The viewfactor calculation can also be automatically repeated a number of times throughout the analysis history (this is user-controlled) if cavity surfaces are moved in space causing the viewfactors to change.
HKS is pleased to acknowledge that the viewfactor calculation technology implemented in ABAQUS was developed by the Atomic Energy Authority of the United Kingdom; see, for example, Johnson (1987). The remainder of this section contains a general description of this technology.
Figure 2.11.5-1 Heat exchange between surfaces by radiation.
natural_image
Pure diagram of a rectangular channel with directional arrows indicating flow or movement, no text or symbols present
Viewfactor between elementary areas
The viewfactor between two elementary areas, A _ { i } and A _ { j } , can be generally written as
Equation 2.11.5-1
F _ {i j} = \int_ {A _ {i}} \int_ {A _ {j}} \frac {c o s \phi_ {i} c o s \phi_ {j}}{\pi R ^ {2}} d A _ {i} d A _ {j},
where R is the distance between the two areas and \phi _ { i } , \phi _ { j } are the angles between R and the normals to the surfaces of the areas (Figure 2.11.5-2). This formula applies when the areas A _ { i } and A _ { j } are small compared with the distance R. If R approaches zero, the viewfactors calculated by the above expression tend to infinity, and, therefore, ABAQUS takes special care of such cases.
Discretization
Cavities are composed of surfaces; and surfaces, in turn, are made up of finite element faces. For the purpose of viewfactor calculations, one can then think of a cavity as a collection of element faces (or facets) corresponding to the finite element discretization around the cavity.
Figure 2.11.5-2 Schematic of viewfactor calculation.
text_image
dA n₁ φ₁ R φ₁ n₁ dA dA
In the two- and three-dimensional cases the element faces composing cavities can be treated as elementary areas and, thus, Equation 2.11.5-1 applies. In axisymmetric cases the element faces represent rings so that the viewfactors involve two ring surfaces looking at each other. This requires an integration over 2¼ where it is important to account for "horizon" effects ( Johnson, 1987).
In so far as the viewfactor calculations are concerned, first- and second-order element faces are treated similarly in the sense that the midside nodes of the faces in the second-order elements are ignored. This means that a pair of four-noded faces looking at each other will produce the same viewfactor as a pair of eight-noded faces with corner nodes coinciding with the nodes of the four-noded faces.
Radiation blocking
Radiation within a cavity implies that every surface exchanges heat with every other surface. The problem is made more complex when solid bodies are interposed between radiating surfaces blocking (or shadowing) off some but not all the possible paths along which heat can be radiated from the facets of one surface to the facets of another surface (Figure 2.11.5-3).
It is inconceivable that the user could handle this complexity in all but the simplest situations. Therefore, by default, ABAQUS automatically checks if blocking takes place for every possible radiation path in a cavity. This requires that the program check if the ray joining the centers of each pair of facets intersects any other facet. For cavities with a large number of facets this can be very time consuming. For this reason ABAQUS allows the user to guide its blocking algorithm by accepting input of which surfaces cause blocking, thus significantly reducing the computational effort required.
Figure 2.11.5-3 Blocking or shadowing example.
text_image
A surfaces blocked from A
If a ray between two facets intersects any other facet, then in the two- and three-dimensional cases that ray is eliminated and no radiative heat transfer takes place between the facets. In the axisymmetric case blocking is much more complicated since each element face in the finite element model represents a ring. This is handled automatically and requires that the program calculate which part of the 2 \pi extent of the ring is blocked.
Radiation symmetries
Use of symmetry can greatly reduce the size of a problem, but--in the case of cavity radiation--it requires that special facilities for definition and handling of symmetries be available. ABAQUS provides capabilities for three different kinds of symmetries: simple reflection symmetry, periodic symmetry, and cyclic symmetry. Reflection symmetry allows one additional image of the model to be created by reflection through a line in two dimensions or reflection through a plane in three dimensions. Periodic symmetry can be used to create multiple images of the model by periodic repetition in two- or three-dimensional space according to a periodic distance vector. Cyclic symmetry creates multiple images of the model by cyclic repetition about a point in two dimensions or by cyclic