# Nonlinear Steady-State Heat Transfer Analysis

Calculates the temperature distribution in a system, in which material properties are a function of temperature.

(1)
$\left({K}_{C}+H\right)T+R{\left(T+{T}_{abs}\right)}^{4}=f$
Where,
${K}_{C}$
Conductivity matrix
$H$
Boundary convection matrix, due to free convection.
$T$
Unknown nodal temperature
$R$
${T}_{abs}$
Absolute temperature scale defined via PARAM, TABS
$f$

The system of equations is solved to find nodal temperature $T$ .

Thermal load vector can be expressed as:(2)
$f={f}_{B}+{f}_{H}+{f}_{Q}+{f}_{R}$
Where,
${f}_{B}$
Power, due to heat flux at boundary specified by QBDY1.
${f}_{H}$
Boundary convection vector, due to convection specified by CONV (automatic free convection definition can be activated via CONVG Bulk/Subcase pair).
${f}_{Q}$
Power vector, due to internal heat generation specified by QVOL.
${f}_{R}$

The matrix on the left-hand side of Equation 1 is singular, unless temperature boundary conditions are specified.

The system of equation (shown above) is solved using the Newton's method. Solution control is provided by defining parameters on the NLPARM Bulk Data Entry. TEMPERATURE(INITIAL) can be used to provide an initial temperature distribution. The temperature results from the nonlinear heat transfer analysis can be used in subsequent structural analysis.

## Nonlinear Steady-State Heat Transfer Analysis Setup

The following steps are a guide to setup a Nonlinear Steady-State Heat Transfer Analysis.
1. Use the solution sequence identifier (ANALYSIS) in the Subcase Information Entry section to select the nonlinear steady-state heat transfer analysis using: ANALYSIS=NLHEAT.
2. The likely initial temperature distribution can be defined using the TEMPERATURE Subcase Information Entry (type=INITIAL). A good initial temperature estimate improves the convergence of the solver.
3. The MATT4 Bulk Data Entry can be used to define temperature dependent thermal material properties.
4. To indicate that a nonlinear solution is required for any subcase, a NLPARM Subcase Information Entry is required. This subcase entry points to a NLPARM Bulk Data Entry that specifies convergence tolerances and other nonlinear parameters.
5. Loads and boundary conditions are defined in the Bulk Data Entry section of the input deck. These should be referenced in the Subcase Information Entry section using SPC and LOAD entries in a subcase. Each Subcase defines a load vector.

1. Radiation is considered as a boundary condition (similar to Convection).
2. Radiant exchange is between the surface element and a black body space node.
3. The ambient space node is defined by an SPC Bulk Data Entry and is identified via the NODAMB field of the RADBC Bulk Data Entry.
4. RADBC is only used when applied to a surface element (CHBDYE Bulk Data).
5. The ambient node is treated as a black body with its own temperature for radiation exchange between the surface element and space.
6. The radiation view factor (FAMB on RADBC entry) defines the geometric association between the element surface and the ambient node.
7. The emissivity (EMIS1 field) and absorptivity (ABSORP field) can be defined on the RADM Bulk Data Entry.
8. The RADM entry is directly referenced by a surface element entry (CHBDYE entry).
9. The two parameters are required for radiation calculations:
• PARAM, TABS: Absolute temperature scale factor
• PARAM, SIGMA: Stefan-Boltzmann constant

## Example: ANALYSIS and NLPARM Usage

SUBCASE  5
ANALYSIS=NLHEAT
SPC=10
ENDDATA