# Nonlinear Transient Heat Transfer Analysis

Calculates the temperature distribution in a system with respect to time.

The applied thermal loads can either be time-dependent or time-invariant; transient thermal analysis is used to capture the thermal behavior of a system over a specific period of time.

The basic finite element equation for nonlinear transient heat transfer analysis is given by:(1) $C\stackrel{˙}{T}+\left[{K}_{C}+H\right]T+R{\left(T-{T}_{abs}\right)}^{4}=f$
Where,
$C$
Heat capacity matrix.
${K}_{C}$
Temperature-dependent conductivity.
$H$
Temperature-dependent boundary convection matrix, due to free convection.
$\stackrel{˙}{T}$
Derivative of the nodal temperature matrix with respect to time.
$T$
The unknown nodal temperature matrix.
$R$
${T}_{abs}$
Absolute temperature scale defined via PARAM, TABS.
$P$
Thermal load vector can be expressed as:(2) $f={f}_{B}+{f}_{H}+{f}_{Q}+{f}_{R}$
${f}_{B}$
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.
${f}_{R}$
Note:
1. For Nonlinear Heat Transfer Analysis, Conductivity (${K}_{C}$), and/or Free Convection Coefficient ($H$) are temperature-dependent.
2. The differential equation is solved by backward Euler method to find nodal temperature $T$ at the specified time steps. The difference between this equation and the Linear Steady-State Heat Transfer Analysis equation is the term, $C\stackrel{˙}{T}$, that captures the transient nature of the analysis.

## Input

Defines how to setup the nonlinear transient heat transfer subcase.

1. Use the solution sequence identifier (ANALYSIS) in the Subcase Information Entry section to select the linear transient heat transfer analysis using: ANALYSIS=NLHEAT.
2. The NLPARM Bulk Data and corresponding subcase entries are required to activate Nonlinear Transient Heat Transfer Analysis.
3. Define the time step intervals at which the solutions will be calculated for transient analysis using the TSTEP Bulk Data Entry. This is referenced in the TSTEP Subcase Information Entry which is used to select the integration type (TSTEP=SID) for transient analysis.
4. The initial conditions for transient heat transfer analysis are selected by the use of the IC Subcase Information Entry. This entry can be used in the Subcase Information section to specify the set identification number of the temperature field defined by TEMP or TEMPD Bulk Data Entries.
5. Use the single point constraint (SPC) Bulk Data Entry to specify the fixed boundary conditions for this analysis.
6. Use the DLOAD Subcase Information Entry to reference the set IDs of DLOAD, TLOAD1 and TLOAD2 Bulk Data Entries. The corresponding TSTIME field on the TLOAD1/TLOAD2 Bulk Data Entries can be used to switch between Total time and Subcase time. Use the TLOAD1 and TLOAD2 Bulk Data Entries to specify:
1. Time dependent thermal loading - The EXCITEID field of the TLOAD1 and TLOAD2 Bulk Data Entries should point to the ID's of QVOL and QBDY1 Bulk Data Entries or a combination of them using LOADADD.
2. Temperature boundary condition - The EXCITEID field of the TLOAD1 and TLOAD2 Bulk Data Entries should point to the ID of the SPCD Bulk Data Entry. Also, the TYPE field in the TLOAD1 and TLOAD2 entries should be set to 1.
7. The MAT4 and MAT5 Bulk Data Entries can be used to define thermal material properties such as thermal conductivity K, heat capacity C, density RHO, convection heat transfer coefficient H and heat generation capability HGEN used in the QVOL Bulk Data Entry.
8. The MATT4 Bulk Data Entry can be used to define temperature-dependent thermal material properties.
9. Heat capacity (CP) is defined on MAT4/MAT5 entries is defined per unit mass. It is multiplied by density (RHO) to calculate heat capacity matrix in transient heat transfer analysis. If RHO is not defined on MAT4/MAT5, then positive density from a structural material entry with matching MID is used. If MAT4/MAT5 entries do not have a corresponding matching structural material, then the default value of 1.0 is used.
10. The THERMAL I/O Option Entry can be used to request nodal temperature output T for transient heat transfer analysis subcases. The FLUX I/O Option Entry can be used to request temperature gradient and flux output for transient heat transfer analysis subcases. Heat flow results are available through the RESULTANT and SECTION entries.
11. The TYPE=TEMP option on the SENSOR Bulk Data Entry is available to define a temperature sensor that turns off loading outside a particular temperature range.

In Step 6(a) of the guide above, the ability to use QBDY1 data to apply heat flux loading is illustrated. This is accomplished as explained in the following steps.

1. The value of the heat flux load is input in the Q0 field of a QBDY1 Bulk Data Entry.
2. The EID# field in the QBDY1 Bulk Data Entry requires the identification number of CHBDYE surface elements. These surface elements should be created on the surfaces of the model to which heat flux loads are to be applied.
3. This is conducted in HyperMesh by creating an interface of type CONDUCTION, selecting all the relevant surfaces and then adding CHBDYE surface elements to those surfaces.
4. These newly created surface elements via the interface group can then be referenced in the EID# field of the QBDY1 Bulk Data Entry.
5. For convection, automatic free convection definition can be activated via CONVG Bulk/Subcase pair.
Note:
1. Shell elements are considered to be membranes in Heat Transfer Analysis. Composite properties are homogenized (1 degree of freedom per grid). The temperature distribution through the thickness of shell elements is not calculated. Only nodal temperature is determined.
2. Non-zero SPC will be considered as zero SPC for transient thermal analysis, except when non-zero SPC are used to specify ambient points for convection. When an ambient point is controlled by TLOAD1/TLOAD2 via SPCD, the corresponding SPC should be zero.

### Automatic Time Stepping

This solution provides automatic time stepping based on the Local Truncation Error (LTE).

Automatic time stepping can be controlled using the MREF field of the TSTEP Bulk Data Entry. The Local Truncation Error (LTE) is calculated as:(3) $\epsilon =\frac{h}{9.09}\sqrt{\frac{\Delta {\stackrel{˙}{T}}^{T}C\Delta \stackrel{˙}{T}}{{T}^{T}CT}}$
Where,
$h$
Time step.
$T$
Nodal temperature matrix.
$C$
Heat capacity matrix.
The time steps are automatically adjusted based on the following conditions (TOL is the user-defined tolerance set on the TSTEP Bulk Data Entry):
$\stackrel{˜}{e}$ > TOL
Reject current step, cutback to half the current time step, and redo the current step.
TOL > $\stackrel{˜}{e}$ > 0.5 * TOL
Accept current step, cutback the next step to half the current time step.
0.5 * TOL > $\stackrel{˜}{e}$ > 1/16 * TOL
No changes.
1/16 * TOL > $\stackrel{˜}{e}$
The next time step is enlarged to 1.25 times the current time step.

The MREF continuation line on TSTEP entry can be used to control automatic time stepping, so the time step $h$ is adjusted according to the LTE of the current step. When error is "large" compared to the tolerance (TOL), $h$ is reduced by half and the current step is re-calculated. The maximum number of such operations within each step is controlled by the TN1 field.

When $h$ is "small" compared to the tolerance (TOL), $h$ is requested to be increased; it is only increased after TN2 contiguous steps with such a request.

### Transient Thermal Subcase Continuation

Nonlinear transient thermal subcase can continue from the temperature solution of a previously solved steady-state or transient thermal subcase, by pointing IC to that subcase’s subcase or TSTRU ID.

### Coupled Thermal-structural Analysis

The temperature results from the final time step of a nonlinear Transient Heat Transfer Analysis can be applied to a structural subcase.

Both TEMPERATURE(LOAD) and TEMPERATURE(MATERIAL) are allowed to reference the subcase ID or temperature result sets from the Nonlinear Transient Heat Transfer Analysis for use in either material property calculations or thermal loading.

If temperature history at multiple time steps should be applied to a structural subcase, One Step Transient Thermal Stress Analysis should be used.