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Transient Heat Conduction Analysis

This section presents the time discretization and iterative solution method for heat conduction analysis of solids using the finite element method (FEM). For the governing equations and boundary conditions of the continuum, see Heat Conduction Equation.

Discretized Equation (Starting Point)

Discretizing the heat conduction equation (heat conduction equation (gov_he_main)) by the Galerkin method gives

\[\begin{equation} K T + M \frac{\partial T}{\partial t} = F \label{eq:2.4.8} \end{equation}\]

where

\[\begin{equation} K = \int\left( k_x \frac{\partial N^T}{\partial x}\frac{\partial N}{\partial x} + k_y \frac{\partial N^T}{\partial y}\frac{\partial N}{\partial y} + k_z \frac{\partial N^T}{\partial z}\frac{\partial N}{\partial z} \right) dV + \int hc N^T N ds + \int hr N^T N ds \label{eq:2.4.9} \end{equation}\]
\[\begin{equation} M = \int \rho c N^T N dV \label{eq:2.4.10} \end{equation}\]
\[\begin{equation} F = \int Q N^T dV - \int q_s N^T dS + \int{hc} T c N^T dS + \int{hcTr} ({T+Tr}) ({T^2 + T r^2}) N^T dS \label{eq:2.4.11} \end{equation}\]
\[\begin{equation} N = (N^1, N^2, \ldots, Ni) \label{eq:2.4.12} \end{equation}\]

Here, \(K\), \(M\), \(F\), and \(N\) are the heat conduction matrix (including convective and radiative boundary contributions), mass matrix, thermal load vector, and shape-function matrix, respectively. The definitions of the material-property symbols (\(\rho\), \(c\), \(k_x, k_y, k_z\), \(Q\), \(hc\), \(hr\), etc.) follow Heat Conduction Equation.

Time Discretization and Iterative Solution

Equation \(\eqref{eq:2.4.8}\) is a nonlinear, transient equation. Using the backward Euler method for time discretization, when the temperature at time \(t=t_0\) is known, the temperature at time \(t=t_0+\Delta t\) is calculated using the following equation.

\[\begin{equation} K_{t=t_0+\Delta t} T_{t=t_0+\Delta t} + M_{t=t_0+\Delta t} \frac{T_{t=t_0+\Delta t} - T_{t=t_0}}{\Delta t} = F_{t=t_0+\Delta t} \label{eq:2.4.13} \end{equation}\]

Consider improving the temperature vector \(T_{t=t_0+\Delta t}^{(i)}\), which approximately satisfies Eq.\(\eqref{eq:2.4.13}\), to obtain the more accurate solution \(T_{t=t_0+\Delta t}^{(i)+1}\).

For this purpose, first express the temperature vector as follows.

\[\begin{equation} T_{t=t_0+\Delta t}= T_{t=t_0+\Delta t}^{(i)} + \Delta T_{t=t_0+\Delta t}^{(i)} \label{eq:2.4.14} \end{equation}\]

The product of the heat conduction matrix and temperature vector, the mass matrix, and related terms are approximated as follows.

\[\begin{equation} K_{t=t_0+\Delta t} T_{t=t_0+\Delta t} = K_{t=t_0+\Delta t}^{(i)} T_{t=t_0+\Delta t}^{(i)} + \frac{\partial \big(K_{t=t_0+\Delta t}^{(i)} T_{t=t_0+\Delta t}^{(i)}\big) } {\partial T_{t=t_0+\Delta t}^{(i)} } \Delta T_{t=t_0+\Delta t}^{(i)} \label{eq:2.4.15} \end{equation}\]
\[\begin{equation} M_{t=t_0+\Delta t} = M_{t=t_0+\Delta t}^{(i)} + \frac{\partial M_{t=t_0+\Delta t}^{(i)}}{\partial T_{t=t_0+\Delta t}^{(i)}} \Delta T_{t=t_0+\Delta t}^{(i)} \label{eq:2.4.16} \end{equation}\]

Substituting Eq.\(\eqref{eq:2.4.14}\), Eq.\(\eqref{eq:2.4.15}\), and Eq.\(\eqref{eq:2.4.16}\) into Eq.\(\eqref{eq:2.4.13}\) and omitting second- and higher-order terms gives the following equation.

\[\begin{equation} \bigg(\frac{M_{t=t_0+\Delta t}^{(i)}}{\Delta t} + \frac {\partial M_{t=t_0+\Delta t}^{(i)} } { \partial T_{t=t_0+\Delta t}^{(i)} } \frac{T_{t=t_0+\Delta t}^{(i)} - T_{t=t_0}}{\Delta t} + \frac{\partial \big(K_{t=t_0+\Delta t}^{(i)} T_{t=t_0+\Delta t}^{(i)}\big)} {\partial T_{t=t_0+\Delta t}^{(i)}} \bigg) \Delta T_{t=t_0+\Delta t}^{(i)} \\\ = F_{t=t_0+\Delta t} - M_{t=t_0+\Delta t}^{(i)} \frac{T_{t=t_0+\Delta t}^{(i)} - T_{t=t_0}}{\Delta t} - K_{t=t_0+\Delta t}^{(i)} T_{t=t_0+\Delta t}^{(i)} \label{eq:2.4.17} \end{equation}\]

The coefficient matrix on the left-hand side is further approximated using the following equation.

\[\begin{equation} K^{(i)} = \frac{M_{t=t_0+\Delta t}^{(i)}}{\Delta t} + \frac{\partial \big( K_{t=t_0+\Delta t}^{(i)} T_{t=t_0+\Delta t}^{(i)} \big)}{\partial T^{(i)}_{t=t_0+\Delta t}} = \frac{M_{t=t_0+\Delta t}^{(i)}}{\Delta t} + K_{T_{t=t_0+\Delta t}}^{(i)} \label{eq:2.4.18} \end{equation}\]

Here, \(K_{T_{t=t_0+\Delta t}}^{(i)}\) is the tangent stiffness matrix.

Ultimately, the temperature at time \(t=t_0+\Delta t\) can be calculated by performing iterative calculations using the following equation.

\[\begin{equation} K^{(i)} \Delta T_{t=t_0+\Delta t}^{(i)} = F_{t=t_0+\Delta t} - M_{t=t_0+\Delta t}^{(i)} \frac{T_{t=t_0+\Delta t}^{(i)} - T_{t=t_0}}{\Delta t} - K^{(i)} T_{t=t_0+\Delta t}^{(i)} \label{eq:2.4.19} \end{equation}\]

For steady-state analysis in particular, iterative calculations are performed using the following equation.

\[ K_T^{(i)} \Delta T_{t=\infty}^{(i)} = F_{t=\infty} - K_T^{(i)} \Delta T_{t=\infty}^{(i)} \]
\[\begin{equation} T_{t=\infty}^{(i+1)} = T_{t=\infty}^{(i)} + \Delta{T}_{t=\infty}^{(i)} \label{eq:2.4.20} \end{equation}\]

In transient analysis, because an implicit method is used for time discretization, the choice of time increment \(\Delta t\) is generally not subject to a restriction on its magnitude. However, if the time increment \(\Delta t\) is too large, the number of iterations required for convergence increases. In general, an excessively large time increment \(\Delta t\) increases the number of iterations. In the implementation, automatic increment control monitors the magnitude of the residual vector, reducing \(\Delta t\) when convergence is slow and increasing \(\Delta t\) when the number of iterations is small (→ see Step Control for details).