1D Transient Heat Conduction (Semi-Infinite Body)

It's a classic problem for verifying the transient analysis capability of thermal analysis solvers, using the temperature response of a semi-infinite body whose surface temperature is instantaneously changed to $T_s$. There is an exact solution involving the complementary error function erfc. It forms the theoretical basis for NAFEMS T1 and T2 benchmarks.

Exactly. It's the first step in Code Verification for heat conduction. A characteristic not found in structural problems is that it can verify the accuracy of both spatial and temporal discretization.

The Fourier heat conduction equation (1D) is

where $\alpha = k/(\rho c_p)$ is the thermal diffusivity. The exact solution for initial condition $T(x,0) = T_i$ and boundary condition $T(0,t) = T_s$ is

erfc is the complementary error function. The penetration depth where the temperature decays to 1% of the surface value is $\delta \approx 3.6\sqrt{\alpha t}$.

Yes. The surface heat flux is

It diverges to infinity as $t \to 0$. This singularity stems from the non-physical boundary condition of an instantaneous temperature change, causing reduced accuracy in the initial few time steps in FEA. Using a Ramp input (linearly increasing the temperature) avoids this singularity.

$T_i = 0$°C, $T_s = 100$°C, $k = 50$ W/(m·K), $\rho = 7800$ kg/m³, $c_p = 500$ J/(kg·K). $\alpha = 1.282 \times 10^{-5}$ m²/s.

Temperature at $x = 0.01$ m for $t = 10$ s:

$\eta = 0.01/(2\sqrt{1.282 \times 10^{-5} \times 10}) = 0.441$

$T = 100 \times \text{erfc}(0.441) = 100 \times 0.534 = 53.4$°C

The NAFEMS T1 benchmark uses equivalent parameters for one-sided heating of steel.

• Forward Euler (Explicit): 1st order accuracy. Has a CFL condition $\Delta t < h^2/(2\alpha)$. Stable if condition is met, but $\Delta t$ becomes very small.
• Backward Euler (Implicit): 1st order accuracy. Unconditionally stable but has large numerical diffusion in the time direction.
• Crank-Nicolson: 2nd order accuracy. Unconditionally stable. However, it can produce overshoot (Gibbs phenomenon-like oscillation) for step changes.
• Galerkin method ($\theta = 2/3$): Suppresses oscillation while maintaining relatively high accuracy.

In Abaqus, the HEAT TRANSFER step defaults to Backward Euler. For TRANSIENT HEAT TRANSFER, the Alpha (AMPLITUDE parameter) can be set. $\alpha = 0$ is Backward Euler, $\alpha = 0.5$ is Crank-Nicolson.

Nastran's SOL 159 (nonlinear transient heat) uses a Newmark-type $\theta$ method, with $\theta = 0.5$ (equivalent to Crank-Nicolson) as the default.

In the semi-infinite body problem, the temperature penetration depth $\delta(t) = 3.6\sqrt{\alpha t}$ increases with time, so concentrate the mesh near the surface.

Recommended settings:

• Surface element size: $h_{min} = \delta(t_{final})/20$
• Geometric progression bias: Coarsen towards the interior with a ratio of 1.5–2.0
• Model length: $L > 5\delta(t_{final})$ to approximate a semi-infinite body
• Time step: $\Delta t = h_{min}^2/(4\alpha)$ as an initial guideline, confirmed with GCI.

Of course. By fixing the spatial mesh and systematically varying $\Delta t$, Richardson extrapolation in the time direction is possible. Similarly, by fixing $\Delta t$ and varying the spatial mesh, spatial GCI can be obtained. Evaluating both independently is the recommended procedure in ASME V&V 20.

Abaqus: Solve using DC1D2 (1D heat conduction element) or DC2D8 (2D). INITIAL CONDITIONS, TYPE=TEMPERATURE for initial temperature,BOUNDARY to fix the surface temperature.

Nastran: SOL 159 + CHBDY elements to define boundary conditions. Specify time-dependent temperature boundary conditions with TLOAD1/TLOAD2.

OpenFOAM: Use laplacianFoam (pure heat conduction). Specify temperature with fixedValue + uniform in boundary conditions.

COMSOL: Heat Transfer in Solids module. Use Time Dependent Study for transient analysis.

OpenFOAM is primarily for CFD, but laplacianFoam is a pure diffusion equation solver, so it can be used directly for heat conduction. However, for solving only solid heat conduction, CalculiX or Code_Aster is more natural. OpenFOAM's chtMultiRegionFoam excels in fluid-solid coupling (Conjugate Heat Transfer: CHT).