2D Steady-State Heat Conduction

It's a problem where temperature changes in two directions. Representative examples include in-plane temperature distribution on circuit boards, cross-sectional temperature fields in molds, and thermal bridge evaluation in building walls.

The 2D steady-state heat conduction equation for isotropic materials is as follows.

If $k$ is constant and there is no heat generation, it reduces to the Laplace equation $\nabla^2 T = 0$.

For a rectangular domain with specified temperatures on each side, a series solution can be obtained using the separation of variables method. For example, for a square domain where three sides are 0°C and one side is $T_0$:

This solution is extremely important as a verification benchmark for FEM codes.

As a convenient method to treat 2D geometric effects in a 1D manner, there is the shape factor $S$.

For example, the heat loss from a buried pipe in the ground can be evaluated with $S = 2\pi L / \cosh^{-1}(D/r)$. Textbooks (Incropera et al.) summarize $S$ for major shapes.

That's right. However, for complex shapes, the shape factor is unknown, so we solve them using FEM. In practice, it's common to back-calculate the shape factor value from FEM results and reuse it for similar designs.

For 2D heat conduction, there are triangular and quadrilateral elements.

The 3-node triangle has a constant temperature gradient within the element, so accuracy is poor unless refined. In practice, 6-node triangles or 8-node quadrilaterals are recommended.

For the same number of nodes, quadrilaterals have higher accuracy. However, triangles (tetrahedra) are easier to generate for automatic meshing of complex shapes. Choosing hex-dominant meshing in Ansys provides a good balance.

Central differencing on a uniform grid is:

If $\Delta x = \Delta y$, it becomes the famous 5-point stencil. This can be implemented in Excel or custom Python code for verification.

Especially for 2D verification, it can be written in a few dozen lines of Excel VBA or Python, so there's no reason not to do it.

The computational cost is orders of magnitude smaller. 2D is optimal for parametric studies running dozens of cases or for problems where temperature gradients in the cross-sectional direction are dominant.

When modeling the cross-section of a multilayer PCB, each layer (copper foil, prepreg, core) is arranged as a strip with different $k$. The equivalent thermal conductivity differs greatly between an L1 layer with 80% copper coverage and an L3 layer with 20%.

PCB substrates are extremely anisotropic materials. 2D cross-sectional analysis evaluates through-plane temperature gradients and via effects, while in-plane direction is corrected using spreading resistance. This is the technique also used in the internal modeling of FloTHERM and Icepak.

Utilizing symmetry can significantly reduce computational scale.

• 1-axis symmetry: 1/2 model, adiabatic condition on the symmetry plane
• 2-axis symmetry: 1/4 model
• Cyclic symmetry: Model only one repeating unit

If the temperature field is mirror-symmetric across the symmetry plane, the temperature gradient crossing the plane is zero, meaning the heat flux is zero. It's a natural consequence when you think about it physically.