Parameters
Grid Size N
20
10 – 50 (N×N cells)
Thermal Conductivity k
50.0 W/m·K
Steel:50 / Al:200 / Cu:385
Heat Generation q'''
0 W/m³
Boundary Conditions
Top
Bottom
Left
Right
Ready
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Max Temperature [°C]
—
Min Temperature [°C]
—
Max Heat Flux [W/m²]
—
Convergence Iterations
Temperature Contour Map (click to select cross-section)
Cross-Section Temperature Profile
Theory
2D steady-state heat conduction with internal heat generation:
$$\nabla^2 T + \frac{q'''}{k} = 0 \quad \Rightarrow \quad \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{q'''}{k} = 0$$FDM discretization with central differences (uniform grid Δx = Δy):
$$T_{i,j} = \frac{T_{i+1,j}+T_{i-1,j}+T_{i,j+1}+T_{i,j-1}}{4} + \frac{q'''(\Delta x)^2}{4k}$$Gauss-Seidel update:
$$T_{i,j}^{(n+1)} = \frac{1}{4}\bigl(T_{i+1,j}^{(n)}+T_{i-1,j}^{(n+1)}+T_{i,j+1}^{(n)}+T_{i,j-1}^{(n+1)}\bigr)+\frac{q'''(\Delta x)^2}{4k}$$
CAE Applications: Pre-processing estimates for Abaqus/Ansys thermal models · PCB and heat sink initial thermal design · Furnace wall and refractory steady-state temperature verification · FEM result cross-validation.