SME
Numerical Methods for Engineers · Problem type
An interior node of a steady 2-D plate
Use this for “A square-grid plate node has neighbours at 100, 60, 40 and 20 °C. What is its steady temperature?”
Chart
This is an example. Type your own values to start a new problem.
T (°C)
=
TN (°C)
+
TS (°C)
+
TE (°C)
+
TW (°C)
4
Formulas
T
=
TN + TS + TE + TW
4
55
=
100 + 60 + 40 + 20
4
Assumptions
•
Steady, with no heat made inside the plate: Laplace’s equation, ∂²T ÷ ∂x² + ∂²T ÷ ∂y² = 0.
•
A square grid (Δx = Δy), so each interior node is the mean of its four neighbours.
•
Gauss–Seidel redoes each node with its neighbours’ newest values until nothing changes.
Step-by-step
WE KNOW
TN = 100 °C, TS = 60 °C, TE = 40 °C, TW = 20 °C
FIND
node temperature (T)
Step 1 · Find node temperature (T)
Use T = (TN + TS + TE + TW) ÷ 4
With no heat made inside, the node settles at the mean of its four neighbours.
T
=
100 + 60 + 40 + 20
4
T
=
220
4
T
=
55
°C
Check
Put the numbers back in. Both sides should match.
55
=
100 + 60 + 40 + 20
4
✓
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