SME
Finite Element Analysis
Meshing and convergence
Use this for “Three meshes, each twice as fine, give tip deflections of 7.80, 8.20 and 8.30 mm. What is the observed order and the converged deflection?”
Table
Fine-mesh deflection (f₁)
(mm)
Observed order (p)
p against f₁, worked out for every f₁ from 8.26 to 8.34 mm
This is an example. Type your own values to start a new problem.
p
=
ln
f₃ (mm)
−
f₂ (mm)
f₂ (mm)
−
f₁ (mm)
÷
ln(
r
)
fₑₓₜ (mm)
=
f₁ (mm)
+
f₁ (mm)
−
f₂ (mm)
r
−
1
e
Error of f₁
calculated · %
Formulas
p
=
ln
(
f₃
−
f₂
) ÷ (
f₂
−
f₁
)
ln(
r
)
2
=
ln
(7.8 − 8.2) ÷ (8.2 − 8.3)
ln(2)
fₑₓₜ
=
f₁
+
f₁
−
f₂
r
p
− 1
8.3333
=
8.3
+
8.3 − 8.2
2
2
− 1
e
=
fₑₓₜ −
f₁
fₑₓₜ
×
100
0.4
=
8.3333 − 8.3
8.3333
×
100
Assumptions
•
Each mesh is r times finer than the last, and all three are fine enough to converge smoothly.
•
The results converge monotonically: the changes keep one sign and shrink.
•
Richardson extrapolation: the error falls as hᵖ, so fₑₓₜ estimates the exact answer.
Step-by-step
WE KNOW
f₃ = 7.8 mm, f₂ = 8.2 mm, f₁ = 8.3 mm, r = 2
FIND
observed order (p), extrapolated deflection (fₑₓₜ), error of f₁ (e)
Step 1 · Find observed order (p)
Use p = ln((f₃ − f₂) ÷ (f₂ − f₁)) ÷ ln(r)
Each refinement cuts the error by rᵖ, so the ratio of the two changes is rᵖ: take logs.
p
=
ln
(7.8 − 8.2) ÷ (8.2 − 8.3)
ln(2)
p
=
ln
−0.4 ÷ (−0.1)
ln(2)
p
=
ln(4)
ln(2)
p
=
1.386
0.6931
p
=
2
Step 2 · Find extrapolated deflection (fₑₓₜ)
Use fₑₓₜ = f₁ + (f₁ − f₂) ÷ (rᵖ − 1)
The
error
left
in
f₁
is
the
last
change
shared
by
rᵖ
−
1:
add
it
to
f₁
.
fₑₓₜ
=
8.3
+
8.3 − 8.2
2
2
− 1
fₑₓₜ
=
8.3
+
0.1
4 − 1
fₑₓₜ
=
8.3
+
0.1
3
fₑₓₜ = 8.3 + 0.03333
fₑₓₜ = 8.3333 mm
Step 3 · Find error of f₁ (e)
Use e = (fₑₓₜ − f₁) ÷ fₑₓₜ × 100
How far the fine mesh is from the extrapolated answer, as a percent of that answer.
e
=
8.3333 − 8.3
8.3333
×
100
e
=
0.0333
8.3333
×
100
e
=
0.003996
×
100
e
=
0.4%
Check
Put the numbers back in. Both sides should match.
2
=
ln
(7.8 − 8.2) ÷ (8.2 − 8.3)
ln(2)
✓
8.3333
=
8.3
+
8.3 − 8.2
2
2
− 1
✓
0.4
=
8.3333 − 8.3
8.3333
×
100
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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