Module 12 · Lesson 12.4
Conditioning and meshing
Where the precision goes, and how to tell a mesh that is converging from one that never will.
Why this matters
Two topics that look unrelated and share a root: both are about the difference between what the model computes and what the structure does, and in both the reassuring-looking symptom is the dangerous one.
Ill-conditioning produces noise that looks like results. A singular mesh produces a sequence that looks like it is converging on something. Neither announces itself.
By the end of this lesson you should be able to
- Name the three causes of ill-conditioning and estimate the digits each costs
- Distinguish a converging refinement sequence from a singular one
- Explain why a peak stress at a singularity is meaningless and what to use instead
- Judge element quality by aspect ratio, skew and Jacobian
- Explain what nodal averaging removes
Where the precision goes
A 64-bit number carries about sixteen significant decimal digits, and a solve spends some of them. The rule from Module 9: a condition number of 10ᵏ costs roughly k of them.
Three things drive it up, and they all draw on the same account:
Stiffness range. A 5 mm stub among 6 m members is stiffer by a factor of order 10⁹ in bending. Use a rigid link or a constraint instead — a constraint forces nodes to move together without contributing an enormous stiffness, which is exactly the behaviour wanted.
Coordinate magnitude. A model in national grid coordinates sits hundreds of kilometres from the origin. Six digits are spent describing where it is before any are left for the millimetre-scale differences between nearby nodes. Work near the origin and keep the survey transformation as a separate, documented step.
Element distortion. Element formulations assume a shape near square or equilateral. Long, thin or skewed elements degrade the conditioning as well as the accuracy.
Mesh convergence, and its exception
The standard discipline: halve the element size and see whether the answer changes. If it does, the previous mesh was inadequate. If it does not, you have converged.
That rule has an exception, and recognising it is the skill:
At a mathematical singularity, refinement never terminates. A re-entrant corner, or a point support on a two-dimensional mesh, has an unbounded elasticity solution. Every refinement finds a larger stress, and the sequence has no limit.
The discriminator is not the values but the ratios between them:
- A converging sequence has shrinking increments. The change from 8 to 16 elements is smaller than from 4 to 8.
- A singular sequence grows at a roughly constant ratio per halving, forever.
One extra refinement settles which you are looking at.
What to do at a singularity
Stop chasing the peak. It is a property of the idealisation, not of the structure.
Use a quantity that converges instead:
- The integrated moment or force over a strip. It converges — though slowly, because the missed fraction falls as h0.46 at a 90° re-entrant corner, so getting within 10 % needs an element about three hundred times smaller than the strip.
- Better: the resultant from equilibrium. The punching shear at a flat-slab column is the difference in column axial force above and below the connection. That comes from statics and has no mesh in it at all — it is exact at any density.
Where a peak is singular, the resultant is not. And where a resultant can be had from equilibrium rather than from the stress field, take it from equilibrium.
Element quality
Three separate defects, measured separately:
Aspect ratio — longest edge over shortest. Above about 4 it is acceptable only where the stress gradient runs along the long side; above 10 it is poor.
Skew — how far a corner has strayed from 90°. Above about 20° it degrades the strain field throughout the element; above 45° it is poor.
Jacobian — the determinant of the mapping from the ideal element to the real one. If it goes negative anywhere the element is folded and is not an element at all.
Averaging hides the evidence
Stress is discontinuous between elements in a displacement-based formulation. The size of the jump is a direct measure of how far the mesh is from resolving the field — it is the mesh's own error estimate, free.
Nodal averaging smooths it away. It makes contours pretty and it deletes the diagnostic.
Check with averaging off. Report with averaging on. A smooth contour proves nothing except that averaging is enabled.
Try it
Mesh convergence tool
Two refinement sequences. One settles; one grows at a constant ratio and never will.
Quantity
The series settles. Halving the element size stops changing the answer, and that is what 'the mesh is fine enough' means — not that the mesh looks nice.
| Elements | Value | Change | Ratio to previous |
|---|---|---|---|
| 1 | 0.0307 | — | — |
| 2 | 0.0308 | 0.14 % | 1.0014 |
| 4 | 0.0308 | 0.04 % | 1.0004 |
| 8 | 0.0308 | 0.01 % | 1.0001 |
| 16 | 0.0308 | 0.00 % | 1.0000 |
| 32 | 0.0308 | 0.00 % | 1.0000 |
- Converges?
- yes
- Limit
- 0.0308
- Adequate at
- 2 elements
change below 2 %
The increments shrink. That is convergence, and the last value is usable.
The discriminator
- Look at the RATIO column, not the value column. A converging sequence has ratios tending to 1; a singular one has a constant ratio above 1.
- One extra refinement settles which you are looking at, and it is the cheapest question you can ask of a mesh.
- For the singular case the honest fix is not a finer mesh. It is a different quantity: the punching shear at a flat-slab column is the difference in column axial force above and below, which has no mesh in it at all.
What this shows: 'The answer changed when I refined it' tells you the previous mesh was inadequate. Whether the new one is adequate depends on how the increments behave, not on the values.
Try it
Mesh quality explorer
Distort a quadrilateral and watch three independent metrics move. Then size a mesh by the two rules.
Slide the top edge sideways to introduce pure skew.
Mesh sizing
- Aspect ratio
- 1.00
- Worst corner off square
- 0.0°
- Smallest internal angle
- 90.0°
- Minimum Jacobian
- 0.2500
- Verdict
- good
Sizing the mesh
- Twice the thickness
- 0.60 m
- Span over ten
- 0.72 m
- Span-to-depth ratio
- 24.0
- Recommended
- 0.60 m
At a span/depth of 24 the two rules give nearly the same size, which is why either is a reasonable start.
What this shows: Aspect ratio, skew and a negative Jacobian are three different defects with three different consequences — and a folded element is not an element at all.
Worked example
Converging, or not?
Given
- Two quantities are tracked through four mesh refinements, each halving the element size
- Quantity A: 12.1, 13.4, 13.8, 13.9 mm
- Quantity B: 41, 56, 77, 106 N/mm²
Find
Which is converging, and what to do about each
Practice
A reduced stiffness matrix has a condition number of 10¹². Using the rule that 10ᵏ costs about k of sixteen significant digits, how many significant digits survive?
Practice
A slab is 300 mm thick and its largest span is 7.2 m. Using the two rules — element width at least twice the thickness, and about ten elements across the major span — what is the recommended element size in metres? Take the smaller of the two.
Check yourself
A flat-slab model's peak moment at a column keeps rising with every refinement. What is the correct design quantity to use for the punching check?
Check yourself
How do you tell a converging refinement sequence from a singular one?
Check yourself
What sets a model's condition number in practice?
Summary
- Stiffness range, coordinate magnitude and element distortion all spend the same precision budget
- A converging sequence has shrinking increments; a singular one has a constant ratio
- At a singularity, use an integrated resultant — better, one from equilibrium
- Aspect ratio, skew and Jacobian are three separate defects
- Check with averaging off; report with it on
This is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions. Module overview and checkpoint