Computational Engineering / Diagnosing computational models

Conditioning and meshing

Where the precision goes, and how to tell a mesh that is converging from one that never will.

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A computational engineering reviewer must check algorithms, convergence claims, statistical assumptions and numerical examples beyond the cases explicitly recorded.
Model Doctor lab case banks are separate from these lesson blocks and need their own full technical audit.
Confirm mesh-quality limits and punching checks for each adopted element and design standard.

These are educational models and exercises. Check their assumptions before using the results. The full technical content and every diagram have not yet been independently verified.

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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

Conditioning

Large stiffness contrasts, near-mechanisms, scaling and element quality can make the assembled equations sensitive to numerical error. A condition number of 10ᵏ gives a rough worst-case loss of k decimal digits, not an exact count of reliable result digits. Coordinate subtraction can also lose precision when a small geometry is stored far from the origin.

Mesh convergence

Track the response quantity needed across systematic refinements using a stated tolerance. Shrinking changes suggest convergence, but a short sequence cannot prove it. Singular fields may have growing peaks; inspect the idealisation and local boundary conditions as well as the trend. The exponent and convergence rate depend on the particular singularity and quantity.

At an idealised singularity, use a physically relevant convergent quantity or improve the local model. Integrated forces and equilibrium resultants can be useful, but their accuracy still depends on global load distribution and the model. A punching check requires the relevant control-perimeter shear, load deductions and moment transfer specified by the design method; a peak nodal moment is not a substitute.

Element quality and stress recovery

Check aspect ratio, skew and Jacobian using limits appropriate to the element formulation and expected field variation. There is no universal aspect-ratio or skew threshold for every mesh.

Compare averaged and unaveraged stress output. Inter-element jumps can reveal discretisation error, but legitimate material or section discontinuities also create jumps. Report the recovery method and retain discontinuities where they are physically meaningful; smooth contours alone do not demonstrate accuracy.

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.

Successive refinements. Watch the change column, and the ratio between values.
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

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.

Width

Height

Top edge offset

Slide the top edge sideways to introduce pure skew.

Mesh sizing

Slab thicknessm

Largest spanm

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 300 mm slab spans 7.2 m. Two illustrative trial mesh spacings are twice the thickness and one tenth of the span. What is the smaller value, in metres? This arithmetic does not set an element-size requirement.

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
Use refinement trends together with the idealisation and a stated response tolerance
At a singularity, use an integrated resultant — better, one from equilibrium
Aspect ratio, skew and Jacobian are three separate defects
Inspect averaged and unaveraged results; document the recovery method and preserve real discontinuities