The answer that depends on how it was divided

Everything computed on this site has been computed on a structure divided into parts. A frame becomes members meeting at nodes; a section becomes strips; an arch becomes a thrust line evaluated at stations; a slab’s collapse load becomes a search over yield-line positions.

The division never appears in the answer. It decides how good the answer is, and the interesting fact is that its effect is not always an approximation at all.

Which free body produced the number #

A member’s stiffness matrix comes from a free body of the member with unit displacements imposed at its ends, and the forces required to hold them. Getting those forces means knowing the shape the member takes, and the standard beam element assumes a cubic.

That assumption is not an approximation. Integrate the beam equation for a member with no load between its ends and the exact deflected shape is a cubic — it is the moment, which is linear, integrated twice. So a beam element with loads only at its nodes reproduces the exact displacements, the exact end forces and the exact moment diagram, and dividing the member into ten does not improve anything because there is nothing to improve.

This is worth stating plainly because it is not true of most numerical methods and it is the reason frame analysis is so much more trustworthy than analysis of anything two-dimensional. A frame model has no mesh error.

Where the error comes back #

It comes back in exactly one place in a frame: within a member carrying load along its length.

The cubic cannot represent the quartic shape a uniformly loaded member takes, so the displacement inside the element is wrong. The end forces are not, because the load is converted to statically equivalent nodal forces and fixed-end moments before the solve, and that conversion is exact. So a single element gives exact reactions, exact end moments and a wrong shape between the nodes — which is why analysis software plots a member’s internal diagram from the end forces and the applied load rather than from the element’s own displacement field.

For everything else — plates, shells, solids — no such luck. There is no shape function that is exact for a plate under a general load, so a plate mesh has a genuine convergence question and a plate answer has an error that falls with the element size at a rate the element’s own order decides.

Why the error has a sign #

The more useful fact about restricting a shape is that it does not produce a random error. It produces one in a known direction.

Assuming a shape is imposing a constraint, and a constraint can only make a structure stiffer. So an energy method that assumes a shape returns a buckling load that is too high, a natural frequency that is too high, and a deflection that is too small — always, and never the other way.

The size of the error is the second useful fact. The quotient is stationary at the true mode, so a first-order error in the shape produces a second-order error in the load: a shape wrong by 30% gives a load wrong by about 80%, and one wrong by a few per cent gives a load wrong by a fraction of a per cent. Guessing the shape is the essay about that property, and it is what makes a crude assumed shape worth using at all.

The sign has a practical consequence that is easy to state and easy to forget: a coarse model is unconservative for stability and for vibration. A stiffness that is too high gives a critical load that is too high and a frequency that is too high, and both errors are in the direction that says the structure is fine.

What a mesh has to be fine enough for #

Where a discretisation does have an error — a plate, a shell, a member with distributed load, a stress field near a hole — the question of how fine is fine enough has a general answer, and it is not about the size of the structure.

It is about the wavelength of what is happening. A shape function of a given order can follow a field that varies slowly over an element and cannot follow one that varies quickly, so the element size has to be compared with the distance over which the answer changes, and nothing else.

That comparison explains most of what a mesh needs to do. A slab under a uniform load has a field varying over the span, so a handful of elements per span is generous. The same slab around a column has a field varying over the effective depth, so the mesh has to be an order of magnitude finer there and only there. A shell has a bending boundary layer at every edge and every discontinuity, whose width is $\sqrt{Rt}$ and which is often a small fraction of the shell — and the whole of the shell’s bending lives inside it.

The same rule explains where a mesh cannot help. Approaching a re-entrant corner the wavelength goes to zero, so no finite element size resolves it and the computed stress climbs without bound as the mesh is refined. The right response is not a finer mesh; it is to stop asking for the peak stress and ask for a quantity that exists — a force across a section, a stress averaged over a length the material cares about.

That is the same distinction a stress concentration draws between a stress and a strength, arriving from the numerical side.

Two bounds and a search #

The other place a discretisation has a direction is plastic collapse, where the two classical theorems make the direction explicit.

A yield-line pattern is a discretisation of the collapse mechanism — searched for rather than quoted, and every pattern gives an upper bound on the collapse load — so searching over positions and taking the minimum converges from above, exactly as the assumed shape did. The search resolution is the discretisation, and here it agrees with the closed-form answer to within the resolution rather than disagreeing with it.

That is two ways of being wrong doing real work. A method that bounds an answer from one side is far more useful than one that approximates it, because a bound plus a factor is a design and an approximation plus a factor is a hope.

The division that changes only the cost #

There is one discretisation decision that changes nothing about the answer and a great deal about the work, and it is worth separating from the others.

Bandwidth is the largest difference between the node numbers at the two ends of any member — a property of the labelling rather than of the structure. A banded factorisation costs about $nb^2$ operations against a dense $n^3/3$, so the numbering decides the work by a large factor and the physics by nothing.

Every solver in existence renumbers before it factorises, which is why nobody thinks about this any more. It is worth knowing anyway, because it is the cleanest available example of a modelling decision that is purely a bookkeeping choice — and the next section is about one that looks identical and is not.

The bookkeeping that was the invention #

A redundant structure can be solved by releasing any set of restraints that leaves it determinate, and the choice makes no difference to the answer. It makes an enormous difference to the matrix.

Release a moment at each interior support and the release is felt only in the two spans either side, so the flexibility matrix is tridiagonal and every equation has three unknowns in it. Release a support instead and a unit reaction anywhere deflects everywhere, so the matrix is full.

That difference is the three-moment equation, and it is the reason continuous beams could be solved on paper for a century. The structure did not change; the bookkeeping did, and the bookkeeping was the invention. The matrix that replaced the hand methods made the choice irrelevant to a machine, which is exactly why it is worth remembering that it was once the whole difficulty.

Two routes, one answer #

The reassuring counterweight to all of this is that independent discretisations of the same problem agree, and agree to a precision that says something about both.

Two methods that share no arithmetic, agreeing to thirteen figures, is evidence about the implementation rather than about the structure. What is evidence about the structure is the third curve on that figure: the deflected shape of the same beam under a single central load, read at the station the moving load occupies, which lands on the same curve to 0.15% — and that is Maxwell’s reciprocal theorem, which neither computation was told about.

One deflection without solving everything is a different discretisation again: it computes one number rather than a field, at a cost that does not grow with the size of the structure.

Where the model stops #

Convergence is not accuracy. A mesh refined until the answer stops moving has converged to the answer of the model, which contains an idealised geometry, idealised supports and idealised material. Refining it further improves nothing, and the remaining error is not visible in any convergence study.

Some quantities converge much more slowly than others. Displacements converge fastest, then internal forces, then stresses, then stress gradients. A mesh that gives a deflection to one per cent may give a peak stress at a re-entrant corner that is wrong by a factor and that gets worse as the mesh is refined, because the exact answer there is infinite.

The load has been discretised too. Converting a distributed load to nodal forces is exact for the end forces of a beam element and is an approximation everywhere else — a pressure on a plate mesh becomes a set of node forces whose equivalent is exact only in the work it does, which is a weaker statement than it sounds.

A discretisation can create a mechanism. A model whose elements are arranged so that some deformation costs no energy has a singular matrix or, worse, a nearly singular one — and a nearly singular matrix gives an answer rather than an error.

The discretisations that are not numerical #

It is worth listing the divisions this collection makes that have nothing to do with a computer, because they obey the same rules.

A section cut into strips. The fibre model integrates a stress distribution by summing over strips, and the error falls with the number of them — except at a sharp boundary such as a neutral axis or the edge of a plastic zone, which no finite number of strips resolves and which is where the integrand changes fastest.

A thrust line evaluated at stations. An arch is checked by finding whether a line of thrust fits inside the ring, and the check is made at a finite number of sections. A line that fits at every station checked may leave the ring between two of them, and the stations have to be closer together where the ring’s geometry changes fastest — the same wavelength rule again.

A truss standing in for a continuum. A cracked concrete beam carrying shear as a truss is a discretisation of a stress field into a small number of struts and ties, chosen rather than computed, and it is an equilibrium solution — so by the lower-bound theorem it is safe whatever the choice, and the choice decides only how much capacity is left on the table.

A load pattern applied at a set of positions. An influence line is evaluated at a finite number of load positions, and the worst position found is the worst of those tried.

In every case the pattern is the same: a continuum is replaced by a finite set, the answer is right where the set is rich enough to contain what really happens, and the direction of the error follows from whether the restriction was on a displacement — which stiffens — or on an equilibrium field, which is safe by a theorem instead.

What the picture cannot show #

A mesh is drawn and a discretisation is not. The stations at which a thrust line is evaluated, the strips a section is cut into, the trial positions of a yield line and the number of terms in an expansion are all discretisations, and none of them appears in any drawing of the structure.

Nor does a convergence plot show the one thing a reader most wants from it, which is whether the sequence is converging to the right answer. A sequence that is monotone, smooth and settling looks identical whether the limit is correct or whether it is the correct answer to a model with a support in the wrong place.

The generalisation #

The habit worth carrying is to ask, of any computed answer, what was assumed about the shape — because that is what a discretisation is.

An element’s shape function, an assumed buckling mode, a yield-line pattern, a thrust line’s segments, a section’s strips: each is a statement that the true field lies in some restricted family. Where the true field is in the family, the answer is exact. Where it is not, the answer is wrong in a direction that the restriction decides — and a restriction on a displacement field always stiffens, which makes the error’s sign knowable without knowing its size.

That is the useful half of numerical analysis for a structural engineer. Not the convergence rates, which the software handles, but the one-sidedness: a model that cannot deform the way the structure can is a model that says the structure is stronger than it is.