What a contact model would have to do

A ladder that has spent five rungs measuring what is wrong owes an account of what it would take to fix, and an opinion about whether to.

What the solve does now #

The current solve is a boundary-value problem with three equality constraints.

A half period of yarn, of known length, has to start at one interlacing and end at the next, with its tangent along the fabric at both ends. Its shape between them is whatever minimises the bending energy, and the minimisation is over the coefficients of a series that satisfies the end conditions identically.

That is a smooth, well-posed, unconstrained minimisation in a dozen variables with three multipliers, and it converges in a handful of Newton steps.

What a contact solve is #

The repair replaces one of those equalities with an inequality, and it is not a small change.

The constraint is that no two points of the fabric’s yarn are closer than a diameter. That is not three conditions; it is a condition at every pair of points, and the set of pairs where it is active is an unknown of the problem.

Such a problem has a name — a variational inequality — and the standard approaches are all substantially heavier than the current one. A penalty method adds a large energy where the constraint is violated and softens the answer. An active-set method guesses which contacts are active, solves, checks, and repeats. A Lagrangian method carries a multiplier per contact, and the number of contacts is not known in advance.

None of them is out of reach and none of them is an afternoon.

What it would cost in yarn #

The geometric cost is the part that can be priced exactly, and it is larger than it looks.

A strand that passes round the standing yarn rather than beside it travels further by about half the standing yarn’s circumference — half of π times a diameter, so 0.26 millimetres for a twenty tex cotton — at each of the two interlacings a stitch has.

That is 0.52 millimetres against a loop length of 3.5: fifteen per cent.

Every dimension this collection computes for a knitted fabric is computed at a fixed loop length, so fifteen per cent of the yarn being reallocated from free run to wrap changes the wale spacing, the course spacing, the thickness, the areal weight and every force.

And in bending #

The wrap follows a radius of about one yarn diameter — the tightest bend anywhere in the fabric, and the same radius as the loop’s own crest.

Bending energy goes as the square of the curvature times the length bent, so a wrap of half a circumference at that radius costs a substantial fraction of what the whole loop currently spends.

So the repaired fabric has more energy in it at the same geometry, and the geometry it settles at will be different. Whether it comes out denser or slacker is not obvious from the sign of one term, which is exactly why it has to be solved rather than estimated.

What it would buy #

Four results this collection does not have, and one it has and should not trust.

A chirality, and therefore a writhe. The model’s course is achiral because a near miss has no handedness, so the fabric has nothing to trade against a twisted yarn’s torque. A threaded crossing has a sense, so the twist–writhe mechanism becomes available and spirality becomes derivable rather than fitted.

A linking number. Two adjacent courses would link once per wale, as a fabric’s do, and the fabric would be a knitted fabric in the only respect that distinguishes one.

A hard extension ceiling. Two threaded loops cannot separate beyond the length of yarn joining them, which is a constraint active from the beginning and tightening throughout — the shape the missing two thirds requires.

And the wale-direction curl. The solve currently runs away to a buckled configuration that the missing loops would physically block, so the calculation is refused rather than made.

The one to distrust is the thickness, which is the collection’s most falsifiable knitted result and is computed from a geometry the repair changes.

What it would not change #

Worth saying, because the list is longer than the previous one.

Every quantity that is an integral over one half period is unaffected to first order: the bending energy of a stitch, the contact force, the fabric’s rigidity in both directions, its thermal resistance, its fibre fraction, the pressure a cuff applies.

Those are the majority of what this collection has computed about a knitted fabric, and they are right for the reason they were always right: a half period does not know what is threaded through it.

So the repair is not a rebuild. It is an addition that changes a few results a great deal and most of them not at all.

The cheaper partial repair #

There is a smaller change that fixes three of the five symptoms and it deserves to be tried first.

The model’s two half periods either side of a crest are exact mirror images, and that symmetry is what makes the course achiral, gives it no writhe, and runs the two strands alongside one another.

In a real fabric they are not mirror images. One arrives at a needle loop’s head and the other leaves it, and a head is a loop with something through it while a sinker loop is not.

Breaking the crest–trough symmetry costs nothing but a longer solve. The configuration space doubles — two half periods rather than one, with a closure condition between them — and the problem stays a smooth equality-constrained minimisation.

That would give the course a handedness, give it a writhe, separate the two strands at a crest, and leave the linking number at nought. Three of five, at a fraction of the cost.

Which one to do #

The recommendation, stated rather than implied.

Do the asymmetric solve first. It is a modest extension of machinery that exists, it is a smooth problem of the kind this collection already solves, and it settles whether the missing chirality is the whole story or only part of it.

If the asymmetric course has a writhe of the right order to explain spirality, the contact solve may not be needed at all for that question.

Then decide about contact on the evidence. The extension ceiling and the curl need threading and nothing else will do, but both are questions about fabric behaviour at extremes rather than about the geometry and forces the collection mostly computes.

And do not do it for the thickness. The thickness is right, it is falsifiable, and it is the collection’s best knitted result. Changing the geometry that produces it should be done deliberately and checked against measurement, not as a side effect of fixing something else.

What a woven contact model would be #

The same question can be asked of the other fabric, and the answer is much shorter — which is worth recording because it says where the effort should go.

A woven cloth’s two systems barely overlap at all: from nothing in an open cloth to four and a half per cent in a dense one. So a woven contact solve would spend its effort enforcing a constraint that is nearly satisfied already, and the answer it produced would differ from the current one by a per cent or two.

That is a poor return, and it is the correct reason not to build one.

It also says something about where this collection’s woven geometry stands relative to its knitted one. Peirce’s construction is nearly self-consistent and the elastica that replaced it is more so; the knitted construction is not, and the difference is the fabric rather than the modelling.

A geometry built on crossing threads is nearly right by construction. One built on parallel ones is not.

Why an active set is the right shape #

There is a practical point about which method to use, and the profile measured earlier settles it.

Only a fifth of a course’s length is inside a diameter of its neighbour. So a constraint imposed at every pair of points would be inactive over four fifths of the domain, and a method that carries a multiplier everywhere would be carrying four fifths of nothing.

An active-set method — guess which contacts are active, solve the resulting equality-constrained problem, check, repeat — is exactly suited to a constraint that is mostly inactive, and the guess is easy to make well here because the active region is where the interlacings are and everybody knows where those are.

A penalty method would be simpler to write and would soften the answer, which for a question about whether two things touch is the wrong kind of approximation.

So the recommendation is an active set with the interlacings as the initial guess, and the profile is what says so.

What the constraint would do to the basis #

One more implementation note, because it is the thing that would go wrong first.

The current solve expands the thread’s tangent in a series of sines that vanish at both ends, so the end conditions are satisfied identically and the minimisation is unconstrained in the coefficients. That is what makes it converge in a handful of steps.

A contact constraint is not expressible in that basis. It is a condition on the curve’s position relative to another curve, at points the basis does not privilege, so it has to be imposed pointwise on the sampled path rather than absorbed into the expansion.

That is a different kind of problem and it is where the smoothness of the current solve would be lost: a contact set that changes between iterations makes the objective non-smooth, and Newton’s method does not like non-smooth objectives.

Recording that now is worth something, because it is the reason a contact solve is not “the same solve with one more constraint” — and somebody starting from that assumption would spend a day finding out.

What was counted, and how #

The yarn cost is half a circumference at each of two interlacings, against the loop length: π times the diameter, against 3.5 millimetres, for the collection’s own default construction.

The bending cost is a curvature of one over a diameter over a length of half a circumference, against the loop’s own bending energy from the solve.

Neither is a solve and both are stated as estimates. This collection has been wrong before about an estimate that added a term to a functional rather than re-minimising it, and the caution is recorded rather than repeated: the numbers here are what the repair would have to find room for, not what it would produce.

Why the collection should not build it now #

An opinion, stated as one, because a specification without a recommendation is a way of deferring a decision.

This collection’s subject is the structure of cloth: how a weave decides what a fabric does, how a loop’s geometry produces its forces, where a number comes from. Its strength is closed-form comparisons that can be checked, and its method is to compute one thing exactly rather than many things approximately.

A contact solve is the other kind of object. It is a numerical machine whose output is a shape, it is hard to check against anything but itself, and the things it would settle — an extension ceiling, a curl radius, a spirality angle — are questions about fabric behaviour rather than about fabric structure.

So the honest recommendation is: build the asymmetric solve, which is small and in the collection’s own idiom, and leave the contact solve alone unless a specific question forces it.

That is not a counsel of laziness. It is a statement that a collection with a method should extend the method rather than adopt a different one because a gap is annoying — and that the four results the repair would buy are, on this collection’s own priorities, worth less than they look.

Where the model stops #

Nothing here is built. The rung is a specification and an opinion.

The yarn cost assumes the wrap is a half circumference. A real interlacing wraps through some angle that the solve would decide, and half a turn is the natural guess rather than a computed value.

And the two repairs are not independent. An asymmetric solve changes where the crossing is and how much yarn reaches it, so the contact repair applied afterwards would be applied to a different fabric.

What five rungs of measurement bought #

It is worth being explicit about what a ladder that fixed nothing produced, because the answer is more than it appears.

A number for the flattening, which was a swept parameter and is now a prediction with a dependence attached.

A ruled-out candidate: contact between courses does not explain the extension ceiling, which was the recorded diagnosis and is wrong.

A separation of two defects that one measurement produced and that would have been treated as one: a section problem and an arrangement problem, needing different repairs.

A hard ceiling on tightness factor from the bending limit, which is a constraint this collection did not have.

A control: the same question asked of a woven cloth gives a different answer, so the knitted result is about knitting.

And a specification for the repair, priced, ordered and with its checks named.

None of that required building anything. All of it required asking whether the model’s output could exist, which is a question that costs a distance calculation and that this collection had never asked of anything.

The generalisation #

The rung is an instance of something this collection does not do often enough and should.

Price a repair before making it, and say whether it is worth it.

The usual pattern in a research collection is that a defect is found and then either fixed or recorded as open. Neither of those is a decision. A defect that is fixed consumed whatever it consumed; one that is recorded as open sits in a list.

What is missing between them is the estimate: what would it cost, what would it buy, and is the ratio good. That estimate is cheap — an afternoon here — and it changes what happens next, because it turns a vague intention to fix something into an ordered plan with a reason attached — the cheaper half first, and here is why.

The most useful output of this rung is not the cost. It is the ordering: the asymmetric solve before the contact solve, and neither of them for the sake of the thickness.

What the repair would have to be checked against #

A model that changes four results needs something to be checked against, and the list is short enough to be worth writing down before anybody builds one.

The thickness. Two yarn diameters, 0.334 millimetres for the collection’s default construction, independent of the gauge. That is the current model’s best prediction and the repaired model has to reproduce it or explain why not.

Munden’s dimensions. The wale and course spacings at three relaxation states are measurements, and a repaired geometry that does not accommodate them at a plausible loop length is wrong.

The linking number. It has to come out at one per wale, as an integer, and the integral is already written.

And the flattening. The repaired geometry should still demand a flattened yarn, because a real one is flattened — but it may demand a different amount, and if it demands none the repair has over-corrected.

Four checks, all of them cheap, all of them available now. A repaired model that passes all four would be a substantially better object than the current one; one that passes three would need an argument about the fourth.

Who found it, and when #

Contact problems in rod mechanics are a mature field and the standard methods are well established, mostly developed for cable, hair and biological filaments.

Knitted fabric models with contact exist in the finite-element literature and are used for drape and impact simulation. They are much larger objects than anything here and they do not produce the closed-form comparisons this collection is built out of.

What is this collection’s own is the pricing: knowing, from its own geometry, exactly how much yarn a threaded interlacing costs and how much bending, so that the decision about whether to build one is made against numbers rather than against an impression.

Where the ladder goes next #

The contact ladder ends here and this work turns to what the two halves have in common. Four unrelated everyday observations — a snarl, a knot, a flattening and a measurement check made several ladders ago — all land at the same end of the same bracket, and putting them together says more than any of them alone.

A fabric reads its own bracket four ways.