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I am interested in approximating the time evolution of 2D curves. Here's an illustration:

enter image description here

An issue that arises when naively making this approximation as illustrated above, is that as one increases the number of "line segments" used to approximate the curve, then one is also introducing extra "flexibility". That is, curves with less approximating line segments are stiffer than curves with many approximating line segments.

I am interested in knowing what the theory that handles analytical solutions of simple continuous cases is called. I realize it would be a topic under solid mechanics/elasticity, but are there any more specific subject headings I might use to look up in textbooks? (an idea I have from my novice intuition is that this is going to be some sort of a PDE problem -- the elasticity is very simple: so I don't think it needs a tensor description?)

I am also interested in knowing if there are any textbooks that explore the sort of numerical approximation I illustrated? (an idea I have from my novice intuition is that it is not finite element method)

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  • $\begingroup$ Is the problem similar to the problem of inflating a balloon? In another case I don't see how the system can have the stiffness that you propose. $\endgroup$ – nicoguaro Jul 18 '15 at 4:55
  • $\begingroup$ @nicoguaro think of it as a "global stiffness"? imagine that we have a world where balloons don't have to be smooth, so they can look like some polyhedra mean to approximate their smooth shape. Rougher polyhedral balloons will expand less ("are stiffer") than smoother polyhedral balloons, if every face has the same material stiffness in both balloons. This is because it seems that the edges add extra degrees of flexibility? I am not sure how to explain it well, so please let me know if this was clearer at all. $\endgroup$ – user89 Jul 18 '15 at 13:38
  • $\begingroup$ I'm asking because I was curious if it was just the surface or it has something in the micro l middle. Regarding the extra "flexibility", I think that you should describe mathematically what you're using, so we get it. Also, I think that the last sentence (the one in parenthesis) can be rephrased to be clearer. $\endgroup$ – nicoguaro Jul 18 '15 at 15:27
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    $\begingroup$ This sounds like a differential geometry question. I am far from an expert, but you might get some guidance from looking up things like "shape derivative". $\endgroup$ – Andrew T. Barker Jul 18 '15 at 22:34
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This sounds like something that could be handled with a level set method, although I've never done enough with them to know how to apply the elasticity. Stan Osher's group at UCLA used to be doing a lot of work in this area, but I'm not sure who the experts are nowadays.

This book by Stan Osher and Ron Fedkiw might be a good start: Level Set Methods and Dynamic Implicit Surfaces

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  • $\begingroup$ I suppose level set methods are a way to approach this problem, but the approach is very different from the approximating method I outlined. I am more interested in approximating methods involving the discretization of the curve. $\endgroup$ – user89 Jul 18 '15 at 13:40

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