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Implementation of psuedo-arclength continuation Trouble with how to evaluate the required derivatives efficiently Rate Topic: -----

#1 Post icon  13 November 2011 - 07:25 PM TokenMonkey 


Quark
Hi there,

I have a quick question regarding pseudo-arclength continuation. As some background, I am a chemical engineer, not a mathematician, applied or otherwise, so my knowledge of numerical methods is limited to the "standard" stuff.

I've been reading up as extensively as I can on pseudo-arclength continuation, but unfortunately, it's all from second-hand sources; I don't have access to Keller's original paper. Here's what I understand at this point:

We want to solve a problem F(x,\lambda)=0. We assume that the solution is known at x^0 and \lambda^0. To avoid the singularity of the Jacobian, and therefore the breakdown of Newton's method, at turning points, x and \lambda both become parameterised by arclength (s), and we end up with an augmented system of equations to solve:

F(x,\lambda)=0
\left(u-u^{0}\right)\mathrm{d}u^{0}/\mathrm{d}s+\left(\lambda-\lambda^{0}\right)\mathrm{d}\lambda^{0}/\mathrm{d}s-\Delta S=0

While this seems simple enough, how does one obtain the derivatives w.r.t s? Not a single text seems to mention this. Ideas that spring to mind are forward differences using, say, cubic splines; however, that seems horrendously inefficient to me. There must be a better way!

Thanks,
TM
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#2 22 November 2011 - 07:46 AM TokenMonkey 


Quark
I found the answer: a procedure is described in Kubicek's Algorithm 502 in ACM Trans. Math. Software. dl.acm.org/citation.cfm?id=355675
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