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Higher Order Spectral Difference Methods

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Old   October 7, 2012, 07:00
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Originally Posted by nikola_m View Post
@sbaffini

Thanks!

The paper you linked is on Hughes Variational Multiscale Approch (interesting thing per se), and they use NURBS based finite elements. A recent Hughes effort is Isogemetric Analysis (cf. isoparametric elements). The idea is to use same functions to model the geometry in CAD and as trial functions in FEM.

I'm aware of Kravchenko and Moins work - they used B-splines for expansion in non-uniform direction in channel flow simulations, if I'm correct.

B-splines (and therfore NURBS) are very closely related with Bernstein polynomials.
Bernstein polynomials are very interesting on their own. Check out my blog (see link below) if you're interested to see more but don't have time to test the code.
Just curious why you want to use Bernstein polynomials ? I can think of some disadvantages, but are there any advantages to it. I have tried bernstein polynomials in a DG framework for convection problems since I was interested to learn about isogeometric analysis.
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Old   October 7, 2012, 17:06
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Originally Posted by praveen View Post
Just curious why you want to use Bernstein polynomials ? I can think of some disadvantages, but are there any advantages to it. I have tried bernstein polynomials in a DG framework for convection problems since I was interested to learn about isogeometric analysis.
Hello! Thanks for the question.

The path that led me to these is a curious one.

I first used Bernstein polynomials to approximate a shape of an "flying saucer" shaped aircraft in an aerodynamic study of Coanda effect. Here is the paper from ICAS 2010 in Nice, France where it was presented.

In presented scheme they seemed powerful in representation of various complicated shapes. Then I was thinking - because solutions of ODE's and PDE's are also some shapes (curves and surfaces), that Bernstein polynomials can approximate them as well.
There were some mathematical results that backed that (e.g. Stone-Weierstrass theorem).

I'm still discovering some of their properties. I even managed to generalize some (formula for non-recursive evaluation of k-th order derivative for polynomials defined on a generalized interval [a,b] is still not published - I will do it when I find time...my PhD thesis is suffering because of my infidelity - I hope it won't ask for a divorce )

I found a couple of papers from 2008, but they are using Galerkin approach (I use collocation), and they solve linear system, where they should only evaluate function...

I really hope I will find time to put everything together and write a paper!
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Old   October 7, 2012, 17:15
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Quote:
Originally Posted by nikola_m View Post
Hello! Thanks for the question.

The path that led me to these is a curious one.

I first used Bernstein polynomials to approximate a shape of an "flying saucer" shaped aircraft in an aerodynamic study of Coanda effect. Here is the paper from ICAS 2010 in Nice, France where it was presented.

In presented scheme they seemed powerful in representation of various complicated shapes. Then I was thinking - because solutions of ODE's and PDE's are also some shapes (curves and surfaces), that Bernstein polynomials can approximate them as well.
There were some mathematical results that backed that (e.g. Stone-Weierstrass theorem).

I'm still discovering some of their properties. I even managed to generalize some (formula for non-recursive evaluation of k-th order derivative for polynomials defined on a generalized interval [a,b] is still not published - I will do it when I find time...my PhD thesis is suffering because of my infidelity - I hope it won't ask for a divorce )

I found a couple of papers from 2008, but they are using Galerkin approach (I use collocation), and they solve linear system, where they should only evaluate function...

I really hope I will find time to put everything together and write a paper!
Hi,
did you analize the spectral properties of such polynomials? What about the equivalent modified wavenumber for the numerical derivative?
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Old   October 7, 2012, 17:35
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@ FM Denaro

Not yet!

Now I'm reading John Boyd's book on Chebyshev and Fourier spectral methods (awesome and cheap book btw) and I collected couple of ideas what tests to do and what problems to try to tackle.

P.S. I'm unfortunately mostly working on my own, without guidance, and I really like this kind of interaction. People around me at work are more into commercial software...
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