[Gmsh] Question about higher order elements

Thu Apr 17 22:45:20 CEST 2014

Thanks! I think I've got it now.

The only thing is that I'm not sure it's useful to assume 
equidistant Lagrange shape functions. Such shape functions would form a 
poorly behaved interpolation basis at high orders due to Runge's phenomenon<http://en.wikipedia.org/wiki/Runge%27s_phenomenon>. 
Basically, you could get bad values while trying to interpolate the 
"equispaced" points $x_i$ to the "Gauss-Lobatto" points $y_k$. One would 
want to use either an orthogonal set of polynomials (modal basis) or 
Lagrange interpolation at the zeros of such a set of polynomials (nodal 
basis -- the unevenly spaced Gauss-Lobatto points are in this category) to 
eliminate this issue.

I think what you mean is that you would compute the Vandermonde matrix 
$K_{ik} = \psi_k(\xi_i)$ where the $\psi_k(\xi)$ is a "good" basis and 
$\xi_i$ are the equispaced points obtained from Gmsh. You then want to know 
$x(\xi) = \sum_{k=1}^N c_k \psi_k(\xi)$ where $x(\xi)$ maps coordinates 
from the master element and $c_i$ is a set of N unknown coefficients. If 
your basis is formed by Lagrange polynomials at arbitrary points (perhaps 
Gauss-Lobatto) $xi_k$, then $c_k = y_k$. Thus, $c_k = y_k = inv(K_{ik}) 
x_i$. Of course, the Vandermonde matrix can be inverted once and then used 
to "reposition" the nodes of each element individually. However, the better 
way to think of it is that you are mapping the physical points (graciously 
generated by Gmsh) to expansion coefficients of an appropriate 
interpolation basis.

Hopefully this clarifies things for anyone else with the same question.

Thanks again!

Nick

On Tuesday, April 8, 2014 6:35:07 AM UTC-4, Jean-François Remacle wrote:

>
> Le 7 avr. 2014 à 21:50, Nicholas Glenn Chisholm <nick.g.... at gmail.com<javascript:>> 
> a écrit :
>
> JF,
>
> I greatly appreciate your reply! I seem to understand what you are saying, 
> but could you clarify one thing for me? Are you suggesting that the 
> equispaced nodes within each element (generated by Gmsh) could be mapped to 
> the Gauss-Lobatto points in a master element implemented within my solver? 
> This is simple and would essentially alter the Jacobian within the element 
> due to the spacial “stretching”. However, I’ve tested this idea for a 
> linear 1D problem, and it seems to severely slow the convergence rate even 
> for elements of moderate order (e.g. P=5).
>
>
> Assume you have an element generated by gmsh with its $N$ verices $x_i$. 
>
> Assume equidistant Lagrange shape functions $L_i(\xi)$ : we have 
>  
> $$x(\xi) = \sum_{i=1}^N L_i(\xi) x_i $$
>
> Assume now your nice shape functions based e.g. on Gauss-Lobatto points 
> $\xi_k$, $k=1,\dots,M$, with
> $M$ not necessary equal to $N$. Then the Gauss Lobato points $y_k$ can be 
> computed as
>
> $$y_k=  \sum_{i=1}^N L_i(\xi_k) x_i,~~~k=1,\dots,M.$$
> Matrix $K_{ik}=L_i(\xi_k)$ is a Vandermonde Matrix that 
> can be computed once and applied to each element.
>
> JF
>
>
> Doing some kind of Lagrange interpolation from data at the equispaced 
> points to the Gauss points seems equally troublesome due to the Runge 
> phenomena.
>
> Thanks again!
> Nick
>
> On Apr 7, 2014, at 2:38 AM, Jean-François Remacle <
> jean-franc... at uclouvain.be <javascript:>> wrote:
>
>
> Le 6 avr. 2014 à 06:08, nick.g.... at gmail.com <javascript:> a écrit :
>
> I'm thinking about using Gmsh for a research project which involves 
> computing fluid flow fields via a spectral/hp-element method (p>5). I was 
> reading through the documentation, and it seems to indicate that higher 
> order nodes are assumed to be equispaced. However, if I am not mistaken, 
> hp-element codes normally place the high-order nodes at the 
> Gauss-Legendre(-Lobatto) points within, for example, a quadrilateral 
> "master" element. This is especially nice if your solver uses a nodal basis 
> for its shape functions. Is there any way to have Gmsh use a Gauss-Lobatto 
> distribution of nodes within a high order element?
>
>
> Hello,
>
> We presently use equidistant Lagrange elements for representing the 
> geometry of the elements. The high order
> points may be optimized in Gmsh, both for reducing the distortion of the 
> elements and for increasing their
> geometrical accuracy.  For more details 
>
> Johnen, A., Remacle, J. F., & Geuzaine, C. (2013). Geometrical validity of 
> curvilinear finite elements. *Journal of Computational Physics*, *233*, 
> 359-372.
>
> Toulorge, T., Geuzaine, C., Remacle, J. F., & Lambrechts, J. (2013). 
> Robust untangling of curvilinear meshes. *Journal of Computational 
> Physics*, *254*, 8-26.
>
> We agree there exist better nodal distributions such as Gauss Lobatto or 
> the electrostatic node distribution of Warburton. Yet, those distributions 
> can
> (or should) be implemented in the solver itself, defining a master element 
> for the interpolation and using our equidistant Lagrange interpolation for
> computing node positions relative to the interpolation. In other words, 
> the geometry and the solution do not have to use the same basis.
>
> Best regards,
>
> JF
>
>
>
>
> Thanks!
> Nick
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>
> ------------------------------------------------------------------
> Prof. Jean-Francois Remacle
> Universite catholique de Louvain (UCL)
> Ecole Polytechnique de Louvain (EPL) - Louvain School of Engineering
> Institute of Mechanics, Materials and Civil Engineering (iMMC)
> Center for Systems Engineering and Applied Mechanics (CESAME)
> Tel : +32-10-472352 -- Mobile : +32-473-909930 
>
>  
>
>
>
>
>  
>
>
> ------------------------------------------------------------------
> Prof. Jean-Francois Remacle
> Universite catholique de Louvain (UCL)
> Ecole Polytechnique de Louvain (EPL) - Louvain School of Engineering
> Institute of Mechanics, Materials and Civil Engineering (iMMC)
> Center for Systems Engineering and Applied Mechanics (CESAME)
> Tel : +32-10-472352 -- Mobile : +32-473-909930 
>
>  
>
>
>
>
>  
>
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