[Nek5000-users] r,s,t index for given point

[nek5000-users at lists.mcs.anl.gov]

Hi Markus,

Functions in the spectral element method are expressed as:

        e                  e
[1]   u (r,s,t) = sum    u     h (r) h (s) h (t)
                    ijk    ijk   i     j     k


Where u^e_ijk are the basis coefficients for element e, with
i,j,k ranging from 0 to N  (1 to lx1 in the code), and the
h_i's are polynomials of degree N on the interval from [-1,1].
Given r,s,t and the set of basis coefficients, one can compute
u^e(r,s,t) using [1].

Note that the h_i's are Lagrangian interpolants on the 
Gauss-Lobatto-Legendre points, {xi_i}, i=0,...,N, such that

       h_i (xi_j)  = delta_ij,   ( = 1 for i=j, 0 otherwise ).

Thus for any point (r,s,t) = (xi_i,xi_j,xi_k), we can associate 
basis coefficient u^e_ijk with u^e at that point.  More specifically,
we can associated the index set (i,j,k) with that point.

However, if the continuous variables (r,s,t) do not coincide with
the quadrature points then there is no associated index --- that is,
the value of u^e(r,s,t) would then depend on _all_ the basis
coefficients, using Eq. [1] above.

The interpolation routine returns the continuous variables - as
are needed for the general interpolation formula [1].

To get the indices, you could check the rst values against the
GLL coordinate points that are stored in WZ in the array zgm1(),
for instance.

Does this help ?

Paul








On Fri, 9 Jul 2010, nek5000-users at lists.mcs.anl.gov wrote:

> Hi,
>
> what is the difference between "index" and "value"?
> Essentially, I am looking for the information postx provides when figuring 
> out history points. I browsed the postx source code, located the routine and 
> implemented it in .usr, but I expect trouble when moving to bigger domains. I 
> also figured there must be something in nek.
>
> So I took the new hpts routine (see attached my modified .usr file), what I 
> am getting for r,s,t for a specific point is:
> 1.00000000000000      -5.551115123125783E-017  -1.00000000000000
>
> The real deal for the same point (from postx) is 4    1    1.
>
> What am I missing? Is there a way to go from the output I have to the r,s,t 
> indices?
>
> Thanks for any help,
> Markus
>
>
> nek5000-users at lists.mcs.anl.gov wrote:
>> 
>> Markus,
>> 
>> those r,s,t values (not indices) are valid  - do they not correspond
>> to your expectations?  Off-hand one might expect that the associated
>> indices would be 1,NY1/2,NZ1
>> 
>> Paul
>> 
>> 
>> On Fri, 9 Jul 2010, nek5000-users at lists.mcs.anl.gov wrote:
>> 
>>> Howdy Neks,
>>> 
>>> is there a function in Nek that for a given point in x,y,z would return 
>>> processor number, element number and r,s,t index of the nearest GLL grid 
>>> point? I was desperately trying to figure out how the interpolation 
>>> routines (f.e. for history points) would help me, but I only got garbage 
>>> (-1, 0 and 1) back for the variables I would expect contain r,s,t. It 
>>> worked with the element number, however.
>>> 
>>> Thanks,
>>> Markus
>>> 
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>>> Nek5000-users at lists.mcs.anl.gov
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