GenerateGrid
Created Montag 08 Februar 2016
This function computes the discretisation of a given interval I of length L into N cells. It allows to distribute the N cells into a fixed number of sub-intervals. Each of these sub-intervals can feature a stretching of its cells' lengths.
1. Purpose of Function
In situations in which large gradients (e.g. temperature, pressure) need to be transferred into a discretised component, e.g. a pipe, it can be numerically advantageous to choose a non-equidistant discretisation, where the discretisation intervals exposed to potential large gradients are chosen smaller.
The function GenerateGrid provides a straight forward way to generate such stretched discretisation.
2. Limits of Validity
Each of the sub-intervals shall have at least 2 cells. Otherwise the scaling of each sub-interval becomes meaningless.
3. Interfaces
4. Nomenclature
5. Governing Equations
Idea:
We want to assign a stretching to the 
cells of a discretised model. This results in a vector 
containing 
sub-interval of the physical length 
.
Implementation
The input 
consists of a vector of 
signed reals, e.g
corresponds to a subdivision of 
into 
sub-intervals of length 
. Accordingly every sub-interval contains (up to rounding)
Here
collects the normalised remainders 
of the integer division at each 
(see source code). This ensures that 
.
For the 
sub-interval the number 
defines a length distribution of cells.
Here
corresponds to equidistant cell spacing of the sub-interval, 
to increasing cell width from beginning to the end of the sub-interval, 
to decreasing cell width from beginning to the end of the sub-interval.
The modulus 
determines the stretching rate of cells. The sub-interval has 
cells
where
with the normalisation constant
6. Remarks for Usage
7. Validation
Example:
This corresponds to the following discretisation:
8. References
9. Version History
09.02.2016 - 0.1 - initial implementation - Johannes Brunnemann, XRG Simulation GmbH