convergence

A Python program to Perform Calculations Associated with a Grid Convergence Study.

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Introduction

This is a port of verify.f90, a program provided as part of the NASA Examining Spatial (Grid) Convergence tutorial.

http://www.grc.nasa.gov/WWW/wind/valid/tutorial/spatconv.html

Additional changes have been made to reflect the recommendations in:

Celik, I. B., Ghia, U., & Roache, P. J. (2008). Procedure for estimation and reporting of uncertainty due to discretization in CFD applications. Journal of fluids Engineering-Transactions of the ASME, 130(7).

This code was written by Mathew Topper between 2009 and 2011 as part of the Supergen Marine Research Consortium project. I was younger then and would do many things differently now. I am working on updating this code to make the main class and functions more easy to use and also integrate into other projects. Watch this space.

Installation

The package requires no dependencies and is tested on the current bugfix and security versions of Python.

The latest stable version of the package can be downloaded from PyPI using pip:

pip install convergence

The development version of the package can be installed by cloning or downloading the source code, and using the command prompt as follows:

cd /path/to/convergence
pip install --editable .

Note that the stable version may not contain all of the features found in the development version but it should be more reliable.

Basic Usage

The most straightforward method of using convergence is to generate a report which details the grid convergence metrics for a given set of grids.

Command Line

The package provides a command line interface. The input data must be a space delimited text file with the first column being the grid spacing and the second column being the metric of interest. An example can be found in the data folder of the source code. The program can then be executed as follows:

grid-convergence /path/to/data/file

By default, the results of the program are written to a file called verify_report.txt in the calling directory. The file name can be changed using the -o or --out command line options.

Scripting

The package can also be used from within a script. Grids are provided in a list of pairs, i.e [(size1, value1), (size2, value2), ...]. Example usage is:

>>> from convergence import Convergence
>>> grids = [(1.000000, 0.970500),
...          (2.000000, 0.968540),
...          (4.000000, 0.961780)]
>>> convergence = Convergence()
>>> convergence.add_grids(grids)
>>> print(convergence) # doctest:+ELLIPSIS
<BLANKLINE>
Number of grids to be examined = 3 ...

Expected Output

The result, contained in the output file when using the command line interface or when printing a Convergence object, will resemble the following:

Number of grids to be examined = 3 

     Grid Size     Quantity 

     1.000000      0.970500 
     2.000000      0.968540 
     4.000000      0.961780 


Discretisation errors for fine grids:

       Grids |     e_approx |     e_extrap |      f_exact |   gci_coarse | 
 =========================================================================
       1 2 3 |     0.002020 |     0.000824 |     0.971300 |     0.003555 | 
 -------------------------------------------------------------------------

       Grids |     gci_fine |            p |          r21 |          r32 | 
 =========================================================================
       1 2 3 |     0.001031 |     1.786170 |     2.000000 |     2.000000 | 
 -------------------------------------------------------------------------


Discretisation errors for coarse grids:

       Grids |     e_approx |     e_extrap |      f_exact |   gci_coarse | 
 =========================================================================
       1 2 3 |     0.006980 |     0.002842 |     0.971300 |     0.012287 | 
 -------------------------------------------------------------------------

       Grids |     gci_fine |            p |          r21 |          r32 | 
 =========================================================================
       1 2 3 |     0.003562 |     1.786170 |     2.000000 |     2.000000 | 
 -------------------------------------------------------------------------


Asymptotic ratio test:

           Grids | Asymptotic ratio | 
 ====================================
           1 2 3 |         0.997980 | 
 ------------------------------------

In the first table the input data is displayed. The second table shows the fine analysis results for each trio of grids and the second table shows the coarse analysis results for each trio. The final table shows the asymptotic ratio.

The headers of the tables have the following meanings:

• Grids: the trio of grids being analysed
• e_approx: approximate relative error
• e_extrap: extrapolated relative error
• f_exact: the estimated value at zero grid spacing
• gci_coarse: coarse grid convergence index
• gci_fine fine grid convergence index
• p: order of convergence
• r21: ratio of the middle to fine grid spacing
• r32: ratio of the coarse to middle grid spacing

Advanced Usage

Known Analytical Result

If there is a known zero spacing value for the convergence study this value can be added to the analysis using the -a or --analytical command line option. To illustrate, the basic example would now become:

grid-convergence /path/to/data/file -a 0.12345

Alternatively, when using the Convergence class, add the analytical value when instantiating the object, using the f_anal argument:

>>> convergence = Convergence(f_anal=0.9713)
>>> convergence.add_grids(grids)
>>> print(convergence) # doctest:+SKIP

Additional headers now appear in the fine and coarse analysis tables with the following meanings:

• e_analytic: analytical relative error
• f_analytic: the analytical value at zero grid spacing
• f_delta: the different between the analytical and estimated zero grid spacing values

Report Attribute Access

Values for the report attributes can be accessed through the Convergence class. A namespace containing the values for each triplet of grids is stored in the items of a Convergence object, ordered from finest to coarsest. For example:

>>> len(convergence)
1
>>> convergence[0] # doctest:+ELLIPSIS
Namespace(...

Values associated the both the fine and coarse grids are available at the first level of the namespace. For example, to get the asymptotic ratio of the finest triplet of grids:

>>> convergence[0].asymptotic_ratio
0.997980422462648

Values associated to either the fine of coarse analysis, are stored under the fine and coarse keys. For example, to examine the extrapolated relative errors:

>>> convergence[0].fine.e_extrap
0.0008239813226325151
>>> convergence[0].coarse.e_extrap
0.002841894765814084

Required Grid Resolution

To determine the required grid resolution for a given GCI value the get_resolution method can be used. For example:

>>> convergence.get_resolution(0.001)
0.9831461212423797

By default the GCI for the fine grid is used. To use the coarse grid estimate pass "coarse" as the second optional argument:

>>> convergence.get_resolution(0.001, "coarse")
0.9831461212423799

Note that the equation in the NASA tutorial is incorrect. If $\text{GCI}^{*}$ is the desired accuracy and $h^{*}$ is the required resolution, then:

$$r = \left( \frac{\text{GCI}_{12}}{\text{GCI}^{*}} \right)^{p^{-1}}$$

and

$$h^{*} = \frac{h_1}{r}$$

License

Copyright 2011 SuperGen Marine Energy Research Consortium
Copyright 2013 SuperGen UK Centre for Marine Energy Research
Copyright 2017-2024 Mathew Topper

This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.

This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with this program. If not, see https://www.gnu.org/licenses/.