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Colebrook-White-Calculation-Time

README.md

Colebrook_White_nonlinear_regression

In this study we compare various existing formulas for calculating pipe friction using Colebrook-White (CW) equation.

These equations are either explicit or implicit.

CW equation is used to calculate Darcy-Weisbach (DW) friction factor for turbulent pipe flow (Re>4000).

There are many papers for calculating f with CW formula & they are kind of obsessed with making the calculation faster but to put matters into perspective, we compare calculation time of f from CW equation with the time it takes to solve a Water Distribution Network (WDN).

Solving WDNs via the gradient descent node method results in formation of a coefficient matrix with the size equal to the number of nodes (junctions) in the network. For each pipe, the elements in matrix corresponding to start & end nodes have a non-zero value, as well as diagonal elements. All other elements are zero. This results in formation of a sparse coefficient matrix. Then the system is converted to Ah=b, wherein A=coefficient matrix, h=unknown heads, and b=right hand side.

In order to simplify this procedure, the nodes in the sparse matrix must be reordered. There are many ways to do this. This is basically from the graph theory. The Python package NetworkX has a function called reverse_cuthill_mckee_ordering that does this task. After performing this, the system can solved much faster.

Here we gathered some input files from EPAnet software with 100 to 140000 nodes. We then computed the time it takes to reorder the matrix A & solve the sparse system 10 times. The reason for this is that most water distribution networks converge after 10 iterations. We also calculated the time it takes to calculate f from CW equations with a reasonable accuracy (for example maximum absolute error of 0.01%). We then compared the time for each one of these. Our results show that solving the system usually takes a lot more time than finding the f values. This is basically true for networks with more than 10000 pipes but even for smaller networks, this can be true if fast C routines are called for calculating f.

References:
A sixteen decimal places' accurate Darcy friction factor database using non-linear Colebrook's equation with a million nodes: A way forward to the soft computing techniques, https://www.sciencedirect.com/science/article/pii/S2352340919310881

In order to speed up calculations, we can use AVX instructions. This is possible by using intrinsic numpy elementwise operations in python. To take advantage of multithreading, numba can be used. If you use simple for loops, then you are using SIMD operations. In fact, you can compare single-thread calculations of numpy if you have a new CPU (core i7 gen 3) and an old one (Core 2 Duo). All of the mid-range and above CPUs from AMD and Intel from 2012 onwards have this capability, either AVX with 128, 256 or 512 bit wide FP instructions. Generally CPUs have integer (arithmetic logic unit) and floating (floating point unit) calculation units (instructions). Taking advantage of FP transistors makes it faster for our application.

about simd:

'numexpr' is another possibility in python but I haven't tested it yet.

If Intel C++ is used, then enabling multithreading & AVX (specific to CPU) can speed up the things to blitz speeds. Very very quick.

My GPU is very old (Nvidia 610m) and doesn't support CUDA I think. But I will probably be able to use OpenCL on it. It's a programming language. If I ever want to enter that, it's a huge time investment.

I have worked with longitudinal dispersion models for rivers but not for pipes. This can be a venue for soft computing calculations.

Papers to look at:

  • Efficient Resolution of the Colebrook Equation, Clammond
  • Optimization of water distribution networks, Samani
  • A gradient method for fuzzy analysis of water distribution networks, Moosavian

Books to look at:

  • Design of water supply pipe networks, Swamee & Sharma

Links to look at:

write an introduction to cuthill-mckee & minimum multiple-degree reordering.

We can put a table for each formula and calculate the number of log calls & simple operators +, *. The power operator in its direct form x^c1 should be avoided if c1 is an integer (sometimes the compiler understands it's an integer but better not to risk it).

Github markup: (to write grave accent use alt+numpad96, to write in italic use two asterists signs).

For generating water distribution network (WDN) of cities, multiple options are available: