Precipitate Dislocation Model from Brown and Ham Exploiting the Signaloid Compute Engine

This example implements the equation for a materials precipitate "cutting" dislocation model from Brown and Ham¹ and shows how metallurgists can gain insight into the uncertainty of a model of a metal alloy's strength. The example highlights how the Signaloid C0 processor allows you to take unmodified programs and track uncertainty through them², getting all the benefits that you would usually only be able to obtain from a hand-crafted (and time-consuming) Monte Carlo evaluation. This demo exemplifies how uncertainties in empirical model parameters affect the uncertainty distribution of the model's output, for a model of a physical process.

Getting started

The correct way to clone this repository to get the submodules is:

	git clone --recursive git@github.com:signaloid/Signaloid-Demo-Metallurgy-BrownHam.git

If you forgot to clone with --recursive and end up with empty submodule directories, you can remedy this with:

	git submodule update --init

Running the application on the Signaloid Cloud Developer Platform

To run this application on the Signaloid Cloud Developer Platform, you need a Signaloid account. You can sign up for a Signaloid account using this link.

Once you have a Signaloid account, you can click the "add to signaloid.io" button at the top of this README.md to connect this repository to the Signaloid Cloud Developer Platform and run the application.

Running the application locally

Apart from using Signaloid's Cloud Compute Platform, you can compile and run this application locally. Local execution is essentially a native Monte Carlo implementation, that uses GNU Scientific Library³ to generate samples for the different input distributions. In this mode the application stores the generated output samples, in a file called data.out. The first line of data.out contains the execution time of the Monte Carlo implementation in microseconds (μs), and each next line contains a floating-point value corresponding to an output sample value.

In order to compile and run this application in the native Monte Carlo mode:

0. Install dependencies (e.g., on Linux):

sudo apt-get install libgsl-dev libgslcblas0

1. Compile natively (e.g., on Linux):

cd src/
gcc -I. -I/opt/local/include main.c utilities.c common.c uxhw.c -L/opt/local/lib -o native-exe -lgsl -lgslcblas -lm

2. Run the application in the MonteCarlo mode, using (-M) command-line option:

./native-exe -M 10000

The above program runs 10000 Monte Carlo iterations. 3. See the output samples generated by the local Monte Carlo execution:

cat data.out

Inputs

The inputs and their distributions are:

• gamma: Uniform($0.15, 0.25$)
• phi: Uniform($0.30, 0.45$)
• Rs: Equal mixure of Gaussian($10^{-8}, 2 \times 10^{-9}$) and Gaussian($3 \times 10^{-8}, 2 \times 10^{-9}$)
• G: Uniform($6 \times 10^{10}, 8 \times 10^{10}$)
• b: $2.54 \times 10^{-10}$ (i.e., constant)
• M: Uniform($1.9, 4.1$)

The parameter gamma is the APB energy with units J/m^2, phi is the precipitate volume fraction, Rs is mean particle radius on plane with units m, G is the shear modulus with units Pa, b is the magnitude of the Burgers vector with units m, and M is the Taylor factor.

Outputs

The output is the cutting stress, $\sigma_c$, where

$$\sigma_c = \left( \frac{M \cdot \gamma}{2.0 \cdot b} \right) \cdot \left( \sqrt{\frac{8.0 \cdot \gamma \cdot \phi \cdot Rs}{\pi \cdot G \cdot b^2}} - \phi \right).$$

Following is an example output, using Signaloid's C0Pro-XS+ core, for the default inputs:

Usage

Example: Precipitate Dislocation Model from Brown and Ham - Signaloid version

Usage: Valid command-line arguments are:
        [-o, --output <Path to output CSV file : str>] (Specify the output file.)
        [-M, --multiple-executions <Number of executions : int> (Default: 1)] (Repeated execute kernel for benchmarking.)
        [-T, --time] (Timing mode: Times and prints the timing of the kernel execution.)
        [-v, --verbose] (Verbose mode: Prints extra information about demo execution.)
        [-b, --benchmarking] (Benchmarking mode: Generate outputs in format for benchmarking.)
        [-j, --json] (Print output in JSON format.)
        [-h, --help] (Display this help message.)
        [-g, --apb-energy <gamma: double> (Default: Uniform(0.15, 0.25))] (Set `gamma` variable.)
        [-p, --precipitate-volume-fraction <phi: double> (Default: Uniform(0.30, 0.45))] (Set `phi` variable.)
        [-R, --mean-particle-radius <Rs: double> (Default: UxHwDoubleMixture(Gauss(1.0e-08, 2.0e-09), Gauss(3.0e-08, 2.0e-09), 0.5))] (Set `Rs` variable.)
        [-G, --shear-modulus <G: double> (Default: Uniform(6.0e+10, 8.0e+10))] (Set `G` variable.)
        [-B, --burgers-vector <b: double> (Default: 2.54e-10)] (Set `b` variable.)
        [-m, --taylor-factor <M: double> (Default: Uniform(1.9, 4.1))] (Set `M` variable.)

Acknowledgements

We learned about the Brown and Ham model from Prof. Hector Basoalto⁴ of the University of Sheffield. We are most grateful to him and his team for guiding us through the ideas and evaluating our initial implementation in this example.

Footnotes

1. Brown, L. M., and R. K. Ham. "Dislocation-particle interactions." Strengthening methods in crystals (1971): 9–135. ↩

2. Running this example on the Signaloid C0-Cloud processor uses less than 1% of the free monthly credits on the Signaloid C0-Cloud Free Tier plan. ↩

3. GNU Scientific Library. ↩

4. M.J. Anderson, F. Schulz, Y. Lu, H.S. Kitaguchi, P. Bowen, C. Argyrakis, and H.C. Basoalto. 2020. On the modelling of precipitation kinetics in a turbine disc nickel based superalloy. Acta Materialia 191 (2020), 81–100. https://doi.org/10.1016/j.actamat.2020.03.058. ↩