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Wang, Y., Zhang, X., & Cai, W. (2019). Spherical Harmonics Method for Computing the Image Stress Due to A Spherical Void. Journal of the Mechanics and Physics of Solids.
Wang, Y., Zhang, X., & Cai, W. (2019). Spherical Harmonics Method for Computing the Image Stress Due to A Spherical Void. Journal of the Mechanics and Physics of Solids.

[https://doi.org/10.1016/j.jmps.2019.01.020 doi:10.1016/j.jmps.2019.01.020]

[https://arxiv.org/abs/1806.11165 arXiv]


The code is tested under Ubuntu 16.04 LTS. with package python 3.6, numpy 1.15, scipy 1.0, matplotlib 3.0, jupyter, and SHTOOLS 4.3. Please see [http://micro.stanford.edu/wiki/Install_SHTOOLS Install_SHTOOLS] for more details on installing required packages
The code is tested under Ubuntu 16.04 LTS. with package python 3.6, numpy 1.15, scipy 1.0, matplotlib 3.0, jupyter, and SHTOOLS 4.3. Please see [http://micro.stanford.edu/wiki/Install_SHTOOLS Install_SHTOOLS] for more details on installing required packages
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or directly from https://gitlab.com/micronano/ShElastic
or directly from https://gitlab.com/micronano/ShElastic


== How to Install SHTOOLS on mc2 ==
== Suggested Installation of SHTOOLS (pip) ==


If you have python3 and pip installed on your machine, you can simply install everything by:
On mc2 cluster, since we can only install packages for user only, we are not allowed to use sudo command to install the required packages. We need to compile the packages by our own.


python3 -m pip install --user numpy scipy matplotlib ipython jupyter pandas sympy nose pyshtools
For compiling shtools, the required packages are fftw3, lapack, and blas. Libraries for lapack and blas are available on mc2, we can use them directly for compiling shtools. For installing fftw3, please refer to [http://micro.stanford.edu/wiki/Install_FFTW3 Install_FFTW3] for more details.


== Generate coefficients ==
After you followed the steps in FFTW3 installation (Serial only), you should have all the fftw3 libraries available:


The coefficients up to lmax = 60 is pre-computed and included in the repository. If you want to use higher mode spherical harmonic basis, please continue reading.
~/usr/include/fftw3.h
~/usr/include/fftw3.f
~/usr/lib/libfftw3.a
~/usr/lib/libfftw3.la
~/usr/lib/libfftw3.so


Please change the first few lines in `ShElastic/scripts/generate_modes.py`:
And also, run the command:


lKfull = 10
ls /usr/lib64
lJfull = lKfull + 3

savepath = '../module/lmax%dmodes'%lKfull
The libraries for lapack and blas should be available here:

/usr/lib64/liblapack.a
/usr/lib64/liblapack.so
/usr/lib64/liblapack.so.3
...
...
/usr/lib64/libblas.a
/usr/lib64/libblas.so
/usr/lib64/libblas.so.3
...

Before we proceed, we need to make sure to have python installed.

=== Install python + scipy + SHTOOLS on local directory ===

We create a local folder .localpython for installing python:

mkdir ~/.localpython

download python source code:

mkdir ~/soft
cd ~/soft
wget http://www.python.org/ftp/python/X.X.X/Python-X.X.X.tgz
tar -xvf Python-X.X.X.tgz
cd Python-X.X.X

compile python from source and install to local directory:

make clean
./configure --enable-shared --prefix=${HOME}/.localpython
make
make install

Add the following lines to ~/.bashrc:

export PATH=${HOME}/.localpython/bin:$PATH
export PYTHONPATH=${HOME}/.localpython/bin
export LIBRARY_PATH=$LIBRARY_PATH:${HOME}/.localpython/lib
export LD_LIBRARY_PATH=$LD_LIBRARY_PATH:${HOME}/.localpython/lib

if python version < 2.7.9, install pip by:

wget https://bootstrap.pypa.io/get-pip.py
python get-pip.py
export PATH=${HOME}/.local/bin:$PATH

Install SciPy Stack using pip on user's directory:

pip(pip3) install --user numpy scipy matplotlib ipython jupyter pandas sympy nose

Install pyshtools using pip:
First, we put the pre-compiled FFTW3 libraries into LIBRARY_PATH:

export LIBRARY_PATH=$LIBRARY_PATH:${HOME}/usr/lib
export LD_LIBRARY_PATH=$LD_LIBRARY_PATH:${HOME}/usr/lib

Then we can install pyshtools through pip on user's directory:

pip(pip3) install --user pyshtools

=== Compile SHTOOLS ===

load the python module (we use python 2 as an example):

module load python/2.7.8 (Whichever is available)

Also make sure you have numpy available:

pip list

Then you are ready to install shtools on your mc2 account.

First, download the shtools source package from [https://github.com/SHTOOLS/SHTOOLS/releases Releases]

cd ~/soft
wget https://github.com/SHTOOLS/SHTOOLS/archive/v3.4.tar.gz
tar -zxvf v3.4.tar.gz
cd SHTOOLS-3.4

Then, indicate the libraries and compile the source code:

export LD_LIBRARY_PATH=$LD_LIBRARY_PATH:"$HOME/usr/lib"
make python2 LAPACK="-L/usr/lib64 -llapack" BLAS="-L/usr/lib64 -lblas" FFTW="-L$HOME/usr/lib -lfftw3"

The output should be something look like [https://shtools.oca.eu/shtools/www/makeout.html this]. Then your package is ready for use. If you are compiling with python 3, just change "python2" to "python3" in the command above.

To load the SHTOOLS module in python, use the following lines:

import sys
sys.path.append('$HOME/soft/SHTOOLS-3.4')
import pyshtools as shtools

where the "$HOME" has to be substituted by your own home folder path.

== Examples for using SHTOOLS in python ==

We consider a simple example where we do the spherical expansion of function <math>x^2</math> on the surface of a unit sphere.

First we import the necessary packages:

from __future__ import print_function # only necessary if using Python 2.x

import numpy as np

import sys
sys.path.append('/home/yfwang09/soft/SHTOOLS-3.4')
import pyshtools as psh

Then we generate the mesh grids for the spherical coordinates (Nx2N grids for latitude and longitude variables <math>\phi, \theta</math>; or <math>\pi/N</math> latitudinal and <math>2\pi/2N</math> longitudinal, both start from 0):

Ngrid = 100
phi = np.linspace(0, np.pi, Ngrid)
theta = np.linspace(0, 2*np.pi, 2*Ngrid)

THETA, PHI = np.meshgrid(theta, phi) # make the grid size as Nx2N

where the grid PHI and THETA both have N rows and 2N columns. Then we translate the Cartesian coordinates into the spherical meshgrids, and generate the target function into spherical coordinates:

Z = np.cos(PHI);
X = np.sin(PHI)*np.cos(THETA);
Y = np.sin(PHI)*np.sin(THETA);

X2 = X**2


Then we can generate the mode files by calling:
Finally we expand the target function into spherical harmonic series:


cd ShElastic/scripts
cilm = np.empty([2,Ngrid/2,Ngrid/2])
python3 generate_modes.py
filename = 'xsquare'
cilm = psh.expand.SHExpandDH(X2, sampling=2)
np.savetxt(filename+'_clm.txt', cilm[0,:,:], header='Row: l, Column: m')
np.savetxt(filename+'_slm.txt', cilm[1,:,:], header='Row: l, Column: m')


The mode files will be saved in the savepath folder.
where cilm represent the coefficients of different spherical harmonic modes. cilm[0, l, m] and cilm[1, l, m] are the cosine and sine modes of spherical harmonics. The parameter sampling=2 means that we are using the grid Nx2N instead of NxN.


== Reproduce results from the paper ==
See the page [https://shtools.oca.eu/shtools/www/man/python/pyshexpanddh.html SHExpandDH] for the detailed description of the expanding function and the page [https://shtools.oca.eu/shtools/www/conventions.html Real Spherical Harmonics] for mathematical definition.


The results showing in the paper can be reproduced by running the jupyter notebooks in `ShElastic/notebook` folder.
== Visualize spherical harmonic modes using scipy ==


Numerical Case 1: `Case01-Uniform_Tensile_Hole.ipynb`
We first import the necessary packages:


Numerical Case 2: `Case02-Void_Dislocation_Interaction.ipynb`
import numpy as np
from scipy.special import sph_harm
import matplotlib.pyplot as plt
from matplotlib import cm, colors
from mpl_toolkits.mplot3d import Axes3D


Numerical Case 3: `Case03-Prismatic_Dislocation_Loop.ipynb`
For a certain pair of parameters (m, l), we can plot the corresponding spherical harmonic mode Y(l, m) on the surface of a unit sphere using scipy and matplotlib packages. Here we use the meshgrid THETA, PHI, X, Y, and Z we generated in the previous section:


Launch the jupyter notebook by:
fvalues = sph_harm(m, l, THETA, PHI).real # Get the values of spherical harmonic functions
fmax, fmin = fvalues.max(), fvalues.min()
fcolors = (fvalues - fmin)/(fmax - fmin) # normalize the values into range [0, 1]


cd ShElastic
Then we plot Y(l, m) on a sphere:
jupyter notebook


After you run through the notebooks, the result files will be saved in `ShElastic/figures`. The figures in the paper can be reproduced by running plotdata.m in matlab.
fig = plt.figure(figsize=plt.figaspect(1.)) # make the axis with equal aspects
ax = fig.add_subplot(111, projection='3d') # add an subplot object with 3D plot
ax.plot_surface(x, y, z, rstride=1, cstride=1, facecolors=cm.Blues(fcolors))
plt.show()


<gallery caption="Fig.1 generated figures for the paper" widths="200px" heights="200px" perrow="3">
For example, the figures for (l, m) = (2, 0), (3, 2), and (1, 1) are shown as below:
Image:Case1_results.png | (a) Case 1
<gallery caption="Fig.1 Visualization of Spherical Harmonics" widths="200px" heights="200px" perrow="3">
Image:sph_harm_Y20.png|(a) <math> Y_{20}(\theta,\phi) </math>
Image:Case2_results.png | (b) Case 2
Image:sph_harm_Y32.png|(b) <math> Y_{32}(\theta,\phi) </math>
Image:Case3_results.png | (c) Case 3
Image:sph_harm_Y11.png|(c) <math> Y_{11}(\theta,\phi) </math>
</gallery>
</gallery>

Latest revision as of 01:06, 20 February 2019

ShElastic is the python code for the paper

Wang, Y., Zhang, X., & Cai, W. (2019). Spherical Harmonics Method for Computing the Image Stress Due to A Spherical Void. Journal of the Mechanics and Physics of Solids.

doi:10.1016/j.jmps.2019.01.020

arXiv

The code is tested under Ubuntu 16.04 LTS. with package python 3.6, numpy 1.15, scipy 1.0, matplotlib 3.0, jupyter, and SHTOOLS 4.3. Please see Install_SHTOOLS for more details on installing required packages

Obtain the ShElastic package

Please download the package using git: 

git clone https://gitlab.com/micronano/ShElastic.git

or directly from https://gitlab.com/micronano/ShElastic

Suggested Installation of SHTOOLS (pip)

If you have python3 and pip installed on your machine, you can simply install everything by: 

python3 -m pip install --user numpy scipy matplotlib ipython jupyter pandas sympy nose pyshtools

Generate coefficients

The coefficients up to lmax = 60 is pre-computed and included in the repository. If you want to use higher mode spherical harmonic basis, please continue reading.

Please change the first few lines in `ShElastic/scripts/generate_modes.py`: 

lKfull = 10
lJfull = lKfull + 3
savepath = '../module/lmax%dmodes'%lKfull
...

Then we can generate the mode files by calling: 

cd ShElastic/scripts
python3 generate_modes.py

The mode files will be saved in the savepath folder.

Reproduce results from the paper

The results showing in the paper can be reproduced by running the jupyter notebooks in `ShElastic/notebook` folder.

Numerical Case 1: `Case01-Uniform_Tensile_Hole.ipynb`

Numerical Case 2: `Case02-Void_Dislocation_Interaction.ipynb`

Numerical Case 3: `Case03-Prismatic_Dislocation_Loop.ipynb`

Launch the jupyter notebook by: 

cd ShElastic
jupyter notebook

After you run through the notebooks, the result files will be saved in `ShElastic/figures`. The figures in the paper can be reproduced by running plotdata.m in matlab.

  • Fig.1 generated figures for the paper
  • (a) Case 1

    (a) Case 1

  • (b) Case 2

    (b) Case 2

  • (c) Case 3

    (c) Case 3

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