<P ALIGN="CENTER">NetworCh manuals: Getting started</P>
<P ALIGN="CENTER"><FONT SIZE="+2" color="darkred"><STRONG>
NetworCh: Introduction and examples</STRONG></font></P>
<DIV>
<P ALIGN="CENTER"><STRONG>Nicolas Bertin</STRONG></P>
</DIV>
<DIV>
<P ALIGN="CENTER">October 2016</P>
</DIV>

<HR>

== Obtaining NetworCh ==
The NetworCh library is maintained through the SVN server of the group. To obtain the latest version, you can use the following commands:

<pre>
mkdir NetworCh.svn
cd NetworCh.svn/
svn co https://micro.stanford.edu/svn/NetworCh .
</pre>

The core of the NetworCh library is mainly written in Matlab and therefore does not require any installation. The Matlab functions are found in subfolder <tt>matlab/</tt>. In order to use these functions from your Matlab installation, do not forget to add the library directory to your Matlab path, or use the following command at the beginning of your scripts:

<pre>
addpath(genpath('<path to NetworCh.svn>/matlab'));
</pre>

A list of the main functions available in the library is provided here: [[NetworCh Documentation]].

== NetworCh basics ==

=== Network representation ===
A network is a collection of vertices (nodes) joined by edges (links). In NetworCh, each network/structure is represented by a list of nodes position <tt>rn</tt> and a list of links <tt>links</tt>. The first three columns of <tt>rn</tt> are reserved for the <tt>(x,y,z)</tt> spatial coordinates of each node. The two first columns of <tt>links</tt> must contain the id of nodes connected by the edges.

For example, a simple network consisting of two nodes A = <tt>(xa,ya,za)</tt> and B = <tt>(xb,yb,zb)</tt> joined by a single segment can be initialized as:

<pre>
% Nodes
rn = zeros(2,3);
rn(1,1:3) = [xa,ya,za];
rn(2,1:3) = [xb,yb,zb];

% Links
links = zeros(1,2);
links(1,1:2) = [1,2];
</pre>

=== Network visualization ===
A network can be plotted with function <tt>plot_structure</tt>:

<pre>
plot_structure(rn,links);
</pre>

Note that function <tt>plot_structure</tt> uses the spatial coordinates <tt>rn(:,1:3)</tt> to position nodes and does not rely and graph drawing algorithm to generate a planar layout of the network.

In many situations, the nodes of the network are contained in a rectangular box (e.g. the simulation box in DD or MD simulations). In NetworCh, such a volume is defined with array <tt>bounds</tt>. Array <tt>bounds</tt> specifies the bounds of the volume in the following manner: <tt>bounds = [xmin,xmax;ymin,ymax;zmin,zmax]</tt>. For example, if we assume that our structure is contained in a cubic simulation box of edge size <tt>L</tt> centered in the origin <tt>(0,0,0)</tt>, it can be visualized using:

<pre>
bounds = L/2*[-1,1;-1,1;-1,1];
plot_structure(rn,links,bounds);
</pre>

By default, periodic boundary conditions will be applied for the visualization when argument <tt>bounds</tt> is specified in function <tt>plot_structure</tt>, such that nodes whose position lies outside the volume defined by <tt>bounds</tt> will be folded back to the volume.

== Importing/Creating networks ==

NetworCh can generate or import networks from different sources.

=== Test structures ===
Several simple test structures are provided within the library to test its functions. Test structures are generated by using the <tt>test_structure</tt> function:

<pre>
[rn,links,bounds] = test_structure(id);
</pre>

where <tt>id</tt> is the index of the test structure to generate.

=== DD structures ===
When working with DD structures, the following additional data convention are used: the fourth column of the nodes position array <tt>rn(:,4)</tt> is reserved for the nodal constraint (e.g. unconstrained or pinned node), while entries <tt>links(i,3:5)</tt> and <tt>links(i,6:8)</tt> contain the components of the Burgers vector and that of the slip plane normal associated with segment <tt>i</tt>, respectively.

'''DDLab''' <br>
The format used in NetworCh to represent networks is the same as that used in DDLab to represent dislocation nodes and segments. Therefore, DDLab dislocation structures can be directly manipulated with the library.

'''ParaDiS''' <br>
DD structures can be imported from ParaDiS <tt>data</tt> files using function <tt>read_nodes</tt>:

<pre>
[rn,links,bounds] = read_nodes(datafile);
</pre>

where <tt>datafile</tt> is the path to the ParaDiS <tt>data</tt> file.

'''DXA analysis (Ovito)''' <br>
DD structures can be imported from DXA files generated with Ovito by using the <tt>read_dxa_file</tt> function:

<pre>
[rn,links,bounds] = read_dxa_file(cafilename);
</pre>

where <tt>cafilename</tt> is the path to the DXA <tt>ca</tt> file.

=== MD structures ===
'''CGSD structures''' <br>
CGSD structures can be imported using function <tt>read_cgsd</tt>

<pre>
[cn,bnds,vol] = read_cgsd(filename);
</pre>

where <tt>filename</tt> contains the path and name (without extension) of the CGSD data files <tt>filename.cn</tt> and <tt>filename.bnd</tt> (Note that both the positions and bonds files must have the same name, only their extension should differ).

Similarly to the other import functions, array <tt>cn</tt> contains the position of the atoms/beads, <tt>bnds</tt> contains the bonds information, and <tt>vol</tt> specifies the bounds of the simulation volume.

=== Networks generation ===
Random networks can be generated using function <tt>random_network</tt>. For instance

<pre>
[rn,links] = random_network(n,P,bounds);
</pre>

will generate a network of <tt>n</tt> nodes randomly positionned within <tt>bounds</tt>, and whose degree distribution (number of connections per node) is specified with array <tt>P</tt>. The algorithm to generate such a network is based on the configuration model, with the additional constraint that links must join nodes that are in the spatial vicinity of one another.

== Exporting networks ==
Networks can be exported in standard graph formats <tt>GML</tt> and <tt>GEXF</tt> for visualization / analysis with other network tools with the following functions:

<pre>
export_gml('filename.gml',rn,links);
export_gexf('filename.gexf',rn,links);
</pre>

Properties can be associated with nodes or links. For instance, the weight of the links can be exported by specifying property <tt>'EdgeWeights'</tt>:

<pre>
export_gml('filename.gml',rn,links,'EdgeWeights',weights);
</pre>

where array <tt>weights</tt> contains the list of weight associated with each link such that <tt>size(weights,1) = size(links,1)</tt>.

NetworCh Manuals

2018-01-31T20:11:42Z

Nbertin: /* Getting started */

= NetworCh library manuals =

NetworCh is a library, mainly written in Matlab, developed to manipulate, visualize, analyze and characterize network structures in various areas of mechanics of materials.

== Getting started ==
[[NetworCh Introduction and examples]] <br>
[[NetworCh Documentation]] <br>
[[Installing and using the igraph package in R]]

== Dislocation dynamics ==
[[DDD structure generation]] <br>
[[DDD irreducible core graph]]

== CGSD ==
[[Shortest paths PBC calculation]]

NetworCh Introduction and examples

2018-01-31T20:10:35Z

Nbertin:

NetworCh Manuals

2018-01-31T19:45:04Z

Nbertin: /* NetworCh library manuals */

= NetworCh library manuals =

NetworCh is a library, mainly written in Matlab, developed to manipulate, visualize, analyze and characterize network structures in various areas of mechanics of materials.

== Getting started ==
[[Introduction and examples]] <br>
[[Documentation]] <br>
[[Installing and using the igraph package in R]]

== Dislocation dynamics ==
[[DDD structure generation]] <br>
[[DDD irreducible core graph]]

== CGSD ==
[[Shortest paths PBC calculation]]

DDD-XRD Matlab implementation and validation

2018-01-31T19:38:25Z

Nbertin: /* Code */

<P ALIGN="CENTER">DDD-XRD approach</P>
<P ALIGN="CENTER"><FONT SIZE="+2" color="darkred"><STRONG>
DDD-XRD: Matlab implementation and validation</STRONG></font></P>
<DIV>
<P ALIGN="CENTER"><STRONG>Nicolas Bertin and Wei Cai</STRONG></P>
</DIV>
<DIV>
<P ALIGN="CENTER">January 2018</P>
</DIV>

<HR>

This page provides Matlab functions to compute the displacement gradient associated with discrete dislocation segments based on the non-singular formulation presented in Bertin and Cai, CMS, 2018.

== Analytical non-singular displacement gradient formulation ==

It can be shown that the deformation gradient <math>G_{ml} = u_{m,l}</math> produced by a dislocation loop <math>C</math> with Burgers vector <math>\vec{b}</math> can be written as (see Bertin and Cai, CMS, 2018):

{|border="0" align="center"
|<math>
\begin{align}
u_{m,l}(\vec{x}) = &-\frac{1}{8\pi} \oint_C b_m \epsilon_{jlk} R_{,ppj} dx'_k -\frac{1}{8\pi} \oint_C b_i \epsilon_{mik} R_{,ppl} \, dx'_k \\
&-\frac{1}{8\pi(1-\nu)} \oint_C b_i \epsilon_{ijk} R_{,mjl} \, dx'_k
\end{align}
</math>
|}

where <math>R = \|\vec{x'}-\vec{x}\|</math> is the norm of the distance vector linking the field point <math>\vec{x}</math> and the coordinate <math>\vec{x'}</math> spanning the dislocation line, <math>\epsilon_{ijk}</math> is the permutation tensor, and <math>\nu</math> is Poisson's ratio of the medium. <math>R_{,ijk} = \partial^3 R / \partial x_i\partial x_j\partial x_k</math> denotes the third derivative of the radius vector wrt. the field coordinate.

Following the isotropic Burgers distribution proposed in Cai et al., JMPS, 2006, the singularity in the above expression can be eliminated by employing the modified radius vector <math>R_a</math> defined as:

{|border="0" align="center"
|<math>
R_a = \sqrt{R^2 + a^2} = \sqrt{R_i R_i + a^2} = \sqrt{(x'_i - x_i)(x'_i - x_i) + a^2}
</math>
|}

where <math>a</math> denotes the dislocation core radius. With this, the displacement gradient produced by a straight dislocation segment with end points <math>\vec{x}_a</math> and <math>\vec{x}_b</math> can be analytically expressed as:

{|border="0" align="center"
|<math>
\begin{align}
u_{m,l}(\vec{x}) = &-\frac{1}{8\pi} b_m \epsilon_{jlk} A_{jk}(\vec{x}) -\frac{1}{8\pi} b_i \epsilon_{mik} A_{lk}(\vec{x}) \\
&-\frac{1}{8\pi(1-\nu)} b_i \epsilon_{ijk} B_{jklm}(\vec{x})
\end{align}
</math>
|}

where line integrals <math>A_{jk}(\vec{x})</math> and <math>B_{jklm}(\vec{x})</math> are given by:

{|border="0" align="center"
|<math>
A_{jk}(\vec{x}) = t_k \left[ 3a^2 d_j J_{05} + 2d_j J_{03} + 3a^2 t_j J_{15} + 2t_j J_{13} \right]
</math>
|}
{|border="0" align="center"
|<math>
\begin{align}
B_{jklm}(\vec{x}) = &t_k \left[ (\delta_{mj}d_l + \delta_{jl}d_m + \delta_{lm}d_j)J_{03} + (\delta_{mj}t_l + \delta_{jl}t_m + \delta_{lm}t_j)J_{13} - 3(d_m d_j d_l)J_{05} \right. \\
&\left. -3(d_m d_j t_l + d_m t_j d_l + t_m d_j d_l)J_{15} - 3(d_m t_j t_l + t_m d_j t_l + t_m t_j d_l)J_{25} -3(t_m t_j t_l)J_{35} \right]
\end{align}
</math>
|}

where <math>\vec{t}</math> denotes the unit dislocation line tangent, and <math>\vec{d} = \vec{x}_0-\vec{x}</math> is the vector linking field point <math>\vec{x}</math> to its orthogonal projection <math>\vec{x}_0</math> on the dislocation line.

When adopting the following segment parametric representation

{|border="0" align="center"
|<math>
\vec{x'} = \vec{x}_0 + s\vec{t}, \; s \in (s_1,s_2)
</math>
|}

an analytical expression for the line integrals <math>J_{ij}</math> is obtained as follows:

{|border="0" align="center"
|<math>
\begin{align}
&J_{03} = \int_{s_1}^{s_2} \frac{1}{R_a^3} ds = \left. \frac{s}{( \vec{d} \cdot \vec{d} + a^2 ) R_a } \right|_{s_1}^{s_2} \\
&J_{13} = \int_{s_1}^{s_2} \frac{s}{R_a^3} ds = \left. -\frac{1}{R_a} \right|_{s_1}^{s_2} \\
&J_{05} = \int_{s_1}^{s_2} \frac{1}{R_a^5} ds = \left. \frac{2s^3}{3( \vec{d} \cdot \vec{d} + a^2 )^2 R_a^3 } \right|_{s_1}^{s_2} + \left. \frac{s}{( \vec{d} \cdot \vec{d} + a^2 ) R_a^3 } \right|_{s_1}^{s_2} \\
&J_{15} = \int_{s_1}^{s_2} \frac{s}{R_a^5} ds = \left. -\frac{1}{3 R_a^3} \right|_{s_1}^{s_2} \\
&J_{25} = \int_{s_1}^{s_2} \frac{s^2}{R_a^5} ds = \left. \frac{s^3}{3( \vec{d} \cdot \vec{d} + a^2 ) R_a^3 } \right|_{s_1}^{s_2} \\
&J_{35} = \int_{s_1}^{s_2} \frac{s^3}{R_a^5} ds = \left. -\frac{2(\vec{d} \cdot \vec{d} + a^2)}{3 R_a^3} \right|_{s_1}^{s_2} -\left. \frac{s^2}{R_a^3 } \right|_{s_1}^{s_2}
\end{align}
</math>
|}

== Matlab implementation ==

=== Code ===

The computation of the non-singular displacement gradient based on the expressions provided in the above (Bertin and Cai, CMS, 2018) is implemented in a Matlab code available here:

[[Media:ddd_xrd_matlab.tar | ddd_xrd_matlab.tar]]

Function <tt>displacement_gradient_seg.m</tt> readily implements the displacement gradient for a straight dislocation segment using the expressions presented in the above. An optimized version (faster) is provided in <tt>displacement_gradient_seg_opt.m</tt>. The calculation of the displacement gradient field for a set of dislocation segments can be performed using function <tt>displacement_gradient_structure.m</tt>. Note that this implementation uses the [http://micro.stanford.edu/~caiwei/Forum/2005-12-05-DDLab/ DDLab] data structure to represent the dislocations.

=== Validation: triangular loop test case ===

The code includes a test case (<tt>test_triangular_loop.m</tt>) that (i) provides an example of how to use the calculation functions, and (ii) performs calculations that validate the non-singular displacement gradient formulation.

This test case does the following:
* generates a random triangular dislocation loop
* computes the displacement gradient field along a line using the non-singular expression provided in Bertin and Cai, CMS, 2018.
* compares it with the displacement gradient field obtained by numerically differentiating the displacement field
* compares the stress obtained from the displacement gradient field with the non-singular stress expression provided in Cai et al., JMPS, 2006

ParaDiS tools

2018-01-31T19:37:09Z

Nbertin:

<H4>NetworCh for ParaDiS</H4>
<UL>
<LI>[[Extract links from DDD]]
</UL>

<H4>DDD-XRD</H4>
<UL>
<LI>[[DDD-XRD Manual]]
<LI>[[DDD-XRD Matlab implementation and validation]]
</UL>

Tutorials

2018-01-31T19:31:04Z

Nbertin: /* Simulation Codes */

== Simulation Codes ==

{| class="wikitable" cellpadding="2"
|-
!width="200" | [[MD++ Manuals]]
!width="200" | [[ParaDiS Manuals]]
!width="300" | [[ParaDiS Workshop Notes]]
|-
!width="200" | [[DDLab Manuals]]
!width="200" | [[VASP Manuals]]
!width="300" | [[How to compile Qbox]]
|-
!width="200" | [[NetworCh Manuals]]
!width="200" | [[How to compile LAMMPS]]
!width="300" | [[How to compile pimc++]]
|-
!width="200" | [[ParaDiS tools]]
!width="200" |
!width="300" |
|}

== Computers ==

{| class="wikitable" cellpadding="2"
|-
!width="200" | [[Unix Basics]]
!width="200" | [[Computer Setup]]
!width="200" | [[Computing Clusters | Parallel Clusters]]
|-
!width="200" | [[Install Ubuntu]]
!width="200" | [[Install FFTW3]]
!width="200" | [[Install HDF5]]
|-
!width="200" | [[Install GSL]]
!width="200" | [[Standard Tcl Library | Tcl Library]]
!width="200" | [[Install the GNU Compiler Collection (GCC)]]
|-
!width="200" | [[Running Hybrid MPI/OpenMP Simulations]]
!width="200" | [[Install SHTOOLS]]
|}

== Scientific Background ==

{| class="wikitable" cellpadding="2"
|-
!width="200" | [[Dislocations]]
|}

== Science Outreach ==

{| class="wikitable" cellpadding="2"
|-
!width="200" | [http://www.youtube.com/watch?v=SgM-Xes16Sw Outreach interview]
!width="200" | [[How to install MD++ in Ubuntu | Install MD++ in Ubuntu]]
!width="200" | [[Introduction to Molecular Dynamics Simulations of Fullerenes | MD of Fullerenes]]
|-
!width="200" | [[Images of Fullerenes]]
!width="200" | [[Android Applications and Information]]
|

|}

<br> <br>

== [[Tutorial:Members_Only | Members Only]] ==

DDD-XRD Manuals

2018-01-31T19:30:50Z

Nbertin: Nbertin moved page DDD-XRD Manuals to ParaDiS tools

#REDIRECT [[ParaDiS tools]]

ParaDiS tools

2018-01-31T19:30:50Z

Nbertin: Nbertin moved page DDD-XRD Manuals to ParaDiS tools

<H4>Codes</H4>
<UL>
<LI>[[DDD-XRD Matlab implementation and validation]]
</UL>

Tutorials

2018-01-31T19:29:47Z

Nbertin: Undo revision 6698 by Nbertin (talk)

== Simulation Codes ==

{| class="wikitable" cellpadding="2"
|-
!width="200" | [[MD++ Manuals]]
!width="200" | [[ParaDiS Manuals]]
!width="300" | [[ParaDiS Workshop Notes]]
|-
!width="200" | [[DDLab Manuals]]
!width="200" | [[VASP Manuals]]
!width="300" | [[How to compile Qbox]]
|-
!width="200" | [[NetworCh Manuals]]
!width="200" | [[How to compile LAMMPS]]
!width="300" | [[How to compile pimc++]]
|-
!width="200" | [[DDD-XRD Manuals]]
!width="200" |
!width="300" |
|}

== Computers ==

{| class="wikitable" cellpadding="2"
|-
!width="200" | [[Unix Basics]]
!width="200" | [[Computer Setup]]
!width="200" | [[Computing Clusters | Parallel Clusters]]
|-
!width="200" | [[Install Ubuntu]]
!width="200" | [[Install FFTW3]]
!width="200" | [[Install HDF5]]
|-
!width="200" | [[Install GSL]]
!width="200" | [[Standard Tcl Library | Tcl Library]]
!width="200" | [[Install the GNU Compiler Collection (GCC)]]
|-
!width="200" | [[Running Hybrid MPI/OpenMP Simulations]]
!width="200" | [[Install SHTOOLS]]
|}

== Scientific Background ==

{| class="wikitable" cellpadding="2"
|-
!width="200" | [[Dislocations]]
|}

== Science Outreach ==

{| class="wikitable" cellpadding="2"
|-
!width="200" | [http://www.youtube.com/watch?v=SgM-Xes16Sw Outreach interview]
!width="200" | [[How to install MD++ in Ubuntu | Install MD++ in Ubuntu]]
!width="200" | [[Introduction to Molecular Dynamics Simulations of Fullerenes | MD of Fullerenes]]
|-
!width="200" | [[Images of Fullerenes]]
!width="200" | [[Android Applications and Information]]
|

|}

<br> <br>

== [[Tutorial:Members_Only | Members Only]] ==

Tutorials

2018-01-31T19:28:08Z

Nbertin: /* Simulation Codes */

DDD-XRD Matlab implementation and validation

2018-01-20T03:09:21Z

Nbertin: /* Code */

<P ALIGN="CENTER">DDD-XRD approach</P>
<P ALIGN="CENTER"><FONT SIZE="+2" color="darkred"><STRONG>
DDD-XRD: Matlab implementation and validation</STRONG></font></P>
<DIV>
<P ALIGN="CENTER"><STRONG>Nicolas Bertin and Wei Cai</STRONG></P>
</DIV>
<DIV>
<P ALIGN="CENTER">January 2018</P>
</DIV>

<HR>

This page provides Matlab functions to compute the displacement gradient associated with discrete dislocation segments based on the non-singular formulation presented in Bertin and Cai, CMS, 2018.

== Analytical non-singular displacement gradient formulation ==

It can be shown that the deformation gradient <math>G_{ml} = u_{m,l}</math> produced by a dislocation loop <math>C</math> with Burgers vector <math>\vec{b}</math> can be written as (see Bertin and Cai, CMS, 2018):

{|border="0" align="center"
|<math>
\begin{align}
u_{m,l}(\vec{x}) = &-\frac{1}{8\pi} \oint_C b_m \epsilon_{jlk} R_{,ppj} dx'_k -\frac{1}{8\pi} \oint_C b_i \epsilon_{mik} R_{,ppl} \, dx'_k \\
&-\frac{1}{8\pi(1-\nu)} \oint_C b_i \epsilon_{ijk} R_{,mjl} \, dx'_k
\end{align}
</math>
|}

where <math>R = \|\vec{x'}-\vec{x}\|</math> is the norm of the distance vector linking the field point <math>\vec{x}</math> and the coordinate <math>\vec{x'}</math> spanning the dislocation line, <math>\epsilon_{ijk}</math> is the permutation tensor, and <math>\nu</math> is Poisson's ratio of the medium. <math>R_{,ijk} = \partial^3 R / \partial x_i\partial x_j\partial x_k</math> denotes the third derivative of the radius vector wrt. the field coordinate.

Following the isotropic Burgers distribution proposed in Cai et al., JMPS, 2006, the singularity in the above expression can be eliminated by employing the modified radius vector <math>R_a</math> defined as:

{|border="0" align="center"
|<math>
R_a = \sqrt{R^2 + a^2} = \sqrt{R_i R_i + a^2} = \sqrt{(x'_i - x_i)(x'_i - x_i) + a^2}
</math>
|}

where <math>a</math> denotes the dislocation core radius. With this, the displacement gradient produced by a straight dislocation segment with end points <math>\vec{x}_a</math> and <math>\vec{x}_b</math> can be analytically expressed as:

{|border="0" align="center"
|<math>
\begin{align}
u_{m,l}(\vec{x}) = &-\frac{1}{8\pi} b_m \epsilon_{jlk} A_{jk}(\vec{x}) -\frac{1}{8\pi} b_i \epsilon_{mik} A_{lk}(\vec{x}) \\
&-\frac{1}{8\pi(1-\nu)} b_i \epsilon_{ijk} B_{jklm}(\vec{x})
\end{align}
</math>
|}

where line integrals <math>A_{jk}(\vec{x})</math> and <math>B_{jklm}(\vec{x})</math> are given by:

{|border="0" align="center"
|<math>
A_{jk}(\vec{x}) = t_k \left[ 3a^2 d_j J_{05} + 2d_j J_{03} + 3a^2 t_j J_{15} + 2t_j J_{13} \right]
</math>
|}
{|border="0" align="center"
|<math>
\begin{align}
B_{jklm}(\vec{x}) = &t_k \left[ (\delta_{mj}d_l + \delta_{jl}d_m + \delta_{lm}d_j)J_{03} + (\delta_{mj}t_l + \delta_{jl}t_m + \delta_{lm}t_j)J_{13} - 3(d_m d_j d_l)J_{05} \right. \\
&\left. -3(d_m d_j t_l + d_m t_j d_l + t_m d_j d_l)J_{15} - 3(d_m t_j t_l + t_m d_j t_l + t_m t_j d_l)J_{25} -3(t_m t_j t_l)J_{35} \right]
\end{align}
</math>
|}

where <math>\vec{t}</math> denotes the unit dislocation line tangent, and <math>\vec{d} = \vec{x}_0-\vec{x}</math> is the vector linking field point <math>\vec{x}</math> to its orthogonal projection <math>\vec{x}_0</math> on the dislocation line.

When adopting the following segment parametric representation

{|border="0" align="center"
|<math>
\vec{x'} = \vec{x}_0 + s\vec{t}, \; s \in (s_1,s_2)
</math>
|}

an analytical expression for the line integrals <math>J_{ij}</math> is obtained as follows:

{|border="0" align="center"
|<math>
\begin{align}
&J_{03} = \int_{s_1}^{s_2} \frac{1}{R_a^3} ds = \left. \frac{s}{( \vec{d} \cdot \vec{d} + a^2 ) R_a } \right|_{s_1}^{s_2} \\
&J_{13} = \int_{s_1}^{s_2} \frac{s}{R_a^3} ds = \left. -\frac{1}{R_a} \right|_{s_1}^{s_2} \\
&J_{05} = \int_{s_1}^{s_2} \frac{1}{R_a^5} ds = \left. \frac{2s^3}{3( \vec{d} \cdot \vec{d} + a^2 )^2 R_a^3 } \right|_{s_1}^{s_2} + \left. \frac{s}{( \vec{d} \cdot \vec{d} + a^2 ) R_a^3 } \right|_{s_1}^{s_2} \\
&J_{15} = \int_{s_1}^{s_2} \frac{s}{R_a^5} ds = \left. -\frac{1}{3 R_a^3} \right|_{s_1}^{s_2} \\
&J_{25} = \int_{s_1}^{s_2} \frac{s^2}{R_a^5} ds = \left. \frac{s^3}{3( \vec{d} \cdot \vec{d} + a^2 ) R_a^3 } \right|_{s_1}^{s_2} \\
&J_{35} = \int_{s_1}^{s_2} \frac{s^3}{R_a^5} ds = \left. -\frac{2(\vec{d} \cdot \vec{d} + a^2)}{3 R_a^3} \right|_{s_1}^{s_2} -\left. \frac{s^2}{R_a^3 } \right|_{s_1}^{s_2}
\end{align}
</math>
|}

== Matlab implementation ==

=== Code ===

The computation of the non-singular displacement gradient based on the expressions provided in the above (Bertin and Cai, CMS, 2018) is implemented in a Matlab code available here:

[[File:ddd_xrd_matlab.tar | ddd_xrd_matlab.tar]]

Function <tt>displacement_gradient_seg.m</tt> readily implements the displacement gradient for a straight dislocation segment using the expressions presented in the above. An optimized version (faster) is provided in <tt>displacement_gradient_seg_opt.m</tt>. The calculation of the displacement gradient field for a set of dislocation segments can be performed using function <tt>displacement_gradient_structure.m</tt>. Note that this implementation uses the [http://micro.stanford.edu/~caiwei/Forum/2005-12-05-DDLab/ DDLab] data structure to represent the dislocations.

=== Validation: triangular loop test case ===

The code includes a test case (<tt>test_triangular_loop.m</tt>) that (i) provides an example of how to use the calculation functions, and (ii) performs calculations that validate the non-singular displacement gradient formulation.

This test case does the following:
* generates a random triangular dislocation loop
* computes the displacement gradient field along a line using the non-singular expression provided in Bertin and Cai, CMS, 2018.
* compares it with the displacement gradient field obtained by numerically differentiating the displacement field
* compares the stress obtained from the displacement gradient field with the non-singular stress expression provided in Cai et al., JMPS, 2006

File:Ddd xrd matlab.tar

2018-01-20T03:04:31Z

Nbertin: Matlab implementation and validation of the non-singular displacement gradient for DDD-XRD calculations. See Bertin and Cai, CMS, 2018.

Matlab implementation and validation of the non-singular displacement gradient for DDD-XRD calculations. See Bertin and Cai, CMS, 2018.

DDD-XRD Matlab implementation and validation

2018-01-20T03:02:58Z

Nbertin: /* Code */

<P ALIGN="CENTER">DDD-XRD approach</P>
<P ALIGN="CENTER"><FONT SIZE="+2" color="darkred"><STRONG>
DDD-XRD: Matlab implementation and validation</STRONG></font></P>
<DIV>
<P ALIGN="CENTER"><STRONG>Nicolas Bertin and Wei Cai</STRONG></P>
</DIV>
<DIV>
<P ALIGN="CENTER">January 2018</P>
</DIV>

<HR>

This page provides Matlab functions to compute the displacement gradient associated with discrete dislocation segments based on the non-singular formulation presented in Bertin and Cai, CMS, 2018.

== Analytical non-singular displacement gradient formulation ==

It can be shown that the deformation gradient <math>G_{ml} = u_{m,l}</math> produced by a dislocation loop <math>C</math> with Burgers vector <math>\vec{b}</math> can be written as (see Bertin and Cai, CMS, 2018):

{|border="0" align="center"
|<math>
\begin{align}
u_{m,l}(\vec{x}) = &-\frac{1}{8\pi} \oint_C b_m \epsilon_{jlk} R_{,ppj} dx'_k -\frac{1}{8\pi} \oint_C b_i \epsilon_{mik} R_{,ppl} \, dx'_k \\
&-\frac{1}{8\pi(1-\nu)} \oint_C b_i \epsilon_{ijk} R_{,mjl} \, dx'_k
\end{align}
</math>
|}

where <math>R = \|\vec{x'}-\vec{x}\|</math> is the norm of the distance vector linking the field point <math>\vec{x}</math> and the coordinate <math>\vec{x'}</math> spanning the dislocation line, <math>\epsilon_{ijk}</math> is the permutation tensor, and <math>\nu</math> is Poisson's ratio of the medium. <math>R_{,ijk} = \partial^3 R / \partial x_i\partial x_j\partial x_k</math> denotes the third derivative of the radius vector wrt. the field coordinate.

Following the isotropic Burgers distribution proposed in Cai et al., JMPS, 2006, the singularity in the above expression can be eliminated by employing the modified radius vector <math>R_a</math> defined as:

{|border="0" align="center"
|<math>
R_a = \sqrt{R^2 + a^2} = \sqrt{R_i R_i + a^2} = \sqrt{(x'_i - x_i)(x'_i - x_i) + a^2}
</math>
|}

where <math>a</math> denotes the dislocation core radius. With this, the displacement gradient produced by a straight dislocation segment with end points <math>\vec{x}_a</math> and <math>\vec{x}_b</math> can be analytically expressed as:

{|border="0" align="center"
|<math>
\begin{align}
u_{m,l}(\vec{x}) = &-\frac{1}{8\pi} b_m \epsilon_{jlk} A_{jk}(\vec{x}) -\frac{1}{8\pi} b_i \epsilon_{mik} A_{lk}(\vec{x}) \\
&-\frac{1}{8\pi(1-\nu)} b_i \epsilon_{ijk} B_{jklm}(\vec{x})
\end{align}
</math>
|}

where line integrals <math>A_{jk}(\vec{x})</math> and <math>B_{jklm}(\vec{x})</math> are given by:

{|border="0" align="center"
|<math>
A_{jk}(\vec{x}) = t_k \left[ 3a^2 d_j J_{05} + 2d_j J_{03} + 3a^2 t_j J_{15} + 2t_j J_{13} \right]
</math>
|}
{|border="0" align="center"
|<math>
\begin{align}
B_{jklm}(\vec{x}) = &t_k \left[ (\delta_{mj}d_l + \delta_{jl}d_m + \delta_{lm}d_j)J_{03} + (\delta_{mj}t_l + \delta_{jl}t_m + \delta_{lm}t_j)J_{13} - 3(d_m d_j d_l)J_{05} \right. \\
&\left. -3(d_m d_j t_l + d_m t_j d_l + t_m d_j d_l)J_{15} - 3(d_m t_j t_l + t_m d_j t_l + t_m t_j d_l)J_{25} -3(t_m t_j t_l)J_{35} \right]
\end{align}
</math>
|}

where <math>\vec{t}</math> denotes the unit dislocation line tangent, and <math>\vec{d} = \vec{x}_0-\vec{x}</math> is the vector linking field point <math>\vec{x}</math> to its orthogonal projection <math>\vec{x}_0</math> on the dislocation line.

When adopting the following segment parametric representation

{|border="0" align="center"
|<math>
\vec{x'} = \vec{x}_0 + s\vec{t}, \; s \in (s_1,s_2)
</math>
|}

an analytical expression for the line integrals <math>J_{ij}</math> is obtained as follows:

{|border="0" align="center"
|<math>
\begin{align}
&J_{03} = \int_{s_1}^{s_2} \frac{1}{R_a^3} ds = \left. \frac{s}{( \vec{d} \cdot \vec{d} + a^2 ) R_a } \right|_{s_1}^{s_2} \\
&J_{13} = \int_{s_1}^{s_2} \frac{s}{R_a^3} ds = \left. -\frac{1}{R_a} \right|_{s_1}^{s_2} \\
&J_{05} = \int_{s_1}^{s_2} \frac{1}{R_a^5} ds = \left. \frac{2s^3}{3( \vec{d} \cdot \vec{d} + a^2 )^2 R_a^3 } \right|_{s_1}^{s_2} + \left. \frac{s}{( \vec{d} \cdot \vec{d} + a^2 ) R_a^3 } \right|_{s_1}^{s_2} \\
&J_{15} = \int_{s_1}^{s_2} \frac{s}{R_a^5} ds = \left. -\frac{1}{3 R_a^3} \right|_{s_1}^{s_2} \\
&J_{25} = \int_{s_1}^{s_2} \frac{s^2}{R_a^5} ds = \left. \frac{s^3}{3( \vec{d} \cdot \vec{d} + a^2 ) R_a^3 } \right|_{s_1}^{s_2} \\
&J_{35} = \int_{s_1}^{s_2} \frac{s^3}{R_a^5} ds = \left. -\frac{2(\vec{d} \cdot \vec{d} + a^2)}{3 R_a^3} \right|_{s_1}^{s_2} -\left. \frac{s^2}{R_a^3 } \right|_{s_1}^{s_2}
\end{align}
</math>
|}

== Matlab implementation ==

=== Code ===

The computation of the non-singular displacement gradient based on the expressions provided in the above (Bertin and Cai, CMS, 2018) is implemented in a Matlab code available here:

[[File:ddd_xrd_matlab.tar | ddd_xrd_matlab.tar]]

Function <tt>displacement_gradient_seg.m</tt> readily implements the displacement gradient for a straight dislocation segment as presented in the above. An optimized version (faster) is provided in <tt>displacement_gradient_seg_opt.m</tt>. The calculation of the displacement gradient field for a set of dislocation segments can be performed using function <tt>displacement_gradient_structure.m</tt>. Note that this implementation uses the [http://micro.stanford.edu/~caiwei/Forum/2005-12-05-DDLab/ DDLab] data structure to represent the dislocations.

=== Validation: triangular loop test case ===

The code includes a test case (<tt>test_triangular_loop.m</tt>) that (i) provides an example of how to use the calculation functions, and (ii) performs calculations that validate the non-singular displacement gradient formulation.

This test case does the following:
* generates a random triangular dislocation loop
* computes the displacement gradient field along a line using the non-singular expression provided in Bertin and Cai, CMS, 2018.
* compares it with the displacement gradient field obtained by numerically differentiating the displacement field
* compares the stress obtained from the displacement gradient field with the non-singular stress expression provided in Cai et al., JMPS, 2006

DDD-XRD Matlab implementation and validation

2018-01-20T03:02:30Z

Nbertin: /* Code */

<P ALIGN="CENTER">DDD-XRD approach</P>
<P ALIGN="CENTER"><FONT SIZE="+2" color="darkred"><STRONG>
DDD-XRD: Matlab implementation and validation</STRONG></font></P>
<DIV>
<P ALIGN="CENTER"><STRONG>Nicolas Bertin and Wei Cai</STRONG></P>
</DIV>
<DIV>
<P ALIGN="CENTER">January 2018</P>
</DIV>

<HR>

This page provides Matlab functions to compute the displacement gradient associated with discrete dislocation segments based on the non-singular formulation presented in Bertin and Cai, CMS, 2018.

== Analytical non-singular displacement gradient formulation ==

It can be shown that the deformation gradient <math>G_{ml} = u_{m,l}</math> produced by a dislocation loop <math>C</math> with Burgers vector <math>\vec{b}</math> can be written as (see Bertin and Cai, CMS, 2018):

{|border="0" align="center"
|<math>
\begin{align}
u_{m,l}(\vec{x}) = &-\frac{1}{8\pi} \oint_C b_m \epsilon_{jlk} R_{,ppj} dx'_k -\frac{1}{8\pi} \oint_C b_i \epsilon_{mik} R_{,ppl} \, dx'_k \\
&-\frac{1}{8\pi(1-\nu)} \oint_C b_i \epsilon_{ijk} R_{,mjl} \, dx'_k
\end{align}
</math>
|}

where <math>R = \|\vec{x'}-\vec{x}\|</math> is the norm of the distance vector linking the field point <math>\vec{x}</math> and the coordinate <math>\vec{x'}</math> spanning the dislocation line, <math>\epsilon_{ijk}</math> is the permutation tensor, and <math>\nu</math> is Poisson's ratio of the medium. <math>R_{,ijk} = \partial^3 R / \partial x_i\partial x_j\partial x_k</math> denotes the third derivative of the radius vector wrt. the field coordinate.

Following the isotropic Burgers distribution proposed in Cai et al., JMPS, 2006, the singularity in the above expression can be eliminated by employing the modified radius vector <math>R_a</math> defined as:

{|border="0" align="center"
|<math>
R_a = \sqrt{R^2 + a^2} = \sqrt{R_i R_i + a^2} = \sqrt{(x'_i - x_i)(x'_i - x_i) + a^2}
</math>
|}

where <math>a</math> denotes the dislocation core radius. With this, the displacement gradient produced by a straight dislocation segment with end points <math>\vec{x}_a</math> and <math>\vec{x}_b</math> can be analytically expressed as:

{|border="0" align="center"
|<math>
\begin{align}
u_{m,l}(\vec{x}) = &-\frac{1}{8\pi} b_m \epsilon_{jlk} A_{jk}(\vec{x}) -\frac{1}{8\pi} b_i \epsilon_{mik} A_{lk}(\vec{x}) \\
&-\frac{1}{8\pi(1-\nu)} b_i \epsilon_{ijk} B_{jklm}(\vec{x})
\end{align}
</math>
|}

where line integrals <math>A_{jk}(\vec{x})</math> and <math>B_{jklm}(\vec{x})</math> are given by:

{|border="0" align="center"
|<math>
A_{jk}(\vec{x}) = t_k \left[ 3a^2 d_j J_{05} + 2d_j J_{03} + 3a^2 t_j J_{15} + 2t_j J_{13} \right]
</math>
|}
{|border="0" align="center"
|<math>
\begin{align}
B_{jklm}(\vec{x}) = &t_k \left[ (\delta_{mj}d_l + \delta_{jl}d_m + \delta_{lm}d_j)J_{03} + (\delta_{mj}t_l + \delta_{jl}t_m + \delta_{lm}t_j)J_{13} - 3(d_m d_j d_l)J_{05} \right. \\
&\left. -3(d_m d_j t_l + d_m t_j d_l + t_m d_j d_l)J_{15} - 3(d_m t_j t_l + t_m d_j t_l + t_m t_j d_l)J_{25} -3(t_m t_j t_l)J_{35} \right]
\end{align}
</math>
|}

where <math>\vec{t}</math> denotes the unit dislocation line tangent, and <math>\vec{d} = \vec{x}_0-\vec{x}</math> is the vector linking field point <math>\vec{x}</math> to its orthogonal projection <math>\vec{x}_0</math> on the dislocation line.

When adopting the following segment parametric representation

{|border="0" align="center"
|<math>
\vec{x'} = \vec{x}_0 + s\vec{t}, \; s \in (s_1,s_2)
</math>
|}

an analytical expression for the line integrals <math>J_{ij}</math> is obtained as follows:

{|border="0" align="center"
|<math>
\begin{align}
&J_{03} = \int_{s_1}^{s_2} \frac{1}{R_a^3} ds = \left. \frac{s}{( \vec{d} \cdot \vec{d} + a^2 ) R_a } \right|_{s_1}^{s_2} \\
&J_{13} = \int_{s_1}^{s_2} \frac{s}{R_a^3} ds = \left. -\frac{1}{R_a} \right|_{s_1}^{s_2} \\
&J_{05} = \int_{s_1}^{s_2} \frac{1}{R_a^5} ds = \left. \frac{2s^3}{3( \vec{d} \cdot \vec{d} + a^2 )^2 R_a^3 } \right|_{s_1}^{s_2} + \left. \frac{s}{( \vec{d} \cdot \vec{d} + a^2 ) R_a^3 } \right|_{s_1}^{s_2} \\
&J_{15} = \int_{s_1}^{s_2} \frac{s}{R_a^5} ds = \left. -\frac{1}{3 R_a^3} \right|_{s_1}^{s_2} \\
&J_{25} = \int_{s_1}^{s_2} \frac{s^2}{R_a^5} ds = \left. \frac{s^3}{3( \vec{d} \cdot \vec{d} + a^2 ) R_a^3 } \right|_{s_1}^{s_2} \\
&J_{35} = \int_{s_1}^{s_2} \frac{s^3}{R_a^5} ds = \left. -\frac{2(\vec{d} \cdot \vec{d} + a^2)}{3 R_a^3} \right|_{s_1}^{s_2} -\left. \frac{s^2}{R_a^3 } \right|_{s_1}^{s_2}
\end{align}
</math>
|}

== Matlab implementation ==

=== Code ===

The computation of the non-singular displacement gradient based on the expressions provided in the above (Bertin and Cai, CMS, 2018) is implemented in a Matlab code available here:

[[File:ddd_xrd_matlab.tar]]

Function <tt>displacement_gradient_seg.m</tt> readily implements the displacement gradient for a straight dislocation segment as presented in the above. An optimized version (faster) is provided in <tt>displacement_gradient_seg_opt.m</tt>. The calculation of the displacement gradient field for a set of dislocation segments can be performed using function <tt>displacement_gradient_structure.m</tt>. Note that this implementation uses the [http://micro.stanford.edu/~caiwei/Forum/2005-12-05-DDLab/ DDLab] data structure to represent the dislocations.

=== Validation: triangular loop test case ===

The code includes a test case (<tt>test_triangular_loop.m</tt>) that (i) provides an example of how to use the calculation functions, and (ii) performs calculations that validate the non-singular displacement gradient formulation.

This test case does the following:
* generates a random triangular dislocation loop
* computes the displacement gradient field along a line using the non-singular expression provided in Bertin and Cai, CMS, 2018.
* compares it with the displacement gradient field obtained by numerically differentiating the displacement field
* compares the stress obtained from the displacement gradient field with the non-singular stress expression provided in Cai et al., JMPS, 2006

DDD-XRD Matlab implementation and validation

2018-01-20T02:59:56Z

Nbertin:

<P ALIGN="CENTER">DDD-XRD approach</P>
<P ALIGN="CENTER"><FONT SIZE="+2" color="darkred"><STRONG>
DDD-XRD: Matlab implementation and validation</STRONG></font></P>
<DIV>
<P ALIGN="CENTER"><STRONG>Nicolas Bertin and Wei Cai</STRONG></P>
</DIV>
<DIV>
<P ALIGN="CENTER">January 2018</P>
</DIV>

<HR>

This page provides Matlab functions to compute the displacement gradient associated with discrete dislocation segments based on the non-singular formulation presented in Bertin and Cai, CMS, 2018.

== Analytical non-singular displacement gradient formulation ==

It can be shown that the deformation gradient <math>G_{ml} = u_{m,l}</math> produced by a dislocation loop <math>C</math> with Burgers vector <math>\vec{b}</math> can be written as (see Bertin and Cai, CMS, 2018):

{|border="0" align="center"
|<math>
\begin{align}
u_{m,l}(\vec{x}) = &-\frac{1}{8\pi} \oint_C b_m \epsilon_{jlk} R_{,ppj} dx'_k -\frac{1}{8\pi} \oint_C b_i \epsilon_{mik} R_{,ppl} \, dx'_k \\
&-\frac{1}{8\pi(1-\nu)} \oint_C b_i \epsilon_{ijk} R_{,mjl} \, dx'_k
\end{align}
</math>
|}

where <math>R = \|\vec{x'}-\vec{x}\|</math> is the norm of the distance vector linking the field point <math>\vec{x}</math> and the coordinate <math>\vec{x'}</math> spanning the dislocation line, <math>\epsilon_{ijk}</math> is the permutation tensor, and <math>\nu</math> is Poisson's ratio of the medium. <math>R_{,ijk} = \partial^3 R / \partial x_i\partial x_j\partial x_k</math> denotes the third derivative of the radius vector wrt. the field coordinate.

Following the isotropic Burgers distribution proposed in Cai et al., JMPS, 2006, the singularity in the above expression can be eliminated by employing the modified radius vector <math>R_a</math> defined as:

{|border="0" align="center"
|<math>
R_a = \sqrt{R^2 + a^2} = \sqrt{R_i R_i + a^2} = \sqrt{(x'_i - x_i)(x'_i - x_i) + a^2}
</math>
|}

where <math>a</math> denotes the dislocation core radius. With this, the displacement gradient produced by a straight dislocation segment with end points <math>\vec{x}_a</math> and <math>\vec{x}_b</math> can be analytically expressed as:

{|border="0" align="center"
|<math>
\begin{align}
u_{m,l}(\vec{x}) = &-\frac{1}{8\pi} b_m \epsilon_{jlk} A_{jk}(\vec{x}) -\frac{1}{8\pi} b_i \epsilon_{mik} A_{lk}(\vec{x}) \\
&-\frac{1}{8\pi(1-\nu)} b_i \epsilon_{ijk} B_{jklm}(\vec{x})
\end{align}
</math>
|}

where line integrals <math>A_{jk}(\vec{x})</math> and <math>B_{jklm}(\vec{x})</math> are given by:

{|border="0" align="center"
|<math>
A_{jk}(\vec{x}) = t_k \left[ 3a^2 d_j J_{05} + 2d_j J_{03} + 3a^2 t_j J_{15} + 2t_j J_{13} \right]
</math>
|}
{|border="0" align="center"
|<math>
\begin{align}
B_{jklm}(\vec{x}) = &t_k \left[ (\delta_{mj}d_l + \delta_{jl}d_m + \delta_{lm}d_j)J_{03} + (\delta_{mj}t_l + \delta_{jl}t_m + \delta_{lm}t_j)J_{13} - 3(d_m d_j d_l)J_{05} \right. \\
&\left. -3(d_m d_j t_l + d_m t_j d_l + t_m d_j d_l)J_{15} - 3(d_m t_j t_l + t_m d_j t_l + t_m t_j d_l)J_{25} -3(t_m t_j t_l)J_{35} \right]
\end{align}
</math>
|}

where <math>\vec{t}</math> denotes the unit dislocation line tangent, and <math>\vec{d} = \vec{x}_0-\vec{x}</math> is the vector linking field point <math>\vec{x}</math> to its orthogonal projection <math>\vec{x}_0</math> on the dislocation line.

When adopting the following segment parametric representation

{|border="0" align="center"
|<math>
\vec{x'} = \vec{x}_0 + s\vec{t}, \; s \in (s_1,s_2)
</math>
|}

an analytical expression for the line integrals <math>J_{ij}</math> is obtained as follows:

{|border="0" align="center"
|<math>
\begin{align}
&J_{03} = \int_{s_1}^{s_2} \frac{1}{R_a^3} ds = \left. \frac{s}{( \vec{d} \cdot \vec{d} + a^2 ) R_a } \right|_{s_1}^{s_2} \\
&J_{13} = \int_{s_1}^{s_2} \frac{s}{R_a^3} ds = \left. -\frac{1}{R_a} \right|_{s_1}^{s_2} \\
&J_{05} = \int_{s_1}^{s_2} \frac{1}{R_a^5} ds = \left. \frac{2s^3}{3( \vec{d} \cdot \vec{d} + a^2 )^2 R_a^3 } \right|_{s_1}^{s_2} + \left. \frac{s}{( \vec{d} \cdot \vec{d} + a^2 ) R_a^3 } \right|_{s_1}^{s_2} \\
&J_{15} = \int_{s_1}^{s_2} \frac{s}{R_a^5} ds = \left. -\frac{1}{3 R_a^3} \right|_{s_1}^{s_2} \\
&J_{25} = \int_{s_1}^{s_2} \frac{s^2}{R_a^5} ds = \left. \frac{s^3}{3( \vec{d} \cdot \vec{d} + a^2 ) R_a^3 } \right|_{s_1}^{s_2} \\
&J_{35} = \int_{s_1}^{s_2} \frac{s^3}{R_a^5} ds = \left. -\frac{2(\vec{d} \cdot \vec{d} + a^2)}{3 R_a^3} \right|_{s_1}^{s_2} -\left. \frac{s^2}{R_a^3 } \right|_{s_1}^{s_2}
\end{align}
</math>
|}

== Matlab implementation ==

=== Code ===

The computation of the non-singular displacement gradient based on the expressions provided in the above (Bertin and Cai, CMS, 2018) is implemented in a Matlab code available here:

[[ddd_xrd_matlab.tar]]

Function <tt>displacement_gradient_seg.m</tt> readily implements the displacement gradient for a straight dislocation segment as presented in the above. An optimized version (faster) is provided in <tt>displacement_gradient_seg_opt.m</tt>. The calculation of the displacement gradient field for a set of dislocation segments can be performed using function <tt>displacement_gradient_structure.m</tt>. Note that this implementation uses the [http://micro.stanford.edu/~caiwei/Forum/2005-12-05-DDLab/ DDLab] data structure to represent the dislocations.

=== Validation: triangular loop test case ===

The code includes a test case (<tt>test_triangular_loop.m</tt>) that (i) provides an example of how to use the calculation functions, and (ii) performs calculations that validate the non-singular displacement gradient formulation.

This test case does the following:
* generates a random triangular dislocation loop
* computes the displacement gradient field along a line using the non-singular expression provided in Bertin and Cai, CMS, 2018.
* compares it with the displacement gradient field obtained by numerically differentiating the displacement field
* compares the stress obtained from the displacement gradient field with the non-singular stress expression provided in Cai et al., JMPS, 2006

DDD-XRD Matlab implementation and validation

2018-01-20T02:59:15Z

Nbertin:

<P ALIGN="CENTER">DDD-XRD approach</P>
<P ALIGN="CENTER"><FONT SIZE="+2" color="darkred"><STRONG>
DDD-XRD: Matlab implementation and validation</STRONG></font></P>
<DIV>
<P ALIGN="CENTER"><STRONG>Nicolas Bertin and Wei Cai</STRONG></P>
</DIV>
<DIV>
<P ALIGN="CENTER">January 2018</P>
</DIV>

<HR>

This page provides Matlab functions to compute the displacement gradient associated with discrete dislocation segments based on the non-singular formulation presented in Bertin and Cai, CMS, 2018.

== Analytical non-singular displacement gradient formulation ==

It can be shown that the deformation gradient <math>G_{ml} = u_{m,l}</math> produced by a dislocation loop <math>C</math> with Burgers vector <math>\vec{b}</math> can be written as (see Bertin and Cai, CMS, 2018):

{|border="0" align="center"
|<math>
\begin{align}
u_{m,l}(\vec{x}) = &-\frac{1}{8\pi} \oint_C b_m \epsilon_{jlk} R_{,ppj} dx'_k -\frac{1}{8\pi} \oint_C b_i \epsilon_{mik} R_{,ppl} \, dx'_k \\
&-\frac{1}{8\pi(1-\nu)} \oint_C b_i \epsilon_{ijk} R_{,mjl} \, dx'_k
\end{align}
</math>
|}

where <math>R = \|\vec{x'}-\vec{x}\|</math> is the norm of the distance vector linking the field point <math>\vec{x}</math> and the coordinate <math>\vec{x'}</math> spanning the dislocation line, <math>\epsilon_{ijk}</math> is the permutation tensor, and <math>\nu</math> is Poisson's ratio of the medium. <math>R_{,ijk} = \partial^3 R / \partial x_i\partial x_j\partial x_k</math> denotes the third derivative of the radius vector wrt. the field coordinate.

Following the isotropic Burgers distribution proposed in Cai et al., JMPS, 2006, the singularity in the above expression can be eliminated by employing the modified radius vector <math>R_a</math> defined as:

{|border="0" align="center"
|<math>
R_a = \sqrt{R^2 + a^2} = \sqrt{R_i R_i + a^2} = \sqrt{(x'_i - x_i)(x'_i - x_i) + a^2}
</math>
|}

where <math>a</math> denotes the dislocation core radius. With this, the displacement gradient produced by a straight dislocation segment with end points <math>\vec{x}_a</math> and <math>\vec{x}_b</math> can be analytically expressed as:

{|border="0" align="center"
|<math>
\begin{align}
u_{m,l}(\vec{x}) = &-\frac{1}{8\pi} b_m \epsilon_{jlk} A_{jk}(\vec{x}) -\frac{1}{8\pi} b_i \epsilon_{mik} A_{lk}(\vec{x}) \\
&-\frac{1}{8\pi(1-\nu)} b_i \epsilon_{ijk} B_{jklm}(\vec{x})
\end{align}
</math>
|}

where line integrals <math>A_{jk}(\vec{x})</math> and <math>B_{jklm}(\vec{x})</math> are given by:

{|border="0" align="center"
|<math>
A_{jk}(\vec{x}) = t_k \left[ 3a^2 d_j J_{05} + 2d_j J_{03} + 3a^2 t_j J_{15} + 2t_j J_{13} \right]
</math>
|}
{|border="0" align="center"
|<math>
\begin{align}
B_{jklm}(\vec{x}) = &t_k \left[ (\delta_{mj}d_l + \delta_{jl}d_m + \delta_{lm}d_j)J_{03} + (\delta_{mj}t_l + \delta_{jl}t_m + \delta_{lm}t_j)J_{13} - 3(d_m d_j d_l)J_{05} \right. \\
&\left. -3(d_m d_j t_l + d_m t_j d_l + t_m d_j d_l)J_{15} - 3(d_m t_j t_l + t_m d_j t_l + t_m t_j d_l)J_{25} -3(t_m t_j t_l)J_{35} \right]
\end{align}
</math>
|}

where <math>\vec{t}</math> denotes the unit dislocation line tangent, and <math>\vec{d} = \vec{x}_0-\vec{x}</math> is the vector linking field point <math>\vec{x}</math> to its orthogonal projection <math>\vec{x}_0</math> on the dislocation line.

When adopting the following segment parametric representation

{|border="0" align="center"
|<math>
\vec{x'} = \vec{x}_0 + s\vec{t}, \; s \in (s_1,s_2)
</math>
|}

an analytical expression for the line integrals <math>J_{ij}</math> is obtained as follows:

{|border="0" align="center"
|<math>
\begin{align}
&J_{03} = \int_{s_1}^{s_2} \frac{1}{R_a^3} ds = \left. \frac{s}{( \vec{d} \cdot \vec{d} + a^2 ) R_a } \right|_{s_1}^{s_2} \\
&J_{13} = \int_{s_1}^{s_2} \frac{s}{R_a^3} ds = \left. -\frac{1}{R_a} \right|_{s_1}^{s_2} \\
&J_{05} = \int_{s_1}^{s_2} \frac{1}{R_a^5} ds = \left. \frac{2s^3}{3( \vec{d} \cdot \vec{d} + a^2 )^2 R_a^3 } \right|_{s_1}^{s_2} + \left. \frac{s}{( \vec{d} \cdot \vec{d} + a^2 ) R_a^3 } \right|_{s_1}^{s_2} \\
&J_{15} = \int_{s_1}^{s_2} \frac{s}{R_a^5} ds = \left. -\frac{1}{3 R_a^3} \right|_{s_1}^{s_2} \\
&J_{25} = \int_{s_1}^{s_2} \frac{s^2}{R_a^5} ds = \left. \frac{s^3}{3( \vec{d} \cdot \vec{d} + a^2 ) R_a^3 } \right|_{s_1}^{s_2} \\
&J_{35} = \int_{s_1}^{s_2} \frac{s^3}{R_a^5} ds = \left. -\frac{2(\vec{d} \cdot \vec{d} + a^2)}{3 R_a^3} \right|_{s_1}^{s_2} -\left. \frac{s^2}{R_a^3 } \right|_{s_1}^{s_2}
\end{align}
</math>
|}

== Matlab implementation ==

=== Code ===

The computation of the non-singular displacement gradient based on the expressions provided in the above (Bertin and Cai, CMS, 2018) is implemented in a Matlab code available here:

[[ddd_xrd_matlab.tar]]

Function <tt>displacement_gradient_seg.m</tt> readily implements the displacement gradient for a straight dislocation segment as presented in the above. An optimized version (faster) is provided in <tt>displacement_gradient_seg_opt.m</tt>. The calculation of the displacement gradient field for a set of dislocation segments can be performed using function <tt>displacement_gradient_structure.m</tt>. Note that this implementation uses the [http://micro.stanford.edu/~caiwei/Forum/2005-12-05-DDLab/ DDLab] data structure to represent the dislocations.

=== Validation: triangular loop test case ===

The code includes a test case (<tt>test_triangular_loop.m</tt>) to (i) provide an example of how to use the calculation functions and (ii) perform calculations that validate the non-singular displacement gradient formulation.

This test case does the following:
* generates a random triangular dislocation loop
* computes the displacement gradient field along a line using the non-singular expression provided in Bertin and Cai, CMS, 2018.
* compares it with the displacement gradient field obtained by numerically differentiating the displacement field
* compares the stress obtained from the displacement gradient field with the non-singular stress expression provided in Cai et al., JMPS, 2006

DDD-XRD Matlab implementation and validation

2018-01-20T01:29:24Z

Nbertin:

<P ALIGN="CENTER">DDD-XRD approach</P>
<P ALIGN="CENTER"><FONT SIZE="+2" color="darkred"><STRONG>
DDD-XRD: Matlab implementation and validation</STRONG></font></P>
<DIV>
<P ALIGN="CENTER"><STRONG>Nicolas Bertin and Wei Cai</STRONG></P>
</DIV>
<DIV>
<P ALIGN="CENTER">January 2018</P>
</DIV>

<HR>

This page provides Matlab functions to compute the displacement gradient associated with discrete dislocation segments based on the non-singular formulation presented in Bertin and Cai, CMS, 2018.

== Analytical non-singular displacement gradient formulation ==

It can be shown that the deformation gradient <math>G_{ml} = u_{m,l}</math> produced by a dislocation loop <math>C</math> with Burgers vector <math>\vec{b}</math> can be written as (see Bertin and Cai, CMS, 2018):

{|border="0" align="center"
|<math>
\begin{align}
u_{m,l}(\vec{x}) = &-\frac{1}{8\pi} \oint_C b_m \epsilon_{jlk} R_{,ppj} dx'_k -\frac{1}{8\pi} \oint_C b_i \epsilon_{mik} R_{,ppl} \, dx'_k \\
&-\frac{1}{8\pi(1-\nu)} \oint_C b_i \epsilon_{ijk} R_{,mjl} \, dx'_k
\end{align}
</math>
|}

where <math>R = \|\vec{x'}-\vec{x}\|</math> is the norm of the distance vector linking the field point <math>\vec{x}</math> and the coordinate <math>\vec{x'}</math> spanning the dislocation line, <math>\epsilon_{ijk}</math> is the permutation tensor, and <math>\nu</math> is Poisson's ratio of the medium. <math>R_{,ijk} = \partial^3 R / \partial x_i\partial x_j\partial x_k</math> denotes the third derivative of the radius vector wrt. the field coordinate.

Following the isotropic Burgers distribution proposed in Cai et al., JMPS, 2006, the singularity in the above expression can be eliminated by employing the modified radius vector <math>R_a</math> defined as:

{|border="0" align="center"
|<math>
R_a = \sqrt{R^2 + a^2} = \sqrt{R_i R_i + a^2} = \sqrt{(x'_i - x_i)(x'_i - x_i) + a^2}
</math>
|}

where <math>a</math> denotes the dislocation core radius. With this, the displacement gradient produced by a straight dislocation segment with end points <math>\vec{x}_a</math> and <math>\vec{x}_b</math> can be analytically expressed as:

{|border="0" align="center"
|<math>
\begin{align}
u_{m,l}(\vec{x}) = &-\frac{1}{8\pi} b_m \epsilon_{jlk} A_{jk}(\vec{x}) -\frac{1}{8\pi} b_i \epsilon_{mik} A_{lk}(\vec{x}) \\
&-\frac{1}{8\pi(1-\nu)} b_i \epsilon_{ijk} B_{jklm}(\vec{x})
\end{align}
</math>
|}

where line integrals <math>A_{jk}(\vec{x})</math> and <math>B_{jklm}(\vec{x})</math> are given by:

{|border="0" align="center"
|<math>
A_{jk}(\vec{x}) = t_k \left[ 3a^2 d_j J_{05} + 2d_j J_{03} + 3a^2 t_j J_{15} + 2t_j J_{13} \right]
</math>
|}
{|border="0" align="center"
|<math>
\begin{align}
B_{jklm}(\vec{x}) = &t_k \left[ (\delta_{mj}d_l + \delta_{jl}d_m + \delta_{lm}d_j)J_{03} + (\delta_{mj}t_l + \delta_{jl}t_m + \delta_{lm}t_j)J_{13} - 3(d_m d_j d_l)J_{05} \right. \\
&\left. -3(d_m d_j t_l + d_m t_j d_l + t_m d_j d_l)J_{15} - 3(d_m t_j t_l + t_m d_j t_l + t_m t_j d_l)J_{25} -3(t_m t_j t_l)J_{35} \right]
\end{align}
</math>
|}

where <math>\vec{t}</math> denotes the unit dislocation line tangent, and <math>\vec{d} = \vec{x}_0-\vec{x}</math> is the vector linking field point <math>\vec{x}</math> to its orthogonal projection <math>\vec{x}_0</math> on the dislocation line.

When adopting the following segment parametric representation

{|border="0" align="center"
|<math>
\vec{x'} = \vec{x}_0 + s\vec{t}, \; s \in (s_1,s_2)
</math>
|}

an analytical expression for the line integrals <math>J_{ij}</math> is obtained as follows:

{|border="0" align="center"
|<math>
\begin{align}
&J_{03} = \int_{s_1}^{s_2} \frac{1}{R_a^3} ds = \left. \frac{s}{( \vec{d} \cdot \vec{d} + a^2 ) R_a } \right|_{s_1}^{s_2} \\
&J_{13} = \int_{s_1}^{s_2} \frac{s}{R_a^3} ds = \left. -\frac{1}{R_a} \right|_{s_1}^{s_2} \\
&J_{05} = \int_{s_1}^{s_2} \frac{1}{R_a^5} ds = \left. \frac{2s^3}{3( \vec{d} \cdot \vec{d} + a^2 )^2 R_a^3 } \right|_{s_1}^{s_2} + \left. \frac{s}{( \vec{d} \cdot \vec{d} + a^2 ) R_a^3 } \right|_{s_1}^{s_2} \\
&J_{15} = \int_{s_1}^{s_2} \frac{s}{R_a^5} ds = \left. -\frac{1}{3 R_a^3} \right|_{s_1}^{s_2} \\
&J_{25} = \int_{s_1}^{s_2} \frac{s^2}{R_a^5} ds = \left. \frac{s^3}{3( \vec{d} \cdot \vec{d} + a^2 ) R_a^3 } \right|_{s_1}^{s_2} \\
&J_{35} = \int_{s_1}^{s_2} \frac{s^3}{R_a^5} ds = \left. -\frac{2(\vec{d} \cdot \vec{d} + a^2)}{3 R_a^3} \right|_{s_1}^{s_2} -\left. \frac{s^2}{R_a^3 } \right|_{s_1}^{s_2}
\end{align}
</math>
|}

== Matlab implementation ==

=== Code ===

- based on DDLab format
- functions descreption

=== Validation: tringular loop test case ===

- what the test case does

DDD-XRD Matlab implementation and validation

2018-01-20T00:53:06Z

Nbertin:

DDD-XRD Matlab implementation and validation

2018-01-20T00:28:13Z

Nbertin: Created page with "<P ALIGN="CENTER">DDD-XRD approach</P> <P ALIGN="CENTER"><FONT SIZE="+2" color="darkred"><STRONG> DDD-XRD: Matlab implementation and validation</STRONG></font></P> <DIV> <P AL..."

ParaDiS tools

2018-01-19T22:14:17Z

Nbertin: Created page with "<H4>Codes</H4> <UL> <LI>DDD-XRD Matlab implementation and validation </UL>"

<H4>Codes</H4>
<UL>
<LI>[[DDD-XRD Matlab implementation and validation]]
</UL>

Tutorials

2018-01-19T22:10:32Z

Nbertin: /* Simulation Codes */

== Simulation Codes ==

{| class="wikitable" cellpadding="2"
|-
!width="200" | [[MD++ Manuals]]
!width="200" | [[ParaDiS Manuals]]
!width="300" | [[ParaDiS Workshop Notes]]
|-
!width="200" | [[DDLab Manuals]]
!width="200" | [[VASP Manuals]]
!width="300" | [[How to compile Qbox]]
|-
!width="200" | [[NetworCh Manuals]]
!width="200" | [[How to compile LAMMPS]]
!width="300" | [[How to compile pimc++]]
|-
!width="200" | [[DDD-XRD Manuals]]
!width="200" |
!width="300" |
|}

== Computers ==

{| class="wikitable" cellpadding="2"
|-
!width="200" | [[Unix Basics]]
!width="200" | [[Computer Setup]]
!width="200" | [[Computing Clusters | Parallel Clusters]]
|-
!width="200" | [[Install Ubuntu]]
!width="200" | [[Install FFTW3]]
!width="200" | [[Install HDF5]]
|-
!width="200" | [[Install GSL]]
!width="200" | [[Standard Tcl Library | Tcl Library]]
!width="200" | [[Install the GNU Compiler Collection (GCC)]]
|-
!width="200" | [[Running Hybrid MPI/OpenMP Simulations]]
!width="200" | [[Install SHTOOLS]]
|}

== Scientific Background ==

{| class="wikitable" cellpadding="2"
|-
!width="200" | [[Dislocations]]
|}

== Science Outreach ==

{| class="wikitable" cellpadding="2"
|-
!width="200" | [http://www.youtube.com/watch?v=SgM-Xes16Sw Outreach interview]
!width="200" | [[How to install MD++ in Ubuntu | Install MD++ in Ubuntu]]
!width="200" | [[Introduction to Molecular Dynamics Simulations of Fullerenes | MD of Fullerenes]]
|-
!width="200" | [[Images of Fullerenes]]
!width="200" | [[Android Applications and Information]]
|

|}

<br> <br>

== [[Tutorial:Members_Only | Members Only]] ==

Before You Start: Known Bugs in ParaDiS

2017-04-27T23:03:37Z

Nbertin:

This page lists the current known bugs in the different ParaDiS public releases.

== General bugs ==

=== Bug 1. Cell ownership ===
As the numbers of
cells and numbers of processors are changed, ownership or nodes/segments can
alter which may result in segments changing the cells into which they
are accounted... and hence alter results. So for a small test case, forces may be perfectly correct in serial but slightly off in parallel using cells (option FULL_N2Forces turned off).

On a large simulation the results should be insignificant, but on small simulations with very few
dislocations, differences are more noticable.

=== Bug 2. Segfault with X-window ===
Not really a bug, but a suggestion. When an X-window cannot be opened sucessfully the code exits with a segmentation fault. Should be able to add a descriptive message and a break instead.

=== Bug 3. Input files created in DOS system ===
When ParaDiS input files (control and data files) are created in a DOS system (MS-DOS or Windows), the input files sometimes cannot be executed even though the files look okay in your editor of your linux system. For example, when you create input files using 'create_dislocation.f' in your DOS (or Windows) system, the hidden character, which is ^M, is embedded at the end of each line in your files. This character is sometimes shown or not shown in your editor. If it is shown, the input files look as the following.

<blockquote style="background: white; border: 0; padding: rem; width: 625px;"><pre>
### ParaDis nodal file (create_dislocation.f)^M
### S.A. (10/29/07)###^M
^M
#File version number^M
2^M
^M
#Number of data file segments^M
1^M
#Minimum coordinate values (x, y, z)^M
-18348.6238532110 -18348.6238532110 -18348.6238532110^M
#Maximum coordinate values (x, y, z)^M
18348.6238532110 18348.6238532110 18348.6238532110^M
^M
#Node count^M
54^M
#Domain geometry (x, y, z)^M
1 1 1^M
...
...
</pre></blockquote>

When the character ^M is not shown, it is frustrating because the input files look perfect, ParaDiS canot execute these inputs with ^Ms and shows the errors such as

<blockquote style="background: white; border: 0; padding: rem; width: 625px;"><pre>
...
Error: Expected '=' in parameter assignment
Fatal: Error obtaining values for parameter ) is produced.
...
</pre></blockquote>

We can remove this hidden character ^Ms simply using the command "<tt>dos2unix</tt>". This command erase all ^Ms in your files. Just run <tt>dos2unix (your-input-file-name)</tt>, Then, the input files can be executed.

=== Bug 4. White space in makefile.setup ===

Make sure there is no white space after your specification of the 'SYS' variable, e.g.

SYS = gcc

If there is a white space following gcc then the code won't compile.

== ParaDiS v2.5.1 ==

=== Bug 1. FCC_0 Mobility law with useLabFrame ===

A bug in ParaDiS v2.5.1 is known to prevent dislocation motion when the mobility <tt>FCC_0</tt> is used and the crystal is rotated using option <tt>useLabFrame = 1</tt> (Note: there is no bug when the crystal is not rotated).

To fix this bug, lines 111-112 in file <tt>src/MobilityLaw_FCC_0.c</tt>

<pre>
if ( (fabs(fabs(node->nx[i]) - fabs(node->ny[i])) > FFACTOR_NORMAL) ||
(fabs(fabs(node->ny[i]) - fabs(node->nz[i])) > FFACTOR_NORMAL) )
</pre>

need to be replaced by the following lines:

<pre>
if ( (fabs(fabs(normX[i]) - fabs(normY[i])) > FFACTOR_NORMAL) ||
(fabs(fabs(normY[i]) - fabs(normZ[i])) > FFACTOR_NORMAL) )
</pre>

=== Bug 2. Cross-slip in FCC ===

A bug in ParaDiS v2.5.1 affects the cross-slip procedure in FCC crystals when cross-slip is enabled (default). The bug arises from the fact that the <tt>shearModulus</tt> parameter controlling the threshold for noise level is not initialized within the <tt>CrossSlipFCC</tt> function.

To fix this bug, the following line must be added after line 78 in file <tt>src/CrossSlipFCC.c</tt>

<pre>
shearModulus = param->shearModulus;
</pre>

Once corrected, lines 74-79 in file <tt>src/CrossSlipFCC.c</tt> should look like

<pre>
eps = 1.0e-06;
thetacrit = 2.0 / 180.0 * M_PI;
sthetacrit = sin(thetacrit);
s2thetacrit = sthetacrit * sthetacrit;
areamin = param->remeshAreaMin;
shearModulus = param->shearModulus;
</pre>

== ParaDiS v2.3.5 ==

=== Bug 1. Gcc with v2.3.5 ===
Segmentation fault occurs with v2.3.5s and v.2.3.5.1 when compiling with GCC on Ubuntu and running codes with FMM. The error when compiling using <tt>-03</tt> optimization in serial or parallel (<tt>cc -O3</tt> or <tt>mpicc -03</tt>). Appears to be specific to <tt>-03</tt> and doesn't occur for <tt>-02</tt> or <tt>-g</tt>, for example. <tt>-03</tt> works properly when using Rijm tables.

This doesn't occur with the older version of ParaDiS in the subversion (~2.2.3 or 2.2.6?).

Details of error:
<blockquote style="background: white; border: 0; padding: rem; width: 625px;"><pre>
Initialize: Control file parsing complete
Generating uniform domain decomposition.

Program received signal SIGSEGV, Segmentation fault.
0x00007ffff732896d in ?? () from /lib/libc.so.6
(gdb) backtrace
#0 0x00007ffff732896d in ?? () from /lib/libc.so.6
#1 0x00007ffff732b424 in calloc () from /lib/libc.so.6
#2 0x0000000000448911 in ReadNodeDataFile ()
#3 0x000000000042dcd4 in Initialize ()
#4 0x000000000040e321 in ParadisInit ()
#5 0x000000000040185a in main ()
</pre></blockquote>

== ParaDiS v2.0 ==

=== Bug 1. Serial mode in v2.0 ===

If you download the standard distribution of ParaDiS <tt>pub-dd3d.v2.0.tar.gz</tt>, you will not be able to compile and run it in SERIAL mode. The fix to this problem is described below.

You need to download these three files (available on [http://paradis.stanford.edu/downloads ParaDiS download site]):

makefile.sys
makefile.setup
ReadConfig.c

This will allow you to use <tt>SYS = i386</tt> or <tt>SYS = cygwin</tt> and <tt>MODE = SERIAL</tt> in <tt>makefile.setup</tt>.

The <tt>SYS = i386</tt> option requires intel compiler <tt>icc</tt>. If you do not have this compiler, you can modify the <tt>makefile.sys</tt> file to use a different compiler.

We are working to enable the <tt>SYS = mac</tt> option. Please come back soon.

If your computer does not allow opening of X-window, then you need to set <tt>enable_window = 0</tt> in your <tt>win.script</tt> file. This is also necessary if you submit your job to a queue in a cluster, because you won't be able to open an interactive window there.

Installing and using the igraph package in R

2017-01-20T22:03:03Z

Nbertin: /* MC2 */

NetworCh manuals: Getting started
= Installing and using the igraph package in R =
The <tt>igraph</tt> package is a powerful library for analyzing, manipulating and vizualizing graphs and networks in <tt>R</tt>. Specifically, it implements advanced functions, such as graph isomorphism, that are not yet available in the NetworCh matlab library.

== Installation ==

=== MC2 ===
<tt>R</tt> is installed on mc2. To run <tt>R</tt>, simply load the module and type <tt>R</tt> in the terminal:
<pre>
module load R/3.1.2
R
</pre>

The <tt>igraph</tt> package is a library that is not natively available with <tt>R</tt>. To install it, go to your <tt>Codes/</tt> directory and create a new directory called <tt>R-packages/</tt>:
<pre>
cd Codes/
mkdir R-packages
</pre>

The installation of the <tt>igraph</tt> package then proceeds within <tt>R</tt> using the <tt>install.packages()</tt> command:
<pre>
install.packages("igraph", lib="~/Codes/R-packages/")
</pre>

The package will be downloaded from the selected mirror and then directly compiled on mc2 (~5 minutes). Once the installtion is finished, the package can be loaded from <tt>R</tt> using the <tt>library()</tt> command and specifying the path to the installation directory:
<pre>
library("igraph", lib.loc="~/Codes/R-packages/")
</pre>

Installing and using the igraph package in R

2017-01-20T22:01:46Z

Nbertin: Created page with "NetworCh manuals: Getting started = Installing and using the igraph package in R = The <tt>igraph</tt> package is a powerful library for analyzing, manipulating and vizualizi..."

NetworCh manuals: Getting started
= Installing and using the igraph package in R =
The <tt>igraph</tt> package is a powerful library for analyzing, manipulating and vizualizing graphs and networks in <tt>R</tt>. Specifically, it implements advanced functions, such as graph isomorphism, that are not yet available in the NetworCh matlab library.

== Installation ==

=== MC2 ===
<tt>R</tt> is installed on mc2. To run <tt>R</tt>, simply load the module and type <tt>R</tt> in the terminal:
<pre>
module load R/3.12
R
</pre>

The <tt>igraph</tt> package is a library that is not natively available with <tt>R</tt>. To install it, go to your <tt>Codes/</tt> directory and create a new directory called <tt>R-packages/</tt>:
<pre>
cd Codes/
mkdir R-packages
</pre>

The installation of the <tt>igraph</tt> package then proceeds within <tt>R</tt> using the <tt>install.packages()</tt> command:
<pre>
install.packages("igraph", lib="~/Codes/R-packages/")
</pre>

The package will be downloaded from the selected mirror and then directly compiled on mc2 (~5 minutes). Once the installtion is finished, the package can be loaded from <tt>R</tt> using the <tt>library()</tt> command and specifying the path to the installation directory:
<pre>
library("igraph", lib.loc="~/Codes/R-packages/")
</pre>

NetworCh Manuals

2017-01-20T21:59:56Z

Nbertin: /* Getting started */

= NetworCh library manuals =

NetworCh is a library, mainly written in Matlab, developed to manipulate, visualize, analyze and characterize network structures in various areas of mechanics of materials.

== Getting started ==
[[How to obtain NetworCh]] <br>
[[Introduction and examples]] <br>
[[Documentation]] <br>
[[Installing and using the igraph package in R]]

== Dislocation dynamics ==
[[DD structure generation]] <br>
[[DD irreducible core graph]]

== CGSD ==
[[Shortest paths PBC calculation]]

== Network analysis ==

Installing and using R with the igraph package

2017-01-09T16:48:22Z

Nbertin: /* MC2 */

NetworCh manuals: Getting started
= Installing and using the igraph package in R =
The <tt>igraph</tt> package is a powerful library for analyzing, manipulating and vizualizing graphs and networks in <tt>R</tt>. Specifically, it implements advanced functions, such as graph isomorphism, that are not yet available in the NetworCh matlab library.

== Installation ==

=== MC2 ===
<tt>R</tt> is installed on mc2. To run <tt>R</tt>, simply load the module and type <tt>R</tt> in the terminal:
<pre>
module load R/3.12
R
</pre>

The <tt>igraph</tt> package is a library that is not natively available with <tt>R</tt>. To install it, go to your <tt>Codes/</tt> directory and create a new directory called <tt>R-packages/</tt>:
<pre>
cd Codes/
mkdir R-packages
</pre>

The installation of the <tt>igraph</tt> package then proceeds within <tt>R</tt> using the <tt>install.packages()</tt> command:
<pre>
install.packages("igraph", lib="~/Codes/R-packages/")
</pre>

The package will be downloaded from the selected mirror and then directly compiled on mc2 (~5 minutes). Once the installtion is finished, the package can be loaded from <tt>R</tt> using the <tt>library()</tt> command and specifying the path to the installation directory:
<pre>
library("igraph", lib.loc="~/Codes/R-packages/")
</pre>

Installing and using R with the igraph package

2017-01-09T15:54:22Z

NetworCh manuals: Getting started
= Installing and using the igraph package in R =
The <tt>igraph</tt> package is a powerful library for analyzing, manipulating and vizualizing graphs and networks in <tt>R</tt>. Specifically, it implements advanced functions, such as graph isomorphism, that are not yet available in the NetworCh matlab library.

== Installation ==

=== MC2 ===
<tt>R</tt> is installed on mc2. To run <tt>R</tt>, simply load the module and type <tt>R</tt> in the terminal:
<pre>
module load R/3.12
R
</pre>

The <tt>igraph</tt> package is a library that is not natively available with <tt>R</tt>. To install it, go to your <tt>Codes/</tt> directory and create a new directory called <tt>R-packages/</tt>:
<pre>
cd Codes/
mkdir R-packages
</pre>

The installation of the <tt>igraph</tt> package then proceeds within <tt>R</tt> using the <tt>install.packages()</tt> command:
<pre>
install.packages("igraph", lib="~/Codes/R-packages/")
</pre>

The package will be downloaded from the selected mirror and then directly compiled on mc2 (~5 minutes). Once the installtion is finished, the package can be loaded from <tt>R</tt> using the <tt>library()</tt> command and specifying the path to the installation directory:
<pre>
install.packages("igraph", lib="~/Codes/R-packages/")
</pre>

NetworCh Manuals

2017-01-09T15:29:54Z

Nbertin: /* Getting started */

= NetworCh library manuals =

NetworCh is a library, mainly written in Matlab, developed to manipulate, visualize, analyze and characterize network structures in various areas of mechanics of materials.

== Getting started ==
[[How to obtain NetworCh]] <br>
[[Introduction and examples]] <br>
[[Documentation]] <br>
[[Installing and using R with the igraph package]]

== Dislocation dynamics ==
[[DD structure generation]] <br>
[[DD irreducible core graph]]

== CGSD ==
[[Shortest paths PBC calculation]]

== Network analysis ==

NetworCh Manuals

2017-01-09T15:29:18Z

Nbertin: /* Getting started */

= NetworCh library manuals =

NetworCh is a library, mainly written in Matlab, developed to manipulate, visualize, analyze and characterize network structures in various areas of mechanics of materials.

== Getting started ==
[[How to obtain NetworCh]] <br>
[[Introduction and examples]] <br>
[[Documentation]]
[[Installing and using R with the igraph package]]

== Dislocation dynamics ==
[[DD structure generation]] <br>
[[DD irreducible core graph]]

== CGSD ==
[[Shortest paths PBC calculation]]

== Network analysis ==