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	timprove documentation - slidergrid - grid of elastic sliders on a frictional s…
	git clone git://src.adamsgaard.dk/slidergrid
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	---
	commit 01f9042427f6b9e5933c803f55b9fdafe5b4a33c
	parent 897289bb9cc3233e556ccd4b863cdfaf4dea29b0
	Author: Anders Damsgaard Christensen <[email protected]>
	Date: Wed, 4 May 2016 13:10:02 -0700

	improve documentation

	Diffstat:
	M doc/doc.pdf \| 0
	M doc/doc.tex \| 295 +++++++++++++++--------------…
	M slidergrid/grid.c \| 29 +++++++++++++++++++++++++++++
	M slidergrid/slider.h \| 1 +

	4 files changed, 175 insertions(+), 150 deletions(-)
	---
	diff --git a/doc/doc.pdf b/doc/doc.pdf
	Binary files differ.
	diff --git a/doc/doc.tex b/doc/doc.tex
	t@@ -19,7 +19,7 @@
	\usepackage[T1]{fontenc} % Font encoding
	\usepackage{charter} % Serif body font
	\usepackage[charter]{mathdesign} % Math font
	-\usepackage[scale=0.9]{sourcecodepro} % Monospaced fontenc
	+\usepackage[scale=0.8]{sourcecodepro} % Monospaced fontenc
	\usepackage[lf]{FiraSans} % Sans-serif font

	\usepackage{listings}
	t@@ -48,15 +48,24 @@ of Oceanography\\University of California, San Diego}\\[3m…
	\maketitle

	\section{Methods}
	-The method is derived from \citet{Schlangen1996}, \citet{Radjai2011} and
	-\citet{Potyondy2004} but is, relative to the cited works, adapted for three
	-spatial dimensions and non-linear properties.
	-
	-The Lagrangian nodes are connected with visco-elastic beam elements which are
	-resistive to relative translational or rotational movement. The kinematic
	-degrees of freedom are determined by explicit integration of Newton's second l…
	-of motion for translation and rotation. For a point $i$ with bonded
	-interactions to nodes $j\in N_c$, the translational accelerations
	+Our approach treats the short-temporal scale behavior of stick-slip as a
	+rigid-body dynamics problem. The material is represented as a discrete number
	+of Lagrangian points (\emph{nodes}) which are mechanically interacting with ea…
	+other and the boundary conditions.
	+
	+The Lagrangian nodes are connected with visco-elastic beam elements.
	+The bonds are resistive to tension and compression, shearing, twisting, and
	+bending, which ensures elastic uniformity regardless of geometric node
	+arrangement \citep{Bolander1998, Radjai2011}. Alternatively, the node
	+interaction could be parameterized as simple springs which exclusively provide
	+resistance to tension and compression, and resistance to shearing, bending and
	+twisting would be introduced by discretizing the elastic material into an
	+irregular network of many more nodes \citep[e.g.][]{Topin2007, Topin2009}. He…
	+we chose the former approach which allows us to keep the number of nodes low.
	+
	+The kinematic degrees of freedom are determined by explicit integration of
	+Newton's second law of motion for translation and rotation. For a point $i$
	+with bonded interactions to nodes $j\in N_c$, the translational accelerations
	($\boldsymbol{a}$) are found from the sums of forces:
	\begin{equation}
	\boldsymbol{a}_i =
	t@@ -85,24 +94,85 @@ torques:
	\boldsymbol{\alpha}_i =
	\sum^{N_c}_j
	\left[
	- \frac{\boldsymbol{t}^\text{s}_{i,j}}{I_i} +
	- \frac{\boldsymbol{t}^\text{t}_{i,j}}{J_{i,i}}
	+ \frac{\boldsymbol{t}^{i,j}_{\bar{x}}}{I^\text{p}_{i,j}} +
	+ \frac{\boldsymbol{t}^{i,j}_{\bar{y}}}{I^\text{n}_{i,j}} +
	+ \frac{\boldsymbol{t}^{i,j}_{\bar{z}}}{I^\text{n}_{i,j}}
	\right]
	\label{eq:n2-ang}
	\end{equation}
	here, $\boldsymbol{t}^\text{s}$ is the torque resulting from shearing motion o…
	the bond, while the torque $\boldsymbol{t}^{t}$ results from relative twisting.
	-$I_i$ is the local moment of inertia at the point, and $J_{i,j}$ is polar mome…
	-of inertia of the bond.
	+$I^\text{n}_{i,j}$ is the bond-normal mass moment of inertia at the point, and
	+$I^\text{p}$ is polar mass moment of inertia of the bond. The above equation
	+implies the simplifying assumption that the nodes are bonded in a configuratio…
	+with geometric symmetry, which is a good approximation inside the grid but
	+slightly worse at the grid edges.


	-At the beginning of each time step the accumulated strain on each inter-point
	-bond is determined by considering the relative motion of the bonded nodes. Th…
	-bond deformation is decomposed per kinematic degree of freedom, andis determin…
	-by an incremental method derived from \citet{Potyondy2004}. The strain can be
	-decomposed into bond tension and compression, bond shearing, bond twisting, an…
	-bond bending. The accumulated strains are used to determine the magnitude of
	-the forces and torques resistive to the deformation.
	+\subsection{Visco-elastic interaction between nodes}
	+The total force ($\boldsymbol{f}$) and torque ($\boldsymbol{t}$) on two nodes
	+($i$ and $j$) is found by determining the stress response of a three-dimension…
	+elastic Timoshenko beam to strain \citet{Schlangen1996, Austrell2004,
	+ Aastroem2013}.
	+In the following the forces and torques are described for node $i$ but are
	+of equal magnitude and with opposite sign for node $j$. The interaction
	+accounts for resistance to tension and compression, shear, torsion, and bendin…
	+The equations below are derived from the stiffness matrix in
	+\citet{Austrell2004}. The components for the three-dimensional force vector o…
	+node $i$ are:
	+\begin{equation}
	+ \begin{split}
	+ f_{\bar{x}}^i & = \frac{EA}{L}
	+ \left( p_{\bar{x}}^{,i} - p_{\bar{x}}^{,j} \right)\\
	+%
	+ f_{\bar{y}}^i & = \frac{12EI_A}{L^3}
	+ \left( p_{\bar{y}}^{,i} - p_{\bar{y}}^{,j} \right)
	+ + \frac{6EI_A}{L^2}
	+ \left( \Omega_{\bar{z}}^{,i} + \Omega_{\bar{z}}^{,j} \right)\\
	+%
	+ f_{\bar{z}}^i & = \frac{12EI_A}{L^3}
	+ \left( p_{\bar{z}}^{,i} - p_{\bar{z}}^{,j} \right)
	+ - \frac{6EI_A}{L^2}
	+ \left( \Omega_{\bar{y}}^{,i} + \Omega_{\bar{y}}^{,j} \right)
	+ \end{split}
	+\end{equation}
	+where $\bar{x}, \bar{y}, \bar{z}$ are the bond-relative coordinates.
	+The linear and angular relative displacement of the nodes is described by $p^*…
	+and $\Omega^*$. $E$ is Young's modulus, $G$ is the shear modulus, $A$ is the
	+beam cross-sectional area, and $L$ is the original beam length. $I_A$ is the
	+area moment of inertia of the beam ($I_A = a^4 12^{-1}$ where $a$ is the beam
	+side length). The torque on node $i$ is:
	+\begin{equation}
	+ \begin{split}
	+ t_{\bar{x}}^i & = \frac{GJ}{L}
	+ \left( \Omega_{\bar{x}}^{,i} + \Omega_{\bar{x}}^{,j} \right)\\
	+%
	+ t_{\bar{y}}^i & = \frac{6EI_A}{L^2}
	+ \left(p_{\bar{z}}^{,j} - p_{\bar{z}}^{,i} \right)
	+ + \frac{4EI_A}{L}
	+ \left( \Omega_{\bar{y}}^{,i} + \frac{\Omega_{\bar{y}}^{,j}}{2}
	+ \right)\\
	+%
	+ t_{\bar{z}}^i & = \frac{6EI_A}{L^2}
	+ \left(p_{\bar{y}}^{,i} - p_{\bar{y}}^{,j} \right)
	+ + \frac{4EI_A}{L}
	+ \left( \Omega_{\bar{z}}^{,i} + \frac{\Omega_{\bar{z}}^{,j}}{2}
	+ \right)\\
	+ \end{split}
	+\end{equation}
	+$GJ$ is the Saint-Venant torsional stiffness, where $J$ called the torsional
	+rigidity multiplier or torsion constant. For a beam with a solid square
	+cross-sectional shape it can be approximated as $J \approx 0.140577 a^4$ where
	+$a$ is the side length \citep{Timoshenko1951, Roark1954, Weisstein2016}.
	+
	+% Torsional constant:
	+% https://en.wikipedia.org/wiki/Torsion_constant
	+% http://mathworld.wolfram.com/TorsionalRigidity.html (K: K_v)
	+% http://physics.stackexchange.com/questions/83148/where-i-can-find-a-torsiona…
	+% St Venant torsion: K_v = 1/G (Austrell et al. 2004, table 3) Does it make s…
	+% chapter4.pdf: K_v = J
	+% https://skyciv.com/free-moment-of-inertia-calculator/
	+

	The deformation and reactive forces are determined relative to the orientation
	of the bond. Common geometrical vectors include the inter-distance vector
	t@@ -137,82 +207,7 @@ orientation, e.g.\ the bond-parallel and bond-shear veloc…
	\right)
	\end{equation}

	-The axial strain is the bond-parallel deformation and is determined as the
	-change in inter-point length relative to the initial distance:
	-\begin{equation}
	- \epsilon_a = \frac{
	- (\boldsymbol{d}_{i,j} - \boldsymbol{d}^0_{i,j}) \cdot n_{i,j}}
	- {\|\|\boldsymbol{d}^0_{i,j}\|\|}
	-\end{equation}
	-The cross-sectional area of a bond ($A_{i,j}$) varies with axial strain
	-($\epsilon_a$) scaled by Poissons ratio $\nu$:
	-\begin{equation}
	- A_{i,j} = A^0_{i,j}
	- - A^0_{i,j}
	- \left(
	- 1 -
	- \left(
	- 1 + \epsilon_a
	- \right)^{-\nu}
	- \right)
	-\end{equation}
	-The mass of point $i$ is defined as the half of the mass of each of its bonds:
	-\begin{equation}
	- m_i = \frac{\rho}{2} \sum^{N_c}_j A^0_{i,j} \|\|\boldsymbol{d}^0_{i,j}\|\|
	-\end{equation}
	-The density ($\rho$) is adjusted so that the total mass of all nodes matches t…
	-desired value.
	-

	-\subsection{Resistance to tension and compression}
	-Bond tension and compression takes place when the relative translational
	-distance between a pair of bonded nodes changes, and is the most important
	-deformational mode in this model. The current axial strain is determined with…
	-second-order central difference scheme. It is determined from the previous
	-point positions and projected future positions:
	-\begin{equation}
	- \Delta d^t_{i,j} = \frac{d_{i,j}^{*,t+\Delta t} - d_{i,j}^{t-\Delta t}}{2}
	-\end{equation}
	-The future point distance in the above ($d_{i,j}^{*,t+\Delta}$) is found by
	-applying a second-order Taylor expansion:
	-\begin{equation}
	- \boldsymbol{p}_i^{*,t+\Delta t} =
	- \boldsymbol{p}_i^{t} +
	- \boldsymbol{v}_i^{t} \Delta t +
	- \frac{1}{2}\boldsymbol{a}_i^{t} \Delta t^2
	-\end{equation}
	-
	-
	-
	-The bond-parallel force is determined from Young's modulus ($E$) and the
	-cross-sectional area ($A_{i,j}$) of the bond:
	-\begin{equation}
	- \boldsymbol{f}^{i,j}_\text{p} =
	- \frac{E A_{i,j}}{\|\| \boldsymbol{d}^0_{i,j} \|\|}
	- \left(
	- \boldsymbol{d}_{i,j} -
	- \boldsymbol{d}^0_{i,j}
	- \right)
	-\end{equation}
	-
	-\subsection{Shear resistance}
	-The bond-shear force is determined incrementally for the duration of the
	-interaction:
	-\begin{equation}
	- \boldsymbol{f}^{i,j}_\text{s} = \int^t \Delta \boldsymbol{f}^{i,j}_\text{s…
	- %\, dt
	-\end{equation}
	-where the increment in shear force is determined from the shear modulus ($G$),
	-the cross-sectional area ($A_{i,j}$) of the bond, and the
	-\begin{equation}
	- \Delta \boldsymbol{f}^{i,j}_\text{s} =
	- \frac{G A_{i,j}}{\|\|\boldsymbol{d}^{i,j}\|\|}
	- \Delta \boldsymbol{d}^{i,j}_\text{s}
	-\end{equation}
	-
	-\subsection{Twisting resistance}
	-
	-\subsection{Bending resistance}

	\subsection{Temporal integration}
	Once the force and torque sum components at time $t$ have been determined, the
	t@@ -278,71 +273,71 @@ stiffness in the system relative to the smallest mass:
	where $\epsilon$ is a safety factor related to the geometric structure of the
	bonded network. We use $\epsilon = 0.07$.

	-The total force ($\boldsymbol{f}$) and torque ($\boldsymbol{t}$) on two nodes
	-($i$ and $j$) with translational ($\boldsymbol{p}$) and angular
	-($\boldsymbol{\Omega}$) positions interconnected with a three-dimensional
	-elastic beam can be expressed as the following set of equations. The
	-interaction accounts for resistance to tension and compression, shear, torsion…
	-and bending. The symmetrical matrix on the right hand side constitutes the
	-\emph{stiffness matrix} \citep{Schlangen1996, Austrell2004}:
	+
	+\appendix
	+\section{Stiffness matrix}
	\begin{equation}
	\begin{bmatrix}
	- f_\text{x}^i\\[0.6em]
	- f_\text{y}^i\\[0.6em]
	- f_\text{z}^i\\[0.6em]
	- t_\text{x}^i\\[0.6em]
	- t_\text{y}^i\\[0.6em]
	- t_\text{z}^i\\[0.6em]
	- f_\text{x}^j\\[0.6em]
	- f_\text{y}^j\\[0.6em]
	- f_\text{z}^j\\[0.6em]
	- t_\text{x}^j\\[0.6em]
	- t_\text{y}^j\\[0.6em]
	- t_\text{z}^j\\
	+ f_{\bar{x}}^i\\[0.6em]
	+ f_{\bar{y}}^i\\[0.6em]
	+ f_{\bar{z}}^i\\[0.6em]
	+ t_{\bar{x}}^i\\[0.6em]
	+ t_{\bar{y}}^i\\[0.6em]
	+ t_{\bar{z}}^i\\[0.6em]
	+ f_{\bar{x}}^j\\[0.6em]
	+ f_{\bar{y}}^j\\[0.6em]
	+ f_{\bar{z}}^j\\[0.6em]
	+ t_{\bar{x}}^j\\[0.6em]
	+ t_{\bar{y}}^j\\[0.6em]
	+ t_{\bar{z}}^j\\
	\end{bmatrix}
	=
	\begin{bmatrix}
	\frac{EA}{L} & 0 & 0 & 0 & 0 & 0 & \frac{-EA}{L} & 0 & 0 & 0 & 0 & 0\\…
	- 0 & \frac{12EI_\text{z}}{L^3} & 0 & 0 & 0 & \frac{6EI_\text{z}}{L^2} &…
	- 0 & 0 & \frac{12EI_\text{y}}{L^3} & 0 & \frac{-6EI_\text{y}}{L^2} & 0 …
	- 0 & 0 & 0 & \frac{GK_\text{v}}{L} & 0 & 0 & 0 & 0 & 0 & \frac{-GK_\tex…
	- 0 & 0 & \frac{-6EI_\text{y}}{L^2} & 0 & \frac{4EI_\text{y}}{L} & 0 & 0…
	- 0 & \frac{6EI_\text{z}}{L^2} & 0 & 0 & 0 & \frac{4EI_\text{z}}{L} & 0 …
	+ 0 & \frac{12EI_{\bar{z}}}{L^3} & 0 & 0 & 0 & \frac{6EI_{\bar{z}}}{L^2}…
	+ 0 & \frac{-12EI_{\bar{z}}}{L^3} & 0 & 0 & 0 &
	+ \frac{6EI_{\bar{z}}}{L^2}\\[0.5em]
	+ 0 & 0 & \frac{12EI_{\bar{y}}}{L^3} & 0 & \frac{-6EI_{\bar{y}}}{L^2} & …
	+ & 0 & 0 & \frac{-12EI_{\bar{y}}}{L^3} & 0 & \frac{-6EI_{\bar{y}}}{L^2}…
	+ 0\\[0.5em]
	+ 0 & 0 & 0 & \frac{GJ}{L} & 0 & 0 & 0 & 0 & 0 & \frac{-GJ}{L} & 0 &
	+ 0\\[0.5em]
	+ 0 & 0 & \frac{-6EI_{\bar{y}}}{L^2} & 0 & \frac{4EI_{\bar{y}}}{L} & 0 &…
	+ & 0 & \frac{6EI_{\bar{y}}}{L^2} & 0 & \frac{2EI_{\bar{y}}}{L} &
	+ 0\\[0.5em]
	+ 0 & \frac{6EI_{\bar{z}}}{L^2} & 0 & 0 & 0 & \frac{4EI_{\bar{z}}}{L} & …
	+ & \frac{-6EI_{\bar{z}}}{L^2} & 0 & 0 & 0 &
	+ \frac{2EI_{\bar{z}}}{L}\\[0.5em]
	\frac{-EA}{L} & 0 & 0 & 0 & 0 & 0 & \frac{EA}{L} & 0 & 0 & 0 & 0 & 0\\…
	- 0 & \frac{-12EI_\text{z}}{L^3} & 0 & 0 & 0 & \frac{-6EI_\text{z}}{L^2}…
	- 0 & 0 & \frac{-12EI_\text{y}}{L^3} & 0 & \frac{-6EI_\text{y}}{L^2} & 0…
	- 0 & 0 & 0 & \frac{-GK_\text{v}}{L} & 0 & 0 & 0 & 0 & 0 & \frac{GK_\tex…
	- 0 & 0 & \frac{-6EI_\text{y}}{L^2} & 0 & \frac{2EI_\text{y}}{L} & 0 & 0…
	- 0 & \frac{6EI_\text{z}}{L^2} & 0 & 0 & 0 & \frac{2EI_\text{z}}{L} & 0 …
	+ 0 & \frac{-12EI_{\bar{z}}}{L^3} & 0 & 0 & 0 & \frac{-6EI_{\bar{z}}}{L^…
	+ & 0 & \frac{12EI_{\bar{z}}}{L^3} & 0 & 0 & 0 &
	+ \frac{-6EI_{\bar{z}}}{L^2}\\[0.5em]
	+ 0 & 0 & \frac{-12EI_{\bar{y}}}{L^3} & 0 & \frac{-6EI_{\bar{y}}}{L^2} &…
	+ & 0 & 0 & \frac{12EI_{\bar{y}}}{L^3} & 0 & \frac{6EI_{\bar{y}}}{L^2} &
	+ 0\\[0.5em]
	+ 0 & 0 & 0 & \frac{-GJ}{L} & 0 & 0 & 0 & 0 & 0 & \frac{GJ}{L} & 0 &
	+ 0\\[0.5em]
	+ 0 & 0 & \frac{-6EI_{\bar{y}}}{L^2} & 0 & \frac{2EI_{\bar{y}}}{L} & 0 &…
	+ & 0 & \frac{6EI_{\bar{y}}}{L^2} & 0 & \frac{4EI_{\bar{y}}}{L} &
	+ 0\\[0.5em]
	+ 0 & \frac{6EI_{\bar{z}}}{L^2} & 0 & 0 & 0 & \frac{2EI_{\bar{z}}}{L} & …
	+ & \frac{-6EI_{\bar{z}}}{L^2} & 0 & 0 & 0 & \frac{4EI_{\bar{z}}}{L}\\
	\end{bmatrix}
	\begin{bmatrix}
	- p_\text{x}^i\\[0.6em]
	- p_\text{y}^i\\[0.6em]
	- p_\text{z}^i\\[0.6em]
	- \Omega_\text{x}^i\\[0.6em]
	- \Omega_\text{y}^i\\[0.6em]
	- \Omega_\text{z}^i\\[0.6em]
	- p_\text{x}^j\\[0.6em]
	- p_\text{y}^j\\[0.6em]
	- p_\text{z}^j\\[0.6em]
	- \Omega_\text{x}^j\\[0.6em]
	- \Omega_\text{y}^j\\[0.6em]
	- \Omega_\text{z}^j\\
	+ p_{\bar{x}}^i\\[0.6em]
	+ p_{\bar{y}}^i\\[0.6em]
	+ p_{\bar{z}}^i\\[0.6em]
	+ \Omega_{\bar{x}}^i\\[0.6em]
	+ \Omega_{\bar{y}}^i\\[0.6em]
	+ \Omega_{\bar{z}}^i\\[0.6em]
	+ p_{\bar{x}}^j\\[0.6em]
	+ p_{\bar{y}}^j\\[0.6em]
	+ p_{\bar{z}}^j\\[0.6em]
	+ \Omega_{\bar{x}}^j\\[0.6em]
	+ \Omega_{\bar{y}}^j\\[0.6em]
	+ \Omega_{\bar{z}}^j\\
	\end{bmatrix}
	\end{equation}
	-$E$ is Young's modulus, $G$ is the shear stiffnes, $A$ is the beam
	-cross-sectional area, and $L$ is the original beam length. $I_\text{y}$ is the
	-moment of inertia normal to the beam in the $\bar{y}$-direction, and
	-$I_\text{z}$ is the moment of inertia normal to the beam in the
	-$\bar{z}$-direction. $K_\text{v}$ is the Saint-Venant torsional stiffness.
	-
	-% Torsional constant:
	-% https://en.wikipedia.org/wiki/Torsion_constant
	-% http://mathworld.wolfram.com/TorsionalRigidity.html
	-% http://physics.stackexchange.com/questions/83148/where-i-can-find-a-torsiona…
	-% St Venant torsion: K_v = 1/G (Austrell et al. 2004, table 3) Does it make s…
	-
	-



	diff --git a/slidergrid/grid.c b/slidergrid/grid.c
	t@@ -34,6 +34,35 @@ slider* create_regular_slider_grid(
	return sliders;
	}

	+slider* create_regular_slider_grid_with_randomness(
	+ const int nx,
	+ const int ny,
	+ const int nz,
	+ const Float dx,
	+ const Float dy,
	+ const Float dz)
	+{
	+ slider* sliders;
	+ sliders = malloc(sizeof(slider)nxny*nz);
	+
	+ int i = 0; int ix, iy, iz, j;
	+ for (iz = 0; iz < nz; iz++) {
	+ for (iy = 0; iy < ny; iy++) {
	+ for (ix = 0; ix < nx; ix++) {
	+ sliders[i].pos.x = dx * ix;
	+ sliders[i].pos.y = dy * iy;
	+ sliders[i].pos.z = dz * iz;
	+ i++;
	+
	+ for (j=0; j<MAX_NEIGHBORS; j++)
	+ sliders[i].neighbors[j] = -1;
	+ }
	+ }
	+ }
	+
	+ return sliders;
	+}
	+
	/* Find neighboring sliders within a defined cutoff distance */
	void find_and_bond_to_neighbors_n2(
	slider* sliders,
	diff --git a/slidergrid/slider.h b/slidergrid/slider.h
	t@@ -34,6 +34,7 @@ typedef struct {
	// Macroscopic mechanical properties
	Float youngs_modulus;
	Float shear_modulus;
	+ Float torsion_stiffness;

	// inter-slider bond-parallel Kelvin-Voigt contact model parameters
	Float bond_parallel_kv_stiffness; // Hookean elastic stiffness [N/m]