Remove unnecesary align and use X instead of chi for ALE coordinates

Author: hkjeldsberg

docs/_config.yml

@@ -17,6 +17,7 @@ execute:
 
 # Define the name of the latex output file for PDF builds
 latex:
+  latex_engine: xelatex
   latex_documents:
     targetname: book.tex
 
@@ -39,6 +40,7 @@ sphinx:
   config:
     html_last_updated_fmt: "%b %d, %Y"
     suppress_warnings: [ "mystnb.unknown_mime_type" ]
+    mathjax_path: https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js
 
 launch_buttons:
   notebook_interface: "jupyterlab"

docs/getting_started.md

@@ -37,7 +37,6 @@ When loading a problem file, OasisMove will start by looking inside the `problem
 look in the current working directory. If the problem you have specified after `problem=` is not located in either locations, an error will occur.
 ```
 
-## References
 
 ```{bibliography} references.bib
 :filter: docname in docnames

docs/index.md

@@ -3,9 +3,9 @@
 [OasisMove](https://github.com/KVSlab/OasisMove) is a high-level/high-performance open-source Navier-Stokes solver
 written in Python/[FEniCS](https://fenicsproject.org/), and is an extension of the computational fluid dynamics (CFD)
 solver [Oasis](https://github.com/mikaem/Oasis). In OasisMove the Navier-Stokes equations are expressed in the arbitrary
-Lagrangian-Eulerian formulation, which is suitable for handling moving domains. Through verification the solver has
+Lagrangian-Eulerian (ALE) formulation, which is suitable for handling moving domains. Through verification the solver has
 shown to follow theoretical convergence rates, begin second order accurate in time, and second and third order accurate
-in space with P1/P1 and P2/P1 finite elements. Through validation the solver has showed good agreement with existing
+in space with $\mathbb{P_1}/\mathbb{P_1}$ and $\mathbb{P_2}/\mathbb{P_1}$ finite elements. Through validation the solver has showed good agreement with existing
 benchmark results, and demonstrates the ability of the solver to capture vortex patterns in transitional and
 turbulent-like flow regimes.
 

docs/movingcylinder.md

@@ -36,18 +36,14 @@ frequency of the oscillations are defined as:
 
 ```{math}
 :label: eq-freq
-\begin{align}
-f =  \frac{St U_{\infty}F}{D}.
-\end{align}
+    f =  \frac{St U_{\infty}F}{D}.
 ```
 
 Furthermore, the cylinder position followed a sinusoidal profile in the transverse direction to the flow:
 
 ```{math}
 :label: eq-sine
-\begin{align}
-    \mathbf x(\mathbf \chi, t) =  (0, A \sin (2 \pi f t )), 
-\end{align}
+    \mathbf x( \mathbf X, t) =  (0, A \sin (2 \pi f t )), 
 ```
 
 where $A = A_{ratio}D$ is the amplitude.
@@ -78,9 +74,7 @@ $C_L$) coefficient defined as:
 
 ```{math}
 :label: eq-force
-\begin{align}
     C_D = \frac{2 F_D}{\rho U_\infty^2 D}, \qquad C_L = \frac{2F_L}{\rho U_\infty^2 D}, 
-\end{align}
 ```
 
 stored in the `forces.txt` file. In {numref}`vel-cylinder` we display the resulting velocity field, where we have added
@@ -127,12 +121,9 @@ Reynolds number by definition:
 
 ```{math}
 :label: eq-re
-\begin{align}
     \text{Re } = \frac{U_{\infty} D}{\nu}
-\end{align}
 ```
 
-## References
 
 ```{bibliography} references.bib
 :filter: docname in docnames

docs/movingtg.md

@@ -17,34 +17,30 @@ absence of body forces, $\mathbf{f} = 0$, namely:
 
 ```{math}
 :label: eq-tg3d
-\begin{align}
-    \mathbf u(\mathbf x,t = 0) &= (u(\mathbf x),v(\mathbf x),w(\mathbf x)\\
+    \mathbf u(\mathbf x,t = 0) &= (u(\mathbf x),v(\mathbf x),w(\mathbf x))\\
     u(\mathbf x) &= + \sin(x) \cos(y) \cos(z) \\
     v(\mathbf x) &= -\cos(x) \sin(y) \cos(z) \\
     w(\mathbf x) &= 0 \\
     p(\mathbf x,t=0) &= \frac{1}{16}(\cos(2x) + \cos(2y)) (\cos(2z) + 2).
-\end{align}
 ```
 
 The domain boundaries are prescribed a movement described by the following mesh motion from Fehn et al.
 {cite}`fehn2021high`:
 
 ```{math}
 :label: eq-tg3d-d
-    \begin{align}
-    \mathbf x(\mathbf \chi,t) &= \mathbf \chi + A_0 \sin \left(\frac{2 \pi t}{T_G}\right)
+    \mathbf x(\mathbf X,t) &= \mathbf X + A_0 \sin \left(\frac{2 \pi t}{T_G}\right)
         \begin{pmatrix}
-        \sin \left(2\pi \frac{\chi_2 + L / 2}{L}\right) \sin\left(2\pi \frac{\chi_3 + L / 2}{L}\right)  \\ 
-        \sin\left(2\pi \frac{\chi_1 + L / 2}{L}\right) \sin\left(2\pi \frac{\chi_3 + L / 2}{L}\right)    \\
-        \sin\left(2\pi \frac{\chi_1 + L / 2}{L}\right) \sin\left(2\pi \frac{\chi_2 + L / 2}{L}\right)    
+        \sin \left(2\pi \frac{X_2 + L / 2}{L}\right) \sin\left(2\pi \frac{X_3 + L / 2}{L}\right)  \\ 
+        \sin\left(2\pi \frac{X_1 + L / 2}{L}\right) \sin\left(2\pi \frac{X_3 + L / 2}{L}\right)    \\
+        \sin\left(2\pi \frac{X_1 + L / 2}{L}\right) \sin\left(2\pi \frac{X_2 + L / 2}{L}\right)    
         \end{pmatrix}\\
-        \mathbf w (\mathbf \chi, t) &= \frac{\partial \mathbf x}{\partial t}
-\end{align}
+        \mathbf w (\mathbf X, t) &= \frac{\partial \mathbf x}{\partial t}
 ```
 
-where $\mathbf \chi = (\chi_1, \chi_2)$ are the ALE coordinates, $A_0=\pi / 6$ is the amplitude, and $T_G=20$ is the
-period. Further parameters include the total simulation time $T$, and $L=1$ describing the height, width, and depth of
-the box mesh. By default, the Reynolds number is set to Re = $1/\nu=1600$.
+where $\mathbf X = (X_1, X_2, X_3)$ are the ALE coordinates, $A_0=\pi / 6$ is the amplitude, and $T_G=20$ is the period.
+Further parameters include the total simulation time $T$, and $L=1$ describing the height, width, and depth of the box
+mesh. By default, the Reynolds number is set to $\text{Re} = 1/\nu = 1600$.
 
 ## (HPC) Simulation in OasisMove
 
@@ -87,7 +83,7 @@ resolution parameters $N_x=32$, $N_y=32$, and $N_z=32$. In contrast to the 2D [m
 the number of tetrahedral cells are now computed using the following formula:
 
 ```{math}
-\text{Number of tetrahedral cells } = 6\times N_x \times N_y \times N_z,
+    \text{Number of tetrahedral cells } = 6\times N_x \times N_y \times N_z,
 ```
 
 resulting in a total of 196 608 cells using the default parameters. To increase the mesh resolution, we can supply these
@@ -98,8 +94,6 @@ we can run the following command:
 $ oasismove NSfracStepMove problem=MovingTaylorGreen3D Nx=50 Ny=50 Nz=50
 ```
 
-## References
-
 ```{bibliography} references.bib
 :filter: docname in docnames
 ```

docs/movingvortex.md

@@ -14,24 +14,20 @@ body forces, $\mathbf{f} = 0$, namely:
 
 ```{math}
 :label: eq-vortex
-\begin{align}
     \mathbf u(\mathbf x,t) &= (-\sin (2\pi  x_2), \sin(2\pi x_1))e^{-4\nu \pi^2 t}\\
     p(\mathbf x,t) &= -\cos(2\pi x_1)\cos(2\pi x_2)e^{-8\nu \pi^2 t}
-\end{align}
 ```
 
 where $\mathbf x = (x_1, x_2)$ are the Euclidean coordinates, and $\nu = 0.025$ $\text{m}^2$/s. Furthermore, the mesh
 velocity $\mathbf w$ is determined by the following displacement field:
 
 ```{math}
 :label: eq-vortex-d
-    \begin{align}
-    \mathbf x(\mathbf \chi,t) &= \mathbf \chi + A_0 \sin \left(\frac{2 \pi t}{T_G}\right) \left(\sin\left(2\pi \frac{\chi_2 + L / 2}{L}\right),\sin\left(2\pi \frac{\chi_1 + L / 2}{L}\right)\right)\\
-    \mathbf w (\mathbf \chi, t) &= \frac{\partial \mathbf x}{\partial t}
-    \end{align}
+    \mathbf x(\mathbf X , t) &= \chi + A_0 \sin \left(\frac{2 \pi t}{T_G}\right) \left(\sin\left(2\pi \frac{\chi_2 + L / 2}{L}\right),\sin\left(2\pi \frac{\chi_1 + L / 2}{L}\right)\right)\\
+    \mathbf w(\mathbf X , t) &= \frac{\partial \mathbf x}{\partial t}
 ```
 
-where $\mathbf \chi = (\chi_1, \chi_2)$ are the ALE coordinates, $A_0=0.08$ is the amplitude, $T_G=4T$ is the period
+where $\mathbf X= (X_1, X_2)$ are the ALE coordinates, $A_0=0.08$ is the amplitude, $T_G=4T$ is the period
 length of the mesh motion, $T$ is the length of one cycle, and $L=1$.
 
 ## Simulation in OasisMove
@@ -84,15 +80,12 @@ $ oasismove NSfracStepMove problem=MovingVortex T=1 dt=0.001
 When adjusting spatial and temporal resolution is important to know relationship between the cell size and the time-step size, which are closely related through the Courant number ($C$) given by the Courant-Friedrichs-Lewy (CFL) condition:
      
 $$
-\begin{align}
-C ={\frac  {u\,\Delta t}{\Delta x}} < C_{\max },
-\end{align}
-$$
+    C ={\frac  {u\,\Delta t}{\Delta x}} < C_{\max },
+$$ (eq:cfl-vortex)
 where $u$ is the velocity magnitude, $\Delta t$ is the time step size, and $\Delta x$ is the length interval. The general consensus is that $C_{\max} = 1$.
 
 ```
 
-## References
 
 ```{bibliography} references.bib
 :filter: docname in docnames

docs/movingwall.md

@@ -15,9 +15,7 @@ which remains parallel to the x-axis, defined as:
 
 ```{math}
 :label: eq-wall-height
-\begin{equation}
     h(t) = h_0 (1 + \epsilon e^{-i\omega t}),
-\end{equation}
 ```
 
 where $\omega$ is the pulsation of the movement, $h_0$ is the mean distance between the symmetry axis $(y = 0)$ and the
@@ -56,7 +54,6 @@ name: vel-wall
 On top, the grid displacement, and on the bottom, the velocity field for the wall-driven channel flow, where the vector arrows have been scaled by the velocity magnitude. 
 ```
 
-## References
 
 ```{bibliography} references.bib
 :filter: docname in docnames

docs/parameters.md

@@ -58,15 +58,12 @@ Ny = 50
 By adjusting `Nx` and `Ny` we adjust the resolution of the computational domain, which for the `DrivenCavity` problem is
 a 2D unit square mesh consisting of triangular elements. The parameters `Nx` and `Ny` determine the number of uniform
 cell intervals in the $x$ and $y$ direction, respectively. These parameters can also be used to compute the totan number
-of triangular cells (in 2D) or tetrehedral cells (in 3D) the computational domain consists of:
+of triangular cells in 2D: or tetrehedral cells (in 3D) the computational domain consists of:
 
 ```{math}
-\begin{align}
-    \text{Number of cells (2D) } &= 2 \times N_x \times N_y \\[0.5em] 
-    \text{Number of cells (3D) } &= 6 \times N_x \times N_y  \times N_z
-\end{align}
+    \text{Number of cells (2D) } &= 2 \times N_x \times N_y             \\
+    \text{Number of cells (3D) } &= 6 \times N_x \times N_y  \times N_z 
 ```
-
 ### `Fluid` parameters
 
 Fluid parameters are related to the fluid properties. In regard to the `DrivenCavity`

docs/restart.md

@@ -36,7 +36,7 @@ the `Checkpoint` folder:
 $ oasismove NSfracStep solver=IPCS_ABCN problem=DrivenCavity restart_folder=results_driven_cavity/data/1/Checkpoint 
 ```
 
-The simulation should now restart at the latest checkpoint, and continue as normal until $T=10.
+The simulation should now restart at the latest checkpoint, and continue as normal until $T=10$.
 
 ### Restarting a finished simulation
 

docs/stenosis.md

@@ -14,7 +14,7 @@ implementation of backflow stabilization is general, and works for both rigid an
 name: stenosis-fig
 ---
 A schematic of the two-dimensional stenosis model with a slight eccentricity of the stenosis, borrowed from 
-Varghese et al. {cite}`varghese2007direct`.
+Varghese et al.
 ```
 
 ## Problem description
@@ -46,22 +46,22 @@ rather common in cardiovascular flows, particularly in blood flow in large vesse
 left atrium. Because backflow is a naturally occurring physiologic phenomenon, careful treatment is necessary to
 realistically model backflow without artificially altering the local flow dynamics. To achieve this we consider the
 first backflow stabilization method first proposed by Bazilevs et al. {cite}`bazilevs2009patient` and rigorously tested
-by Moghadam et al. {cite}`esmaily2011comparison`. In short, the method modifies the weak formulation of the
+by Moghadam et al. In short, the method modifies the weak formulation of the
 Navier-Stokes equations by adding a backflow stabilization term for the Neumann boundaries, in this case the outlet.
 Specifically, the following convective traction term is added to the weak form on each Neumann boundary to be
 stabilized:
 
 ```{math}
 :label: eq-weak
--\beta \int_{\partial \Omega} (\mathbf u \cdot \mathbf n)\_\mathbf u \cdot \mathbf v \, \text{d}s,
+    -\beta \int_{\partial \Omega} (\mathbf u \cdot \mathbf n)\_\mathbf u \cdot \mathbf v \, \text{d}s,
 ```
 
 where $\beta$ is a positive coefficient between 0.0 and 1.0, $\mathbf u$ is the velocity vector, $\mathbf v$ is the
 velocity test function, $\partial \Omega$ denotes the boundary, and $(\mathbf u \cdot \mathbf n)\_$ is defined as:
 
 ```{math}
 :label: eq-weak2
-(\mathbf u \cdot \mathbf n)\_ \,=\, \frac{\mathbf u\cdot \mathbf n - \| \mathbf u \cdot \mathbf n\|}  {2}.
+    (\mathbf u \cdot \mathbf n)\_ \,=\, \frac{\mathbf u\cdot \mathbf n - \| \mathbf u \cdot \mathbf n\|}{2}.
 ```
 
 In OasisMove, the backflow stabilization method is applied by supplying the
@@ -120,8 +120,6 @@ The velocity field for the stenosis problem without (top) and with (bottom) back
 arrows that have been scaled by the velocity magnitude. 
 ```
 
-## References
-
 ```{bibliography} references.bib
 :filter: docname in docnames
 ```