Upgrade reference format in documentation (continued) (#1200)

Description
References in documentation are changed according to PR #1196.

doc/source/examples/incompressible-flow/2d-backward-facing-step/2d-backward-facing-step.rst

@@ -300,7 +300,7 @@ where the final value of :math:`x_r` is :math:`2.893`. We notice from the graph
 
 .. image:: image/Reynolds100-error-analysis.png
 
-The reference value used in the error analysis is taken from Erturk (2008) `[1] <https://doi.org/10.1016/j.compfluid.2007.09.003>`_.
+The reference value used in the error analysis is taken from Erturk (2008) [#erturk2008]_.
 
 
 Higher Reynolds Number (:math:`Re=1000`)
@@ -324,11 +324,11 @@ Validation and Comparison
 Reattachment Length
 ~~~~~~~~~~~~~~~~~~~
 
-In this section, the solutions obtained with Lethe are compared with data that can be found in the scientific literature (Erturk (2008) `[1] <https://doi.org/10.1016/j.compfluid.2007.09.003>`_, Armaly and al. (1983) `[2] <https://doi.org/10.1017/S0022112083002839>`_ and Velivelli and Bryden (2015) `[3] <https://doi.org/10.1016/j.advengsoft.2014.11.006>`_). Several studies include datasets of :math:`x_r/h = f(Re)` (reattachment length) either experimentally or numerically. The next figure illustrates some of them in comparison with *Lethe*.
+In this section, the solutions obtained with Lethe are compared with data that can be found in the scientific literature (Erturk (2008) [#erturk2008]_, Armaly and al. (1983) [#armaly1983]_ and Velivelli and Bryden (2015) [#velivelli2015]_). Several studies include datasets of :math:`x_r/h = f(Re)` (reattachment length) either experimentally or numerically. The next figure illustrates some of them in comparison with *Lethe*.
 
 .. image:: image/xr-comparison.png
 
-First, the results provided by Lethe are identical or so to all of the three selected studies for low Reynolds numbers (:math:`Re \leq 400`). After that point, both results form *Lethe* and from Erturk (2008) `[1] <https://doi.org/10.1016/j.compfluid.2007.09.003>`_ diverge from the experimental data of Armaly and al. (1983) `[2] <https://doi.org/10.1017/S0022112083002839>`_. According to `[1] <https://doi.org/10.1016/j.compfluid.2007.09.003>`_, this error is due to 3D effects that are more potent as the flow becomes more and more turbulent. Furthermore, there is also a less significant but clearly noticeable error between *Lethe* and Erturk (2008) `[1] <https://doi.org/10.1016/j.compfluid.2007.09.003>`_: the fact that certain tolerances have been set higher for higher Reynolds number cases might have underestimated the reattachment length. Also, first order elements have been used throughout the whole simulation process. Using second order elements for velocity, for instance, could yield better results for higher Reynolds numbers, however, at a higher computational cost. The following table illustrates the error at :math:`Re = 600` for first and second order velocity elements.
+First, the results provided by Lethe are identical or so to all of the three selected studies for low Reynolds numbers (:math:`Re \leq 400`). After that point, both results form *Lethe* and from Erturk (2008) [#erturk2008]_ diverge from the experimental data of Armaly and al. (1983) [#armaly1983]_. According to [#erturk2008]_, this error is due to 3D effects that are more potent as the flow becomes more and more turbulent. Furthermore, there is also a less significant but clearly noticeable error between *Lethe* and Erturk (2008) [#erturk2008]_: the fact that certain tolerances have been set higher for higher Reynolds number cases might have underestimated the reattachment length. Also, first order elements have been used throughout the whole simulation process. Using second order elements for velocity, for instance, could yield better results for higher Reynolds numbers, however, at a higher computational cost. The following table illustrates the error at :math:`Re = 600` for first and second order velocity elements.
 
 +---------------+----------------+----------------+
 | Order         | :math:`x_r/h`  | Error          |
@@ -369,11 +369,11 @@ Possibilities for Extension
 References
 ----------
 
-`[1] <https://doi.org/10.1016/j.compfluid.2007.09.003>`_ E. Erturk, “Numerical solutions of 2-D steady incompressible flow over a backward-facing step, Part I: High Reynolds number solutions,” *Comput. Fluids*, vol. 37, no. 6, pp. 633–655, Jul. 2008, doi: 10.1016/j.compfluid.2007.09.003.
+.. [#erturk2008] \E. Erturk, “Numerical solutions of 2-D steady incompressible flow over a backward-facing step, Part I: High Reynolds number solutions,” *Comput. Fluids*, vol. 37, no. 6, pp. 633–655, Jul. 2008, doi: `10.1016/j.compfluid.2007.09.003 <https://doi.org/10.1016/j.compfluid.2007.09.003>`_\.
 
-`[2] <https://doi.org/10.1017/S0022112083002839>`_ B. F. Armaly, F. Durst, J. C. F. Pereira, and B. Schönung, “Experimental and theoretical investigation of backward-facing step flow,” *J. Fluid Mech.*, vol. 127, pp. 473–496, Feb. 1983, doi: 10.1017/S0022112083002839.
+.. [#armaly1983] \B. F. Armaly, F. Durst, J. C. F. Pereira, and B. Schönung, “Experimental and theoretical investigation of backward-facing step flow,” *J. Fluid Mech.*, vol. 127, pp. 473–496, Feb. 1983, doi: `10.1017/S0022112083002839 <https://doi.org/10.1017/S0022112083002839>`_\.
 
-`[3] <https://doi.org/10.1016/j.advengsoft.2014.11.006>`_ A. C. Velivelli and K. M. Bryden, “Domain decomposition based coupling between the lattice Boltzmann method and traditional CFD methods – Part II: Numerical solution to the backward facing step flow,” *Adv. Eng. Softw.*, vol. 82, pp. 65–74, Apr. 2015, doi: 10.1016/j.advengsoft.2014.11.006.
+.. [#velivelli2015] \A. C. Velivelli and K. M. Bryden, “Domain decomposition based coupling between the lattice Boltzmann method and traditional CFD methods – Part II: Numerical solution to the backward facing step flow,” *Adv. Eng. Softw.*, vol. 82, pp. 65–74, Apr. 2015, doi: `10.1016/j.advengsoft.2014.11.006 <https://doi.org/10.1016/j.advengsoft.2014.11.006>`_\.
 
 
 

doc/source/examples/incompressible-flow/2d-flow-around-cylinder/2d-flow-around-cylinder.rst

@@ -31,7 +31,7 @@ All files mentioned below are located in the example's folder (``examples/incomp
 -----------------------
 Description of the Case
 -----------------------
-We simulate the flow around a fixed cylinder with a constant upstream fluid velocity. The following schematic describes the geometry with its relevant quantities (taken from the article by Blais *et al.* `[1] <https://doi.org/10.1016/j.compchemeng.2015.10.019>`_):
+We simulate the flow around a fixed cylinder with a constant upstream fluid velocity. The following schematic describes the geometry with its relevant quantities (taken from the article by Blais *et al.* [#blais2016]_):
 
 .. image:: images/geometry-description.png
     :alt: The geometry
@@ -283,4 +283,4 @@ Possibilities for Extension
 Reference
 ----------
 
-`[1] <https://doi.org/10.1016/j.compchemeng.2015.10.019>`_ 	B. Blais, M. Lassaigne, C. Goniva, L. Fradette, and F. Bertrand, “A semi-implicit immersed boundary method and its application to viscous mixing,” *Comput. Chem. Eng.*, vol. 85, pp. 136–146, Feb. 2016, doi: 10.1016/j.compchemeng.2015.10.019.
+.. [#blais2016] \B. Blais, M. Lassaigne, C. Goniva, L. Fradette, and F. Bertrand, “A semi-implicit immersed boundary method and its application to viscous mixing,” *Comput. Chem. Eng.*, vol. 85, pp. 136–146, Feb. 2016, doi: `10.1016/j.compchemeng.2015.10.019 <https://doi.org/10.1016/j.compchemeng.2015.10.019>`_\.

doc/source/examples/incompressible-flow/2d-lid-driven-cavity-flow/lid-driven-cavity-flow.rst

@@ -21,8 +21,8 @@ Files Used in This Example
 All files mentioned below are located in the example's folder (``examples/incompressible-flow/2d-lid-driven-cavity``).
 
 - Base case parameter file (:math:`Re=400`): ``cavity.prm``
-- Experimental data file from Ghia `et al.` (1982) `[1] <https://doi.org/10.1016/0021-9991(82)90058-4>`_: ``ref-2d-ghia-u.txt``
-- Experimental data file from Erturk `et al.` (2005) `[2] <https://doi.org/10.1002/fld.953>`_: ``ref-2d-erturk-u.txt``
+- Experimental data file from Ghia `et al.` (1982) [#ghia1982]_: ``ref-2d-ghia-u.txt``
+- Experimental data file from Erturk `et al.` (2005) [#erturk2005]_: ``ref-2d-erturk-u.txt``
 - Higher-Reynolds case parameter file (:math:`Re=7500`): ``Reynolds_7500/cavity.prm``
 - Postprocessing Python script for the :math:`Re=400` case: ``post_process_Reynolds_400.py``
 - Postprocessing Python script for the :math:`Re=7500` case: ``Reynolds_7500/post_process_Reynolds_7500.py``
@@ -299,7 +299,7 @@ Increasing the number of cells by a factor 4 (to :math:`\approx` 65k cells) allo
 Possibilities for Extension
 ----------------------------
 
-- **Validate at even higher Reynolds numbers:** The Erturk `[2] <https://doi.org/10.1002/fld.953>`_ data within the example investigates this case up to a Reynolds number of 20000.  It is an interesting exercise to simulate these more complex cases using the adjoint time-stepping ``steady_bdf`` scheme.
+- **Validate at even higher Reynolds numbers:** The Erturk [#erturk2005]_ data within the example investigates this case up to a Reynolds number of 20000.  It is an interesting exercise to simulate these more complex cases using the adjoint time-stepping ``steady_bdf`` scheme.
 
 - **High-order methods:** Lethe supports higher order interpolation. This can yield much better results with an equal number of degrees of freedom than traditional second-order (Q1-Q1) methods, especially at higher Reynolds numbers. 
 
@@ -310,6 +310,6 @@ Possibilities for Extension
 References
 -----------
 
-`[1] <https://doi.org/10.1016/0021-9991(82)90058-4>`_ U. Ghia, K. N. Ghia, and C. T. Shin, “High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method,” *J. Comput. Phys.*, vol. 48, no. 3, pp. 387–411, Dec. 1982, doi: 10.1016/0021-9991(82)90058-4.
+.. [#ghia1982] \U. Ghia, K. N. Ghia, and C. T. Shin, “High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method,” *J. Comput. Phys.*, vol. 48, no. 3, pp. 387–411, Dec. 1982, doi: `10.1016/0021-9991(82)90058-4 <https://doi.org/10.1016/0021-9991(82)90058-4>`_\.
 
-`[2] <https://doi.org/10.1002/fld.953>`_ E. Erturk, T. C. Corke, and C. Gökçöl, “Numerical solutions of 2-D steady incompressible driven cavity flow at high Reynolds numbers,” *Int. J. Numer. Methods Fluids*, vol. 48, no. 7, pp. 747–774, 2005, doi: 10.1002/fld.953.
+.. [#erturk2005] \E. Erturk, T. C. Corke, and C. Gökçöl, “Numerical solutions of 2-D steady incompressible driven cavity flow at high Reynolds numbers,” *Int. J. Numer. Methods Fluids*, vol. 48, no. 7, pp. 747–774, 2005, doi: `10.1002/fld.953 <https://doi.org/10.1002/fld.953>`_\.

doc/source/examples/incompressible-flow/2d-naca0012-low-reynolds/2d-naca0012-low-reynolds.rst

@@ -249,7 +249,7 @@ The following average pressure and velocity fields are obtained for an angle of
 .. image:: image/average_velocity.png
 
 
-It is already noticeable that the higher the angle of attack, the greater the pressure gradient. Following this observation, the lift coefficient :math:`C_L` is expected to increase with the angle of attack until stall condition is reached. The variation of the lift and drag coefficients are given below with a comparison to the work of Kouser *et al.* `[1] <https://doi.org/10.1177/17568293211055656>`_. Both coefficients are computed using the following formula:
+It is already noticeable that the higher the angle of attack, the greater the pressure gradient. Following this observation, the lift coefficient :math:`C_L` is expected to increase with the angle of attack until stall condition is reached. The variation of the lift and drag coefficients are given below with a comparison to the work of Kouser *et al.* [#kouser2021]_. Both coefficients are computed using the following formula:
 
 .. math::
         C_L = \frac{F_L}{0.5\rho_{\infty} u_{\infty}^2 S} \; \; \; \; \; C_D = \frac{F_D}{0.5\rho_{\infty} u_{\infty}^2 S}
@@ -258,7 +258,7 @@ with :math:`F_L` and :math:`F_D`, respectively, the lift and drag forces. Those
 
 .. image:: image/cl_cd_results_plot.png
 
-The results obtained fit the drag and lift coefficients found by Kouser *et al.* `[1] <https://doi.org/10.1177/17568293211055656>`_. Note that the value given for the :math:`C_D` and :math:`C_L` coefficients are Root Mean Squared (RMS) values. The time span considered is 25s long (between 15 :math:`\text{s}` and 40 :math:`\text{s}`). The first 15 seconds were not considered to let the system reach a pseudo-steady state.
+The results obtained fit the drag and lift coefficients found by Kouser *et al.* [#kouser2021]_. Note that the value given for the :math:`C_D` and :math:`C_L` coefficients are Root Mean Squared (RMS) values. The time span considered is 25s long (between 15 :math:`\text{s}` and 40 :math:`\text{s}`). The first 15 seconds were not considered to let the system reach a pseudo-steady state.
 
 One can also see the low-velocity zones on the upper part of the airfoil, which corresponds to the recirculating zone: the ``noslip`` condition on the NACA imposes a zero velocity condition on the fluid. The following streamline representation helps to see the movements of the fluid inside the recirculating zone: 
 
@@ -285,7 +285,7 @@ In order to retrieve the frequency of the vortex shedding, one can look at the f
 
 .. image:: image/plot_cl_time.png
 
-The best mathematical tool available to make a spectral analysis is a Fourier transform, which is performed below, with literature results (Kouser *et al.* (2021) `[1] <https://doi.org/10.1177/17568293211055656>`_) for comparison:
+The best mathematical tool available to make a spectral analysis is a Fourier transform, which is performed below, with literature results (Kouser *et al.* (2021) [#kouser2021]_) for comparison:
 
 .. image:: image/fft_cl_comparison.png
 
@@ -300,17 +300,17 @@ Possibilities for Extension
 
 - **Going 3D**: the mesh can be extruded into the third dimension. Some modifications will be required in the boundary conditions, and getting the correct boundaries id is not trivial. However, with periodic boundary conditions set on the sides of the box, spanwise effects can be taken into account, which should yield much better results.
 
-- **Validate for higher Reynolds numbers**: Literature is available for comparison at :math:`Re=10000` at Yamaguchi *et al.* (2013) `[2] <https://doi.org/10.1299/jsmeicjwsf.2013.4._1201-1_>`_ and :math:`Re=23000` at Kojima *et al.* (2013) `[3] <https://doi.org/10.2514/1.C031849>`_.
+- **Validate for higher Reynolds numbers**: Literature is available for comparison at :math:`Re=10000` at Yamaguchi *et al.* (2013) [#yuta2013]_ and :math:`Re=23000` at Kojima *et al.* (2013) [#kojima2013]_.
 
 
 ----------
 References
 ----------
 
-`[1] <https://doi.org/10.1177/17568293211055656>`_ T. Kouser, Y. Xiong, D. Yang, and S. Peng, “Direct Numerical Simulations on the three-dimensional wake transition of flows over NACA0012 airfoil at Re=1000,” *Int. J. Micro Air Veh.*, vol. 13, p. 17568293211055656, Jan. 2021, doi: 10.1177/17568293211055656.
+.. [#kouser2021] \T. Kouser, Y. Xiong, D. Yang, and S. Peng, “Direct Numerical Simulations on the three-dimensional wake transition of flows over NACA0012 airfoil at Re=1000,” *Int. J. Micro Air Veh.*, vol. 13, p. 17568293211055656, Jan. 2021, doi: `10.1177/17568293211055656 <https://doi.org/10.1177/17568293211055656>`_\.
 
-`[2] <https://doi.org/10.1299/jsmeicjwsf.2013.4._1201-1_>`_ Y. Yuta, O. Tomohisa, and M. Akinori, “1201 Pressure Distribution on a Naca0012 Airfoil at Low Reynolds Numbers,” *Proc. Int. Conf. Jets Wakes Separated Flows ICJWSF*, vol. 2013.4, p. 1201-1 - 1201-5 , 2013, doi: 10.1299/jsmeicjwsf.2013.4._1201-\1_.
+.. [#yuta2013] \Y. Yuta, O. Tomohisa, and M. Akinori, “1201 Pressure Distribution on a Naca0012 Airfoil at Low Reynolds Numbers,” *Proc. Int. Conf. Jets Wakes Separated Flows ICJWSF*, vol. 2013.4, p. 1201-1 - 1201-5 , 2013, doi: `10.1299/jsmeicjwsf.2013.4._1201-\1_ <https://doi.org/10.1299/jsmeicjwsf.2013.4._1201-1_>`_\.
 
-`[3] <https://doi.org/10.2514/1.C031849>`_ R. Kojima, T. Nonomura, A. Oyama, and K. Fujii, “Large-Eddy Simulation of Low-Reynolds-Number Flow Over Thick and Thin NACA Airfoils,” *J. Aircr.*, vol. 50, no. 1, pp. 187–196, Jan. 2013, doi: 10.2514/1.C031849.
+.. [#kojima2013] \R. Kojima, T. Nonomura, A. Oyama, and K. Fujii, “Large-Eddy Simulation of Low-Reynolds-Number Flow Over Thick and Thin NACA Airfoils,” *J. Aircr.*, vol. 50, no. 1, pp. 187–196, Jan. 2013, doi: `10.2514/1.C031849 <https://doi.org/10.2514/1.C031849>`_\.
 
 

doc/source/examples/incompressible-flow/2d-taylor-couette-flow-nitsche/2d-taylor-couette-flow-nitsche.rst

@@ -2,7 +2,7 @@
 Taylor-Couette Flow Using Nitsche Immersed Boundary
 ========================================================
 
-This example revisits the same taylor-couette flow problem in :doc:`../2d-taylor-couette-flow/2d-taylor-couette-flow`, 
+This example revisits the same taylor-couette flow [#bird2006]_ problem in :doc:`../2d-taylor-couette-flow/2d-taylor-couette-flow`,
 now using immersed boundaries to represent the inner cylinder. This example demonstrates some of the capabilities of Lethe to simulate the flow around complex geometries without meshing them explicitly with a conformal mesh, but instead by using the Nitsche immersed boundary method available within Lethe.
 
 
@@ -384,4 +384,4 @@ Possibilities for Extension
 Reference
 ------------
 
-[1] R. B. Bird, W. E. Stewart, and E. N. Lightfoot, *Transport Phenomena*, vol. 1. John Wiley & Sons, 2006.
+.. [#bird2006] \R. B. Bird, W. E. Stewart, and E. N. Lightfoot, *Transport Phenomena*, vol. 1. John Wiley & Sons, 2006\.

doc/source/examples/incompressible-flow/2d-taylor-couette-flow/2d-taylor-couette-flow.rst

@@ -2,7 +2,7 @@
 Taylor-Couette Flow
 ==================================
 
-This example showcases another classical fluid mechanics problem, the Taylor-Couette flow. This example introduces the usage of analytical solution and monitors the convergence of the CFD solver by using progressively refined meshes.
+This example showcases another classical fluid mechanics problem, the Taylor-Couette flow [#bird2006]_. This example introduces the usage of analytical solution and monitors the convergence of the CFD solver by using progressively refined meshes.
 
 
 ---------
@@ -293,4 +293,4 @@ Possibilities for Extension
 References
 ------------
 
-[1] R. B. Bird, W. E. Stewart, and E. N. Lightfoot, *Transport Phenomena*, vol. 1. John Wiley & Sons, 2006.
+.. [#bird2006] \R. B. Bird, W. E. Stewart, and E. N. Lightfoot, *Transport Phenomena*, vol. 1. John Wiley & Sons, 2006\.

doc/source/examples/incompressible-flow/2d-transient-flow-around-ahmed-body/2d-transient-flow-around-ahmed-body.rst

@@ -42,7 +42,7 @@ The following schematic describes the simulation.
 * bc = 1 (u = 1; flow in the x-direction)
 * bc = 2 (Slip boundary condition)
 
-The basic geometry for the Ahmed body is given below, as defined in Ahmed et al. `[1] <https://www.jstor.org/stable/44434262>`_, with all measures in mm.
+The basic geometry for the Ahmed body is given below, as defined in Ahmed et al. [#ahmed1984]_, with all measures in mm.
 
 .. image:: images/ahmed-geometry.png
     :alt: Geometry detailed description
@@ -236,4 +236,4 @@ Possibilities for Extension
 ----------
 Reference
 ----------
-`[1] <https://www.jstor.org/stable/44434262>`_ Ahmed, S. R., et al. “Some Salient Features of the Time -Averaged Ground Vehicle Wake.” *SAE Transactions*, vol. 93, 1984, pp. 473–503. http://www.jstor.org/stable/44434262.
+.. [#ahmed1984] \Ahmed, S. R., et al. “Some Salient Features of the Time -Averaged Ground Vehicle Wake.” *SAE Transactions*, vol. 93, pp. 473–503, 1984. http://www.jstor.org/stable/44434262\.