edit markdown files for markdown/math errors

Author: cdarve245

documentation/svfsi/cep/propagation/readme.md

@@ -2,9 +2,11 @@
 
 The propagation of electrical signal in the heart is governed by a reaction-diffusion like equation,
 
-$$\frac{\partial V}{\partial t} + \frac{I\_{ion} - I\_{app}(t)}{C\_m}  = \nabla \cdot\left( \mathbf{D}\nabla V \right) $$ 
-$$ \mathrm{in~} \Omega^E\times(0,T] \nonumber$$
-$$(\mathbf{D}\nabla V) \cdot \mathbf{N}=0 \mathrm{~on~} \partial\Omega^E\times(0,T]$$
+$$\frac{\partial V}{\partial t} + \frac{I\_{ion} - I\_{app}(t)}{C\_m}  = \nabla \cdot\left( \mathbf{D}\nabla V \right) $$
+
+$$ \mathrm{in~} \Omega^E\times(0,T] \nonumber $$
+
+$$ (\mathbf{D}\nabla V) \cdot \mathbf{N}=0 \mathrm{\~on\~} \partial\Omega^E\times(0,T] $$
 
 $V$ is the local transmembrane potential. $C\_m$ is the local membrane capacitance per unit area. $I\_{ion}$ and $I\_{app}$ are the ionic current flux (current per unit area) and applied external current flux, respectively. Here, $\mathbf{D}$ dictates the propagation velocity of the electrical signal and has the similar physical meaning as the diffusivity. It is calculated as
 
@@ -16,16 +18,16 @@ $\sigma$ is the conductivity tensor and has the unit $S/m$. $\chi$ is the ratio
 
 The above equation is the mono-domain description of the cardiac electrophysiology, i.e. we don't solve the intra- and extra-cellular electrical signal propagation separately. The mono-domain and multi-domain conductivities are connected through the following relation,
 
-$$  \sigma = \frac{\sigma\_i\sigma\_e}{\sigma\_i + \sigma\_e} $$
+$$ \sigma = \frac{\sigma_i\sigma_e}{\sigma_i + \sigma_e} $$
 
 where $\sigma\_i$ and $\sigma_e$ are the intra- and extra-cellular conductivity tensors <a href="#ref-1">[1]</a>. It is commonly assumed that the conductivity is transversely isotropic,
 
-$$ \mathbf{\sigma} = \sigma\_f \mathbf{f}\otimes \mathbf{f} + \sigma\_s (\mathbf{I}-\mathbf{f}\otimes \mathbf{f}) $$
+$$ \mathbf{\sigma} = \sigma_f \mathbf{f}\otimes \mathbf{f} + \sigma_s (\mathbf{I}-\mathbf{f}\otimes \mathbf{f}) $$
 
 where $\sigma\_f$ and $\sigma\_s$ are the conductivities along the fiber direction and in the transverse plane. $\mathbf{f}$ is the fiber direction vector.
 
 In <strong>svFSI</strong>, we directly specify $\mathbf{D}$ in the input file. We choose a slightly different form to enforce the transverse isotropy of the conductivity tensor,
 
-$$  \mathbf{D} = D\_{iso}\mathbf{I} + \sum\_{n=1}^{nsd}D\_{ani,n}\mathbf{fN}\_n\otimes\mathbf{fN}\_n $$
+$$ \mathbf{D} = D\_{iso}\mathbf{I} + \sum\_{n=1}^{nsd}D\_{ani,n}\mathbf{fN}\_n\otimes\mathbf{fN}\_n $$
 
-Here, $nsd$ is the dimension, and $\mathbf{fN\_n}$ is the local orthonormal coordinate system built by fiber direction and sheet direction. To connect with the previous expression, we have $D\_f=D\_{iso}+D\_{ani}$ and $D\_s=D\_{iso}$.
+Here, $nsd$ is the dimension, and $\mathbf{fN\_n}$ is the local orthonormal coordinate system built by fiber direction and sheet direction. To connect with the previous expression, we have $D\_f=D\_{iso}+D\_{ani}$ and $D\_s=D\_{iso}$.

documentation/svfsi/fluid/refs/readme.md

@@ -1,19 +1,18 @@
 ## Reference
 
 <p><a id="ref-1">
-[1] Chandran KB, Rittgers SE, Yoganathan AP. <strong>Biofluid mechanics: the human circulation.</strong> CRC press; 2006 Nov 15.  </a></a></p>
+[1] Chandran KB, Rittgers SE, Yoganathan AP. <strong>Biofluid mechanics: the human circulation.</strong> CRC press; 2006 Nov 15.  </a></p>
 
 <p><a id="ref-2"> <a href="https://doi.org/10.1063/1.2772250">
-[2] Boyd, Joshua, James M. Buick, and Simon Green. <strong>Analysis of the Casson and Carreau-Yasuda Non-Newtonian Blood Models in Steady and Oscillatory Flows Using the Lattice Boltzmann Method</strong>. Physics of Fluids 19, no. 9 (September 2007): 093103.  </a></a></p>
+[2] Boyd, Joshua, James M. Buick, and Simon Green. <strong>Analysis of the Casson and Carreau-Yasuda Non-Newtonian Blood Models in Steady and Oscillatory Flows Using the Lattice Boltzmann Method</strong>. Physics of Fluids 19, no. 9 (September 2007): 093103.</a></a></p>
 
 <p><a id="ref-3"> 
-[3] Vedula V, Lee J, Xu H, Kuo CC, Hsiai TK, Marsden AL.<strong> A method to quantify mechanobiologic forces during zebrafish cardiac development using 4-D light sheet imaging and computational modeling.</strong> PLoS computational biology. 2017 Oct 30;13(10):e1005828.</a></a></p>
+[3] Vedula V, Lee J, Xu H, Kuo CC, Hsiai TK, Marsden AL.<strong> A method to quantify mechanobiologic forces during zebrafish cardiac development using 4-D light sheet imaging and computational modeling.</strong> PLoS computational biology. 2017 Oct 30;13(10):e1005828.</a></p>
 
 <p><a id="ref-4"> 
-[4] Mittal R, Seo JH, Vedula V, Choi YJ, Liu H, Huang HH, Jain S, Younes L, Abraham T, George RT. <strong>Computational modeling of cardiac hemodynamics: current status and future outlook.</strong> Journal of Computational Physics. 2016 Jan 15;305:1065-82.  </a></a></p>
+[4] Mittal R, Seo JH, Vedula V, Choi YJ, Liu H, Huang HH, Jain S, Younes L, Abraham T, George RT. <strong>Computational modeling of cardiac hemodynamics: current status and future outlook.</strong> Journal of Computational Physics. 2016 Jan 15;305:1065-82. </a></p>
 
 <p><a id="ref-5"> <a href="https://doi.org/10.1115/1.4048032">
-[5] Kong, F., and Shadden, S. C. (2020). **Automating Model Generation for Image-based Cardiac Flow Simulation**. ASME. J Biomech Eng. </a> </p>
+[5] Kong, F., and Shadden, S. C. (2020). <strong>Automating Model Generation for Image-based Cardiac Flow Simulation.</strong> ASME. J Biomech Eng. </a> </a></p>
 
-
-<p><br><br><br><br><br></p>
+<p><br><br><br><br><br></p>

documentation/svfsi/fsi/appendix_creating_fluid_geometry/readme.md

@@ -1,6 +1,6 @@
 ### Creating the geometry for the fluid domain
 
-For most cardiovascular modeling applications, the geometry of the fluid domain is generated by segmenting blood vessels out of medical image data. This process is described on the main SimVascular documentation: http://simvascular.github.io/docsModelGuide.html. We now want delete all the caps off the model. Once you have done that, export it by right-clicking the model from the left-hand menu and selecting ``Export as Solid Model''. When SimVascular prompts you for a name and location for the exported model, make sure to add an .stl extension to make sure the exported model is in .stl format. We will perform the next step in Meshmixer, and Meshmixer only takes in .stl format surfaces.
+For most cardiovascular modeling applications, the geometry of the fluid domain is generated by segmenting blood vessels out of medical image data. This process is described on the main SimVascular documentation: http://simvascular.github.io/modeling.html. We now want delete all the caps off the model. Once you have done that, export it by right-clicking the model from the left-hand menu and selecting ``Export as Solid Model''. When SimVascular prompts you for a name and location for the exported model, make sure to add an .stl extension to make sure the exported model is in .stl format. We will perform the next step in Meshmixer, and Meshmixer only takes in .stl format surfaces.
 
 <figure>
   <img class="svImg svImgMd" src="/documentation/svfsi/fsi/imgs/SV_Export_as_stl1.png" style="width:100%;height:auto;max-width: 30vw;">

documentation/svfsi/fsi/appendix_creating_solid_mesh/readme.md

@@ -1,6 +1,6 @@
 ### Create the solid domain mesh
 
-Similar to the way we create the mesh for the fluid domain in SimVascular (see: http://simvascular.github.io/docsMeshing.html) we can now create the mesh for the solid domain.
+Similar to the way we create the mesh for the fluid domain in SimVascular (see: http://simvascular.github.io/meshing.html) we can now create the mesh for the solid domain.
 
 1.  Import the .stl of the structural domain geometry into SimVascular
 

documentation/svfsi/fsi/mesh_adam_bl/readme.md

@@ -2,7 +2,7 @@
 
 To run an FSI simulation we need a mesh for both the structural domain and the fluid domain. These two meshes must have their interface nodes coincide exactly in order to satisfy the interfacial conditions that result from conservation of mass and momentum. The coincident nodes of the fluid mesh are mapped onto the corresponding nodes on the structural mesh and the solution of velocity, displacement, pressure, etc. are treated as equal in the structural and fluid domains.
 
-The fluid domain geometry for patient-specific anatomies are generated using the usual [SimVascular modeling pipeline](http://simvascular.github.io/docsModelGuide.html). To create the geometry for the structural domain, we will make use of the [boundary layer meshing feature](https://simvascular.github.io/docsMeshing.html#tetgenboundarylayer) in the `Meshing` module. The usual case for boundary layer meshing involves extruding this thin layer of elements *inwards*, starting from the walls and going into the direction of the vessel centers. To make a wall mesh, we will instead use the boundary layer meshing feature to extrude elements *outwards* to effectively make a new mesh with a specified thickness that surrounds our fluid domain. This new mesh will form the geometry of our structural domain.
+The fluid domain geometry for patient-specific anatomies are generated using the usual [SimVascular modeling pipeline](http://simvascular.github.io/modeling.html). To create the geometry for the structural domain, we will make use of the [boundary layer meshing feature](https://simvascular.github.io/meshing.html#tetgenboundarylayer) in the `Meshing` module. The usual case for boundary layer meshing involves extruding this thin layer of elements _inwards_, starting from the walls and going into the direction of the vessel centers. To make a wall mesh, we will instead use the boundary layer meshing feature to extrude elements _outwards_ to effectively make a new mesh with a specified thickness that surrounds our fluid domain. This new mesh will form the geometry of our structural domain.
 
 Before we use the boundary layer meshing to extrude outwards, it is extremely important that we thoroughly smooth our model. The outward extrusion of elements has the potential to cause some elements to extrude into each other and overlap, which will cause the extrusion to fail. This is especially true at bifurcations, where the extruded elements from the two daughter branches could easily run into each other near the junction if not properly smoothed. We will therefore make thorough use of the local smoothing tools in the Models module before meshing. Below is an example of a bifurcation that would normally occur after lofting segmentations together without smoothing. If we try to extrude a boundary layer mesh from this, the elements at the sharp corner will intersect with each other and cause it to fail.
 
@@ -48,9 +48,9 @@ Now that we have a smoothed model, we can open the "Meshes" module. Right-click
   <figcaption class="svCaption" >SV Meshing.</figcaption>
 </figure>
 
-We must first select an appropriate "Global Max Edge Size" for our model. This edge size will determine the thickness of our wall mesh. We will remesh the volume of the fluid and solid domains later, so select the edge size to be *double* what your desired thickness is. We select this to be double to cut down the meshing time. If you are not sure how to select the thickness of your model, choosing a thickness that is 10% of the mean radius in your model is a reasonable assumption used throughout the literature. Below the edge size selection, you should see a box to select boundary layer meshing. First, click the checkbox next to "Boundary Layer Meshing" to turn it on. Below this, you should notice three spaces to select parameters of the boundary layer meshing. Since we are producing this boundary layer for the wall mesh, we can use the same settings for these. "Portion Edge Size" determine the overall thickness of our boundary layer mesh, as a fraction of the "Global Max Edge Size" selected above. Since we selected a "Global Max Edge Size" to be double our desired thickness, we want this parameter to be 0.5. Next, you will have to choose the "Number of Layers" in your boundary layer mesh. Increasing this number will increase the accuracy of your structural domain calculations but also increase the number of elements and thus increase your cost. A reasonable number for this parameter is 2. Last, we must select the "Layer Decreasing Ratio", which is a parameter that allows subsequent layers to be a smaller size than the one before it. Since this parameter does not matter too much for creating a wall mesh, we can select this to be 1.0 to make it so all our layers are the same size.
+We must first select an appropriate "Global Max Edge Size" for our model. This edge size will determine the thickness of our wall mesh. We will remesh the volume of the fluid and solid domains later, so select the edge size to be _double_ what your desired thickness is. We select this to be double to cut down the meshing time. If you are not sure how to select the thickness of your model, choosing a thickness that is 10% of the mean radius in your model is a reasonable assumption used throughout the literature. Below the edge size selection, you should see a box to select boundary layer meshing. First, click the checkbox next to "Boundary Layer Meshing" to turn it on. Below this, you should notice three spaces to select parameters of the boundary layer meshing. Since we are producing this boundary layer for the wall mesh, we can use the same settings for these. "Portion Edge Size" determine the overall thickness of our boundary layer mesh, as a fraction of the "Global Max Edge Size" selected above. Since we selected a "Global Max Edge Size" to be double our desired thickness, we want this parameter to be 0.5. Next, you will have to choose the "Number of Layers" in your boundary layer mesh. Increasing this number will increase the accuracy of your structural domain calculations but also increase the number of elements and thus increase your cost. A reasonable number for this parameter is 2. Last, we must select the "Layer Decreasing Ratio", which is a parameter that allows subsequent layers to be a smaller size than the one before it. Since this parameter does not matter too much for creating a wall mesh, we can select this to be 1.0 to make it so all our layers are the same size.
 
-Below these parameter are three checkboxes. The "Extrude Boundary Layer Inward from Wall" checkbox will extrude the boundary layer mesh inwards if selected. Since we want to extrude the boundary layer mesh *outwards*, we will uncheck this box. The next box is a setting for "Use Constant Boundary Layer Thickness", which will attempt to make the entire boundary layer the same thickness if selected. We recommend leaving this box unchecked so the boundary layer mesh can adaptively change thickness in areas of tricky geometry. The last checkbox, "Convert Boundary Layer to New Region/Domain", is very important. We want to check this so that we will have a way to separate the fluid and structural domains later.
+Below these parameter are three checkboxes. The "Extrude Boundary Layer Inward from Wall" checkbox will extrude the boundary layer mesh inwards if selected. Since we want to extrude the boundary layer mesh _outwards_, we will uncheck this box. The next box is a setting for "Use Constant Boundary Layer Thickness", which will attempt to make the entire boundary layer the same thickness if selected. We recommend leaving this box unchecked so the boundary layer mesh can adaptively change thickness in areas of tricky geometry. The last checkbox, "Convert Boundary Layer to New Region/Domain", is very important. We want to check this so that we will have a way to separate the fluid and structural domains later.
 
 Now, you are ready to run the mesher. Click "Run Mesher" near the top right of the "SV Meshing" window to run the mesher. If the meshing was successful, you should see a window pop up to inform you of the statistics of your mesh. If the meshing was unsuccessful, it is likely that your model may need to be smoothed more to avoid intersecting elements. Once you are successful in producing this boundary layer mesh, right-click it from the SV Data Manager, and click "Export Mesh-Complete" and choose a location to send the mesh.