Convert to Principal Curvature (#132)

* Convert to Principal Curvature

Replace the gaussian curvature test for topography resolution with the max of the abs of the two principal curvatures. This removes an issue of zero curvature in one of the directions. New mesh benchmarks were needed because the meshes are different with the different curvature tests.

* add license

* Doc Editing and file removal

Edit the doc's for typos discovered along the way, plus some additional notes. Remove files associated with gaussian curvature since they are no longer needed

* typo fix

* point to MIT textbook in the docs for principle curvatures

* Update assumptions-and-algorithms.md

Fix a typo.

* Update appendix.md

Add description of elliptic arc block.

* Create WorkflowTools.md

Add a page that describes two add-ons that can be used to generate control files. Other tools can be added as desired.

* add new docs file to the build

* move the Workflow Tools docs section after the User Guide

---------

Co-authored-by: Andrew Winters <[email protected]>

Author: DavidAKopriva

Benchmarks/BenchmarkData/SeaMount.txt

@@ -1,8 +1,8 @@
            2
-         728        1005
- 8.24881E-01
- 1.99965E+00
- 1.65152E-01
- 7.74288E-01
- 2.33881E-01
- 2.01487E-01
+         564         800
+ 8.66819E-01
+ 1.60272E+00
+ 2.17557E-01
+ 8.53463E-01
+ 1.76174E-01
+ 2.60118E-01

Benchmarks/BenchmarkData/SeaMountCubic.txt

@@ -1,8 +1,8 @@
            2
-         528         755
- 8.38454E-01
- 1.51042E+00
- 2.16931E-01
- 8.74839E-01
- 1.96438E-01
- 2.65867E-01
+         412         610
+ 8.94582E-01
+ 1.42363E+00
+ 2.88336E-01
+ 9.04681E-01
+ 1.45161E-01
+ 3.41464E-01

Benchmarks/PlotFiles/Benchmarks/SeaMount.tec

@@ -0,0 +1,15 @@
+# Workflow Tools
+
+Two tools are available to help generate control files from which to generate a mesh with HOHQMesh.
+
+## HOHQMesh.jl
+
+<img src="https://user-images.githubusercontent.com/25242486/174798986-6b900fa8-840c-4c04-bc61-00f0749af1be.png" width="400">
+
+[HOHQMesh.jl](https://trixi-framework.github.io/HOHQMesh.jl/stable/) is a simple graphical front end to HOHQMesh that allows one to draw and edit the geometry, mesh and visualize in one package. It also includes binaries for the mesher so that it is not necessary to compile HOHQMesh oneself. Tutorials on how to use HOHQMesh.jl can be found [here](https://trixi-framework.github.io/HOHQMesh.jl/stable/).
+
+## svg-to-HOHQMesh
+
+<img src="https://github.com/trixi-framework/HOHQMesh_examples/blob/a8610caf1c5047741ee341462cb3e2f9d34471c6/gallery/jack-o-lantern-small.png?raw=true" width="350">
+
+[svg-to-HOHQMesh](https://github.com/FluidNumerics/svg-to-HOHQMesh) is a tool created by [FluidNumerics](https://www.fluidnumerics.com) that enables one to draw the geometry in a program like [inkScape](https://inkscape.org) and then write a control file to be meshed. A demo is included on the [gitHub](https://github.com/FluidNumerics/svg-to-HOHQMesh) site, and on [YouTube](https://www.youtube.com/watch?v=xvZwEHcwYOc). The pumpkin mesh in the HOHQMesh [gallery](https://trixi-framework.github.io/HOHQMesh/Gallery/) was generated with svg-to-HOHQMesh.

Benchmarks/PlotFiles/Benchmarks/SeaMountCubic.tec

@@ -49,21 +49,33 @@ The data file will have the number of nodes as the first line, followed by the d
 Defining a Straight Line:
 
       \begin{END_POINTS_LINE}
-	 	name   = <name>
-	 	xStart = [x,y,0]
+	 	   name   = <name>
+	 	   xStart = [x,y,0]
            xEnd   = [x,y,0]
       \end{END_POINTS_LINE}
 
 Defining a Circular Arc:
 
       \begin{CIRCULAR_ARC}
-	 	name 		 = <name>
-		units 	 =degrees/radians(Optional.Default:radians)
-	 	center 	 = [x,y,0]
-           radius 	 = r
-		start angle = Tstart
-		end angle   = Tend
+	 	   name 	 = <name>
+		   units 	 = degrees/radians(Optional.Default:radians)
+	 	   center 	 = [x,y,0.0]
+          radius 	 = r
+		   start angle = Tstart
+		   end angle   = Tend
       \end{CIRCULAR_ARC}
+Defining an Elliptic Arc:
+
+		  \begin{ELLIPTIC_ARC}
+			   name    = <name>
+			   center  = [x, y, 0.0]
+			   xRadius = rX
+			   yRadius = rY
+			   start angle = Tstart
+			   end angle   = Tend
+		      units 	   = degrees/radians (Optional. Default:radians)
+		      rotation     = <angle> (Optional. Default 0.0)
+	   	  \end{ELLIPTIC_ARC}
 
 Chaining curves:
 

Benchmarks/StatsFiles/Benchmarks/SeaMount.txt

@@ -14,4 +14,5 @@ This is a collection of comments on some of the approximations and assumptions m
 
 6. The sizer controls spread the region out using a Gaussian profile. The square root (for distance) and exponentials used are very expensive. Meshing times for meshes with sizer controls are significantly longer than those without. Since the actual shape of the regions is not important, consider using a quadratic polynomial approximation to a Gaussian instead to perhaps speed things up. In practice, of all the horrendous algorithms, the size controls seem to be the only ones that are costly.
 
-7. Sizing due to topography is chosen according to its Gaussian curvature. The Gaussian curvature is approximated by second order finite differences, and is described in the document [approximation of Gaussian curvature](ApproximationOfGaussianCurvature.md).
+7. Sizing due to topography is chosen according to its principal curvatures. Background details on principle curvature are available in [Section 3.4](https://web.mit.edu/hyperbook/Patrikalakis-Maekawa-Cho/node30.html) of the online textbook "Shape Interrogation for Computer Aided Design and Manufacturing" by N. M. Patrikalakis, T. Maekawa, and W. Cho. The formulas to compute the principle curvatures for explicit surfaces
+are found in [Section 3.5.1](https://web.mit.edu/hyperbook/Patrikalakis-Maekawa-Cho/node32.html) of the same textbook. In HOHQMesh these formulas are approximated by second order finite differences.