Fast Spline based optimization of edges with Matthek‘s spline and spiro spline for finalizing best results

[stefano2734]

1. Step:
   Matthek’s Triangles spline:
   This is a theme for snappy hex meshed and the volume fluid optimizer in openfoam.
   And also for the stress optimization of the solid solver.

See
http://bibliothek.fzk.de/mattheck/mattheck_postersammlung.pdf

and in Detail
https://publikationen.bibliothek.kit.edu/220061912/3814714

Look in the poster and you see the triangle method with 3 Triangles to construct spline points.
With this points a good edge curve is the result without FEM for beams, shells and solids or CFD for fluids.

there is a pdf Poster in german with stress Relation of matthek spline by triangles to circle arc .
https://publikationen.bibliothek.kit.edu/220076489

so an Optimization of edges of the calculation room at the beginning is here simple possible.

A result can be a model by splines by intelligent interpolation of the mesh surface.

2. Step:
   Spiro Splines by Levien is here an interesting solution and better than bezier splines.

Circular Arcs are simple but not the best curves in nature and in technics.
Bezier Splines are limited in precision and mathematics.
Spiro splines are Derivats of spirals.

See phd Levien at https://levien.com/phd/phd.html

There are here two things

1. Improvement of bezier splines:
   In this PhD are some pages for better bezier splines
   See page 135 and more with error picture in page 141.

2. The new spiro spline from Levien:
   The spiro spline is more than a better bezier spline.

See also https://www.levien.com/spiro/

with some code for his phd and his font program.
and for more clearness:
https://raphlinus.github.io/curves/2018/12/21/new-spline.html

The code in Inkscape with the spiro path tool was improved in last years.
So Inkscape source code is here perhaps a better source with spiro splines also for this project.

So Openfoam can calculate optimized meshes with splines surfaces for better smooth surface results back to CAD.
With both splines at beginning and end the evolution of best contours is much faster and elegant possible.