Triangular mesh optimization.

``` on your machine and you're good to go. For commercial use, please contact support@mondaytech.com. ### Using optimesh Example call: ``` optimesh in.e out.vtk ``` Output:  The left hand-side graph shows the distribution of angles (the grid line is at the optimal 60 degrees). The right hand-side graph shows the distribution of simplex quality, where quality is twice the ratio of circumcircle and incircle radius. All command-line options are documented at ``` optimesh -h ``` ### Showcase  The following examples show the various algorithms at work, all starting from the same randomly generated disk mesh above. The cell coloring indicates quality; dark green is bad, yellow is good. #### CVT (centroidal Voronoi tessellation) |  |  |  | | :---------------------------------------------------------------------------------------------------------------------------: | :------------------------------------------------------------------------------------------------: | :------------------------------------------------------------------------------------------------: | | uniform-density relaxed [Lloyd's algorithm](https://en.wikipedia.org/wiki/Lloyd%27s_algorithm) (`--method lloyd --omega 2.0`) | uniform-density quasi-Newton iteration (block-diagonal Hessian, `--method cvt-block-diagonal`) | uniform-density quasi-Newton iteration (default method, full Hessian, `--method cvt-full`) | Centroidal Voronoi tessellation smoothing ([Du et al.](#relevant-publications)) is one of the oldest and most reliable approaches. optimesh provides classical Lloyd smoothing as well as several variants that result in better meshes. #### CPT (centroidal patch tessellation) |  |  |  | | :-------------------------------------------------------------------------------------------------------------------------------------: | :----------------------------------------------------------------------------------------------: | :----------------------------------------------------------------------------------------------: | | density-preserving linear solve ([Laplacian smoothing](https://en.wikipedia.org/wiki/Laplacian_smoothing), `--method cpt-linear-solve`) | uniform-density fixed-point iteration (`--method cpt-fixed-point`) | uniform-density quasi-Newton (`--method cpt-quasi-newton`) | A smoothing method suggested by [Chen and Holst](#relevant-publications), mimicking CVT but much more easily implemented. The density-preserving variant leads to the exact same equation system as Laplacian smoothing, so CPT smoothing can be thought of as a generalization. The uniform-density variants are implemented classically as a fixed-point iteration and as a quasi-Newton method. The latter typically converges faster. #### ODT (optimal Delaunay tessellation) |  |  |  | | :------------------------------------------------------------------------------------: | :----------------------------------------------------------------------------------------------: | :--------------------------------------------------------------------------------------------------: | | density-preserving fixed-point iteration (`--method odt-dp-fp`) | uniform-density fixed-point iteration (`--method odt-fixed-point`) | uniform-density BFGS (`--method odt-bfgs`) | Optimal Delaunay Triangulation (ODT) as suggested by [Chen and Holst](#relevant-publications). Typically superior to CPT, but also more expensive to compute. Implemented once classically as a fixed-point iteration, once as a nonlinear optimization method. The latter typically leads to better results. ### Using optimesh from Python You can also use optimesh in a Python program. Try ```python import optimesh # [...] create points, cells [...] points, cells = optimesh.optimize_points_cells( points, cells, "CVT (block-diagonal)", 1.0e-5, 100 ) # or create a meshplex Mesh import meshplex mesh = meshplex.MeshTri(points, cells) optimesh.optimize(mesh, "CVT (block-diagonal)", 1.0e-5, 100) # mesh.points, mesh.cells, ... ``` If you only want to do one optimization step, do ```python points = optimesh.get_new_points(mesh, "CVT (block-diagonal)") ``` ### Surface mesh smoothing optimesh also supports optimization of triangular meshes on surfaces which are defined implicitly by a level set function (e.g., spheres). You'll need to specify the function and its gradient, so you'll have to do it in Python: ```python import meshzoo import optimesh points, cells = meshzoo.tetra_sphere(20) class Sphere: def f(self, x): return 1.0 - (x[0] ** 2 + x[1] ** 2 + x[2] ** 2) def grad(self, x): return -2 * x # You can use all methods in optimesh: points, cells = optimesh.optimize_points_cells( points, cells, "CVT (full)", 1.0e-2, 100, verbose=False, implicit_surface=Sphere(), # step_filename_format="out{:03d}.vtk" ) ``` This code first generates a mediocre mesh on a sphere using [meshzoo](https://github.com/nschloe/meshzoo/), and then optimizes. Some results: |  |  |  | | :-------------------------------------------------------------------------------------: | :------------------------------------------------------------------------------------------: | :--------------------------------------------------------------------------------------------: | | CPT | ODT | CVT (full Hessian) | ### Which method is best? From practical experiments, it seems that the CVT smoothing variants, e.g., ``` optimesh in.vtk out.vtk -m cvt-uniform-qnf ``` give very satisfactory results. (This is also the default method, so you don't need to specify it explicitly.) Here is a comparison of all uniform-density methods applied to the random circle mesh seen above: (Mesh quality is twice the ratio of incircle and circumcircle radius, with the maximum being 1.) ### Why optimize? | | | | | :----------------------------------------------------------------------------------------: | :-------------------------------------------------------------------------------------------------: | :----------------------------------------------------------------------------------------: | | Gmsh mesh | Gmsh mesh after optimesh | [dmsh](//github.com/nschloe/dmsh) mesh | Let us compare the properties of the Poisson problem (_Δu = f_ with Dirichlet boundary conditions) when solved on different meshes of the unit circle. The first mesh is the on generated by [Gmsh](http://gmsh.info/), the second the same mesh but optimized with optimesh, the third a very high-quality [dmsh](https://github.com/nschloe/dmsh) mesh. We consider meshings of the circle with an increasing number of points: |  |  |  | | :-----------------------------------------------------------------------------------------: | :-----------------------------------------------------------------------------------: | :-------------------------------------------------------------------------------: | | average cell quality | condition number of the Poisson matrix | number of CG steps for Poisson problem | Quite clearly, the dmsh generator produces the highest-quality meshes (left). The condition number of the corresponding Poisson matrices is lowest for the high quality meshes (middle); one would hence suspect faster convergence with Krylov methods. Indeed, most CG iterations are necessary on the Gmsh mesh (right). After optimesh, one saves between 10 and 20 percent of iterations/computing time. The dmsh mesh cuts the number of iterations in _half_. ### Access from Python All optimesh functions can also be accessed from Python directly, for example: ```python import optimesh X, cells = optimesh.odt.fixed_point(X, cells, 1.0e-2, 100, verbose=False) ``` ### Relevant publications - [Qiang Du, Vance Faber, and Max Gunzburger, _Centroidal Voronoi Tessellations: Applications and Algorithms_, SIAM Rev., 41(4), 1999, 637–676.](https://doi.org/10.1137/S0036144599352836) - [Yang Liu et al., _On centroidal Voronoi tessellation—Energy smoothness and fast computation_, ACM Transactions on Graphics, Volume 28, Issue 4, August 2009.](https://dl.acm.org/doi/10.1145/1559755.1559758) - [Long Chen, Michael Holst, _Efficient mesh optimization schemes based on Optimal Delaunay Triangulations_, Comput. Methods Appl. Mech. Engrg. 200 (2011) 967–984.](https://doi.org/10.1016/j.cma.2010.11.007)