mesh

Mesh generation and adaptation

• L. Chen and J. Xu. Optimal Delaunay triangulations. Journal of Computational Mathematics, 22(2):299-308, 2004.

This is the first paper we propose a new concept: optimal Delaunay triangulation (ODT) for the mesh adaptation.

• L. Chen. Mesh smoothing schemes based on optimal Delaunay triangulations. In13th International Meshing Roundtable, pages 109-120, Williamsburg, VA, 2004. Sandia National Laboratories.

We then apply the ODT to the mesh smoothing to improve the mesh quality. The algorithms are developed for any dimensions but the numerical examples are restricted to 2-D. Later on, a French research group did the 3-D numerical examples using our approach. See ftp://ftp-sop.inria.fr/geometrica/alliez/vtm.pdf

• L. Chen, P. Sun, and J. Xu. Optimal anisotropic simplicial meshes for minimizing interpolation errors in $L^p$ norm.Mathematics of Computation, 76(257):179-204, 2007.

This is a paper devoted to the anisotropic interpolation error analysis on ODT although the concept ODT is not explicitly stated out.

• L. Chen. New analysis of the sphere covering problems and optimal polytope approximation of convex bodies. Journal of Approximation Theory, 133(1):134-145, 2005.

We apply ODT to obtain sharp estimates of some constants in the sphere covering problem and optimal polytope approximation of convex bodies

• L. Chen. On minimizing the linear interpolation error of convex quadratic functions and the optimal simplex.East Journal on Approximations, 14(3), 271--284, 2008.

This is a paper to explicitly write out some constants which are important in the theory of approximation.

• Alliez, Pierre and Cohen-Steiner, David and Yvinec, Mariette and Desbrun, Mathieu. Variational tetrahedral meshing. ACM Transactions on Graphics. 24(3):617--625, 2005.

 ...our results indicate that in practice, our minimization procedure generates well-shaped 
tets inside the domain, with better radius ratio distribution curves than any of the tet
meshes we came across.  

• Tournois, Jane and Wormser, Camille and Alliez, Pierre and Desbrun, Mathieu. Interleaving Delaunay refinement and optimization for practical isotropic tetrahedron mesh generation. ACM Trans. Graph.. 28(3):1--9, 2009.

 Among the large body of work in mesh opti mization, the
Optimal Delaunay Triangulation approach (ODT for short) stands out, as
it casts both geometric and topological mesh improvement as a single,
unified functional optimization [Chen and Xu 2004; Chen 2004] that
tries to minimize in R4 the volume be- tween a paraboloid and the
linear interpolation of the mesh vertices lifted onto the paraboloid.