Surface Remeshing by Local Hermite Diffuse Interpolation

2nd Symposium on Trends in Unstructured Mesh Generation, University of Colorado, Boulder, August 1999

MESHING
RESEARCH
CORNER

LG2MS, Universit, de Technologie de CompiSgne, Centre de Recherche de Royallieu, 60206 CompiSgne Cedex
alain.rassineux@utc.fr

CGES, Ecole des Mines de Paris, 33, rue Saint-Honor,, 77305 Fontainebleau

Abstract
We propose a method to build a three dimensional adapted surface mesh with respect to a mesh size map driven by surface curvature. The data needed to optimize the mesh have been reduced to an initial mesh. The building of a local geometrical model but continuous over the whole domain is based on a local Hermite diffuse interpolation calculated from the nodes of the initial mesh and from the normal vectors to the surface. The whole mesh optimization procedure can be described as follows:

Singularities such as sharp edges, contour lines and singular points are identified.

The geometrical support is build by a weighted least squares approximation method on a local window denoted as diffuse approximation. In our case, interpolating weights have been chosen. The objective is to determine a local surface equation (second degree) using the nodes of the initial mesh and the normals to the surface calculated from the mesh. The interpolating nodes belong to the set of elements sharing at least one node with the element that includes the point where we calculate the interpolation.

Local coordinates are calculated by projection on an average plane. Our goal is to measure the maximun bending of the surface by calculating the principal curvatures given by the fundamental forms on a Monge patch z=f(x,y). This measure is linked to our geometrical error estimator. The diffuse interpolation leads to the minimization of a criterion which is provided by nodal interpolation and colinearity of normal vectors. The minimization leads to solve a linear system in order to determine the coefficients of the surface.

The determination of a local equation enables us to locate nodes on the surface and on the contours with respect to the curvature during a refinement process. It also allows us to control the coarsening of the mesh. An error estimator measuring the accuracy with which the mesh describes the geometry has been used. The error is given by determining an approximate deformation gradient tensor between a reference configuration and the current configuration.

Mesh optimization procedures based on optimum mesh size are carried out in an iterative process. The method involves extracting sub-shells from the surface mesh which are then remeshed to improve their quality.

References

Nayroles B., Touzot G., Villon P.r L'approximation diffuse _, C.R.Acad. Sci, Paris 313, s,rie II, pp 293-296, 1991

A. Rassineux. 'Generation and optimization of tetrahedral meshes by advancing front technique', International journal for numerical methods in engineering, vol 41, pp 651-674, 1998.