A novel quasi-statistical approach to improve the quality of triangular
meshes is presented. The present method is based on modeling of an
event of the mesh improvement. This event is modeled via modeling of a
discrete random variable. The random variable is modeled in a tangent
plane of each local domain of the mesh. One domain collects several
elements with a common point. Values of random variable are calculated
by modeling formula according to the initial sampling data of the
projected elements with respect to all neighbors of the
domain. Geometrical equivalent called potential form is constructed for
each element of the domain with a mesh quality parameter value equal to
the modeled numerical value. Such potential forms create potential
centers of the domain. Averaging the coordinates of potential centers
of the domain gives a new central point position. After geometrical
realization over the entire mesh, the shapes of triangular elements are
changed according to the normal distribution. It is shown
experimentally that the mean of the final mesh is better than the
initial one in most cases, so the event of the mesh improvement is
likely occurred. Moreover, projection onto a local tangent plane
included in the algorithm allows preservation of the model volume
enclosed by the surface mesh. The implementation results are presented
to demonstrate the functionality of the method. Our approach can
provide a flexible tool for the development of mesh improvement
algorithms, creating better-input parameters for the triangular meshes
and other kinds of meshes intended to be applied in finite element
analysis or computer graphics.