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Unstructured hexahedral non-conformal mesh generation

dinsdag, 20 december, 2005 - 16:00
Campus: Brussels Humanities, Sciences & Engineering campus
auditorium P. Janssens
Konstantin Kovalev

Mesh generation is a process of spatial subdivision of a generally complex domain into
simple-shaped sub-volumes of pre-defined topology. These sub-volumes are usually
referred to as mesh elements or cells. Elements of a mesh are connected to each other,
they do not intersect with each other and cover the entire domain.

The problem of generation of computational meshes arose as a pre-requisite for
numerical analysis of physical phenomena. When computational power of new
computers increased sufficiently for modeling physics using discrete approaches such as
finite-difference, finite-volume and finite element methods, spatial discretization emerged
as a critical issue. On one hand, all requirements of numerical methods for computational
meshes should be met during mesh generation. For example, these requirements may
include limitations on types of elements allowed in the mesh, on mesh quality, or on
connectivity between mesh elements. On the other hand, meshing tools should be capable
of successful treatment of the widest possible class of geometries. These two factors
provide a perpetual challenge for creators of mesh generation tools for numerical

The recent evolution of numerical analysis and design methods, combined with quickly
growing range of simulated phenomena and the constantly increasing complexity of
involved geometries, provide a high demand for powerful automatic meshing tools.
Meshing became a necessary component of industrial design loops and a decrease in its
cost is as important as that of other components. Moreover, meshing fills the gap between
geometry processing in CAD systems and numerical analysis. Therefore, automation is
the ultimate direction of evolution of up-to-date mesh generation tools.

Although the problem of automatic tetrahedral mesh generation has practically been
solved and is successfully employed in the industry, there is still a high degree of interest
in fully automatic hexahedral mesh generation. An obvious reason is that hexahedral
elements are naturally suitable for numerical analysis for both finite-volume and finiteelement
methods. Besides, for a certain number of nodes, fewer hexahedra are required to
mesh a domain as compared to tetrahedra. It means that less memory is used for the mesh
and solution data storage and a shorter time is required to compute the solution.
Therefore, automatic hexahedral unstructured mesh generation remains to be one of the
biggest challenges among industrial demands for numerical design.

In recent years attempts to solve the problem of automated unstructured hexahedral
meshing have been focused on the few principal approaches reviewed in the following
sections. Among others, the so-called Overlay method is considered as one of the most
robust automatic tools capable of dealing with geometries of practically arbitrary
complexity. This approach is considered as one of the most promising directions towards
full automation of hexahedral meshing process. Certain critical aspects of this approach
constitute the subject of this thesis.

Untangling concave cells. The term untangling describes a transformation of a cell shape
from concave to convex. Generally, the issue of improvement of mesh cell quality
represents a combination of two operations applied to mesh cells, one of which is
untangling. Its goal is to convert concave cells into convex ones via finding new
appropriate positions for their vertices. Accordingly, the second operation is
optimization, which is aimed at improving cell quality according to a certain quality
measure. The problem of untangling consists in finding any convex cell shape, while the
goal of optimization is to find a convex cell shape of best quality. The former can be
solved via linear programming methods that are generally fast and not demanding.

Optimization of mesh cells. As the untangling process recovers only invalid (concave)
cells, it is clearly insufficient for a global mesh quality improvement. Even if all mesh
cells are valid, the low quality of mesh cells (for instance, too small dihedral angle or
another quality measure) may degrade not only the convergence rate of computations but
also the accuracy of final results. Mesh optimization procedure is aimed at direct
improvement of mesh cell quality. It is based on repositioning mesh vertices to minimize
a certain functional. This functional expresses the deformation energy of a cell with
respect to shape of an undeformed cell. Generally, this functional represents a non-linear
function of mesh vertex coordinates. Unlike the untangling procedure, solving an
optimization problem is usually quite time-consuming.

Viscous mesh generation. Modern methods for viscous computations as well as integrated
physical models often require high quality of computational meshes. This includes
smoothness of viscous grids as well as control of position of the first mesh point close to
wall, depending on the type of a turbulence model. Viscous layer insertion methodology
developed in this thesis addresses the above issues. It combines the Laplacian smoothing
with the untangling and optimization procedures in order to inflate Euler mesh cells
adjacent to solid boundaries (i.e. increase their normal sizes). The inflated cells are
refined tangentially until a desired number of anisotropic cell layers is inserted with their
vertices being redistributed in normal direction.

Mesh deformation. There exists a wide class of physical phenomena modeling of which
requires flow analysis in domains of shapes that vary during computation. This class
encompasses multiple coupled problems with moving or deforming interfaces, fluidstructure
interaction problems and many others. Solving such problems demands that a
computational mesh follows deformation of domain boundaries. Therefore, two main
possibilities exist. The first one is to fully re-mesh the domain every time its shape is
modified. The second is to deform an existing mesh according to deformation of its
boundary. However, in many phenomena mentioned above, relative deformation of mesh
boundary is small and full re-meshing appears to be too costly. Instead, deforming an
already existing mesh may be quick and reliable provided that appropriate tools are
available. Development of such a tool for unstructured hexahedral non-conformal meshes
is addressed in the thesis. The proposed methodology preserves mesh topology and
modifies only positions of vertices. Displacements of boundary points are propagated to
volume mesh points. Optimization loops are incorporated in order to ensure high mesh