Achieving Finite Element Mesh Quality via Optimization of the Jacobian Matrix Norm and Associated Quantities, Part 1 - A Framework for Surface Mesh Optimization Page: 4 of 23
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issue that is reserved for a later paper is that of mesh tangling. The goal in the
present paper is to propose and analyze reasonable objective functions and compare
how they perform in benign situations to identify those which are most promising for
application to more difficult problems. This paper only considers schemes in which
the nodes are moved with fixed connectivity. The topic of edge-swapping in which
mesh connectivity is changed as well as nodal positions will be considered within the
present framework in a later paper'. Another important issue, control of the mesh on
the boundary, will also be considered in a subsequent paper.
The outline of this paper is as follows: Section 2 describes the basic building
blocks used in constructing 2D nodally-based objective functions. Section 3 discusses
a number of objective functions, most of them previously known either from varia-
tional grid generation or structured grid optimization, but interpreted in a new light
using matrix norms. Section 4 on implementation issues considers the gradients of
the building blocks given in section 2. Gradients of complete objective functions are
given in Appendix I. Section 4 also discusses various global and local approaches to
mesh optimization. Section 5 gives results of several test problems for the objective
functions. The paper finishes with section 6, a summary and conclusions.
2. Building Blocks for Nodally-Based Objective Functions
In finite element meshing, the goal of optimization is often stated to be that of im-
proving element quality. As with finite element analysis, the focus is on the mesh
elements. Such a perspective leads to formulation of mesh optimization functions in
terms of element quality metrics such as skew, taper, aspect ratio, and warpage for
quadrilateral elements  or aspect ratio and maximum/minimum angle for trian-
gular elements. Although optimization of element quality is the main goal, the direct
approach of formulating objective functions in terms of element quality has a number
of drawbacks. First, geometric descriptions of element quality can lead to non-convex
objective functions. Lack of convexity generally means that the existence of a min-
imum is not guaranteed. Another potential problem with element-based objective
functions is that they are cumbersome to implement in a local fashion and difficult to
define generally enough to include both triangular and quadrilateral elements. In spite
of these difficulties, element-based mesh optimization schemes are feasible. This paper
takes an indirect approach which focuses not on the element quality but on the quality
of the geometric quantities centered about each node of the mesh. This approach is
motivated by the way in which the partial differential equations of structured grid
generation are solved using finite difference methods, in which a node-centered stencil
plays the key role. Capitalizing on work done in , it is shown here that discrete
objective functions for finite element meshes can be formulated in terms of quantities
that are analogous to, but not the same as, the Jacobian matrix and metric tensor
of structured meshing. Convexity is naturally obtained using nodal-based objective
functions because neighbor nodes create conflicting requirements which can only be
satisfied in a least-squares sense. Another significant advantage of the node-centered
approach is that it applies equally well to quadrilateral, triangular, and mixed ele-
ment meshes. It is an open question to determine the conditions under which a mesh
1 In unstructured mesh optimization techniques, the term "smoothing" is widely used to refer to
moving the nodes of the mesh with fixed connectivity in order to obtain high quality meshes. In
structured meshing the term smoothing usually refers to some notion of ellipticity of an underlying
PDE. The proper term for general node movement schemes should be optimization, with smoothing
reserved for optimization methods having a connection with ellipticity.
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Knupp, P.M. Achieving Finite Element Mesh Quality via Optimization of the Jacobian Matrix Norm and Associated Quantities, Part 1 - A Framework for Surface Mesh Optimization, article, January 18, 1999; Albuquerque, New Mexico. (digital.library.unt.edu/ark:/67531/metadc682780/m1/4/: accessed January 17, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.