Page 163 - Basic Structured Grid Generation
P. 163
6
Variational methods and adaptive
grid generation
6.1 Introduction
Despite the availability of a variety of algebraic and differential models for grid genera-
tion, many practitioners experience substantial difficulties when applying these models
to new problems. The grids generated may turn out to be badly skewed (with large
departures from orthogonality), compressed, or expanded, folding may occur, and
sometimes even convergence fails. Ever since grid generation methods began to be
studied seriously in the late 1960s, variational approaches have been used in attempts
to attain a deeper understanding of the limitations and strengths of the various differ-
ential models. In this chapter we present an introduction to variational methods and
show how they may be used to control grid ‘quality’.
The quality of the grids used to discretize the physical domain when solving par-
tial differential equations very much affects the accuracy of the numerical solutions
obtained. Given that cost constraints allow only a limited number of grid nodes to be
introduced into a solution domain, one aspect of quality is related to the density of grid
nodes in regions where there are large gradients of field variables. A grid-generating
scheme should be able to allocate more grid nodes where large gradients occur (and
fewer where field variables vary smoothly). Moreover, in transient problems where the
solution develops in time, regions of high gradients may not be known apriori,and
there is therefore a need for some mechanism which can automatically detect regions
of high gradients as they arise, so that grid density can be increased there. In other
words, a grid generation procedure may be coupled to the numerical solution, produc-
ing grids which may be called solution-adaptive,or dynamically adaptive. Algorithms
which automatically concentrate and disperse grid nodes in this way are called adap-
tive. Adaptive methods may also be used in steady-state problems when regions of
high gradients are not known in advance.
As well as the optimum spacing of grid points, the quality of grids depends on
factors such as cell-areas (or volumes) and the angles between grid lines (how far
a grid departs from orthogonality). An ideal structured grid would be an orthogonal
grid with grid-node density able to cope with sharp solution gradients. In many cases,
however, the geometrical complexity of the physical domain makes the construction of
such a grid difficult or impossible, and a compromise has to be reached. The attraction