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Produced by the NASA Center for Aerospace Information (CASI)
Intrinsic Adaptive Mesh Techniques

(11ESE-CE-175375) INTRINSIC ADAPTIVE BESB B84-17941

TECHNIQUES Final Report, i A;r. - 31 Oct.

1983 (Columbia Uuiv. ) 6 p aC ,02/HF 110 1

CSCL 12A unclas

G3/64 18249

Final Report

Principal Investiga'--s: Peter R. Eiseman and C.K. Chu

(April 1, 1983 to October 31, 1983)

Columbia University

New York, NY 10027

Grant Number 

NASA NAG-1-355
1.

During the grant period, an alternating direction adaptive

grid movement code was developed and, in addition, the thesis

research of Gordon Erlebacher was directed on adaptive triangular meshes. Subsequently, the thesis research was successfully completed and Gordon Erlebacher joined t'1e Computational

Methods Branch at NASA Langley Research Center. The alternating

direction code was also established on the NASA Langley computer

system and is available for use there.

The development of the alternating direction code was

based upon a theoretical study initiated under AFOSR sponsorship

(AFOSR-82-0176) and continued under this NASA grant. The basic

theory along with some modest applications were presented at

the AIAA Computational Fluid Dynamics Conference in Danver< MA

in June,1983 as AIAA paper 1937 on pages 339-348 of the proceedings [1]. The paper has also been accepted for publication

in the AIAA Journal, pending some modest revisions.

In essence, grid points are moved on an "abstract surface"

above physical space by means of alternating coordinate directions.

The abstract surface is formed with the salient solution properties if they can be extracted by a priori physical reasoning;

or otherwise, in the absence of such reasoning, by the use of

error estimates in some chosen norm. Upon formulation, all

important driving properties for adaptive purposes are consolidated into one object - the abstract surface. At a basic level,

a uniform distribution of surface points is equivalent to

gradient resolution. This arises from a projection back down

into physical space. At a higher level, a more accurate view

of the abstract surface is obtained when changes in surface

direction are also resolved. The appropriate measure for direction changes is normal curvature. It is defined as the rate

of change of surface tangent planes as a surface coordinate curve

is transversed in uniform increments of arc length. Such

variations in surface tangents clearly pick up the desired

direciton changes which can become extremely tilted about the
2.

specific coordinate curve employed. Should only curve information be used, the tilting effect of the tangent planes would

be discarded, resulting in a severe loss in the detection of

surface direction changes. The curves used in the algorithm

are coordinate curves on the surface. Taken one at a time,

the points on each are redistributed in cycles which sweep

through each direction in succession. This alternation of

directions and splitting into one dimensional pieces is similar

in spirit to that of ADI numerical schemes.

The alternating direction algorithm represents a substantial synthesis of the generalization beyond previous methods.

In one-dimension, White (2) and Ablow and Schecter t3) used

the arc length of the actual solution by deriving an auxiliary differential equation for movement which subsequently

was simultaneously solved by implicit methods. The differential equations added complexity particularly in the case of

Ablow and Schecter when curvature was used. Even in onedimension, the use of an abstract surface and the explicit

decoupling of movement is advantageous. Not only does the

abstract surface allow a greater freedom of choice to simplify

the movement process, but also it can be smoothed independently of the solution. In cases where the actual solution

contains discontinuities, the computation of curvature would be

greatly in error; thereby, rendering it useless as a clustering property. To smooth the actual solution in order to

retrieve curvature clustering would be a cure which could

kill the solution. By contrast, smoothing the abstract surface

gives the desired movement while retaining the solution. In 

multidimensional problems, to my knowledge, there have been no

prior studies that use a solution surface formulation. If the

abstract surface, however, is trivially taken to correspond

with physical space, then the alternating direction method

reduces to that of Gnoffo [41 when gradient clustering is

used as a property in the weight function. Rhen further

specialization is given by restricting movement to just one

direction, the method of Dwyer et al. (51 is obtained where
r I 1 . 3 

unlike Gnoffo, second derivatives of the solution are also

used in the weight function. Since the second derivatives use

only curve information, the critical tilting measurements of

tangent planes are lost.

Returning to the use of general abstract solution generated

surfaces, the alternating direction development served as the basis

for the development of the adaptive triangular mesh algorithm

with Gordon Erlebacher for his thesis(61. In his thesis, Dr.

Erlebacher formed an abstract surface for a plasma equilibrium

study in a toroidally symmetric tokamak configuration. Over

the tokamak cross section, he prescribed the abstract surface as

a linear combination of toroidal current and inverse flux

gradient squared. The current part resolved the plasma boundary

which could be taken as a free boundary. The flux gradient part

resolved the magnetic axis. By contrast, the solution itself,

which for historical reasons was a first try, had failed to

resolve any of the desired properties. The mesh movement on

the abstract surface was accomplished with a geometrically

constructed molecule which was used in a point iterative sense.

Due to the use of a general connectivity triangular mesh, curvature was mor6 conveniently represented with an approximation of

mean curvature as a Laplacian suitably normalized with the

magnitude of the gradient, rather than by the direct use of

normal curvatures. The finite difference representation of

gradient and Laplacian as exemplified by Fritts and Boris (7) was

also improved upon and analyzed in the sense of local truncation

errors.

In the alternating direction algorithm, the effect of

normal curvature was examined through a sequence of test cases.

In the first instance, a circular bump was examined with a

conflicting rectalinear mesh topology. The use of normal

curvature was seen there to cause coordinate curves to wrap

around the base and top rim of the bump in correspondence with

the regions where direction changes occurred. Upon projection,

the curvature regions were added on to both sides of the

gradient regions; thereby, giving a broader band of resolution.
4.

Since the use of normal curvature caused coordinate curves to 

follow along surface folds as in the case of the bump, it seemed 

reasonable that it could also be used to automatically push

coordinate curves into alignment with disturbances such as

shock waves. As a second sequence of test cases, the alignment

r

issue was examined with an artifically prescribed moving bow

wave in front of a biconvex airfoil. The motion was from a

vectical position to one which bent over and extended downsti`am

from the ai-foil. Grid movement with and without normal curvature was then examined for the same evolutionary abstract

surface. In comparison between these two cases, grid alignment appeared only with curvature and there it appeared quite

well; substantiating our claim.

When a nontrivial boundary in physical space must be

resolved, a lifted form of it appears in the abstract solution

generated surface as a boundary. Since normal curvature measures direction changes of surface tangent planes, changes

within such planes are undetected and are those which belong

to direction changes of curves within the surface. These latter

changes, however, are measured by geodesic curvature. In particular, geodesic curvature is required to detect the above

mentioned surface boundary. An initial examination was begun

on the combined effect of both normal and geodesic curvature.

The primary properties of each were observed in the case of a

disturbance over an airfoil with an O,type grid. However, the

correct balancing between the two has yet to be established in

that case. With rectalinear topologies, by contrast, the

balance is readily done.

- _ -- -- - _ _
5.

REFERENCES

Ill Eiseman, P.R. (1983) AIAA CFD Conf., Danvers, MA, 339-3

[2) White, A.B., (1982) SIAM J. Num. Anal. 19, 683-697.

[31 Ablow, C.M. and S. Schecter (1978) J. Comp. Phys. 27,

351-362.

[41 Gnoffo, P. (1982) Numerical Grid Generation, Ed. by

J.F. Thompson, North Holland , 819-836.

[51 Dwyer, H.A., M.D. Smooke and R.J. Kee (1982) Numerical

Grid Generation, Ed. by J.F. Thompson, North Holland,

339-356.

[61 G. Erlebacher, (1984) Columbia University Ph.D. Thesis.

[7l Fritts, M.J. and J.P. Boris (1979) J. Comp. Phys. 31,

173-215.