The Galerkin weighted residual technique using linear triangular weight functions is employed to develop finite difference formula in cartesian coordinates for the Laplacian operator, first derivative operators and the function for unstructured triangular grids. The weighted residual coefficients associated with the weak formulation of the Laplacian operator are shown to agree with the Taylor series approach on a global average. In addition, a simple algorithm is presented to determine the Voronoi (finite difference) area of an unstructured grid

Baumeister, Kenneth J.

English

NASA Technical Reports Server

NASA Technical Memorandum 101298
Discretization Formulas for
Unstructured Grids
(I_SA-T_-IO12s8) LISCI_IZAa]t.h
ECL_ 0hS_I_Cc_I01il;/_ 61_IES (I_AS_) _CR/SULA5
12 PCSCL 20D _-102_2
Kenneth J. Baumeister
Lewis Research Center
Cleveland, Ohio
Unclas
G3/34 0166358
Prepared for the
Second International Conference on Numerical Grid Generation
in Computational Fluid Dynamics
cosponsored by the National Aeronautics and Space Administration,
the U.S. Air Force Office of Scientific Research, and the University of Miami
Miami Beach, Florida, December 5-8, 1988
https://ntrs.nasa.gov/search.jsp?R=19890000871 2020-03-20T06:00:43+00:00Z
DlSCRETIZATION FORMULAS FOR UNSTRUCTURED GRIDS
Kenneth J. Baumeister
National Aeronautics and Space Administration
Lewis Research Center
Cleveland, Ohio 44135
ABSTRACT
The Galerkin weighted residual technique using linear tri-
angular weight functions is employed to develop finite differ-
ence formula in cartesian coordinates for the Lap]acian
operator, first derivative operators and the function for
unstructured triangular grids. The weighted residual coeffici-
ents associated with the weak formulation of the Laplacian
operator are shown to agree with the Taylor series approach on
a global average. In addition, a simple algorithm is presented
to determine the Voronoi (finite difference) area for an
unstructured grid.
INTRODUCTION
In developing unstructured finite difference equations,
Jameson If, Eq. (4.]0)] has applied the standard weighted
residual Galerkin method to obtain a time dependent discretiza-
tion of the Euler equations for an unstructured triangular
mesh. Erlebacher [2] has utilized variational methods on a
pointwise basis to establish a difference operator for the
Laplacian operator for a central difference cell similar to
that shown in Fig. I. The present paper will utilize the
global Galerkin weighted residual technique to also generate
slmp]e closed form finite difference approximations for the
Laplaclan operator, first derivative operators, and the func-
tlon itself for the general central cell in Fig. l, as well as
the boundary element cell also shown in Fig. I. The boundary
element cells are required in acoustics and electromagnetic
theory in establishing the far field radiation conditions.
GALERKIN WEIGHTED RESIDUAL FORMULATION
Consider the following partial derivatives:
_/ " 611/dP " 132 6____. 133 6_____ 134¢ (I)
• 8x ' Gy '
¢ is a scaler or potentlal quantity and the 6's are variable
property coefficients. To obtain the finite difference expres-
slons for the terms in Eq. (I), the continuous domain D is
first divided into discrete triangular areas Ae staked out by
nodal (grid) points P as shown in Fig. I. The number of con-
nected areas (called a cell) needed to define the difference
equation for node o is labeled M.
T-
LpO
L01
_L_/
711111111111//1111111111111111111111111111111,
(p) (Xl"Y]_(1) TOTAL(2)
_l_j (x2'Y2)_ AREA
AT
(x3,Y 3)
X_'_)I (3) (3) (P)
BOUNDARY ( J ) (k)
CELL GENERAL CENTRAL CELL
P-R+1 P-PI
'//////////////////////////////////////////I
FIGURE 1. - GLOBAL FESH COfiFIGURMIOH.
!
The difference equations developed in Appendices A and B
take on the form
(62s$/sx)
(13365/6Y)
(645)
= kodPO + kl¢ 1 + k2¢ 2 + ... + kn¢ n + ... + kp4pp
(2)
_ [-9W i • 619 _] dA = _o¢o + c_i¢I + =2¢2 + ... +=p¢p
AT
(3)
where the Laplaclan operator is converted to the later form by
means of the weak formulation [3, p. 443] of the weighted
residual approach. The values of the coefficients k and
depend only on the location of the grid points and the proper-
ties 6 The superscript • represents either the derivative
in x (* = x) or y (* = y) and the variable itself (* = ¢)
The coefficients k* for ¢ and its first derivatives are
included in Appendix C.
2
The major effort of the present work is to establish the
relationship between the Laplacian operator and the _ coeffi-
cients from the method of weighted residuals. An expression
for the Laplacian of the form
V2¢ = f(_l, a2 .... ) (4)
is desired. Using the result from Appendix D for a linear
shape function, the Laplacian can be expressed as
V2¢ : _0¢° + _I¢I + _2¢2 + "'" + _PCP + r
AT
3
(5)
where AT is the total area of the cell. The _ values in
Eq. (5) are given as
an = -[B(nm)/4Anm][lyn_ - Yn)lYo - Yn_) + _x n - Xn_)IXn_ - Xol ]
x _I- 6n,P_M)- [IS(n)/4An] [CYn- Yn+)<Yn+ - Yo) + _Xn+- Xn)
x IXo- Xn+)]ll - 6p+l_n,P_Ml (6)
M
_o = __--_ [B (e)
e:1
/4Ae][lYe - Ye+l 2 + _Xe+ - Xe)2 ] (7)
£ : 3 IS(1)V¢ • n bc T + A_T I_(M)9¢ • n bc T 6M,P-I
(8)
where the subscripts are defined in Appendlx C. The term E
Is the contribution from the natural boundary condition that
comes automatlcally from the weak formulatlon of the method of
weighted residuals and contributes only on the boundary of the
domain.
DISCUSSION OF RESULTS
Uniform Grids
For the conventional six-point hexagon difference grid as
shown in Fig. 2, the difference equations for the Laplacian,
first derivatives, and the function as calculated from Eqs. (2)
and (5) are in agreement with the standard difference operators
that appear in the literature [4, p. II14]. Similarly, the
very popular six-node system shown in Fig. 3 also shows agree-
ment. For brevity only the Laplacian operator will be shown in
Fig. 3 and on the remaining figures.
As is well known in the literature [3, p. 105], the method
of weighted residual and the conventional Taylor series expan-
sion often yield different results. For the conventional four-
point square difference grld as shown in part A of Flg. 4, the
difference equation for the Laplacian Is d|fferent by a factor
of 3/2 from the conventional Taylor series representation of
the Laplacian operator [4, p. Ill4]. Similarly, the eight-node
system shown in part B of Fig. 4 differs by a factor of 3/4.
However, since the method of weighted residual (finite element
theory) is guaranteed to converge, these are acceptable formu-
lae provided they are applied to a consistent global mesh and
not a single arbitrary cell.
Relationship Between FE and FD Methods
For uniform grids, of the type displayed in Figs. 2 to 4,
the finite difference and finite element expressions for the
Laplacian operator will be in agreement with the conventional
Taylor series difference equation provided that the bias con-
stant B, defined by the following area rule, is identical to
zero.
WEIGHTED
RESIDUAL
CELL BOUNDARy 7
h "/_ I/
k_ , _ II"I .I i
} -(5)
'-FINITE DIFFERENCE
CELL BOUNDARY
B=O
¢ =
FIGURE2. - UNIFORM 6 MODEHEXAGON6RIDSYSTEM.
_ _'x_ _FINITE DIFFERENCE
'i'x_ CELL BOUNDARY
\: I4
--_ v2 ¢ = h2
(5) (6)
FIGURE 3. - UNIFORM G NODE SLANTED GRID SYSTEM.
¢1 +¢2 +¢4"¢5 - _¢o
4
F-,-H
(q) (3) (2)
(3_(I) (5) _ _ (I)
(_) (6) (7) (8)
(A) (B)
B,C4_OBAL 1
- - x 2h2 ',' (-l/q) qh2 - 0
2
X
/
V
,/
(C)
FIGURE 4. - GLOBAL CONVERGENCE EXAMPLE WITH RECTANGULAR ELER£NTS.
B = 3Av/A T - l (9)
Av is the Voronoi neighborhood [4] or the area normally
employed in the finite difference analysis. If B _ O, the
coefficient in front of the finite element expression will dif-
fer from the expression for the Taylor series expression by a
factor of I + B. If B = O, a single pattern of uniform ele-
ments can normally be placed in a regular domain.
Global Relation To Taylor Series
Although the difference equations for the Laplacian opera-
tor in Fig. 4 differ in each cell from the standard Taylor
series difference equation, on a global average the Taylor
series and finite element equation will average out to the same
value. In this case the area average of the eight-node ele-
ment combined with the four-node element will be such that the
Taylor series will be valid. As shown in Fig. 4, the average
global value of B is equal to zero. (B is a measure of the
difference between the finite element and Taylor series as just
defined.) A similar situation occurs for triangular grids,
although the results are not shown.
Nonuniform Grids
Figure 5 displays the difference equations for a nonuni-
form five-node cell. The Laplaclan operator shown in Fig. 5
is only valid when taken in conjunction with the other grid
systems that surround it in a given domain. It is not to be
thought of as a Taylor series approximation for the cell shown
in Fig. 5.
91 _L2_
-I I I i i I
- I 3 5 7 9
AXIAL COORDINATE. X
'[2.?'07_) 1 +_2 + 19.385¢3+ E.q61_ 4 " 11.077_)5+ 1.38.- ql.615¢01172¢ = 0.86q 80.88
FIGURE 5. - NO_UNIFORM G NODE GRID SYSTEM.
Voronoi (Finite Difference) Area
The determination of the Voronoi area plays a major role
in the mesh generation by triangulation of arbitrary points
[2]. The coefficients used to generate the Laplacian operator
can also be used to easily determine the Voronoi area without
geometric construction;
AV: 4 * 4 + 4 * "'"
+ 4
(lO)
where _'s are given by Eqs. (6) and (7). The rationale for
development of this simple algorithm to predict the Voronoi
area comes from the fact that replacing AT/3 by Av in
Eq. (5) fortuitously satisfies the Taylor series approximation
for uniform grids. (However, it does not |n general for non-
Av when B = 0.) Therefore, If ¢uniform grids, and AT/is assumed to vary as x + y2]/4 then the Laplacian for uni-
form grids will have a value of unity and Eq. (lO) results.
To validate this hypothesis, Eq. (lO) was checked against the
nonuniform cell shown in Fig. 6, as well as a large variety of
other nonuniform cells, and found to always be in exact agree-
ment with geometrical calculations. For certain complex cells,
the Voronoi boundaries can cross as shown in Fig. 6. In these
cases, the algorithm will consider reversed areas as negative,
in that it senses the direction of the path of the Voronoi
region. Again the algorithm agrees with the geometric
calculation.
9(2) AT = 10
%=2-,,=-2
(q) (6) \', ""
- f _('_)1 { I I
- 13 5 7 9
AXIAL COORDINATE, X
FIGURE G. - C_a_.CULATIOHOF VORONOI _ FOR
G NODE GRID WITH REVERSAL.
Local Convergence
If B = O, Eq. (5) combined with Eq. (9) for central cells
yields
_2¢ = (o_0¢0 + _i¢i + _2¢2 + ... + _pCp)/Av (ll)
where Av is the Voronoi area of the cell and the m values
are given in Eqs. (6) and (7). Equation (ll) was checked
against a number of nonuniform grids (six or less nodes) where
B _ 0 and found to satisfy the test functions l, x, y, x2
and _2. However, the test function xy was not satisfied'
when B _ O. For the special case, as shown in Fig. 5, where
B ~ O, Eq. (ll) nearly satisfied the xy function test.
Therefore, Eq. (ll) may be valid on a pointwise bases for the
special condition of B = O. More testing will be required
for va]idat_on.
CONCLUSIONS
The Galerkin weighted residual technique using linear
triangular weight functions is employed to develop finite dif-
ference formulae in cartesian coordinates for the Laplacian
operator, first derivative operators and the function for
unstructured triangular grids.
REFERENCES
I. Jameson, A.: Successes and Challenges in Computational Aero-
dynamics, AIAA Paper 87-I184, AIAA 8th Computational Fluid
Dynamics Conference, Honolulu, Hawaii, June 9-11, 1987.
2. Erlbacher, G.: Finite Difference Operators on Unstructured
Triangular Meshes, Paper in Lecture Notes in Physics,
Vol. 238, Springer-Verlag, NewYork, 1985, pp. 22-53, edi-
tors Frltts, M. J., Crowley, W. P., and Trease, H.
3. Lapldus, L. and Pinder, G. F.: Numerical Solution of Par-
tial Differential Equations in Science and Engineering,
John Wiley & Sons, New York, 1982.
4. Rektorys, K., Ed.: Survey of Applicable Mathematics, MIT
Press, Cambridge, MA, 1969.
APPENDIX A - DERIVATION GALERKIN WEIGHTED RESIDUAL EQUATIONS
In accordance wlth the method of weighted residuals,
weight functions Wo in the form of a linear pyramid are
introduced (one for each grid point), such that the integra-
tion of the product of each term in Eq. (1) times the weight
Wo over the cell area shown in Fig. l yields
Xf ' XI
e=l Ae
AT
[_,, (e) }]
-v,,o , [v(e_] {¢(e) dA
(Al)
e=l A
AT e
(A2)
where the subscript x stands for the x derivatives of the
known shape functions and N stands for the linear interpola-
tion functions whose x and y dependence is presented in
most finite element texts [3, p. Ill, Eq. (3.1.4)]. The equa-
tions for the derivative of ¢ with respect to y and the
function itself are similar to Eq. (A2). The Laplacian opera-
tor in Eq. (AI) has utilized the weak formulation of the
weighted residual formulation [3, p. 443].
APPENDIX B - AVERAGED OPERATORS
Assuming averaged finite difference values over the
weighting function area allows the function and first deriva-
tives to be written explicitly. For example,
' 1
e=l A
e
(BI)
In this case, the integral of the weight Wo over the
cell area is equal to AT/3, which was determined with the aid
of the standard area integration formula [3, p. ll2]. Equa-
tion (BI) as well as the equation for the Laplacian, the func-
tion and the derivative of the function with respect to y
can now be easily evaluated using conventional finite element
theory to obtain the difference equations given in Appendix C
and Eqs. (6) and (7) in the body of the report.
APPENDIX C - COEFFICIENTS FOR FUNCTION AND FIRST DERIVATIVES
To develop the difference equations, the Kronecker delta
is defined as
6c,p = O, _ _ p: or 6_,p = l, _ = _ (Cl)
and the following circular indices are also defined
n- = n - 1 + P6n l (C2)
n+ = n + I - P6n, P (C3)
e+ = e + l - M6e, P (C4)
nm= n - l + M6n, l (C5)
with
Ae = [I12] xo(Yj - Yk) + [I12] xj(y k - yo ) + [I/2] xk(Yo - yj)
(C6)
[3, p. 110, Eq. (3.1.2)]. Using this notation, the coeffici-
ents for the following operators can be written as follows:
x derivative operators
+ [13_n)/2AT](Yn+ - Yo)( l - 6p+l_n,p_M) (C7)
M
e=l
9
y derivative operators
]
KY = [B_Rm)/2AT](XR_- Xo)(1- 6n,P_M)
+ (B_n)/2AT)IXo - Xn+)[l- 6p+l_n,P_M] (C9)
M
kY = _ IF&_3)/2ATI[Xe+- Xe] (ClO)
¢ functlon
ken = [B(4nm){Anm/4AT)]<] - 6n,P-M)
+ IB_ n) IAn/4AT)]{I- 6p+l_n,P_M) (Cll)
M
ko@: (I/2 AT ) _-_ ^(e)Ael_4
e=l
(C12)
APPENDIX D - SOLUTION OF THE WEIGHTED RESIDUAL EQUATIONS
Consider the partial differential equation of the form
92¢ + 6@ = 0 (DI)
6x
Employing the weak solution [3, p. 443] for the four-node cell
in part A of Fig. 4 yields
(¢I + ¢2 + ¢3 + ¢4 - 4¢0) + hi(el - ¢3 )/3] : 0 (D2)
To put Eq. (D2) in a more familiar form which coincides with
the standard Taylor series finite difference approach, Eq. (D2)
Is multlplled by 3/A T, which corresponds to 3/2h L for the
geometry shown In Fig. 4. Thus, Eq. (D2) becomes
312 [(¢I + ¢2 + ¢3 + ¢4 - 4¢o)/h2] + (¢I - ¢3 )/2h = 0 (D3)
The 3/A T term was chosen as the multiplying constant because
it will always convert the function and its first derivatives
so that they match the Taylor series expression. Consequently,
the AT/3 term appears in Eq. (5) in the body of this report.
lO
Report Documentation Page
National AeronaulPcS and
Space AOmlmst rahon
1. Repo_ No
NASA TM-I01298
4. T_tleand Subtme
Dlscretizatlon
7. Author(_)
Kenneth J.
Formu Ias
Baumelster
2. Government Accession No.
for Unstructured Grids
3. Rec=p_enrs Catalog No.
5. Report Date
6. Performing Organ=zabon Code
8. Performmg Organization Report No.
E-4094
10. Work Unit No.
505-62-2]
11. Contract or Grant No.
13. Type of Report and Period Covered
Technlcal Memorandum
14. Sponsoring Agency Code
g. Performing Organizatron Name and Address
Natlonal Aeronautlcs and Space
Lewis Research Center
Cleveland, Ohio 44135-3191
Admlnistratlon
12. Sponsoring Agency Name and Address
Natlonal Aeronautics and Space
Hash|ngton, D.C. 20546-0001
Administratlon
15 Supplementa_ Notes
Prepared for the Second International Conference on Numerical Grid Generation in
Computatlonal Fluid Dynamics cosponsored by the Natlonal Aeronautics and Space
Admlnlstratlon, the U.S. Air Force Office of Sclentlflc Research, and the Univer-
sity of Miami, Miami Beach, Florida, December 5-8, 1988.
16. Abstract
The Galerkln weighted residual technique using llnear triangular weight func-
tions is employed to develop finite difference formula In cartesian coordinates
for the Laplaclan operator, first derivative operators and the function for
unstructured triangular grids. The weighted residual coefficients associated
with the weak formulation of the Laplaclan operator are shown to agree with the
Taylor series approach on a global average. In addition, a simple algorithm is
presented to determine the Voronoi (finite difference) area for an unstructured
grid.
17. Key Words (Suggested by Author(s))
Finite difference, Finite element,
Laplacian, Varonoi, Area rule
19. Secunty Classlf. (of this report)
.Unclassified
18. DIstribubon Statement
Unclassified - Unlimited
i Subject Category 34
20. Securl_ Classlf (of this page) 21. No of pages
Unclassified 12
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