a general finite difference method for arbitrary meshes

Computers & Structures, Vol. 5, pp. 45-58. Pergamon Press 15’75. Printed in Great Britain

A GENERAL FINITE DIFFERENCE METHOD FOR ARBITRARY MESHESt

NICHOLAS PERRONES and ROBERT KAO§

Catholic University of America, Washington, D.C., U.S.A.

(Received 23 January 1974)

Abstract-A two-dimensional finite-difference technique for irregular meshes is formulated for derivatives up to the second order. The domain in the vicinity of a given central point is broken into eight 45 degree pie shaped segments and the closest finite-difference point in each segment to the center point is noted. By utilizing Taylor series expansions about a central point with a unique averaging process for the points in the four diagonal segments, good approximations to all derivatives up to the second order and including the mixed derivatives are obtained. For square meshes the general derivative expressions for arbitrary meshes which were determined reduce to the usual finite difference formulae. In one example problem the Poisson equation is solved for an irregular mesh. In a second example for the first time a problem with a geometric nonlinearity, namely large deflection response of a flat membrane, is solved with an irregular mesh. The solutions compare very favorably with results obtained previously. Some discussion is given on possible approaches for determination of finite difference derivatives higher than the second.

INTRODUCTION

The astonishingly rapid development of high speed computer technology has had a tremendous impact on the area of numerical methods. Complex problems which were beyond reach of analysis in pre-computer days are now routinely attacked numerically, and usually quite successfully.

One viewpoint is that there are two broad avenues of approach in the quest for numerical solutions: variational techniques used in conjunction with energy formulations and direct solution of the governing differential equations. Connection between both approaches is evidenced by the equivalence between boundary value problems of differential equations on one hand and problems of calculus of variations on the other.

In one variational approach, the Rayleigh-Ritz method, simpler approximating extremum problems substitute for the variational problems; hence, only a finite number of parameters need be determined. A major difficulty of this method is in the construction of coordinate functions, the combination of which is used as a solution function; these functions usually are required to satisfy prescribed boundary conditions.

To remove this obstacle a modified form of the Rayleigh-Ritz method, the finite element method, arose by considering a generic small piece of a domain and subsequently tacking all these elements together. In this modified version, the finite element method could be used to deal with complex structures with various boundary conditions.

In the direct differential equation approach, there are many numerical methods. One which stands out as being universally applicable to both linear and nonlinear problems is the method of finite differences.

For one-dimensional problems, this method has been used with great success for either regular or irregular mesh spacings. However, in two-dimensional analyses, the application of the method is still largely limited to regular meshes. Among other things, this limitation poses some difficulty when applied to problems with irregular boundaries. Consequently, the development of finite differences for irregular meshes is highly desirable.

It has been observed by Zienkiewicz[l] that one of the obvious critical differences between finite difference and finite element techniques is the ability of the latter to treat irregular domains.

Finite differences have recently been combined with the energy formulation through the use of Green’s formula so that derivatives can be expressed in terms of functions at intermediate

The research reported on here was supported by the Office of Naval Research (Contract NR 064-452) and the National Science Foundation (GK 23747).

SAdjunct Professor of Mechanics and also Director of Structural Mechanics Program, Office of Naval Research, Arlington, Virginia.

#Research Assistant Professor, Department of Civil and Mechanical Engineering. C.A.S., Vol. 5. No. I-D

45

46 NICHOLAS PERRONE and ROBERT KAO

nodal points [2,3]. This technique was applied to a one-dimensional shell analysis in [4,5] and to two-dimensional shell analyses in[6,7]. The major disadvantage of this approach is the difficulty arising from the determination of the second order derivatives; its application to general problems still remains to be seen.

Use of two-dimensional Taylor’s series expansion in irregular meshes can be considered as an alternative to the above method. In[8-101, a 6 control point scheme was adopted for obtaining derivatives of up to the order of two. But only few applications to the solutions of problems were made since this technique is offset by two disadvantages: unpredictable singularity in the derivative coefficient matrix and limited accuracy of derivatives obtained.

In this paper, we continue the investigation of irregular finite difference techniques and suggest a scheme which can not only avoid a singularity in the coefficient matrix of derivatives but also improve the accuracy of derivatives obtained.

In the next section, the procedure for obtaining derivatives for a two-dimensional irregular mesh up to the second order is described. A scheme is then proposed to avoid the singularity in derivative coefficient matrix and to improve the accuracy of derivatives obtained. As an aid to programmers, some key steps in the computer program are outlined. The technique is then applied to solve a linear Poisson equation and a non-linear large deflection square membrane problem. Some prospects on problems with higher order derivative terms are examined. In the final section, a discussion and conclusions are presented.

DIFFERENCE COEFFICIENTS FOR IRREGULAR MESHES

In this section, the technique for obtaining derivative expressions at a given point in terms of functions at its neighbors is described[&lO]. For any sufficiently differentiable function f(x, y) in a given domain, its Taylor series expansion about a point, say (x0, yO) can be written as

(1)

where

f = f(x, y), f0 = f(xo, y,,), h = x -x0, k = y - y. and A = d(h2 + I?).

The last term in this equation gives the error order of magnitude in f caused by premature termination of the series.

Equation (1) indicates that five independent equations are needed to obtain the five different derivatives at point (x0, yO), i.e. the values of f at five neighboring points are required.

After writing equation (1) for each of five neighbors of point (x0, yO) and putting them in a matrix form, we immediately arrive at

h, kl h,2/2 k1=/2 h,k,

h2 k2 htl2 kz% h2kz

h3 k, h,2/2 k,2/2 h&3

hq k, h:l2 kc+% Lb

h, ks h;l2 k,2/2 h&x

or symbolically

[Al {OfI = ifI 5X55X1 5x1

=

fl -fo

fz-fo

f3 -fo

f4 - fo

fs-fo

(24

(2b)

where fi = f(xi, yi), hi = Xi - ~0, ki = y, - y. for i = 1,2, . . ., 5.

A general finite difference method for arbitrary meshes 47

Derivatives {Of} in equations (2) can be obtained by various methods for solving linear algebraic systems[ll, 163 such as the method of Gaussian elimination, the technique of matrix inversion, etc. provided that [A] is non-singular. The matrix [A] is said to be singular if its determinant is equal to zero.

Obviously, if the function f(x, y) is a polynomial of order two, then the derivatives obtained from equations (2) are exact. On the other hand, if the function is a polynomial of order higher than two or any other function, the derivatives obtained are just approximate and their accuracy will depend on higher order derivatives as well as the mesh size utilized. Moreover, an examination of the error in f(x, y) suggests that, in addition to mesh size utilized, relative position among the nodal points is also a decisive factor affecting derivative approximations.

The main obstacles to the success of previous attempts [8-lo] are the difficulties in avoiding a singularity or an ill conditiont in matrix [A] of equations (2) and in obtaining acceptable derivatives. Singularity or ill condition in matrix [A] of equations (3) makes it impossible to obtain derivatives; poor derivative approximations might yield an unsatisfactory result for a given problem or even cause divergence.

In this paper, general remedies to these two shortcomings are suggested and will be discussed separately in the following sections.

TECHNIQUE TO AVOID SINGULARITY IN DERIVATIVE MATRIX

As stated earlier, if the determinant of a square matrix vanishes, then the matrix is said to be singular. Some important situations resulting in a singular matrix are those for which rows or columns of the matrix are linearly connected by certain relations [ 111.

In this connection, some typical examples for matrix [A] of equations (2) being singular will be given here. The first example is as shown in Fig. I(a) in which all 6 nodal points including p. are located on the same line. The singularity problem for this situation is attributed to a proportionality arising in [A] between columns 1 and 2. In the second (Fig. lb) and third (Fig. lc) examples, all five neighbor points of p. are respectively lined up horizontally and vertically in terms of PO local coordinate system.

These configurations require a linear relationship to occur in [A] between columns 1 and 5 for Fig. l(b), and columns 2 and 5 for Fig. l(c). In the final example (Fig. Id), there is also a linear relationship in [A] between columns 1 and 3 with a proportionality factor of A/2. Clearly, all of these possibilities should rarely occur, but should such an arrangement somehow emerge, it should be easily recognized and trivially rectified.

Realizing the importance of nodal point distribution, we select a 6 control point scheme for arbitrary mesh to avoid singularity in [A]: the area around p. is divided into eight different zones each of which is a pie-shaped segment with central angle of 45deg, Fig. 2. Segments I and III are bisected by the x’-axis (horizontal line through po); segments II and IV are bisected by the y’-axis (vertical line through po); segments V-VIII as well as the coordinate directions are bisected by 45deg lines through po. Points 1-4 are located in segments I-IV, and point 5 is situated in one of the segments V-VIII. Points l-5 are the closest to the center point for each respective segment wherein they are located.

Before going on to discuss Fig. 2 in connection with the singularity problem, we observe that careful examination of [A] in equations (2a) indicates that the rare singularity usually is not due to linear dependency in rows but rather in columns. Moreover, a further review of Fig. 2 and equations (2a) reveals that the only possibility of having a singularity problem in [A] will be a proportionality between column 5 and either columns 1 or 2. For example, if singularity is due to columns 1 and 5 in [A] of equation (2a), then there must exist a situation such that

hl hz h, h4 hs 1 -=-=-=-=-=- htk, hzkz hsk, h,k, h,k, k’

According to Fig. 2, k2 and k4 can not have the same sign, and the only chances for the above relation to be true are: (a) both ht and hq to be zero and the other three points be lined up horizontally related to po-coordinate system; (b) one of hz and h4 to be zero and the other point be

tThe matrix [A] is said to be ill conditioned when its condition number cond (A) = /IA\/. llA-‘11 is large[l6], where J/All is a norm of matrix [A]. Singularity is essentially an extreme form of an ill conditioned matrix.

NICHOLAS PERRONE and ROBERT KAO

Fig. la. Singularity in [A] from proportionality in columns 1 and 2, equations (2a).

Fig. lb. Singularity in [A] from proportionality in columns 1 and 5, equations (2a).

Fig. lc. Singularity in [A] from proportionality in columns 2 and 5, equations (2a).

I Y

Dt. I pt.4 t .

co,A) (A,A,

II % pt. 3 -X (O,O) (A,O)

pt. 2 (0,-A, pt. 5 .

‘(A.-A)

Fig. Id. Singularity in [A] from proportionality in columns 1 and 3, equations (2a).

together with points 1, 3 and 5 on the same horizontal line parallel to po-coordinate system. A similar argument follows for the case of

k, kz k, kzt ks 1 _=-=-=-- --=- h,k, hzkz h,k, h4k4 hsk, h’

A general finite difference method for arbitrary meshes

pt.Scan be pts. A, 8. C or D

Fig. 2. Six control point scheme for an arbitrary mesh.

One arrangement of Fig. 2 to deny both possibilities is to have the fifth point not be lined up with points 1 and 3 horizontally and with points 2 and 4 vertically related to po-coordinate system. And this can easily be accomplished by inserting a simple conditional check in the computer program by comparing ks with k1 and kS, and hs with h2 and L

An ill condition of [A] of equations (2a) may arise if the fifth point nearly lines up with points 1 and 3 horizontally or with points 2 and 4 vertically related to po-coordinate system. In this connection, the conditional check just described above can also provide some indications on the degree of this unwanted allignment of mesh points; thus, a criterion may be added in order to determine whether or not a redistribution of nodal points is required. With this conditional check on the fifth point, Fig. 2 should produce a determinate nodal set.

Actually, the open angle in each segment can be varied; for example, segment I and II can be greater or less than 45deg while segment V can also be greater or less than 45deg. From experience, it is observed that if the area of each segment around its center line is smaller, better accuracy of derivatives would usually be expected.

Once possibility of singularity or ill-conditioning in [A] is dismissed, we can proceed to discuss how to improve the derivatives obtained based on the scheme shown in Fig. 2.

A SCHEME TO IMPROVE ACCURACY OF DERIVATIVES

One way of assessing the accuracy of derivative approximations obtained from equations (2) (Fig. 2) is to employ a square mesh as shown in Fig. 3 and to compare known values for a given function such as the following

f(x, y) = sin (7r.x) cos ( > ;Y 9 -lSX,YSl

with computed values obtained by central finite differences. This comparison indicates that the values obtained by usual central finite differences agree closely with the actual values; but those obtained by equations (2) (using points l-5 of Fig. 3) provide a poor approximation for the mixed

6 2 5 -

3 0

!lFj

( I

b-a

f

b=a

7 6 4

I a a

Fig. 3. Square md


derivative a*fJdxay, whereas the remaining computed derivatives are identical with those obtained by usual central finite differences.

The comparison is indeed very interesting and an understanding of the cause of the result should be quite useful in improving the current technique for obtaining better derivative approximations. For this reason, we substitute the coordinates of points l-5 of Fig. 3 into equations (2a). It turns out that the first and third equations are sufficient to solve for @Jax and a2f0/dx2, and the second and fourth for af,,/ay and a%/ay’; these four expressions so obtained are identical with what are obtained by usual central finite differences. We next substitute these four expressions in the fifth equation of equations (2a) and solve for the mixed derivative

a2fo f5-f2+fo-f, -= axay a2

while usual central finite differences give

a’fo fs-f6+f7-f8 -= axay 4a2 ’ (4)

Now, it should be obvious from equations (3) and (4) that the values of the mixed derivative at point 0 differ when computed via usual central finite differences or the current approach. Actually, in the current 6 control point scheme, we do not include points 6, 7 and 8 at all. It appears that the only way in which the current approach will provide a better mixed derivative expression is to include these three points in some way.

With this idea in mind, we write equations (2a) again but replace point 5 in the fifth equation by point 6 and solve for the mixed derivative; we repeat this procedure two more times replacing point 5 by points 7 and 8, respectively. It is significant that averaging all four sets of mixed derivative expressions (calculated by using points 5,6,7, or 8 in turn as the “fifth” point) yields the same result as equation (4).

Based on this experiment, a 9 control point scheme similar to Fig. 2 is now proposed for arbitrary meshes as shown in Fig. 4; in fact, this figure is the same as Fig. 2 except that points A, B, C and D are replaced by points 5,6,7 and 8, respectively. The regular square mesh of Fig. 3 with associated usual finite difference derivatives turns out to be one special case of Fig. 4 and results in identical derivative expressions. All derivatives associated with this scheme take the average value from the solutions of four sets of 5 X 5 equations, each set containing the same first four equations as those of equations (2a) while the fifth equation is related to points 5,6,7 and 8, respectively.

For the square mesh of Fig. 3 the averaging process is somewhat superfluous for the four non-mixed derivatives as they have the same value independently of the choice of the fifth point (which is, in turn, 5,6,7 or 8). However, for an arbitrarily spaced mesh distribution, the averaging process should influence, to some extent, all derivative values.

x’. v’ are coordinates . pt. 2

#‘I/

local to PO @

“_^ _ 0pt.5

Fig. 4. Nine control scheme for an arbitrary mesh.


DERIVATIVE EXPRESSIONS

51

The derivative expressions for point 0 corresponding to Fig. 4 now can be written as:

afo/ax = colfo+ Cl’fl + * * * + cl”fS afo/ay = coZfo + c12fl + - * . + c8*f8

a’fo/ax* = co’fo+ c,2f, + * * . + cs’fs a*fo/ay* = co4fo + c,4f, + - * * + cs4f8 a2fo/axay = co’fo + c,5f, + . - * + cs5fse

(5)

As just discussed in the previous section, the coefficients in these equations are obtained from four sets of solutions of equations (2) and may be given as follows:

c~=-~~,~,b2,

c’L=;i b;, 1=1,...,4 for k=l,2,...,5 ” I

(6)

elk = ; b .&’ 1 = 5,. . ., 8

where b;, is an element of n -th set 5 x 5 matrix [B] = [A]-‘; these four sets of matrices are all related to points l-4 except that the fifth points in these matrices are points 5, 6, 7 and 8, respectively.

It may be advisable to explain briefly how to obtain equations (6). For this purpose, the derivation of afo/ax is taken here as an example. From equations (2), the expressions for this derivative corresponding to four sets of these equations may be written as follows:

n=l

$’ = b :,cf, - fo) + b :d.fz - fo) + * * . + b :5cf5 - fo)

=-$,b:f+b:,f,+*..+b:,f,+b:,f5

n=2

$=-2, b:mfo+ b:,f,+

n=3

,=-$,b:mfo+b:,f,+ afo

n =4

+ b :dfd + b :sfei (7)

+ b :dfd + b :5f,

afo ,=-&:mfo+b:,f,+...+b;~f~+b;sff..

Taking an average value on these four sets of equations will yield the expression as those given in equations (5) and (6).

Numerical results show that the derivative approximations obtained by the 9 point scheme (equations (5), Fig. 4) are better than those from the 6 point scheme (equations (2), Fig. 2) when compared with theoretical solutions. Moreover, as a consequence of the discussion in the previous section, the usual central finite difference is a special (and verifying) case of equations (5) when a square mesh is used. Hence, equations (5) and their corresponding arbitrary nodal point configuration, Fig. 4, will be adopted for the analysis in this paper.


COMPUTER PROGRAMMING

For a problem defined in a given domain, there are, in general, three preliminary steps needed for the method proposed herein before a solution attempt can be initiated: (1) generation of mesh points, (2) choice of the 8 nearest points, and (3) determination of derivative coefficients in equations (5) for each point in the domain.

Figure 5 represents a scheme of mesh points generated for some example problems treated in this paper. In general, distribution of mesh points depends on the behavior of the function in each individual zone; for the area in which function changes are rapid, more nodal points are usually required. An automatic mesh generation technique[12] may be used for complex structures to save manpower, computer time and schedule time.

I- I

4 Fig. 5. Irregular meshes on one quarter of a square region.

After mesh points in the domain are arranged we select the 8 nearest points for each point in this domain. The choice of these 8 points must meet the requirements as described before. This can be done either automatically by computer or manually.

For convenience, we may define these points as L = INEAR (I, 1): the Jth nearest point associated with point I (Fig. 4) is point L. After INEAR (I, .I) is obtained for every point in the domain, we can move to determine derivative coefficients in equations (5). These coefficients may be defined symbolically as C(I, .I, K) in which K is referred to the 5 different types of derivatives as given in equations (5).

Both INEAR (I, J) and C(I, .I, K) are obtained once and for all for a given mesh point arrangement. Once these quantities are defined, one can proceed to obtain numerical solutions.

We now physics, the

SOLUTION TO POISSON EQUATION

apply the technique discussed to solve a fundamental problem in mathematical Poisson equation,

with boundary conditions

+(x,y)=O at x,y=kl. (9)

The term on the right hand side of equation (8) is chosen such that the theoretical solution


associated equations (8) and (9) can be in the form of

In view of double symmetry along x = 0 and y = 0, we need consider only one quarter of the domain as shown in Fig. 5. In this figure, distribution arbitrary, 64 interior points (from points 1 to 64) and 17 boundary points (from points 65 to 81). Due to conditions

procedure, as in the previous section, consists of of the 8 nearest points, INEAR (I, J), and determination derivative coefficients

replacing derivatives in equation (8) with their discrete analogs, (5), we obtain a system of whose matrix is in the form of

[Al (41 = IPI 64x6464~1 64x1 (11)

where [A] is a square matrix in terms of C(I, J, K), and (4) and (p} are the column matrices related to 4 values and the values on the right side of equation (8) at each of interior points, respectively.

Solution of equations (11) can be obtained by established methods such as relaxation method[l3,15], inversion of matrix, etc. The result obtained here compared with theoretical solution, equation (lo), is very good: the maximum error is of the order of 0.79 per cent at point 28 (x = 0.08; y = 0.83) (the numerical value 0.261789 vs theoretical 0.263858); the minimum error is of 0.056 per cent at point 60 (x = 0.6; y = 0.89) (0*101055 vs 0~101111); overall absolute average error is 0.605 per cent.

The computer time for obtaining the 8 nearest points for all interior points is about 25 set of PDP-10 CPU time, and that for derivative coefficients as well as the solution of equations (11) by inversion of matrix is about 20 sec.

The current neighbor point search technique is involved in comparing distances between a given point and all other points in the domain. However, this technique can be improved by comparing with only a limited number of nodal points and hence neighbor point search time can be reduced greatly.

For a regular mesh 5 control points ((0, O), (A, 0), (-A, 0), (0, A) and (0, -A)) are sufficient to yield the derivatives which appeared in equation (8) since no mixed derivative is included. Additionally, in view of the well behaved nature of the solution function, equation (lo), a solution was attempted based on a 6 control point scheme of an arbitrary mesh. The result obtained is quite acceptable although its accuracy is not as good as that obtained based on a 9 control point scheme.

I a

a

1

Fig. 6. Geometry of a finite square flat membrane.


LARGEDEFLECTI~NOFSQUAREFLATMEMBRANE

The second example problem treated is a doubly symmetric square flat membrane undergoing large deformations, Fig. 6; this problem was solved previously in[ 141 using a nonlinear relaxation technique in conjunction with a square finite difference mesh.

The differential equations in nondimensional quantities are reported as follows [ 141:

2u,,, + (1 - V)u,YY + (1+ V)& + (1 + V)W,YW,XY + 2W,XW,XX + (1 - V)W,XW,YY = 0 (13)

(1 - vhxx + 2v,n + (1 + U)&y + (1 + V)w,Xw,xy + 2w,Yw,YY + (1 - V)w,Yw,XX = 0 (14)

where ( ),X = (a/8x)( ), ( ),, = (a/ay)( ), E, v are material constants, h is the thickness of the membrane and p uniform transverse loading, u, v and w are displacement components in x, y and z directions, and

K = 2(1- V’).

Pertinent nondimensional terms are given in equations (1.5) and associated dimensions in Fig. 6.

x = 5/a, y = qjb

( > z/3

U=va E .

Boundary conditions are assumed to be fixed:

w=u=v=O at x,y=?l. (16)

Due to double symmetry only one quarter of the domain is considered; the same mesh points (Fig. 5) used in the previous example are utilized here again.

In this problem, we have 3 x 64 nonlinear algebraic equations to be solved; a matrix of 192 x 192 is required in solution process. To cut down the size of matrix, an alternative approach is adopted: three sets of equations (resulting from equations (12-14), respectively) are solved separately so that three displacement matrices of size 64 x 64 are required. More detail about this alternative strategy is given in[14,15]. To obtain solutions to d system of nonlinear equations, a modified Newton-Raphson techniquet is employed.

Before going to an irregular mesh (Fig. 5) we begin with a regular mesh of 8 x 8 which was used previously in[14]. The current solution, using the 9-point general scheme of Fig. 4, is i’dentical with that reported in[14].

No additional difficulty is encountered in obtaining the solutions for irregular meshes (Fig. 5). The solutions are displayed in Fig. 7 in which the deflections are plotted against center lines along with the results obtained by regular meshes in [ 141. A good correlation between these two sets of results is observed. The computer time taken for each of the solutions (regular and irregular meshes) is about the same, i.e. about 5 min PDP-10 CPU time.

The convergence criterion used is that after obtaining u and v solutions from equations (13) and (14) (they are linear in u and v, respectively, for the alternative strategy employed) if, for one

tin this modified version, the matrix elements are recalculated only after a given number of iterations.

A general finite difference method for arbitrary meshes 5.5

,Normot deflection curve, w

A-along x-ox/s, irregular meshes(ftg.5)

o-regulormeshes&8),obtoined in this paper and ref. [14]

x-v alongy-axis (irregulor,fig.5)

A-U along x-0x1s

-u or v along xory axis (regularBxB),obtained in this paper and ref.[l41

0 0.1 0.2 0.3 O-4 0.5 O-6 0.T O-8 0.9 I.0

x-or y-oxis

Fig. 7. Deflection profiles along symmetric lines of a finite square flat membrane (Fig. 6).

iteration of the w problem (equation (12)), the average absolute change of the functions at all nodes is less than O$Ot, then the co~esponding solution is said to be acceptable. Otherwise, the same iteration procedure is repeated until this criterion is satisfied.

The convergence criterion used for the regular mesh situation is that a complete w, u and u solution is considered to have been obtained if, after yielding u and v solutions, a single iteration of the w problem produces an average absolute change of the functions at all nodes which is less than O-0005. In fact, it was not successful to use this criterion for obtaining irregular mesh solutions. However, the convergence in w iteration, equation (12), is very rapid (u and u are held as constants) and the average absolute change of w functions can be brought down to as small as desired.

This reflects that the difficulty in satisfying better accuracy for the first iteration of w equations (equation (12)) (when the iteration procedure is reverted back from u and v iterations) is not due to nonlinearity in w equations but rather due to instability in u and Y solutions (equations (13) and (14)). This instability is arising from less accuracy in u and u derivatives than that in w because u and v behave like sin (mr) cos {7ry/2) and cos (7rx/2) sin (?ry), respectively, while w is like cos (TX/~) cos (ry/2); it is equivalent to saying that the w function is smoother than u and v functions.

PROBLEMS WITH HIGHER ORDER DERIVATIVES

The technique discussed herein is used only to obtain the derivatives up to the second order. For problems such as shells and plates, derivative terms involved are usually up to the fourth order. In general, 14 neighbor points are required to obtain all derivatives up to the fourth order at a given point. Selection of these nodal points may become cumbersome.

Instead of attacking the problems by solving governing differenti~ equations directly, one may go along an alternative route by using an energy formulation in conjunction with the method of finite differences. In the energy expression, the highest order of the derivatives is only one-half of what is encountered in the differential equations; actually no derivatives of order higher than two appear in the energy expression (such as shells and plates). Hence, the derivative expressions derived herein, equations (5), are sufficient for this purpose.

This finite difference energy method was applied to solve a one-dimensional shell problems by Bushneil[4,5], and to solve a two-dimensional reactor diffusion equation using rectangular meshes by Forsythe and Wasow[8]. In the combination of energy formulation and finite differences of arbitrary meshes, a difficulty one may encounter is how to effectively and precisely define the integral area. Other difficulties may arise from the need of introducing fictitious points beyond the boundary when the integral area aIong boundary points is to be defined. Overcoming these di~~u~ties constitutes a major problem area for future research.


In[4,7], schemes were suggested to define the integral area using different contra1 points for u (or v) and w functions; similar approaches could be adopted with the variable mesh techniques discussed in this paper.

DISCUSSION AND CONCLUSIONS

A two-dimensional finite difference technique for irregular meshes is proposed herein to obtain derivatives up to the second order. The domain in the vicinity of a given central point is divided into eight 45 degree pie shaped segments and the closest finite difference point in each segment is selected. By the use of Taylor series expansions about a central point, with an averaging process on the four points in the diagonal segments, good approximations are found for derivatives up to the second order including the mixed derivative. When applied to square meshes, these general derivative expressions for arbitrary meshes reduce to the usual finite difference formulae.

The technique is then utilized to solve a Poisson equation and a flat membrane problem involving geometric nonlinearity, both for irregular meshes. Solutions compare very closely with results obtained previously. It should be noted that the large deflection membrane problem represents the first nonlinear problem, to the knowledge of the authors, solved with an irregular finite difference mesh field.

In the application of the current technique to an irregular mesh field, two key factors should be noted which may very well determine whether the procedure can be successful in attempting a solution for a given problem. These two factors are: (a) avoidance of a singularity problem in matrix [A] of equations (2), and (b) ability to obtain good derivative approximations.

A singularity problem makes it impossible to obtain derivative approximations. The proposed 6 control point scheme (Fig. 2) plus an added conditional check on the fifth point of Fig. 2 as described earlier are designed to avoid the singularity problem in [A].

Good finite difference approximations are fundamentally essential when good accuracy of numerical results for a given problem is desired. Good derivative approximations are especially needed when solving nonlinear problems. In many nonlinear problems, the solution procedure usually involves repeated iterations; poor derivative approximations may cause this iteration procedure to be divergent. On the other hand, one can always (except for singularity situations) find a solution for a linear problem; however, its accuracy certainly very much depends on how close are the derivative approximations.

To obtain good derivative approximations, nodal point distributions around a central point as well as the strategy by which derivative approximations are obtained should be carefully considered. The 9-point scheme (Fig. 4) proposed herein tends to have nodal points roughly uniformly distributed around the central point, and takes advantage of the unique averaging process with which the error terms in each set of equations, equations (2), can be partially balanced out. Thus over all error can be greatly reduced.

It may be mentioned here that regular meshes seem to yield better accuracy of derivative approximations than those for irregular meshes. This suggests that for a problem involving a curved boundary, advantage can be taken by using regular meshes in the interior of the domain and irregular meshes along the boundary.

For problems with higher order derivatives, either more nodal points could be employed around each central point, or the current 9-point scheme could be utilized in conjunction with an energy formulation. The former approach may be cumbersome but is possible. Research aimed towards these directions should be highly desirable.

REFERENCES

[ 11 0. C. ZIENKIEWICZ, The finite element method: from intuition to generality, Appl. Me& Rev. 249-256 (March 1970). 121 R. COURANT and D. HILBERT, Method of Mathematical Physics. Vol. 1, Inter-science, New York (1937). [3] R. S. VARGA, Matrix Iteratioe Analysis. Prentice-Hall, New Jersey (1962). [4] D. BUSHNELL, Finite-difference energy method versus finite element models: two variational approaches in one

computer program, Paper presented at International Symposium on Numerical and Computer Methods in Structural Mechanics, Urbana, III. (September 1971).

[5] D. BUSHNELL, Analysis of buckling and vibration of ring-stiffened, segmented shells of revolution, ht. J. Solid Structures, 6, 157-181 (1970).

[6] B. BUDIANKY and D. ANDERSON, Numerical shell analysis-nodes without elements, 12th International Congress of Applied Mechanics, Stanford University, (August 26-31, 1968).

A general finite difference method for arbitrary meshes 51

[7] D. E. JOHNSON, A difference-based variational method for shells, Int. J. Solid Structures, 6, 69%724 (1970). [8] G. FORSYTHE and W. WASOW, Finite-difference Methods for Partial Differential Equations. John Wiley, New York

(19f(o) [9] L. COLLATZ, The Numerical Treatment of Differential Equations. Springer, Berlin (1966).

1101 P. S. JENSEN. A finite difference techniaue for variable mids. uresented at Conference on Com&er Oriented Analysis of . _ Shell Structures, Lockheed Missiles ani Space Company, <alo Alto, Ca. (August 1970). .

_ ~

[11] R. FRAZER, W. DUNCAN and A. COLLAR, Elementary Maltices. Cambridge University Press (1960). [t2] H. A. KAMEL and H. K. EISENSTEIN, Automatic mesh generation in two and three dimensional interconnected

domain, Paper presented at the Symposium on High Speed Computing of Elastic Structures, IUTAM, Liege, Belgium (August 23-28, 1970).

1131 N. PERRONE and R. KAO, A general nonlinear relaxation technique for solving nonlinear problems in mechanics, J. Appl. Mech. 38, 2 (June 1971). -

[14] R. KAO and N. PERRONE, Large deflection of flat arbitrary membranes, J. Comput. Struct. 2, 535-546 (1972). [15] N. PERRONE and R. KAO, Large deflection response and buckling of partially and fully loaded spherical caps, AIAA

J. 8, 12, 2130-2136 (December 1970). [16] G. S. FORSYTHE and C. S. MOLER, Computer Solution of Linear Algebraic Systems. Prentice-Hall, New Jersey

(1%7).

a general finite difference method for arbitrary meshes

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