Some useful techniques for testing infinite elements P. Bet tess , C. E m s o n and K. Bando

Department of Civil Engineering, University College Swansea, Singleton Park, Swansea SA2 8PP, UK (Received June 1982)

The problems of testing non-standard finite elements are discussed. It is pointed out that the patch test may be inapplicable. Other techniques, short of a full scale analysis are needed for testing infinite elements. A method for testing such elements is proposed. The crux of the method is that a one-dimensional problem is posed which is made artificially to contain the essentially two- (or three-) dimensional decay of the solution. This procedure is illustrated with a range of examples, both static and periodic.

Key words: infinite elements, testing, differential equations

I n t r o d u c t i o n

In recent years numbers of special elements have been introduced in order to extend the scope of the finite element method. Such elements have included crack tip elements, 1'2 global elements s and more recently infinite elements. 4-6 The specialized nature of these elements is reflected in the special difficulties they present at the testing stage. For ordinary finite elements the patch test is a well known and highly respected test method. It has proved invaluable in evaluating and improving the design of various finite elements. However, when it comes to the special elements it does not appear to be very useful. Clearly the usual hope that increased element subdivision will bring increased accuracy is neither true nor meaning- ful for some of these special elements.

Suppose then that a new formulation has been derived for some special situation. How is it to be tested? One possibility is that the prototype element can be slotted into a finite element package and tested, preferably on a problem for which there exists an analytical solution. This is a hazardous undertaking. Almost invariably the first results obtained are garbage. The developer can then do only two things. He can work backwards from the erroneous end results, checking solution, assembly and so on back to the element matrix, and he can check forwards, from the integration abscissae and weights, the shape function, and so on up to the element matrix. If he is lucky all the coding bugs will thus be crushed between two millstones and the correct, or highly accurate solution will pop out. In the case of the first infinite element for wave problems, 7 debugging of the element routine was not achieved until a complete element matrix had been evaluated by hand. This was not a procedure one would choose to repeat. Moreover, it is extremely cumbersome and uneconomical, meaning that

for every new formulation dreamed up in the bath, the researcher must spend many hours of algebra, coding, debugging and running programs, only to find at the end that there is a conceptual error, which nullifies the special formulation.

There is no really general solution to this difficulty, but it has been our experience in the recent development of some infinite elements that the formulation can be tested fairly exhaustively by synthesizing special differential equations for the formulation to solve. Often these new differential equations are not those which, it is hoped, will be eventually solved by the infinite elements, but other equations which are simpler, and preserve the properties of the final differential equation. The process of synthe- sizing and using such differential equations will now be described.

Tes t d i f fe ren t ia l equa t ions fo r inf in i te e l e m e n t s

Static infinite element, exponential decay The first example is that of a simple ordinary differential

equation, used to test a basic infinite element. It is supposed that an infinite element is being developed for exterior potential problems governed by V2¢ = 0 and subject to ¢ ~ 0 as r ~ ~. Such problems have a Green's function of In(r), for two dimensions. For all realistic two-dimensional problems, however, the field is dominated by a dipole solu- tion, which decays as 1/r. In this case the decay function type of infinite element is used and the decay function is chosen to be of exponential type. Two tests are applied. First the formulation is tested to see if it will solve differen- tial equations which it should model, i.e. a quasi patch test (an infinite patch perhaps). Second, it is tested on a one- dimensional 1/r type problem to see how well it will model

0307-904X/82/060436-5/$03.00 436 Appl. Math. Modelling, 1982, Vol. 6, December © 1982 Butterworth & Co. (Publishers) Ltd.

this behaviour. It is important to note that it is of no help to test the element on the one-dimensional version o f Laplace's equation, since solutions of this equation do not exhibit the characteristic decaying behaviour o f the two- and three-dimensional solutions. The element was given nodes at x = 0 and 1. The shape functions for a finite element between these nodes are simply:

M1 = 1 - - x and M 2 = X (1)

Corresponding decay functions are:

e -x and e(1-x) (2)

The resulting int'mite element shape functions are thus:

N1 = (l - - x ) e -x and N2 = x - e ( l -x) (3)

Now a differential equation and boundary conditions are required to test these shape functions. Clearly the solution can be o f the form e -x, and this should be solved for exactly. So a differential equation is synthesized for which this is the solution:

dN = u u(0) = 1 (4)

dx 2

clearly fits the bill. Many other possibilities exist. Another example is d2u/dx 2 = e -x. The corresponding variational problem is to make:

1 S rl<'"l:+.:]<,.,< 2 L L d x d

x=O

(5)

stationary. Now if the boundary condition is inserted:

u = (1 - -x ) e -x + u2x-e ( l -x)

du - - = ( - - 2 + x ) e -x + u 2 ( 1 - - x ) e O - x ) dx

Minimization now yields:

(6)

(7)

f x (1 - -x )e l -2x dx +u2=fx 2 e 20 -x ) dx

o o

+ i (1 - -x)( - -2 + x ) e 1-2x dx

o

+ u 2 t ( l - - x ) 2 e l - x dx = 0 (8)

0

A little algebra gives the solution u2 = l/e, which is correct. This shows that the formulation works. It can now be tested on similar equations. One test is to take a slightly different differential equation, for example:

d2u " = o~2u and u(0) = 1 (9)

dx 2

with solution u = e -~x. As c~ departs from 1.0, we would expect the approximation to depart from the exact solution. The effects are shown in Table 1 and Figure 1. For a large range of values of c~ the solution obtained by the e -~x element, u2 = 2/e(1 + a 2) remains accurate. The exponential has many advantages as a decay function, but it might be

Techniques for testing inf ini te elements: P. Bettess et al.

Table 1 Comparison of approximate and exact solutions as decay parameter ~ is varied

e 21e( l+~ 2)

0.1 0.9048374 0.7284741 0.2 0.8187308 0.7074605 0.5 0.6065307 0.5886071 1.0 0.3678794 0.3678794 1.5 0.2231302 0.2263873 2.0 0.1353353 0.1471518 4.0 0.0183156 0.0432799

10.0 0.0000454 0.0072847

1.0

0.8

~'~ O.6

0.4

0.2.

0.1 0.2 0.5 1.0 1..%2.0 4.0 Gt

Figure I u 2 as a function of a. (o), numerical solutions; ( - - ) , exact solutions

10.0

argued that it would not handle the dipole distribution, which is essentially l[r. This can be tested by synthesizing another appropriate differential equation. Of the many possibilities, we choose:

d2u 2u = 0 u(1) = 1 (10)

dx 2 x 2

which has a solution u = 1Ix. The corresponding functional is:

7r = + dx (11) t LdxJ - ~ -

]

Consider 2 nodes at x = 1 and 2. The exponential shape functions are now:

M, = (2 - - x ) M2 = (x -- 1)

N1 = (2 - - x ) e ( l -x) N2 = (x -- 1) e (2-x) (12)

The satisfaction of the boundary condition gives:

u = ( 2 - - x ) e O-x) + u2(x -- l) e (2-x) (13)

du - - = e ( 1 - x ) _ (2 - -x ) e O-x) dx

-I- u2[e (2 - x ) _ (x -- 1) e (2 -x)]

= e -x e[x(1 - - u 2 e ) - - 3 + 2uze] (14)

[ du]2=e-2X e2[x2(1-- 2u2e + u]e2 ) dxJ

+ (--3 + 2u2e)(1--u2e)x + (--3 + 2u2e) 2]

(15)

App l . Ma th . Mode l l i ng , 1982, Vo l . 6, December 4 3 7

Techniques for testing infinite elements: P. Bettess et al.

u = e O-x ) [(2 - - x ) + u=e(x -- 1)1

= eO-~)[x(u2e -- 1) + 2 --u2e]

u 2 = e (2-=~) [ x=(u2e - 1)2+ 2 ( 2 - u2e ) (u2e - 1)x

+ (2 - u2e)21 (16)

After completing the integrations u2 = 0.45926 is obtained, which is reasonably close to the correct value of 0.5. Figure 2 shows the analytical expression and the shape function.

Wave problems, r -1 decay

Infinite elements have also been developed for exterior wave problems. 7 These have to model out-going progressive waves, and also the decay of the amplitude with increasing radius. In addition the periodic nature of the wave must be included. This makes the general form of the shape function fairly complicated. It is complex valued, and can include a polynomial, a decay function and a periodic component. Because of the complexity the development of the element matrix, and its debugging, is fairly difficult. More recently a new wave infinite element has been developed, based on a mapping approach due to Zien- kiewicz. 6 During the development of this element, it was found useful to test the concepts using an ordinary differential equation. Consider Helmholtz 's equation in polar coordinates with axial symmetry, in three dimensions. This takes the form:

V2~b+k2~b----O2¢+2-SC+k2¢-- = 0 (17) Or 2 r Or

where k is the wave number. In addition, the Sommerfeld radiation condition s must

be satisfied. With i representing the square root o f - - l , this condition is:

lirn + ik~b = 0 (18)

To test the infinite element formulation, it is first assumed that the outgoing wave is spherically symmetrical, which enables us to derive a sharper form of the Sommerfeld condition. Let:

f l ( r - - ct) f2(r + et) ¢ = - -t (19)

r r

This is the d 'Alembert solution of equation (17). f] repre- sented outgoing and ]'2 incoming waves. Now if f2 is

1.00(

0.7,5

0.5(3

O.2_ ¢

0

0 0

0 0

I I 1.0

Figure 2

2.0 3O x

Inf ini te shape funct ions and analytical solution

4.0

excluded,f] can be eliminated. It is now straightforward to derive a spherical radiation condition:

a ¢ + 1 I 0q~ - - - . ~ + - - - = 0 ( 2 0 ) Or r c Ot

although this is not used directly here. The solution:

e i kr ~b = • e iuJt (21)

r

(where co is the frequency) is an outgoing wave satisfying the radiat ion condi t ion and Helmhol tz ' equation. I t is possible to test an element to model such a wave in axi- symmetric form. Here, however, it was done by setting up a special ordinary differential equation in ID as follows.

The solution which we are trying to model exactly is:

eik-r = (22)

r

Differentiating twice:

d q ) - - e i k r i k e i k r [._~rl k] - - - + ~ = ~ + i

dr r 2 r (23)

d2¢ de - - Iv ] ~b J = - - + i k + - dr2 dr r 2

leads to the equation to be solved:

= k = ~ ( 2 4 ) dr 2 r

The corresponding functional for this equation is:

l f [ [ d ~ b ] = - - [ 2 2ik ~b=]dr ~r = 2 J LL drJ Lr = r k2] (25)

1

A single quadratic mapped infmite element is used extending from r = 1, through node at r = 2, to infinity. The boundary condition at r = 1 is:

ei(l) ~, = (26)

1

and at infinity, 42 = O. The variation of the function over the element can now

be expressed as:

{b = N I ( ~ ) q~l + N2(~) q~2 (27)

The shape functions N(~) arise from the method described elsewhere,e,0 and are written as:

NI(~) = M](~j) • e ikr. e -is

= 0.5(--~ + ~2). eikr. e-iS

N2(~) = M2(~) • e i ~ . e -is = (1 -- ~2). e i ~ . e-i,~ (28)

The third shape function can be neglected due to the zero boundary condition at infinity. The factor a is to ensure zero phase at ~= -- 1, which gives conformity with adjacent finite elements.

Making the functional in equation (25) stationary is now equivalent to solving the expression:

*dr+ > r k= NTN, dr=0

' ] ( 2 9 )

438 Appl. Math. Modell ing, 1982, Vol. 6, December

where ~ are the nodal values. This is a matrix equation o f the form:

[K + f ] [$] = 0 (30)

There are only two simultaneous equations here, and since one nodal value is given as a boundary condition, the solution is:

~ = -- ~b~- (31) (K~ + f~)

The mapping used in the infinite element is:"

2 ~ = 1 - - -

with inverse:

2 r =

1 - ~

from which:

2 dr =

The terms form:

( 1 - - ~ ) 2 d~

(32)

in the matrix equation above are therefore o f the

Kq =

+ I

a~ a~ 2 exp - -2 ik d~ 1--~2

- -1

+1

--1

1 x - - ~ ( 1 exp --1_~ -- 2ik d~ (33)

To perform these integrals, a special integration routine was devised. 6,1° This numerical procedure was applied here, and leads to the four quantities:

K21 = (0.920058 -- i1.20313)- e -2i

Kz2 = (--1.48660 + i l .07547) , e -2i

f2~ = (0.136354 + i0.276630)- e -2i (34)

f22 = (--0.626224 + i0.777532)- e -2i

(where k, the wavenumber, is taken as unity). From these quantities it is found that:

~b2 = 0.270151 + i0.420735

= 0 .5 . e il'O (35)

and hence, using:

¢ = Nlq~l + N2¢2 (36)

the value at the centre node, at ~ = 0 (r = 2) is 0.5 • e i2"0, to six decimal places. This is identical to the exact solution to within the accuracy of the calculation, showing that the method is giving very good solutions to problems decaying as r -1.

lVave element, r -~/2 decay The previous section was concerned with solutions

decaying as r -1 and the testing o f a new infinite element to model this. In 2D problems, however, it is more usual to

Techniques for testing infiniteelements: P. Bettess et al.

encounter wave problems whose far field solution decays more as r -u2. It was therefore required to modify the given infinite element to model r -u2 decays. To test this, a partial differential equation was devised whose solution is:

eikr 4 = - - 1.1/2

this equation being:

i3

(37)

(38)

Displaced origin method. Two methods were examined in order to improve the new infinite elements, and allow them to model r -u2 type o f decay. The first of these in- volves moving the pole o f the mapping function, 6,9 such that over part of the element the r -1 decay due to the shape functions resembles r -u2. The mapping function used becomes:

6 6 4 + 2~ ~ = I - - - - r = 2 = - -

r + 2 1 - - g 1 - - ~

6 dr - (1 -- g)2 d~ (nodes at r = 1, r = 4 and r=oo) (39)

This leads to terms in the matrix equation of the form:

+1 f aNiaNi(1--~) 2 [ [12~ ]

K, j = 0~ a~ 6 exp LI--~ -- 4 i k - - 2ik d~

- -1

=

+1 6

f f(r) .N~N i (1 -- ~)u

--1

r 12ik 2ik] dg x exp | I -- g -- 4ik - -

[3 k2-i f(r) = 4 (4 + 2~) 2 (4-4-2-~)]

On solving as before, this leads to:

~b2 = 0.4204 e i0"08 e -i6 (where k = 1.0)

and hence, using ¢ = Nl~bl + N2¢2:

¢(~ = 0 , r = 4) = 1.0 e i l '0 ei3"°~2

= 0.4204 e i4"°8

The exact solution is 0.5 exp (i4.0), but the above is a considerable improvement over the method without the modification, which would have given the value 0.25 exp (i4.0) at the same point.

(40)

(41)

Modified shape function method. The second method involved modifying the shape function in order to model r -u2 types o f decay. The new shape function used is:

r r ]I/2 N(~) = M(~) . [ ; J • eikr.e i~ (42)

where e and r0 are to ensure continuity with finite elements at ~ = -- 1. This was tested on the same differential equa- tion as above (equation 38), following the same procedure.

Appl. Math. Modelling, 1982, Vol. 6, December 439

Techniques for testing infinite elements: P. Bettess et aL

The mapping used is the original one (unmodified):

2 2 = 1 - - - - r = (43)

r 1 - - ~

On solving this, the value at the centre node, ~ = 0, (r = 2) is found to be:

~b = 0.5 (2) 1/2 e i2"0 (to 5 decimal places) (44)

This is now identical to the exact solution (o witltin the accuracy of the calculation. At this point therefore it was decided that a suitable method had been found to model problems with r -1/2 types of decay in 2D.

The next stage of testing the new infinite element would be to set up a larger mesh of finite and infinite elements, and solve a nontrivial problem for which a reliable solution is known. The chances of errors occurring are then greatly reduced, as it is by then known that the computer code for the infinite elements is correct, and that the formulation up to that point is satisfactory.

C o n c l u s i o n s

It has been shown that simple one-dimensional test problems can be devised and used to test and evaluate new infinite element formulations. These tests do not simply consist of solving the target equation in one-dimensional form. Instead they transfer the essential features of the multi-dimensional solutions to one dimension. By using such tests new formu- lations can be studied very rapidly, using a programmable calculator or a very small computer , in a matter of hours. On the other hand full two- or three-dimensional infinite element development may take months, with the perennial at tendant problem of separating errors of concept and formulation from errors of coding.

Other simple tests could be set up for other special element types. The development of special tests of course need not be confined to one dinaension.

A c k n o w l e d g e m e n t s

The authors thank SERC for financial support through grant number GR/A/62057. One of the authors (K. Bando) also thanks Kajima Corporat ion for financial assistance, and for granting him a period of leave.

R e f e r e n c e s

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2 Tracey, D. M. 'Finite elements for determination of crack tip elastic stress intensity factors', Eng. Fract. Mech. 1971, 3,255

3 Mote Jr., C. D. 'Global-local finite element', Int. J. Num. Meth. Engng. 1971.3,565

4 Bettess, P. 'Infinite elements', Int. J. Num. Meth. Engng. 1977, 11,53

5 Ungless, R. L. 'An infinite finite element', IVlASc Thesis, Uni- versity of British Columbia, 1973

6 Zienkiewicz, O. C. et al. 'Numerical methods for unbounded field problems and a new infinite element formulation', Winter Am1. MeetingASCE, Washington, DC, 15-20 November, 1981. Appl. Mech. Div. [ASME) 1981, AMD-146, 115

7 Bettess, P. and Zienkiewicz, O. C. 'Diffraction and refraction • ~ of surface waves using finite infinite elements', Int. J. Num.

Meth. Engng. 1977, 11, 1271 8 Sommerfeld, A. Jahresber. Deutschen Math. Fereinigung 1912,

12, 312 9 Zienkiewicz, O. C. et al. 'A novel boundary infinite element',

Dept. Civil Engineering Report C/R/401/82 (to be published in hit. J. Num. Meth. Engng.)

10 Bettess, P. et al. 'A new mapped infinite element for exterior wave problems', Report C/R/403/82, to be published in 'Numerical methods in coupled systems' (ed. R. Lewis), Wiley

440 Appl. Math. Modelling, 1982, Vol. 6, December