Finite element techniques for problems of unbounded domains

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL. 19, 1209-1226 (1983)

FINITE ELEMENT TECHNIQUES FOR PROBLEMS OF UNBOUNDED DOMAINS

FRANCISCO MEDINA

URSIJohn A. Blume & Associates, Engineers, Berkeley, California, U.S.A.

ROBERT L. TAYLOR Department of Civil Engineering, University of California, Berkeley, California, U.S.A

SUMMARY A general approach for using infinite elements within the context of the finite element method is presented. Based upon Gauss-Laguerre quadrature, a scheme for numerically computing the infinite element characteristic matrices is proposed. The general formulation of the element shape functions is outlined, including that for infinite elements that model the far field of hyperbolic systems. The technique is illustrated with a classical example in elasticity: a vertically loaded circular rigid plate.

INTRODUCTION

Many interesting problems occur in regions whose spatial dimensions are very small compared with those of their surrounding media. In order to characterize the field behaviour, the surrounding medium is assumed to extend to infinity. This assumption has led to the develop- ment of a number of continuum, discrete and hybrid formulations to represent the unbounded domain. Closed-form solutions to the continuum field equations (often referred to as continuum formulations) may be used efficiently when the problem geometry and the medium constitutive law are simple. Discrete formulations are used when complex geometries and medium conditions are considered, yielding necessarily to finite element techniques. However, until recently finite element techniques were not available to directly treat problems in unbounded domains. In effect, finite elements are able to model the region of interest (near field, nf) fairly well, but they are unable to model the surrounding medium (far field, ff). Therefore, an artificial boundary must be introduced (far-field boundary, f f b ) . This often leads to inaccuracies in the computed solutions. This difficulty may be overcome by formulating the problem in a hybrid fashion. Hybrid formulations model the nf with finite elements and replace the ff with a continuous or a discrete representation obtained by separate methods or criteria. The success of hybrid formulations is varied, being somewhat limited by the assumptions made on the ff.

Nevertheless, by using infinite elements,”* a rational approach has emerged as an alternative method of dealing with this class of problems within the context of the finite element method. Infinite and finite elements? may be used together to model the ff and nf, respectively, in problems for which the domain is considered infinite. The ff is then represented in the analysip by means of infinite elements, which characterize ff subdomains as finite elements do nf subdomains. Although the concept of f fb as such is no longer valid in this case, it is convenient to define the interface between finite and infinite elements as ffb.

t The concept of an infinite element refers to an element which has at least one side located at infinity. On the other hand, a finite element is completely bounded.

0029-5981/83/081209-18$01.80 @ 1983 by John Wiley & Sons, Ltd.

Received May 1982 Revised July 1982

1210 F. MEDINA A N D R . L. TAYLOR

The formulation of this technique does not deviate from the classical finite element method: discretize the continuum, assume shape functions, minimize errors by means of weighted residuals or variational principles, obtain a set of algebraic equations that characterize the problem, etc. Infinite elements should meet the same requirements as finite elements in addition to two others: finiteness and r a d i a t i ~ n . ~ The latter, known as the Sommerfeld radiation condition, is essential to the solution of propagation problems. This technique, using finite and infinite elements, therefore permits a direct one-step solution to problems defined in unbounded domains, while preserving the flexibility of the classical element method.

INFINITE ELEMENTS

By using infinite elements, the finite element method can be extended to solve problems of unbounded domains. A different formulation is not required, and infinite elements should satisfy the same conditions as finite elements, in addition to special requirements imposed on their shape functions.

In theory, infinite elements should satisfy compatibility and completeness condition^.^ In practice, either or both conditions may be relaxed depending upon the problem under consideration. Assume that the field variable functions appearing under the integrals in the element equilibrium equations contain derivatives up to order p. Then, an element is (a) compatible with respect to an adjacent element, when at the element interface (common boundary), the field variable in both elements is expanded with the same shape functions, which are continuous up to derivatives of order p -1 ; and (b) complete when the shape functions are continuous and smooth within the element up to derivatives of order p, as the element size (at the limit) shrinks to zero.

Elements satisfying conditions (a) and (b) are said to be ~onfo rming .~ Conforming elements yield approximate numerical solutions having a finite norm of order p, i.e.

11411:,, = 11411: + Ila4/ax 11: + Ila4/ay 11: + lla4/az 11: + Ila24/ax 7: + . . . + Ila"/az "11: < co (1)

where 4 is the field variable and 11 denotes the Euclidian norm. For example,

In addition to conditions (a) and (b) above, infinite elements must satisfy finiteness requirements. An element satisfies (c) the finiteness requirement when all the shape function derivatives up to order p vanish at infinity. Moreover, in propagation problems, infinite elements must also satisfy radiation requirements. An element satisfies (d) the radiation requirement when the shape functions up to derivatives of order p yield outgoing waves at large distances from the energy source, and when the shape functions vanish at infinity. These two requirements must be met. Indeed, when the finiteness requirement is not met, the integrals appearing in the element equilibrium equations cannot be computed. On the other hand, when the radiation requirement is not met, part of the energy is not radiated outward from the ffb, which leads to spurious results. Note that finiteness and radiation are satisfied if and only if equation (1) is satisfied. Furthermore, if adjacent infinite and finite elements are compatible, the infinite element is called consistent.

In general, the better the element shape functions resemble the form of the expected solution, the coarser the element mesh may be. Since at the limit piecewise polynomials may approximate any function, finite element shape functions may always be polynomials. Hence,

PROBLEMS OF UNBOUNDED DOMAINS 1211

for bounded-domain problems, satisfactory answers may always be obtained by systematically refining the element mesh. This is not the case with infinite elements. The infinite element approximation accuracy in the infinite direction is obtained by (a) selecting realistic shape functions,’ i.e. shape functions containing approximately the form of the expected solution in the infinite direction; or (b) increasing the number of shape functions (element nodes) in the infinite direction. If (a) is satisfied, (b) is not necessary and may lead to undesirable results6 if imposed.

On the assumption that solutions to problems of unbounded domains do not change dramatically in the ff if the nf conditions are changed, realistic shape functions may be obtained from known closed-form solutions to simple problems. Hence, the infinite element shape functions may be obtained from solutions to unbounded-domain problems having a single disturbance (boundary condition or geometry). This concept is illustrated later for elasticity.

Parametric infinite elements require the use of a mapping function, different from the element shape functions. This mapping function is used to map the element from the local co-ordinates to the global co-ordinates. For example, for a three-dimensional infinite element, the mapping function that maps the element from the global co-ordinate system (x, y, z ) onto the local co-ordinate system (5, q, 5) for node j , is

Mj(t, 7 7 9 0 =Ljit)L,(q, 5) (3) where L, is the mapping function in the infinite direction and L j is the Lagrange polynomial for node j . On the other hand, the shape function defines the numerical approximation of the field variable within the element. In particular, for a three-dimensional infinite element, a shape function for node j , may be expressed as

N , ( ~ , T , 5) =f,tt)Lj(t7,5) (4)

where f,, is an attenuation function in the infinite direction.

NUMERICAL INTEGRATION

The integrals appearing in the element equilibrium equations may be conveniently evaluated by numerical integration. Gauss-Laguerre quadrature formulae’ are generally suitable when dealing with infinite domains, whereas Gauss-Legendre quadrature may be preferable for the finite directions. In certain problems, however, Gauss-Laguerre quadrature is not optimum for handling the numerical integration in the infinite direction, and specially designed quadratures should be devised.* Nevertheless, special quadratures are not always feasible and Gauss-Laguerre quadrature must then be used with a large number of integration points in the infinite direction. For these cases, a general method of obtaining the element characteristic matrices is presented below.

Let X denote a typical element matrix. Thus, r r r

where J is the Jacobian of the transformation from the global co-ordinate system to the local co-ordinate system. Once the integration in the finite directions has been performed, the above equation may be expressed and numerically approximated by

1212 F. MEDINA A N D R. L. TAVLOR

where Wf and 5f are the Gauss-Laguerre weights and integration points, respectively ( I = 1,2 , . . . , Lint). Lint is the number of Gauss-Laguerre points required to numerically integrate equation ( 5 ) in direction 5, given a certain error bound. It is then necessary to establish a criterion for selecting L,,,, a criterion that certainly depends on the element shape functions.

On the other hand, since the element shape functions satisfy finiteness (and for propagation problems, finiteness and radiation), the integral appearing in equation (6) is well defined. Hence, the sum that approximates this integral converges. In fact, given a tolerance it may converge before the last term is added. It is convenient to stop the computations once convergence is reached. To establish a convergence criterion, define the sum given by equation (6)ask and the partial sum up to I = L s Lint as XL, i.e.

Then, kL converges to XLn, as L approaches Lint. Eventually, there is a term for which f = L r e d <Lint and whose contribution to the sum may be considered negligible. This last statement may be expressed as

where IJ.II denotes some matrix norm. Once the above equation is satisfied, the computation of the integral may be stopped. Hence, equation (6) yields

APPLICATION I: ELASTOSTATIC PROBLEMS

Elastostatic problems are problems of equilibrium whose solutions are characterized by systems of elliptic equations. Various types of infinite elements capable of modelling the ff for this class of problems have been developed. Element shape functions obtained from closed- form solutions to point loadings on unbounded media are found to give good results. For example, a parametric infinite element, as shown in Figure 1, with a mapping function for node j ,

6.9-12

6.10.12

M,(',;,~)=(1+5)Li(;,5) (10)

(0 s 8 < OO), may be defined to model the homogeneous ff of unbounded media. The shape function isI2

which defines a consistent element, orlo

which defines an inconsistent element. When the ff6 is hemispherical, equation (12) is equivalent to equation (1 l ) , thus defining a consistent element.

The number of Gauss-Laguerre points required to numerically integrate equation (6) may be estimated" from the error made by evaluating I," (1 +()-' d t numerically (see Figure 2).


Figure 1. Three-dimensional parametric infinite element. Definition of co-ordinates: x = {x y z}, point in the space; { u ( x ) u(x) w ( x ) } ~ , displacements at point x

I I 1 I I I I I I I I 2 3 4 5 6 7 8 9 1 0 15 20

NUMBER OF GAUSS-LAGUERRE INTEGRATION POINTS, L,,,

Figure 2. Error in the numerical evaluation of I," (1 +[)-* d[ as a function of the number of Gauss-Laguerre integration points'*

1214 F. MEDINA AND R. L. TAYLOR

Two Gauss-Legendre integration points in the finite direction and seven Gauss-Laguerre integration points in the infinite direction yield sufficiently good results. l 2

APPLICATION 11: ELASTODYNAMIC PROBLEMS

Elastodynamic problems are problems of propagation whose solutions are characterized by systems of hyperbolic equations. Energy is propagated throughout the media in the form of elastic waves, that usually have different simultaneous components. Infinite elements capable of transmitting these components must then be developed. To this effect, the following procedure is suggested.

Multi- wave propagation

For elastic media through which Q different elastic waves propagate, the displacement component u may be expressed as

q = l

where f:(x) is the wave component q corresponding to displacement u as a function of the co-ordinate x. a: is a constant. An infinite element with Q nodes may then be defined such that displacement u at node j is approximated by

u ( x i ) = ui = fu(xi)au

( j = 1,2 , . . . , Q), or in matrix form

u = Fuau

where

and

Substituting equation (15) into equation (13) gives

u ( x ) =f,(x)F;'u~N,(x)u (18)

where N,(x) contains the element shape functions for displacement u at co-ordinate x. Note that

t 19) Nu (x) = f, (x)F,'

The shape functions for displacement components u and w may be similarly obtained, using in general different functions for each displacement component. Thus, within infinite element e


yielding ""'"'1 (21) aN,/ax o o aN,/ay o

[ o [Be]== o aN,/ay o aN,/ax aN,/az

o aN,/az o aNw/ay NJax

which relates strains with nodal displacements for three-dimensional rectangular co-ordinates. As it is constructed, the infinite element defined above does not meet compatibility, and it is inconsistent for arbitrary wave components f , (y = u, u, w ) . It is readily seen that completeness, finiteness and radiation should be imposed on the wave components individually rather than on the shape functions.

Following this procedure, frequency-dependent infinite elements capable of simultaneously transmitting Rayleigh, shear and compressional waves are d e ~ e l o p e d ' ~ to model the f f of semi-infinite media. Wave components are obtained from approximations to closed-form solutions to harmonic single-wave propagation in unbounded media. These wave components, given in Table I, are combined to form the shape functions of the parametric three-node infinite elements shown in Figure 3. The mapping function for these elements is given by equation (10).

Table I. Displacement wave components for frequency-dependent 3-node infinite elements?

Displacement Rayleigh wave Shear wave Compressional ( Y ) ( f 3 ( f 3 wave (f:)

U

U

W

U

t Parameter definitions:

r, z =cylindrical co-ordinates ( r = J ( x 2 + y2) , for rectangular co-ordinates x, y, z ) .

R = J ( r Z + I '), radial co-ordinate.

6 =element parametric co-ordinate in the infinite direction.

k o =wave propagation number (Q = R : Rayleigh; S: shear; P: compressional).

S, p = J(k i - k&).

Nevertheless, by modelling the f f with a convenient combination of infinite elements, which are able to transmit one wave component only, reasonable results may still be obtained."

Single- wave propagation

When there is only one wave component propagating through the medium, simpler, compatible and consistent infinite elements may be defined. For example, the propagation of shallow water waves' or elastic torsional are treated by modelling the f f with rather simple

1216 F. MEDINA AND R . L. TAYLOR

( b )

Figure 3. Three-node parametric infinite elements: (a) axisymmetric; (b) three-dimensional

infinite elements. The shape function for these elements may be of the f01-m'~

(22) -(l+ikR,)€ Nj(S, ~ , 5 ) = e L j h , 0 where k is the wave propagation number and Ro is a characteristic distance for the elements, say R (6 = 0, q = 0,5 = 0). The mapping function is given by equation (10).

Numerical integration

The number of Gauss-Laguerre points required to numerically integrate equation (61, Lint, may be e ~ t i m a t e d ' ~ from the error made by evaluating I," ((1 +t) e O *) d( numerically (see Figure 4). Figure 4 shows that Lint becomes very large for large values of kRo. However, the computation of the element matrices may be stopped once equation (8) is satisfied, which defines a reduced Gauss-Laguerre quadrature. Good results are obtained13 using two Gauss- Legendre integration points in the finite direction and a reduced Gauss-Laguerre quadrature in the infinite direction. The criterion that reduces Lint is based upon the convergence of the Euclidian norm of the diagonal of the element dynamic stiffness matrix (see Appendix I). Then equation (8) yields

- ( l + i k R ) 2


DIMENSIONLESS FREOUENCY OF OSCILLATION, fl

Figure 4. Number of Gauss-Laguerre integration points required to integrate numerically 1," ((1 +5) e-"+'n'' ) * d5 as a function of n for different error bound^'^

TEST PROBLEM AND DISCRETIZATION

As an illustration, a vertically loaded rigid circular plate on a homogeneous, isotropic, elastic semi-infinite medium is considered (see Figure 5 ) . The numerical solutions are obtained using axisymmetric element meshes.

Two types of element meshes are used in this study: sectorial and rectangular. The sectorial meshes are constructed from the mesh shown in Figure 6(a) by successively removing the outer rings of finite elements (e.g. the mesh shown in Figure 9(a) is generated after removing the three outer finite element rings from the mesh shown in Figure 6(a). Two rectangular meshes are defined: a coarse one, Figure 6(b), and a refined one, Figure 6(c).

(a) (b)

Figure 5. Rigid circular plate vertically loaded on a homogeneous, isotropic, elastic semi-infinite medium: (a) static case; (b) dynamic case


(b) (C)

Figure 6. Axisymmetric element meshes to model a rigid circular plate on a homogeneous semi-infinite medium: (a) base sectorial mesh; (b) rectangular coarse mesh; (c) rectangular refined mesh

The rigid plate and surrounding media in the n f are modelled with a reasonably small number of finite elements. The f f is modelled using the number of infinite elements specified by the nf discretization on the ff6 (i.e. the use of infinite elements does not increase the total number of nodes of the discretization). The nf characteristic matrices are computed using Gauss-Legendre quadrature with two integration points per direction.+ The f f characteristic matrices are computed using Gauss-Laguerre quadrature as indicated in previous sections.

NUMERICAL EXAMPLE I: ELASTOSTATICS

Figure 7 shows the cases studied using sectorial meshes and the maximum error of the numerical vertical displacement solutions on the semi-infinite medium surface (plane z = 0) . The

f This constitutes fully reduced q ~ a d r a t u r e ' ~ for which spurious zero energy modes exist in individual elements. For the problems considered, however, these modes are not possible in the solution.


2 26 269 203

2 w 137 147 103'

\ Rigid / /b >\

\ \

I. \ \ \

A 0

A 0 .

0 8 I3

X . O

I I I I 1 2 3 4

DISTANCE OF FAR-FIELD BOUNDARY ( f fb) FROM ORIGIN, R,/a

(C)

Figure 7. Vertically loaded rigid circular plate on a homogeneous, isotropic, elastic semi-infinite medium; (a) geometric definitions; (b) numerical cases studied; (c) maximum surface (plane z = 0) vertical displacement solution error

maximum error of the numerical vertical displacement solution on plane z = 0 is defined as

where wj and Gj are the exact and numerical vertical displacement values, respectively, for node j , and where w(0 ,O) is the exact vertical displacement of the plate. The errors obtained by using the attenuation functions defined by equations (1 l), (12), and several other attenuation functions, are plotted in Figure 7(c).


As expected, the error decreases as the distance from the origin to the f fb increases, as shown in Figure 7(c). The attenuation functions defined by equations (1 1) and (12) give the least errors for all the cases treated. Subsequently, only infinite elements generated by these shape functions are considered. It should be noted that results may improve by adding parameter^"^.'^*^^ to the attenuation functions. As also shown in Figure 7(c), increasing the ratio R o / a beyond 2.5 does not decrease the error but does increase the solution cost. In fact, increasing this ratio from 1.47 to 2.00 decreases the error by only 2.7 per cent but increases the solution cost considerably. In effect, the computational effort made in computing the additional element stiffness matrices increases 24 per cent (see Table 11, which shows the

Table 11. Elapsed CPU time ratio between the computation of the stiffness matrix for some axisymmetric elements and the computation of the stiffness matrix for a 9-node axisymmetric finite element

Elapsed time

Integration

ratio 0.239 0.550 0.806 1 0.361 0.532

points 2 x 2 2 x 2 2 x 2 2 x 2 2 x 7 2 x 7

relative computational effort made in computing some popular elements). On the other hand, the effort made in solving the new system of equations increases 33 per cent. It should be noted that the results shown in Figure 7(c) pertain to the location of the ffb and not to convergence properties. In fact, for convergence it is required to refine both the nf and the ff meshes.

Although satisfactory displacement solutions may be achieved with relatively few degrees-of- freedom, the corresponding stress solutions do not perform as well. Numerical stress and displacement solutions on plane z = 0 were obtained for all the meshes shown in Figure 7. It was observed that the numerical stress solution is highly influenced by the discretization characteristics, while the numerical displacement solution is strongly influenced by the ratio between the distance from the origin to the ffb and the plate radius (Rola) . Some of the cases treated are shown in Figure 8. In Figure 8(c), the numerical solutions are shown with the exact solutions.16 It is observed that the sectorial mesh gives a better displacement solution and generally a worse stress solution than the rectangular meshes. Increasing the size of the sectorial mesh did not improve the stress solution.

NUMERICAL EXAMPLE 11: ELASTODYNAMICS

A spherical cavity subjected to harmonic pressure in a homogeneous, isotropic, elastic infinite medium17 was extensively studied. The study focused on determining the appropriate error bound needed to compute Linr (see Figure 4), and the tolerance required for the convergence of the infinite element dynamic stiffness matrices, as defined by equation (23). Although the problem is one-dimensional, it was axisymmetrically modelled with a number of infinite elements. For error bounds ranging from and tolerances ranging from lo-' to to

PROBLEMS OF UNBOUNDED DOMAINS

IP

1221

6a,&,la = 2

6 C

I 0 0 0.5 1 0 1.5 2 0

DISTANCE FROM Z-AXIS, r/a

1 I

(4 Figure 8. Vertically loaded rigid circular plate on a homogeneous, isotropic, elastic semi-infinite medium (v = 1/4):

(a) geometric definitions; (b) numerical cases studied; (c) vertical contact stress and surface vertical displacement

the variations in the numerical complex displacement solutions along the cavity were below 0.094 per cent. Thus, the lower bounds of these ranges present safe choices.

Multi- wave propagation

computed for discrete values of the dimensionless frequency Numerical solutions for a rigid circular plate subjected to harmonic vertical loading are

a. = aks (25) where a is the plate radius and ks is the medium shear wave propagation number. The error in the numerical approximation is computed to be the maximum plate vertical displacement error in absolute value, i.e.

where w and G are the exact and numerical values, respectively, of the plate vertical complex displacement, also called the plate vertical compliance.


- m -

A" = C,(a,)V

w 0 1 2 z

I I 0 0 1 0 2.0

DIMENSIONLESS FREQUENCY, a, = aw'V,

(b) Figure 9 . Rigid circular plate on a homogeneous, isotropic, elastic semi-infinite medium ( Y = 1/3) subjected to

harmonic vertical loading: (a) element mesh; (b) compliance function

Table 111. Numerical solution errors for the vertical compliance of a rigid circular plate on a

semi-infinite medium

Infinite element Rayleigh wave component,?

f ? Error, E ( O h )

f r e-'"' (Table I) 8.8 e-' 10.3 e-r/Ro 1 1 . 1 e-6 11.9 e - ( l z l + r l / R ~ ~ 16.4 e - t l z l+r )kn 16.8

t f f (see Table I) may be defined as a product of functionssuch that f r ( r , i , k ~ ) = f r ( r , z , k ~ ) e - ' ~ ' .


GUPTA. 81 a/ 2'

DAY 8 FRAZIER" 1,020

DIMENSIONLESS FREQUENCY, a, = a d ,

(b)

Figure 10. Rigid hemispherical body embedded in a homogeneous, isotropic, elastic semi-infinite medium subjected to harmonic t o r ~ i o n : ' ~ (a) element mesh; (b) compliance function

The sectorial mesh defined by a ratio R o / a of 2, as shown in Figure 9(a), was selected to study the effect of different infinite element shape functions on the plate compliance. The shape functions differed only by the Rayleigh wave component. Table I11 lists the errors obtained using various Rayleigh wave components and the corresponding error obtained using the Rayleigh wave components defined in Table I. A comparison of the values in Table I11 shows that using the wave components defined in Table I gives the least error. For these wave components, the numerical solution for the plate vertical compliance is shown in Figure 9(b), along with the exact solution.'* This numerical solution was compared to numerical solutions obtained using different matrix norms (see Appendix I) as the basis for applying the convergence


DIMENSIONLESS FREQUENCY, a, = a d ,

Figure 11. Spherical cavity embedded in a homogeneous, isotropic, elastic infinite medium ( Y = 0) subjected to harmonic pressure: (a) element mesh; (b) compliance function

criterion given by equation (8) to the infinite element dynamic stiffness matrices. It was found that the variations in the numerical plate compliances were below 0.021 per cent. Thus, the use of I I . I l d is justified.

Single- wave propagation

The use of infinite elements generally yields inexpensive solution^^'^ ',13 to single-wave propagation problems. Two representative example problems are considered:

Rigid hemispherical body. A rigid hemispherical body embedded in a semi-infinite medium and subjected to harmonic torsion is solved. As shown in Figure 10(a), the rigid body is modelled by four finite elements and the f f is modelled by four infinite elements. The


compliance function obtained for the foundation is shown in Figure 10(b), along with the exact s o l ~ t i o n ' ~ and two other available numerical solutions.zO.zl

Elastic spherical cavity. An elastic spherical cavity embedded in an infinite medium and subjected to harmonic pressure is solved. As shown in Figure l l ( a ) , the f f is modelled by two infinite elements. The compliance function obtained for the cavity is shown in Figure l l (b ) , along with the exact ~o lu t ion . '~ In this case the system is characterized by 5 degrees-of-freedom and the error is 11.4 per cent.

In both cases, the solutions obtained agree fairly well with the exact solutions, despite the small number of degrees-of-freedom used to characterize the systems.

CONCLUSIONS

The finite element method may be used to solve problems of unbounded domains by merely defining appropriate infinite elements to model the far field. This technique preserves the flexibility of the classical finite element method. It is a one-step method that permits the far field to be incorporated directly into the solution scheme and that reduces the size of the near field. However, the definition of infinite elements generally presents certain disadvantages: the selection of element shape functions requires some insight into the solution behaviour; special quadrature formulae should be used; and if nodes are moved on the far-field boundary, the element characteristic matrices must be recomputed.

ACKNOWLEDGEMENTS

J. E. Luco,'* S. M. Dayz0 and S. Gupta" kindly provided original data. This work was partially supported by National Science Foundation grants ENV76.04264 and PFR79.08261, and by URS/John A. Blume & Associates, Engineers.

APPENDIX I

Let X be a square complex matrix of elements x,, and order n. The Euclidian norm of the diagonal of X is defined as

It is easy to prove that the definition above satisfies

I IlXlld - IlYlld I IIX - Ylld (31)

where Y is a square complex matrix of order n and a is a scalar. Computing 1 1 . Ild requires 2n real multiplications and 2n - 1 real additions, i.e. thc computational effort is of order 2n. In Table IV there is a comparison of computational efforts for some matrix norms and ( I . l l d . Although 11 * I l d does not define a matrix norm, it may be shown that if there exists s such that

max lxsCl= bssI 132) s.1

1226 F. MEDINA A N D R. L. TAYLOR

Table IV. Computational effort for norms of square complex matrices (based upon real variable floating point arithmetic)

Order of computational effort

Norm Operationsi Square roots Comparisons storage Additional

t One operation is defined as a multiplication plus an addition. $ k is the number of iterations required to compute the eigenvalue needed

llXlld is bounded by some other conventional norms operating on X. Therefore, a sequence (2, = XI} of square matrices of order n satisfying equation (32) converges to X if and only if ~ ~ X & converges to 0.

REFERENCES

1. R. F. Ungless, ‘An infinite finite element’, M.A.Sc. thesis, Univ. of British Columbia (1973). 2. P. Bettess, ‘Infinite elements’, Int. J. nurn. Merh. Engng, 11, 53-64 (1977). 3. Y. C. Fung, Foundations of Solid Mechanics, Prentice-Hall, Englewood Cliffs, N.J., 1965. 4. K. H. Huebner, The Finife Element Method for Engineers, Wiley, New York, 1975. 5 . A. R. Mitchell and R. Wait, The Finite Element Method in Partial Differential Equations, Wiley, New York, 1977. 6. P. P. Lynn and H. A. Hadid, ‘Infinite elements with l / r ” type decay’, Inf . J. nurn. Meth. Engng, 17,347-355 (1981). 7. F. B. Hildebrand, Infroduction to Numerical Analysis, 2nd edn, McGraw-Hill, New York, 1974. 8. P. Bettess and 0. C. Zienkiewicz, ‘Diffraction and refraction of surface waves using finite and infinite elements’,

9. P. Bettess, ‘More on infinite elements’, Int. 1. nurn. Mefh. Engng, 15, 1613-1626 (1980). Inf. J. nurn. Mefh. Engng, 11, 1271-1290 (1977).

10. G. Beer and I. L. Meek, ‘‘‘Infinite domain” elements’, Int. J. num. Mefh. Engng, 17, 43-52 (1981). 11 . Y. K. Chow and I. M. Smith, ‘Static and periodic infinite solid elements’, Inr. J. nurn. Meth. Engng, 17, 503-526

12. F. Medina, ‘An axisymmetric infinite element’, Int. J. num. Merh. Engng, 17, 1177-1185 (1981). 13. F. Medina, ‘Modelling of soil-structure interaction by finite and infinite elements’, EERC/LICB-80/43, Univ.

14. 0. C. Zienkiewicz, The Finite Element Method, 3rd edn, McGraw-Hill, London, 1978. 15. C. Emson and P. Bettess, ‘Applications of infinite elements to external electromagnetic field problems’, in

NurnericdMethods for Coupled Problems (E. Hinton, P. Bettess and R. W. Lewis, Eds.), Pineridge Press, Swansea, U.K., 1981.

16. H. G. Poulos and E. H. Davis, Elastic Solutions for Soil and Rock Mechanics, Wiley, New York, 1974. 17. K. F. Graff, Waue Motion in Elastic Solids, Ohio State Univ. Press, 1975. 18. J. E. Luco and R. A. Westmann, ‘Dynamic response of circular footings’, Proc. A.S.C.E. 97 (EMS), 1381-1395

(197 1 ) . 19. J. E. Luco, ’Torsional response of structures for SH waves: the case of hemispherical foundations’, B.S.S.A. 66,

20. S . M. Day and G. A. Frazier, ‘Seismic response of hemispherical foundation’, Proc. A.S.C.E. 105 (EMI), 2 9 4 1

21. S. Gupta, T. W. Lin, J. Penzien and C. S. Yeh, ‘Hybrid modelling of soil-structure interaction’, UCBIEERC-

(1981).

of California, Berkeley (1980).

109-123 (1976).

( 1976).

80/09, Univ. of California, Berkeley (1980).

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