dispersion analysis of finite element semidiscretizations of the two-dimensional wave equation

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL. 18, 11-29 (1982)

DISPERSION ANALYSIS OF FINITE ELEMENT

WAVE EQUATION SEMIDISCRETIZATIONS OF THE TWO-DIMENSIONAL

ROBERT MULLEN Department of Civil Engineering, Case Western Reserve University, Cleveland, Ohio, U.S.A.

TED BELYTSCHKO

Department of Civil Engineering, The Technological Institute, North western University, Evanston, Illinois, U.S.A.

SUMMARY

The dispersive properties of finite element semidiscretizations of the two-dimensional wave equation are examined. Both bilinear quadrilateral elements and linear triangular elements are considered with diagonal and nondiagonal mass matrices in uniform meshes. It is shown that mass diagonalization and underintegration of the stiffness matrix of the quadrilateral element markedly increases dispersive errors. The dispersive properties of triangular meshes depends on the mesh layout; certain layouts introduce optical modes which amplify numerically induced oscillations and dispersive errors. Compared to the five-point Laplacian finite difference operator, rectangular finite element semidiscretizations with consistent mass matrices provide superior fidelity regardless of the wave direction.

INTRODUCTION

The numerical solutions of linear wave propagation problems by semidiscretizations, such as the finite element or finite difference method, are often dispersive when the analytical solutions are not. Krieg and Key’ have examined this problem in a classical study of one-dimensional constant strain elements, which treated both the effects of mass approximations and time integrators. Belytschko and Mullen* extended this analysis to higher order, one-dimensional elements and showed the existence of an optical branch in the spectrum which increases the noise associated with the propagation of discontinuities. Holmes and Belytschko3 numerically showed the development of additional noise with changes in mesh size; Bazsnt4 has analysed the magnitudes of these reflections analytically. Chin and Hedstrom’ have obtained closed form solutions for the behaviour of a one-dimensional semidiscretization at discontinuities which show the magnitude and extent of numerically induced oscillations and their response to viscosities.

The behaviour of semidiscretized wave propagation solutions in two dimensions has received less attention, although the potential for dispersive errors is increased because of the effects of the direction of the wave. Goudreau6 has calculated the frequency spectrum for the bilinear displacement, rectangular element and the five-point Laplacian finite difference operator. Holmes’ has given numerical evidence of the significant effects of mesh layouts on wave propagation solutions for two-dimensional constant strain elements. Weinberger879 has investigated the frequency bounds for irregular, bounded regions semidiscretized by finite difference methods.

In this paper, the dispersive errors for plane waves in two-dimensional meshes will be examined. Attention will be focused on finite element semidiscretizations, but the five-point

0029-5981/82/010011-19$01.90 @ 1982 by John Wiley & Sons, Ltd.

Received 8 May 1980 Revised 12 February 1981

12 R. MULLEN AND T. BELYTSCHKO

Laplacian finite difference operator emerges as a special case. Both consistent, lumped, and averages of the lumped and consistent mass matrices will be treated for quadrilateral and triangular elements.

GOVERNING EQUATIONS

We will consider the two dimensional wave equation

-+--- a2u a2u 1 a x 2 a y 2 - c 2 ii

where u is the unknown variable, c the wavespeed, and superposed dots denote time deriva- tives. This equation governs numerous physical problems. For example, in linear elasticity, equation (1) corresponds to the antiplane shear wave equation with u the antiplane displacement, while in linear acoustics equation (1) governs the pressure.

The plane wave time harmonic solution to equation (1) can be expressed as

u = A exp[iK(r. p-ct)] (2)

where i = 4-1, K is the wave number, r a position vector, p the unit normal to the wavefront, and A the amplitude of the wave. The angle 8 will be defined by

r . p = y sin 8 + x cos 8 (3)

The finite element semidiscretization of equation (2) is obtained by taking a weak form of equation (l), over fl as follows:

find u E H' Vv E H '

av au av au 1 --+--+?~ii ax ax ay ay c

Within each element u and v will be approximated by

= uf ( t )Wx, Y )

v = U r ' ( t ) N I ( X , Y )

(4)

where a summation is implied over repeated subscripts; uf are the nodal values of u at the nodes of an element, NI the shape functions for the element. The element nodal variables are related to the global matrices uI by a Boolean connectivity matrix Lfj, where

U ; = L t u j (6)

A similar relation holds between vI and of. By using equations (5) and (6) with (4), we obtain

MIjiij + K ~ j u j = 0

with

DISPERSION ANALYSIS OF FINITE ELEMENT SEMIDISCRETIZATIONS 13

Here k;, and rnb are the element spatial and temporal matrices, respectively; in solid mechanics, they are called the element stiffness and mass matrices, and we will use this nomenclature henceforth.

We now consider a regular mesh with nodes equispaced by A x and Ay along the x and y axes, respectively. We denote the nodal value at x = rn Ax, y = n Ay, by Solutions to the semidiscretized equations (7) will be sought, as in Reference 9, in a form derivable from equation (2):

= A exp [iK (Ax m cos 0 + Ay n sin 0) - iwt]

The phase velocity cp of the semidiscretized equations is given by

w

K cp =-

Once a particular element, and hence the shape functions Nr are chosen, the phase velocity for various meshes can be obtained by equations (8)-(13).

QUADRILATERAL ELEMENTS

The shape functions for the bilinear quadrilateral element are

N -1

N -1

N -1

N -1

1 - 4(1- O(1 - 77)

3 - 4 (1+0(1+ 77)

4 - 4(1- 5x1 + 77)

2 -4(1+6)(1-77)

The element matrices miJ and k;, as defined by equations (10) and (11) are given in Appendix I; the mass matrix m;, here is nondiagonal and is called consistent. In an unbounded uniform mesh, each of equation (7) are the same and given by

where y = Ay/Ax. Substituting equation (12) into (15), we obtain

14

where

R. MULLEN AND T. BELYTSCHKO

fi = c o s [ K A x ( c o s 8 + y sin 8 ) ]

f z = cos [ K Ax (cos 8 - y sin 8) ]

f 3 = c o s [ K Axy sin f?]

f 4 COS [ K AX COS 81

The following will also be defined for use in the sequel:

f s = cos [ K Ax (3 cos 0 + y sin f?)]

f 6 = cos [ K Ax (3 cos f? - y sin O)] (17b)

By use of equation (13) the dispersion relation becomes

(18)

The above expression is plotted for various values of 8 in Figure 1 and various values of y in Figure 2. From these results, it is clear that the phase velocity for the numerical solution

3 - 1 2 4 ( 1 + ~ 2 ) - 6 ( 1 + y 2 ) ( f ~ + f ~ ) + 1 2 ( ~ 2 - 2 ) f ~ - 1 2 ( 2 y 2 - l ) f ~ 8 + f l + f 2 + 4f3 + 4f4

2--[ c K ~ A X

2 .ooo

1.750

1 300

1.250

CP - C

1 .ooo

-750

a500

250

0.000 0.

e = 450

I 250 .5m -750 1. M

Figure 1. Dispersion of bilinear quadrilateral element with consistent mass as a function of relative wave number for various propagation directions


2*om 1.750 i 1.250 1 7 y = 0.75

CP - C

1 .ow y = 0.5

y = 0 . 2 5

0.om 4 0.000 .250 300 -750 1

2 8 7r

30

Figure 2. Dispersion of bilinear quadrilateral element with consistent mass as a function of propagation direction for various aspect ratios when the relative wave number ( K Ax) is 2.0

for short wavelengths varies markedly with the direction of propagation. Moreover, the more the element aspect ratio differs from unity, the greater this effect becomes.

Two commonly used procedures in finite element analysis are mass lumping and reduced integration. The effect of these alterations on the dispersion in linear quadrilateral elements will now be investigated. The typical differential equation associated with node m, n will be written

(19) 2 B I , i i m + r , n + J + P D&rn+r,n+, = 0

The value of the matrices B and D for various mesh constructions are given in Table I. For a diagonal mass matrix, the resulting dispersion function is

which is plotted for various values of 8 in Figure 3. As observed by other the dispersion error of the diagonal mass results in a slower phase velocity than the analytic value, while the consistent mass formulation results in a faster phase velocity than the analytic value.


Table I. Typical differential equations

BlfUm+t.n+f +P2Dlf~,+,.,+f = 0 with mass = a[consistent] + (1 - a ) [diagonal]

Quadrilateral element; full integration

1 i Bl, D8f

0 0 16a+( l - a )36 8(yz+1) 0 *l 4a 2(yZ-2)

*1 0 4a 2(1- 277 Z t l *1 a 4 1 + Y’) *1 T l a 4 1 + Y2)

Quadrilateral element; reduced integration

1 i Blf Dt,

0 0 16@+(1-a)36 4(yZ+1) 0 *1 4a 2(Y2 - 1)

*l 0 4a 2(-y2+ 1) *1 *1 f f -41 + Y’) *1 71 a 4 1 + Y’)

Five-point Laplacian finite difference operator

D,,

*1 0 0 -Y

~. i i Bz,

_ _

0 0 1 2(1+ 72’

0 *1 0 -1

One-directional triangular mesh ( /3 = __ ;f;). .

0 0 12a+( l - a )24 4(1+yz) 0 *l 2a -2

*1 *l 2a 0 *1 0 2a -2y’

Hexagonal triangular mesh

0 0 6~1+12(1-a) (3+4y2) *1 0 a -2y2+4 *+ *1 a -1 *$ 71 a -1


Table I. (Continued from previous page)

‘ X ’ triangular element mesh

Nodes where ( m + n ) even

r i 8, Dr,

0 0 96(1-a)+48a 4(1+y2) 0 i 1 6a -2

Zt1 i 1 6a 0 i l T l 6a 0

*1 0 6a -2y2

Nodes where (m + n ) odd

r i Btl a, _ _ -

0 0 24a+48(1-a) 4(1+y2) 0 i1 6a -2

i 1 0 6a -2vz

‘Drum’ triangular element mesh

Nodes where rn is even

i i 0 0 36a + (1 - a)72 4( 1 + y 2 ) 0 i 1 6a -2

i1 -1 6a 0 *l 0 6a -2 y 2

Nodes where rn is odd

1 i 4 Dl,

0 0 3 6 a + ( l - a ) 7 2 4(1+y2) 0 +l 6a -2

*1 0 6a -272 i l 1 6 a 0

The maximum magnitude of the error for the lumped mass is about twice that of the consistent mass.

Another measure of the error introduced by a finite element semidiscretization is the range of the dispersion in different directions of propagation. This error will affect the distortion in the shape of a cylindrical or spherical wave front. This error cannot be corrected by a compensatory error in the numerical integrator as sometimes occurs in the one-dimensional

In this respect, the consistent mass formulation is slightly superior.

18

0.000 0.000

R. MULLEN AND T. BELYTSCHKO

1 .P50 .500 .750 1 .ooo

I .so0

1.250

1 .ooo

CP - C

.750

.500

a250

e = 0 0

e = 200

8 = 30"

e = 45"

0 .ooo

K A X TI

10

Figure 3. Dispersion of bilinear quadrilateral element with lumped mass as a function of relative wave number for various propagation directions

1 .so0

1.250

1 .on0

C P - C

.750

.so0

.250

\

K U x __ TI

Figure 4. Dispersion of underintegrated bilinear quadrilateral element with lumped mass as a function of relative wave number for various propagation directions

DISPERSION ANALYSIS OF FINITE ELEMENT SEMIDISCRETIZATIONS

Table 11. Dispersion functions

Mass = consistent] + (1 - a ) [diagonal]

Quadrilateral element; full integration:

6C(1 +y2)(4-fl-f2)-(4-2y2)f3-(4y2-2)f41 (1 - a ) 18 + a [8 + f l + f 2 + 4f3 + 4f4]

Quadrilateral element; reduced integration:

c, = __ y~ Ax 1

Five-point Laplacian finite difference operator:

C

YK Ax c, =- [2(1+ yZ)-2f3-2yZf4]

One-directional triangular element mesh:

3[4(1+ 7’) -4f3 -4y2f4] 6(1-a)+cu(3+fl+f3+f4)

Hexagonal mesh of triangular elements:

YK Ax ( 6(1-a)+a(3+f4+f5+f6) c 3[3+4y2-(4y2- l)f4-2(f~+f6)]

c, =-

‘ X ’ triangular element mesh:

c -b + J(bZ - 4ad) c, =-

a = [a(48 + 12f1 + 12fz) +96(1 -a)][48 -24a1-[12a(f3 +f4)I2 Y K Ax 2a

b = 36{(4 + 4yz)[(a - 1) 144 - a (72 + 12flf 12f2)] - 96( f3 +f4)(f3 +f4y2))

d =362[16(1 +yZ)’-16(f3+y2f4)’]

‘Drum’ triangular element mesh:

c -b*J(b2-4ad) c, = -

a = [12(1 -a ) + a (6 + 2f3)]’ - 4a2fj (2 + 2f3)

YK Ax 2a

b = 6{2[12( 1 - a ) + d =36{[4f3-4(1+y2)]2-16y4fj}

(6 + 2f3)][4f3 - 4(1+ ?’)I- 8ay2f? (2 + 2f3)}

19

The use of one-point integration for this element has been proposed by several research- ers.’1312 Underintegration procedures have been recommended to improve computational efficiency and to reduce the locking effect in this element. The typical differential equation for the underintegrated quadrilateral element is given in Table I for both consistent and diagonal mass formulations. The resulting dispersion functions are given in Table 11. The dispersion in the underintegrated element is the same as the exactly integrated element for waves propagating perpendicular to the element sides for any mass matrix; otherwise the dispersion errors are increased by underintegration. The dispersion function for underintegrated stiffness and lumped mass matrices are plotted in Figure 4.


TRIANGULAR ELEMENTS

Linear displacement triangular elements can be arranged in several uniform mesh patterns. Four different mesh patterns will be considered. The first mesh considered is shown in Figure 5. The typical differential equation and associated dispersion function are given in Tables I and 11, respectively. The dispersion function is also plotted in Figure 5. The dispersion of waves propagating along the x and y axes is similar to that of the bilinear quadrilateral and the one-dimensional, linear displacement element.2 The differential equation for this mesh, when the mass matrix is diagonal, is identical to the five-point Laplacian finite difference operator. The dispersive errors are comparable to those of the lumped mass quadrilateral element but markedly greater than of the consistent mass quadrilateral.

2 -000

1.750

1.500

1 .P50

CP - C

1.000

.750

-500

250

0.000 0 .250 s o 0 .750 1.

K&

7T

I0

Figure 5. Dispersion of ‘one-directional’ triangular element mesh as a function of relative wave number for various propagation directions (lumped mass dispersion corresponds to five-point finite difference)

The dispersion functions in all the previous examples are periodic in 8 with a period of 7 ~ 1 2 , but the period of the dispersion function of the hexagonal mesh shown in Figure 6 is ~ / 3 , because the mesh is invariant under 60 degree rotations. The phase velocity for this mesh is shown in Figure 6. For this mesh, the dispersion error at high frequencies is slightly larger than for the quadrilateral element, but the spread in the phase velocity is smaller in this mesh than in any other mesh considered in this paper. While this mesh seems to be


.son .

21

C I

O.GU0 250 sno .750 1. 0.000 L :

n

Figure 6. Dispersion of 'hexagonal' triangular element mesh as a function of relative wave number for various propagation directions

superior, the necessity to change element size along a straight boundary will introduce spurious reflection as shown in Reference 4.

The other two mesh configurations are shown in Figures 8 and 10. The analysis of these meshes yields different equations at alternate nodes. For the mesh shown in Figure 8 the typical differential equations are:


6 .OOO

5.000

4 .ooo

3 000

W A X - C

2 .ooo

1 .ooo

0 .ooo 0

I

K n x Tr -

Figure 7 . Frequency of the ' X ' triangular element mesh as a function of relative wave number for various propagation directions

and for (m + n ) odd

u m , , , = A 2 e x p i [ ~ ( m Ax c o s 8 + n AysinO)-wt] (24) Substitution of equations (23) and (24) into equations (21) and (22) yields a set of homogeneous equations which have a non-trivial solution only if the determinant of the characteristic matrix vanishes, i.e.

where = 6c/ y Ax. The resulting quartic equation's solution for positive w is shown in Figure 7. The existence

of two branches in the solution to the frequency equation has also been observed in higher order elements for one-dimensional problems.' In the analysis of wave propagation in crystal structures, the two branches are called the acoustical (lower) and optical (higher) branches. lo

The gap in frequencies between the two branches is called a stopping band. Waves with frequencies in the stopping band decay exponentially.

The dispersion curves for this mesh are shown in Figure 8, but the optical branch is only plotted for large wave numbers because the phase velocity becomes very large for small wave numbers. The phase velocity is less in the semidiscretized solution than in the continuous equation for certain values of K Ax and 8, which contradicts the usual notion that consistent mass formulations always increase the phase velocity. The reason for this is the decrease in


2.000

1.750

1 .5m

1.250

C P - C

1 .ooo

0 .p50 .ooo 0 .ooo 1 2 5 0 ,500 .750 1.

KAX ~

Tr

I0

Figure 8. Dispersion of ‘X’ triangular element mesh as a function of relative wave number for various propagation directions

the cutoff frequency as the angle of propagation approaches 45 degrees. Because of the periodicity of the discrete spectrum, waves shorter than the wavelength at the cutoff frequency are characterized by decreasing frequencies with increasing wave number. Hence, their phase velocities decrease and the group velocities are negative.

The existence of an optical branch for a semidiscretization results in an increase of the noise in the solution. This has been shown in higher order, one-dimensional elements.* It is again later illustrated in Figure 11.

The last mesh considered is shown in Figure 9. This mesh also contains two typical difference equations for even and odd values of m + n. The differential equations and phase velocity are given in Tables I and 11, respectively. The phase velocity is plotted in Figure 10. In this mesh the optical and acoustic branches intersect below the cutoff frequency. The optical branch obtained by equations (25) may also be a fictitious result of the analysis method as in Figures 9 and 10, where an optical branch occurs for 0 = 0. For the purpose of explaining this, consider the one-dimensional linear displacement element. The typical differential equation as given


in Reference 2 is

If different solutions are assumed for even and odd nodes, even though the differential equations are identical at each node, the resulting expressions for frequency as a function of wave number are

12c2 sin2 ( K A x / 2 ) 3 + 2a COS'(K A x / 2 ) - 2a

" 2 =

8 = 90"

6.000

5.000

3 .ooo

2 .ooo

1 .ooo

0 .a00 O

K& ll

Figure 9. Frequency of the 'drum' triangular element mesh as a function of relative wave number for various propagation directions

and

" 2 = 12c2 cos' ( K Ax/2)

3 + 2 a sin2(K A x / 2 ) - 2 a

Equation (27) is identical to the result when the same solution is assumed at each node, while equation ( 2 8 ) is the same as equation (27) with the relative wave number K Ax replaced by K Ax + 7 ~ . The phase velocity computed by equation (28) would result in a spurious optical branch in the phase velocity curves. In order to investigate the results given in Figure 9 for spurious optical branches, the graphs of equation ( 2 5 ) are examined for identical form with a shift along the relative wave number axis. The optical branch for waves propagating at 0 degrees can be seen to be a spurious optical branch by examining Figure 9.


The large dispersion errors in this mesh, along with the intersection of the optical and acoustical branches, make this mesh appear to be the poorest mesh, of the ones considered, for wave propagation problems. Numerical evidence of this was shown by H01mes.~ The maximum eigenvalue associated with the mesh in Figure l l ( a ) is 1-98clhx. The maximum eigenvalue in an unbounded mesh is 2.0clAx. The mesh in Figure l l ( b ) has a maximum eigenvalue of 1.78clA.x for the same computational effort. Figure 11 also shows the increased noise that the mesh in Figure l l ( b ) introduces in a wave.

2.000

1.750

1 .so0

1.250

C P - C

1 .ooo

.750

.500

-250

0.000 0.

\

\ \

0

KAX ~

TT

Figure 10. Dispersion of the ‘drum’ triangular element mesh as a function of relative wave number for various propagation directions

In order to study the directional dependence of the phase velocity, a cylindrical wave was propagated through the mesh shown in Figure 12. The differential equations resulting from the diagonal mass formulation were integrated with a Courant number of 0.25. The phase velocity computed from the time to peak response at a node compared to the calculated values is given in Table IV. The difference between calculated and observed phase velocities is due to the frequency errors introduced by the numerical integration of the differential equations.


12.000

9.000 t k-

" 1 _1

- 6.000

3.000

0.000 0. - I 10.000 20.000 30.000

12.m

9.000 > + u a J w

-

' 6.000

3 .ooo

0.000 0.

TIME

(a)

I I 10.000 20.000 30.000

T I M E

(b)

Figure 11. Numerical results of velocity time histories for 'drum' and one-directional meshes

The use of the average of the lumped and consistent mass has been shown to result in a more accurate representation of the spectrum in one-dimensional problems.2 All of the previous meshes were also analysed using an average of the lumped and consistent mass. The dispersion for K Ax = 2 is given for various meshes in Table 111. While the error in the phase velocity is reduced by the average mass, the variation in the wavespeed with direction of propagation is

Figure 12. Mesh used for cylindrical wave propagation problem


Table 111. Dispersion c,/c at K Ax = 2.0

Mesh Mass ratio (a) Y 0" 45" 90"

Quadrilateral (Figure 1)

Quadrilateral (Figure 4) underintegrated

One-directional (Figure 5) triangular mesh

Hexagonal (Figure 6) triangular mesh

'X' triangular mesh (Figure 8)

Optical branch of above

Drum mesh (Figure 10)

Optical branch of above

1.0 0.5 0.0

1.0 C.5 0.0

1.0 0.5 0.0

1.0 0.5 0.0 1.0 0.5 0.0

1.0 0.5 0.0

1.0 0.5 0.0

1.0 0.5 0.0

1.0 0.5 0.0

1.0 1.0 1.0

1.0 1.0 1.0

1.0 1.0 1.0

1.0 1.0 1.0 0.866 0.866 0,866

1.0 1.0 1.0

1.0 1.0 1.0

1.0 1.0 1.0

1.0 1.0 1.0

1.158 0.9627 0.8415

1.158 0.9627 0.8415

1.158 0.9627 0.841 5 1.116 0,9720 0.8723 1.129 0.9832 0.8823 1.143 0.9293 0.7992

2.05 1 1.513 1,269

1.104 0,9627 0.8415

0.6021 0.5687 0.5403

1.084 0.8944 0.7788 0.9719 0.8021 0.6985

1.465 1.101 0.9187

1.158 0.9836 0.8702 1.126 0.9813 0.8810 1.162 0.9670 0.8457

1,807 1.443 1.239

1.228 1.041 0.9187

1.427 1.157 1.000

1.158 0,9627 0.8415 1.158 0.9627 0.8415 1.158 0.9627 0.8415

1.158 0,9627 0.8415 1.123 0.9794 0.8796

1.143 0.9293 0.7992

2.05 1 1.513 1.269

1.134 0.9597 0,8415

2.480 1.615 1.307

sometimes increased. It should also be noted that the monotonic nature of the convergence of the wavespeed may be lost when the average mass is used.

SUMMARY AND CONCLUSION

The dispersive properties of semidiscretizations of the two-dimensional wave equation have been analysed. It has been shown that the phase velocity varies significantly with the angle of propagation, particularly for certain arrangements of constant strain triangles.

Table I V

0" c, observed c, analytical C

0 0.8464 0.8415 1.0 45 0,8311 0.7788 1.0 90 0.8464 0.8415 1.0


Among the combinations studied, quadrilateral elements with a consistent mass appear to give the best performance. Mass lumping (diagonalization) decreases the performance of the semidiscretization, particularly in directions that do not coincide with mesh lines, and this is further exacerbated by reduced integration.

When compared to the standard five-point Laplacian finite difference operator, the quadrilateral finite element with a consistent mass is definitely superior. Although consistent mass matrices entail extra computational effort, this increase in accuracy for wave propagation effects may, in certain problems, justify the cost.

Triangular elements, even with a consistent mass, in general do not perform as well as quadrilaterals. Furthermore, the arrangement of elements should be chosen with care, because it may introduce an optical mode, which introduces substantial noise. The best arrangement of triangles appears to be hexagonal, which has the further benefit that it almost removes the directional dependence of the phase velocity, which can be useful in maintaining wave front definition.

APPENDIX I

Quadrilateral elements

Mass matrix:

lumped mass matrix:

1 0 0 1 0

stiffness matrix:

C 2

6Y k ' = -

2 y 2 + 2 -2y2+1 -1-y 2

2y2 + 2 y2 - 2 sym. 2y2+2

underintegraged stiffness matrix:

y2-2 -y2-1

-2y2+1 2 y 2 + 2

y2+1 1-y2 -y 2 - 1 y2-1

y 2 + 1 -y2+1 y 2 + 1

k'=-


Triangular element

Mass matrix:

stiffness matrix:

where xii = xi -xi. 1 k ‘ = - Y:1 +x:3 Y 3 1 Y 1 2 + ~ 1 3 ~ 2 1 2 A x 2 y 1 sym.

29

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1. R. D. Krieg and S. W. Key, ‘Transient shell response by numerical time integration’, Int. J. num. Meth. Engng,

2 . T. Belytschko and R. Mullen, ‘On dispersive properties of finite element solutions’, in Modern Problems in Elastic

3. N. Holmes and T. Belytschko, ‘Postprocessing of finite element transient response calculations by digital filters’,

4. Z. P. Bazant, ‘Spurious reflection of elastic waves in nonuniform finite element grids’, Computer Meth. Appl .

5 . R. C. Y. Chin and G. W. Hedstrom, ‘A dispersion analysis for finitedifference schemes: tablesof generalized Airey

6. G. L. Goudreau, ‘Evaluation of discrete methods for linear dynamic response of elastic and viscoelastic solids’,

7. N. Holmes, ‘Characteristics of transient finite element solutions’, Ph.D. thesis, Dept. of Materials Engng, Univ.

8. H. F. Weinberger, ‘Upper and lower bounds for eigenvalues by finite difference methods’, Comm. Pure A p p l .

9. H. F. Weinberger, ‘Lower bounds for higher eigenvalues’, Pac. J. Math., 8, 339-368 (1958).

7(3), 273-286 (1973).

Wave Propagation (J. Miklowitz et al., Eds.), Wiley, 1978, pp. 67-82.

Computers Struct., 6 , 211-216 (1976).

Mech. Engng, 16, 91-100 (1978).

functions’, Math. Comp., 32, 1163-1170 (1978).

Ph.D. thesis, Dept. of Civil Engng, Univ. of California, Berkeley (1970).

of Illinois at Chicago Circle, Chicago (1976).

Math., 9, 613-623 (1956).

10. L. Brillouin, Wave Propagation in Periodic Structures, Dover Press, New York, 1946. 11. T. Belytschko and J. M. Kennedy, ‘A fluid-structure finite element method for the analysis of reactor safety

12. D. S. Malkus and T. J. R. Hughes, ‘Mixed finite element methods-reduced and selective integration techniques: problems’, Nucl. Eng. Des., 38, 71-81 (1976).

a unification of concepts’, Comp. Meth. Appl. Mech. Eng., 15, 63-81 (1978).

dispersion analysis of finite element semidiscretizations of the two-dimensional wave equation

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