Computrrs & S~rucrures Vol. 16, No. 6. pp. 69141%. 1983 Printed in Great Britain.

cnl45-7949/83/%0691-06so3.0010 Pergamon Press Ltd.

AN ANALYSIS OF AN UNCONDITIONALLY STABLE EXPLICIT METHOD

ROBERT MULLEN

Department of Civil Engineering, Case Institute of Technology, Case Western Reserve University, Cleveland, OH 44106, U.S.A.

and

TED BELYTSCHKO

Department of Civil Engineering, The Technological Institute, Northwestern University, Evanston, IL 60201, U.S.A.

(Received 17 February 1982; receioed for publication 22 April 1982)

Abstract-Recently, several semi-implicit methods have been proposed for the time integration of the structural dynamics equations which are unconditionally stable yet explicit in their algorithmic structure. While these methods seem to violate the basic premise of the Courant requirement that the speed of information flow in the discrete model must not exceed that in the continuous problem, it is here shown that this is not the case. However, an analysis of the phase velocities of waves shows that the real flow of information of short wavelengths barely exceeds one spatial mesh per time step in these methods. Thus these semi-implicit methods appear to be useful primarily for problems dominated by very low frequencies, which is borne out by some estimates of computational costs made here.

1. INTRODUCTION

In most numerical solutions of transient structural or continuum mechanics problems, the governing hyper- bolic partial differential equations are lirst discretized in space. This procedure is called a semidiscretization; either finite element or tinite difference methods can be used to semidiscretize the governing partial dzerential equa- tions. The semidiscretization will reduce the problem to a system of ordinary differential equations in time which in turn must be integrated to complete the solution process.

There are two basic classes of methods for integrating the ordinary differential equations resulting from the semidiscretizations: explicit and implicit. The explicit methods allow the displacements at the next time step to be found in terms of accelerations and displacements at the current time step. If a diagonal mass matrix is used, no simultaneous system of equations need to be solved.

In the implicit methods, the equations for the dis- placements at the next time step involve the accelerations at the next time step, so the determination of displace- ments involves the solution of a simultaneous system of equations. Since the solution of a simultaneous system of equations usually consumes the majority of the computer time associated with solving mechanics problems, the explicit methods are considerably more efficient per time increment than the implicit methods. Although the im- plicit methods usually require considerably more com- putational effort per time step than explicit methods, the time step in most implicit methods is restricted in size only by accuracy requirements. The time step in explicit methods, on the other hand, is restricted by numerical stability requirements which may result in a time incre- ment much smaller than that needed for the requisite accuracy, thus increasing the cost of the explicit method.

The stability restriction on the time increment in expl- icit integration methods applied to hyperbolic equations was first noted by Courant, Friedrichs and Lewy in 1928[1]. They stated that the time step, At of an explicit integration method must be less that (Ax/c) where c is

the wave speed (slope of the characteristic lines) and Ax is the distance between nodes. Introducing the Courant number r

CAt r=x

YA

then the Courant-Friedrich-Lewy condition that

Tdl (1.2)

corresponds to the requirement that the domain of dependence of the discretized model totally encompass that of the original equation. Alternatively, this can be restated in terms of the rate of information flow in the discrete model: a necessary condition for stability is that the rate of numerical information flow must be greater than or equal to the rate of information flow in the continuous problem. While more rigorous mathematical studies of the stability are available which provide shar- per conditions,[2-4], the basic idea of Courant, Frie- drichs and Lewy has provided insight into the general properties of explicit integration methods.

Recently, Trujillo[5] has presented a temporal in- tegration method which, while explicit in computational structure, was shown to be stable for any positive time step. This type of method is often called semi-implicit. The method appears to contradict the Courant-Frie- drichs-Lewy requirement on the domain of influence, for in an explicit scheme the domain of influence is expected to be finite, so unconditional stability is not anticipated. In order to reconcile this contradiction, a careful study of the Trujillo method was made. The character of the information flow in the method along with the relation- ship between the discrete and continuous problem was determined. It is shown that the domain of dependence and hence the information flow in this method, is in fact infinite, just as in implicit methods. However, a dis- persion analysis reveals that the phase velocity of high

CAS Vol. 16. No. b-A 691

692 ROBERT MULLEN and TED BELYTSCHKO

frequency waves is severly retarded by the method, so that the real flow of information of high frequency waves barely exceeds one spatial mesh per time step; thus the flow of information for high frequency waves resembles that in explicit methods.

In Section 2 a review of conventional explicit methods is given. Section 3 presents the information flow charac- ter of the Trujillo method along with a dispersion analy- sis. Section 4 contains an estimate of the computational efficiency of the Trujillo method compared to con- ventional methods. Results are summarized in Section 5.

2. CONVFBTlONAL EXPLICIT MElNODS

The semidiscretized equation of motion for an un- damped structure can be written

MiitKu=f (2.1)

where the mass and stiffness matrices are denoted by M and K, respectively. The vector f is the forcing term, and u is the vector of generalized displacements. Superscript dot denotes time derivatives.

As an example of a conventional explicit method, consider the central difference method. The central difference formulas are

u(t t At) = u(t) t A?& t :Ar) (2.2)

ri(t t:At) = ri(t -:At)+ Atii(t). (2.3)

Alternatively, if we let superscripts j indicate the time, jAt, eqns (2.2) and (2.3) can be written

U j+l = Ui + Afij+(l/z)

(2.4)

U . i+(t/z) = kj-(l/2) + Atfij.

(2.5)

Equations (2.4) and (2.5) can be combined to obtain an expression containing only displacements and ac- celerations

1 ii’=-@ r+’ - 2n’ + ni-‘1. (2.6)

An expression identical to eqn (2.6) results if the New- mark p-method[6] with f3 = 0 is used, although the Newmark p-method involves velocities only at integral time increments.

Inserting the discrete integration formula, eqn (2.6), into the semidiscrete equation of motion, eqn (2.1) will yield the fully discrete system:

U j+’ = At2M-‘(_Ku’ + jj) + 2U’ - &‘. (2.7)

As can be seen from the above equation, integration by the explicit central difference method does not require the solution of any simultaneous equations if the mass matrix is diagonal. Even if a nondiagonal mass matrix is used, it would only need to be inverted once, so explicit methods are computationally more efficient per time step compared to implicit methods.

Wile a diagonal mass matrix rarely occurs in a con- sistent formulation of the semidiscrete equations, diagonal mass matrices can be generated by modifying the numerical integration used to compute the mass matrix[7l. The convergence of a diagonal mass ap proximation has been proved by Fujii[2]. The errors

introduced by mass lumping have been examined by Belytschko and Mullen[8] and Key[9], among others.

The asymptotic stability of the central difference method can be examined by either Fourier or energy (Liapunov) methods. Fujii[2] has given a proof of stabil- ity in energy for the central difference method, which was extended for nonlinear materials by Oden and Frost [3]. The stability of the central difference method is also a special case of the proof given by Hughes and Liu[lO]. These proofs show that the time step in the central difference method is limited by

A&--_ (2.13)

for an undamped system, where w,,,~ is the maximum frequency in the system. This frequency can be esti- mated by maximizing the Rayleigh quotient, which yields for a one-dimensional two node element that

(2.14)

where c is the wave speed. From eqns (2.13) and (2.14), the stability restriction on the time increment becomes

At+ (2.18)

Equation (2.18) corresponds to the Courant-Friedrichs- Lewy [l] condition that the time increment must be less than the wave’s traversal time across an element.

In order to give an intuitive explanation of the stability restriction, we will consider the information flow through a one-dimensional mesh. The discrete equation at node n for a uniform, one-dimensional mesh of linear elements integrated by the central difTerence method from eqn (2.7) can be seen to be given by

nn m+l _ 2n,m + Unm-l = ( ) e 2(u:+1 - 2u.” t u :_,)

(2.19)

where subscripts indicate the node number and Ax is the element length. From eqn (2.191, it can be seen that only the displacements at nodes n - 1 to n t 1 will affect the node n at the next time step. This limited flow of information is shown in Fii. 1. The stability limit can be thought of as a requirement that the physical information

t Y x-Ax x+2Ax x+3Ax

Fig. 1. Explicit integration information flow.

An analysis of an unconditionally stable explicit method 693

t+.qAt

t+zAt (w---w--) ii ir li JL

X x+Ax x+2Ax x+JAx

Fig. 2. Implicit integration information flow.

flow rate (wave speed) not exceed the computational information flow rate Ax/At.

The information flow in an implicitly (trapazoidal rule) integrated mesh is shown in Fig. 2. The horizontal in- formation flow paths in the implicit method results from the coupling of the acceleration and displacements at the next time step. This coupling requires the solution of a system of simultaneous equations, but also produces an infinite information flow rate; hence the unconditional stability of these methods is not unexpected.

While the stability limits computed from the infor- mation flow arguments are not as sharp as limits cal- culated by more rigorous methods and provide only necessary conditions, the information flow can be used to examine methods that will enlarge the range of stable time increments.

3. SEWLMPLlClT MElNOD

The Trujillo method for integrating eqn (2.1) without damping can be written

@+.!cKL) i*+(lm= (M_;K’) i’

-+‘+;(f+l+p) (3.1)

U i+w2j = #&I +p+w2>+ i’)

(3.2)

( ~f:~U)~j+l= (~_Tpu ui+(1/2) ) _+j+W2) +$f+‘+f’ (3.3)

where K” and KL are the strictly upper and lower triangular matrices, respectively, resulting from the symmetric splitting

K=KLtK” KL=(K”)= (3.5)

and h = 2At; the superscript j now gives the time by t = jh = 2jAt. The use of the matrix splitting given above can be recast as a method of taking advantage of the serial nature of the explicit calculation sequence. Once one degree of freedom is advanced to the next time increment, the new value can be used in the calculation

for the adjacent nodes in the integration formula. The information flow rate is Ax/At in one direction (in the direction of the node numbers) and infinite in the other direction. This is shown in Fii. 3. Since the order that the degrees of freedom are advanced is reversed in the next time step, an intinite information flow rate results in the other direction, so the domain of dependence is infinite. Thus, while the method is explicit, the domain of dependence of the numerical scheme is infinite and its unconditional stability does not conflict with the notions of Courant. However, the situation is actually more complex, as will be seen from the subsequent examina- tion of the phase velocity of waves in a mesh when integrated by the TrujiUo method.

In order to analyze the dispersive error of the Trujillo method when applied to one-dimensional, linear wave propagation, we note that by eliminating the half-step terms u1+w2), ji+w2),

the homogeneous form of eqn (2.1) can be written as

The forcing term has been removed, for in linear prob- lems, the dispersive property of the solution is in- dependent of the forcing term.

In order to examine the dispersive properties of eqns (3.6) and (3.7), a specific semidiscretization of the governing partial dtierential equations must be used. For simplicity, a one-dimensional field of evenly spaced, linear elements will be considered. The mass matrix will be assumed diagonal. This physically corresponds to a series of bar elements with constant stiffness and mass in all elements.

Away from the boundaries a typical equation of motion of the homogeneous semidiscrete system is

2

ii, =+-Um_,+2u,- uln+1 I (3.8)

where subscripts indicate the node of the spatially dis- crete system. Defining the following dBerence operators

vu icU i+l_U i m m m (3.9)

t+sAt

t+zAt

At

t X x+Ax x+2Ax x+3Ax

Fig. 3. Information flow in TrujiUo method.

694 ROBERT MULLEN and TED BELYTSCHKO

Au,’ = u,j+’ t u,,,‘. (3.10)

Equation (3.6)-(3.8) can be combined to give

~(vum’-vu~~-l)+;(vu~+,-vu~~,-vuml+u~~-l)

+$(Vu’,_,-vu’-‘,-vu,‘+vu:‘)

4 t @4m’ -vu’,,, -vuptvu’,:,-vu’,_,tvu,’

+ vu:.!, -vu,‘-‘)

=~(Au’,_,+Au’_~,-~Au,~-*Au,~-’

t Au!,,+, t Au;:,). (3.11)

The solution to eqn (3.11) can be obtained by letting

u,‘=Ae i(ojh--2aoAxx) (3.12)

where i is the imaginary basis, o is the radial frequency and a is the wave number. By substituting eqn (3.12) into eqn (3.11) and combining terms the following relation between o and a can be obtained.

r4[2 cos oh - 2 - 2(cos wh - 1) cos KAX]

- 4r2[3 cos KAX - 3 t (cos KAX - 1) cos oh1

t4(c0swh-1)=0 (3.13)

where

K = 27ra. (3.14)

Solving eqn (3.13) for o and recalling the definition of phase velocity

W CP =G (3.15)

the phase error (dispersion) in the Trujillo method is

5ae=- 1 c 4araAx

x cos_l 2r4[1 - cos (2?raAx)] t 12r*[cos (2aaAx) - 11 t 4 2r4[1 - cos (2TaAx)]- 4rZ[cos (2raAx) - 11 t 4 ’

(3.16)

Curves showing the solution of the above transcendental equation for various values of the Courant number r are shown in Fig. 4.

Similar curves for the Newmark p-method, whose analysis is given in [8], are shown in Fig. 5. For values of r < 1, the dispersion in the Trujillo method compares favorably with the Newmark g-methods. For values of r greater than one, the dispersion in the Trujillo method is much greater than in the Newmark family. In Fig. 6 the dispersion is shown as a function of the Courant number, r, for various values of a. The dashed line represents the Courant limit based on the discrete phase velocity, that is, it represents the locus of time steps and mesh sizes so that c,At =Ax. For long wavelengths, the Trujillo method is accurate above the Courant limit, but as the wavelength is decreased the wave speeds in the discrete

KAX 71

Fig. 4. Phase error in the Trujillo method for various Courant numbers.

%3 I I 0.60 0180

1

0.20 0.40 I.00

K AX 7

Fig. 5. Phase error in the Newmark p-method for various Courant numbers.

mesh are retarded until they are almost at the Courant limit regardless of the size of the time step. Thus, short waves will progress at about one spatial step per time step regardless of the actual wave speed or At/Ax. It is also clear that the semi-implicit method has a much larger dispersion error than the conventional implicit integration methods.

4. COMPUTATIONAL EFFICIENCY OF TRUJlLLO METEOD

In the selection of an integration method, the primary criterion is that minimal computational effort be achieved for a given accuracy. A comparison of the Trujillo method with other integration methods in computational efficiency and dispersion error must be made in order to permit rational selection of an integration method for a given problem. When compared to the Newmark fi method with /3 =: (another unconditionally stable

An analysis of an unconditionally stable explicit method 695

-. 1 oO.OC I.00 2.00 I 3.00 1 4.00 , 5.cO 1

COURANT NUMBER

K rlX=lT -

2.00 4.00 6.00 8.00 10.00

COURANT NUMBER. r Fig. 6. Phase error as a function of Courant number, r, for various wave numbers in the Trujillo method; the Courant line is

Fig. 7. Phase error as a function of Courant number for various

dashed and corresponds to c,At = Ax. wave numbers in the Newmark @method.

method), the value of the Courant number that will result in the same dispersion error is given in Table 1.

For long wavelengths, the Trujillo and Newmark fl methods exhibit the same dispersion error at comparable Courant numbers, but for smaller wave numbers the Newmark /3 method is superior by a factor of two. Thus, for the same error in the low frequency response, the time step used in the Trujillo method must be smaller than that used in Newmark fi method for similar dis- aersion errors.

here N is the number of degrees of freedom of the system of equations and b is the semi-bandwidth of the mesh. The computational effort needed to multiply a banded matrix by a vector is

E = (2b - l)N. (4.2)

For an upper or lower semi-banded matrix the expres- sion is

E=bN. (4.3) = Once the time step that will give comparable accuracy is estimated, the cost of each method for a given ac- Using the previous estimates, the computational effort

curacv can be comouted. As an estimate of com- per time step for the Trujillo method is

put&i&al effort, the iumber of multiplications per time increment will be examined. For a symmetric banded

E = N(4b t 6). (4.4)

matrix, the number of multiplications, E, needed for For the Newmark B-method the computational effort per inversion is approximated bv time steb is

I

E = ;Nb2 (4.1) Nb2

E==tN(4bt3) (4.5)

Table 1. Maximum Courant number for a given dispersion error and wave number

P 1.0

.95

.90

.85

.80

.75

.70

.65

.60

.55

.50

.45

.40

.35

.30

.25

r

0.4

0.7

1.0

1.1

1.2

1.3

1.3

1.5

3* = f Trujlllo

0.3

0.4

0.5 0.1

0.6 0.5

0.7 0.7

0.8 0.9

0.9 1.0

1.0 1.1

1.1 1.2

1.3 1.3

1.5 1.4

1.8 1.5

2.1 1.6

2.4 1.7

2.9 2.0

3.6 2.2

‘.? - 0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.1

2.4

2.7

3.1

3.5

4.7

4.9

5.0

73 -

KAX = 0.25~

rujlllo

0.6

0.9

1.1

1.2

1.4

1.5

1.6

1.7

1.9

2.0

2.2

2.3

2.6

2.8

3.1

0.7

1.3

1.7

2.1

2.5

2.9

3.4

3.9

4.5

5.2

6.0

6.9

8.1

9.6

KAX - 0.1s

.l

1.3

1.6

1.9

2.1

2.3

2.5

2.6

2.8

3.0

3.2

3.5

3.7

4.1

4.5

4.9

,.t -

.l

2.7

3.9

5.0

6.1

1.2

0.4

9.1

-

* = Newmark B-method

6% ROBERT MULLEN and TED BELYTSCHKO

Table 2. Guidelines for use of semi-implicit (Trujillo) method

WHEN USE

FOR BANDED STORAGE

b2 TcM Newnark 6, 6 = l/4

Trujillo

FOR BLOCK STORAGE

b2 < M (80 - Bb + 34) Newark 6, 6 =94

b2 > M (4D - Bb + 14) Trujillo

where M is the number of time increments used in the problem. From the above cost and accuracy estimates the criteria for selection of an integration method can be summarized as given in Table 2. The bounds are not decisive for all values of b; for certain values of semi- bandwidth &her method is clearly preferable.

If the multiplications are performed by blocks, the relative efficiency of the Trujillo method is improved. The number of computations for the Trujillo method is now given by

where

E = N(2D t 10) (4.6)

D=(Ctl)*NDOF (4.7)

and C is the number of nodes connected to each node and NDOF is the number of degrees of freedom per node. For example, for a regular mesh of quadrilaterals, C = 8 and NDOF = 2, while for a regular mesh of hexa- hedra, C = 26 and NDOF = 3.

5. SUMMARY

The existence of an unconditionally stable explicit integration method which is often called semi-implicit, can be reconciled with the Courant-Friedrichs-Lewy Hypothesis by a careful examination of the information flow in the method. However, in semi-implicit methods the phase velocity in the discrete mesh is reduced so the higher frequency waves propagate only about a single spatial node per time step regardless of the actual wave

speed in the material. This large dispersive error, as compared to the Newmark p-method, allows the Trujillo method to be more efficient for a given accuracy only for a limited range of problem parameters.

REFERplCEs

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2.

3.

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I. 8.

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10.

R. Courant, K. Friedrichs and H. Lewy, Uber die partiellen Differenzengl-eichungen der mathemtischen Physik. Ma&matische Ann&n, 100,32-74 (1928). H. Fujii, Finite Element Schemes: Stability and Convergence. Advances in Computational Methods in Structural Mechanics and Design (Edited by J. T. Gden), pp. 201-218. University of Alabama Press (1972). J. T. Gden and R. B. Frost, Convergence accuracy and stability of finite element approximations for a class of nonlinear hyperbolic equations. Jnt. J. Num. Meth. Engng 6, 357-365 (1973). R. D. Richtmyer and K. W. Morton, Difference Methods for Initial Value Problems. Interscience. New York (l%n. D. M. Trujillo, An unconditionally itable explicit aliorithm for structural dynamics. Int. J. Num. Meth. Engng 11, 1579- 1592 (1977). N. Newmark, A method of computation for structural dynamics. .I. Engng Mech. Div. Proc. of ASCE, 85, 67-94 (1959). T. J. R. Hughes, Private Communication, 1977. T. Belytschko and R. Mullen, On dispersive properties of finite dlement solutions. Modem Problems in Elastic Wave Prooanation (Edited bv J. Miiowitz and J. D. Achenbach). W&y:New York (1978). S. W. Key, H0NDG-a finite element computer program for the large deformation dynamic response of axisymmetric solids. Sandia Laboratories SLA-74-0039, 1974. T. J. R. Hughes and W. K. Liu, Implicit-explicit finite ele- ments in transient analysis: stability theory. J. Appl. Mech. 45, 371-374 (1978).