on accuracy, stability and efficiency of the newmark method with incomplete solution by multilevel...

*Correspondence to: Jacob Fish, Department of Civil Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180-3590, U.S.A. E-mail: "[email protected]

Contract/grant sponsor: O$ce of Naval Research; Contract/grant number: N00014-97-1-0687Contract/grant sponsor: National Science Foundation; Contract/grant number: CMS-9712227Contract/grant sponsor: Sandia National Laboratories; Contract/grant number: AX-8516

CCC 0029-5981/99/260253}21$17.50 Received 26 October 1998Copyright ( 1999 John Wiley & Sons, Ltd. Revised 15 February 1999

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING

Int. J. Numer. Meth. Engng. 46, 253}273 (1999)

ON ACCURACY, STABILITY AND EFFICIENCY OFTHE NEWMARK METHOD WITH INCOMPLETE

SOLUTION BY MULTILEVEL METHODS

JACOB FISH* AND WEN CHEN

Civil, Mechanical and Aerospace Engineering, Rensselaer Polytechnic Institute, ¹roy, N> 12180, ;.S.A.

SUMMARY

The use of an incomplete iterative solver aimed at approximating solution of equilibrium equation resultingfrom the "nite element semidiscretization in space is studied. Our approach is motivated by the fact that:(i) the e!ective sti!ness matrix is well conditioned because of the stabilizing e!ect of the diagonallydominated mass matrix, (ii) the dominance of the temporal error in implicit computations with large stepsize, and (iii) the utilization of a class of iterative methods where the primary computational cost isassociated with a construction of the preconditioner rather than with an iterative process. One of theprimary goals of the present manuscript is to control the temporal and the equilibrium solution errors andto give their a posteriori estimates. Numerical experiments reveal that the computational cost of three cyclesconducted with the Generalized Aggregation Multilevel solver [1}4] is comparable to a single backsubstitution on the source grid, while the resulting equilibrium solution error is negligible compared with thelocal temporal error. Copyright ( 1999 John Wiley & Sons, Ltd.

KEY WORDS: multilevel; stability; accuracy; Newmark method

1. INTRODUCTION

Direct time integration schemes are often used to integrate equations of motion resulting from thespatial semidiscretization using "nite element method. Implicit time integration methods necessi-tate solution of the linearized system of equations at each time step. Typically, the integrationprocess involves factorization of the overall e!ective sti!ness matrix plus forward and back-substitutions with factorization being the main computational cost. For linear problems withconstant time stepping, which is the best case scenario for implicit methods, the sti!ness matrix isfactorized only once prior to the time integration process, and subsequently, only forward- andback-substitutions are carried out at each time steps. For large-scale 3-D problems the computa-tional cost of factorization is very high and the storage requirement could be prohibitively large

owing to "ll-ins encountered in the factorization process. On the other hand, iterative solvers arein general not suitable for problems with numerous right-hand sides because they often mustrestart from scratch for each new right-hand side. However, unlike for multiple load staticanalyses iterative methods might be better fared against direct methods for dynamic analysesbecause of the following:

(i) The e!ective sti!ness matrix for dynamic analysis is better conditioned than for staticanalysis because of the stabilizing e!ect of a diagonally dominated mass matrix inparticular in the case of small time step size.

(ii) Since the temporal error is often dominant, convergence tolerance for solving the dynamicequilibrium equations can be relaxed provided that the accuracy and stability require-ments are satis"ed.

(iii) The iterative method of choice should be such that most of the computational e!ort iscarried out prior to the iterative process.

The Generalized Aggregation Multilevel (GAM) solver [1}4], which is based on the philo-sophy of multilevel methods, has the aforementioned quali"cations. In the GAM scheme the sizeof the coarse model and thus the complexity of the preconditioner is adaptively controlled by thecut-o! frequency parameter, c, to yield an optimal performance for a given problem. In the limitwhen c is equal to the maximal eigenvalue of the e!ective sti!ness matrix, the GAM solvercoincides with the direct method. As a motivation for the present work, Figure 1 illustrates theperformance of the implicit (Newmark) and explicit (central di!erence) integrators for the di!usercasing problem modelled with 75 717 dofs. In the implicit scheme, both the state-of-the-art directsolver [16] and the GAM solver with various convergence tolerances have been employed forsolving equilibrium equations at each time step. The time step, *t"0)02 s, for all implicitmethods considered has been selected on the basis of temporal accuracy requirements. For theexplicit method the time integration step, *t"1)5]10~7 s, was de"ned by stability requirement.For details on stability and accuracy of explicit and implicit time integration schemes we refer to[5, 14, 15].

Based on the results shown in Figure 1, several observations can be made. The "rst and perhapsnot surprising, is the fact that there exists an intermediate time interval, 6]10~3)t)0)3 s (forthe problem considered), where the iterative solver (GAM) with exact solution of discreteequilibrium equations (up to the tolerance of 10~6 in the relative residual norm) is more e$cientthan both the explicit method and the implicit method with the direct solution [16] of linearsystem of equations. The second observation is somewhat unexpected. It can be seen that thecomputational cost of a single GAM cycle consisting of pre-smoothing, restriction to the coarsemodel, back substitution in the auxiliary coarse model, prolongation back to the source grid and"nally post-smoothing, is less than that of a single back substitution on the source grid.Moreover, the CPU time of the implicit method with three GAM cycles is comparable to that ofa single back substitution on the "ne scale. This suggests that if the implicit method with up tothree GAM cycles per time step is adequate in terms of accuracy, its useful range of applicationcan be extended to 2]10~4)t)R s, which contains not only structural vibration problems,but also low velocity impact applications. Storage savings are even more signi"cant [1}4].

For implicit methods with approximate solution of equilibrium equations, an error due to thisapproximation is introduced in addition to the usual spatial and temporal discretization errors.The obvious question arises as to what is the proper convergence tolerance that should be set inorder to maintain an adequate accuracy of the computational results. One of the primary goals of

254 J. FISH AND W. CHEN

Copyright ( 1999 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng. 46, 253}273 (1999)

Figure 1. Di!user casing: solver CPU time

the present manuscript is to control the temporal and the equilibrium solution errors and to givetheir a posteriori estimates. To this end we brie#y summarize related research activities in this"eld.

Hageman and Young [6] suggested that in the solution of time-dependent problems using aniterative solver, a good choice for the convergence tolerance should be 1

10of the value of the local

time discretization error for that time step. However, no theoretical analysis was given to support

INCOMPLETE SOLUTION BY MULTILEVEL METHODS 255


this choice. Shakib et al. [7] demonstrated by a numerical example that for an iterative processbased on the Generalized Minimum Residual (GMRES) method, the convergence rate is insensi-tive to the accuracy of the solution at each iteration. These "ndings advocate the use of fairly largevalues of convergence tolerance.

One approach to estimate the local temporal error is to compare the results of two di!erentstep sizes, or to compare the results given by two integration methods of di!erent orders [8].Unfortunately, for both methods, the cost for the error estimate is high. Zienkiewicz and Xie [9]proposed a simple error estimator by comparing the Newmark solution with the exact solutionobtained from the Taylor series. Zeng et al. [10] obtained the same result in a more intuitive way.Hulbert and Jang [11] presented an error estimator for the generalized*a method which can bereduced to the identical formulae for the Newmark method [9, 10]. Wiberg and Li [12]developed a local temporal error estimator by a post-processing technique. Hulbert and Hughes[13] studied the in#uence of truncated initial acceleration values on the accuracy of the solutionand showed that by setting the initial acceleration values to zero, the global accuracy of thesolution will be reduced to the "rst order.

The remainder of the manuscript is organized as follows. A brief description of the Newmarkintegrator and the GAM solver are given in Section 2. A posteriori error estimator is developed inSection 3. Numerical examples in Section 4 conclude the paper. Attention is restricted to theNewmark integration scheme, even though the proposed methodology can be extended to otherintegrators.

2. PRELIMINARIES

2.1. ¹ime integration formulae

The semidiscrete system of equilibrium equations governing the linear dynamic response canbe represented as

MdG#Cd0 #Kd"F (1)

d(0)"d0, d0 (0)"d0

0(2)

where M, C and K are the mass, damping and sti!ness matrices, respectively. F is the vector of anexternally applied load which may vary with time; d is the vector of displacement unknowns; thesuperposed dots indicate di!erentiation with respect to time. For simplicity, attention is restrictedto positive de"nite M, C, K matrices.

The Newmark scheme is based on the following two assumptions:

dn`1

"dn#*tv

n#*t2 [(1

2!b) a

n#ba

n`1] (3)

vn`1

"vn#*t[(1!c) a

n#ca

n`1] (4)

where dn, v

nand a

nare the displacement, velocity and acceleration vectors at time t

n, respectively;

*t the time step size; b and c parameters which determine the stability and accuracy character-istics of the integration scheme under consideration. For the solution of the displacements,velocities and accelerations at time t

n`1"t

n#*t, the equilibrium at time t

n`1is considered

using recurrent formulae

K< dn`1

"F<n`1

, dn`1

3Rm, F)n`1

3Rm (5a)



where the m]m symmetric positive-de"nite matrix

K< "K#b0M#b

1C (5b)

is referred to as the e!ective sti!ness matrix, and

F<n`1

"Fn`1

#M (b0dn#b

2vn#b

3an)#C(b

1dn#b

4vn#b

5an) (5c)

as the e!ective load vector. After dn`1

is calculated, the velocities and accelerations at time tn`1

can be evaluated as

an`1

"b0(d

n`1!d

n)!b

2vn!b

3an

(6)

vn`1

"vn#b

6an#b

7an`1

(7)

where b0}b

7are constants given by

b0"

1

b*t2, b

1"

cb*t

, b2"

1

b*t, b

3"

1

2b!1, b

4"

cb!1

b5"

*t

2 Acb!2B; b

6"(1!c)*t, b

7"c*t (8)

When c*0)5 and b"(0)5#c)/4 are chosen, the integration scheme is unconditionally stable. Inparticular, when c"0)5 and b"0)25, the integration scheme is known as the trapezoidal rule orthe constant-average-acceleration method.

2.2. ¹he generalized aggregation multilevel solver

The dynamic equilibrium equations at each time step will be solved using the GeneralizedAggregation Multilevel (GAM) solver [1}4]. The GAM solver belongs to the class of multilevelmethods, which possess an optimal rate of convergence by which the computational workrequired to obtain "xed accuracy is proportional to the number of discrete unknowns. Theprincipal idea of multilevel methods consists of capturing the oscillatory response of the systemby means of smoothing, whereas remaining lower frequency response is resolved on the auxiliarycoarse model. Prior to focussing on the speci"cs of the GAM solver we brie#y outline the basicprinciple of the multilevel methods.

Consider the linear system of equations given by equation (5a). Coarse model functions aredenoted with subscript 0. For example, d

0(n`1)denotes the solution in the coarse model, where

d0(n`1)

3Rl, l(m. We also denote the prolongation operator from the coarse model to the "negrid by Q:

Q :RlPRm (9)

The restriction operator from the "ne to the coarse model is conjugated with the prolongationoperator, i.e.

QT :RmPRl (10)

In this section, superscripts are reserved to indicate the iteration (cycle) count. The residualvector in the ith cycle is

rin`1

"F)n`1

!K) din`1

(11)

where din`1

is the solution approximation at the ith cycle.



A single cycle of the two-level method for the solution of equation (5a) is as follows:

1. Starting with the current approximation, d1 in`1

, perform l1

preconditioned iterations on the"ne grid to obtain di

n`1

din`1

"d1 in`1

#D~1(F)n`1

!K) d1 in`1

) (12)

l1

is chosen so that the high-frequency components of the error associated with din`1

will bereduced to such a level that they can be adequately resolved on the coarse model. D isa preconditioner for smoothing. Any preconditioned iterative procedure which has goodsmoothing properties and requires little computational work, can, in principle, be used asa smoother in the multilevel process.

2. Restrict the "ne grid residual onto the coarse model to obtain the coarse model residual

ri0(n`1)

"QTrin`1

(13)

3. Carry out coarse model correction

di0(n`1)

"K~10

ri0(n`1)

"K~10

QTrin`1

(14)

where K0"QTK) Q is the coarse model sti!ness matrix.

4. Prolongate di0(n`1)

from the coarse model to the "ne grid to obtain the "ne grid correction,Qdi

0(n`1), and evaluate partial solution obtained after the "ne grid correction

d3 in`1

"din`1

#Qdi0(n`1)

(15)

5. Perform l2

post-smoothing iterations of the form

di`1n`1

"d3 in`1

#D~1(F)n`1

!K) d3 in`1

) (16)

to obtain the new approximation to the solution of equation (5a). The parameter l2

is chosenso that any high-frequency error components introduced by the prolongation are signi"cantlyreduced.

This cycle is repeated k times until the approximate solution dkn`1

has converged to the exactsolution d

n`1. The convergence is assumed when

R"

Erkn`1

E

EFKn`1

E)g (17)

where E E denotes the Euclidean norm of a vector, and g is the convergence tolerance.The major di!erence between various multilevel schemes is in the choice of smoothing and

prolongation/restriction operators. While the geometric multigrid approach constructs the pro-longation operator from auxiliary coarser grids, the GAM solver accomplishes the same goal onthe basis of information available in the source grid. In the GAM scheme the coarse model isdirectly constructed from the source grid by grouping "nite elements into either non-overlappingor over-lapping subdomains referred to as aggregates, and then for each aggregate assigninga reduced number of modes with an intent of e!ectively capturing the lower frequency response ofthe source system [1}4].

On each aggregate, AiE, a low order polynomial function (constant or linear "eld) is used to

approximate the solution (typically for Poisson or elasticity equations with constant coe$cients).For problems where eigenfunctions are oscillatory, such as in the case of elasticity with oscillatory



coe$cients [17] or Helmholtz equation, an analytical solution with either periodic boundaryconditions or on unbounded domains is used instead.

An alternative to selecting analytical functions on AiE

is to conduct a local eigenvalue analysison each aggregate K< i/i"ji diag(K) *)/i with zero Neumann boundary conditions on Ai

Eand to

select the eigenmodes for which ji)c. diag(K) i) denotes the diagonal of the aggregate e!ectivesti!ness matrix K) i. The value of c controls the e!ectiveness of the aggregated model. In the limit ascPmax

i(ji), the auxiliary coarse model captures the response of the source system for all

frequencies and therefore the two-level procedure converges in a single iteration even withoutsmoothing. For best performance of the iterative process with single right-hand side the value ofparameter c is typically selected in the range of 10~1}10~3. The optimal value depends on theproblem type (3-D elasticity, shells, Helmholtz). Typically, 6}50 modes satisfying ji)c areselected. For problems with multiple right-hand sides (or multiple time steps) the optimal value ofc should be higher so that most of the computational e!ort would be spent prior to the iterativeprocess.

3. A POS¹ERIORI LOCAL ERROR ESTIMATION

For a posteriori error estimation, attention is restricted to linear problems. For such systems theNewmark method can be written as

Xn`1

"AXn#LF

n`1(18)

where

Xn"Md

nvn

anNT (19)

is the Newmark approximation of the displacement, velocity and acceleration at time step tn;

A and L are the ampli"cation matrix and the load operator, respectively. For details on thesematrices we refer to [14, 15].

Let the exact solutions at tnbe denoted as

X(tn)"Md (t

n) v(t

n) a (t

n)NT (20)

and the Newmark approximation which accounts for the equilibrium solution error (to be furtherreferred to as the incomplete Newmark integrator) denoted as

X3n"Md3

nv8n

a8nNT (21)

Given the initial values X3n~1

at tn~1

, if the equilibrium equations at tn

are solved exactly, theNewmark approximation at t

nwith exact semidiscrete equilibrium solution is denoted by

X@n"Md@

nv@n

a@nNT (22)

The local temporal error (or the local truncation error) of the Newmark method at time tnis

de"ned as

qn"Mq

dnqvn

qan

NT"AX(tn)#LF(t

n`1)!X(t

n`1) (23)

where

F (tn`1

),Fn`1

(24)



The local error due to the solution approximation at tnis given by

en"Me

dnevn

eanNT"X3

n`1!X@

n`1"X3

n`1!AX3

n!LF

n`1(25)

The global error at tn, i.e. the di!erence between the incomplete Newmark integration and the

exact solutions, becomes

En"X3

n!X(t

n) (26)

From equations (23), (25) and (26), the global error at time tn`1

can be derived as

En`1

"X3n`1

!X(tn`1

)"AEn#e

n#q

n(27)

3.1. Stability

Using equation (27) repeatedly, we obtain the global error at time tn:

En"AnE

0#

n~1+i/0

An~1~i(qi#e

i) (28)

The "rst term on the right-hand side vanishes since E0"0. Thus, the global error reduces to

En"

n~1+i/0

An~1~i(qi#e

i) (29)

If the time integration scheme is stable, the following two conditions are satis"ed:

(i) o (A))1. o (A)"max D jiD is the spectral radius of A.

(ii) The eigenvalues of A of multiplicity greater than one are strictly less than one in modulus.

Consequently, the global error Enis bounded if the integration scheme is stable. This implies

that the stability of the integration scheme will not be violated even if the equilibrium equations ateach time step are solved approximately.

3.2. Local error estimates

From equation (27), it can be seen that the local error per time step, denoted as en, comes from

two sources: the temporal error qnand the equilibrium solution error e

n. The total local error in

the displacements per time step is denoted as

edn"e

dn#q

dn(30)

where qdn

and edn

are the temporal and the equilibrium solution errors in the displacements,respectively. Further exploiting the fact that the residual corresponding to the exact solution ofequilibrium equations is zero, we have

rkn`1

"K) (d@n`1

!d3 kn`1

)"!K) ekdn

(31)

where ekdn

is the equilibrium solution error in the displacements in the kth cycle.The equilibrium solution error, ek

dn, can be computed from equation (31). Unfortunately, if an

iterative method is employed the factorization of K) is not available. Thus in Section 3.2.1 we focuson the estimation of ek

dn.



3.2.1. Equilibrium solution error estimator. From equations (12)}(16), an approximation to thesolution of equation (5a) after the ith cycle (with a single pre-smoothing) can be expressed as

di`1n`1

"Gdin`1

#k (32)

in which

G"(I!D~1K) ) (I!C~1K) )"I!P~1K) (33)

k"P~1F<n`1

(34)

P~1"D~1#C~1!D~1K) C~1 (35)

C~1"QK~10

QT (36)

where G and P are termed as the iteration matrix and the multilevel pre-conditioner, respectively;C is the coarse model pre-conditioner and I is the m]m identity matrix.

It can be easily shown that d@n`1

is a solution to the related system

(I!G)dn`1

"k (37)

if and only if d@n`1

is also the unique solution to equation (5a), i.e.

d@n`1

"K) ~1F)n`1

(38)

Substitution of equation (37) into equation (32) leads to

ei`1dn

"Geidn

(39)

where

eidn"di

n`1!d@

n`1(40)

is the equilibrium solution error associated with the approximation din`1

in the ith cycle. Wefurther de"ne the pseudoresidual vector, di

dn, as

didn,Gdi

n`1#k!di

n`1"di`1

n`1!di

n`1"P~1ri

n`1(41)

Inserting equation (37) into equation (41) yields

didn"(G!I)ei

dn(42)

Based on equation (39), the equilibrium solution error after the nth cycle can be represented by

endn"(G)ne0

dn(43)

From the above equation, it follows that

Eendn

E)E (G)nEEe0dn

E)(EGE)nEe0dn

E (44)

We consider convergent iterative processes for which the norm of the iteration matrixEGE(1 [6].

We will now focus on establishing the relationship between pseudoresidual and equilibriumsolution error. Suppose that

(I!G)x"0 (45)

for an arbitrary vector x. Then

ExE"EGxE)EGEExE (46)



Since EGE(1, the above inequality is in contradiction unless x"0. But since x is arbitrary,(I!G) must be non-singular and thus (I!G)~1 exists [6]. From equation (42), the equilibriumsolution error can now be expressed as

eidn"!(I!G)~1di

dn(47)

Based on equations (39) and (47), it follows that

Eei`1dn

E)EGEEeidn

E)EGEE (I!G)~1EEdidn

E (48)

Let

B"(I!G)~1 (49)

or

B"I#BG (50)

so that

EBE"EI#BGE)1#EBGE)1#EBEEGE (51)

from where it follows that

EBE)1

1!EGE(52)

Substituting equation (52) into equation (48) yields the estimate for the equilibrium solutionerror after the ith cycle:

Eei`1dn

E)EGE

1!EGEEdi

dnE (53)

Since the pseudoresidual vector didn

is readily available at the end of the ith cycle, equation (53)can be used to estimate the equilibrium solution error at a little computational cost.

It remains to estimate the norm of the iteration matrix EGE. From equation (42) and (43), weget

dndn"(G!I)en

dn"(G!I) (G)ne0

dn"(G)n(G!I) e0

dn"(G)nd0

dn(54)

from where we have

Edndn

E)E(G)nEEd0dn

E)(EGE)nEd0dn

E (55)

The above inequality leads to the lower bound of EGE

EGE*AEdn

dnE

Ed0dn

EB1@n

(56)

Since EGE(1, the actual value of EGE is somewhere between the lower bound given by (56) andunity. In our numerical experiments we have considered a geometric average between the lowerand upper bounds, i.e.

ME"A

Edndn

E

Ed0dn

EB1@(2n)

(57)

is used as an approximation for EGE.



3.2.2. Local temporal error estimator. Zienkiewicz and Xie [9] and Zeng et al. [10] proposeda simple local temporal error estimator for the Newmark method based on the series expansion,which we adopt in the present manuscript. Assuming that all values at time t

nare exact, the local

temporal error in the displacements can be estimated as

qdn"(b!1

6) *t2(a

n`1!a

n) (58)

In the case of the incomplete Newmark integrator only a8n`1

and a8nare available. Recall that in

the Newmark scheme after equation (5a) is solved for dn`1

, the acceleration vector an`1

isevaluated by using equation (6). If equation (5a) is solved approximately, an equilibrium solutionerror e

dnis introduced into the displacement vector and only the numerical approximation,

d3n`1

"d@n`1

#edn

, can be obtained. Substitution of d3n`1

into equation (6) introduces the equilib-rium solution error into the acceleration vector, i.e.

ean"b

0edn

(59)

Consequently, the local time discretization error in the displacements can be estimated as

qdn"(b!1

6) *t2(a@

n`1!a8

n)"(b!1

6) *t2(a8

n`1!e

an!a8

n) (60)

Inserting equation (59) into equation (60) yields

qdn"Ab!

1

6B *t2(a8n`1

!a8n)#A

1

6b!1B e

dn(61)

Substituting equation (61) into equation (30) yields an estimate for the total local error in thedisplacements

edn"Ab!

1

6B *t2(a8n`1

!a8n)#

1

6bedn

(62)

Taking the norm of equation (62), we obtain a posteriori total local error estimate

Eedn

E)K b!1

6 K*t2 Ea8n`1

!a8nE#

EGE6b(1!EGE)

Edkdn

E (63)

It is more meaningful to track the relative local error de"ned as

e3dn"

Eedn

E(Ed3 E )

.!9

(64)

where (Ed3 E ).!9

is the maximum value of the corresponding norm of the displacement recordedduring the computation.

4. NUMERICAL EXAMPLES

In order to evaluate the e!ectivity of the proposed error estimator and the performance of theincomplete Newmark integrator in dynamic analysis, we consider two numerical examples. The"rst one is a model problem of a simply supported beam excited in transverse motion witha harmonic point load acting at the midpoint. The second example is a complex industry problem



Figure 2. Simply supported beam: equilibrium solution errors with step size *t"0)04 s: (a) g"0)1; (b) g"0)01

of the Di!user Casing shown in Figure 1. For all problems considered the prolongation operatoris based on the Generalized Aggregation Method [1}4].

4.1. A simply supported beam

The beam of length ¸"1 m, cross section 100]10 mm, Young's modulus E"2)0]1011 N/m2,Poisson's ratio k"0)3 and mass density o"8000 (kg/m3) is excited in transverse direction bya harmonic point load P"500 sin(5t) (N) acting at the mid-point. The damping e!ect is ignored.The beam is discretized with 20]5 Mindlin shell elements in the axial and width directions,respectively, totaling to 630 dofs.

First, we study the e!ectivity of the equilibrium solution error estimator. The equilibriumsolution error is estimated using equation (53). The exact equilibrium solution error is obtainedby solving equation (31) with a direct solver. Figure 2 shows the exact and estimated equilibriumsolution errors with two di!erent convergence tolerances g"0)1 and 0)01. Figure 3 illustrates theexact and estimated equilibrium solution errors when only two and three cycles are performed at



Figure 3. Simply supported beam: equilibrium solution errors with step size *t"0)04 s: (a) 2 iteration cycles per timestep; (b) 3 iteration cycles per time step

each time step. The results show that the proposed equilibrium solution error estimator yieldssatisfactory estimate.

Secondly, we study the e!ectiveness of the total local error estimator. The exact equilibriumsolution error is de"ned by

e%9dn"d3

n`1!d%9

n`1(65)

where d3n`1

is the displacement given by the incomplete Newmark integrator and d%9n`1

is the exactsolution using the numerical solution at time t

n, i.e. d3

nand v8

nas the initial conditions. Since the

analytical solution for this problem is unknown, we evaluate an approximation to d%9n`1

bysubdividing each time step into numerous substeps and integrating the equations of motion fromtn

to tn`1

numerically using a direct solution of the equilibrium equations at each substep.Comparisons between the exact and estimated local errors with convergence tolerances of

g"10~6 and 10~2 are shown in Figure 4. For this problem, it takes 20 GAM cycles to attain theconvergence tolerance of 10~6. These results suggest that the local error is not signi"cantlya!ected as the convergence tolerance of the iterative solver varies from 10~6 to 10~2, since inthese cases, the temporal error dominates the total local error and the equilibrium solution error



Figure 4. Simply supported beam: local errors with step size *t"0)04 s: (a) g"10~6; (b) g"10~2

is relatively small (see Figure 2 for the magnitude of the equilibrium solution error). These"ndings suggest the use of a fairly large convergence tolerance in the computation withoutadversely a!ecting the computational accuracy.

It is of interest to investigate the accuracy of the computational results in the case that onlya few GAM cycles are performed at each time step. The exact and the estimated local errors of theGAM solver with two and three cycles per time step and that of the direct solver are illustrated inFigure 5. Figure 6 compares the mid-point displacements calculated by the GAM solver with oneand two cycles at each time step with the exact solution. The results show that the performance ofthe incomplete Newmark integrator is satisfactory for the problems considered. With only a fewcycles per time step, the residual is su$ciently reduced and satisfactory accuracy from theengineering standpoint can be obtained.

4.2. Di+user casing

The di!user casing with gates for casting (Figure 1) is discretized using the 4-node tetrahedralelements. Lumped mass matrix is adopted and the damping e!ect is ignored. The material proper-ties of the di!user are E"2)0]1011 N/m2, k"0)3 and o"8000 kg/m3. The computational



Figure 5. Simply supported beam: local errors with step size *t"0)04 s: (a) 2 iteration cycles per time step; (b) 3 iterationcycles per time step; (c) direct solver

e$ciency of the incomplete Newmark integrator have been compared with the Newmarkintegrator employing direct solution of equilibrium equations in Figure 1. Additional results ofincomplete Newmark integrator with di!erent convergence tolerances for equilibrium equationsare presented in Tables I and II.

If the convergence tolerance of the GAM solver is set very small, e.g. 10~6, the computationalcost on the iterative process will increase signi"cantly and the total CPU time of the GAM solverwill surpass that of the direct solver after only 20 time steps. Figures 7}10 suggest that sucha small convergence tolerance is not needed to maintain an adequate computational accuracy.



Figure 6. Simply supported beam: responses of the midpoint: (a) 1 iteration cycle per time step; (b) 2 iteration cycles pertime step; (c) direct solver

Figures 7 and 8 show the exact and estimated equilibrium solution errors using GAM solver withtwo and three cycles per time step for *t"0)02 s and 0)04 s. The corresponding exact andestimated total local errors of the Newmark integrator with approximate and exact equilibriumsolutions are plotted in Figures 9 and 10. The results suggest that with three GAM cycles per timestep, the equilibrium solution error is negligible and the total local error is comparable to thatcalculated by the direct solver.

It can be seen from Tables I and II that by decreasing the time step size, the number ofiterations required to reach a speci"c convergence tolerance at each time step will also decrease.



Table I. Solver CPU time in second (*t"0)02 s)

GAM solver GAM solverComputational stages Direct solver g"10~6 g"0)1

Before time integration 2606)7 193)8 193)8Back-substitution per time step 23)1 * *

Iterative process per time step * 155)0 56)5Number of iterations required per time step * 18 7

Table II. Solver CPU time in second (*t"0)002 s)

GAM solver GAM solverComputational stages Direct solver g"10~6 g"0)1

Before time integration 2601)6 186)5 186)5Back-substitution per time step 23)6 * *

Iterative process per time step * 124)0 35)1Number of iterations required per time step * 15 5

Figure 7. Di!user casing: equilibrium solution errors with step size *t"0)02 s: (a) 2 iteration cycles per time step;(b) 3 iteration cycles per time step



Figure 8. Di!user casing: equilibrium solution errors with step size *t"0)04 s: (a) 2 iteration cycles per time step;(b) 3 iteration cycles per time step

This is due to the fact that decreasing time step size increases the contribution of the mass matrixin the e!ective sti!ness matrix and this improves the conditioning of the e!ective sti!ness matrix,which leads to faster convergence.

5. SUMMARY, CONCLUSIONS AND FUTURE WORK

First, a Newmark integration scheme with an approximate solution of equilibrium equation hasbeen studied. The discrete equilibrium equations are solved approximately by the GeneralizedAggregation Multilevel (GAM) method. Numerical experiments indicated that with only two orthree GAM cycles per time step, the equilibrium solution error becomes signi"cantly smaller thanthe local temporal error. This is either because the system is well conditioned (for small step size)or the temporal error is very large (for large step size).

Secondly, a posteriori error estimator was developed to estimate the local temporal error ofthe Newmark integration scheme employing an approximate solution of discrete equilibrium



Figure 9. Di!user casing: local errors with step size *t"0)02 s: (a) 2 iteration cycles per time step; (b) 3 iteration cyclesper time step; (c) direct solver

equations at each time step. The local error estimator has been found to be in good agreementwith the exact local error for all problems considered.

In the present manuscript we have not attempted to estimate the e!ect of approximation ofequilibrium equations on the global temporal error. The issue of balancing the spatial discretiza-tion, temporal and equilibrium solution errors have not been addressed as well. These issues willbe investigated in our future work.



Figure 10. Di!user casing: local errors with step size *t"0)04 s: (a) 2 iteration cycles per time step; (b) 3 iteration cyclesper time step; (c) direct solver

ACKNOWLEDGEMENTS

This work was supported by the O$ce of Naval Research through grant number N00014-97-1-0687, the National Science Foundation under grant number CMS-9712227, and the SandiaNational Laboratories under grant number AX-8516.

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