ENERGY ESTIMATES OF THE FINITE ELEMENT SOLUTIONS FOR TRANSIENT RESPONSE

Bull. Kyushu Inst. Tech.

(Math. Natur. Sci.) No. 31, 1984, pp. 15---26

ENERGY ESTIMATES OF THE FINITE ELEMENT SOLUTIONS FOR TRANSIENT RESPONSE PROBLEMS OF ELASTIC BEAMS

By

Kazuo IsHIHARA

(Received Nov. 14, 1983)

1. IntreduCtion

In the dynamic problem of the elastic beam theory, the defiection w(x, t). at place x and time t is governed by the following equation (see Fig. 1):

04w 02w

(1•1) p ot2 +EI o.4 =f(x, t), OÅqxÅqL, OÅqt5To.

Here p is the mass density, EI is the bending rigjdity of the beam,f(x, t) is the applied load, L is the total beam length, and To is a fixed positive number. On the boundary x==O, x= L and at time t=O, we consider the clamped or simply supported boundary condition

ow

(1•2) w== bi?-==O, on x==O, x=L, OÅqtSTo(clampedbeam), 02w

= O, on x :O, x= L, OÅqtE To (simply supported beam) ,

(1.3) W== ox2

and the initial condition

O (1.4) w(x, O) == uo(x), ot w(x, O) =:vo()cf), OÅqxÅq L.

The transient response analysis of beams is important in str.uctural mechanics.

During the last several years, many developments have been made in the finite element methods. In particular, the finite element formulation with piecewise cubic Hermite polynomials for the beam problems is well established in structural engineering ([7]).

The object of this paper is to show energy estimates of the finite element solutions for (1.1)-(1.4) by using piecewise cubic Hermite pplynomials and piecewise constant functions, from a mathematical point of view. We also give pumerical results to indicate the effec- tiveness of our theories. For related results from an engineering point of view, see [6].

We also refer to [2, 8], in which piecewise linear finite elements are used.

.

16 Kazuo lsHmARA

pt LS

Fig. 1. Clamped beam subjected to applied load.

2. Preliminaries

In this section, we shall describe some notations. The rotation e of the beam is defined by

ow e==

ox '

The interval (O, L) is denoted by 9. Let L2(2) be the real space of square integrable functions on 9. For a natural number s, we use the Sobolev space Hs(9) which consists of real-valued functions which together with their generalized derivatives up to the s-th order belong to L2(9). Set

(u, v)o== S,Lu(x)v(x)dx, llttll =(u,u)6/2, for u,veL2(2),

Åq{u, w}, {v, z}Åro=(u, v)o+(w, z)o, lll{u, w}lll ==Åq{u, w}, {u, w}År6/2,

for {u, w}, {v, z}GL2(9)Å~L2(9), ca (u, v) = EI( gftu, , tgev, ),, f., ., ,. H2(g),

lluHHs(o)=(t"., gk4. 2)i/2, foruEHs(s;}),

Hs(2) ={u e Hs(9); ddi, u(O) == S.i, u(L) == o, i---- o, 1,..., s- 1}.

'

The space Hos(9) is equipped with the norm ll • llH.(g). We assume that uo(x), vo(x) are suMciently smooth and thatf(x, t) E L2(9), OÅqtS. To.

For the transient response problem (1.1)-Åq1.4), we introduce the following variational form:

Find w(x, t) e tf'", OÅqt S. To such that

(2.1)

Energy Estimates of the, Finite Element Solutions 17

p(Oo2tlll , w), +va(w, ip{) =(f, w)o forall iP(et,e',

a

w(x,O)==uo(x), Dt w(x,O)=vo(x), xG9.

Here

,et" = H3(9) (clamped beam) or 75'" =H2(9) nH6(9) (simply supported beam).

Ow

to ut in (2.1), we have the following energy inequality By equating ot

(2.2) p O ^oW t 2 + va(w, w) s c{" vo l12+ va (uo, uo) + j8 11f i1 2d t}, OÅqtS To, where c is a positive constant. it is noted that -l;p OoWt 2and -S- va(w, w) represent the

'

kinetic energy and the strain energy ofthe beam, respectively. Whenf(x, t) =- O, we obtain the energy conservation law

0W 2

P zii- +9(w, w)=pllvol12+es(uo,uo), OÅqt;:$To,

In the sequel, we make the assumption that therg exists a unique solution w(x, t) gf (2.1), which is sufficiently smooth.

3. Finite element step-by-step method

In order to construct the finite element scheme, we divide the interval 9 :(O, L) into a finite number of subintervals {2i} (i=1, 2,..., m) in such a way that

'

O==xoÅqx1Åq•••.Åqxi-1ÅqxiÅq•••Åqx.=:L, 9i=(xi-.1,xi).

Let

hi == xi --- xid i, h =max {hi; 1 -Åq- i S. m}, h. =min {hi; 1Si Åq.. m} .

' For the finite element decomposition, we assume that there exists a positive constant co

coS.h*lhS.1.

As the basis functions, we use piecewise cubic Hermite polynomials {bLll•, bÅí?l} and piece- wise constant functions {b(hO,l}, O5iÅq.. m, which are defined in 2i as follows ([9]):

bM• -i(x) =(1 - x-i)2(1 + 2x--i), b(,', l•(x)=:5i2, (3 --- 2x-,),

' ' -

b(h2, l•-i(x)=hix-i(1-J-ci)2, bkl•(x) == h,x-2, (;-,c, --- 1), i

18 Kazuo lsHmARA

bigl.,(.)=i 1' Xi-iS-xg(xi-,+x,)12,

{ O, , (x,-,+x,)/2Åqx;Slx,, bÅí91(x) .,, I O' Xi-i S- XÅq(Xi-,+x,)12,

(x,.,+x,)/2$.xEx,, u,

bÅík}-(x)=O, x,-,S.xS.x,, j---O,1,...,i-2,i+1,.,.,m, k=O,1,2, where

x-, = : (x - x, - ,)/h,.

Define finite element spaces

m

W" == {wh ; wh == 2 (wh,ibÅít l• + eh,ibi? l•)} c H2(9) ,

i= o

VV8 = {wh e Wh ; wh(O) =: w,(L) =O} c(H2(2) n H6(n)) , fi7e={w,E w6; -iS"- w,(o)= -Åí.t w,(L)=o}c H3(n),

m

Uh ={uh; u,= ]Åí uh ibÅíOl}cL2(9),

vt i=o

where wh,i, eh,i, uh,i are nodal parameters, wh,i :wh(xi), eh,i--- liillE- wh(xi), uh,i'-"'uh(xi)• We

also define a lumping operator .g. with a parameter ct ÅrO by .fer.: Wh ---. Uh Å~ uh,

mm

m

.gi.I ( E] (wh,ibS,R+eh,ib`h2,l))={ 2 wh,ibL91, 2 -vt6ihieh,ibL91}.

i=o

i=o i=o

Let TÅrO be a time mesh such that To=pT, where p is a natural number. By putting w(n) == w(x, nT), we set forward and backward difference operators in time as follows:

Dtw(n)= W("+i)'w(") , D---tw(.).. w(n) ---w(n-i) .

TT

Now, we formulate the finite element consistent scheme for (2.1) in such a way that Find win+i) e Y"h at time t=(n+ 1)T (n=1, 2,..., p- 1) such that

(3.1)

p(D,b,w(hn), Wh)o+ va(wi"), Wh) +BT2va(D,b,wS"), Wh) =: (f., qh)o

for all ut,e77'h,

wÅíO) = tT.i, (uo(xi) bil l• + zil.T uo(xi)bÅí?l) ,

wii) =wÅíO)+ T ,EM:., (v,(x,)bi9 + -ifY v,(.x,)bÅí?l) .

Energy Estimates of the Finite Element Solutiens 19

Here fi is a nonnegative parameter, f. ==f(x, nT) and

3e"h == IZ78 (clamped beam) or v"h == W8 (simply supported beam) . For the finite element tumped scheme, we have the following way:

Find w- (h"'i) e ';e"h at time t=(n +1 )T (n = 1, 2,..., p- 1) such that

(3.2)

pÅqD,D'- ,.gr.iiJLn), .spl.!lthÅro+es(w-(hn), Wh)+6T2va(D,D'-',WÅí"), yOh)= (f., yefh)o

for all uthe ie7'h,

iV iO) == wkO), w- (,i) = w(hi ).

Let di =(wh,i-i, wh,i, eh,i-i, eh,i)T be the nodal displacement vector of the element 9i.

Here wh,i, eh,i are viewed as the approximate deflection and rotation of the beam at the point xi, and T denotes the transpose. Then from (3.1) and (3.2), stiffness, consistent mass and lumped mass matrices ki, mf•, ml of the element 2i corresponding to di are wgll es- tablished ([6, 7]). They are given by

k, =EI

12 12 6 6

7tllr. '-h9. h?• h?•

12 6 6

h9• h? h?

42

hi hi

symmetric 4

hi

'

mf• ==p

13h, 9h, 11h? --13h?

35 70 210 420

l3h, 13hl -11hl

35 420 210

h?. -- h9•

105 140

symmetric h?

105

20 ' • Kazuo lsHiHARA

hi o '0 'b ' ''

2 '

. e,. h,i o o. ,

-

mP• ==P '

o o cth?' o

2 '

o o o cth?•

2

By K and M, we denote the global stiffness matrix and thg global consistent (or lumped) mass matrix, respectively. Then (3.1) and (3.2) are written in matrix form as

M vn+i--21ivii+vn-i +Kv.+BT2K v"+i-h21iv2"+v';-i ..fn,

where v" is the unknown global displacement vector and f" is the global force vector at time t= nT. It is noted that the lumped scheme (3.2) with 6==O is expticit, so that we need not solve a simultaneous system of linear equations at each stage.

We say that the finite element scheme is stable if its solution satisfies the discrete energy inequality, anagolous to (2.2), for example,

p1l D-twÅín)ll2+ ,fai?(wÅín), wÅín))

;s C, {p li D,wiO)lf2+ ,fiz, (wLo), wko)) + "S' TIf f, ll 2}, n ,,. 2,3,... ,p,

i=1

for the consistent scheme, where Ci is a positive constant.

4. Stability and convergence

In this section, we derive stability condition and convergence results in the sense of energy estimates. Firstly, some lemmas are prepared.

LEMMA 1. For any whGWh, it holds that

va (wh, wh) ::l; kL pH wh l1 2, va(wh, wh) ,Åq. i4.2 p lli .Jir. wh ll1 2,

7, = 8400EIIp, h == (48 + 121ct)EIIp.

PRooF. Let m

w, = i,., (w,,,bÅíl l• + e,,,b t? l•), d, --- (w,,, in i , w,,i, e,,, - i, e,,,)T.

Consider the matrix eigenvalue probl.ems

Energy Estimates of the Finite Element Solutions 21

kiy==Amf,y, kiz :4me• z,

respectively. Then we have the following eigenvalues A,==A,=o, z,,., la2,IO..-tlil!., A,,,,,, 8h4.l.I!O Epl,

, (48+121ct) EI - 41ct EI

C,=4,=O, 43- hf. p, S4= hf. p,

respectively. Hence, from the properties of the eigenvalues, we get dT•kidi;$ 84hOf.O EpldT•mf•d,,

d]kidi;s (48+h.,121ct) EpidT•me•d,.

Therefore, direct matrix computations yield

ta(wh, wh)= tT.,d]kidiIIS X.; tt/,d]mf•d,= h7.; pllw,ll2,

ts("h,Wh)S i4i ,$,dl'm9•di'--- h74: plll.se.1whiil2. '

This completes the prooÅí

Remark 1. If the lumped mass parameter is employed with ct==11696, it holds that Yl =72•

LEMMA 2. Let cllO, fÅrO be constants. Suppose that OÅqiÅq1 and y.År--...O, IS nEp satisfy

n

(4.1) y. S- c + i., iyi, 1 ;-S n ;IS p•

Then

y.5cl(1-i)", 1Sn5p.

PRooF. From (4. 1) with n == i, we have yi :S cl(1 •- i). Assume that yi :!i c/(1 - T-)`, 1 S. i Åq= n. Again, from (4.1) we find

Yn+i S'C+ t?SYi+iYn+i S'C+C t"i il(1'i)i+fYn+1

=iyn+i+cl(1-f)n,

fromwhiÅëhfollows ,•- -• -• . - ,,.t

22 Kazuo lsHiHARA

Yn+i S- c• 1(1-i)'"+i.

Hence, by mathematical induction, the proof is complete.

LEMMA 3([9]). Assume thatweH3(9). Let wheWh be the interpolating function

of w such that

wh== tl.li,(w(xi)bSll+ dd. w(xi)bÅí?l•).

Then there exists a positive constant 6 which is independent of h and w such that Il w - w, li .2(.) s e!h ll ,•v ll .3 (.)•

We are now in a position to prove the following theorem concerning the stability condition for the consistent scheme.

THEoREM l. TIie consistent scheme (3.1) is unconditi,onalty stabte ilf 62.114, or stable under the condition

k,ÅqvimV.in,.-.,,•

ilfO,Åq..6Åq114, in the sense that thefol'lowing energ'y inequality holds: '

pllb,wi')Il2+ va(wÅír), wÅír))

.Åq., Ci {p ll D,wiO'li2+ va (w kO', wLO') + "2' ` Til f. Il 2}, r= 2, 3,,.., p,

n=1

where Ci is a positive constant.

PRooF. Choosing uth==D,wÅí")+D-,wi") in (3.1), multiplying T and summing from n=1 to n=r-1, for any ÅíÅrO we can obtain

p ll D--,wX") ll2 + va (wL'), wk")) + 6T2va (D-,wÅí"', b, itv h'))

l:; pIID,wLO)ll2+ va(wLO), wLO)) + JBT2va(D,wiO), D,.LO)) + -i} va(wk", w"hr))+ -iSilt. es(b,wir), b,wir))

tL

+ -ill- •ca(wXo), vvEo)) + -iillii;- va(D,wxo), D,ws,o)) ' '

'

r-'1 'r

+.;.,TIIfnli2+.;,TpIID-twÅí")ll2, 2;.Sr.Åq".-p.

Combining the above inequality and Lemma 1 lead$ to " ''' •'••J •••-.

Energy Estimates of the Finite Element Solutions 23

pllD-,wl')II2+ (1 - -Itl-)va(wÅír), wkr))

s. pllD,wlO)II2+ (1 + -ili-) va(wÅío), wko))

+max Io, zSi,- -61 Tk7,' pllb,wi')ll2+(t, +B) TZX.i pllD,wiO)Il2

+ "E':' T li f. li 2+ S Tp ll D-,wÅí") l1 2, 2 S. r ;-S p,

n=1. n=1

from which follows

[i - max IO, -2i --- 61 -Ti4.7 '- ] p II b,wLr) fl 2 + (1 - -ill-) va (w Lr), wÅír))

:S (1+ (t, +fi) Ti,7.i l pllD,wS,O)ll2+ (1+ -g-)va(w(,O), wiO))

+ '2'iTlif.Il2+ Åí T{pIIb,vvln)ll2+va(wl"), wL"))}, 2;Sr;.Sp.

n=1 n=1

Therefore, if

(4.2) 1- max Io, -Sk,- -6] Ti,Y.i Åro, 1- -Il- Åro,

then, from Lemma 2, there exists a positive constant Ct such that p ll btwf,') ll 2 + va (w Lr), wÅír))

r-1

;IEI Ci {P ll DtWÅíO) ll2+ es (WIO), WLO)) + .;, TII fn Il 2}, 2 l'S ' S' P'

In the case where 6År114, (4.2) is satisfied for any T, h* by choosing 6== -ih?- . In the case where OS.65114, (4.2) is satisfied if the following quadratic equation in e

i-(t, -B) TiX.i -i-g

has a positive root 6 such that OÅqsÅq2. This is satisfied for any T, h* if6=114, or for the condition

rk, Åq 7ht-vSzA- drVtin' vi!46-

if O S. 6Åq 114. Thus, the proof is complete.

Similarly, we have the fo11owing theorem for the lumped scheme.

24 Kazuo lsHiHARA

THEoREM 2. The lumped scheme (3.2) is unconditionally stable if6).1/4, or stable under the condition

hl. Åq .,/,,t,i. Vt-E'}-v,!46

ifO;$ fiÅq1!4, in the sense that thefollowing energy inequality holds:

p lil D-, iv ir) Ill 2 + va ( vv Sr), va; (,r))

r- 1

;-SC2{PMDt,-'V(hO'1112+va(VVÅíO), M-'ÅíO))+ E) Tllf.II2}, r =2, 3,...,p,

n=1

where C2 is a positive constant.

Next, we state error estimates which assert that the finite element solutions converge to the exact solution of (2.1) in the sense of the energy norm as h and T tend to zero under the stability conditions. Since these results are obtained by the same arguments as used in [2, 3] with the help of Lemma 3, we omit the proofs.

THEoREM 3. Let w(x, t), wX")(x) be the solutions of (2.1), (3.1), respectively. if the stability condition of Theorem 1 is satisfied, then there exists a posi,tive constant C3 whieh is independent of h and T sueh that

p ll bte. ll2 + va (e., e.) ;:;l C3(h2 + T2), n = 2, 3,..., p,

where

en = w(x, nT) -- wtn)(x) .

THEoREM 4. Let w(x, t), iVÅí")(x) be the solutions of (2.1), (3.2), respectively. lf the stability condition of Theorem 2 is satis,fied, then there exists a positive constant C4 which is independent of h and T such that

pll D-te. ll 2 + va (?., E.) ;.S C4(h2 + T2), n = 2, 3,.. ., p,

where

e. = w(x, nT) - w-- (hn)(x) .

Remark 2. It is easily verified that over the space Y", the semi-norm Vva(•, •) is a norm, equivalent to the norm II • IIH2(g), i•e•,

Cs llull H2 (g) ;IEi Vva (u, u) ;:ll C6 11ull H2 (n), for all u E ')fr",

where Cs and C6 are positive conStants independent of u. By combining the above ine- quality and Theorems 3 and 4, there exist positive constants C7 and Cs which are independ- ent of h and T such that

ll en ll H2(g) ;II; C7v' h2 +T2, ll e. II H2e) ;S CsVh2 + T2, n=2, 3,..., p,

Energy Estimates of the Finite.Element Solutions 25

under the stability conditions of Theorems1 and 2. Furthermore, from the Sobolev inequality ([10])

suplu(x)l5CgllullHi(g), forall ueH6(S2), XEO

there exist positive constants Cio and Cii which are independent of h and T such that sup Ie.(x)l :El CioVh2 + T2, sup le.(x)l ;:S Ci iVh2 +T2, n =2, 3,..., p,

xEn xE9 under the stability conditions of Theorems 1 and 2.

5. Numericalexample

To test the stability conditions derived in the preceding section, we show here numerical results for the following model problem of the simply supported beam (see Fig. 2).

Problem :

The exact solution of Problem is w(x t) == sin (zx) • sm (n t)

We divjde 9=(O, 1) into 5 uniform elements with equal length (h =114), and choose 02w

04w ot2 + o;,c4 ==O, OÅqXÅq1, OÅqt;:SO.Ol,

02w

==O, on x=O, x==1, OÅqtSO,Ol,

W=

0X2

w(x, O)= O, oO t ",(x, O) == n2 sin (nx), OÅqxÅq 1.

)-

Fig. 2. Simply supported beain (EI= p==L==1.0).

Table 1. Numerical results at the center xi == 112 of the beam for Problem

Consistent Lumped (cr=11696) Lumped (cr=1/12) Lumped (a=1) Exact

t=nT wEn)a/2) ipEn)(112) i;,E")(112) iPE")(112) w(112, t)

O.OOI O,O09869 O.O09869 O.O09869 O.O09869 O.oo9869

O.oo2 0.01972 O.O1973 O.O1974 O.O1974 O.O1974

O.O03 O.02946 O.02956 O.02960 O.02960 O.02960

O,O04 0.03889 O.03931 O.03946 O.03946 O.03947

O.oo5 0.04772 O.04899 O.04929 O.04931 O.04933

O.O06 0.05561 O.05858 O.05909 O.05915 O.05918

O.oo7 0.06227 O.06807 O.06886 O.06898 O.06903

O.O08 O.06749 O.07742 O.07859 O.07880 O.07887

O.oo9 0.07121 O.08663 O.08827 O.08863 O.08871

O.OIO O.07349 O.09567 O.09787 O.09845 O.09854

26 ' Kazuo lsHiHARA

T=O.OO05 (p=200), 62tO. In Krieg and Key [6], the lumped mass parameter is employed with ct=1112, from an engineering point of view. Our choices for ct are 11696, 1112, 1.0g iso that the stability conditions ofTheorems 1 and 2 are satisfied. Table 1 gives the numerical results of the deflection at the center (xi-- l/2) of the beam in comparison with the exact solutions. Based on these results, it seems that the lumped mass scheme is both practical and efficient. These numerical experiments demonstrate the validity of our results.

The computations were done on the MELcoM-CosMo 8oo III computer at Kyushu Institute of Technology, by using single-precision arithmetic.

References

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Numer. Methods Engrg., 7 (1973), 273-286.

[7] H. C. Martin, Introduction to matrix methods of structural analysis, McGraw-Hill, New York, 1966.

[8] T. Miyoshi, A mixed finite element method lbr the sotutions Qf" lburth order partial di:t7rerential equations, Kumamoto J. Sci. (Math.), 9 (1972), 87-116.

[9] M.H.Schultz, Splineanalysis, Prentice-Hall,NewJersey,1973.

[10] S.L. Sobolev, Applications qffunctional analysis in mathematical physics, Amer. Math. Soc., Transl., Math. Mono. 7, 1963.

[111 S.TimoshenkoandS.Woinowsky-Krieger, Theoryofplatesandshells, McGraw-Hill,NewYork, 1959.

[12] K. Washizu, Variationalmethods in elastieity andptastieity, Pergamon, Oxford, 1968.

[13] O. C. Zienkiewicz, The,finite element method, third edition, McGraw-Hill, London, 1977.

Department of Mathematics,

Kyushu Institute of Technology