On certain approximations in the finite-element method

-5

1975

ARCH1EF

Nomenclature

= value of approximate

solu-tionatz =

g, G = exact, approximate

fwsc-fions (g = ce) nìiiiínitni value of g

1, ja = functionals of variational

principles M1, M2, Jf _{= positive numbers} 58

_{/ MARCH 1971}

liitrou cl io n

IN the aipIualioit of tise flnite-elenwnt method [1,

2] Ito cuniplex problems il i.s often Ilsefill to make zspproxintatiolss that are outside of the I Iit'ory of varialional calculus, _For

ex-ample, iii many dynamic tiialyses il. is useful to lump themass of

the system by some rule of experience. The 1(51 ut is a diagonal mass matrix that is easier tu handle thaii a colisisteuil mass matrix 13] which is coupled. iniilarly, tise stillness matrix for als

ele-ment is generally derived with certain simplifying assumptions.

The cross-sectional area, moment of inertia, material, etc., usually are assumed to be constants over ais element. These

approxima-Numbers in brackets de,.ig,iatc Ileferences at end ofpaper.

Contributed by the Applied i\Ic,lsnnks Division and presented at

the Winter Annual Meeting, New York, N. Y., November 29-December 3, 1970, of THE AMERICAN SOCIETY _{OF MECHANICAL}

ENGINEERS.

Discussion of this paper should be addressed to the Editorial De-partment, ASME, Tnjte Engineering ('enter, 345 East 47th Street, New York, N. Y. 101)17, and will he accepted until April 20, 1971.

Disusiun received after the closirlg date will be returned. Manu-script received by ASME Applied Mechanics Division, October 20,

lt-liti; final revision, January 30, 1970 Paper No. 70-WA/APM-34.

On Certain Approximations in the

Finite-Elemeiit Method

Approximations made outsideof the variational principle used in the finite-element

iiiet hod are examined for a restricted problem in elasticity. _{They are s/town to be riorous}

front the standpointofa generalized variational principle in which Lagrange multipliers are utilized. The convergence oflite Ritz solution to 1h e exact solution is demonstrated. The bound is shown to he a functionofthe qualityofboth the displacement functions and oilier approximate f unclions in tite a nalysis.

Lab.

y.

Scheepsbouwkunde

Technische Hogeschool

Deift

= extremal function of I

(exact solution)

ii0 = extremal function of I i5f

= first variations of

fune-tionals

Xg À = Lagrange multiplier

func-tions

Transactions o the ASME

at ('

ssu

di

X

X r O. XrL

Fig. i(a) Boundary-value problem

110115 are desirable 5juni I mvmeducc tite complexity of derivations luid simplify coding and (lcbugging of the digital programs. From

experience we know that they IISU:llly du not affect the quality of

the approximate solution. However, there still remain several questions. Why do we obtaiui a good approximate solutiot_{I to it}

coulIplex problem when we usiake approximations outside nf the theory of variational calculus? _{how do we obtain the sollutioml?} What are tise error relatioisships betweefl the approximate solis-lions and the exact solution to the problem? _{This work attempts}

to partially answer these questions by examining a restricted

problem in the light of the approximation theory. An Investigation

We examine here the problem of tise rod loaded by a body force

q shown in Fig.

i(a), l'bis probleun is the simplest mixed

boundary-value problem in elasticity. We

assume that g =

ae > 0, the product of the area and elastic modulus, is a

piece-n = piece-number of elemepiece-nts ipiece-n

finite-element mesh

q, Q = exact, approximate

body-force fwsctions

U = approxi mate displacement

function

= Ritz function

R. W. MCLAY

Associate Professor, Department of Mechanical Engineering, University of Vermont,

X:O X Fig. 1(6) Example function g

X:O. _)t=L

Fig. 1(c) Example function q

e diitreri i al >Je fi r ti ni ro id ib at q is a piecewise continuous

Irr. F;XilIii)i finiti iiiris for q, q are shown in Figs. i (b, e).

fl

t\ati soliti loi i to titeproblem posed in Fig. I will sal isfy i Li erer ii irtiequation

d

f

ii

\

-

(g

J

+

q = O

dx\

(LV,

at p ii Is at which (dg/iI.r),q are coin ri nous arai t he t ranisit ion

11115

=

itt

iris ¿ where either or both (dg/dx), q

arediscorithuions.

Tri zìdlitioii

lii irder to obtain roiripprcxi ni atesol uit ji ut tot his prinbleni lu'

using the flruite-elenieiit rnethuiil we make lise of the following f unit ronrui

11

'il

_{= 2j>}

g Uix) dx

-

qUdx

.J(i

whiri. U isan apptoximritefunction_{satisfying theessential}

eon-(ijtioiiï

I

UC()

_{wilinini ihs elements,}

-, U

= at element boundaries, (5) 3

_UrO

_{= O.} If we choose U as

U =

a

₊

(asi - a) (

J (6)

-Witiuiuia fInite element where a0 a+ are vali

lesof the approximate

solitI ioni at, the left arid right elenienitboundaries x0

respec-tively, it cari be shown_{[4-Ji] t hat the}

_flitz

_{solution obtained with} (J arid the funìctional_{of equation (4) coiiverge to the exact}

soin-lion _{, on refinementof the finite-element mesh.}

Let us

desig-nate the flitz soltition as U11. The sense of convergencet>f itSfollows:

fL(du

j)2

dx <

!',

JUR - uJ

(L2)h/2

(7)

Where n is the nuniiruirer ofcl>rireiiis in lite analysis and21f1,M2are

Journal of Applied Mechanics

(4)

positive ritniniliers. _{1(luiit tue ?l>vel?l?lirenut if eciurrtjoni (7)is (nIy} luruilrin' to tine work >f i efenrnii loi i t will riot, he prescrit ed here.

We i uext ileveli ilIi nie i iiI eint, rif this work, a _t,u1(1g of

ap-poixi nrii inrisheytuiiil I lun ii illoweuli,v the essential corel t ionsof

t Ii> originalfinitiii tuai lu tii irtt.ioit (4). Suppose that itis

meurt-veu uienit to _{evntl ini le tine ini (grals ofequation (4) di IC lo i he}

corn-l>leity of generati rig t lii fiiii ilions g, q. We crini _{sim Ill fy the}

problem byapprox juunaiirg these functions. _{Titis eroi he ji isti fled}

by the tise of Lagrriruge niuiil i pIers. _{l'ue funnuctiottal of equation} (4) then becomes

i

1"

IdU\

¡'L

= -

_2J0

I

G ( -

_\dx/

I

(Ix -

I QUdx

J0

+

j

iXg(G - g) + À5(Q - q)Jdx, (S) where G,Q_{are fuiictionuii ripproxirurit inng g, q, respectively, arid XQ} À5 aie Lagrange muli iplier functions imposi rig approximate

equality of the fiiiretiouu l)airs. _{Note that this approach brings}

the approximation of the functions g, q within the variational principle. We can further extend nur discussion of equation (S)

h iimiting (lie variations of (lie approximate functions G, _Qsuch that the third integral is zero. On doiuug this I becomes

1

fL

G

(?!)2

i/x

L

QUdx. (9)

-

dx

Itenauise l is not t lu?' annre _{J the extreunril funictiout u0 of I}

will sat isfy a differeni t l in lcr_{(II? ial ii uni nu rial ural troriditionis}

than

tliuir.i' 1 '1'}us linI,i >Iluutit >tiof l(; 15

nl /

diu1\

-- ( (i

- - J

+

(J =

Lr

nix,

'a-i t bui t lie elenueints Iul k, t he natural conudiiioiis are il,u G----"

_=G---'

dx x= dx (1111

G--'1

_dx

=0.

a

I'he theory of Lrigr:uuige nuuuil t pliers shows _{us that the}

ripproxi-inritiouis in eqnialioru (S) (arid (9) ris well) are correct. Yet, if u9 and 0G satisfy ditfereiitt Euler equrrtiu)uas and natural conditions

t hey cannot he the same furìctiont. _{I luw then do we obtain a} soiu-tioni to ihe original prohulinui?

The answer to this qnuist unni Iic irr the study of convergence of

tire solutions, which will mow be developed.

_{This study will}

show that the flitz soluttinir Uz associated with Io approaches 0G while u0 approaches the exact solution _{u, both approaches being} uniform.

_{Tire first derivatives itt each ease approach in the}

ineaii.

Let us make the essential condii ions of J anni the suinte, =

uJ0 = o.

Then u0 is admissible to 'u riunii, sinarilruiiv, u is admissible to It;. Oir substitution anid by nirikiuig use of tire fact that älg₌ = O for ali admissible futtictions it is possible to show that

L _fn/ui0 2

1(u) - 1(in) = f g

-

dx 0,

Ï0(u) - IG(ilG)

=

f G

(n/uu dx 0. 2 dx dx

MARCH 1971 / 59

X Xi (IO) dU g__i? dx X = - ₍₂₎

If we now further i'it riet ourselves to approximate functions

G > (I, we have a minimum ninuciple. It is interesting that the ftiincliouirul I; of equation _{I)) is ri nuinuimuimn principle aiuti yet is}

i nrplied I y thefuinuctin >tutul 1,; tf equi at ioni (M) which

is riot it mi

ni-mii ni priru ripie. Ni ile :tIu u th uit le of equal inri (9)15 not tItesame

nnuiuuniiuinn principle as I, uf equal 'iii (4).

= O,

('iii?

= t). (3)

TItt' basic teqiinlit tes of eilt _{loti (13) are a st itt erneut of the}

utili nomi pri itt-i pies, t lie i t it egut ts bei tig tite sou it id variations of

thet'e--.lie(-t ive fitmietionals.

_{We next add tltee}

two inequalities

ttt obt titi

J9(u1) - ¡12(u0) -4- 1(ti,)

_t0(u)

i

(due

(lU'\2 L _(due

_(/ti\2

= -

tii-

-

_i

u'.r+

2

_{\ dx}

_dxj

₂

\

dx dx L

¡j

/ì\2

g ii 0G I (IX > 0. 2

\dz

_dxj

We expa t al arid rea rialige_{CIII 1:1 t ion(14)to obtai ti}

j

(g - (;)

(duo)

dx

jL

_{(q -}

_Q)u0dx 2

+

¡ 1L

(( - g)

(dy

/

1L

_{- q)udx}

I Lg (du0 dx 0. (13)

2J0

\dx

(IX/

Next, we rearrange equation (15),

i

("

/du02

de 2\

rL

-

_(gG)(

_Jdx

_{(qQ)(u0--u)dx}

2

\

_dx

_{dx /}

¡-'L _{1d h}

, ¡

f_2_Li()

_{dx0. (16)}

2J0

\dx

dx,

We develop two basic inequalities by making tise of equations (3) and (3).

lue -

nul =

IftItiQ

I----

\dx

(Ix/

/u\

-Jdx

< L" [

çL

(duo

,/,,\2

dxl

-

L.J

\dx

il.r/

_J

r

rLld1i0

du \2

J

XI

I

j____P)

dx

[J0 \dx

dx/

-

f1.

(q -

Q)(u0 - u0)dx

<[fL

1/2 _L (q Q)2dx]

[f (u0 -

_uQ)2dx]

-

o (17)

Also, the right-hat id n_{tot iii ter of eq u a t lt t t (16)}

can be bouinidej

below with the knowledge that._{g miu> (i. Thins}

i

('

_(du0

du\2

gv--- I

(IX 2

\dx

_dx/

I / J ( (/(1t(/ (.0 0. (21)

2J

\dx

_dx/

On combining eulliat jolis (1G),_{( 19), (20), and (21), we obtain}

I

'

(due

du)2

_{dx ì}(due;

du)2

tlx

2 j0

\dx

dx

_{2 j0}

_{dx dx}

f(g - G)

(--

dx

1L

_{(q Q)(u0}

-.[f1_

G)2

+Q)2dx]1/2

r r (du0

du'\-Xl

I

1 --

I dx

[J0 \dx

cLr/

+ [f

L

(q -

_Q)2d1/

_{[f (u0 -}

_tio)2dx]'.

₍₂₂₎

Next, we consider outiv (lu-

_{left and right members of}

_{t (te}

in-equality of eqtration (22) and niake use of equation (18) to obtain _tif

fu (du-

-Adx

_lit' 2

\

dx drJ .

[

f (g - G)2

(pia

+

)2

(1.r]

[r-('

(/u)2

_th E

r

i'

r

_r1

_¡du2, a0

\2

11/.

+ '

Li

t' - Q i dx]

[j

-

)

(IX]

rL

_r

_rL

_la

_{du \ 2} o _o (Ix

J

(U -

u0)2dx L _{\ dx}q ) dx dx

dx,

_{lu getterai, tite fiuuirtittis}

it(;( u _{satisfy tlitfereut ditTen-ittial}

. /

2 eqiiat lotis; thus the expu-e-ittui of tue mean difference uf t lie

< ji

₍

_-°

_i

_{dx. (13) derivatives in equation (2) is}

not zero ioud we can divido t hiough

\ dx

dx /

_{by it.}

_{Also, since n0 atid}

ti _{sat i'fv (lie respective clifferctttia!}

/

(hiC

(Ill \

2

By returtiitig to equation (16), rearranging tite ieft-hnd lute- equations we know that

₊

y")

is bounded above by 2If.

grais, unid niakiitg tise of tite _{Se.hwartz-Iltiitiakovsky intequalit.y}

liv using these facts_{we obtain tite firmai iiiequalit y tif t his}

in-we obtain the following expressions:

vest igation,

11L

(g - G)

(2

-

iix

_[1L(th10

_du)2

d.r]

<[.i13

1L

_(g

_G)2dÏ]'

=

_{f (g - G) (}

+

(

-

) dx

₊

2I

r

1L(q Q)2dx]/2 ₍₂₄₎ <

i[

r

_{(g - G)2 (}

₊

_ax

2LJo

dx dx

g, [j0

(23)

The inequality f equation

(24) elemoiistrates the meant cou-vergence of to _{as the fuinictionis G, Q approach g,}

_{q, r}

dx dx

(19) _spectively.

By equation (17) we cani show uittifornt convergence of the displacements.

Discussion

Two fitta! expressiu)ns at-e needed to discuss the quest loti raised

(20) in the Introduction. These are obttiuted by ulsitug the _triangle inequalities

ELf'

[j

i-J /

MARCH 1971

liarisactious of the ASME

<

1dudug

_dx

-

(ix dx

dx

I- ti

(,r, -

1101 -f

iI; - u,l,

r ¡'

f(l(TR duQ 2 /J

j.!.

du\

'/2

Li0

!,,

dx -

ej;)

IIX]

[

¡

- -i;)

dx]

L (i

du,\2

2/3

+[Í

dx

I dx

dx/

_]

(25)

13v contlittting equations(7)atol (24) with (25), we obtain

jUR

uI

()hí2

+

[(.

j.L

(g - G)2dx )'/z

+ 2L

(f (q -

Q)2i.r)]

wli it'h _{t bese ilpprt Ix iitt:tt ions are pari i cri larly mua ,rtttttt is shell}

liti tlysis using vu ri ti jot ial pri n ripies. 'l'i re fu net h u tais associated

wit h shell problems are very complex, rer1ltirittg sittiplihittition of tine geometrical expressions itt t,rder to lie feasible for

program-ming.

The author thanks tite reviewer for suggest iltg that he indicate t lie existence of a ttrtrnber of pitblished papers ott tite subject of

hitnite-element couvergetice. Ile also expresses appreciatiott for

tite prompt reply oit the review. Acknowledgment

This work was supported by NASA N(ffi 46-001-008 S2 68-16. Tite author wiiìes to thank P. E. Grafton of the Boeing Company Aerospace Group fuir suggesting the research topic.

References

i Martin, H. C., Introduction to Matrix .tfethods of Strnct,iral

Analysis, McGraw-Hill, New York, 1966.

2 _{l'rzemieziietki, J. S., Theory f ,tfati'ix Striietiii'al Analyitt,} McGraw-Hill, New York, 196S.

3 _{Guyan, R. J., 'Redurtiot, of Stillness and Mass Matrices,"} AJAA Journal, Vol. 3, No. 2, Feb. 11)65 j , 380.

4 MeLay, R. WC., 'An Investigatioti Lutto the Titeory of tite

Dis-liittiernent Method of Analysis for I.iue'rr Elasticity," Ihi) thesis, Department of Etigiticering Mer1itntic... Uutiversitv of "uVisconsin,

M:rdison, Wis., Aug. 1963.

5 Key, S. M., ('rnttvergeiuce Ittvc'stigation of the I)irect Stiff-ncr_Sn Methoil," PhD e-i., [)ettriiuietii of Aeronautics nuit

Astro-1trirttít'-, University of \Vashittgtoii, Serti Iv, \Vttsh., iInir. 1966.

Tong, I'., :triul l'ian, T. IL. IL., ''Tie ('onvergence of the

Finite-1']etiieri t Method itt Sitiviitg Li trt':t t 11 ;n r ir I 'rol ,lenis, '' fnternaiional Joirrno/ of Solûls and S!rnrlures, 'ol. 3, ltui7, pp. 865--879.

7 Birkoff, O., $n'}tultz, M. LI., iii Varga, R. S., 'Piecewise

lierntite interi,ol:iti,ri it, Otte and Two \':ti'iabies With Aptiications to I'artial Differenti:,! Eu:ttiotto," ,\'nim ,ische Mathemat'i/r, Voi. 1),

1968, p. 232 -256.

S Oliveira, E. it. A., ''Tlteoretic:tl i"oundations of tite Finite-Flettiert Met hod,'' I',tri',al ¡0/0/ .100 d i of .Soiitis and ,S'lr' wi ri's,

4, 1968, pp. 112U t t52.

9 Ii ulnte, B. L.. ''irrt ert ii: ti titi iv hit z A pproxi nation.'' Jour-nalo.fMat/rernatics a,'!.lJ,'e/nar,je.', Vol LS. 1968, pit. 337-342.

It) Johnson, M. \\'., J,'., 'riti MeLay, It. W'., "Convergence of the

Firnite-1'1ettient Metl>d jut lit, l'iteory f Elasticity,'' JOUIINAL OF

Ai't't.ii, MECHANICS, \'i,l. 35,TItANS.A*\EE, Vii. 90, Series E., No.

2, Joue 1968, iii. 274 278.

Ii Fix, G., and St rang, (t., ''Fourier Artal','sis of tite

Fittite-Ele-meint Methods in ltiiz-(i:tlerkin 'I'heory,'' .S/r,dies in App/ted Math-e-no/leS, \'ol. 48, No. 3. 191W, trI,. 26,'- 273.

r r'-

(dUR

_du\

1''

fj[\l/2

I I I

- I

(lXi

I

-LJo

\dx

dx/

J

\2

Journal of Applied Mechanics

_{MARCH 1971 /61}

+

L [eu

j.L

(g - G)hdx

_{+ 2L}

(1L(q

-

Q)2dx)']

(26)

Equation (26) illustrates the uniform convergence of the Ritz

soluti, it U, to the exact solution Q and mean convergence of the first dcrivativei. Note that the qttili1 y tif tite solution depends on bot it the ¡tiimber of elemen i s it t e t tush n and on t he q tt;ilit y

of the approximate fioul jotis G tii,l ÇL We also see I hat tito funet i ut dGjdx need not converge t vEx. This property itf

i lie va 'ial i tital principle is used whet i we idealize each elemet ti in an antilysis t.o be constant in t lie ervs-setI it liai area and Young's

moduli s. In this case dG'dx = I) in each element.

Equal ion (26) suggest s t hut a cit isit citi formulation reip tires the qitility of the approximations of ('.', Q to lie uf theMil1i( order

as thai uf the straiti appriximul ivttt. l'itt siniplv, if strains :ite

approximated as piecewise ci;nt titi funv'i iuv t hen it is sufficient that G,Q also be piecewise cotisi uit finid ions.

Thi si udy of a rest ii'ied prtui luto sii uws t hat many approxi-niatiov rs made ott using variai ii teil pri i cip us are rig( rous, ai-thotigi tt hey do ifltl appear sit si tltel'li tiny. The knowledge of

titis is of practical import tinco to the analyst ; lie cati go so far as

to define fuori ions numet'itally in i he eqttatiotts of the Rit z