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hr. J. Solids Srrucrures 1968. Vol. 4 pp. 929 to 952. Pergamon Press. Printi in Great Britain

THEORETICAL FOUNDATIONS OF THEFINITE ELEMENT METHODEDUARDO R. DE ARANTES E OLIVEIRA

Applied Mathematics Division, Laboratdrio National de Engenharia Civil, Lisboa, Portugal

Abstract-The finite element method is nowadays the most general and one of the most powerful tools for theanalysis of structures.

It is also a general mathematical technique and the main concern of the paper is to present it in this light.Functional Analysis is used as the ideal frame for a general abstract formulation.

The ability to predict convergence to the exact solution of a sequence af approximate solutions obtainedfrom patterns of finite elements with decreasing size is fundamental in the application of the method.

In case conformity between elements is obtained, the finite element method is a particular case of Ritzsmethod, so that convergence can be ensured as far as completeness is achieved.

A general completeness criterion is justified in the paper. Such criterion requires that the field componentsand all their derivatives, of order not higher than the highest order of derivative entering into the energy densityexpression, can take up any constant value within the element.

It is finally proved that such criterion is also a general convergence criterion, i.e. a sufficient condition forconvergence even if conformity is not achieved.

NOTATIONAll the symbols will be defined where they are introduced.The following general conventions are adopted :1. Matrices (or vectors) will be denoted by bold face symbols : q. H.2. The dummy index convention will be used : Aijx, = A, xl +Azjx, +3. A derivative will be denoted by a comma followed by indices indicating the variables with respect to which

the function is differentiated. The order is indicated by a superscript in parentheses :1( duiA... = dxjax,...axr

4. A sequence will be denoted by its general term between braces: {U,}.

1. INTRODUCTIONTHE finite element method is nowadays the most general and one of the most powerfultools for the analysis of structures.

Although it was developed for structural analysis it is really a general mathematicaltechnique, and the main concern of this paper is to present it in this light. Mikhlinsbook [l] was used as a basis for such purpose.

The important problem of the convergence to the exact solution of a sequence ofapproximate solutions generated by elements with decreasing size will be given specialattention. Experience seems indeed to indicate that the best control of the approximationerror consists in examining the behaviour of a sequence of that kind. It has also beenobserved that no reasonable approximate solutions are likely to be generated if the typeof finite element used is such that convergence to the exact solution is not obtainable.

929


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930 EDIJARDO. DE ARANTES OL.WEIRABefore convergence to the exact solution was given the attention it deserves, there wasa tendency to make monotonic convergence play the fuudame~tal role. Monotonic

convergence is nowadays no more considered so important. It has indeed been demon-strated [2] that conformity (a condition for monotonic convergence) does not alwaysspeed up the convergence to the exact solution, i.e. less approximate solutions have beenobtained for some problems when the monotonic convergence requirements were verifiedthan when they were not.The capacity for convergence to the exact solution of some kinds of elements has beenalready examined in the case where continuity is preserved [3]. The author himself presenteda first proof [4] of the known criteria [2,5] which is also valid for cases where continuityis violated.if continuity is not violated, the finite element method becomes a parti~ulari~tion ofthe classical Ritz method. This connection with the Ritz method has been observed veryoften but very seldom studied in detail and explored.It is very important to notice however that, if continuity is violated, the finite elementmethod is not a simple application of the Ritz method. A section of this paper will bedevoted to demonstrating that convergence to the exact solution is still possible even inthose cases.

Let A be a linear bounded operator defined for a dense linear subset M of a real Hilbertspace H. Assume the operator .ii to be symmetrj~ and positive definite [ 11.Let (u, C) denote the scalar product of two elements of H. Let ]j~l] denote the norm ofan element in H.This paper is concerned with the solution of the equation

ilu = j fl)that is, in the determination of the element u which the operator A transforms into f;u and,jare elements of H.

Equality (I) is meaningful if element Mbelongs to M. It is possible however that noelement of M can correspond to an arbitrary elementfof H ; this is what is meant by statingthat equation (1) can have no solution in M.It can be shown [l] that, if equation (1) has a solution, this will be unique. It can alsobe demonstrated that the solution of equation (1) minimizes the functionalF(u) = Au, u)- 2 u, j-1 2)

and conversely, that the element which minimizes F in N satisfies equation (1).If A is positive-bounded-below, and not merely positive definite, that is, if

(Au,u) 5 y2(u,) (3)y being a real constant, then the field of definition of A can be extended so that equation (1)has a solution for an arbitrary elementfof H.The extended field of definition belongs to a new Hilbert space, NA, which is a densesubset of H, defined as the completion of the Hilbert space which results from associating


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Theoretical foundations of the finite element method 931

with the elements of M the scalar product[U, u] = (Au, u). (4)

This scalar product, which will be denoted by square brackets, is called the energyproduct. The norm in H, is termed the energy norm and will be denoted by bold verticalrules :

IuI = Jb, ul. (5)The energy norm of the difference of two elements is the distance between both :

d(u, u) = I u-~1. (6)The square of the energy norm is termed energy [11.If u0 is the solution to equation (l), then

Au, = f: (7)Functional (2) can thus be expressed as

F(u) = [u, u] - 2[u, uo]. (8)It can further be transformed into

F(u) = [(u-u,), (u - &Jl - [%I, ol = Iu %12 I&)(. (9)Expression (9) makes it clear that the minimum value of F in H, is obtained for

t4 = Ug.A sequence of elements {u,,}, belonging to the field of definition of a functional F,

is termed minimizing [l] for F iflim F(u,,) = F, (10)I l oO

F,, being the exact lower bound of F.As

equation (10) impliesFO = F(u,) = -j u012 (11)

lim Iu,,-- uOl = 0. (12)n-+mEquation (12) means that any sequence which is minimizing for F converges in energy

to the exact solution. Energy convergence is characterized by the fact that the distancebetween each term of the sequence and its limit tends to zero [l].

3. PARTICULARIZATION TO VECTOR FIELDSLet Sz be an open, connected and bounded domain with a finite number of dimensions.

Let S be its boundary, which is supposed to be composed by a finite number of closed,smooth or piecewise smooth stretches.

Let n be the closed domain resulting from the combination of Q and S.Take for space H the space of the real vector fields (with a fixed number of components)

whose moduli are quadratically summable over a. The scalar product of a pair of elements,


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932 EDUARDO R DE ARANTES E OLIVEIRA

u and 0, will be given by the Lebesgue integral :

u and v being column vectors containing the components of u and D.The number of components of the vectors is independent of the number of dimensionsof the domain.Equation (1) can be written in a more explicit form:

Au = f (14)A being a matrix of operators. These operators are from now on assumed to be differential.

The fields belonging to M are not supposed to satisfy all the boundary conditions ofthe problem. Those which are necessarily satisfied by every field in M and by each fieldin H,, are termed principal buundury conditions. The remaining ones are called natural~oundffry conditions.

Any field belonging to A4 is supposed to meet homogeneous principal boundaryconditions. Besides, both the field and the derivatives involved in A must be continuous.These derivatives will not however generally be continuous for every field in H,.The energy product between elements belonging to M can be computed by the use ofequations (13) and (4) :[u,c] = s,A#-v dh (19

An energy product involving elements in H, not belonging to M can be computed asthe limit of the energy product of a sequence of pairs of elements belonging to M.

It is assumed that the expression (1.5) for the energy product can be transformed,by suitable partial integration, into[u, u] = j (Ru)~L(Rv) dQ.R (16)

L is a square, symmetric and definite positive matrix, R a di~erential operator.The energy of any element u in H, is given by

[u, u] = / (Ru)L(Ru) dQ.R (17)The expression under the integral sign receives the name of energy density.Assume that Ru involves derivatives of component tli with order not greater than pi.

The derivatives of order (pi- 1) or less are termed principai derivatives.As the energy [u, U] of any field u belonging to H, must be finite, (Ru) has to be boundedalmost everywhere in R for every field in H,. The field components and their principalderivatives must thus be continuous almost everywhere in Q.in what follows, f will be supposed such that the exact solution falls into the subsetC, c N, of the fields whose components are continuous everywhere in a, together withtheir principal derivatives. These continuity properties will be referred to in the text asprincipal continuity conditions.


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4. APPLICATION TO LINEAR THEORY OF ELASTICITYElastic theories involve three kinds of magnitudes : stresses, strains and displacements,

whose vectors will be denoted by B, E and u.These magnitudes are related by three kinds of field equations which can be symbolizedas follows :(a) Equilibrium equations :

Ea = X. (18)(b) Strain-displacement relations :

(c) Stress-strain relations :Du = E (19)

CT= HE. 20)E and D are differential operators, X is the vector of the body force density components,H is a symmetric positive definitive matrix.Equations (18) (19) and (20) are valid on 0. On the boundary S, the equilibrium

equations become :Na = p. 21)

N is a matrix whose elements depend on the orientation of the normal vector at agiven boundary point. p is the vector of the tractions applied to the boundary.The analysis of the equilibrium of elastic bodies reduces to finding the solution of thesystem of field equations (18), (19) and (20) which satisfies certain boundary conditions.The simplest and most important types of boundary conditions can be expressed directlyin terms of displacements or tractions applied to the boundary. Let S1 and S2 denote theportions of the boundary where tractions or displacements are respectively prescribed.Operators E and D and matrix N are such that the following relation holds if II iscontinuous :

I oT(Du) dR = (E#u dR + (Na)% dS. (22)n I R ssIn this relation, vectors B and u are not necessarily related by the stress-strain relations

(20).Combining equations (18) (19) and (20) we obtain :EHDu = X.Combination of equations (19) (20) and (21) yields :NHDu = p.

Comparing equations (23) and (24) with equation (14), there results :

(23)

(24)

A=EHDf=X for points in R (25)

A=NHDf=p for points on S. 26)


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934 EDUARDO R. E ARANTES E OLIVEIRA

It can easily be shown that operator A defined by equations (25) and (26) has theproperties which were indicated in Section 2, if the displacement boundary conditions areenough to eliminate rigid body motion.Using equation (15) we obtain :

[u, r] = j-n (EHDu)~v dR + s, (NHDu)~v dS.Equation (22) allows the transformation of (27) into :

[u, V] = 1 (DmTH(Dv) dR.nOperator R coincides thus with D and matrix L with H.*Functional F becomes :

F= sTHs dR - 2R s XTu dQ - 2n s pTu dSs

(27)

(28)

(29)i.e. twice the total potential energy, if the displacement boundary conditions are supposedto be homogeneous.

The theorem of the minimum total potential energy, which states that the exact solutionis the one, from all the compatible elastic fields, which makes the total potential energy aminimum, is thus a particularization of the theorem which affirms that the solution ofequation (1) makes F a minimum in the space of the fields with finite energy.The formulation which has been presented is quite general as it is valid not only forlinear two and three-dimensional elasticity but also for linear theories of plates, shells andbeams.

In the case of a plate, for instance, vector u contains the transverse displacement andtwo rotations, vector E contains the curvatures and the transverse shear strains, vector ITcontains the bending and twisting moments and the transverse shearing forces.Operator D involves derivatives of the first order. The principal derivatives are thus oforder zero. This means that the elements of C, are elastic fields with displacementcomponents continuous everywhere in 0.The principal boundary conditions, which are supposed to be homogeneous, are thoseinvolving linear combinations of the displacement components. The natural boundaryconditions are expressed in terms of stresses.A very frequent simplification in the analysis of plates, shells and beams consists inneglecting the transverse shear deformation.This makes it possible to reduce the number of the unknowns to one (the normaldisplacement) in the theory of plates, and to three (the normal displacement and the

tangential displacements) in the theory of thin shells.* The equation

5 (Ru)~L(Rv) dR = s (Au)~v dR + I (Au)~v dSn n swhich results from the combination of (15) and (16), performs thus in the general theory of the role of the workequation (22).


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The simplified theories represent by themselves also a particularization of the generalproblem formulated in Section 2. The only field components are now the independentunknowns [l].

The rotations become in the simplified theory first derivatives of the normal displace-ment. The corresponding energy density involves thus first derivatives of the tangentialdisplacements and second derivatives of the normal displacement, so that the principalderivatives are the derivatives of first order of the normal displacement and the derivativesof order zero of the tangential displacements.The energy will be finite if the normal displacement and its first derivatives, as well asthe tangential displacements, are continuous everywhere in a. As the first derivatives ofthe normal displacement are the rotations, the elements of C, are still the elastic fieldswith all the displacement components continuous everywhere in Iz.Similar conclusions could easily be derived for beams.

The principal continuity conditions are thus the same both in the simplified theoriesand in the corresponding theories where the transverse shear deformation is not neglected.

5. EQUIVALENT PROBLEMConsider the domain R subdivided into a number of subdomains, R, a, R3,. . . , fle,. . .

and let He be a real functional Hilbert space whose elements are fields defined on thegeneral closed subdomain &. The scalar product between any pair of elements, ue and L~,belonging to He is defined by(Ue,ue) = s ueTve dfi + ueTve dS = ueTve da. (30)oe I Se I ne

Let H, be another Hilbert space (index n refers to a certain degree of subdivision of 0into subdomains). Each element a, E H, may be regarded as a piecewise defined field.It represents, however, not truly a single field defined on a but a set of fields ue (one persubdomain), belonging to the diKerent spaces He. Such fields are called subfields of u,.The scalar product in H, is defined by

(%4J, = cbe?uel (31)ewhere ue and ue are the subfields of u, and v,, and 5 denotes a summation over the wholeset of subdomains.Let Me be a dense linear subset of He. Every field in Me is assumed to coincide, on theclosed subdomain &, with an arbitrary field of M.Let Me be the field of definition of a linear, bounded and symmetric differential operator,defined to be such that, within $7,

and, on Se,Aeue = Au (32)

(Aeue)=vedR = (Ru)~L(Rv) dR - I (Ae~e)TvedaRe (33)ue and ve being arbitrary fields belonging to Me.


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936 EDUARDO R. E ARANTES E OLIVEIRA

Make[IF, Lqe = sRQTL(Rv )R = s (A'u~)~v~a = (Acne, I:~)~.Re Ae (34)

Call M, the dense linear subset of H, whose elements have subfields belonging to thesets M , nd consider a linear subset of M, such that, given any pair of its elements, U, andu, with subfields ue and ve, the magnitude[u,, I ,=c [Ue,UT

e(35)

can be properly chosen as their scalar product. Let H,, e the completion of the Hilbertspace obtained by associating the scalar product (35) (termed energy product) to suchsubset.

The distance between any two elements in H, , will be given by&(n,, %) = J[(% - t$l), 14, - di n = I &- 1' 1, . (36)

Consider now the linear subspace of H, , k , hose elements are such that their subfieldsu@, corresponding to adjacent subdomains, take equal values at points lying on thecommon interface.

Let H, and HA be the field of definition and the range of an operator B, such that, ifB,u, = u:, (37)

the sum of the values taken on the interface of two adjacent subdomains by the correspond-ing subfields of u, is equal to the sum of the same values respecting u;. The effect of theoperator B, s thus to distribute that sum equally between the subdomains in contact.Let VA elong to Hk nd let tP denote its subfields. Wt can write

(u,, v;), = C (ue, vy = ; s,.eTPdlii = ; s,. ufeTP da2= C (cP, tP)e = (u;, u;)n B,u,,;)n. (38)

Let M, be the field of definition of a new operator, A, , uch that

Anun= Bnfn (39)in whichf, is an element of H, with subfieldsf = Aeue.

By virtue of equations (38) and (39), we haveA,u,, I(,) = (B,f,, v;), = (&, u;), = x AT, de), = 1 [ue,~1~ [u,,(,],,. (40)e e

Consider now the subset of H, whose elements fulfill the condition that all the corre-sponding subfields coincide, within their respective subdomains, with a given field ubelonging to H, (the same for all the subdomains). It is clear that such subset is containedin Hb. all H, he Hilbert space obtained by associating the scalar product [u,, v,], to


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such subset. As[a,, Ll,= c [ae, 0 = 1 s (Ru~)~L(R~=)dRe e W (41)

= s (Ru)=L(Rv) dQ = [u, u]n[u,, v,], is a proper scalar product. No contradiction is thus introduced if HA n s assumedto be contained in H,,.Call Co, the subset of H, corresponding to the subset C, of H, .Consider the equation

J&l = f; (42)The solution of equation (42) in M, is generally not unique and the operator A, not

positive definite. If, however, the field of definition of A, is restricted to HAn, then theoperator A, becomes positive definite. Indeed, as H, is contained in H, (40) permits towrite

(A,,u,, u,), = [a,,, 4, > 0. (43)Consider the functional

F, (a,) = [u,, ~, 1, - 2( ~, >f ; ) , . (44)Let uO,,denote the solution of (42) in H,,. Then

AnuOn= f;. (45)If u, belongs to Ha, then

(GM, = (G AJ+,), = [u,, dn. (46)Introducing(46) in (44), there resultsF,(%) = [a,, Ul,- 2[um Uonln= b - kh), (%I %U - [%nr hnl. (47)

Expression (47) makes it clear that uO,,minimizes F,, and that any element whichminimizes Fn in HAn must coincide with uO,,. The solution of (42) in H,_ is thus unique.Assume now that fi vanishes on the subdomain interfaces, and let f be any field in Hwhich takes the same values, within the subdomains R, as the subfields off;. Let u, bethe element in H, which corresponds to the element u of H,. Then

(u,, f;), = 1 (ue, f) = C J_ ueTfe diie= ;I uTfdSZ+ /n:fdS = (u,f).e R s

By virtue of (41) and (48)

(48)

Fn(u,) = [u, ~1- 2(4 f) = F(u).Let ub be the field of H, which corresponds to uO,,.By virtue of (9)

W,) = [(ub -a,), (ub - wd - [u,, ~01.

(49)

(50)


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938 EDUARD~ R DE ARANTES E OLIVEIRA

As F,,(u,) = F(u) and uOminimizes F&u,) in HA,, ub minimizes F(u) in H, . But equation(50) shows that F can only be minimized by ub if ub coincides with uO. Thus the solutionof equation (42) in HA, ,oincides, within each subdomain, with the solution of equation (I)in H, .This means that the problem of the solution ofequation (42) is equivalent to the problemof the solution of equation (1).The concepts introduced along this Section represent a generalization of the conceptsintroduced in the preceding ones. It is convenient to interpret such generalization in termsof three-dimensional Elasticity.

Operator A has the same meaning for subdomain Q as operator A for the globaldomain, Sz. While A is defined by expressions (25) and (26), respectively for points locatedwithin Szand on S, operator A is defined by the same expressions for points located withinZ and on S.

.f; represents an external force distribution acting on each subdomain of thebody. Assuming that f; belongs to H; is equivalent to assume that the total forceacting on the interface between two adjacent subdomains is equally distributed betweenboth.

The value taken byf; on the subdomain boundaries are thus half the surface densityvalues of the forces acting on those interfaces, These values must vanish, if the body forcevolume density is to be bounded everywhere in +Q.To solve equation (42) means to determine an elastic field which verifies equation (23)within each subdomain and equation (24) on Si and whose stresses present, on the sub-domain interfaces, the discontinuities required to equilibrate the external forces appliedon such interfaces. Any solution of equation (42) equilibrates thus the external forcedistribution symbolized byf;.The solution of equation (42) becomes also compatible if it belongs to HAn, ecause thedisplacement boundary conditions are then respected on SZ and the continuity of thedisplacements is preserved across the subdomain boundaries. The corresponding subfieldscoincide thus, within each subdomain, with the solution of equation (1).

6. THE FINITE ELEMENT METHODThe finite element method is a general technique of numerical analysis which provides

an approximate solution for equation (1).In this method, domain n is considered to be decomposed into a finite number ofsubdomains and families of fields are considered which have different analytical expressionsinside each subdomain.A finite element is a closed subdomain, @, together with the family of fields which areallowed to occur within it. This family is a linear combination with coefficients q; of afinite number of unit modes, so that each field of the family corresponds to ascribingparticular values to the parameters qe.The values of the field components and its principal derivatives, at a certain number ofpoints on the boundary of the elements, called nodes or mogulpoints, are as a rule chosen asparameters.The type of an element refers to its general shape, nodal point specification and to theallowed fields, analytically defined by expressing a general field ue in terms of the parameters


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and the coordinates with respect to a given frame :If = p(X1) x2, . .)q (51)

q is the vector of the parameters.*Elements qt of matrix cpare supposed to be continuous and have continuous deriv-atives of order (pi- l), or less, in the closed domain fi occupied by the finite element e.The unit modes are defined by the columns of cp.

We suppose that each field component depends only on its own values at the nodes andon the values taken by its principal derivatives also at the nodes. Thus, if q; correspondsto the field component uf or one of its derivatives at any node of the element, all themagnitudes cpi for which k i will be equal to zero.

If q; corresponds to a derivative of order s of uf, qrj will take the formcgj(Xl, x2,. . .) I) = (l)ljqj Xl x2t I, . . .1 1 (52)

in which 1 is a typical dimension of the element, for instance its maximum diameter, and$t is a function which does not depend on the absolute dimensions of the element. This isnecessary in order that equation (51) can be homogeneous.The different finite elements are compatibilized through the specification of reduced

continuity condit ions. These require that the values of the field components and theirprincipal derivatives be the same at coincident nodes of adjacent elements and equal theprescribed ones at the nodes located on S2, the portion of S where the principal boundaryconditions are specified.

A point of the domain is said to be a node oft he system if it is a node for one or moreelements.Let qn be the vector of the field components and their principal derivatives at every

node of the system but those which are located on S2. The reduced continuity conditionscan be expressed by writing for each elementq = Tq,, (53)

where matrix T depends on the topology of the system.Equations (53) show that the knowledge of qn is enough for the definition of the fieldwithin every element of the system.The reduced continuity conditions are generally not sufficient to make the fieldcomponents and their principal derivatives continuous across the element boundaries.This depends on the type of the element.If the type is such that the reduced continuity conditions are sufficient to ensurecontinuity of the components and their principal derivatives across the element boundaries,the piecewise defined fields generated by the system of finite elements are said to beconforming. Every conforming field thus belongs to Con,i.e. to the subset of H, correspond-ing to Co.If the continuity requirements are violated across the element boundaries, the fieldsare said to be non-conforming.Let U,, be the subset of H, containing the elements whose subfields are defined byequation (51) and compatibilized through the reduced continuity conditions. Any element

* The allowed ields need not be introduced by giving the expression of the field components directly in termsof their own nodal values and the nodal values of their principal derivatives. They can indeed be given in termsof equal number of arbitrary parameters which in turn can be expressed in terms of those nodal values (see [9]).


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940 EDUARDO R. DE ARANTES E OLIVEIRA

u, E U, can be expressed, within R, byue = wq* (54)

whereW = ~IT. (55)

Take for space H,, (see Section 5) the space spanned by U, and H,,. The distancebetween any pair of elements belonging to H,, is defined by expression (36). The discussionof completeness and convergence will be based on that concept of distance. The distancebetween any element u, in U, and any element u in H, will indeed be measured by thedistance between U, and the element in HA corresponding to u.

The approximate solution, a,,, which the finite element method provides for equation(42) and thus for equation (l), is determined by making the functional F,, stationary in U,,.Such solution could be the exact one if nOn was contained in U,. As, generally, it is not,the solution yielded by the finite element method is only approximate.

Introducing (51) and (35) in (44), we obtain :F, = 1 [q?Kq- 2qeTQ]

whereK = sRcp)TL(Rcp) dR (57)W

Introducing now (53) in (56) we obtainF,, = q;(c TeTKT)q, - 2q,T(x TQ). (59)

Makinge e

K, = 1 TKT (60)Q, = 1 TeTQe

there resultsFn = s,TK,sn-2d'Qw (62)

The stationary conditions for F, are obtained by equating to zero the derivatives ofF, with respect to the mutually independent parameters Qni. It results in the system of linearequations

Knq, = Qn (63)Introducing (57) in (60) and using (55), we obtain

K,=x J (RW)TL(RW) dRe R (64)

Q, = 1 s_,f d=l. (65)e W


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Matrix K, is non-singular whenever the columns of W are linearly independent. AsL is definite positive, K, is also definite positive.

If K, is non-singular, the parameters qni can be uniquely determined by solving thesystem of equations (63). Let qOnbe the vector of the parameters which verify equation (63).Functional F. can be expressed as

F,, = q,TK,a - 2q$K,q,, = (qn- q,#K,(q, - q,,) - q&K,q,, (66)As K, is definite positive, the first term in the right-hand side of (66) is positive unless

q,, equals q,,. This proves that the solution of (63) minimizes F,, in U,.Let now uin and u2,, be two elements belonging to U,. Let qln and qzn, be the vectors

of the corresponding parameters. The energy product of pi,, and uzn can be given by[Un9 %lln = ce [ ne(Ru;)~L(Ru;) dR = q;, 1s (RW)=L(RW) dR qzne Re

(67)

wheref;,, and&, are the right hand sides (of equation (42)) which aln and u2,, correspond to(as approximate solutions).It results from (67) that the functional F, may take in U, the following expression

FAun)= [u,, , - 2(u,>I-,&= [urn, - 2[umGJ, (68)which makes it clear that u,, minimizes F, in U,. Such expression will be used in Section 10.

7. THE RITZ METHODThe method just described is justified if it can generate a sequence of fields convergingto u0 (the solution of equation (l)), when successive subdivisions are considered with

elements of invariant type but decreasing size.Conditions to be met by matrix cp n order that this convergence may be ensured canbe established if it is remarked that the finite element method is related to the well-knownRitz method [l, 61.

The Ritz method is a technique for generating a minimizing sequence for a givenfunctional, say F. This technique, which can be used whenever H is a separable space,is based [6] on the determination of a sequence of families, {V,}, satisfying the followingconditions :(a) the sequence is complete in energy with respect to a class C c H, containing u0

(completeness requirement) ;(b) the nth family depends on a finite number, N, of arbitrary parameters;


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(c) every element which can be obtained by ascribing arbitrary values to the parametersbelongs to C, (conformity requirement).In what concerns condition (a), it is remembered that a sequence of families of elementsis said to be complet e i n energy with respect to a given class C c H, if it is possible, for a

specified E > 0, to find an integer N such that, in each family with order n > N, thereexists an element u,, which satisfies the inequality.

d(u, u,,) < e (69)where u is any element of C.The terms of the minimizing sequence { V,,j are obtained by minimizing F in eachfamily V,.

The elements of the nth family are generally given as a linear combination withcoefficients qni of N linearly independent fixed elements Y,i which are termed coordinateelements :

N

%I = c Y,%, = I%i=l (70)qn being the vector of the coefficients qni nd I,,he matrix with columns Yni.

Family V, becomes thus a linear N-dimensional space. Introducing (70) in (2), we obtain :W4 = dK,q, - 2Qh (71)

in whichK, = (RI,)TL(RY,) dn (72)

and vector Q, is defined byQ, = I_ ,Tf da (73)n

The values of the parameters which make F stationary can be determined by solvingthe system of linear equationsCL = Qw (74)

K, is a non-singular matrix if the coordinate elements are linearly independent [l]. Thesystem has thus a unique solution which provides the unique stationary point of F in V,,.The finite element method can be considered as a technique for the application of theRitz method only if the piecewise defined fields are conforming (conformity requirement).Only thus can indeed condition (c) be respected. The sets V, are then the subsets of H,corresponding to the subsets U, c H,,.In order that convergence to the exact solution may be obtained, it is thus only necessaryto meet condition (a), that is completeness. It will be seen later on how this can be obtained.Comparing (70) with (54) it can be concluded that the coordinate fields used in thefinite element method are defined by

\y, = $T = @, within element e. (75)The analytical expression of the coordinate fields varies thus from element to elementand this piecewise definition is the main characteristic of the finite element method.


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It is also important to notice that such piecewise definition and the reduced continuityconditions allow matrix K, and vector Q, to be assembled from simpler matrices, K andQ, connected with the finite elements themselves (see equations (60) and (61)). This isone of the most interesting features of the method.

8. MONOTONIC CONVERGENCEAssume a sequence of families, { I/n{,fulfilling the conditions (b) and (c) stated in thepreceding section and suppose that the nth family contains all the families with smaller

order. As u,, makes F a minimum in V,, we have:F(v ,,) 5 F(u,J 5 F(t ,J 5 . 5 F(u,,) 5 . F(u,). (76)

By Bolzanos theorem [7], the sequence {F(u,,)} converges to a limit which cannot besmaller than F(u,). It is remarked that this conclusion is valid even if condition (a) ofSection 7 is not obeyed. If it is obeyed, then we know that the limit is F(u,).

As the inequalityF(r,,) - F(a,,) 7 0 (77)holds, for m < n, equation (9) yields

I&l-%I 7 I&l,-4. (78)This means that the distance to the exact solution decreases when n increases.Convergence is said to be monotonicMonotonic convergence does not ensure convergence to the exact solution. On theother hand, convergence to the exact solution is not necessarily monotonic.Consider now a sequence of approximate solutions generated by finite elements with

decreasing size.Conformity and the requirement that the family of fields corresponding to a givensubdivision contains the families corresponding to elements with larger sizes have beenproposed by Melosh [8] as sufficient conditions for monotonic convergence of suchsequence.However, as the approximate solution minimizes F,, in U,, regardless of conformitybeing respected, the requirement that each family of fields contains the families correspond-ing to elements with larger sizes stands by itself as a sufficient condition for convergence.We can indeed write the set of inequalities (76) once this condition is fulfilled.

9. COMPLETENESS CRITERIONIn what concerns convergence to the exact solution, we know that the Ritz methodgenerates a minimizing sequence and that a minimizing sequence actually converges in

energy to the exact solution.Convergence to the exact solution can thus be ensured if conformity and completenessare both achieved. We shall see however that completeness is the truly importantrequirement.Before proceeding further we remark that completeness of a sequence of families withrespect to a set C c H, has a meaning provided we can compute the distance betweenevery field of each family and any element in C (see Section 6).


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A general criterion for completeness will be stated and justified in this section. Thiscriterion was presented in a recent book [9] by Zienkiewicz but it has not yet been justifiedas far as we know.

Let (pi - 1) be the maximum order of the principal derivatives for component ui.We wish to demonstrate that completeness will be obtained if the general analytical

expression for UT, within element e (see equation (51)), is given* as a polynomial with anumber of arbitrary coefficients equal to the number of unit modes corresponding to theelement. Furthermore this polynomial expression must contain a complete polynomial ofthe pith degree all the terms of which are affected by independent arbitrary coefficients.The terms of higher degree can vanish whatever the values taken by those coefficients.

We remark that, if this is the case, the field component U: or any of its derivatives oforder pi or less can take any arbitrary constant value throughout the element if suitablevalues are ascribed to the parameters. In order that the derivative UT,,,,,, assumes anarbitrary constant value I/ in R, it is then indeed only necessary that the coefficient whichmultiplies the monomial (x:x; . . . ) in uQ be equal to V/r r - 1). . . s(s - 1). . . , all the remain-ing coefficients being equal to zero.The right hand side of equation (1) has been constrained in Section 3 to be such thatsolution u,, belongs to C,, so that the derivatives of order pi of solution u0 are boundedbut not necessarily continuous.

In the next Sections we assume furthermore that the exact solution falls into a subsetof Co, Ci, such that the derivatives of order (pi + 1) of the field component Ui are continuouswithin each element. Discontinuities of the pith and (pi + 1)th derivatives are still allowedat points which always remain on element boundaries as the size of the elements isprogressively reduced.

Let Ci,, be the subset of H,,, corresponding to the subset C, or HA.Any field U, belonging to Ci, can thus be represented inside R by the following

Taylors expansion of its subfield components :Us = Up O)+U~,j U). Xj-X9)+ . . . +did;lt ~).Xj-Xg). . X,-X~) +

1 (79)1+ -uy,$+:(oJ. (xj-xy)(x,-x,,. . . (x,-xxp)(Pi+ l)0 and Oi are points of sz. Oi depends on the coordinates of the point where up is to bedetermined.

Let us consider now a polynomial field with components

u:= uf(0)+u~,j(O).(Xj-X~)+ . . +$y:l,(o).(xj-xy)...(xk-x~)I

which we call tangent field to u at 0.As all the derivatives of order (pi + 1) are bounded inside R, (79) and (80) yield

(80)

(81)uf-U;l < &. v, (P)PcflI .

See footnote at page 939.


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in which Vi is an upper bound for all the (pi+ 1)th derivatives and I is the maximumdiameter of element e. d is the total number of the (pi + 1)th derivatives.

By considering similar expansions for the derivatives of Ui, it is possible to derive thefollowing general inequality :IG,yk-4$ ..kl < p,_;+) . pJl . (pyJ-r+ 1

I

for r 7 pi.As operator R involves derivatives of up of order pi or less, we have :

[R(u-u;)lTLIR(u-u;)] < V(l=j2for I sufficiently small. V, is a positive number.Thus :

(83)

[(Lie- de), (Lie Ufe)le < V,(Pycr. (84)If tangent fields u~ are considered for every subdomain P, piecewise defining a fieldui in Q, we obtain, by using (35)

(85)in which 1, denotes the maximum value of I in the whole set of elements.This means that the distance between any field in Ci and the tangent field ui, piecewisedefined by (80) tends to zero with the size of the finite elements.

Consider now a type of finite element generating a sequence of families of fields whosecompleteness is to be investigated.Call uSe the field within the finite element e such that the values of its components andtheir principal derivatives at the nodal points are respectively equal to the values of thecomponents of the field U, E Ci, and corresponding partial derivatives at the same points.Suppose the general criterion to be satisfied. u~can thus be one of the fields which canoccur within the finite element. Let this field correspond to values q: of the parameters :

On the other handu= = qq.

ufe = qfqf.

(86)

(87)From (86) and (87) we obtain 1 y - upi = 1 pFj(qF qfe)l

or, considering (52)(88)

I u;= ure1 = c (I) . I l/t(qF - qj)j (89)in which s is the order of the field derivative to which parameter qj corresponds.But

(90)


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946 EDUARDO R. DE ARANTESE OLIVEIRA

and

For this reason,h@ , - ~$ r ,l = l4#...1(C$ - 4j)l

(91)

As the absolute dimensions of the element do not appear explicitly in the functions$yj, these functions remain bounded as the size of the element decreases.The same happens to the derivatives ~~+G~~ja(xJ). .8(x,/P), for r 7 pi, because thefunctions CJI~~nd their derivatives of order Cpi- 1) or less were in Section 5 supposed to be

continuous. Assume the moduli of all these magnitudes remain below a positive number V,.Then

(93)On the other hand, as the components of u/e and their principal derivatives take thesame values at the nodes as the corresponding magnitudes in ue, equation (82) permits us

to write:14:- l < p,_f+l),, . (le)pl-s+

I

when parameter q; corresponds to a derivative of order s.Equations (93) and (94) hold even if the pith derivatives of u{ are discontinuous in R.This is an important remark because sometimes [IO] the element itself is considered

subdivided into parts and the field admits different analytical expressions within each part.Our proof remains valid however even if the derivatives of order pi are not continuousacross the internal boundaries of the element.

As the parameters cannot correspond to derivatives of order larger than (pi - l), s cannotbe larger than (pi - 1) and equation (94) yieldslq:-q$-el < ;v, ly~+l. (95)

Introducing equations (95) in (93) we obtain :

N being the total number of parameters corresponding to element e.This equation is still valid for r = 0 if the derivatives of order zero are interpreted asthe field components themselves.The similarity between equations (96) and (82) allows a jump straight to the inequality :(lu:,-uJy < v,1,2sz (97)


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V, being a positive number and n; denoting the piecewise defined field which coincideswith ufe within a general element e.

Equation (97) means that the distance between ui and u; tends to zero with I,,.Combining (97) and (85) we conclude that the distance between u, and ui tends also tozero when the size of the element decreases, so that, as u, is an arbitrary element of Cln,the completeness proof is finally achieved.

10. CONVERGENCE DISCUSSIONConsider any type of finite element which can generate a sequence {U,} of families of

generally non-conforming fields complete in energy with respect to Ci .We wish to investigate if the sequence of approximate solutions {uan} obtained byminimizing F, in each family U, converges in energy to the exact solution.We know already that completeness implies convergence to the exact solution if it isassociated with conformity. It will be concluded in this Section that completeness with

respect to C1 is a sufficient condition for convergence, regardless of conformity beingobtained.Let u,, be the field in U, which presents the same nodal values of the field componentsand corresponding principal derivatives as uon (the solution of equation (42) in HA,).As completeness is ensured, it is possible to determine N such that, for n > N,

d,(% K?)< s (98)E being a positive and arbitrarily small number.As F. is continuous, we can find such that

F&-J = F,(uo,) f 8 (99)Ebeing also positive and arbitrarily small.As u,, belongs to U,, and u,, (the approximate solution yielded by the finite elementmethod) minimizes F,, in U,,

F,(u,,) 7 M,,) (100)and

F.(u,,,) 7 F,,(uo,) + 6.Let now f;, be an element of H; defined by

f:, = A,%,.

(101)

(102)As u,, is an approximate solution to equation (42), fi, generally does not coincidewith f;.Let u,, denote the solution of the equation

Anun= fcin (103)in H,,. Assume that u,, belongs to C,,. This assumption will later be discussed.Let ubn be the field in U, whose components and corresponding principal derivativestake the same nodal values as u,,. As a,, belongs to Cln, and the sequence {U,} is complete


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948 EDUARD~ R. DE ARANTES E OLIVEI RA

with respect to Cr, it is possible to find N 1 such that, for n > N r,U&l, Uhn) < 8 (104)

E being a positive number, arbitrarily small.As A, is a continuous operator, it is also possible to determine E such that (104) implies

IIAn~cn - Anubn I/ < c (105)E being positive and arbitrarily small.

Let.f;, = Anubn. (106)

As u,, is a solution to equation (103)A,u,, = Sk,. (107)

The inequality (105) can thus be transformed intoII f,, -.fd,ll < E. (108)

By virtue of (67) as ubn and u,, both belong to U,,,[d,(u,,, hJIZ = k4m - hm), km - A (109)= ((u,, - 14bn),K, -.f;;fJ ),

As, by Cauchys inequality,((a,,--ub.), (fb,-.fb,)) 7 /I ,--bnlI,, Ilfbn-.Gll, < ~II n-~bnIIn (110)

we obtaind,,(u,,> ubn) < &I/ u,,, - 4 .I. (111)

Combining (104) with (1 ll), there resultsk&z,, u,,) 7 d,(u,,, G,) + d&n, GJ < 2 (112)

whereI: = t; + &/I U,, - U&J (113)

As F, is a continuous functional, it is then possible, given E > 0, to determine siv suchthat

f+JI(&J = in f s. (114)As IA,, belongs to H,_,

in 5 ~nbh). (115)Thus

F,(uo,) r 6 7 Fn(u,,). (116)Combining (99) (100) and (116), there results

(117)


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and thus

in which E~ < E+E~.~&&I) = F(G) + svi (118)

But, as u,, and u,, both belong to U,, (68) permits to writeF,(u,,) - F(%) = [(%I uen), (u,,- k?n) l .r 41( % , ~, ) 12P.

Combining (119) and (98), we obtain finally(119)

d&z,> n,z,,,) 7 &(a,,, u,,) + d,(n,,, uo,,) < s + Jsvi. (120)Expression (120) shows that u,, converges to uOn.It remains to prove that u,,, belongs to C,,. This assumption was indeed used to obtain

(104).Our reasoning will be based on a theorem which is known to be valid for Poissons

equation Au =f (121)Such theorem states that u has continuous second order derivatives in a domain Q

wheneverfis a Holder continuous function in R [ 121.A corresponding theorem is lacking which refers to the general problem with which the

present paper is concerned. We are thus not sure that the derivatives involved in theoperator A are continuous wheneverfis Holder continuous. It seems however very reason-able to expect the theorem to be true, at least as far as linear elastic theories are concerned.It will thus be admitted that, at least in case of Elasticity, the Holder continuity of the bodyforce density implies the continuity of the displacement derivatives involved in the operator.

u,, denotes the solution to equation (103) in H,,. This means, in terms of Elasticity,that u,, represents the compatible field which equilibrates the same external forces asu,, . Such forces are of two kinds : body forces distributed within each subdomain and forcesdistributed on the subdomain interfaces and on S. u,, will belong to Cl,,, i.e. its (pi+ 1)thderivatives will be continuous within X, if the derivatives involved in the operator arecontinuous within R, and thus if the body force density corresponding to u,, is Holdercontinuous within R.

The problem now consists in proving that the body force density corresponding toU is Holder continuous within R, no matter how large is n. We shall not attempt toiliestigate the general conditions in which such a statement is true.

Sometimes, however, the proof is trivial. This is namely the case if the type of theelement is such that the body force density vanishes or is forced to a prescribed boundedand continuous polynomial variation within each element, no matter the values of theparameters.

If such a common situation arises, there remains no doubt that the completenesscriterion is also a convergence criterion, even if conformity is not achieved. But, if it isachieved, the finite element method becomes a particularization of the Ritz method, andcompleteness will ensure convergence in any case.

Our reasoning can be adapted to cases [lo] in which the elements are subdivided intoparts and the allowed fields have different analytical expressions within each part. Thebody force density is then generally not continuous within the element taken as a whole.Convergence will however easily be proved if each part is treated as a separate element.


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150 EIXJARDO R DE ARANTES E OLIVEIRA

11. CONCLUSIONSThe finite element method has been presented as an analytical technique which canbe applied to a very broad class of problems.Functional Analysis provides the frame for an abstract formulation in which some

generalized concepts, like energy and distance, with a physical or geometrical origin, playa fundamental role. Particularly, the definition of distance between two fields, as the squareroot of the energy of their difference, is an extremely convenient basis for the discussion ofconvergence.

The description of the finite element method also required the introduction of theconcept of principal derivatives, Continuity of the field components and their principalderivatives is necessary if the energy density is to be finite. In two and three-dimensionalElasticity, for instance, as the continuity of the displacement components is enough toensure a finite energy, the principal derivatives are of order zero.The fact that the finite element method is based on the decomposition of the globaldomain into subdomains, made it convenient to transform the initial problem into anequivalent one. In Elasticity, for instance, the initial problem consists in the determinationof an elastic field which verifies the field equations everywhere within the domain, whilethe transformed problem consists in the determination of a set of fields respecting the fieldequations within each subdomain and verifying compatibility and equilibrium conditionson the subdomain interfaces. In both cases the conditions imposed on the external boundarymust be fulfilled.In the finite element method, the principal continuity requirements are replaced byreduced continuity conditions which may imply the fulfillment of the principal continuityconditions everywhere in the domain. If they do, the piecewise defined fields are said to beconfo~ing but non-conforming if it happens otherwise.

This is an opportunity to remark that conformity has been with difficulty obtained forplate and shell elements [lo]. Such difficulty results from the fact that rotations are usuallyregarded as derivatives of the transverse displacements. This has not however to be so(see Section 4) and conformity can be easily obtained if the rotations are considered astrue displacements.When conformity is achieved, the finite element method becomes a particularizationof the Ritz method. Such particularization is characterized by the piecewise definition ofthe field, which, together with the reduced continuity conditions, allows matrix K, andvector Q, (the stiffness matrix and the force vector in elastic problems) to be assembledfrom simpler matrices and vectors which refer to each finite element.

The matrix analysis which is developed in this paper for the determination of K, andQ, is a generalization of the displacement method of Structural Analysis. The forcemethod [4] could also be used.Completeness is a sufficient condition for convergence to the exact solution in theRitz method, i.e. if conformity is achieved.A general completeness criterion is justified in Section 9. It is proved that such criterionensures completeness with respect to a set C1 containing the fields whose derivatives oforder up to (pi+ 1) are continuous within the subdomains corresponding to the elements.The principal derivatives are of order pi - I, so that the criterion does not ensure complete-ness with respect to the set of all the fields with finite energy.In two and three-dimensional Elasticity, the (pi+ 1)th derivatives are second order


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derivatives. Their continuity implies the continuity of the body force distribution density.The continuity of the body force distribution density is not however a very strong restric-tion, as discontinuities of the first and second derivatives of the displacements are stilladmitted at points which remain on element boundaries when the size of the elements isprogressively reduced. The solution of problems in which external forces are distributed onelement interfaces is thus not excluded.Completeness with respect to C, is a sufficient condition for convergence wheneverconformity is achieved. However, it was shown in Section 10, that completeness impliesconvergence in any case, i.e., even when conformity is not achieved, if the body force densityremains continuous and bounded within each element as n tends to infinity.

Along the whole paper the principal boundary conditions (which correspond todisplacement boundary conditions in Elasticity) were supposed to be homogeneous. If theyare not homogeneous, the finite element method can however still be applied. All that mustbe done is to make the values of the field components and their principal derivativescoincide with the prescribed values, at the nodes which are located on SZ. As the size ofthe elements tends to zero, we obtain approximate solutions tending to a solution whichobeys the field equation inside the domain (in case the convergence criterion has beenrespected) and the prescribed boundary conditions on the boundary. This is of course theexact solution.It remains to indicate that the formulation presented in this paper is not the onlypossible one.

In the present paper, the nodal values of the field components and their principalderivatives are indeed chosen as parameters in terms of which the reduced continuityconditions are to be expressed.It is however possible, in elastic problems, to take as parameters the resultantsand moments of the forces distributed on the element boundaries. The analysis startsthen from reduced equilibrium conditions, which are directly expressed in terms ofsuch parameters [ll]. The interest of this second formulation is that it can generateequilibrated solutions while the first formulation leads to compatible ones (if conformityis achieved).

This second formulation may be generalized, as well as the first, to cover the generalproblem with which the present paper is concerned. It will be shown in a next paper howthis can be done and which criterion may be used to ensure convergence to the exactsolution.

REFERENCES[I] S. G. MIKHLIN, Variational Methods of Mathematical Physics. Pergamon Press (1963).[2] G. P. BAZELEY, Y. K. CHEUNG, B. M. IRONS and 0. C. ZIENKIEWICZ, Triangular elements in plate bending.

Conforming and nonconforming solutions. Conf. Matrix Methods in Structural Mechanics, Wright-Patterson AFB, Ohio, 1965.

[3] S. W. KEY, A convergence investigation of the direct stiffness method. Ph.D. Dissertation, University ofWashington (1966).[4] E. R. ARANTES OLIVEIRA. hrrodzqcio b teoria das estruturas. Lisboa, Authors edition (Dec. 1966).

[5] B. IRONS and K. DRAPER, Inadequacy of nodal connections in a stiffness solution for plate bending. AIAAJnl3, 965 1963).

[6] L. V. KANTOROVICH and V. I. KRYLOV, Approximate Methods of Higher Analysis. Noordhoff (1958).[7j E. T. WHITTAKER and G. N. WATSON. A Course of Modern Analysis. Cambridge University Press (1965).[S] R. J. ELOSH, Basis for the derivation of matrices for the direct stiffness method. AIAA Jnl 1, 1631 (1963).


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[9] 0. C. ZIENKIE WICZ and Y. K. CHELJ NG,The Finite Element Method in Structural and Continuum Mechanics.McGraw-Hill (1967).

[lo] R. W. CLOUGH and J . L. TOCHER, Finite element stiffness matrices for analysis f plate bending. Conf.Matrix Methods in Structural Mechanics, Wright-Patterson AFB, Ohio, 1965.

[I ] E. R. ARANTES OLEVE IRA. FormulaC6es bisicas do mttodo dos elementos finitos. 2,s Jornadas Luso-Brasileiras de Engenharia Civil, Rio de Janeiro-S. Paul0 (Agosto 1967).

[12] R. COL RANT~~~ D. HILBERT. Methods o/Mathematical Physics, Vol. I I . Interscience (1962).

Received 13 June 1967 ; revised 22 March 1968)

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