Chapter 1 Pages 22-52

8/3/2019 Chapter 1 Pages 22-52

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Chapter 1Basic Concepts

1.1 Fundamental conceptsCo nsider a very simple differential equ atio n applying in a one-dimensional dom ainx, from x =O to x = 1, Le.,

d2u- + j u 2 u - b = O i n xd x2

(1 .1 )u is the function which governs the equation and we usually need to find it usinga numerical technique which gives a n approx imate so lution. i2s a kno wn positiveconstant and b is a known function of x.Th e solution of equ ation (1 l ) ca n be found by assuming a v ariation Sor uconsisting of a series of know n shape s (o r functio ns) multiplied by u nkno wncoefficients. These coefficients can then be found by forcing (1.1) to be satisfied ata series of points. This is the basis of the collocation (or point collocation to beprecise) metho d an d is essentially wh at on e does w hen using finite differences. Infinite elements instead the solution is found using the concept of distribution oferro r within the dom ain . This is som ew hat a process of 'smoothing' and it is thennot surprising that finite clcment solutions tend to have less 'noise' than finitedifference ones.Th e concept of distribution o r weighting OS a differential equation is not onlyvalid for approximate solutions but it is a fundamental mathematical concept,which can be used in countless engineering applications. Enginecrs for instanceare very familiar with the principle of virtual work which is usually formulated interms of work done by interna1 and cxtcrnal forces. They are usually unawarehowever that the first 'demonstration' of the principle was proposed by Lagrangeusing the con cepts of distributions, apply ing what a re now called the 'Lag rangia n'multipliers. These concepts are also essential to study the behaviour of thedifferential equations, and in particular thc type of boundary conditions theyrequire and which are consistent with them.To und erstand what these concepts mean before proposing any approxirnation,one can consider a noth er function w, arbitrary except for being continuous in thedomain x and whose derivatives are continuous up t o a requircd dcgree (the degreeof con tinuity will vary with the pro blem as will be show n sh ortly ). O ne c an nowmultiply thc whole of equa tion ( 1 . 1 ) by this w function and integrate on the doma in.u as follows:


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1 l . Fundamental Concepts 15This opera tion is called an 'inncr' produ ct in mathernatics and althoug h do es notimply any new concepts, allows us to investigate the properties of the governingequ ation . This is don e by integrating by par ts terms w ith derivatives in the ab oveexpression. In this case une can only 'manipulate' in this manner the first term,i.e. d 2 u / d x 2 ,which gives

Notice that the integration by parts has produced two terms, one in the domainwith first derivatives of u and w, nd the other on the boundaries (which in thiscase are simply the two points x = O , x = 1).Furthermore, if the w function has sufficient degree of continuity one canintegrate by p arts again to obtain

Expression (1.4) is of course equivalent to (1.3) but here not only has onepassed al1 derivatives to the newly defined w function but the two terms at x =Oand x = 1 give us an insight into the boundary conditions required to solve theproblem. In this case,du

u or - needs to be known at x = O and x = ldx (13 )Notice that the w function which in principle was an arbitrary function witha certain degree of continu ity can be m ade t o satisfy certain b ound ary cond itions ifone wishes to d o so. In the principle of virtual displacements for instance, arbitra ryfunctions of this type are defined as virtual displacements but they are assumedto satisfy the homogeneous version of the displacement boundary conditions, i.e.they are set identically to zero a t any points where the displacernents are prescribedeven if those displacements (represented by u ) are not se t to zero, i.e. w =O onthe parts of the boundary where u is given. This is done in order to eliminateterms of the type - which give rise to a type of 'work' one does n ot w ish to[:: 1have. In general however one can assume that w and dw /dx can have valuesdifferent from zero o n the boundaries a nd this m akes expression (1.4 )mo re general.The concept of an arbitrary function w used a s a distribu tion function is relatednot only to virtual functions and consequently t o virtual work but a lso to the ideaof Lagrangian multipliers. These arc functions of th e kv type defined in order tosatisfy certain equations. 'l'hey will be defined better in what follows.Although equation (1.4} gives the user an insight into the type of boundaryconditions rcquircd to solve the problem, these conditions have not yet beenexplicitly incorpora ted in to the problem . In ord er t o do so let us consider that the


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Chapter 1 Basic Conccptsboundary conditions are as follows:

where the derivatives of u are now defined as q and the terms with bars representknown values of the function and its derivatives. It is usual to cal1 the first typeof conditions in (1.6) essential' and those like q involving derivatives as 'natural' .Substituting those values into (1.4) gives

It is now interesting to try to return to the original expression (1.2) byintegrating by parts again, but this time passing the derivatives for w to u. Thefirst integration gives,

Notice that only the term in [ u dw/dx ] ,= , disappears.Furthermore the following expression results after carrying out a secondintegration,

Once again only one term disappears, in this case (Iqw],=, - Notice thatq = du ldx as defined earlier. - Grouping the terms together one now arrives at aninteresting expression, different from the original formula (1.4) i.e.

This expression implies that one is trying to enforce not only satisfaction of the


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1.2. The Poisson7sEquatian 17differential equation in x but the two boundary condit iom. The kv and dwidxfunctions can be seen as Lagrangian multipliers.Furthermore nothing has yet been said about approximations; the aboveexpressions are valid for exact solutions as well. In other words the proceduredescribes a general tool for tl~envestigation uf differential equalions.1.2 The Poisson's EquationAn important equation in engineering analysis is the so-called Poisson equationwhich for two dimensions can be written as

a2( ) dZ( ) .where V2( ) =- t IS called the Laplace operato r. x , and x, are the two8x? ex?coordina tes and b i s a k h w n func tion of x,,x,. LL is the domain on which theequ ation applies and is assumed to be bounded by T. he outward normal to theboundary is defined as n (figure 1.1).Th e Poisson equa tion o r its homogeneous form (i.e. b =O) which is the Laplaceeq ua tion , governs many types of engineering problems, such as seepage an d aquiferanalysis, heat conduction, diffusion processes, torsion, fiuid motion and others .Consequently it is a very important equation in engineering analysis.

Figure 1.1 Dom ain under Consideration for Poissun Equatiuion Basic Definitions


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18 Chapter 1 Basic ConceptsHe re one can also introdu ce the idea of multiplying equ atio n (1.12) by anarbitrary w function, continuous up to thc second derivative. This gives,

Integrating by p arts the terms in x, and x, gives

In this case the integration by parts of the two terms produces the derivativeof u with respect to the normal, i.e. duldn which will later on be called q , i.e.q = ?u,ian.Integrating by parts again, one obtains,

Expression (1.1 6) is equal to (1.13) and hence one can write,

where the term in b has been eliminated as it appears on the two sides of thecquat ion.Eq ua tion (1.17) can also be expressed in the form known as the Green 'stheorem, i.e.

Although this theorem is in many cases given as the starting point for manyengineering applications, including boundary clement formulations, it is muchmore iliuminating to use the concept of distribution as it illustrates the degree of~ on ti n u it y equired of the functions and the importance of the accurate treatmentof the bou nd ary con ditions. In this regard let us now consider that the i- boundaryof the Q domain under study is divided into two parts, T, and T'; (T'= 1; f T,)such that,


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1.2. The Poisson's EquationHence equation (1.16) can now be written as,

Once again one can integrate by parts to retrieve the original Laplacian V 2 uin orde r to see how the irnpo rtance of th e boundary conditions affect the equation.Integrating by parts once we have,

One can split the first integral on T nto two terms (one on TI nd the other onT,), he seco nd of which ca n be cancelled with the last integral in (1.21). This gives

Integra ting again by pa rts th e following expression is obtainedS { ( V 2 u )w- w ) dRR

The first integral in T can again be written as a summation of two integrals, oneon Fl and the other on F,. The one on TI an be cancelled with the integ ral on rlof q w in the above equ ation. This gives


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20 Chapter 1 Rasic ConceptsThis f / ,la can be written as,

Once gain this expression shows that one is trying to satisfy a differentialequation in the domain plus two types of boundary conditions, the 'essential'conditions u = u on r, plus the 'natural' conditions q = q on T2. his is very muchwhat has been sho wn in equ atio n (1.10) with the only exception th at the sign ofthe last term is different in both expressio ns. This is because in (1 lo ) the derivativeswere take n with respect to x rather t ha n with respect to the norma l, as they are now.1.3 Approximate SolutionsAlthough the previous sections have introduced the concept of distributions, theform ulations apply irrespective of the type of solutio n one finds, i.e. they are validfor exact as well as approximate solutions. This section however will investigatewhat happens when the concept of an approximate solution is introduced in theform ulat ion . In engineering practice the exact solution c an only be known in a fewsimple cases and it is hence important to see how the solution behaves when oneintroduces an approximation. Let us consider now that the function u defines anappro xim ate rather than the exact solution. In this case one can write for instance,

where u i are unknown coefficients and the $ i are a set of linearly independentfunctions which are known. cci are generalized coefficients although in somecases they can be associated with nod al values of the variable under conside ration.In general in engineering problems, one prefers to use nodal values as they havea clear physical meaning and this is done in finite elements, finite differences orthe boundary element method. In such cases the approximation for u can bewritten as

where 4j are a set of linearly independent functions which are sometimes calledinterpola tion functions. u, are the nod al values of the field variable o r its derivative(or more generally the nodal value of any variable with physical meaning directlyrelated to u or its derivatives).Introducing thc approximation for u into the govcrning differential equationone finds that the equation is no longer identically satisfied except for the case inwhich (1.26) or (1.27) can represent the exact so lution. This produces a n error orresidual function which will soon be defined.


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1.3. Approximate Solutions 21For instance, introducing an approximate value of u into equation (1 .1 ) onegenerally finds that

d 2 u- + j . 2 u - - b # 0 i n xd x 2

Th e same will generally oc cur with the bo und ary con ditions correspo nding to thisequ ation, i .e.

One can now introduce the concept of an error function or residual whichrepresents the errors occurring in the domain or on the boundary due tu nun-satisfaction of the above equations. The error function in the dornain is called Rand is given by

and on the boundary one has,R , = u - U

and R 2 = q - qAlthough the above case is a particular and relatively simple equation the sameoccurs for any other problem. If one considers the Poisson's equation (1.12) forinstance, the error function in the dom ain is

and the e rrors for the bou nda ry co nditions (equation (1.19)) are defined by

The numerical methods used in engineering try to reduce these errors to aminimum by applying different techniques. This reduction is carried ou t by forcingthe errors to be zero a t certain points, regions or in a mean sense. This operationcan be generally interpreted as distributing these errors. Thc way in which thisdistribution is carricd ou t produ ces different types of error d istribution techniqueswh ich, in gene ral, force the integrals OS the residuals weighted by a certain functionto be ze ro. Rccausc of this they ar e called weighted residual techniqu es.


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Chap ter 1 Basic Concepts1.4 Weighted Residual TechniquesTh e solution of the b oun dary value problem defined by equations (1.28) and (1.29),(1.32) and (1.33) or similar sets for other problems can be a ttempted by choosingan approxim ation for the function 11. One can then have three types of method:

( i ) If the assumed approximate solution identically satisfies al1 boundarycond itions but not the governing equations in R, one has a purely 'doma in'method.(ii) If the approximate solution satisfies the field or governing equations butnot the boundary conditions one has a 'boundary' method.( i i i ) If the assumed so lutio n satisfies neither the field equ ation no r the bo und aryconditions. one has a 'mixed' method.

Let us first assume that the functions cP j which are defined to approximate u,satisfy al1 bou ndary cond itions. On e then has a residual R function in the dom ainas the field equations are generally not identically satisfied. The idea is now tomake R as small as possible by setting its weighted residual equal to zero forvarious values of the weighting functions, i C / j , such that

These functions have to be linearly independent.Notice that ano the r way of writing (1.34) in a form th at is more compact a ndeasy to operate with, is by defining a new function w, such that

where p j are ar bit rar y coefficients. Wence equ ation (1.34) can now be written ina more cornpact form as,

Different types of weighting functions $ j (or w ) will define different app rox im atemethods. Equation (1.34) or ( 1 2 5 ) will pro duce a systern of algebraic equationsfrom which the unknown values of the ai or ui coefficients used in u (equation(1.26) o r (1.27)) can be obtained.'I'he app rox ima tion can always be improved by increasing the number of Nfunctions used. ( N is the number of terms in the approximate solution equal tothe number of weighting functions required.)App roxim ate method s based o n equation (1.36) are called weighted residualmeth ods an d , given an ap prox imate solu tion, the meth od will vary in accordancewith the functions used as weighting functions. In what follows a few will bereviewed.


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1.4. Weighted Residual Techniques 23(i ) Subdomain CollocationEor this method the domain fl is divided in M subdomains and the integral ofthe err or in each of them is set to z ero. T hc weighted functions a re simply chosen as,

(E indicates belonging to and Qj is the 'j' subd om ain). Equation (1.34) becomes,

(ii) Galerkin MethodIn thc case of Galerkin's method the weighting functions are the same as theappoximating functions, .e.

hence equation (1.34) becomes,

Using the same definition as in (1.35) this can be written as,[ R W ~ Q = O ,n

with,w = I r , 4 , + 8 2 $ 2 + , . + B N $ N

This method is the starting point of many finite element form uiations for whichthe symm etry of t$j= / / j coupled to inherently symmetric field equations, lead tosymmetric algebraic matrices.

(iii) Point Collocation MethodIn this case N points x,, .Y,, . . ., x , are chosen in th e domain and the residual isset to zero at these poinls This operation can be interpreted as defining weightingfunctions in terms of Dirac dcltas, Le.


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24 Chapter 1 Rasic ConceptsA(x - j) at point x - j has an infinite value but is such that its integral givesunity, i.e.

The Dirac function can be interpreted as the limit of a regular function when itsbase tends to zero. .Hence equa tion (1.34) can no w be written as,

which simply says that the error function is zero at a series of points, i.e.

The method consists of setting the residual or error function equal to zero atas many points as there are unknown coefficients in the approximate solution.Th e distribution of the collocation points is in principle a rbitr ary , but in practicebetter results are obtained if they are uniformly distributed.

Example 1.1As an illustration of how to use weighted residuals, consider the followingdifferential or field equation in the one dimensional domain x (where x variesfrom x = O to x = l ) , i.e.

with homogeneous boundary conditions, i .e.

(No tice tha t equa tion (a ) is a particula r case of equation (1.1) when i. O andb = - x . )Th e exact solution of (a) can be found by integration and gives,

Let us now a ttem pt to solve (a ) using th e weighted residual techniques describedabove, starting by defining an approximate solution which satisfies the boundaryconditions and can be written as


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1.4. Weighted Residual Techniques 25On e can use Herm itian polynomials lar d j but since only two of them satisfy thehornogeneous boundary conditions, only these tw o will be used, i.e.

2 = 2141 +where411 = x - 2 x 2 + .x3$ z = x3 - 2

T he residual o r err or function in this case is obtained by su bstituting (e) intoequation (a) which gives,

= z , ( h x - )+ a,(6x - )+ x (8 )Let us now reduce (g) using the various techniques previously described.

(i) Subdomain CollocationConsider the dmain divided into 2 equal parts, one from O to $ and the otherfrom $ t o 1 . In this case one can write,

112'fR d x = j [ r 1 ( 6 x - 4 ) + a 2 ( 6 x - 2 ) + x ] dx= Oo oand

which produce the following system of equations- . 2 ~ ~ . 2 5 ~ ~0.125 = O0 . 2 5 ~ ~1 . 2 ~ ~0.375 = O

from which one can obtain,a l =4; g - -12 - 3

Substituting ( j) into ( e )gives the following result

Notice th at the exact solution (c) has been obtained since the assum ed sha pes ofu arc able to rcprcscnt jt.


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Chapter 1 Basic Concepts(ii) GalerkinIn this case the weighting functions are,

@ 1 = 41

$ 2 = 4 2

and the w eighted residual expressions are

which produces the following algebraic equa tions in al nd a,.

This also results in

(iii) Point CollocationHere one forces the residual to be zero at a series of points. Consider in this casethat R is zero at the two points x = 0.25 and x = 0.75. This gives

with the same results for a, and a,, .e .

Notice that this case is rather trivial and the same results have been obtainedfor al1 rhe meth ods. In general this will not be true when the exact solu tion can no tbe reproduced by the proposed value of u and onc will find different resultsdepending on the method used.Example 1.2Let us now study another equation using point collocation such that in this casewe will obtain an approximate rather than thc exact solution.


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1.4. Weighted Residual TechniquesConsider the equation (1.1), with EW Z = 1 and x = - b : i.e.

and the hornogeneous bouadary conditions, u = O at x = O and x = l .The exact solution of (a) can be easily obtained by integration and givessin xu= - - xsin 1

Instead of using (b) we will try to approximate it defining a solutionu = a 1 4 ,+ a Z $ , + a , $ , t- . . . (C1

where the +i re terms of a polynomial in x, ; .e .4 , = x , 4 , = x 2 . . . (d 1

In order to satisfy the bo undary conditions exactly, equation (c) has to give,

which implies that,

Hence a , = O an d a, can be expressed in function of the oth er ai arameter, i.e.a , - - ( a , + a , + . . . )

Substituting a , r an d (g) nto (c) one c an write,

Defining now a new set of unknown parameters ai uch that,E 1 = - U 3 - U 4 ; a,=-a, . . .

one can wrile.


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28 Chapter 1 Basic ConceptsThis function satisfies the boundary conditions in u and has the degree ofcontinu ity req uired by the derivatives in eq uati on (a ), hence it is said to be'admissible'. We will also see that the 'distance' between the approximate andexact solution decreases when the num ber of terms in ( j) increases and this implies

that the approximate formulation u is 'complete', i.e. tends to represen t the exactsolution better and better when the number of terms increases.In order to apply the point collocation technique we will restrict ourselves totwo terms in the ( j ) expression, i.e.

Sub stituting this function into the governing equ ation (a ) one finds the followingresidual, i.e.

Collocation can now be interpreted as setting R r at two points, say x = $ andx = 4.This can also be expressed in terms of Dirac delta functions applied at thesetwo points, i.e. the weighting fu nction is,

The weighted residual integrals are represented by

or simply,

Substituting these values of x into (1) one obtains two e quations in a l and a ,.They can be written in matrix form as follows,

The solution of this system gives

Thc approximate value of u - equation ( k ) -- can now be written as,


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1.4. Weighted Residual TechniquesTable 1.1 Pesults for Point Collocadonx u (exact) u (approximate) R0.10 0.018641 0.019078 -0.0099530.30 0.051194 0.052258 +0.0020270.50 0.069746 0.071428 + O.OooO00.70 0.065585 0.065806 - .0248840.90 0.030901 0.032350 - .081474

Notice that the error function can now also be fully defined in terms of x, bysubstituting a, an d a, into (1). Th is gives,

These results can be tabulated in table 1.1 where they are compared against theexact solution for u. Notice that the values of R are identically zero a t x = andx = but th at this does not mean th at the solution for u is exact at those points.Example 1.3Let us apply Galerkin's technique to equa tion (1.1) for which i 2 1 and b = - xwith homogeneous bou nda ry conditions u - a t x = O and x = 1 . The approximatesolution will be the same as in example 1.2., i.e.which can be written as

where 4 , and 4, are the shape functions (4 , = x(1 - ) ; #, = x Z ( t- )). T h eresidual is the same as previously, i.e.

Th e w eighting function w in Galerkin is assumed t o have the same shap e functionas the approximate solution (b), Le.

Th e coefficients P , and ,, are arbitrary.


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30 Chapter 1 Basic ConceptsThe weighted residual statement is,

which produces two integral expressions as 8 , and 8 , are arbitra ry, .e.

or simply,1 1~ 4 ,x = O and R4, d x = Oo o

Substituting ( c) an d the functions 4 , and $, into (g) gives

After integratio n this gives the following system

Notice that the matrix is symmetric because the equation is of an even order andthe ap proxim ate an d wcighting functions are the same. Solving (i) gives

Substituting these values in to ( a) produces the appr oxim ate solution for u, .e.

On e can also find the residual function R (equation c) which is now

The results for u and R are given in table 1.2 where they a re com pared againstthe exact solution of u. Notice th at alth ough the solution is overall more acc uratethan in the case of using the collocation technique, one now needs to carry outsome integrations as shown in formula (h). This operation was not reyuired forthe case of point collocation.


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1.5.Weak Formulat ionsTable 1.2 Results for Galerkinx u (exact) u (approximate) R0.1 0.018641 0.018853 - .0269450.3 0.05 1194 0.051 162 +0.0004850.5 0.069746 0.069444 + 0.0138880.7 0.065582 0.065505 t 0.0050700.9 0.030901 0.03 1146 - .0341651.5 Weak FormulationsThe fundamental integral statements of the boundary element and the finite elementmethods can be interpreted as a combination of a weighted residual statementand a process of integration by parts that reduces or 'weakens' the order of thecontinuity required for the u function.If one returns to equation (1.12) with b =O for simplicity, .e.

one can write formula (1.25) as,

or in terms of residual functions,

A special case of this equation is the case for which the function u exactlysatisfies the 'essential' boundary conditions, u = u on TI, hich results in R , - 0.In this case equation (1.49) becomesA more usual form of this expression can be obtained by integrating by parts oncewhich gives


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32 Chapter 1 Uasic ConceptsIt should be pointed o ut th at equ ation (1.52) could also be obtained byintegrating by parts over the do ma in the weighted residual statemen t for V 2 u andthen introducing the boundary conditions, i.e. starting with

one can integrate by parts once to produce the following expression,

Introducing then the corresponding boun dary conditions in i- ( r= T , + T2)resultsin equation ( I .52).T he last term in equ atio n (1.52) is usually forced to be identically eq ual tozero by the requirement th at the w functions have t o satisfy the Lag rangian versionof the essential boundary conditions, or condition on r,, i.e. w - on T ,. Thisgives a relationship well known in finite elements, i.e.Eq uat ion (1.55) is usually interp reted in terms of virtual work o r virtual power,by associating w with a virtual function. Notice that the integral on the left handside is a measure of the interna1 virtual work an d th e one o n the right th e virtualwork do ne by the externa1 forces q. Eq uati on (1.55) is the starting point of mostfinite element schemes for Laplacian problems and is usually called a 'weak'variational fo rmulatio n. The 'weakness' c an be interpreted as due to two reasons,(i) the o rde r of u function continuity has been reduced as its derivatives are nowof a lower o rde r (i.e. first rather th an second o rde r); (ii) satisfaction of the n atu ralbou ndary con ditions is don e in an ap prox imate rather tha n exact manner, whichreduces the accuracy of bo un dar y values of this variable. (N otice tha t R 2 s generallydifferent frorn zero .)The b oun dary element formulation can be interpreted as introducing a furtherformal step in the process of integration by parts on the derivatives of u , andconsequently weakening the continuity requirements for u.If one s tarts again from eq uati on (1.48) and integrates by par ts as before, themore complete expression obtained is as follows:

( a u aw au a w ) aw+-- d ~ = - ! w d T - S q w d r - S ( u . . ) - d ra d x , d x , d x , dx2 r2 r l r l dnIntegrating again in order to eliminate al1 derivatives in u on the left hand sideintegral, one finds,


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1.5. Weak F'ormulations 33

This is the starting stateme nt for the Bou ndary Element forrnulation of the L aplaceequation. The same equation can be obtained starting from the integral of thcweighted residual over the dornain R (equation (1.5 3)), ntegrating by parts twiceand then introducing thc boundary conditions. The processes have already beenshow n from a no ther field equ ation in formu lae (1.1 3 ) to (1.16)and then (1.19)and(1.201, the only difference now being that b is zero.Consider now equation ( l .1) again to illustrate how a weak form ulation ca nbe used and the domain and boundary element statements are obtained. Let usstarl with equation ( 1 . l o ) which was deduced from (1 .1 ) by a process of integraiiunsby parts and application of boundary conditions, i.e.

which ca n also be expressed in a more co mp act form in function ofresid uals, .e.

Th e function u will now be assumed to satisfy exactly the 'essential' boundaryconditions u = u a t x = O. In this case (1.58 ) becomes,[S+ (i?u - b)w d x = [ ( q -j)wJ.=1or in terms of (1.59), simply

Integrating by parts e qu atio n (1.60) one can write

If the weighting function w is forced to satisfy the homogeneous version of theessential boundary conditions at x =O, equation (1.62) becomes,

which is analogous to equ ation (1.55) obtaine d for the Laplace field equ atio n.Notice th at equa tion (1.59) can also be o btaincd by applying the boundarycond itions into stateme nt (1.3) and that this slatem ent was simply obtain ed byintegrating by parts weighted residual expression (1.2).


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34 Chapter 1 Basic ConceptsThe Boundary Element type governing statement for the example underdiscussion is found by carrying ou t two consecutive integrations by parts of (1.10)and this gives the previously obtained formula (1.7), i.e.

This expression could also have been obtained by carrying out a doubleintegration by parts of the weighted residual equ ation (1.2) an d applying afterwardsthe boundary conditions.It is worth noting that both in this one dimensional example and the twodimen sional Laplace equations, a Finite Element type statement has been obtainedafter the first integration by parts (equations (1.3) and (1.14)), an d Bou ndaryElement type integral equa tion after the second in tegration (equations (1.4) an d( l .15)).

Example 1.4In order to understand the effect of weak formulations on the satisfaction ofboun dary conditions, we will now consider again equ ation (1 . l ) but assume thatthe boun dary conditions are of two types, i .e.

at x = O - + u = O ('essential' c on dit io n)duat x = 1 -+ q =- q ( 'natural ' condition)dx

The expression previously used for the approximate values of u can not nowbe applied as the boundary conditions are different. Let us consider again thestarting expression,

and satisfy exactly the essential condition, at x -O, i.e.

but not the natural condition.Hence the approximate solution is now,

where a l= u Z , 2 = a j . . .


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1.5. Weak Formulations 35The residual will now be diflerent from the one in the previous examples, i.e.

The weighted residual statement has to include now the natural boundarycondition R, residual which is not identically satisfied, i.e.

or in expanded form,

O ne can now solve equation (g j n its present form or reduce the order ofderivativesin the domain and the number of terms o n the right hand side by integrating byparts the dZu/dx2 erm. This gives

Notice that in Galerkin the weighting function w has the same shapes as theapproximation for u (equation (d)). Hence for tw o terms,u =a ,+ , + a242andw = BI+l + P 2 + 2where 4, = x and + , = x Z .

Substituting these vaiues into (h ) on e finds,


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36 Chapter 1 Basic Concep tsAs the p , and p2 terms are a rbitr ary this implies satisfaction of the followingtwo equations,

Integrating the abo ve equations an d w riting the results in matrix form one inds,

The values of a , and a, are,

Notice that an error will now appear when we try to compute the value of q a tx = 1, i.e.

and hence this value will never be equal to the applied 4, i.e.

This peculiar result is characteristic of weak forrnulations such as those usedin finite elements. Because of this approx ima te satisfaction of the n atu ral bou ndaryconditions, f.e. solution s used in engineering practice tend to give poo r results forsurface fluxes or tractions. Th e resulting errors in man y cases 'pollute' the resultsto such an extent that the finite element solutions are unreliable for many casesof stress or flux concentration except when using very fine meshes.Results for u and R are given in table 1.3 for the case in which q = O. Th e exactsolution is

sin xu=--- XCOS 1

Table 1.3 Results for Weak Formulation and Calerkin Methodx u (exact) u (approximate R0. 1 0.084773 0.094271 "-0.6695 190.3 0.246953 0.256899 - .3069010.5 0.387328 0.384975 0.0211750.7 0.492328 0.478499 0.3146990.9 0.549794 0.537471 0.573671


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1.6 . Boundary and Domain Solutions 371.6 Boundary and Domain SolutionsIn section 4 thc wcighted residual technique was classified into boun dary, do ma inand mixed methods. Boundary methods were defined as those for which theassumed approximate solution satisfies the governing or field equation in sucha way that the only unknowns of the problem remain on the boundary. Thesatisfaction of the field equation rnay be of its homogeneous form or a specialform with a singular right hand side.

In the process of double integration described earlier one had transferred thederivatives of the approximate solution u to the weighting function w and so thecondition s previously imposed o n the former apply now t o the latter. A boundarymethod can be obtained by choosing a weighting function w in either of thefollowing two ways, i.e.(i) By selecting a function w which satisfies the governing equation in itshomogeneous form, or(ii) By using special types of functions which satisfy those equations in a w a ythat it is still possible to reduce the problems to the boundary only. Thebest known of the functions applied as right hand side of the equation inthe second method are the Dirac delta functions which give simply a valuea t a point when integrated over the domain.I t is imp orta nt however t o realize that other functions could also be proposedand may be very appropriate for other cases, provided that they can be reduced

to the boundary.We will now apply both techniques to our simple equation (1.1), i.e.

or its weighted residual statement,

The firsr approach implies th at a solution is known such that

without taking into account the actual boundary conditions of the problem.Hencc statement (1.66) reduces ta

This approach is associated with the method called Trefftz.


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3 8 Chapter 1 Basic ConceptsT he second approach is based on a function w such that

where Ai indicates the Dirac function such thatX +&(singular at the x i point with S A i dx = 1

Notice that in this case

where u i represen ts the value of the function u at the point x i . In this case equa tion(1.66) becom es,

When the x i point is chosen on the bo un da ry, then equ ation (1.72) gives arelationship between boundary variables.The second approach is the one usually applied in boundary elements wherethe function w is called the 'fundamental' solution of the governing equation, orsolution of (1.69). Notice t hat this solution is obtained without taking into c on-sideration the bou ndary conditions of the problem.Domain solutions are obtained from weighted residual statements when theassumed approximate soluti?ns do not satisfy the governing cquations One canreturn lo equ ation (1.1) which after integration by parts gives the followingstatcmcnt,

This is a finite element type equation for which the las1 term can be found to bezero at the boundary points where q = duldx is imkno wn , by the requirement thatw = O therc. Substituting an approximate solution u in terms of unknowncoefficients and kno wn weighting functions leads to a system of equa tions to solvethe problem. Notice tha t in the case of finite elements the unknown function u isexplicitly defined over al1 the domain.Although the abovc remarks refer to the starting one dimensional eqiiation(1.1 they also app ly for the case of the Poisson equ ation (1.12) an d the associatcdweighted residual statem ents (equa tions (1.14) an d (1.16)). Similar considerationscan be made for many other types of field equations.


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1.6. Boundary and Domain Solutions 39Example 1.5Let us now return to the same equation as defined in Example 1.1 and try tosolve it using a weak formulation and considering boundary as well as domaintechniques. The field equation is

with boundary conditions,u = O a t x = O a n d x = l

(i ) Boundary Solution. Homogeneous AppruachA weighting function which will satisfy the hom ogeneous version o feq ua tion (a ), .e.

is the simple functionw = a lx + a, with dwldx = a l ( d )

Equation (1.68) can be written for the case 2. - and b = - x a s ,After substituting above the bo und ary conditions (b) and the expressions for

w and dwldx as given by (d) equation (e) becomes1 x ( a l x + a,) dx + q , (a l + a,) - ,a , = Oo (f )

As the above equation has to be satisfied for any arbitrary values of al and a, ,it gives the following two expressions

and hence,

These values of q a t x = O and x = 1 which are now the problcm unknowns,are in this case the exact values.


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40 Chapter 1 Basic Concepts(11) Boundary Solution. Singular ApproachThe weighting function in this case is chosen such that

A solution of equ ation (i) regardless of bou ndary con ditions is

Once the boundary conditions are applied, equation (1.72) becomes

Taking into consideration that w, O and substituting the other values of w asgiven by (j) one finds,

Notice that only one unknown (q,) remains, since one of the boundary stressesdisapp eared because of the variation of the weighting function w . Th e value of q 1can be determined by taking the coordinate x i = 1 , i.e.

which is the exact value in this case.Any value of u inside the domain can be computed from (l), i.e.

which is also the exact solution.If instead of the fundamental solution given by (j) one had chosen a funda-mental solution that also satisfies the boundary conditions, then no unknownswould exist either in the domain or at the boundaries and the value of u at anypoint would be obtained by a single integratio n. Consider for instance the solution,


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1.6. Boundary and Dornain Solutions 41'rhis function satisfies ( i ) and the boundary condit ions w = O a t x = O and x = l(MI,, - w I = O) . Thus equation (k ) gives

resulting in

which is the exact solution.Fundamental solutions that satisfy the boundary conditions as well as thegovcrning equations are called Green's functions.

(iii) Domain SolutionTh e weighted residual statemen t used here is the on e resulting after one integrationby parts has been carried out (equation (1.73)), i.e.

The proposed approxima te solution is the one in Example 1.1 which satisfies thebounda ry conditions, i.e.u = a141+ a242with 4, =x - x2+ x3

where 4 , and 4, are the Hermitian cubic polynomials.Substituting these values into (S) and using Galerkin, i.e.


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42 Chapter 1 Basic Conceptsone can write,

which after integrating and solving also gives the exact solution, i.e.

It is worth noticing that if the approximation used for the 'weak' formulationis the same as the one used for the original domain weighted residual equationthe result will be the same in al1 cases. The advantage of the 'weak' formulation isthat the order of the derivatives of u is in this case reduced and hence the orderof derivability required by the assumed approximate solution is also reduced.

1.7 Concluding RemarksThis chapter has presented the Bou ndary Element M ethod as a weighted residualtechnique. This approach permits relation of the method to other numericaltechniques and gives an easy way of introducing boundary elements.For simplicity one dimensional problems have been discussed throughout thechap ter to present the relationship between different integral stateme nts and alsobetween approxima te techniques. The presentation was then extended to potentialproblems governed by the Laplace or Poisson's equations, which will be used inthe next chapter. Some authors prefer to deduce the boun dary integral equationsfrom Green's theorem instead. Notice that this theorem has also been prescntedhere (equation (1.18)) where it was shown that it can be derived from Lagrangianmultipliers or basic residual type statements.Later on a similar app roa ch will be discussed for elasticity problems as shownin Chapter 3 . The beauty of weighted residuals is that they are simple to use andcan be applied for a widc range of' problems, including some very complex non-linear and time dependent cases which are not discussed in this book.


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ExercisesExercises

d2u1.1. Solve -- -t.u -+ x = O with boundary conditions u(0)=u(1) = O using a trial functiond x 2of the form u = a, + a l x + a , x 2 and point collocation for 1:= 1!2. Plot the so lu tio nan d com pare it with that of example 1.2 of the text and the exact solution given byequation (b) of that example.

d 2 u1.2. Solve - xp(u) from x = O to x = Iwith boun dary cond itions u(0)=u ( ] ) O usingdx2the same trial function and collocation point of exercise 1.1.

1.3. Solve V a u O in the plane domain U 6 < 1, O 6 y < w with boundary conditions

u(0, Y)= 4 1 , Y )= 0u(x, m )=ou l x , O) = x ( 1 - )

X

using a trial function of the form u = A(y)x(x- 1) and point collocation for x = 1 / 2 ,O< y < co as collocation point.

1.4. Solve exercise 1.3 using the G alerk in meth od with th e integral an d weighting fun ctiononly along the x axis.

1.5. Solve the equatio n = O from x = O to x = 1 with bound ary conditionsu(0)= O and u ( 1 )= 1 using u =no + a 1 x+ a,x2 as trial function and subdomaincollocation with only on e subdomain (x from O t o 1) .

1.6. The equation of the vertical displacement of a cable suspended between two pointsd2uis +p (x )= O where p( x ) s the ratio between the distributed load an d the horizontald x 2force at the extremes.Use the weak formulation and the homogeneous approach of boundary solutionto compute the slope at the extremes for a cable that extends from x = O t o x = 1with boundary conditions u(0)= u(1)=O. The function p [ x ) is given by

1.7. Using the same equation of exercise 1.6 and the singular approach of boundarysolution compute the value of u at the mid-point.


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44 Chapter 1 Basic Concepts1.8. The samc a s exercisc 1.7 using a fund ame ntal solution tha t also satisfies the bou ndar yconditions (Green's function).1.9. Solve the same equation of excrcise 1.6 by means of a domain solution procedure

and Galerkin. Use the following approxim ate solutionu = a , sin xx + a, sin 3xx

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Chapter 1 Pages 22-52