Finite Element Method - Generalization of the finite element concents galerkin - weighted residual and variational approaches _03

Finite Element Method - Generalization of the finite element concents galerkin - weighted residual and variational approaches _03 The description of the laws of physics for space- and time-dependent problems are usually expressed in terms of partial differential equations (PDEs). For the vast majority of geometries and problems, these PDEs cannot be solved with analytical methods. Instead, an approximation of the equations can be constructed, typically based upon different types of discretizations. These discretization methods approximate the PDEs with numerical model equations, which can be solved using numerical methods. The solution to the numerical model equations are, in turn, an approximation of the real solution to the PDEs. The finite element method (FEM) is used to compute such approximations.

Posing the problem to be solved in its most general terms we find that we seek an

unknown function u such that it satisfies a certain differential equation set

in a ‘domain’ (volume, area, etc.) R (Fig 3.1), together with certain boundary conditions

B(u) = { ::;;} = o (3.2)

on the boundaries r of the domain (Fig 3.1)

The function sought may be a scalar quantity or may represent a vector of several variables Similarly, the differential equation may be a single one or a set of simulta- neous equations and does not need to be linear It is for this reason that we have resorted to matrix notation in the above

The finite element process, being one of approximation, will seek the solution in the approximate form

M U = Niai = Na

i = 1

(3.3)

Fig 3.1 Problem domain R and boundary r

where Ni are shape functions prescribed in terms of independent variables (such as the

coordinates x, y , etc.) and all or most of the parameters ai are unknown

We have seen that precisely the same form of approximation was used in the displacement approach to elasticity problems in the previous chapter We also

noted there that ( a ) the shape functions were usually defined locally for elements or

subdomains and (b) the properties of discrete systems were recovered if the

approximating eqations were cast in an integral form [viz Eqs (2.22)-(2.26)]

With this object in mind we shall seek to cast the equation from which the unknown parameters ai are to be obtained in the integral form

(3.4)

in which Gj and gj prescribe known functions or operators

These integral forms will permit the approximation to be obtained element by element and an assembly to be achieved by the use of the procedures developed for

standard discrete systems in Chapter 1, since, providing the functions Gj and gj are integrable, we have

where W is the domain of each element and re its part of the boundary

Two distinct procedures are available for obtaining the approximation in such

integral forms The first is the method of weighted residuals (known alternatively as the Galerkin procedure); the second is the determination of variational functionals

for which stationarity is sought We shall deal with both approaches in turn

If the differential equations are linear, Le., if we can write (3.1) and (3.2) as

A(u) = Lu + p = 0 in 0 ( 3 6 )

B ( u ) r M u + t = O o n r (3.7)

The reader not used to abstraction may well now be confused about the meaning of

the various terms We shall introduce here some typical sets of differential equations

for which we will seek solutions (and which will make the problems a little more

where u 4 indicates temperature, k is the conductivity, Q is a heat source, 4 and ij

are the prescribed values of temperature and heat flow on the boundaries and n is the

direction normal to r

In the above problem k and Q can be functions of position and, if the problem is

non-linear, of 4 or its derivatives

Example 2 Steady-state heat conduction-convection equation in two dimensions:

with boundary conditions as in the first example Here u , ~ and uy are known functions

of position and represent velocities of an incompressible fluid in which heat transfer

occurs

Example 3 A system of three first order equations equivalent to Example 1:

(3.12)

in R and

B(u) = q5 - 6 = 0 on

= q n - q = O o n r ,

where qn is the flux normal to the boundary

Here the unknown function vector u corresponds to the set

This last example is typical of a so-called mixed formulation In such problems the

number of dependent unknowns can always be reduced in the governing equations by suitable algebraic operations, still leaving a solvable problem [e.g., obtaining Eq (3.10) from (3.12) by eliminating qx and q,,]

If this cannot be done [viz Eq (3.10)] we have an irreducible formulation

Problems of mixed form present certain complexities in their solution which we shall discuss in Chapters 11-13

In Chapter 7 we shall return to detailed examples of the above field problems, and other examples will be introduced throughout the book The three sets of problems will, however, be useful in their full form or reduced to one dimension (by suppressing

the y variable) to illustrate the various approaches used in this chapter

Weiahted residual methods

3.2 Integral or 'weak' statements equivalent to the

A(u) # 0 at any point or part of the domain Immediately, a function v can be found which makes the integral of (3.13) non-zero, and hence the point is proved

Integral or ‘weak’ statements equivalent to the differential equations 43

If the boundary conditions (3.12) are to be simultaneously satisfied, then we require

that

VTB(u) d r [GIB1 (u) + Z12B2(~) + .] d r = 0 (3.15)

Jr

for any set of functions V

Indeed, the integral statement that

la vTA(u) dR + lr VTB(u) d r = 0 (3.16)

is satisfied for all v and V is equivalent to the satisfaction of the differential equations

(3.1) and their boundary conditions (3.2)

In the above discussion it was implicitly assumed that integrals such as those in Eq

(3.16) are capable of being evaluated This places certain restrictions on the possible

families to which the functions v or u must belong In general we shall seek to avoid

functions which result in any term in the integrals becoming infinite

Thus, in Eq (3.16) we generally limit the choice of v and V to bounded functions

without restricting the validity of previous statements

What restrictions need to be placed on the functions? The answer obviously

depends on the order of differentiation implied in the equations A(u) [or B(u)]

Consider, for instance, a function u which is continuous but has a discontinuous

slope in the x-direction, as shown in Fig 3.2 which is identical to Fig 2.4 but is repro-

duced here for clarity We imagine this discontinuity to be replaced by a continuous

variation in a very small distance A (a process known as ‘molification’) and study the

behaviour of the derivatives It is easy to see that although the first derivative is not

defined here, it has finite value and can be integrated easily but the second derivative

tends to infinity This therefore presents difficulties if integrals are to be evaluated

numerically by simple means, even though the integral is finite If such derivatives

are multiplied by each other the integral does not exist and the function is known

as non-square integrable Such a function is said to be C, continuous

In a similar way it is easy to see that if nth-order derivatives occur in any term of A

or B then the function has to be such that its n - 1 derivatives are continuous (Cn- I

continuity)

On many occasions it is possible to perform an integration by parts on Eq (3.16)

and replace it by an alternative statement of the form

C ( V ) ~ D ( U ) dR + E(V)TF(u) d r = 0 (3.17)

Jr

In this the operators C to F usually contain lower order derivatives than those occur-

ring in operators A or B Now a lower order of continuity is required in the choice of

the u function at a price of higher continuity for v and V

The statement (3.17) is now more ‘permissive’ than the original problem posed by

Eqs (3 I), (3.2), or (3.16) and is called a weak form of these equations It is a somewhat

surprising fact that often this weak form is more realistic physically than the original

differential equation which implied an excessive ‘smoothness’ of the true solution

Integral statements of the form of (3.16) and (3.17) will form the basis of finite

element approximations, and we shall discuss them later in fuller detail Before

doing so we shall apply the new formulation to an example

Fig 3.2 Differentiation of function with slope discontinuity (Co continuous)

Example Weak form of the heat conduction equation - forced and natural boundary conditions Consider now the integral form of Eq (3.10) We can write the statement

is automatically satisfied by the choice of the functions q5 on ro

Equation (3.18) can now be integrated by parts to obtain a weak form similar to

Eq (3.17) We shall make use here of general formulae for such integration (Green's formulae) which we derive in Appendix G and which on many occasions will be

Integral or 'weak' statements equivalent to the differential equations 45

without loss of generality (as both functions are arbitrary), we can write Eq (3.20) as

where the operator V is simply

We note that

(a) the variable 4 has disappeared from the integrals taken along the boundary r4

and that the boundary condition

dn

B($) 1 k- + = 0

on that boundary is automatically satisfied - such a condition is known as a

natural boundary condition - and

( b ) if the choice of 4 is restricted so as to satisfy the forced boundary conditions

q5 - 6 = 0, we can omit the last term of Eq (3.23) by restricting the choice of v

to functions which give u = 0 on r4

The form of Eq (3.23) is the weak f o r m of the heat conduction statement equivalent

to Eq (3.17) It admits discontinuous conductivity coefficients k and temperature 4

which show discontinuous first derivatives, a real possibility not admitted in the differential form

3.3 Approximation to integral formulations: the

weighted residual Galerkin method

If the unknown function u is approximated by the expansion (3.3), i.e.,

Inserting the above approximations into Eq (3.16) we have

Sa? [ J a wTA(Na) dR + Jr wTB(Na) d r 1 = 0

and since Saj is arbitrary we have a set of equations which is sufficient to determine the parameters a, as

(3.25)

or, from Eq (3.17),

/ a C(wj)TD(Na) dR + E(wj)TF(Na) d r = 0

If we note that A(Na) represents the residual or error obtained by substitution of

the approximation into the differential equation [and B(Na), the residual of the

boundary conditions], then Eq (3.25) is a weighted integral of such residuals The

approximation may thus be called the method of weighted residuals

In its classical sense it was first described by Crandall,' who points out the various forms used since the end of the last century More recently a very full expose of the method has been given by Finlayson.2 Clearly, almost any set of independent func-

tions w, could be used for the purpose of weighting and, according to the choice of

function, a different name can be attached to each process Thus the various common choices are:

1 Point ~ o l l o c a t i o n ~ wj = Si, where Si is such that for x # xi; y # y j , w, = 0 but

Ja w, dR = I (unit matrix) This procedure is equivalent to simply making the

residual zero at n points within the domain and integration is 'nominal' (inciden-

tally although w, defined here does not satisfy all the criteria of Sec 3.2, it is never-

theless admissible in view of its properties)

2 Subdornain c o l l ~ c a t i o n ~ wj = I in R, and zero elsewhere This essentially makes the integral of the error zero over the specified subdomains

Approximation to integral formulations: the weighted residual Galerkin method 47

3 The Galerkin method (Bubnov-Galerkin).”6 wj = N j Here simply the original

shape (or basis) functions are used as weighting This method, as we shall see,

frequently (but by no means always) leads to symmetric matrices and for this

and other reasons will be adopted in our finite element work almost exclusively

The name of ‘weighted residuals’ is clearly much older than that of the ‘finite element

method’ The latter uses mainly locally based (element) functions in the expansion of

Eq (3.3) but the general procedures are identical As the process always leads to equa-

tions which, being of integral form, can be obtained by summation of contributions

from various subdomains, we choose to embrace all weighted residual approximations

under the name of generalizedfinite element method Frequently, simultaneous use of

both local and ‘global’ trial functions will be found to be useful

In the literature the names of Petrov and Galerkin’ are often associated with the

use of weighting functions such that wj # Nj It is important to remark that the

well-known finite difference method of approximation is a particular case of colloca-

tion with locally defined basis functions and is thus a case of a Petrov-Galerkin

scheme We shall return to such unorthodox definitions in more detail in Chapter 16

To illustrate the procedure of weighted residual approximation and its relation to

the finite element process let us consider some specific examples

Example 1 One-dimensional equation of heat conduction (Fig 3.3) The problem here

will be a one-dimensional representation of the heat conduction equation [Eq (3 l o ) ]

with unit conductivity (This problem could equally well represent many other

physical situations, e.g., deformation of a loaded string.) Here we have

d2d

A ( @ ) = T + Q = O ( O < x Q L )

with Q = Q ( x ) given by Q = 1 (0 < x < L / 2 ) and Q = 0 ( L / 2 < x < L ) The bound-

ary conditions assumed will be simply q!~ = 0 at x = 0 and x = L

In the first case we shall consider a one- or two-term approximation of the Fourier

series form, i.e.,

( 3 2 8 )

with i = 1 and i = 1 and 2 These satisfy the boundary conditions exactly and are

continuous throughout the domain We can thus use either Eq (3.16) or Eq (3.17)

for the approximation with equal validity We shall use the former, which allows

various weighting functions to be adopted In Fig 3.3 we present the problem and

its solution using point collocation, subdomain collocation, and the Galerkin method.?

As the chosen expansion satisfies a priori the boundary conditions there is no need

to introduce them into the formulation, which is given simply by

The full working out of this problem is left as an exercise to the reader

( 3 2 9 )

t In the case of point collocation using i = 1 ( x i = L / 2 ) a difficulty arises about the value of Q (as this is

either zero or one) The value of was therefore used for the example

Fig 3.3 One-dimensional heat conduction (a) One-term solution using different weighting procedures

Approximation to integral formulations: the weighted residual Galerkin method 49

Of more interest to the standard finite element field is the use of piecewise defined (locally based) functions in place of the global functions of Eq (3.28) Here, to avoid imposing slope continuity, we shall use the equivalent of Eq (3.17) obtained by integrating Eq (3.29) by parts This yields

The boundary terms disappear identically if wj = 0 at the two ends

The above equations can be written as

K a + f = O

where for each 'element' of length Le,

Le

f e J = - j0 wjQdx with the usual rules of addition pertaining, i.e.,

(3.34)

(3.35)

where Qe is the value for element e

Assembly of a typical equation at a node i is left to the reader, who is well advised to carry out the calculations leading to the results shown in Fig 3.4 for a two- and four- element subdivision

Some points of interest immediately arise if the results of Figs 3.3 and 3.4 are compared With smooth global shape functions the Galerkin method gives better overall results than those achieved for the same number of unknown parameters a

with locally based functions This we shall find to be the general case with higher order approximations, yielding better accuracy Further, it will be observed that the linear approximation has given the exact answers at the interelement nodal points This is a property of the particular equation being solved and unfortunately does not carry over to general problem^.^ (See also Appendix H.) Lastly, the