In the introduction to this chapter we defined the irreducible and mixed forms and indicated that on occasion it is possible to obtain more than one 'irreducible' form.
To illustrate this in the problem of heat transfer given by Eqs (1 1.2) and (1 1.3) we introduced a penalty function a in Eq. (1 1.6) and derived a corresponding single governing equation (1 1.7) given in terms of q. This penalty function here has no obvious physical meaning and served simply as a device to obtain a close enough approximation to the satisfaction of the continuity of flow equations.
On occasion it is possible to solve the problem as an irreducible one assuming a priori that the choice of the variable satisfies one of the equations. We call such forms directly constrained and obviously the choice of the shape function becomes difficult.
We shall consider two examples.
The complementary heat transfer problem
In this we assume apriori that the choice of q is such that it satisfies Eq. (1 1.3) and the natural boundary conditions
V T q = -Q in R and qn = qn on r4 (11.56)
Thus we only have to satisfy the constitutive relation (1 1.2), Le.,
k-'q + V4 = 0 in R with 4 = 4 on (1 1.57) A weak statement of the above is
jn 6qT(k-'q + V4) dR - Jr, 6qn(4 - 6) d r = 0 (11.58)
in which 6q, represents the variation of normal flux on the boundary.
Use of Green's theorem transforms the above into
If we further assume that VTSq 0 in R and Sg, = 0 on r4, i.e., that the weighting functions are simply the variations of q, the equation reduces to
(1 1.60) This is in fact the variation of a complementary flux principle
Numerical solutions can obviously be started from either of the above equations but the difficulty is the choice of the trial function satisfying the constraints. We shall return to this problem in Sec. 1 1.7.2.
The complementary elastic energy principle
In the elasticity problem specified in Sec. 11.4 we can proceed similarly, assuming stress fields which satisfy the equilibrium conditions both on the boundary rr and
in the domain R.
Thus in an analogous manner to that of the previous example we impose on the permissible stress field the constraints which we assume to be satisfied by the approximation identically, i.e.,
STa + b = 0 in R and t = i on r I (1 1.62) Thus only the constitutive relations and displacement boundary conditions remain to be satisfied, Le.,
D-'a - S u = 0 in R and u = U on rU (1 1.63) The weak statement of the above can be written as
In SoT(D-'o - S u ) dR + 1, StT(u - u) d r = 0 (1 1.64)
which on integration by Green's theorem gives
Again assuming that the test functions are complete variations satisfying the homo- STSo = 0 in R and St = 0 on rr (11.66)
geneous equilibrium equation, i.e.,
we have as the weak statement
The corresponding complementary energy variational principle is
(1 1.67)
(11.68) Once again in practical use the difficulties connected with the choice of the approx- imating function arise but on occasion a direct choice is po~sible.~'
Complementary forms with direct constraint 303
11.7.2 Solution using auxiliary functions
Both the complementary forms can be solved using auxiliary functions to ensure the satisfaction of the constraints.
In the heat transfer problem it is easy to verify that the homogeneous equation
is automatically satisfied by defining a function $ such that q = - a$ q = - - a$
ay ax
(11.69)
(11.70) Thus we define
q = L $ + q o and 6q=L6$ (1 1.71)
where qo is any flux chosen so that
VTqo = -Q
and
(11.72)
(1 1.73) the formulations of Eqs (1 1.60) and (1 1.61) can be used without any constraints and, for instance, the stationarity
will suffice to so formulate the problem (here s is the tangential direction to the boundary).
The above form will require shape functions for
In the corresponding elasticity problem a similar two-dimensional form can be Now the equilibrium equations
satisfying C , continuity.
obtained by the use of the so-called Airy stress function $.36
are identically solved by choosing
a = L $ + o o where
and a. is an arbitrary stress chosen so that
(1 1.75)
(1 1.76)
(1 1.77)
(1 1.78) STao + b = 0
Again the substitution of (1 1.76) into the weak statement (1 1.67) or the comple- mentary variational problem (1 1.68) will yield a direct formulation to which no additional constraints need be applied. However, use of the above forms does lead to further complexity in multiply connected regions where further conditions are needed. The reader will note that in Chapter 7 we encountered this in a similar problem in torsion and suggested a very simple procedure of avoidance (see Sec. 7.5).
The use of this stress function formulation in the two-dimensional context was first made by de Veubeke and Zienkiewicz3’ and E l i a ~ , ~ ~ but the reader should note that now with second-order operators present, C1 continuity of shape functions is needed in a similar manner to the problems which we have to consider in plate bending (see Volume 2).
Incidentally, analogies with plate bending go further here and indeed it can be shown that some of these can be usefully employed for other problems.39