USE OF STRESS FUNCTIONS IN FINITE ELEMENT FORMULATION OF FLOW PROBLEMS

Recent developments in the applications of the finite element method to fluid flow problems are briefly reviewed and the advantages and drawbacks of various formulations are highlighted. In a broad sense the salient features of all finite element formulations in Continuum Mechanics are then examined with the objective of identifying the "location" of approximation errors in each formulation. It is concluded that in the methods developed thus far, the errors of approximation arise in the differential equations of equilibrium and/or in differential equations governing the kinematics of the problem. Invariably the algebraic constitutive equations are satisfied identically.

Recognising that the equilibrium equations and the compatibility requirement for the kinematic variables are "exact", whereas the constitutive equations are invariably subject to experimental errors, it is concluded that it would be highly desirable to satisfy the equilibrium and compatibility differential equations identically and locate the approximation errors in the algebraic constitutive equations. The remainder of this investigation is devoted to the development of such a finite element procedure.

In Chapter III, it is shown that the equilibrium equations, including the nonlinear effect of convective acceleration terms may be identically satisfied. This is accomplished through the introduction of a modified form of the Airy stress function. For the case of 2D flows the condition of incompressibility is satisfied through the introduction of the stream function. Thus the stream function and the stress function constitute the essential variables of this formulation. The satisfaction of the constitutive equations is effected through a least squares minimisation procedure. Two finite element models are then developed--one triangular and one rectangular--and several test cases are analysed for steady and unsteady flows.

In Chapter IV a variation of the recommended procedure is developed. Here again the equilibrium and compatibility equations are satisfied identically. However the condition of incompressibility is approximated by a least squares procedure. The desirable feature of this formulation is the retention of the velocity components as essential variables of the formulation. This procedure is illustrated by a numerical example and a number of conclusions are drawn as to its merits and short comings.