The Finite Element Method - Mechanics of Structures: Variational and Computational Methods - page 366

kinematical continuity, the trial solution of Eq. (6.47) is chosen such that at any node shared
by more than one element, a particular variable of v is the same physical variable regardless
from which element the node is approached.
The functions N which make u equal to certain variables at prescribed points are called
interpolation functions .Bydefinition, for two-dimensional problems, [e.g., see Eq. (6.19c)]

At node i : N i (ξ

i ,

η ) = 1

i

(6.48)

At node j : N i (ξ

j ,

η ) = 0

j

Equation (6.48) can serve as the basic definition of interpolation functions. It is because of
these characteristics that we refer to the functions N as “shape functions.” The expression
of Chapter 4, Eq. (4.47) for the deflection w of a beam element that begins at ξ = 0 and ends

at ξ = 1 can be written as

w = Nv = w a N 1 + θ a N 2 + w b N 3 + θ b N 4 (6.49)

Since, as explained in Section 4.4.2, the N i of this expression are Hermitian polynomials, the
polynomials N 1 and N 3 satisfy conditions similar to Eq. (6.48), as well as their derivatives
being zero at points a and b . The polynomials N 2 and N 4 are zero at points a and b . For
certain two-dimensional problems, the derivative conditions would appear as

d
dx N i

At node i : (ξ

i ,

η ) = 1

i

(6.50)

d
dx N i (ξ j ,

At node j : η j ) = 0 i, j = 2 , 4

It should be observed that the quantities N i represent the contribution of a nodal unit
displacement to the total deflection.

6.5.2 Convergence

Presumably, for successful finite element solutions, the interpolation functions should
lead to an analysis that monotonically converges to the exact solution as the size of the
elements tends to zero, i.e., the accuracy of the solution increases as the finite element
mesh is continuously refined. Convergence to the correct solution is critical to the proper
use of a finite element analysis and is the topic of numerous papers and topics, e.g.,
see Bathe (1998). To achieve monotonic convergence, the element must be complete and
compatible (or conforming ) [Bathe, 1996; Zienkiewicz, 1977]. The requirement for complete-
ness means that the displacement functions must be able to represent the rigid body dis-
placements and constant strain states. Compatibility assures that no gaps occur within
the elements and between the elements when the system of elements is assembled and
loaded.
Consider these characteristics in more detail. For completeness,

1. The trial functions should be able to represent displacements that the element under-
goes as a rigid body without developing stress. For example, consider a cantilevered
beam with a concentrated force acting at the midpoint. Since stresses will not be gen-
erated beyond the location of load application, the trial functions for the elements at
the free end must be able to permit the elements to translate and rotate stress free.
2. The displacement functions of an element must be such that the strain in each element
approaches a constant value in the limit as the element approaches a very small size.