Page 181 - Finite Element Modeling and Simulations with ANSYS Workbench

166 Finite Element Modeling and Simulation with ANSYS Workbench

such that

A = A + A + A L T (5.9)
D
L
Iterations continue until solution x converges, that is

k
x (k+1 ) − x () ≤ε (5.10)
x () k

where ε is the tolerance for convergence control.
Iterative solvers with moderate selections of the tolerance are usually much faster than
direct solvers in solving large-scale models. However, for ill-conditioned systems, direct
solvers should be applied to ensure the accuracy of the solutions.

5.5 Nature of Finite Element Solutions
FEA model is a mathematical model of the real structure, based on many approximations.
An observation is made on the FEA models and solutions:

• Real structure—infinite number of nodes (physical points or particles), thus infi-
nite number of DOFs
• FEA model—finite number of nodes, thus finite number of DOFs

In particular, one can argue that the displacement field is controlled (or constrained) by
the values at a limited number of nodes (Figure 5.4).
Therefore, we have the so-called stiffening effect:

• FEA model is stiffer than the real structure.
• In general, displacement results are smaller in magnitudes than the exact values.

Hence, the FEM solution of displacement is a lower bound of the exact solution.

4
Recall that on an element: u = N u

= 1
FIGURE 5.4
Elements in an FEA model.