1. MECH593 Introduction to Finite Element Methods
Nonlinear Problems
Error and Convergence

2. Nonlinear Problems Types of nonlinearity in structural mechanics:
material nonlinearity:
contact nonlinearity
geometric nonlinearity
Example:
Weak form:
Approximation:

3. Nonlinear Problems
Example:
Note: Kij depends on u!

4. Iterative Schemes
Direct iterative method (successive-substitution method)
Advantage: explicit method, easy to implement
Disadvantages: (1) conditional convergence
(2) slow convergence rate, at most linear
Example:

5. Iterative Schemes Newton – Raphson (N-R) method Idea:
Advantage: If converges, the rate is quadratic.
Disadvantages: Convergence is not guaranteed.

6. Nonlinear Problems
Example:
Weak form:

7. Bending of Euler-Bernouli Beam
Governing equations:
Weak form:

8. Bending of Euler-Bernouli Beam
Approximation:
Element equation:

9. Errors in FEM Types of errors: Discretization errors
number of elements
domain approximation
number of nodes per element
the nature of shape functions
integration rule, function evaluation methods
Numerical errors
round-off errors
manipulation errors

10. Numerical Errors Ill-conditioning system: or is small, but is large
Example: P k1 k2

11. Measures of Errors “Sup-metric/ - norm”: “Energy norm”: “L2 norm”:
2m is the order of the differential equation being solved.
“L2 norm”:
Convergence and rate of convergence:
p: rate of convergence

12. Improvement Approaches
h - refinement:
h: element size
p - refinement:
p: the degree of the highest complete polynomial in the
approximation of the field quantity
r - refinement:
r: rearrange
Adaptive mesh refinement – error estimation