1. MECH4450 Introduction to Finite Element Methods Chapter 9 Advanced Topics II - Nonlinear Problems Error and Convergence

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

3. Nonlinear Problems Example: Note: K ij 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 Advantage: If converges, the rate is quadratic. Disadvantages: Convergence is not guaranteed. Idea:

6. Nonlinear Problems Example: Weak form:

7. 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

8. Numerical Errors Ill-conditioning system: is small, but or is large Example: P k1k1 k2k2

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

10. 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