MECH593 Introduction to Finite Element Methods Nonlinear Problems Error and Convergence

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

Nonlinear Problems Example: Note: Kij depends on u!

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:

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

Nonlinear Problems Example: Weak form:

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

Bending of Euler-Bernouli Beam Approximation: Element equation:

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

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

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

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