Nonlinear Finite Element Methods
- LO1: Discretize PDEs for model problems in continuum mechanics (examples are heat conduction and elasticity) using finite elements
- LO 2: Solve nonlinear PDEs through linearization and Newton-Taphson iterations
- LO 3: Integrate time-dependent PDEs with explicit and implicit methods
- LO 4: Recognize other nonlinear problems and solving strategies in solid mechanics
- LO 5: Use commercial or open source finite element packages to solve nonlinear problems in solid mechanics
Most problems in engineering dealing with continua and structures lead to nonlinear PDEs. Nonlinearities arise from geometry and constitutive relations. This course starts by covering topics in tensor analysis and the basic solution of nonlinear systems of equations. Then, a review of continuum mechanics focuses on large deformations, weak forms of the balance laws, and basic nonlinear material models. The spatial discretization is focused on, starting with the model problem on nonlinear heat conduction as a scalar problem before focusing on elasticity. Quasi-static solution methods are described, emphasizing the consistent linearization of the system of equations and the notion of stability. Time-dependent solution methods are then considered. Towards the end, material nonlinearities are discussed, with emphasis on plasticity. The course ends with a brief description of structural elements.
Prerequisites:Introductory course in finite element (ME489 or equivalent). Introductory graduate level knowledge of solid mechanics or elasticity is encouraged.
Applied / Theory:
Homework:Bi-weekly homework, 50% of grade
Projects:Final project will be the application of the nonlinear finite element solver developed in class to a problem of interest for the student or group. Presentation 15% of grade, report 25% of grade.
Exams:One midterm, 10% of grade
Textbooks:Nonlinear Finite Element Methonds. Wriggers. Springer, 2008
Non-linear Finite Element Analysis of Solids and Structures. Borst, Crisfield, Remmers, and Verhoosel. Wiley, 2012
An Introduction to Nonlinear Finite Element Analysis. Reddy. Oxford Press, 2015
All available online for Purdue students