Nonlinear finite elements

Welcome to this learning project about Nonlinear finite elements!

Learning Project Summary

• Project code:
• Suggested Prerequisites:
  • Linear algebra
  • Partial differential equations
  • Introduction to finite elements
  • Continuum mechanics
• Time investment: 6 months
• Assessment suggestions:
• Portal:Engineering and Technology
• School:Engineering
• Department:Mechanical engineering
• Stream
• Level: First year graduate

Content summary

This is an introductory course on nonlinear finite element analysis of solid mechanics and heat transfer problems. Nonlinearities can be caused by changes in geometry or be due to nonlinear material behavior. Both types of nonlinearities are covered in this course.

Goals

This learning project aims to.

• provide the mathematical foundations of the finite element formulation for engineering applications (solids, heat, fluids).
• expose students to some of the recent trends and research areas in finite elements.

Contents

Syllabus and Learning Materials

1. Mathematical Preliminaries
   1. Set and function notation
   2. Vectors
   3. Matrices
   4. Tensors
   5. Partial differential equations
   6. Variational calculus
2. Finite element basics
   1. Weak form of PDEs
   2. Linear time-dependent heat equation
   3. Finite element basis functions
   4. Time integration.
3. Nonlinear finite element basics
   1. Nonlinear heat equation - one dimension
   2. Basic nonlinear continuum mechanics of solids
   3. Total and updated Lagrangian approaches.
4. Nonlinear deformation of beams
5. Nonlinear deformation of plates and shells
   1. Basic linear plate and shell elements
   2. Nonlinear plates and shells
   3. Time-dependent deformation of shells
6. Nonlinear material behaviors
   1. Material nonlinearities
   2. Objective rates
   3. Nonlinear Elasticity
   4. Plasticity
   5. Viscoplasticity
   6. Viscoelasticity.
7. Verification and Validation.

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