# Nonlinear finite elements

Welcome to this learning project about nonlinear finite elements!

## Introduction

Finite element analysis (FEA) is a computer simulation technique used in engineering analysis. It uses a numerical technique called the finite element method (FEM) to solve partial differential equations. There are many finite element software packages, both free and proprietary. Development of the finite element method in structural mechanics is usually based on an energy principle such as the virtual work principle or the minimum total potential energy principle. *

### Project metadata

* Suggested Prerequisites:
** Linear algebra
** Partial differential equations
** Introduction to finite elements
** Continuum mechanics
* Time investment: 6 months
* Portal:Engineering and Technology
* School:Engineering
* Department:Mechanical engineering
* 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.
Here's a short quiz to help you find out what you need to brush up on before you dig into the course:
* Assessment Quiz

## Contents

* Syllabus and Learning Materials
* Syllabus and Learning Materials
* Mathematical Preliminaries
* Set notation
* Functions
* Vectors
* Matrices
* Tensors
* Partial differential equations
* Variational calculus
* Linear finite element basics
* An example: Axially loaded bar
* Strong form: governing differential equation
* Weak form: integral equation
* Approximate solution: Galerkin method
* Approximate solution: Finite element method
* More examples: Some model problems
* Weak form: Weighted residual methods
* Choosing a weight function
* Bubnov Galerkin methods
* Finite element basis functions
* Example of a finite element approximation
* A time-dependent problem: the heat equation
* Steady state heat conduction
* Weak form of the steady state heat equation
* Weak form of the time-dependent heat equation
* Finite element approximation of Poisson equation
* Finite element approximation of the heat equation
* Time integration of the heat equation.
* Nonlinear finite element basics
* Nonlinearities in solid mechanics
* Nonlinear deformation of an axial bar
* Weak form and finite element approximation
* The Newton-Raphson method
* The Newton method applied to finite elements
* The Newton method applied to the axially loaded bar
* Lagrangian and Eulerian descriptions of motion
* Lagrangian finite elements
* Motion from the Lagrangian point of view
* Total Lagrangian approach
* Updated Lagrangian approach
* An example: Effect of mesh distortion
* Solution procedure
* Special case: Natural vibrations
* Nonlinear deformation of beams
* Euler-Bernoulli beams
* Timoshenko beams
* Buckling of beams
* Nonlinear deformation of plates and shells
* Basic linear plate and shell elements
* Nonlinear plates and shells
* Time-dependent deformation of shells
* Basic continuum mechanics
* Kinematics
* Motion, displacement, velocity, acceleration
* Stresses and strains in one and two dimensions
* Strains and deformations in three-dimensions
* Polar decomposition
* Spectral decompositions of kinematic quantities
* Volume change and area change
* Time derivatives and rate quantities
* Objectivity of kinematic quantities
* Stress measures and stress rates
* Stress measures
* Deviatoric and volumetric stress
* Objective stress rates
* Balance laws
* Overview of balance laws
* Balance of mass
* Balance of linear momentum
* Balance of angular momentum
* Balance of energy
* Entropy inequality
* Constitutive models
* Objectivity of hyperelastic relations
* Rate form of hyperelastic relations
* Nonlinear elasticity
* Plasticity
* Viscoplasticity
* Viscoelasticity.
* Finite element formulation in three dimensions.
* Updated Lagrangian formulation
* Verification and Validation.
* Assignments, tests and quizzes
* Assignments
* Homework 1 Problem set
* Solutions
* Homework 2 Problem set
* Solutions
* Homework 3 Problem set
* Solutions
* Homework 4 Problem set
* Hints
* Solutions
* Homework 5 Problem set
* Solutions
* Homework 6 Problem set
* Hints
* Solutions
* Homework 7 Problem set
* Hints
* Solutions
* Homework 8 Problem set
* Solutions
* Homework 9 Problem set
* Solutions
* Homework 10 Problem set
* Solutions
* Homework 11 Problem set
* Solutions
* Tests and Quizzes
* Quiz 1
* Solutions

## Textbooks and References
### Textbooks

* An Introduction to Nonlinear Finite Element Analysis by J. N. Reddy, Oxford University Press, 2004, ISBN 019852529X.
* Nonlinear Finite Elements for Continua and Structures by T. Belytschko, W. K. Liu, and B. Moran, John Wiley and Sons, 2000.
* Computational Inelasticity by J. C. Simo and T. J. R. Hughes, Springer, 1998.
* The Finite Element Method: Linear Static and Dynamic Finite Element Analysis by T. J. R. Hughes, Dover Publications, 2000.