# Nonlinear finite elements

Welcome to this learning project about **nonlinear finite elements**!

## Introduction

[edit | edit source]**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

[edit | edit source]Quality resource: this resource is a featured learning resource. |

Completion status: this resource has reached a high level of completion. |

Educational level: this is a tertiary (university) resource. |

Subject classification: this is an engineering resource. |

- Suggested Prerequisites:
- Time investment: 6 months
- Portal:Engineering and Technology
- School:Engineering
- Department:Mechanical engineering
- Level: First year graduate

### Content summary

[edit | edit source]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

[edit | edit source]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:

## Contents

[edit | edit source]## Textbooks and References

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### Textbooks

[edit | edit source]- 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.

### References

[edit | edit source]**Finite Element Analysis**:- K-J. Bathe (1996), Finite Element Procedures, Prentice-Hall.
*Useful repository of information on nonlinear finite elements.* - J. N. Reddy (1993), An Introduction to the Finite Element Method, McGraw-Hill.
*This book is referred to a number of times in one of the texts.* - O. C. Zienkiewicz and R. L. Taylor (2000), The Finite Element Method: Volume 2 Solid Mechanics, Butterworth-Heinemann.
*Another excellent repository of information of nonlinear finite elements geared toward the Civil Engineers.* - J. Bonet and R. Wood (2008), Nonlinear Continuum Mechanics for Finite Element Analysis, Cambridge University Press.
*This is a particularly good reference for the general continuum mechanics and kinematics involved, with a detailed discussion on material nonlinearities.* - S. C. Brenner and L. R. Scott (2007), The mathematical theory of finite element methods, vol. 15 of Texts in Applied Mathematics, Springer-Verlag,
*Classical book on the mathematics foundation of finite element methods*.

- K-J. Bathe (1996), Finite Element Procedures, Prentice-Hall.
**Continuum Mechanics**:- R. M. Brannon (2004), Large Deformation Kinematics.
*An excellent introduction to kinematics.* - R. M. Brannon (2004), Rotation.
*Almost everything you ever wanted to know about rotations.* - A.J.M Spencer (2004), Continuum Mechanics, Dover Publications.
*An excellent introduction to continuum mechanics. You should own a copy for your personal library.* - L.E. Malvern (1969), Introduction to the Mechanics of a Continuous Medium, Prentice-Hall.
*A good reference for continuum mechanics.*

- R. M. Brannon (2004), Large Deformation Kinematics.
**Mathematics**:- R. M. Brannon (2004), Elementary Vector and Tensor Analysis for Engineers.
*This free online book is the best introduction I have seen for vector and tensor analysis for nonlinear mechanics.* - R. M. Brannon (2004), Curvilinear Coordinates.
*Very useful if you wish to follow the literature on the nonlinear deformation of shells.* - B. Daya Reddy (1998), Introductory Functional Analysis: With applications to boundary value problems and finite elements. , Springer-Verlag.
*Excellent book for engineers who want to understand the terminology used in the finite element literature and how error analysis is done.* - A.P.S. Selvadurai (2000), Partial Differential Equations in Mechanics 1,2. Springer.
*Excellent introductory text on partial differential equations with engineers in mind.*

- R. M. Brannon (2004), Elementary Vector and Tensor Analysis for Engineers.

### Reading List

[edit | edit source]- Taylor, R.L., Simo, J.C., Zienkiewicz, O.C., and Chan, A.C.H, 1986,
**The patch test - a condition for assessing FEM convergence**,*International Journal for Numerical Methods in Engineering*, 22, pp. 39-62. - Simo, J.C. and Vu-Quoc, L., 1986,
**A three-dimensional finite strain rod model. Part II: Computational Aspects,***Computer Methods in Applied Mechanics and Engineering,***58**, pp. 79-116. - Ibrahimbegovic, A., 1995,
**On finite element implementation of geometrically nonlinear Reissner's beam theory: Three-dimensional curved beam elements,***Computer Methods in Applied Mechanics and Engineering,***122**, pp. 11-26. - Buchter, N., Ramm, E., and Roehl, D., 1994,
**Three-dimensional extension of non-linear shell formulation based on the enhanced assumed strain concept,***Int. J. Numer. Meth. Engng.*,**37**, pp. 2551-2568. - Rouainia, M. and Peric, D., 1998,
**A computational model for elasto-viscoplastic solids at finite strain with reference to thin shell applications,***Int. J. Numer. Meth. Engng.*,**42**, pp. 289-311.