View source for Nonlinear finite elements

[[Image:FAE_visualization.jpg|thumb|360px|Visualization of how a car deforms in an asymmetrical crash using finite element analysis.]]Welcome to this learning project about '''nonlinear finite elements'''!
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== Introduction ==

'''Finite element analysis''' (FEA) is a [[Wikipedia:computer simulation|computer simulation]] technique used in engineering analysis. It uses a numerical technique called the [[Wikipedia:finite element method|finite element method]] (FEM). 
There are many finite element [[Wikipedia:software package|software packages]], both [[Wikipedia:free software|free]] and [[Wikipedia:proprietary software|proprietary]]. Development of the [[Wikipedia:finite element method in structural mechanics|finite element method in structural mechanics]] is usually based on an [[Wikipedia:energy principles in structural mechanics|energy principle]] such as the [[Wikipedia:virtual work|virtual work]] principle or the [[Wikipedia:minimum total potential energy principle|minimum total potential energy principle]]. {{wps}}

===Project metadata===
{{featured}}
{{complete}}
{{tertiary}}
{{engineering}}
* Suggested Prerequisites:
** [[Linear algebra]]
** [[Partial differential equations]]
** [[Introduction to finite elements]]
** [[Continuum mechanics]]
* Time investment: 6 months
* [[Wikademia:Major portals|Portal]]:[[Portal:Engineering and Technology|Engineering and Technology]]
* [[Wikademia:Schools|School]]:[[Engineering|Engineering]]
* Department:[[Mechanical engineering|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:
*[[Nonlinear finite elements/Quiz0|Assessment Quiz]]


==Contents==

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=== Syllabus and Learning Materials ===

#Mathematical Preliminaries
## [[/Sets|Set notation]]
## [[/Functions|Functions]]
## [[/Vectors|Vectors]]
## [[/Matrices|Matrices]]
## [[/Tensors|Tensors]]
## [[/Partial differential equations|Partial differential equations]]
## [[/Calculus of variations|Variational calculus]]
#Linear finite element basics 
## [[/Axially loaded bar|An example: Axially loaded bar]]
##* [[/Axial bar strong form|Strong form: governing differential equation]]
##* [[/Axial bar weak form|Weak form: integral equation]]
##* [[/Axial bar approximate solution| Approximate solution: Galerkin method]]
##* [[/Axial bar finite element solution| Approximate solution: Finite element method]]
## [[/Model problems|More examples: Some model problems]]
##* [[/Weighted residual methods|Weak form: Weighted residual methods]]
##* [[/Choosing weight function|Choosing a weight function]]
##* [[/Bubnov Galerkin method|Bubnov Galerkin methods]]
##* [[/Finite element basis functions|Finite element basis functions]]
##* [[/Model finite element approximation|Example of a finite element approximation]]
## [[/Linear heat equation|A time-dependent problem: the heat equation]]
##* [[/Steady state heat conduction|Steady state heat conduction]]
##* [[/Weak form of Poisson equation|Weak form of the steady state heat equation]]
##* [[/Weak form of heat equation|Weak form of the time-dependent heat equation]]
##* [[/Solution of Poisson equation|Finite element approximation of Poisson equation]]
##* [[/Solution of heat equation|Finite element approximation of the heat equation]]
##* [[/Heat equation time integration|Time integration of the heat equation]].
#Nonlinear finite element basics
## [[/Nonlinearities in solid mechanics|Nonlinearities in solid mechanics]]
## [[/Nonlinear axially loaded bar|An example: Nonlinear deformation of an axially loaded bar]]
##* [[/Nonlinear axial bar weak form|Weak form for the nonlinear bar]]
##* [[/Newton Raphson method| The Newton-Raphson method]]
##* [[/Newton method for finite elements| The Newton method applied to finite elements]]
##* [[/Newton method for axial bar| The Newton method applied to the axially loaded bar]]
<!-- ## Nonlinear heat equation - one dimension -->
## [[/Lagrangian and Eulerian descriptions|Lagrangian and Eulerian descriptions of motion]]
## [[/Lagrangian finite elements|Lagrangian finite elements]]
##* [[/Motion in Lagrangian form|Motion from the Lagrangian point of view]]
##* [[/Total Lagrangian approach|Total Lagrangian approach]]
##* [[/Updated Lagrangian approach|Updated Lagrangian approach]]
##* [[/Effect of mesh distortion|An example: Effect of mesh distortion]]
## [[/Solution procedure|Solution procedure]]
## [[/Natural vibration|Special case: Natural vibrations]]
#Nonlinear deformation of beams
## [[/Euler Bernoulli beams|Euler-Bernoulli beams]]
## [[/Timoshenko beams| Timoshenko beams]]
## [[/Buckling of 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
### [[/Kinematics - motion and displacement|Motion, displacement, velocity, acceleration]]
### [[/Stress and strain in one and two dimension|Stresses and strains in one and two dimensions]]
### [[/Kinematics|Strains and deformations in three-dimensions]]
### [[/Kinematics - polar decomposition|Polar decomposition]]
### [[/Kinematics - spectral decomposition|Spectral decompositions of kinematic quantities]]
### [[/Kinematics - volume change and area change|Volume change and area change]]
### [[/Kinematics - time derivatives and rates|Time derivatives and rate quantities]]
### [[/Kinematics - objectivity|Objectivity of kinematic quantities]]
## Stress measures and stress rates
### [[/Stress measures|Stress measures]]
### [[/Deviatoric and volumetric stress|Deviatoric and volumetric stress]]
### [[/Objective stress rates|Objective stress rates]]
## Balance laws
### [[/Overview of balance laws|Overview of balance laws]]
### [[/Balance of mass|Balance of mass]]
### [[/Balance of linear momentum|Balance of linear momentum]]
### [[/Balance of angular momentum|Balance of angular momentum]]
### [[/Balance of energy|Balance of energy]]
### [[/Entropy inequality|Entropy inequality]]
## Constitutive models
### [[/Objectivity of constitutive relations|Objectivity of hyperelastic relations]]
### [[/Rate form of hyperelastic laws|Rate form of hyperelastic relations]]
### [[/Nonlinear elasticity|Nonlinear elasticity]]
### Plasticity
### Viscoplasticity
### Viscoelasticity.
# Finite element formulation in three dimensions.
## [[/Updated Lagrangian formulation|Updated Lagrangian formulation]]
#Verification and Validation.

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===Assignments===

* [[/Homework1|Homework 1 Problem set]]
** [[/Homework1Solutions| Solutions]]
* [[/Homework2|Homework 2 Problem set]]
** [[/Homework2Solutions| Solutions]]
* [[/Homework3|Homework 3 Problem set]]
** [[/Homework3Solutions| Solutions]]
* [[/Homework4|Homework 4 Problem set]]
** [[/Homework4Hints| Hints]]
** [[/Homework4Solutions| Solutions]]
* [[/Homework5|Homework 5 Problem set]]
** [[/Homework5Solutions| Solutions]]
* [[/Homework6|Homework 6 Problem set]]
** [[/Homework6Hints| Hints]]
** [[/Homework6Solutions| Solutions]]
* [[/Homework7|Homework 7 Problem set]]
** [[/Homework7Hints| Hints]]
** [[/Homework7Solutions| Solutions]]
* [[/Homework8|Homework 8 Problem set]]
** [[/Homework8Solutions| Solutions]]
* [[/Homework9|Homework 9 Problem set]]
** [[/Homework9Solutions| Solutions]]
* [[/Homework10|Homework 10 Problem set]]
** [[/Homework10Solutions| Solutions]]
* [[/Homework11|Homework 11 Problem set]]
** [[/Homework11Solutions| Solutions]]

===Tests and Quizzes===
*[[/Quiz1 | Quiz 1]]
** [[/Quiz1Solutions| Solutions]]

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==Textbooks and References==
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===Textbooks===
* [http://books.google.com/books?id=UaPS5_5vdyUC&pg=PP1&ots=swwXWjzzU9&dq=An+Introduction+to+Nonlinear+Finite+Element+Analysis&sig=es9-LASwGNYTzePySvQsA6Jf3MU An Introduction to Nonlinear Finite Element Analysis] by J. N. Reddy, Oxford University Press, 2004, ISBN 019852529X.    
* [http://books.google.com/books?id=ydYEAAAACAAJ&dq=Nonlinear+Finite+Elements+for+Continua+and+Structures Nonlinear Finite Elements for Continua and Structures] by T. Belytschko, W. K. Liu, and B. Moran, John Wiley and Sons, 2000.
* [http://books.google.com/books?id=ftL2AJL8OPYC&pg=PP1&ots=32nr-XGXw1&dq=Computational+Inelasticity&sig=wJ_8caCfdM1k0BblUGbtNPuEKJI Computational Inelasticity] by J. C. Simo and T. J. R. Hughes, Springer, 1998.
* [http://books.google.com/books?id=yarmSc7ULRsC&pg=PP1&ots=48xUoRUEj7&dq=The+Finite+Element+Method:+Linear+Static+and+Dynamic+Finite+Element+Analysis&sig=QLHLMdZKlfAdhPkQRsKcLfL8jFk The Finite Element Method: Linear Static and Dynamic Finite Element Analysis] by T. J. R. Hughes, Dover Publications, 2000.

===References===
     
* '''Finite Element Analysis''':              
** K-J. Bathe (1996), [http://books.google.com/books?id=buMbAAAACAAJ&dq=Finite+Element+Procedures Finite Element Procedures], Prentice-Hall. ''Useful repository of information on nonlinear finite elements.''        
** J. N. Reddy (1993), [http://books.google.com/books?id=QD1xAQAACAAJ&dq=An+Introduction+to+the+Finite+Element+Method 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), [http://books.google.com/books?id=wmDQ_3qzvmIC&pg=PP19&ots=LWZ-8vetGO&dq=The+Finite+Element+Method:+Volume+2+Solid+Mechanics&sig=ywpfBmPXhKy1yDLhHA0kDrBVI18 The Finite Element Method: Volume 2 Solid Mechanics], Butterworth-Heinemann.  ''Another excellent repository of information of nonlinear finite elements geared toward the Civil Engineers.''           
* '''Continuum Mechanics''':               
** R. M. Brannon (2004), [http://www.mech.utah.edu/~brannon/Deformation.pdf Large Deformation Kinematics.] ''An excellent introduction to kinematics.''        
** R. M. Brannon (2004), [http://www.mech.utah.edu/~brannon/rotation.pdf Rotation.] ''Almost everything you ever wanted to know about rotations.''        
** A.J.M Spencer (2004), [http://books.google.com/books?id=AJdfQL0rgrgC&pg=PP1&ots=6bP8zrIt6z&dq=Continuum+Mechanics&sig=PQWngIZre1NjQRABOxvdZP80gXM Continuum Mechanics], Dover Publications. ''An excellent introduction to continuum mechanics.  You should own a copy for your personal library.''        
** L.E. Malvern (1969), [http://books.google.com/books?id=NbtfAAAACAAJ&dq=Introduction+to+the+Mechanics+of+a+Continuous+Medium Introduction to the Mechanics of a Continuous Medium], Prentice-Hall. '' A good reference for continuum mechanics.''          
* '''Mathematics''':      
** R. M. Brannon (2004), [http://www.mech.utah.edu/~brannon/Tensors.pdf 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), [http://www.me.unm.edu/~rmbrann/curvilinear.pdf  Curvilinear Coordinates.] '' Very useful if you wish to follow the literature on the nonlinear deformation of shells.      
** B. Daya Reddy (1998), [http://books.google.com/books?id=aAVzpzQdJc8C&pg=PP1&ots=ihO_OKxw64&dq=Introductory+Functional+Analysis:+With+applications+to+boundary+value+problems+and+finite+elements&sig=dc2lxvX4sYcRUKoPAnobzpSKl6Y 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), [http://books.google.com/books?id=diMcAdSClSoC&pg=PP1&ots=LcgQc5F_R-&dq=Partial+Differential+Equations+in+Mechanics&sig=jC5anykazA-czGeAWNAAGFMJ8n8 Partial Differential Equations in Mechanics 1,2.] Springer. ''Excellent introductory text on partial differential equations with engineers in mind.''

===Reading List=== 

*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.

===External links===
* [http://freshmeat.net/projects/gmsh/?branch_id=51601&release_id=272840 Gmsh] - GPL'd finite element analyzer



[[Category:Finite element analysis]]
[[Category:Computational solid mechanics]]
[[Category:Civil engineering]]
[[Category:Introductions]]
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