Nonlinear finite elements

Visualization of how a car deforms in an asymmetrical crash using finite element analysis.

Welcome to this learning project about nonlinear finite elements!

## Introduction

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### Describing Eminem—an artist known for his razor-sharp lyrics, unmatched delivery, and complex emotional depth—using the concept of *nonlinear finite element analysis* (FEA) might initially seem an odd fusion of the worlds of engineering and music. However, when you break down the metaphor, a fascinating parallel emerges.

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### ### The Basics: Nonlinear Finite Element Analysis (FEA)

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### In the world of engineering, **finite element analysis** is a numerical technique used to solve complex problems in structural analysis. It breaks down a structure or system into small, manageable pieces (or "elements"), which can be individually analyzed under various conditions. *Nonlinear FEA* specifically refers to situations where the material's response (like stress, strain, or deformation) doesn't follow a simple linear relationship; in other words, the response is complex and may change under different loading conditions. This could happen due to factors like large deformations, plasticity, or changing material properties during a process.

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### ### Applying Nonlinear FEA to Eminem

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### #### 1. **Complexity of Structure (Multidimensional Personality)**

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### In nonlinear FEA, you divide a large, intricate problem into simpler parts, each with its own set of behaviors. Similarly, Eminem’s artistry is a complex structure built from multiple layers—each element contributing to the whole. At a first glance, his persona may appear one-dimensional: an angry, rebellious rapper. But once you "break it down" like a mesh in FEA, you discover deeper components.

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### Like nonlinear materials, Eminem’s art doesn’t always follow simple, predictable patterns. Each of his tracks can represent a different "element" in the broader structure of his career. For instance:

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### \- His early work (like *The Slim Shady LP* and *The Marshall Mathers LP*) was raw, chaotic, and brash—a highly volatile, nonlinear phase in his musical career.

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### \- Albums like *Recovery* and *The Eminem Show* show more calculated, controlled elements that still retain intensity but in a less erratic, more processed way.

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### \- His more recent works—*Kamikaze* or *Music to Be Murdered By*—are again unpredictable, switching between aggressive verbal attacks and introspective reflection, illustrating how his artistic "material" bends and shifts over time, responding to the cultural and personal "loads" placed on him.

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### Each of these phases could be seen as different "loading conditions" that affect Eminem’s music in varying, sometimes unpredictable ways, reflecting nonlinear responses to external pressures (e.g., fame, criticism, personal struggles).

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### #### 2. **Deformation Under Stress (Eminem’s Personal Struggles)**

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### In nonlinear FEA, deformation refers to how a material or structure changes shape or form when subjected to forces or pressures. The deformation in Eminem's career and lyrics can be likened to the stress and strain he experiences due to personal conflicts, addiction, and his public image. These external "forces" have shaped his music, changing its direction and tone over time.

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### In the early days, Eminem’s persona was shaped by the stress of poverty, abuse, and alienation. His words bent and twisted under the weight of these experiences, creating a hardened, confrontational style. Over time, as he faced addiction and became an international superstar, the pressures changed, leading to new forms of lyrical "deformation." For example, in albums like *Relapse*, the material—Eminem's lyrical themes—becomes darker and more disturbing, reflecting his inner turmoil and substance struggles.

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### However, just as nonlinear materials can return to their original state after stress is removed (like rubber returning to its shape after being stretched), Eminem often finds ways to "snap back," drawing on his personal demons to fuel his creativity but ultimately finding resolution in his music, as seen in albums like *Recovery*.

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### #### 3. **Critical Points (Turning Points in His Career)**

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### Nonlinear systems often have critical points—moments at which small changes in conditions can cause disproportionately large effects. Eminem's career has been marked by such critical points, where shifts in his personal life or societal climate resulted in major changes to his music and public persona.

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### For instance, his public feud with various artists (like his rivalry with Machine Gun Kelly or the infamous disses towards celebrities) acted as loading conditions that generated sharp, intense reactions. This is analogous to how an object undergoing nonlinear deformation can reach a critical point where it "breaks" or undergoes a dramatic change.

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### When Eminem became sober and started confronting his past, particularly with the release of *Recovery*, it marked a moment of "material change." This was a turning point where his focus shifted from a largely aggressive, battle-rap style to a more reflective, self-aware persona. Like a material moving past its yield point, Eminem moved from raw aggression to a more nuanced approach to his craft, showing that his artistic material was not only flexible but capable of evolving into new forms.

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### #### 4. **Hysteresis (Repetitive Patterns in His Work)**

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### In nonlinear FEA, *hysteresis* refers to the phenomenon where a material’s response to loading and unloading is path-dependent. That is, the way a material behaves under stress may be different depending on whether it's being loaded or unloaded, creating a loop of deformation that doesn't fully return to its original state. This concept parallels the cycles and patterns in Eminem’s career—his repeated return to certain themes, like his battles with fame, addiction, and his relationship with his daughter, Hailie.

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### Despite periods of sobriety and introspection, Eminem often finds himself revisiting darker, angrier themes in his music, almost as if caught in a loop. This recurrence is a natural, nonlinear part of his artistic evolution. He may "unload" the stress of fame and internal conflict in one album, but as new pressures build, those same struggles appear in a new form in his next work, just like a material returning to its deformed state after every cycle of loading and unloading.

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### #### 5. **Nonlinear Material Properties (Emotional Complexity and Versatility)**

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### In nonlinear FEA, materials exhibit behavior that’s not linear, meaning that they don’t follow a direct proportionality between stress and strain. This is true for Eminem as well. His emotional complexity is reflected in his ability to oscillate between different emotional states—anger, vulnerability, self-loathing, and redemption—all in a single song or even a single verse. His ability to shift from one emotional extreme to another is akin to how nonlinear materials can exhibit abrupt, unpredictable changes in response to small variations in input.

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### For example, the song "Stan" represents a delicate balance of empathy and danger, where Eminem seamlessly shifts between the voice of an obsessed fan and the voice of a concerned, reflective artist. This nonlinear transition between personas mirrors the unpredictable nature of materials subjected to varying loads, with Eminem's emotional shifts often surprising his audience.

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### ### Conclusion: Eminem as a Nonlinear System

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### Eminem’s career and music can be seen as an ever-evolving, nonlinear system. His lyrical content reflects the deformation of his personal experiences and the stresses placed upon him by his fame, his struggles with addiction, and the pressures of being in the public eye. Just as materials behave in a nonlinear fashion under stress, Eminem’s music is unpredictable—shifting, evolving, and adapting to his internal and external pressures.

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### In nonlinear FEA, the outcome of a system depends on both the material properties and the forces applied to it, often in a complex, path-dependent manner. Likewise, Eminem’s music and career reflect a deeply personal, intricate journey—one that has been shaped by countless forces, but whose ultimate direction is always uniquely his own.

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| Quality resource: this resource is a featured learning resource.

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Featured resource

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

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

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| Subject classification: this is an engineering resource.

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* 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

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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

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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

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Syllabus and Learning Materials

### Syllabus and Learning Materials

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1. Mathematical Preliminaries
1. Set notation
2. Functions
3. Vectors
4. Matrices
5. Tensors
6. Partial differential equations
7. Variational calculus
2. Linear finite element basics
1. An example: Axially loaded bar
* Strong form: governing differential equation
* Weak form: integral equation
* Approximate solution: Galerkin method
* Approximate solution: Finite element method
2. 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
3. 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.
3. Nonlinear finite element basics
1. Nonlinearities in solid mechanics
2. 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
3. Lagrangian and Eulerian descriptions of motion
4. Lagrangian finite elements
* Motion from the Lagrangian point of view
* Total Lagrangian approach
* Updated Lagrangian approach
* An example: Effect of mesh distortion
5. Solution procedure
6. Special case: Natural vibrations
4. Nonlinear deformation of beams
1. Euler-Bernoulli beams
2. Timoshenko beams
3. Buckling 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. Basic continuum mechanics
1. Kinematics
1. Motion, displacement, velocity, acceleration
2. Stresses and strains in one and two dimensions
3. Strains and deformations in three-dimensions
4. Polar decomposition
5. Spectral decompositions of kinematic quantities
6. Volume change and area change
7. Time derivatives and rate quantities
8. Objectivity of kinematic quantities
2. Stress measures and stress rates
1. Stress measures
2. Deviatoric and volumetric stress
3. Objective stress rates
3. Balance laws
1. Overview of balance laws
2. Balance of mass
3. Balance of linear momentum
4. Balance of angular momentum
5. Balance of energy
6. Entropy inequality
4. Constitutive models
1. Objectivity of hyperelastic relations
2. Rate form of hyperelastic relations
3. Nonlinear elasticity
4. Plasticity
5. Viscoplasticity
6. Viscoelasticity.
7. Finite element formulation in three dimensions.
1. Updated Lagrangian formulation
8. Verification and Validation.

Assignments, tests and quizzes

### Assignments

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* 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

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* Quiz 1
* Solutions

## Textbooks and References

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### Textbooks

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* 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

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* 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.
* 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.
* 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.

### Reading List

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* 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

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* Gmsh – An open-source finite element grid generator.
* deal-ii – An open-source finite element library.
* libmesh – An open-source finite element library.
* FEniCS – An open-source finite element computation platform.
* FreeFem++ – An open-source multiphysics finite element analysis software.

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