Nonlinear finite elements

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

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]

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.

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]

Syllabus and Learning Materials

Syllabus and Learning Materials[edit | edit source]

  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
    2. More examples: Some model problems
    3. A time-dependent problem: the heat equation
  3. Nonlinear finite element basics
    1. Nonlinearities in solid mechanics
    2. Nonlinear deformation of an axial bar
    3. Lagrangian and Eulerian descriptions of motion
    4. Lagrangian finite elements
    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

Textbooks and References[edit | edit source]

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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]

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.

External links[edit | edit source]

  • 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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