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]

Finite element analysis (FEA) is a computer simulation technique used in engineering analysis. It uses a numerical technique called the finite element method (FEM). 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]

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Content summary[edit]

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]

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]

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Syllabus and Learning Materials

Syllabus and Learning Materials[edit]

  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. An example: Nonlinear deformation of an axially loaded 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.
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Assignments, tests and quizzes

Textbooks and References[edit]

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Textbooks[edit]

References[edit]

Reading List[edit]

  • 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]

  • Gmsh - GPL'd finite element analyzer
  • RecurDyn Full Flexible - RecurDyn’s true Finite Element Method combinded with it´s Multi Body Dynamics - FEMBD mechanical system simulation returns more precise dynamic motion results including stress analysis in one single simulation step. It also features flexible body contacts and non-linear deformations.

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