Course Description:

The finite element method is a widely used numerical technique that is commonly applied to solve a large class of engineering problems. The two part course is specially designed to best equip engineers to maximize their understanding and application of this great tool in various areas of engineering. In addition, this set of courses presents the fundamentals of the finite element method and modeling techniques for structural analysis applications.
Part 1 of this course provides a robust understanding of the finite element method and assures that analyses are performed correctly, confidently and efficiently. Theory of the finite element method and applications, using several approaches is covered in Part 1. The direct method, the Principle of Minimum Potential Energy, various methods of Weighted Residual and a Variational approach are introduced. The formulations of several element types are discussed. The isoparametric formulation of the elements stiffness matrices and numerical integration are covered. This demonstrates how software solves large scale problems and provide an understanding which aids with setting up analyses and post-processing results, and sets the basis for more advanced uses, which are covered in Part 2. An introduction to non-linear finite element analyses and modal analyses is provided at the end of Part 1. The concepts covered in this course are enforced with practical problems and examples, and discussion on common errors and typical use of the finite element method in the aerospace industry. Lastly, the concepts of verification and validation, their requirements and methods are presented throughout the course.
Part 2 of the course builds upon the fundamentals covered in Part 1 and presents a well-balanced discussion on the theory and application of finite element modeling techniques for an array of disciplines. In Part 2 non-linear analyses, structural dynamics, stability, contacts, analysis of composite materials, and use of the finite element modeling in damage tolerance and fracture mechanics are discussed in greater detail.

Course Includes:

  • A course notebook prepared by your instructors
  • Supplemental material to enhance the subject matter
  • Refreshments served each morning and afternoon

Course Textbook:

Logan, Daryl L., A First Course in the Finite Element Method, Cengage Learning- Global Engineering Publisher, Fifth Edition, 2012

Course Outline:

Day 1

  • Introduction (1 Hour)
    • Why FEA?
    • No fear FEA – When done right!
    • History
    • Applications – various examples including mechanical, thermal, fluid, electromagnetic, and bio engineering
    • Need for verification and validation of the results
    • Introduction to types of errors with examples
  • Review of Mathematical Fundamentals – Linear Algebra (1 Hour)
    • Vectors and matrices
    • Matrix equations and operations
    • Eigenvalues and eigenvectors
  • Introduction to Finite Element Method (1.5 Hours)
    • Illustrations of nodes, degrees of freedom (DOF) and element (type, order)
    • Numbering convention and notations
    • Element to global stiffness matrix assembly
    • Boundary Conditions enforcement and model solution
    • Interpretation of results
  • Finite Element Analysis Using Direct Method (1.5 Hours)
    • Applying direct method to derive the stiffness matrix for a bar/rod element
      • Shape functions
    • Solving a multi uniaxial-bar (1-D truss) problem
  • Structural Analysis (2 Hours)
    • Design requirements
    • Application of principles of mechanics
    • Structural behavior subject to forces
    • Equilibrium of forces
    • Stress and strain analysis

Day 2

  • Examples (2 Hours)
    1. Solving a case with prescribed forces
    2. Solving a case with prescribed displacements
    3. Assignment
    • Element rotations – Add DOF
    • 2-D Truss problem – Show the importance of putting matrices together
    • Examples – Classical and FE solutions of a bridge
  • Finite Element Analysis Using Minimum Potential Energy (2 Hours)
    • Introduction of the concept
    • Derivation of the stiffness matrix for a bar/rod element
    • Example – show similarity with the direct method
  • Introduction to Beam Element (3 Hours)
    • Bernoulli and Timoshenko beams
      • Formulations
      • Shape functions
    • Extending the concept of minimum potential energy to a beam element
    • Example – Assembly of beam element stiffness matrix

Day 3

  • Finite Element Analysis Using Variational Methods of Weighted Residuals and Galerkin’s Method (3.5 Hours)
    • Introduction of the concept
    • Applying the concepts to derive the stiffness matrix for a bar/rod element
    • Example – Comparison to direct and minimum potential energy methods
    • Applying the concepts to derive the stiffness matrix for a beam element
    • Example – Comparison to direct and minimum potential energy methods
    • Class Exercise – Solve a spring problem in tension with MPE using classical analysis and FEA
  • Introduction of 2-D Solid Elements (3.5 hours)
    • Triangle element
      • Constant strain elements
      • Element limitations
    • Quadrilateral element through various examples
    • Applications: Use FEMAP to solve a problem using 3 element types: constant strain, linear strain, and equilateral
    • Problems with tri-element

Day 4

  • Isoperimetric Formulation (2.5 Hours)
    • Mapping of coordinate systems from physical to natural and vice versa
    • Numerical integration (integration points) – Midpoint Rule, Simpson Rules, Newton Cotes, Gaussian Quadrature, and Gaussian Points
    • Example 1 – 4 element model with loads applied, solve natural vs. physical coordinates
    • Example 2 – Calculate shape functions for a distorted quadrilateral element with straight sides
    • Example 3 – Calculate shape functions for a distorted quadrilateral element with curved sides
  • Introduction to 3-D Solid Elements (1.5 Hours)
    • Show element types and formulations and shape functions
    • Solve example problems
    • How to go from CAD to FEM
    • Class Exercise
  • Introduction to Shell Elements (2 Hours)
    • Introduce and discuss element types and formulations and shape functions
      • Kirchhoff and Mindlin elements
    • Class Exercise
  • Introduction to Higher Order Elements (1 Hour)

Day 5

  • Practical Considerations Using Multi-Element Static Problem (1.5 Hours)
    • Applications – combination of road/beam and plates/shells
    • Use of symmetry to reduce computational cost
      • Reflective vs period symmetry
      • Examples
      • Floor grid models
      • Full fuselage model
      • Application for loads analysis
      • Introduction to sub-modeling – further discussion in Part 2’s course
    • Example – stress and failure analyses
  • Introduction to Non-linear Analyses (1 Hour)
    • Sources of non-linearity
    • Quick mathematical background;
    • FEMAP demonstration
  • Modal Analysis (1.5 Hours)
    • Introduction to Natural Frequency
      • Importance of natural frequency
      • Computation techniques using FEM
        • Instances of real life events
        • Example – Vibration of beams
    • Introduction to buckling
  • Verification and validation of results (3 Hours)
    • Requirement
    • Means
    • Definitions from NAFEMS
  • Closing (0.5 Hour)