Finite Element Methods Finite Element Analysis

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Finite Element MethodsFinite Element Analysis
FEM or FEA

• Original lecture material by Major Pat Heffernan,
  CD, PhD, PEngDepartment of Mechanical
  EngineeringRMC
• Adapted for Mech422 Nov 2002

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References

• Cook, R.D., Malkus, D.S., Plesha, R.E., and Witt,
  R.J., 2002. Concepts and Applications of Finite
  Element Analysis, 4th Ed, Wiley, NY, USA.
• Huebner K.H., et al., 2001. The Finite Element
  Method For Engineers, 4th Ed, Wiley, NY, USA.

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Introduction

• What is the Finite Element Method (FEM) and how
  does it relate to CAD/CAM?
• What is CAD/CAM?
• Design
• Manufacturing
• Why use CAD/CAM?
• Design Manufacture New Products
• Less Time
• Lower Cost
• Enhance Overall Productivity

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MCAE
Mechanical Computer Aided Engineering

• Allows for concurrent engineering
• Analysis/Manufacturing Processes/Jigs can be done
  in parallel
• More design alternatives can be explored
• Reduces life-cycle costs
• Founded on Solid Modelling (not 2-D drawings)
• Solid Model used for
• Mass Property calculations
• Interference Studies
• Stress Analysis
• Manufacturing Planning
• If necessary, creation of conventional 2D drawings

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Product Cycle
Design and Manufacturing
with CAD/CAM overlaid
FEM
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Finite Element Method
A numerical analysis technique for obtaining
approximate solutions to engineering problems.

• Discrete elements approximating a continuum
• Used when the geometry is too complex for an
  analytical solution
• Continuum is divided into a finite number of
  elements
• Each element represents a portion of the
  structure
• Compatibility must be maintained at inter-element
  boundaries
• Equations are generated to represent the system
  and a numerical solution is determined
• Accuracy is largely dependent on mesh size

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Finite Element Method (contd)

• 2D or 3D Elements
• 2D or 3D Problems
• Ideal for complex problems that are difficult to
  solve analytically
• Computation time is a large concern
• Suitable for many applications (stress analysis,
  dynamic analysis, fluid flow, heat transferetc.)

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Approximation to Complex Shapes

• A fairly simple analytical solution but how much
  more complex of a geometry can be reasonably
  handled?
• What if we grooved the shaft to handle a rubber
  o-ring?

P
x,u
P
x,u
Complexities may be geometric or material in
nature.
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The Analysts Dilemma
Approximate the problem
Simplifying Assumptions
or
Approximate the solution
Numerical (FEM) Solution
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Numerical Solutions
(among many)

• Finite Difference
• an array of gridpoints
• write difference equations between points
• pointwise approximation

• Finite Elements
• an array of small, interconnected subregions
• system is an assemblage of finite elements
• piecewise approximation

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FEM A Brief History
(a very brief history)

• Mid-1800s basic concepts
• 1920, Maney (USA) and Ostenfeld (Denmark) basic
  ideas of truss and frame analysis using
  displacements as unknowns.
• 1932, Cross Moment Distribution Method.
• 1950s, Digital Computer Suited matrix format
  methods. Allowed solution of a large number of
  simultaneos equations.1960, Clough - first paper
  published using term Finite Element Method
• 1965, First conference on finite elements.

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FEM A Method

• Steps
• Discretize the Continuum
• Select Interpolation Functions
• Find the Element Properties
• Assemble the Element Properties to Obtain the
  System of Equations
• Impose the Boundary Conditions
• Solve the System of Equations
• Make Additional Computation if Desired

Already donefor most commonelements
Formulate a solution for the element and then
assemble it to find the global stiffness
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General-Purpose Programs

• Products ANSYS, ADINA, SAP, EASE, I-DEAS,
  Unigraphics
• Modular Construction
• Pre-processor definition of geometry, materials,
  loading, boundary conditions.
• Processor construction and solution of
  mathematical model.
• Post-processor display of predicted
  displacements and stresses.
• Between steps 1 and 2, the program will access a
  library of elements to generate mathematical
  models for the elements and applied loadings.

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Types of Elements

• General-purpose programs contain huge libraries
  of elements. Elements can be derived for special
  functions. Some typical elements incl
• Frame Elements
• Basic Plane Stress Elements
• Plate Bending Element
• Thin Shell Element
• Axisymmetric Element

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SAP 2000 Student

• 3D models
• 2D elements
• Static Analysis
• Great Learning Tool
• In Real Life mostly Civil applications.

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SDRC I-DEAS
(or any real life CAD/CAM Integrated Package)

• Integrated Design Engineering Analysis Software
• US 5k/yr academic/US 40k/yr commercial
• To facilitate concurrent engineering
• focus on Mechanical Engineering Product
• Multiple Modules
• Design (Solid Modelling)
• Drafting (Seen in 2nd Yr)
• Simulation (FEM)
• Manufacturing

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Displacement Stress Analysis

• Finite Element Modelling (Simulation)
• Stress/Deflections given loads/constraints
• Also Heat Transfer and Potential Flow
• Optimization sub-module