MCE 466 Introduction to Finite Element Methods

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MCE 466 Introduction to Finite Element Methods

• Instructor David G. Taggart
• Dept. of Mechanical Engineering
• University of Rhode Island

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Introduction

• Finite Element Method a numerical method for
  obtaining approximate solutions to field problems
• Field Problems A problem in which the behavior
  in a well defined region, R, is described by a
  governing set of differential equations with
  certain quantities prescribed on the boundary, C.
  (Also called a boundary value problem BVP)

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Introduction (cont.)

• Boundary Value Problem
• Examples
• Stress analysis (includes system of springs, bars
  under axial loads, trusses, beams, frames, 2-D
  3-D continua, plate bending)
• Stability analysis (buckling), structural
  dynamics, plasticity, viscoelasticity
• Heat transfer
• Fluid flow
• General PDE problems

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Finite Element Approach

• Divide the region into may discrete sub-regions
  (discretization) of finite size (elements).
  Select certain points (nodes) at which the
  solution is desired. Assume a functional form
  for the solution between the nodes (interpolation)

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Important points to remember

• Finite element analyses provide approximate
  solutions in nearly all cases
• Results can be very accurate or totally
  meaningless (easy to make mistakes)
• Method is very susceptible to user error
• Stress averaged results may conceal inaccuracies
  in the results
• Modern commercial codes make it very easy to get
  answers
• Responsibility of user
• Anticipate results (perform approximate
  calculations by hand)
• Explain discrepancies
• Be aware of limitations

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MCE 466 Course Structure

• Appropriate balance of FEA theory and use of
  commercial codes (primarily Abaqus)
• Text problems hand calculations with support of
  Matlab as needed
• Several Abaqus - based computer assignments
• Project individual or teams of 2

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History of Finite Elements

• 1943 R. Courant (mathematician) proposed a
  torsion solution using triangular subregions (not
  implemented since no computers were available)
• Early 50s digital computers which solved large
  systems of equations became available. Aerospace
  companies bega solving structures problems
  (mostly unpublished)
• 1960 term Finite Element Method first coined
• Early 60s new elements formulated, method
  became accepted
• Mid 60s Applied to heat transfer and fluid
  flow problems
• Late 60s, early 70s large general purpose
  programs became commercially available (NASTRAN,
  ANSYS, SAP)
• Early 80s Interactive graphics made using FEA
  easier
• Late 80s Inexpensive PC versions became
  available
• 90s and beyond Widespread application in
  numerous industries, Integration with CAD
  packages.
• Most recent emphasis multiphysics event
  simulation

"Concepts and Applications of Finite Element
Analysis," by R. D. Cook, D. Malkus and M. Plesha
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Review Matrix Algebra (see Appendix A, B)

• We will see that the finite element method
  requires solving large systems of linear
  equations using matrix algebra tools.
• First, review matrix multiplication
• In general

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Review Matrix Algebra (see Appendix A, B)

• To review some basic matrix algebra, consider the
  following system of equations
• or

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Matrix Algebra (cont.)

• Matrix form

Matlab output a 2 1 1 4 b
6 17 determinant 7 solution
1.0000 4.0000
Matlab code a2 11 4 b617 determinantdet(
a) solutiona\b
Note a ? 0
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Matrix Algebra (cont.)

• Now consider a different system of equations

or
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Matrix Algebra (cont.)
Matrix form
Matlab code
Matlab output
a4 22 1 b166 determinantdet(a) solution
a\b
a 4 2 2 1 b 16
6 determinant 0 Warning Matrix is
singular to working precision. solution
Inf -Inf
Note a 0
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Matrix Algebra (cont.)

• System of three equations (Matlab format)
• Code
• a-1 1 23 -1 13 3 1
• b262
• solutiona\b
• Output
• a
• -1 1 2
• 3 -1 1
• 3 3 1
• b
• 2
• 6
• 2

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Matrix algebra applied to FEA

• For displacement based stress analysis

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Other matrix terminology

• Banded matrix
• If all non-zero terms are contained within a band
  along the diagonal, the matrix is said to be
  banded
• Sparse matrix
• If a matrix has relatively few non-zero terms (as
  is common in FEA), the matrix is said to be
  sparse
• Singular matrix
• If the determinant of the matrix equals zero, the
  matrix is said to be singular. As we saw, if A
  is singular, then the system of equations
  Axb has no unique solution.

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Other matrix terminology (cont.)

• Transpose
• The transpose of a matrix is found by
  interchanging rows and columns, e.g.
• Transpose of a product

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Steps in the Finite Element Method

• Discretize the region and select element type
• Select a displacement function
• Define the strain/displacement and stress/strain
  relations
• Derive the element equations
• Direct Stiffness Method
• Energy Methods
• Method of Weighted Residuals (Galerkins method)
• Assemble global equations and impose boundary
  conditions
• Solve for unknown nodal displacements
• Solve for element strains and stresses
• Interpret results

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Demo of a Commercial FEA Package

• Determine maximum stress for the following
  problem