The University of North Carolina

1
Finite Elements

• A Theory-lite Intro
• Jeremy Wendt
• April 2005

2
Overview

• Numerical Integration
• Finite Differences
• Finite Elements
• Terminology
• 1D FEM
• 2D FEM 1D output
• 2D FEM 2D output
• Dynamic Problem

3
Numerical Integration

• Youve already seen simple integration schemes
  particle dynamics
• In that case, you are trying to solve for
  position given initial data, a set of forces and
  masses, etc.
• Simple Euler ? rectangle rule
• Midpoint Euler ? trapezoid rule
• Runge-Kutta 4 ? Simpsons rule

4
Numerical Integration II

• However, those techniques really only work for
  the simplest of problems
• Note that particles were only influenced by a
  fixed set of forces and not by other particles,
  etc.
• Rigid body dynamics is a step harder, but still
  quite an easy problem
• Calculus shows that you can consider it a
  particle at its center of mass for most
  calculations

5
Numerical Integration III

• Harder problems (where neighborhood must be
  considered, etc) require numerical solvers
• Harder Problems Heat Equation, Fluid dynamics,
  Non-rigid bodies, etc.
• Solver types Finite Difference, Finite Volume,
  Finite Element, Point based (Lagrangian), Hack
  (Spring-Mass), Extensive Measurement

6
Numerical Integration IV

• What I wont go over at all
• How to solve Systems of Equations
• Linear Algebra, MATH 191,192,221,222

7
Finite Differences

• This is probably the easiest solution technique
• Usually computed on a fixed width grid
• Approximate stencils on the grid with simple
  differences

8
Finite Differences (Example)

• How we can solve Heat Equation on fixed width
  grid
• Derive 2nd derivative stencil on white board
• Boundary Conditions
• See Numerical Simulation in Fluid Dynamics A
  Practical Introduction
• By Griebel, Dornseifer and Neunhoeffer

9
Finite Elements Terminology

• We want to solve the same problem on a
  non-regular grid
• FEM also has some different strengths than Finite
  Difference
• Node
• Element

10
Problem Statement 1D

• STRONG FORM
• Given f OMEGA ? R1 and constants g and h
• Find u OMEGA ? R1 such that
• uxx f 0
• ux(at 0) h
• u(at 1) g
• (Write this on the board)

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Problem Statement (cont)

• Weak Form (AKA Equation of Virtual Work)
• Derived by multiplying both sides by weighting
  function w and integrating both sides
• Remember Integration by parts?
• Integral(fgx) fg - Integral(gfx)

12
Galerkins Approximation

• Discretize the space
• Integrals ? sums
• Weighting Function Choices
• Constant (used by radiosity)
• Linear (used by Mueller, me (easier, faster))
• Non-Linear (I think this is what Fedkiw uses)

13
Definitions

• wh SUM(cANA)
• uh SUM(dANA) gNA
• cA, dA, g defined on the nodes
• NA , uh, wh defined in whole domain
• Shape Functions

14
Zoom in

• Weve been considering the whole domain, but the
  key to FEM is the element
• Zoom in to The Element Point of View

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Element Point of View

• Dont construct an NxN matrix, just a matrix for
  the nodes this element effects (in 1D its 2x2)
• Integral(NAxNBx)
• Reduces to widthslopeAslopeB for linear 1D

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Now for RHS

• We are stuck with an integral over varying data
  (instead of nice constants from before)
• Fortunately, these integrals can be solved by
  hand once and then input into the solver for all
  future problems (at least for linear shape
  functions)

17
Change of Variables

• Integral(f(y)dy)domain T Integral(f(PHI(x))PH
  Ixdx)domain S
• Write this on the board so it makes some sense

18
Creating Whole Picture

• We have solved these for each element
• Individually number each node
• Add values from element matrix to corresponding
  locations in global node matrix

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Example

• Draw even spaced nodes on board
• dx h
• Each element matrix (1/h)1 -1 -1 1
• RHS (h/6)2 1 1 2

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Show Demo

• 1D FEM

21
2D FEM 1D output

• Heat equation is an example here
• Linear shape functions on triangles ? Barycentric
  coordinates
• Kappa joins the party
• Integral(NAxKappaNBx)
• If we assume isotropic material, Kappa KI

22
2D Per-Element

• This now becomes a 3x3 matrix on both sides
• Anyone terribly interested in knowing what it
  is/how to get it?

23
Demo

• 2D FEM - 1D output

24
2D FEM 2D Out

• Deformation in 2D requires 2D output
• Need an x and y offset
• Doesnt handle rotation properly
• Each element now has a 6x6 matrix associated with
  it
• Equation becomes
• Integral(BATDBB) for Stiffness Matrix
• BA/B a matrix containing shape function
  derivatives
• D A matrix specific to deformation
• Contains Lame Parameters based on Youngs
  Modulus and Poissons Ratio (Anyone interested?)

25
Demo

• 2D Deformation

26
Dynamic Version

• The stiffness matrix (K) only gives you the final
  resting position
• Kuxx f
• Dynamics is a different equation
• Muxx Cux Ku f
• K is still stiffness matrix
• M diagonal mass matrix
• C aM bK (Rayliegh damping)

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Demo

• 2D Dynamic Deformation

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Questions