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### Finite Elements

A Theory-lite Intro

Jeremy Wendt

April 2005

The University of North Carolina – Chapel Hill

COMP259-2005

Overview

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

The University of North Carolina – Chapel Hill

COMP259-2005

Numerical Integration

- You’ve 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 Simpson’s rule

The University of North Carolina – Chapel Hill

COMP259-2005

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 it’s center of mass for most calculations

The University of North Carolina – Chapel Hill

COMP259-2005

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

The University of North Carolina – Chapel Hill

COMP259-2005

Numerical Integration IV

- What I won’t go over at all:
- How to solve Systems of Equations
- Linear Algebra, MATH 191,192,221,222

- How to solve Systems of Equations

The University of North Carolina – Chapel Hill

COMP259-2005

Finite Differences

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

The University of North Carolina – Chapel Hill

COMP259-2005

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

The University of North Carolina – Chapel Hill

COMP259-2005

Finite Elements Terminology

- We want to solve the same problem on a non-regular grid
- Draw Grid on Board

- Node
- Element

The University of North Carolina – Chapel Hill

COMP259-2005

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)

- u – unknown values
- f – known values “forces”

The University of North Carolina – Chapel Hill

COMP259-2005

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(f*gx) = f*g - Integral(g*fx)

- Derived by multiplying both sides by weighting function w and integrating both sides

The University of North Carolina – Chapel Hill

COMP259-2005

Galerkin’s 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)

The University of North Carolina – Chapel Hill

COMP259-2005

Definitions

- wh = SUM(cA*NA)
- uh = SUM(dA*NA) + g*NA
- cA, dA, g – defined on the nodes
- cA = 1 (I think)
- dA = value of unknown at node
- g = bdry condition

- NA , uh, wh – defined in whole domain
- NA - Shape Functions
- wh – weighting function

The University of North Carolina – Chapel Hill

COMP259-2005

Zoom in

- We’ve been considering the whole domain, but the key to FEM is the element
- Zoom in to “The Element Point of View”

The University of North Carolina – Chapel Hill

COMP259-2005

Element Point of View

- Don’t construct an NxN matrix, just a matrix for the nodes this element effects (in 1D it’s 2x2)
- Integral(NAx*NBx)
- Reduces to width*slopeA*slopeB for linear 1D

The University of North Carolina – Chapel Hill

COMP259-2005

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)

The University of North Carolina – Chapel Hill

COMP259-2005

Change of Variables

- Integral(f(y)dy)domain = T = Integral(f(PHI(x))*PHIx*dx)domain = S
- Write this on the board so it makes some sense

The University of North Carolina – Chapel Hill

COMP259-2005

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

The University of North Carolina – Chapel Hill

COMP259-2005

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]]

The University of North Carolina – Chapel Hill

COMP259-2005

2D FEM 1D output

- Heat equation is an example here
- Linear shape functions on triangles Barycentric coordinates
- Kappa joins the party
- Integral(NAx*Kappa*NBx)
- If we assume isotropic material, Kappa = K*I

The University of North Carolina – Chapel Hill

COMP259-2005

2D Per-Element

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

The University of North Carolina – Chapel Hill

COMP259-2005

2D FEM – 2D Out

- Deformation in 2D requires 2D output
- Need an x and y offset
- Doesn’t handle rotation properly

- Need an x and y offset
- Each element now has a 6x6 matrix associated with it
- Equation becomes
- Integral(BAT*D*BB) for Stiffness Matrix
- BA/B – a matrix containing shape function derivatives
- D – A matrix specific to deformation
- Contains Lame` Parameters based on Young’s Modulus and Poisson’s Ratio (Anyone interested?)

The University of North Carolina – Chapel Hill

COMP259-2005

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)

- Muxx + Cux + Ku = f

The University of North Carolina – Chapel Hill

COMP259-2005

Good Sources

- Papers with a graphics slant:
- Matthias Mueller: http://www.matthiasmueller.info/
- Ron Fedkiw (et.al): http://graphics.stanford.edu/~fedkiw/

- Books on FEM and Numerical Methods:
- Finite Element Method: Linear Static and Dynamic Finite Element Analysis by Thomas J.R. Hughes
- Numerical Simulation in Fluid Dynamics by Griebel, Dornseifer, Neunhoeffer
- Computational Fluid Dynamics by T.J. Chung

- Classes on PDEs and Numerical Methods/Solutions:
- Math 191, 192 (I took from David Adalsteinsson) , 221, 222 (both from Michael Minion)

The University of North Carolina – Chapel Hill

COMP259-2005

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