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# Finite Elements - PowerPoint PPT Presentation

Finite Elements. A Theory-lite Intro Jeremy Wendt April 2005. Overview. Numerical Integration Finite Differences Finite Elements Terminology 1D FEM 2D FEM 1D output 2D FEM 2D output Dynamic Problem. Numerical Integration.

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### Finite Elements

A Theory-lite Intro

Jeremy Wendt

April 2005

The University of North Carolina – Chapel Hill

COMP259-2005

• 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

• 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

• 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

• 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

• What I won’t go over at all:

• How to solve Systems of Equations

• Linear Algebra, MATH 191,192,221,222

The University of North Carolina – Chapel Hill

COMP259-2005

• 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

• 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

• 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

• 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

• 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)

The University of North Carolina – Chapel Hill

COMP259-2005

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 1D FEM

The University of North Carolina – Chapel Hill

COMP259-2005

• 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

• 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 - 1D output

The University of North Carolina – Chapel Hill

COMP259-2005

• Deformation in 2D requires 2D output

• Need an x and y offset

• Doesn’t handle rotation properly

• 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

• 2D Deformation

The University of North Carolina – Chapel Hill

COMP259-2005

• 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)

The University of North Carolina – Chapel Hill

COMP259-2005

• 2D Dynamic Deformation

The University of North Carolina – Chapel Hill

COMP259-2005

• 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

The University of North Carolina – Chapel Hill

COMP259-2005