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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

Finite Elements

A Theory-lite Intro

Jeremy Wendt

April 2005

The University of North Carolina – Chapel Hill

COMP259-2005


Overview
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
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
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
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
Numerical Integration IV

  • 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


Finite differences
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
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
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

The University of North Carolina – Chapel Hill

COMP259-2005


Problem statement 1d
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)

The University of North Carolina – Chapel Hill

COMP259-2005


Problem statement cont
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)

The University of North Carolina – Chapel Hill

COMP259-2005


Galerkin s approximation
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
Definitions

  • wh = SUM(cA*NA)

  • uh = SUM(dA*NA) + g*NA

  • cA, dA, g – defined on the nodes

  • NA , uh, wh – defined in whole domain

  • Shape Functions

The University of North Carolina – Chapel Hill

COMP259-2005


Zoom in
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
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
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
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
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
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


Show demo
Show Demo

  • 1D FEM

The University of North Carolina – Chapel Hill

COMP259-2005


2d fem 1d output
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
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


Demo

  • 2D FEM - 1D output

The University of North Carolina – Chapel Hill

COMP259-2005


2d fem 2d out
2D FEM – 2D Out

  • 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


Demo

  • 2D Deformation

The University of North Carolina – Chapel Hill

COMP259-2005


Dynamic version
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)

The University of North Carolina – Chapel Hill

COMP259-2005


Demo

  • 2D Dynamic Deformation

The University of North Carolina – Chapel Hill

COMP259-2005


Questions
Questions

The University of North Carolina – Chapel Hill

COMP259-2005


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