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

• 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

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

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

Numerical Integration IV

• What I won’t go over at all:– How to solve Systems of Equations

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

Finite Differences

• This is probably the easiest solution technique

• Usually computed on a fixed width grid

• Approximate stencils on the grid with simple differences

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

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

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)

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)

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)

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

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”

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

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)

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

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

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]]

Show Demo

• 1D FEM

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

2D Per-Element

• This now becomes a 3x3 matrix on both sides– Anyone terribly interested in knowing what it

is/how to get it?

Demo

• 2D FEM - 1D output

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(BA

T*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?)

Demo

• 2D Deformation

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)

Demo

• 2D Dynamic Deformation

Questions