Lecture26
May 05, 2008
Lecture 26: Putting it all together #5...
Galerkin Finite Element Methods for ODE BVPs
Outline
1) Galerkin Finite Elements: Review
2) A specific Example: Piecewise linear elements
3) Computational Issues and Algorithms
A) Calculation of integrals:
Basis functions vs Element View
B) Element Matrices
C) Global Assembly
4) Matlab Demo
5) The End (of the beginning)
Galerkin Finite Elements
Example problem: Find best fit piecewise linear function ũ(x), that
satisfies -u''+u = f(x) (u(0)=α, u(L)=β) in a least-squares sense
ui-1
u
xi-1
i
xi
x
u
i+1
xi+1
Lecture26
May 05, 2008
Galerkin Finite Elements
Basis Function view: linear combination of "hat functions"
ui-1
ui
ui+1
xi
xi+1
1
0
xi-1
x
Galerkin Finite Elements
Element view: union of piecewise linear
elements (with local element basis functions)
ui-1
ui
ui+1
xi
xi+1
1
0
x
i-1
Element i
hi
Lecture26
May 05, 2008
Galerkin Finite Elements
Galerkin FEM as a least squares problem:
The Continuous problem: find u(x) that satisfies
The Discrete problem: find ũ(x) such that the residual r(x) is
orthogonal to the basis functions
Galerkin Finite Elements
Evaluating the integrals: 3 critical integrals to evaluate
1) Components of the stiffness matrix Kij
2) Components of the Mass matrix Mij
3) Components of the force vector f
i
Lecture26
May 05, 2008
Galerkin Finite Elements
Putting it together, basis function view vs "element view": the Mass matrix
1
0
xi-1
xi
xi+1
for basis function ϕi want:
Galerkin Finite Elements
Putting it together, basis function view vs "element view": the local element
Mass matrix Me: consider the contributions from one element i
N1(t)
Element basis functions: N1(t), N2(t)
N2(t)
xi
-1
t
xi+1
1
Need _____ integrals for element mass matrix
Lecture26
May 05, 2008
Galerkin Finite Elements
Putting it together, basis function view vs "element view": the local element
Mass matrix Me: consider the contributions from one element i
N1(t)
Element Mass Matrix:
N2(t)
xi
-1
t
xi+1
1
Galerkin Finite Elements
Putting it together, basis function view vs "element view":
Element Stiffness Matrix:
N (t)
1
N2(t)
N'2(t)
N'1(t)
Lecture26
May 05, 2008
Galerkin Finite Elements
Putting it together, basis function view vs "element view":
Element Force vector:
N1(t)
N2(t)
-1
t
1
Lecture26
May 05, 2008
Galerkin Finite Elements: Matlab
function [Ke,Me] = getElementMatrix(xCoords)
% GETELEMENTMATRIX - returns element matrices for an individual element
%
% [Ke,Me] = getElementMatrix(xCoords)
%
% xCoords:- coordinates of 1D element [ xMin xMax ]
% Ke: - element stiffness matrix (assuming linear elements)
% Me: - element mass matrix (assuming linear hat elements)
%
N1 = @(t) (1-t)/2; % linear hat function
N2 = @(t) (1+t)/2;
N1p = @(t) -.5*ones(size(t)); % derivative dN1/dt (not really a function)
N2p = @(t) .5*ones(size(t));
% quadrature points and weights (2-point Gauss-legendre)
tQ = [ -1 1 ]'/sqrt(3);
wQ = [ 1 1 ];
h = xCoords(2)-xCoords(1); %element width
Ke = zeros(2);
Me = Ke;
%stiffness matrix
Ke(1,1) = wQ*(N1p(tQ).*N1p(tQ));
Ke(1,2) = wQ*(N1p(tQ).*N2p(tQ));
Ke(2,1) = Ke(1,2);
Ke(2,2) = wQ*(N2p(tQ).*N2p(tQ));
Ke = Ke*2/h;
%Mass matrix
Me(1,1) = wQ*(N1(tQ).*N1(tQ));
Me(1,2) = wQ*(N1(tQ).*N2(tQ));
Me(2,1) = Me(1,2);
Me(2,2) = wQ*(N2(tQ).*N2(tQ));
Me = Me*h/2;
Galerkin Finite Elements: Matlab
function [fe] = getElementForceVector(func,xCoords)
% getElementForceVector - returns element force vector for an individual element
%
% [fe] = getElementForceVector(func,xCoords)
%
% func:-function handle to RHS f(x)
% xCoords:- coordinates of 1D element [ xMin xMax ]
%
% fe: - element force vector
%
h = xCoords(2)-xCoords(1); %element width
N1 = @(t) (1-t)/2; % linear hat function
N2 = @(t) (1+t)/2;
x = @(t) xCoords(1) + h*(t + 1)/2; % affine transformation
% quadrature points and weights (2-point Gauss-legendre)
tQ = [ -1 1 ]'/sqrt(3);
wQ = [ 1 1 ];
h = xCoords(2)-xCoords(1); %element width
%Force vector matrix
fe(1) = wQ*(func(x(tQ)).*N1(tQ));
fe(2) = wQ*(func(x(tQ)).*N2(tQ));
fe = fe'*h/2;
Lecture26
May 05, 2008
Galerkin Finite Elements: Global Matrix assembly
1
0
xi-1
xi
xi+1
Galerkin Finite Elements: Matlab
function [A,f] = assembleGlobalProblem(xCoords,func,alpha,beta)
% assembleGlobalProblem - loops over elements and assembles global matrix
% and right-hand side and sets Dirichlet Boundary conditions
%
%
[A,f] = assembleGlobalProblem(func,xCoords,alpha,beta)
%
% xCoords:- coordinates of all elements
% func:-function handle to RHS f(x)
% alpha: - dirichlet condition at xCoords(1);
% beta:- dirichlet condition at xCoords(end);
%
% A: Global stiffness+Mass matrix K+M
% f: Global force vector
%
N = length(xCoords); % numbier of points
nels = N - 1; % number of elements
A = spalloc(N,N,3*N);
f = zeros(N,1);
for k = 1:nels % loop over elements and assemble global matrix
kEl = k:k+1; % index of element k
[Ke,Me] = getElementMatrix(xCoords(kEl));
fe = getElementForceVector(func,xCoords(kEl));
A(kEl,kEl) = A(kEl,kEl) + Ke+Me;
f(kEl) = f(kEl)+fe;
end
%fix dirichlet boundary conditions
A(1,1:2)=[ 1 0 ];
A(end,end-1:end)=[ 0 1 ];
f([ 1 end ]) = [ alpha ; beta ]';
Lecture26
May 05, 2008
Galerkin Finite Elements: The future
Linear hats on a line is the simplest problem...
but Finite Elements is a very rich subject...
Possible extensions:
1) higher order elements (quadratic, cubic, "spectral", mixed)
2) Higher spatial dimensions: meshing in 2 (and 3-D)
Finite element calculation of viscous fluid flow around a "dolfin"
Galerkin Finite Elements: The future
Extensions:
3) Space-time PDE's
4) Multi-physics/Multi-Scale models
5) Peta-scale high-performance computation
6) Advanced software design
Point:
If you're interested, computational science and modeling can be a very
rewarding mix of Mathematics, Computer Science and domain specific
science and engineering.
We've just scratched the surface in this course...but understanding
these building blocks is where you start...the fun is just beginning...