Lecture 9: Finite Elements Sauro Succi

FEM: non-spherical cows Coordinate-free: Unstructured

FEM for fluids

The Finite Element Method

Less intuition, more systematic, solid math Foundations (functional analysis) Strong vs Weak Convergence

Pointwise (strong) formulation Local interpolation around x=x_j: Looses accuracy on non-uniform meshes Awkward on unstructured lattices

Compute gradient below?

Variational (weak) formulation Hilbert space L2: Global statement For any g, find f_N such that:

Expansion on basis function Convergence in Hilbert space Projection on Hilbert space

Expansion on basis function Operators to Matrices

Examples of matrices: Mass, Stiffness, Advection. using linear hat functions

FEM matrix operators

Finite-support basis function

Useful identities

Mass matrix Uniform mesh:

Mass matrix: smoother

Advection matrix Uniform mesh:

Diffusion matrix Uniform mesh:

Self-advection matrix: triad Uniform mesh:

FEM operators

Matrix assembly Element-wise

FEM operators + Strong math back-up - Expensive (matrix algebra) + Very systematic + Fluid/Solid coupling

Summary FEM +: Powerful math backup (weak convergence) Systematic programming Geometrical flexibility FEM -: Matrix algebra anyway (lumping) Heavy duty Mainstream for solid mech, not fluids

1d example: assembly 4 matrix elements per interval; 2 intervals per node= 8 matrix elements/node

FEM: cows are cows Coordinate-free: Unstructured

Boundary conditions Element-wise

Triangle basis function

Matrix assembly Element-wise: connectivity

       Matrix assembly

Linear Algebra Direct Methods: Minimize bandwidth Optimal Numbering (NP complete) Iterative Methods: Sparse matrix algebra: A*x+y

Optimal Numbering

Some app’s from the web + sample code fem.f

Finite-support basis function