Lecture 9: Finite Elements Sauro Succi. FEM: non-spherical cows Coordinate-free: Unstructured

Lecture 9: Finite Elements

Sauro Succi

FEM: non-spherical cows

Coordinate-free: Unstructured

FEM for fluids

The Finite Element Method

The Finite Element Method

Less intuition, more systematic, solid mathFoundations (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 programmingGeometrical flexibility

FEM -: Matrix algebra anyway (lumping)Heavy dutyMainstream 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

Triangle basis function

Matrix assembly

Element-wise: connectivity

Matrix assembly

Linear Algebra

Direct Methods: Minimize bandwidthOptimal 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

Finite-support basis function

Finite-support basis function