lecture 9: finite elements sauro succi. fem: non-spherical cows coordinate-free: unstructured

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Lecture 9: Finite Elements Sauro Succi

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Page 1: Lecture 9: Finite Elements Sauro Succi. FEM: non-spherical cows Coordinate-free: Unstructured

Lecture 9: Finite Elements

Sauro Succi

Page 2: Lecture 9: Finite Elements Sauro Succi. FEM: non-spherical cows Coordinate-free: Unstructured

FEM: non-spherical cows

Coordinate-free: Unstructured

Page 3: Lecture 9: Finite Elements Sauro Succi. FEM: non-spherical cows Coordinate-free: Unstructured

FEM for fluids

Page 4: Lecture 9: Finite Elements Sauro Succi. FEM: non-spherical cows Coordinate-free: Unstructured

The Finite Element Method

Page 5: Lecture 9: Finite Elements Sauro Succi. FEM: non-spherical cows Coordinate-free: Unstructured

The Finite Element Method

Less intuition, more systematic, solid mathFoundations (functional analysis)

Strong vs Weak Convergence

Page 6: Lecture 9: Finite Elements Sauro Succi. FEM: non-spherical cows Coordinate-free: Unstructured

Pointwise (strong) formulation

Local interpolation around x=x_j:

Looses accuracy on non-uniform meshes

Awkward on unstructured lattices

Page 7: Lecture 9: Finite Elements Sauro Succi. FEM: non-spherical cows Coordinate-free: Unstructured

Compute gradient below?

Page 8: Lecture 9: Finite Elements Sauro Succi. FEM: non-spherical cows Coordinate-free: Unstructured

Variational (weak) formulation

Hilbert space L2: Global statement

For any g, find f_N such that:

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

Expansion on basis function

Convergence in Hilbert space

Projection on Hilbert space

Page 10: Lecture 9: Finite Elements Sauro Succi. FEM: non-spherical cows Coordinate-free: Unstructured

Expansion on basis function

Operators to Matrices

Page 11: Lecture 9: Finite Elements Sauro Succi. FEM: non-spherical cows Coordinate-free: Unstructured

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

Page 12: Lecture 9: Finite Elements Sauro Succi. FEM: non-spherical cows Coordinate-free: Unstructured

FEM matrix operators

Page 13: Lecture 9: Finite Elements Sauro Succi. FEM: non-spherical cows Coordinate-free: Unstructured

Finite-support basis function

Page 14: Lecture 9: Finite Elements Sauro Succi. FEM: non-spherical cows Coordinate-free: Unstructured

Useful identities

Page 15: Lecture 9: Finite Elements Sauro Succi. FEM: non-spherical cows Coordinate-free: Unstructured

Mass matrix

Uniform mesh:

Page 16: Lecture 9: Finite Elements Sauro Succi. FEM: non-spherical cows Coordinate-free: Unstructured

Mass matrix: smoother

Page 17: Lecture 9: Finite Elements Sauro Succi. FEM: non-spherical cows Coordinate-free: Unstructured

Advection matrix

Uniform mesh:

Page 18: Lecture 9: Finite Elements Sauro Succi. FEM: non-spherical cows Coordinate-free: Unstructured
Page 19: Lecture 9: Finite Elements Sauro Succi. FEM: non-spherical cows Coordinate-free: Unstructured

Diffusion matrix

Uniform mesh:

Page 20: Lecture 9: Finite Elements Sauro Succi. FEM: non-spherical cows Coordinate-free: Unstructured

Self-advection matrix: triad

Uniform mesh:

Page 21: Lecture 9: Finite Elements Sauro Succi. FEM: non-spherical cows Coordinate-free: Unstructured

FEM operators

Page 22: Lecture 9: Finite Elements Sauro Succi. FEM: non-spherical cows Coordinate-free: Unstructured

Matrix assembly Element-wise

Page 23: Lecture 9: Finite Elements Sauro Succi. FEM: non-spherical cows Coordinate-free: Unstructured

FEM operators

+ Strong math back-up

- Expensive (matrix algebra)

+ Very systematic

+ Fluid/Solid coupling

Page 24: Lecture 9: Finite Elements Sauro Succi. FEM: non-spherical cows Coordinate-free: Unstructured

Summary

FEM +: Powerful math backup (weak convergence)Systematic programmingGeometrical flexibility

FEM -: Matrix algebra anyway (lumping)Heavy dutyMainstream for solid mech, not fluids

Page 25: Lecture 9: Finite Elements Sauro Succi. FEM: non-spherical cows Coordinate-free: Unstructured

1d example: assembly

4 matrix elements per interval;2 intervals per node=8 matrix elements/node

Page 26: Lecture 9: Finite Elements Sauro Succi. FEM: non-spherical cows Coordinate-free: Unstructured

FEM: cows are cows

Coordinate-free: Unstructured

Page 27: Lecture 9: Finite Elements Sauro Succi. FEM: non-spherical cows Coordinate-free: Unstructured
Page 28: Lecture 9: Finite Elements Sauro Succi. FEM: non-spherical cows Coordinate-free: Unstructured
Page 29: Lecture 9: Finite Elements Sauro Succi. FEM: non-spherical cows Coordinate-free: Unstructured

Boundary conditions

Element-wise

Page 30: Lecture 9: Finite Elements Sauro Succi. FEM: non-spherical cows Coordinate-free: Unstructured

Triangle basis function

Page 31: Lecture 9: Finite Elements Sauro Succi. FEM: non-spherical cows Coordinate-free: Unstructured

Triangle basis function

Page 32: Lecture 9: Finite Elements Sauro Succi. FEM: non-spherical cows Coordinate-free: Unstructured

Matrix assembly

Element-wise: connectivity

Page 33: Lecture 9: Finite Elements Sauro Succi. FEM: non-spherical cows Coordinate-free: Unstructured

Matrix assembly

Page 34: Lecture 9: Finite Elements Sauro Succi. FEM: non-spherical cows Coordinate-free: Unstructured

Linear Algebra

Direct Methods: Minimize bandwidthOptimal Numbering (NP complete)

Iterative Methods: Sparse matrix algebra: A*x+y

Page 35: Lecture 9: Finite Elements Sauro Succi. FEM: non-spherical cows Coordinate-free: Unstructured

Optimal Numbering

Page 36: Lecture 9: Finite Elements Sauro Succi. FEM: non-spherical cows Coordinate-free: Unstructured
Page 37: Lecture 9: Finite Elements Sauro Succi. FEM: non-spherical cows Coordinate-free: Unstructured

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

Page 38: Lecture 9: Finite Elements Sauro Succi. FEM: non-spherical cows Coordinate-free: Unstructured

Finite-support basis function

Page 39: Lecture 9: Finite Elements Sauro Succi. FEM: non-spherical cows Coordinate-free: Unstructured

Finite-support basis function

Page 40: Lecture 9: Finite Elements Sauro Succi. FEM: non-spherical cows Coordinate-free: Unstructured

Finite-support basis function