Homogenisation theory for partial differential equations

1
Micro/macro analysis of a composite
Casa Day 22/11/06
2
Motivation
Aloha Airlines Flight 243
Honolulu International Airport, Hawaii
No visual exterior inspection (nor required)
At 24000 feet loud clap
Flight attendant swept out of the cabin
1 dead, 65 injured
5 million damage
3
Peel Test
Force
Displacement (constant speed)
Nickel layer
45º
Substrate
Peel-Test Support
4
Composites
5
Elasticity problem for a composite material
Inclusions
Matrix
Material Interface
6
Modelling
Matrix
Inclusions
7
Homogenisation
8
Homogenisation
9
Homogenisation
10
Homogenised equations
Boundary Conditions
11
Macro results
12
Micro/Macro
13
Domain decomposition
14
Dirichlet Neumann scheme
15
Finite element approximation
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Finite element approximation
Schur complement system

• symmetric
• positive definite
• ill conditioned

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Algorithm
Dirichlet problem
Neumann Problem
Relaxation step
18
Algorithm
is a preconditioner for
Theorem
This process converges if
where is the maximum eigenvalue of
.
19
Heterogeneous domain decomposition
20
Heterogeneous domain decomposition
21
Overview
homoge- nisation
D-N
Interface conditions?
22
Example
Macro domain
Reference solution
Meso domain
23
Example
Absolute error for the horizontal displacement
Macro domain
Meso domain
24
Future Work
- Interface conditions discussion - Crack
propagation