Periodically

distributed

Overview

2-D elasticity problem

Overview

2-D elasticity problem

Something else…

Periodically

distributed

Overview

Periodic material (everywhere)

One-dimensional problem

One-dimensional problem

Something else… Periodic with period

Periodically

distributed

Something else…

Periodic with period

2-D elasticity problem

Leave for later (latest slides)…

Ch

rono

log

ica

l ord

er

One-dimensional problem

Coefficients

- classical example -

One-dimensional problem

Exact solution

( )

One-dimensional problem

Exact solution

FEM approx. (h = 0.2)

One-dimensional problem

Exact solution

FEM approx. (h= 0.05)

Exact solution

FEM approx. (h= 0.01)

One-dimensional problem

Step size h must be taken smaller than !!!

Conclusion:

Homogenisation

Multiple scale method – ansatz:

Homogenisation

average of

(in a certain sense)

Can be shown…

Homogenisation

approximation for

Complicated to solve…

Easy to solve…

average of

(in a certain sense)

Homogenisation

Captures essential behaviour of

but loses oscillations…

Homogenised

solution :

Homogenisation

Recover the oscillations…

Cell Problem

+ Boundary corrector

Approximate by

Homogenisation

Approximate by (C= boundary Corrector)

Error

Remove simplification...

Periodic material (everywhere)

Simplifications:

One-dimensional problem

0 0.1 1

Domain decomposition

0 0.1 1

0 0.1 0.1 1

0.15

Iterative scheme

(Schwarz)

Domain decomposition

0 0.1 1

0 0.1 0.1 1

0.15

Iterative scheme

(Schwarz)

? ?

Domain decomposition

?

Domain decomposition

Initial

guess

0 0.1 1

?

?

Domain decomposition

Initial guess

0 0.1 1

Homogenised

solution

Periodic with period

Domain decomposition

Domain decomposition

Approximation for k=1 Error

k=1

Domain decomposition

Approximation for k=2 Error

k=1

k=2

Domain decomposition

Approximation for k=3 Error

k=1

k=2

k=3

Hybrid approach

0 0.1 1

0 0.1 0.1 1

0.15

Iterative scheme

(Schwarz)

Aproximate with homogenisation

Error reduction in the Schwarz scheme

Hybrid approach – stopping criterion

Error reduction in the Schwarz schemeError reduction in the Hybrid scheme

Hybrid approach – stopping criterion

Hybrid approach – stopping criterion

Error reduction in the Hybrid scheme

Hybrid approach – stopping criterion

Error reduction in the Hybrid scheme

Error

reduction…

Hybrid approach – stopping criterion

Error reduction in the Hybrid scheme

(after a few

iterations…)

smaller…

Hybrid approach – stopping criterion

Error reduction in the Hybrid scheme

(after a few

iterations…)No error

reduction…

Hybrid approach – stopping criterion

Error reduction in the Hybrid scheme

(after a few

iterations…)Stopping

criterion:

Hybrid approach

Error

Linear elasticity

Young’s modulus

Poisson’s ratio

Young’s modulus

Poisson’s ratio

Linear elasticity

Young’s modulus

Poisson’s ratio

Periodic

Linear elasticity

Periodic

Schwarz

Homogenisation

Homogenisation

-0.5 0.5

0.5 Young’s modulus

Poisson’s ratio

Young’s modulus

Poisson’s ratio

Homogenised solutionHomogenised corrected

solution

Homogenisation

Exact solution

(horizontal component)

Homogenisation ErrorExact solution

(horizontal component)

Hybrid approach

Young’s modulus

Poisson’s ratio

Young’s modulus

Poisson’s ratio

Young’s modulus

Poisson’s ratio

Hybrid approach

Horizontal component of the exact solution Vertical component of the exact solution

Initial guess: disregard inclusions…

Hybrid approach

Horizontal component of the initial guess Vertical component of the initial guess

Hybrid approach

Horizontal component of the corrected Vertical component of the correctedhomogenised function homogenised function

Hybrid approach

Some references

Extras

Homogenisation

Linear elasticity

Extra: Homogenisationfu Α

00 u0Α

01 uu 10 ΑΑ

fuuu 012 210 ΑΑΑ

Solvability condition for : fu 0Α

0(y)d yfY

),(),(),()( 22

10

x

xux

xux

xuxu

Extra: HomogenisationInstead of , we now havefu Α

00 u0Α

01 uu 10 ΑΑ

fuuu 012 210 ΑΑΑ

)(),(0 xuyxu

Homogenised

Equation

Cell problem:assume that

j

j

x

xuyyxu

)(

),(1 i

ijj

y

yay

)(

0A

)()(2

xfxx

xua

jjij

, where

Y

k

j

ikijij yy

yyayaa )

)(

)()()((

Extra: Homogenisation

Extra: Homogenisation

Extra: Homogenisation

Bounds for the error of homogenisation

Error hybrid approach (length overlapping)

Error hybrid approach (length overlapping)

Bound for the error of hybrid approach

Composites

Start off easy...

Periodic material (everywhere)

Simplifications:

One-dimensional problem

Domain decomposition

Iterative scheme (Schwarz)

Hybrid approach

Iterative scheme (Schwarz) Aproximate with homogenisation

Hybrid approach – stopping criterion

Stopping

condition:

Hybrid approach – stopping criterion

Error reduction in the Schwarz scheme

Hybrid approach – stopping criterion

Error reduction in the Schwarz scheme

Hybrid approach – stopping criterion

Error reduction in the Schwarz scheme

Hybrid approach – stopping criterion

Error reduction in the Schwarz scheme

No error reduction!!!

Hybrid approach – stopping criterion

Error reduction in the Schwarz scheme

Stopping

criterion

Hybrid approach – stopping criterion

Error reduction in the Hybrid scheme

No error

reduction…

Hybrid approach – stopping criterion

Error reduction in the Hybrid scheme

Stopping

Criterion

maior que

Mas como…

Hybrid approach – stopping criterion

Error reduction in the Hybrid scheme

Stopping

criterion<

Homogenisation

approximation for

Complicated to solve…

Easy to solve…