periodically distributed overview 2-d elasticity problem
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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…