Periodically distributed Overview 2-D elasticity problem

Overview 2-D elasticity problem Something else… Periodically distributed

Overview Periodic material (everywhere) One-dimensional problem Something else… Periodic with period Periodically distributed Something else… Periodic with period 2-D elasticity problem Leave for later (latest slides)… Chronological order

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

Domain decomposition Iterative scheme (Schwarz)

Domain decomposition Iterative scheme (Schwarz) ?

Domain decomposition ?

Initial guess ? ?

Domain decomposition Initial guess Homogenised solution Periodic with period

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 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

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

Young’s modulus Poisson’s ratio Young’s modulus Poisson’s ratio

Homogenised solutionHomogenised corrected solution Homogenisation Exact solution (horizontal component)

Homogenisation Error Exact 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 corrected homogenised function homogenised function

Hybrid approach

Some references

Extras

Homogenisation

Linear elasticity

Extra: Homogenisation Solvability condition for :

Extra: Homogenisation Instead of, we now have Homogenised Equation Cell problem:assume that, where

Extra: Homogenisation

Bounds for the error of homogenisation

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…