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Periodically distributed

Overview. Periodically distributed. 2-D elasticity problem. Overview. Periodically distributed. 2-D elasticity problem. Something else…. Overview. Periodic material (everywhere). One-dimensional problem. Chronological order. Something else….

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Periodically distributed

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  1. Overview Periodically distributed 2-D elasticity problem

  2. Overview Periodically distributed 2-D elasticity problem Something else…

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

  4. One-dimensional problem - classical example - Coefficients

  5. One-dimensionalproblem Exact solution ( )

  6. One-dimensional problem Exact solution FEM approx. (h = 0.2)

  7. One-dimensionalproblem Exact solution FEM approx. (h= 0.05)

  8. One-dimensionalproblem Conclusion: Step size h must be taken smaller than !!! Exact solution FEM approx. (h= 0.01)

  9. Homogenisation Multiple scale method – ansatz:

  10. Homogenisation Can be shown… average of (in a certain sense)

  11. Homogenisation Complicated to solve… approximation for average of (in a certain sense) Easy to solve…

  12. Homogenisation Homogenised solution : Captures essential behaviour of but loses oscillations…

  13. Homogenisation Approximate by Recover the oscillations… + Boundary corrector Cell Problem

  14. Homogenisation Approximate by (C= boundary Corrector) Error

  15. Removesimplification... Simplifications: Periodic material (everywhere) One-dimensional problem 0 0.1 1

  16. Domain decomposition 0 0.1 1 Iterative scheme (Schwarz) 0.15 0 0.1 0.1 1

  17. Domain decomposition 0 0.1 1 Iterative scheme (Schwarz) 0.15 ? ? 0 0.1 0.1 1

  18. Domain decomposition ?

  19. Domain decomposition ? ? Initial guess 0 0.1 1

  20. Domain decomposition Initial guess Periodic with period 0 0.1 1 Homogenised solution

  21. Domain decomposition

  22. Domain decomposition Approximation for k=1 Error k=1

  23. Domain decomposition Approximation for k=2 Error k=1 k=2

  24. Domain decomposition Approximation for k=3 Error k=1 k=2 k=3

  25. Hybrid approach 0 0.1 1 Iterative scheme (Schwarz) 0.15 Aproximate with homogenisation 0 0.1 0.1 1

  26. Hybrid approach – stopping criterion Error reduction in the Schwarz scheme

  27. Hybrid approach – stopping criterion Error reduction in the Hybrid scheme Error reduction in the Schwarz scheme

  28. Hybrid approach – stopping criterion Error reduction in the Hybrid scheme

  29. Hybrid approach – stopping criterion Error reduction in the Hybrid scheme Error reduction…

  30. Hybrid approach – stopping criterion Error reduction in the Hybrid scheme (after a few iterations…) smaller…

  31. Hybrid approach – stopping criterion Error reduction in the Hybrid scheme (after a few iterations…) No error reduction…

  32. Hybrid approach – stopping criterion Error reduction in the Hybrid scheme (after a few iterations…) Stopping criterion:

  33. Hybrid approach Error

  34. Linear elasticity Young’s modulus Poisson’s ratio Young’s modulus Poisson’s ratio

  35. Linear elasticity Young’s modulus Poisson’s ratio

  36. Homogenisation Linear elasticity Periodic Periodic Schwarz

  37. Homogenisation Young’s modulus Poisson’s ratio 0.5 Young’s modulus Poisson’s ratio -0.5 0.5

  38. Homogenisation Exactsolution (horizontal component) Homogenised corrected solution Homogenised solution

  39. Homogenisation Error Exactsolution (horizontal component)

  40. Hybrid approach Young’s modulus Poisson’s ratio Young’s modulus Poisson’s ratio Young’s modulus Poisson’s ratio

  41. Hybrid approach Horizontal component of the exact solution Vertical component of the exact solution Initial guess: disregard inclusions…

  42. Hybrid approach Horizontal component of the initial guess Vertical component of the initial guess

  43. Hybrid approach Horizontal component of the corrected Vertical component of the correctedhomogenised function homogenised function

  44. Hybrid approach

  45. Some references

  46. Extras

  47. Homogenisation

  48. Linear elasticity

  49. Extra: Homogenisation Solvability condition for :

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