2. Periodically distributed Overview 2-D elasticity problem

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

4. 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

5. One-dimensional problem Coefficients - classical example -

6. One-dimensional problem Exact solution ( )

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

8. One-dimensional problem Exact solution FEM approx. (h  = 0.05)

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

10. Homogenisation Multiple scale method – ansatz:

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

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

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

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

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

16. Remove simplification... Periodic material (everywhere) Simplifications: One-dimensional problem 0 0.1 1

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

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

19. Domain decomposition ?

20. Initial guess 0 0.1 1 ? ?

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

22. Domain decomposition

23. Approximation for k=1 Error k=1

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

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

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

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

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

29. Error reduction in the Hybrid scheme

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

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

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

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

34. Hybrid approach Error

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

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

37. Periodic Linear elasticity Periodic Schwarz Homogenisation

38. -0.5 0.5 0.5 Young’s modulus Poisson’s ratio Young’s modulus Poisson’s ratio

39. Homogenised solutionHomogenised corrected solution Homogenisation Exact solution (horizontal component)

40. Homogenisation Error Exact solution (horizontal component)

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

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

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

44. Hybrid approach Horizontal component of the corrected Vertical component of the corrected homogenised function homogenised function

45. Hybrid approach

46. Some references

47. Extras

48. Homogenisation

49. Linear elasticity

50. Extra: Homogenisation Solvability condition for :

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

52. Extra: Homogenisation

54. Bounds for the error of homogenisation

55. Error hybrid approach (length overlapping)

56. Bound for the error of hybrid approach

57. Composites

58. Start off easy... Periodic material (everywhere) Simplifications: One-dimensional problem

59. Domain decomposition Iterative scheme (Schwarz)

60. Hybrid approach Iterative scheme (Schwarz) Aproximate with homogenisation

61. Hybrid approach – stopping criterion Stopping condition :

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

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

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

65. Hybrid approach – stopping criterion Error reduction in the Schwarz scheme No error reduction!!!

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

67. Hybrid approach – stopping criterion Error reduction in the Hybrid scheme No error reduction…

68. Hybrid approach – stopping criterion Error reduction in the Hybrid scheme Stopping Criterion maior que Mas como…

69. Hybrid approach – stopping criterion Error reduction in the Hybrid scheme Stopping criterion <

70. Homogenisation approximation for Complicated to solve… Easy to solve…