1. 1 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 NUMERICAL METHODS IN APPLIED STRUCTURAL MECHANICS Lecture notes: Prof. Maurício V. Donadon

2. 2 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Solution of equilibrium equations in dynamic analysis

3. 3 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Direct integration methods System of equilibrium equations in linear dynamics The direct integration consists of integrating the equilibrium equations using a numerical step-by-step procedure. The “direct ” term means that prior to the numerical integration, no transformation of the equations into a different form is carried out.

4. 4 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Direct integration methods Central difference method Houbolt method Wilson-Θ method Newmark method

5. 5 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 The central difference method

6. 6 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 The central difference method System of equilibrium equations in linear dynamics Aproximation for aceleration and velocities

7. 7 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 The central difference method Aproximation for aceleration and velocities Substitution of the equations above into the equilibrium equation leads to the following expression:

8. 8 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 The central difference method Alternative way of presenting the equilibrium equation Defining the auxiliary matrices, with,

9. 9 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 The central difference method Alternative way of presenting the equilibrium equation Using the definitions presented previously we obtain, the following system of equations

10. 10 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 The central difference method Having {U} t+Δt determined, compute acelerations and velocities at time t The calculation of {U} t+Δt involves {U} t & {U} t-Δt. Therefore a starting procedure must be used to compute the solution at time Δt. The equation below is used for this purpose:

11. 11 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 The central difference method Critical time step computation to ensure numerical stability of the algorithm:

12. 12 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 CDM flowchart implementation

13. 13 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 The central difference method - Example M1M1 k2k2 U 1 (t), F 1 (t) k1k1 U 2 (t), F 2 (t) M2M2

14. 14 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 The central difference method - Example Time step computation:

15. 15 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 The central difference method - Example

16. 16 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 The central difference method - Example

17. 17 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 The Houbolt method

18. 18 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 System of equilibrium equations in linear dynamics Aproximation for aceleration and velocities The Houbolt method

19. 19 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Substitution of the approximations for velocity and aceleration leads to the following form: The Houbolt method

20. 20 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Defining the auxiliary matrices, with, The Houbolt method

21. 21 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Defining the auxiliary matrices, The Houbolt method Resultant system of equations to be solved,

22. 22 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Having {U} t+Δt determined, compute acelerations and velocities at time t+Δt The Houbolt method The calculation of {U} t+Δt involves {U} t, {U} t-Δt & {U} t-2Δt Therefore a starting procedure must be used to compute the solution at time Δt and 2Δt. The CDM may be used for this purpose

23. 23 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Critical time step computation to ensure numerical stability of the algorithm: The Houbolt method There is no critical time-step limit to integrate the equilibrium equations using the Houbolt method! However, the starting procedure requires the same critical time step used in the CDM!

24. 24 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 HM flowchart implementation

25. 25 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 The Houbolt method - Example Time step computation:

26. 26 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 The Houbolt method - Example

27. 27 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 The Houbolt method - Example

28. 28 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 The Wilson-Θ method

29. 29 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 System of equilibrium equations in linear dynamics The Wilson-Θ method (linear acceleration method)

30. 30 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Assumed acceleration for the time interval t to t+ΘΔt: The Wilson-Θ method (linear acceleration method)

31. 31 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Assumed acceleration for the time interval t to t+ΘΔt: The Wilson-Θ method (linear acceleration method) Integrating the expression above we obtain:

32. 32 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 The Wilson-Θ method (linear acceleration method) Using the previous expressions at time t+ΘΔt we obtain: From which we can solve for acceleration and velocities in terms of {U} t+ΘΔt :

33. 33 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 The Wilson-Θ method (linear acceleration method) The expressions below are then substituted into the dynamic equilibrium equation

34. 34 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Defining the auxiliary matrices, with, The Wilson-Θ method (linear acceleration method)

35. 35 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Leads to the following expression for {U} t+ΘΔt With {U} t+ΘΔt compute acceleration, velocity and displacement at t+Δt: The Wilson-Θ method (linear acceleration method)

36. 36 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Critical time step computation to ensure numerical stability of the algorithm: The Wilson-Θ algorithm is conditionally stable for Θ values greater than 1.37, typically 1.40! The Wilson-Θ method (linear acceleration method)

37. 37 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 W-Θ flowchart implementation

38. 38 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 The Wilson-Θ method - Example Time step computation:

39. 39 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 The Wilson-Θ method - Example

40. 40 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 The Wilson-Θ method - Example

41. 41 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 The Newmark method

42. 42 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 The Newmark method System of equilibrium equations in linear dynamics Aproximation for velocity and displacements

43. 43 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 The Newmark method Solving (II) for acceleration at t+Δt and then substituting the acceleration at t+Δt into (I) and the result into the equilibrium equation we obtain,

44. 44 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 The Newmark method Where,

45. 45 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 The Newmark method with,

46. 46 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 The Newmark method Having {U} t+Δt, velocity and acceleration are computed as follow,

47. 47 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Newmark flowchart implementation

48. 48 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 The Newmark method - Example Time step computation:

49. 49 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 The Newmark method - Example

50. 50 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 The Newmark method - Example

51. 51 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Error analysis

52. 52 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Erro analysis - Example Time step computation:

53. 53 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Error analysis

54. 54 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Error analysis

55. 55 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Error analysis Relative solution error definition:

56. 56 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Error analysis

57. 57 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Analysis of Direct Integration Methods

58. 58 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 The cost of a direct integration analysis is directly proportional to the number of time steps required in the solution; Time step size ↔ Accuracy & Computional Cost! Direct integration methods

59. 59 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Direct integration approximation & Load operators Central difference method:

60. 60 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Direct integration approximation & Load operators Wilson-theta method

61. 61 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Direct integration approximation & Load operators Wilson-theta method

62. 62 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Direct integration approximation & Load operators Newmark method

63. 63 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Stability analysis of direct integration methods An integration method is conditionally stable if the solution does not increase without limit for any  t and initial condition, particularly for larger  t/T. The method is conditionally stable if a condition is satisfied only if  t/T lower than a certain value defined as stability limit.

64. 64 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Stability analysis of direct integration methods An integration method is conditionally stable if the solution does not increase without limit for any  t and initial condition, particularly for larger  t/T. The method is conditionally stable if a condition is satisfied only if  t/T lower than a certain value defined as stability limit.

65. 65 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Stability analysis of direct integration methods

66. 66 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Stability analysis of direct integration methods