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

2 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Solution methods for eigenproblems

3 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Eigenproblem To obtain non-trivial {φ} so that {[K]-λ[M]}{φ} = 0 det {[K]-λ[M]} = 0 Associated eigen-pairs {[K]- i [M]{φ i }= 0 Solution methods for eigenproblems

4 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Vector iteration methods Transformation methods Polynomial iteration techniques Methods based on the Sturm sequence properties of the characteristic polynomials Solution methods for eigenproblems

5 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Vector iteration methods

6 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Vector iteration methods Assuming {φ}= {x 1 } and λ=1, where {x 1 } is arbitrary Eigenproblem

7 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Vector iteration methods

8 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Inverse iteration method Forward iteration method Iteration methods with “shifting” Rayleigh quotient iteration Matrix deflation Gram-Shmidt Orthogonalization Vector iteration methods

9 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Inverse iteration method Vector iteration methods

10 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Inverse iteration method Vector iteration methods Convergence criterion

11 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Inverse iteration method – Fundamental Equation: Vector iteration methods Convergence analysis for inverse iteration method

12 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Vector iteration methods Convergence analysis for inverse iteration method

13 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Vector iteration methods Convergence analysis for inverse iteration method

14 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Vector iteration methods Convergence analysis for inverse iteration method

15 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Vector iteration methods Convergence analysis for inverse iteration method

16 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Vector iteration methods Convergence analysis for inverse iteration method

17 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Inverse iteration method - Example Vector iteration methods

18 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Inverse iteration method - Example Vector iteration methods

19 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Vector iteration methods Inverse iteration method - Example

20 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Forward iteration method Vector iteration methods

21 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Forward iteration method Convergence criterion Vector iteration methods

22 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Forward iteration method – Fundamental Equation: Vector iteration methods Convergence analysis for forward iteration method

23 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Convergence analysis for forward iteration method Vector iteration methods

24 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Vector iteration methods Convergence analysis for forward iteration method

25 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Convergence analysis for forward iteration method Vector iteration methods

26 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Forward iteration method - Example Vector iteration methods

27 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Forward iteration method - Example Vector iteration methods

28 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Vector iteration methods Forward iteration method - Example

29 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Shifting in vector iteration methods “Shifting”

30 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Shifting in vector iteration methods

31 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Shifting in vector iteration methods

32 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Shifting in vector iteration methods

33 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Shifting in vector iteration methods

34 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Shifting in vector iteration methods “Shifting” – Convergence rate in inverse iteration

35 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Shifting in vector iteration methods Shifting in inverse iteration method - Example

36 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Shifting in vector iteration methods Shifting in inverse iteration method – Example Eigenvalues λ i = η i + μ

37 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Shifting in inverse iteration method - Example Shifting in vector iteration methods

38 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Shifting in vector iteration methods “Shifting” in forward iteration

39 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Shifting in vector iteration methods “Shifting” in forward iteration

40 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Vector iteration methods Rayleigh quotient iteration Improves the convergence rate in inverse iteration method with “shifting” A new “shift” is evaluated in each iteration which improves the converge!

41 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Vector iteration methods Rayleigh quotient iteration

42 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Vector iteration methods Rayleigh quotient iteration

43 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Rayleigh quotient iteration method - Example Vector iteration methods

44 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Rayleigh quotient iteration method - Example Vector iteration methods

45 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Vector iteration methods Matrix deflation

46 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Vector iteration methods Matrix deflation

47 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Vector iteration methods Matrix deflation

48 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Vector iteration methods Matrix deflation

49 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Vector iteration methods Gram-Schmidt Orthogonalization

50 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Vector iteration methods Gram-Schmidt Orthogonalization

51 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Vector iteration methods Gram-Schmidt Orthogonalization

52 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Vector iteration methods Gram-Schmidt Orthogonalization - Example

53 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Vector iteration methods Gram-Schmidt Orthogonalization - Example

54 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Vector iteration methods Gram-Schmidt Orthogonalization - Example

55 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Transformation Methods

56 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Transformation methods Basic properties of transformation methods:

57 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Transformation methods

58 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Transformation methods

59 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Transformation methods Jacobi method Generalized Jacobi method Householder-QR-Inverse Iteration solution

60 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 The Jacobi method Developed for the solution of standard eigenproblems [A]{φ}=λ{φ}, with [A] symmetric; Stable method that enables the computation of negative, positive and null eigevalues The transformation matrices [P k ] are orthogonal Transformation methods

61 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 The Jacobi method – Transformation Matrix Transformation methods

62 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 The Jacobi method Transformation methods

63 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 The Jacobi method Transformation methods

64 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 The Jacobi method Although the transformation [P k ] [A k ] [P k ] reduces an off diagonal element in [A k ] to zero, this element will again became nonzero during the transformations that follow!! Based on the previous comment, which element of [A k ] has to be reduced to zero???? One choice is to zero the largest off-diagonal element in [A k ], however this procedure is time consuming!!! An efficient way to choose the element of [A k ] to be reduced to zero is the use of the Threshold Jacobi Method Transformation methods

65 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 The Jacobi method The Threshold Jacobi method consists of sequentially checking the off-diagonal elements of [A k ] applying the rotation to those elements that are larger than the threshold for that sweep. Transformation methods The threshold is defined in terms of a coupling measure between degrees of freedom p and q, defined as follows:

66 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 The Jacobi method CONVERGENCE CRITERIA Transformation methods

67 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon Transformation methods Jacobi Method - Example

68 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 The Generalized Jacobi method Developed for the solution of general eigenproblems [K]{φ}=λ [M]{φ}; Stable method that enables the computation of negative, positive and null eigenvalues [M] must be positive definite [K] and [M] are simultaneously diagonalized Used in the Subspace Iteration Method Transformation methods

69 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 The Generalized Jacobi method -Transformation Matrix Transformation methods

70 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 The Generalized Jacobi method -Transformation Matrix Transformation methods

71 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 The Generalized Jacobi method -Transformation Matrix Transformation methods

72 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 The Generalized Jacobi method -Transformation Matrix Transformation methods

73 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 The Generalized Jacobi method CONVERGENCE CRITERIA Transformation methods

74 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon Transformation methods Generalized Jacobi Method - Example

75 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Householder-QR-Inverse Iteration (HQRI) solution Restricted to standard eigenproblem [K]{φ}=λ{φ} Allows the computation of positive, negative and null eigenvalues Transformation methods Steps in the HQRI solution method: 1.Householder transformations are employed to reduce [K] into tridiagonal form 2.QR iteration yields all eigenvalues 3.Using the inverse iteration, the required eigenvectors of the tridiagonal matrix are calculated and transformed to obtain the eigenvectors of [K]

76 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 HOUSEHOLDER REDUCTION Householder-QR-Inverse Iteration (HQRI) solution Transformation methods

77 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 HOUSEHOLDER REDUCTION Householder-QR-Inverse Iteration (HQRI) solution Transformation methods

78 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Householder-QR-Inverse Iteration (HQRI) solution Transformation methods HOUSEHOLDER REDUCTION

79 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Householder-QR-Inverse Iteration (HQRI) solution Transformation methods HOUSEHOLDER REDUCTION

80 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon Transformation methods Householder Reduction - Example

81 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Householder-QR-Inverse Iteration (HQRI) solution Transformation methods QR ITERATION Applied to the tridiagonal matrix obtained by Householder transformation of [K] It can be also applied to the original matrix [K] to compute all eigenvalues and eigenvectors The transformation of [K] into tridiagonal form prior to the QR iteration improves the efficiency of solution

82 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon Householder-QR-Inverse Iteration (HQRI) solution Transformation methods QR ITERATION

83 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Householder-QR-Inverse Iteration (HQRI) solution Transformation methods QR ITERATION

84 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Householder-QR-Inverse Iteration (HQRI) solution Transformation methods QR ITERATION

85 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Householder-QR-Inverse Iteration (HQRI) solution Transformation methods QR ITERATION

86 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon Transformation methods HQRI - Example

87 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Polynomial iteration techniques

88 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Polynomial iteration techniques These techniques operates on the characteristic polynomial p(λ) = det ([K]- λ[M]) to extract the zeros of the polynomial They are classified into Explicit and Implicit Polynomial iteration techniques

89 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Explicit Polynomial Iteration Polynomial iteration techniques Explicit polynomial iteration technique has been completely abandoned for the solution of the eigenvalue problems. A basic defect of the method is that small errors in the coeficients ak, cause large errors in the roots of the polynomial

90 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Implicit Polynomial Iteration Polynomial iteration techniques For [K]- λ[M] symmetric The fatorization is carried out using Gauss Elimination Method

91 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Implicit Polynomial Iteration Polynomial iteration techniques

92 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Implicit Polynomial Iteration Polynomial iteration techniques

93 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Convergence criterion Implicit Polynomial Iteration Polynomial iteration techniques

94 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 p( ) = det([K] – [M])     k-1 kk  k+1 Secant iteration Method Polynomial iteration techniques

95 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Implicit Polynomial Iteration Method - Example Polynomial iteration techniques

96 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Implicit Polynomial Iteration Method - Example Polynomial iteration techniques

97 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Subspace Iteration Method

98 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Allows the computation of a few eigenvalues and eigenvectors of large finite element eigenproblems Subspace iteration method The method essentially consists of the following steps: 1.Establish q starting iteration vectors, with q > p, where p is the number of eigenvalues and eigenvectors to be calculated 2.Use simultaneous inverse iteration on q vectors to extract the “best” eigenvalue and eigenvector approximations from the q iteration vector

99 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Subspace iteration method Subspace iteration algorithm

100 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Subspace iteration method Starting iteration vectors An efficient way to define the starting vector [X] k is by assuming its first column equals to the diagonal of [M]. Values +1 are then assigned to the elements of the remaining columns with smallest ratio kii/mii. All elements of each column apart from the one with smallest ratio kii/mii are zero!!!!

101 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Lanczos Transformation Method

102 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Very effective procedure for the solution of p eigenvalues and eigenvectors of finite element eigenproblems (with p < n, where n is the eigenproblem order) Lanczos method The basic steps of the Lanczos method transform, in theory, the generalized eigenproblem [K]{φ}=λ [M]{φ} into a standard form with a tridiagonal coefficient matrix

103 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Steps summary of Lanczos Transformation: Lanczos method

104 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Steps summary of Lanczos Transformation: Lanczos method

105 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Steps summary of Lanczos Transformation: Lanczos method

106 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon AE-256 Lanczos method One of the methods decribed previously can be used to obtain the eigenvalues of [T n ] (For instance, the Inverse Iteration Method combined with the Gram-Schmidt orthogonalization can be used for this purpose!).

107 Instituto Tecnológico de Aeronáutica Prof. Maurício Vicente Donadon Lanczos method Lanczos transformation - Example