The Finite Element Method

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1 Page he Finite Element Method for the Analysis of Non-Linear and Dynamic Systems Prof. o. Dr. Michael Havbro bo Faber Dr. Nebojsa Mojsilovic Swiss Federal Institute of EH Zurich, Switzerland

2 Contents of oday's Lecture Page 2 Solution of Equilibrium Equations in Dynamic Analysis ransformation Methods - he Jacobi Method - he Generalized Jacobi Method - he Householders QR Inverse Iteration Solution (HQRI) - he QR iteration - Calculation of eigenvectors

3 he Objective - Mode Superposition Page 3 Modal Generalized Displacements he direct integration methods necessitate that the finite element equations are evaluated for each time step he bandwidth of the matrixes M, C and depend on the numbering of the finite element nodal points In principle we could try to rearrange the nodal point numbering but this approach is cumbersome and has limitations Instead we transform the equations into a form which in terms of numerical effort is less expensive - by a change of basis

4 he Objective - Mode Superposition Page 4 Change of Basis to Modal Coordinates he following transformation is introduced: U () t = PX () t P: n x n square matrix X(t): time dependent vector of order n MX () t + CX () t + X () t = R () t MX () t = P MP, CX () t = P CP, X () t = P P, R () t = P R

5 he Objective - Mode Superposition Change of Basis to Modal Coordinates he question is how to choose P? MX () t + CX () t + X () t = R () t MX () t = PMP, CX () t = PCP, X () t = PP, R () t = PR A good choice is to take basis in the free vibration solution neglecting damping, i.e.: MU + U = 0 U = Φ sin ω ( t t ) which has a solution of the form 0 Page 5 Φ 2 = ω MΦ ( ωφ),( ωφ),...,( ωφ) n n Mode shape vectors

6 he Objective - Mode Superposition Page 6 Change of Basis to Modal Coordinates Any of the solutions satisfy MU + U = 0 ( ωφ),( ωφ),...,( ωφ) n n he n solutions may be written as: 2 2 Φ MΦΩ, Φ = Φ= Ω ; Φ MΦ = I 2 ω 2 2 ωn Φ= [ Φ Φ,..., Φ Ω =, 2, n ]; 2 ω n with: [ ]

7 he Objective - Mode Superposition Page 7 Change of Basis to Modal Coordinates Now using U() t = ΦX() t in MX () t + CX () t + X () t = R () t MX () t = PMP, CX () t = PCP, X () t = PP, R () t = PR we get X Φ CΦX Ω X Φ R 2 () t + () t + () t = () t with X = Φ M U ; X = Φ M U

8 ransformation Methods Page 8 Introduction We need to perform the transformations Φ Φ= Φ MΦ = Λ I he transformation may be pursued by iteration = P P M = P M P 2 3 = P22P2 = k+ Pk kp k 2 M3 = P2M2P 2 with M M = k+ PkM kp k = = M

9 ransformation Methods Page 9 Introduction P he aim being to select such as to bring k k closer to diagonal form i.e. : Λ M I k+, k+, k whereby there is: Φ = P P 2 P l k, M in practice we don t need convergence to Λ M I k+, k+, k

10 ransformation Methods Page 0 Introduction Λ, I In practice we don t need convergence to only to diagonal form; diag( ), M diag( M ), k k+ r k+ r If l is the last iteration there is: ( l ) + r ( l ) r Λ = diag( ) M + Φ = P P P 2 l diag( ) ( l ) M + r

11 ransformation Methods Page Introduction Based on these consideration a number of iteration schemes have been proposed here we will consider the Jacobi and the Householder-QR-method

12 ransformation Methods Page 2 he Jacobi Method (M = I) his method can be applied to calculate negative, zero and positive eigenvalues! Φ= λφ ith jth column θ = sinθ cosθ = k P + k k P cos θ sin where θ is selected such that element (i,j) in k+ becomes zero P k ith row jth row

13 ransformation Methods P P = k+ k k Page 3 he Jacobi Method where θ is selected such that element (i,j) in k+ becomes zero, i.e.: tan 2θ = k ( k 2k ) ij ( k) ( k) for k ( k) ( k) ii kjj ii kjj θ θ = for k = k 4 ( k) ( k) ii jj this zeroing can be performed for any (i,j) but after a new element is zeroed the other become non-zero again

14 ransformation Methods P P = k+ k k Page 4 he Jacobi Method A strategy is therefore to always zero the off diagonal elements furthest away from the diagonal but this is time consuming (the searching process) Another approach is simply to systematically go through all elements - sweep after sweep but this will repeatedly lead to zeroing of elements which are already almost equal to zero

15 ransformation Methods P P = k+ k k Page 5 he Jacobi Method A threshold Jacobi method may be formulated such that only elements larger than a certain threshold are zeroed i.e. convergence is achieved when: k k ( l+ ) ( l) ii ii -s 0 i=,2..., n (zero convergence) ( l+ ) kii ( l + ) 2 kij -s 0 all i, j; i< j ( l+ ) ( l+ ) ii kjj ( ) k (coupling convergence)

16 ransformation Methods P P = k+ k k Page 6 he Jacobi Method he procedure is summarized as: ) Initialize the threshold for the mth sweep (0-2m ) 2) For all i,j with i<j calculate the coupling factor ( ) ( k l+ ) 2 ij k k ( l+ ) ( l+ ) ii jj check if it is larger than the current threshold and only then apply transformation 3) Check for (zero) convergence and if fulfilled check for coupling convergence Convergence can be proved! (in practice s = 2)

17 ransformation Methods Page 7 he Generalized Jacobi Method (M I) Operates directly on and M Φ= λmφ P k ith jth column α = γ α and γ are selected such as to diagonalize and M simultaneously l ith row jth row

18 ransformation Methods he Generalized Jacobi Method (M I) Φ = λmφ ith jth column α Pk = γ Page 8 ith row jth row Performing the multiplications PP k k k PMP k k k and requiring that : ( k + ) ( k + ) kij mij = = 0 we get two equations to determine α and γ: αk + ( + αγ) k + γk = 0 ( k) ( k) ( k) ii ij jj αm + ( + αγ ) m + γ m = 0 ( k ) ( k ) ( k ) ii ij jj

19 Page 9 ( k) ( k) ( k) ransformation Methods αk + αγ k γk he Generalized Jacobi Method (M I) We may solve the two equations from: k = k m -m k ( k) ( k) ( k) ( k) ( k) ii ii ij ii ij k = k m -m k ( k) ( k) ( k) ( k) ( k) jj jj ij jj ij k = k m -k m ( k ) ( k ) ( k ) ( k ) ( k ) ii jj ii ii + ( ) + = 0 ii ij jj αm + ( + αγ) m + γm = 0 ( k) ( k) ( k) ii ij jj ( k ) ( k ) ( k ) ( k ) 2 k jj ( k) ii ( k) ( k) ; α, x sign( k ) kii kjj k k k γ = = = + + x x 2 2 the iteration is performed as before but we must now check that the coupling factors are zero and that the off-diagonal elements are zero for both and M

20 ransformation Methods Page 20 he Householder-QR-Inverse Iteration Solution We consider here Φ = λφ he HQRI solution method stands for the following steps: ) Householder transformations are employed to reduce the matrix to tridiagonal form 2) QR iteration yields all eigenvalues 3) Using inverse iteration the eigenvectors of the tridiagonal matrix are calculated these vectors are then transformed into the eigenvectors of

21 ransformation Methods Page 2 he Householder-QR-Inverse Iteration Solution he Householder transformations: We transform to tridiagonal form by n-2 2 transformations = P P k+ k k k, k =, 2..., n 2 = P =I-θw w, θ = 2 ww k k k k Reflection matrix k

22 ransformation Methods Page 22 he Householder-QR-Inverse Iteration Solution he Householder transformations: We may determine w k from: 0 0 P=, w = w 0 P k k = k 0 k k 0 k k P = P P = = 0 P k 0 P P k P P 2

23 ransformation Methods Page 23 he Householder-QR-Inverse Iteration Solution 0 0 P=, w = w 0 P he Householder transformations: We would like 2 in the form: 2 k 0 0 = k k = k 0 k k 0 k k P = P P = = 0 P k 0 P P k P P 2 (I-θ w w ) k = ± k e 2 whereby he procedure is now continued by considering 2 as in the previous w = k +sign( k ) k e 2 2

24 ransformation Methods Page 24 he Householder-QR-Inverse Iteration Solution he QR iteration he basic step is to decompose into the form: = QR and then RQ = Q Q P P which is of the same form as: 2 R and Q may be found from: Pnn, P3, P2, = R Q = P 2,P 3, P nn, =

25 ransformation Methods Page 25 he Householder-QR-Inverse Iteration Solution he QR iteration We may now iterate: = Q R k k k = R Q k + k k Λ and Q Q Q Φ k k+ k k, k

26 ransformation Methods Page 26 he Householder-QR-Inverse Iteration Solution he QR iteration As we started out with the QR method should have been used on the tridiagonalized matrix If we do that we find the eigenvectors of the triangularized matrix i.e. n- or Denoting the ith eigenvector of, by Ψi we may transform back to the eigenvector of by: Φ = PP P Ψ i 2 n 2 i

27 Very Condensed Summary Page 27 Non-Linear Finite Element Method Basic principle p we sub-divide the loading in time steps, linearize the equlibrium equations (tankent stiffness matrix) and iterate for the solution (Newton Rapson/Modified NR, quasi Newton) Cauchy stress tensor, Green-Lagrange strain tensor, Almansi strain tensor, second Piola irchoff stress tensor otal Lagrange formulation reference to original configuration Updated Lagrange reference to present configuration Material non-linear L=UL Analysis of truss and beam elements

28 Very Condensed Summary Page 28 Non-Linear Finite Element Method ISO-parametric elements 2D: Axisymmetric element, plane strain, plane stress 3D: Solid elements, beam and axi-symmetric shell elements Constitutive relations for non-linear materials Elasto-plastic Prandtl-Reuss van Mises (hardening/steel) Drucker-Prager (rock/soil) hermoelastoplasticity, visco-plasticity, creep Elasto-plasticity velocity) Contact problems Rate based formulations (Jaumann stress rate

29 Very Condensed Summary Page 29 Dynamic Finite Element Method Direct integration subdivide tiem into steps, assume a certain variation of the motion - and integrate Explicit reference to t (no factorization) Imlicit reference to t+δt (factorization) Central differences, Houbolt, Wilson θ, Newmark Modal analysis (with and without damping/rayleigh damping) Precision and convergence

30 Very Condensed Summary Page 30 Dynamic Finite Element Method Material non-linear dynamics iterate within each time step based on a linearization of the stiffness matrix Eigenvalue problems - vector iteration method - transformation methods

Identification Methods for Structural Systems. Prof. Dr. Eleni Chatzi Lecture March, 2016

Identification Methods for Structural Systems. Prof. Dr. Eleni Chatzi Lecture March, 2016 Prof. Dr. Eleni Chatzi Lecture 4-09. March, 2016 Fundamentals Overview Multiple DOF Systems State-space Formulation Eigenvalue Analysis The Mode Superposition Method The effect of Damping on Structural