Eigenvector Expansion Uncoupled Equations of Motion Truncated Sum Definition Elastic Forces Example Truncated sum A truncated sum uses only M < N of the lower frequency modes x ( t ) ≈ � M<N ψ i q i ( t ) , i =1 and, under wide assumptions, gives you a good approximation of the structural response. The importance of truncated sum approximation is twofold: ▶ less computational effort: less eigenpairs to calculate, less equation of motion to integrate etc ▶ in FEM models the higher modes are rough approximations to structural ones (mostly due to uncertainties in mass distribution details) and the truncated sum excludes potentially spurious contributions from the response. Giacomo Boffi Truncated Sums, Matrix Iteration
Eigenvector Expansion Uncoupled Equations of Motion Truncated Sum Definition Elastic Forces Example Elastic Forces Until now, we showed interest in displacements only, but we are interested in elastic forces too. We know that elastic forces can be expressed in terms of displacements and the stiffness matrix: f S ( t ) = K x ( t ) = Kψ 1 q 1 ( t ) + Kψ 2 q 2 ( t ) + · · · . From the characteristic equation we know that Kψ i = ω 2 i Mψ i substituting in the previous equation f S ( t ) = ω 2 1 Mψ 1 q 1 ( t ) + ω 2 2 Mψ 2 q 2 ( t ) + · · · . Giacomo Boffi Truncated Sums, Matrix Iteration
Eigenvector Expansion Uncoupled Equations of Motion Truncated Sum Definition Elastic Forces Example Elastic Forces, 2 The high frequency modes contribution to the elastic forces, e.g. f S ( t ) = ω 2 1 Mψ 1 q 1 ( t ) + · · · + ω 2 20 Mψ 20 q 20 ( t ) + · · · , when compared to low frequency mode contributions are more important than their contributions to displacement, because of the multiplicative term ω 2 i . From this fact follows that, to estimate internal forces within a given accuracy a greater number of modes must be considered in a truncated sum than the number required to estimate displacements within the same accuracy. Giacomo Boffi Truncated Sums, Matrix Iteration
Eigenvector Expansion Uncoupled Equations of Motion Truncated Sum Definition Elastic Forces Example Example: problem statement m 1 x 1 k 1 m 2 k 1 = 120 MN / m , m 1 = 200 t , x 2 k 2 k 2 = 240 MN / m , m 2 = 300 t , m 3 k 3 = 360 MN / m , m 3 = 400 t . x 3 k 3 1. The above structure is subjected to these initial conditions, � � x T 0 = 5 mm 4 mm 3 mm , � � x T ˙ 0 = 0 9 mm/s 0 . Write the equation of motion using modal superposition. 2. The above structure is subjected to a half-sine impulse, 2 . 5 MN sin π t p T ( t ) = � � 1 2 2 t 1 , with t 1 = 0 . 02 s . Write the equation of motion using modal superposition. Giacomo Boffi Truncated Sums, Matrix Iteration
Eigenvector Expansion Uncoupled Equations of Motion Truncated Sum Definition Elastic Forces Example Example: structural matrices k 1 = 120 MN / m , m 1 = 200 t , m 1 k 2 = 240 MN / m , m 2 = 300 t , x 1 k 1 k 3 = 360 MN / m , m 3 = 400 t . m 2 x 2 k 2 m 3 x 3 k 3 The structural matrices can be written 1 − 1 0 with k = 120 MN = k K , K = k − 1 3 − 2 m , 0 − 2 5 2 0 0 = m M , M = m 0 3 0 with m = 100000 kg. 0 0 4 Giacomo Boffi Truncated Sums, Matrix Iteration
Eigenvector Expansion Uncoupled Equations of Motion Truncated Sum Definition Elastic Forces Example Example: adimensional eigenvalues We want the solutions of the characteristic equation, so we start writing that the determinant of the equation must be zero: � � � � ω 2 � = 0 , � � � K − Ω 2 M � K − k/m M � = � rad � 2 Ω 2 . with ω 2 = 1200 s Expanding the determinant � � 1 − 2Ω 2 − 1 0 � � � � 3 − 3Ω 2 − 1 − 2 = 0 � � � � 5 − 4Ω 2 0 − 2 � � we have the following algebraic equation of 3rd order in Ω 2 � � Ω 6 − 11 4 Ω 4 + 15 8 Ω 2 − 1 24 = 0 . 4 Giacomo Boffi Truncated Sums, Matrix Iteration
Eigenvector Expansion Uncoupled Equations of Motion Truncated Sum Definition Elastic Forces Example Example: table of eigenvalues etc Here are the adimensional roots Ω 2 i , i = 1 , 2 , 3 , the dimensional i = 1200 rad 2 eigenvalues ω 2 s 2 Ω 2 i and all the derived dimensional quantities: Ω 2 Ω 2 Ω 2 1 = 0 . 17573 2 = 0 . 8033 3 = 1 . 7710 ω 2 ω 2 ω 2 1 = 210 . 88 2 = 963 . 96 3 = 2125 . 2 ω 1 = 14 . 522 ω 2 = 31 . 048 ω 3 = 46 . 099 f 1 = 2 . 3112 f 2 = 4 . 9414 f 3 = 7 . 3370 T 1 = 0 . 43268 T 3 = 0 . 20237 T 3 = 0 . 1363 Giacomo Boffi Truncated Sums, Matrix Iteration
Eigenvector Expansion Uncoupled Equations of Motion Truncated Sum Definition Elastic Forces Example Example: eigenvectors and modal matrices With ψ 1 j = 1 , using the 2nd and 3rd equations, � 3 − 3Ω 2 � � ψ 2 j � � 1 � − 2 j = 5 − 4Ω 2 − 2 ψ 3 j 0 j The above equations must be solved for j = 1 , 2 , 3 . The solutions are finally collected in the eigenmatrix 1 1 1 . +0 . 648535272183 − 0 . 606599092464 − 2 . 54193617967 Ψ = +0 . 301849953585 − 0 . 678977475113 +2 . 43962752148 The Modal Matrices are 362 . 6 0 0 M ⋆ = × 10 3 kg , 0 494 . 7 0 0 0 4519 . 1 76 . 50 0 0 × 10 6 N K ⋆ = 0 477 . 0 0 m 0 0 9603 . 9 Giacomo Boffi Truncated Sums, Matrix Iteration
Eigenvector Expansion Uncoupled Equations of Motion Truncated Sum Definition Elastic Forces Example Example: initial conditions in modal coordinates 5 +5 . 9027 q 0 = ( M ⋆ ) − 1 Ψ T M 4 mm = − 1 . 0968 mm , 3 +0 . 1941 0 +4 . 8288 mm mm q 0 = ( M ⋆ ) − 1 Ψ T M ˙ 9 = − 3 . 3101 s s 0 − 1 . 5187 Giacomo Boffi Truncated Sums, Matrix Iteration
Eigenvector Expansion Uncoupled Equations of Motion Truncated Sum Definition Elastic Forces Example Example: structural response These are the displacements, in mm x 1 = +5 . 91 cos(14 . 5 t + . 06) + 1 . 10 cos(31 . 0 t − 3 . 04) + 0 . 20 cos(46 . 1 t − 0 . 17) x 2 = +3 . 83 cos(14 . 5 t + . 06) − 0 . 67 cos(31 . 0 t − 3 . 04) − 0 . 50 cos(46 . 1 t − 0 . 17) x 3 = +1 . 78 cos(14 . 5 t + . 06) − 0 . 75 cos(31 . 0 t − 3 . 04) + 0 . 48 cos(46 . 1 t − 0 . 17) Giacomo Boffi Truncated Sums, Matrix Iteration
Eigenvector Expansion Uncoupled Equations of Motion Truncated Sum Definition Elastic Forces Example Example: structural response These are the displacements, in mm x 1 = +5 . 91 cos(14 . 5 t + . 06) + 1 . 10 cos(31 . 0 t − 3 . 04) + 0 . 20 cos(46 . 1 t − 0 . 17) x 2 = +3 . 83 cos(14 . 5 t + . 06) − 0 . 67 cos(31 . 0 t − 3 . 04) − 0 . 50 cos(46 . 1 t − 0 . 17) x 3 = +1 . 78 cos(14 . 5 t + . 06) − 0 . 75 cos(31 . 0 t − 3 . 04) + 0 . 48 cos(46 . 1 t − 0 . 17) and these the elastic/inertial forces, in kN x 1 = +249 . cos(14 . 5 t + . 06) + 212 . cos(31 . 0 t − 3 . 04) + 084 . cos(46 . 1 t − 0 . 17) x 2 = +243 . cos(14 . 5 t + . 06) − 193 . cos(31 . 0 t − 3 . 04) − 319 . cos(46 . 1 t − 0 . 17) x 3 = +151 . cos(14 . 5 t + . 06) − 288 . cos(31 . 0 t − 3 . 04) + 408 . cos(46 . 1 t − 0 . 17) Giacomo Boffi Truncated Sums, Matrix Iteration
Eigenvector Expansion Uncoupled Equations of Motion Truncated Sum Definition Elastic Forces Example Example: structural response These are the displacements, in mm x 1 = +5 . 91 cos(14 . 5 t + . 06) + 1 . 10 cos(31 . 0 t − 3 . 04) + 0 . 20 cos(46 . 1 t − 0 . 17) x 2 = +3 . 83 cos(14 . 5 t + . 06) − 0 . 67 cos(31 . 0 t − 3 . 04) − 0 . 50 cos(46 . 1 t − 0 . 17) x 3 = +1 . 78 cos(14 . 5 t + . 06) − 0 . 75 cos(31 . 0 t − 3 . 04) + 0 . 48 cos(46 . 1 t − 0 . 17) and these the elastic/inertial forces, in kN x 1 = +249 . cos(14 . 5 t + . 06) + 212 . cos(31 . 0 t − 3 . 04) + 084 . cos(46 . 1 t − 0 . 17) x 2 = +243 . cos(14 . 5 t + . 06) − 193 . cos(31 . 0 t − 3 . 04) − 319 . cos(46 . 1 t − 0 . 17) x 3 = +151 . cos(14 . 5 t + . 06) − 288 . cos(31 . 0 t − 3 . 04) + 408 . cos(46 . 1 t − 0 . 17) As expected, the contributions of the higher modes are more important for the forces, less important for the displacements. Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Inverse Iteration Matrix Iteration with Shifts Rayleigh Methods Part II Matrix Iteration Procedures Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Inverse Iteration Matrix Iteration with Shifts Rayleigh Methods
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Inverse Iteration Matrix Iteration with Shifts Rayleigh Methods Introduction Dynamic analysis of MDOF systems based on modal superposition is both simple and efficient ▶ simple: the modal response can be easily computed, analitically or numerically, with the techniques we have seen for SDOF systems, ▶ efficient: in most cases, only the modal responses of a few lower modes are required to accurately describe the structural response. Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Inverse Iteration Matrix Iteration with Shifts Rayleigh Methods Introduction The structural matrices being easily assembled using the FEM , the modal superposition procedure is ready to be applied to structures with thousands, millions of DOF ’s! Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Inverse Iteration Matrix Iteration with Shifts Rayleigh Methods Introduction The structural matrices being easily assembled using the FEM , the modal superposition procedure is ready to be applied to structures with thousands, millions of DOF ’s! But wait, we can know how to compute the eigenpairs only when the analyzed structure has very few degrees of freedom... Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Inverse Iteration Matrix Iteration with Shifts Rayleigh Methods Introduction The structural matrices being easily assembled using the FEM , the modal superposition procedure is ready to be applied to structures with thousands, millions of DOF ’s! But wait, we can know how to compute the eigenpairs only when the analyzed structure has very few degrees of freedom... We will discuss how it is possible to compute the eigenpairs of arbitrarily large dynamic systems using the so called Matrix Iteration procedure (and a number of variations derived from this fundamental idea). Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis A Possible Procedure Matrix Iteration Procedure Convergence of Matrix Iteration Procedure Second Mode Analysis Higher Modes Inverse Iteration Matrix Iteration with Shifts Rayleigh Methods
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Idea Procedure Inverse Iteration Convergence Matrix Iteration with Shifts Rayleigh Methods Equilibrium First, we will see an iterative procedure whose outputs are the first, or fundamental, mode shape vector and the corresponding eigenvalue. When an undamped system freely vibrates with a harmonic time dependency of frequency ω i , the equation of motion, simplifying the time dependency, is K ψ i = ω 2 i M ψ i . In equilibrium terms, the elastic forces are equal to the inertial forces when the systems oscillates with frequency ω 2 i and mode shape ψ i Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Idea Procedure Inverse Iteration Convergence Matrix Iteration with Shifts Rayleigh Methods Proposal of an iterative procedure Our iterative procedure will be based on finding a new displacement vector x n +1 such that the elastic forces f S = K x i +1 are in equilibrium with the inertial forces due to the old displacement vector x n , f I = ω 2 i M x n , that is K x n +1 = ω 2 i M x n . Premultiplying by the inverse of K and introducing the Dynamic Matrix , D = K − 1 M x n +1 = ω 2 i K − 1 M x n = ω 2 i D x n . Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Idea Procedure Inverse Iteration Convergence Matrix Iteration with Shifts Rayleigh Methods Proposal of an iterative procedure Our iterative procedure will be based on finding a new displacement vector x n +1 such that the elastic forces f S = K x i +1 are in equilibrium with the inertial forces due to the old displacement vector x n , f I = ω 2 i M x n , that is K x n +1 = ω 2 i M x n . Premultiplying by the inverse of K and introducing the Dynamic Matrix , D = K − 1 M x n +1 = ω 2 i K − 1 M x n = ω 2 i D x n . In the generative equation above we miss a fundamental part, the square of the free vibration frequency ω 2 i . Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Idea Procedure Inverse Iteration Convergence Matrix Iteration with Shifts Rayleigh Methods The Matrix Iteration Procedure, 1 This problem is solved considering the x n as a sequence of normalized vectors and introducing the idea of an unnormalized new displacement vector, ˆ x n +1 , x n +1 = D x n , ˆ note that we removed the explicit dependency on ω 2 i . Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Idea Procedure Inverse Iteration Convergence Matrix Iteration with Shifts Rayleigh Methods The Matrix Iteration Procedure, 2 The normalized vector is obtained applying to ˆ x n +1 a normalizing factor, F n +1 , x n +1 = ˆ x n +1 , F n +1 1 x n +1 = ω 2 i D x n = ω 2 F = ω 2 but i ˆ x n +1 , ⇒ i Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Idea Procedure Inverse Iteration Convergence Matrix Iteration with Shifts Rayleigh Methods The Matrix Iteration Procedure, 2 The normalized vector is obtained applying to ˆ x n +1 a normalizing factor, F n +1 , x n +1 = ˆ x n +1 , F n +1 1 x n +1 = ω 2 i D x n = ω 2 F = ω 2 but i ˆ x n +1 , ⇒ i If we agree that, near convergence, x n +1 ≈ x n , substituting in the previous equation we have x n x n +1 ≈ x n = ω 2 ω 2 i ˆ x n +1 ⇒ i ≈ . ˆ x n +1 Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Idea Procedure Inverse Iteration Convergence Matrix Iteration with Shifts Rayleigh Methods The Matrix Iteration Procedure, 2 The normalized vector is obtained applying to ˆ x n +1 a normalizing factor, F n +1 , x n +1 = ˆ x n +1 , F n +1 1 x n +1 = ω 2 i D x n = ω 2 F = ω 2 but i ˆ x n +1 , ⇒ i If we agree that, near convergence, x n +1 ≈ x n , substituting in the previous equation we have x n x n +1 ≈ x n = ω 2 ω 2 i ˆ x n +1 ⇒ i ≈ . ˆ x n +1 Of course the division of two vectors is not an option, so we want to twist it into something useful. Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Idea Procedure Inverse Iteration Convergence Matrix Iteration with Shifts Rayleigh Methods Normalization First, consider x n = ψ i : in this case, for j = 1 , . . . , N it is x n +1 ,j = ω 2 x n,j / ˆ i . When x n ̸ = ψ i it is possible to demonstrate that we can bound the eigenvalue � x n,j � x n,j � � ≤ ω 2 min i ≤ max . x n +1 ,j ˆ x n +1 ,j ˆ j =1 ,...,N j =1 ,...,N Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Idea Procedure Inverse Iteration Convergence Matrix Iteration with Shifts Rayleigh Methods Normalization First, consider x n = ψ i : in this case, for j = 1 , . . . , N it is x n +1 ,j = ω 2 x n,j / ˆ i . When x n ̸ = ψ i it is possible to demonstrate that we can bound the eigenvalue � x n,j � x n,j � � ≤ ω 2 min i ≤ max . x n +1 ,j ˆ x n +1 ,j ˆ j =1 ,...,N j =1 ,...,N A more rational approach would make reference to a proper vector norm, so using our preferred vector norm we can write x T ˆ n +1 M x n ω 2 i ≈ , x T ˆ n +1 M ˆ x n +1 Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Idea Procedure Inverse Iteration Convergence Matrix Iteration with Shifts Rayleigh Methods Normalization First, consider x n = ψ i : in this case, for j = 1 , . . . , N it is x n +1 ,j = ω 2 x n,j / ˆ i . When x n ̸ = ψ i it is possible to demonstrate that we can bound the eigenvalue � x n,j � x n,j � � ≤ ω 2 min i ≤ max . x n +1 ,j ˆ x n +1 ,j ˆ j =1 ,...,N j =1 ,...,N A more rational approach would make reference to a proper vector norm, so using our preferred vector norm we can write x T ˆ n +1 M x n ω 2 i ≈ , x T ˆ n +1 M ˆ x n +1 (if memory helps, this is equivalent to the R 11 approximation, that we introduced studying Rayleigh quotient refinements). Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Idea Procedure Inverse Iteration Convergence Matrix Iteration with Shifts Rayleigh Methods Proof of Convergence, 1 Until now we postulated that the sequence x n converges to some, unspecified eigenvector ψ i , now we will demonstrate that the sequence converge to the first, or fundamental mode shape, n →∞ x n = ψ 1 . lim 1. Expand x 0 in terms of eigenvectors an modal coordinates: x 0 = ψ 1 q 1 , 0 + ψ 2 q 2 , 0 + ψ 3 q 3 , 0 + · · · . 2. The inertial forces, assuming that the system is vibrating according to the fundamental frequency, are f I,n =0 = ω 2 1 M ( ψ 1 q 1 , 0 + ψ 2 q 2 , 0 + ψ 3 q 3 , 0 + · · · ) � � ω 2 ω 2 ω 2 1 + ω 2 1 = M 1 ψ 1 q 1 , 0 2 ψ 2 q 2 , 0 + · · · . ω 2 ω 2 1 2 Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Idea Procedure Inverse Iteration Convergence Matrix Iteration with Shifts Rayleigh Methods Proof of Convergence, 2 3. The deflections due to these forces (no hat!, we have multiplied by ω 2 1 ) are � � ω 2 ω 2 x n =1 = K − 1 M ω 2 1 + ω 2 1 1 ψ 1 q 1 , 0 2 ψ 2 q 2 , 0 + · · · , ω 2 ω 2 1 2 (note that every term has been multiplied and divided by the corresponding eigenvalue ω 2 i ). Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Idea Procedure Inverse Iteration Convergence Matrix Iteration with Shifts Rayleigh Methods Proof of Convergence, 3 4. With ω 2 j M ψ j = Kψ j , substituting and simplifying K − 1 K = I , � � 1 � ω 2 x n =1 = K − 1 1 Kψ 1 q 1 , 0 + ω 2 1 � ω 2 � 1 1 Kψ 2 q 2 , 0 + ω 2 2 � � 1 � ω 2 1 Kψ 3 q 3 , 0 + · · · ω 2 3 ω 2 ω 2 ω 2 1 1 1 = ψ 1 q 1 , 0 + ψ 2 q 2 , 0 + ψ 3 q 3 , 0 + · · · , ω 2 ω 2 ω 2 1 2 3 Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Idea Procedure Inverse Iteration Convergence Matrix Iteration with Shifts Rayleigh Methods Proof of Convergence, 4 5. applying again this procedure � � � ω 2 � 2 � ω 2 � 2 � ω 2 � 2 1 1 1 x n =2 = ψ 1 q 1 , 0 + ψ 2 q 2 , 0 + ψ 3 q 3 , 0 + · · · , ω 2 ω 2 ω 2 1 2 3 6. applying the procedure n times � � ω 2 � n � ω 2 � n � ω 2 � n � 1 1 1 x n = ψ 1 q 1 , 0 + ψ 2 q 2 , 0 + ψ 3 q 3 , 0 + · · · . ω 2 ω 2 ω 2 1 2 3 Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Idea Procedure Inverse Iteration Convergence Matrix Iteration with Shifts Rayleigh Methods Proof of Convergence, 5 Going to the limit, n →∞ x n = ψ 1 q 1 , 0 lim because � � n ω 2 1 lim = δ 1 j ω 2 n →∞ j Consequently, | x n | x n | = ω 2 lim 1 | ˆ n →∞ Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Purified Vectors Sweeping Matrix Higher Modes Inverse Iteration Matrix Iteration with Shifts Rayleigh Methods
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Purified Vectors Inverse Iteration Sweeping Matrix Matrix Iteration with Shifts Rayleigh Methods Purified Vectors If we know ψ 1 and ω 2 1 from the matrix iteration procedure it is possible to compute the second eigenpair, following a slightly different procedure. Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Purified Vectors Inverse Iteration Sweeping Matrix Matrix Iteration with Shifts Rayleigh Methods Purified Vectors If we know ψ 1 and ω 2 1 from the matrix iteration procedure it is possible to compute the second eigenpair, following a slightly different procedure. Express the initial iterate in terms of the (unknown) eigenvectors, x n =0 = Ψ q n =0 and premultiply by the (known) ψ T 1 M : ψ T 1 M x n =0 = M 1 q 1 ,n =0 solving for q 1 ,n =0 q 1 ,n =0 = ψ T 1 M x n =0 . M 1 Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Purified Vectors Inverse Iteration Sweeping Matrix Matrix Iteration with Shifts Rayleigh Methods Purified Vectors If we know ψ 1 and ω 2 1 from the matrix iteration procedure it is possible to compute the second eigenpair, following a slightly different procedure. Express the initial iterate in terms of the (unknown) eigenvectors, x n =0 = Ψ q n =0 and premultiply by the (known) ψ T 1 M : ψ T 1 M x n =0 = M 1 q 1 ,n =0 solving for q 1 ,n =0 q 1 ,n =0 = ψ T 1 M x n =0 . M 1 Knowing the amplitude of the 1st modal contribution to x n =0 we can write a purified vector, y n =0 = x n =0 − ψ 1 q 1 ,n =0 . Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Purified Vectors Inverse Iteration Sweeping Matrix Matrix Iteration with Shifts Rayleigh Methods Convergence (?) It is easy to demonstrate that using y n =0 as our starting vector | y n | y n | = ω 2 n →∞ y n = ψ 2 q 2 ,n =0 , lim lim 2 . | ˆ n →∞ because the initial amplitude of the first mode is null. Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Purified Vectors Inverse Iteration Sweeping Matrix Matrix Iteration with Shifts Rayleigh Methods Convergence (?) It is easy to demonstrate that using y n =0 as our starting vector | y n | y n | = ω 2 n →∞ y n = ψ 2 q 2 ,n =0 , lim lim 2 . | ˆ n →∞ because the initial amplitude of the first mode is null. Due to numerical errors in the determination of fundamental mode and in the procedure itself, using a plain matrix iteration the procedure however converges to the 1st eigenvector, so to preserve convergence to the 2nd mode it is necessary that the iterated vector y n is purified at each step n . Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Purified Vectors Inverse Iteration Sweeping Matrix Matrix Iteration with Shifts Rayleigh Methods Purification Procedure The purification procedure is simple, at each step the amplitude of the 1st mode is first computed, then removed from the iterated vector y n q 1 ,n = ψ T 1 My n /M 1 , � � I − 1 ψ 1 ψ T y n +1 = D ( y n − ψ 1 q 1 ,n ) = D ˆ 1 M y n M 1 Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Purified Vectors Inverse Iteration Sweeping Matrix Matrix Iteration with Shifts Rayleigh Methods Purification Procedure The purification procedure is simple, at each step the amplitude of the 1st mode is first computed, then removed from the iterated vector y n q 1 ,n = ψ T 1 My n /M 1 , � � I − 1 ψ 1 ψ T y n +1 = D ( y n − ψ 1 q 1 ,n ) = D ˆ 1 M y n M 1 M 1 ψ 1 ψ T 1 Introducing the sweeping matrix S 1 = I − 1 M and the modified dynamic matrix D 2 = DS 1 , we can write y n +1 = DS 1 y n = D 2 y n . ˆ Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Purified Vectors Inverse Iteration Sweeping Matrix Matrix Iteration with Shifts Rayleigh Methods Purification Procedure The purification procedure is simple, at each step the amplitude of the 1st mode is first computed, then removed from the iterated vector y n q 1 ,n = ψ T 1 My n /M 1 , � � I − 1 ψ 1 ψ T y n +1 = D ( y n − ψ 1 q 1 ,n ) = D ˆ 1 M y n M 1 M 1 ψ 1 ψ T 1 Introducing the sweeping matrix S 1 = I − 1 M and the modified dynamic matrix D 2 = DS 1 , we can write y n +1 = DS 1 y n = D 2 y n . ˆ This is known as matrix iteration with sweeps . Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Inverse Iteration Matrix Iteration with Shifts Rayleigh Methods
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Inverse Iteration Matrix Iteration with Shifts Rayleigh Methods Third Mode Using again the idea of purifying the iterated vector, starting with the knowledge of the first and the second eigenpair, y n +1 = D ( y n − ψ 1 q 1 ,n − ψ 2 q 2 ,n ) ˆ with q n, 1 as before and q 2 ,n = ψ T 2 My n /M 2 , substituting in the expression for the purified vector 1 − 1 � � M 1 ψ 1 ψ T M 2 ψ 2 ψ T y n +1 = D ˆ I − 1 M 2 M y n � �� � S 1 Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Inverse Iteration Matrix Iteration with Shifts Rayleigh Methods Third Mode Using again the idea of purifying the iterated vector, starting with the knowledge of the first and the second eigenpair, y n +1 = D ( y n − ψ 1 q 1 ,n − ψ 2 q 2 ,n ) ˆ with q n, 1 as before and q 2 ,n = ψ T 2 My n /M 2 , substituting in the expression for the purified vector 1 − 1 � � M 1 ψ 1 ψ T M 2 ψ 2 ψ T y n +1 = D ˆ I − 1 M 2 M y n � �� � S 1 The conclusion is that the sweeping matrix and the modified dynamic matrix to be used to compute the 3rd eigenvector are 1 M 2 ψ 2 ψ T S 2 = S 1 − 2 M , D 3 = D S 2 . Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Inverse Iteration Matrix Iteration with Shifts Rayleigh Methods Generalization to Higher Modes The results obtained for the third mode are easily generalised. It is easy to verify that the following procedure can be used to compute all the modes. Define S 0 = I , take i = 1 , 1. compute the modified dynamic matrix to be used for mode i , D i = D S i − i 2. compute ψ i using the modified dynamic matrix; 3. compute the modal mass M i = ψ T M ψ ; 4. compute the sweeping matrix S i that sweeps the contributions of the first i modes from trial vectors, S i = S i − 1 − 1 M i ψ i ψ T i M ; 5. increment i , GOTO 1. Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Inverse Iteration Matrix Iteration with Shifts Rayleigh Methods Generalization to Higher Modes The results obtained for the third mode are easily generalised. It is easy to verify that the following procedure can be used to compute all the modes. Define S 0 = I , take i = 1 , 1. compute the modified dynamic matrix to be used for mode i , D i = D S i − i 2. compute ψ i using the modified dynamic matrix; 3. compute the modal mass M i = ψ T M ψ ; 4. compute the sweeping matrix S i that sweeps the contributions of the first i modes from trial vectors, S i = S i − 1 − 1 M i ψ i ψ T i M ; 5. increment i , GOTO 1. Well, we finally have a method that can be used to compute all the eigenpairs of our dynamic problems, full circle! Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Inverse Iteration Matrix Iteration with Shifts Rayleigh Methods Discussion The method of matrix iteration with sweeping is not used in production because 1. D is a full matrix, even if M and K are banded matrices, and the matrix product that is the essential step in every iteration is computationally onerous, 2. the procedure is however affected by numerical errors, so, after having demonstrated that it is possible to compute all the eigenvectors of a large problem using an iterative procedure it is time to look for different, more efficient iterative procedures. Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Inverse Iteration LU Decomposition Back Substitution Matrix Iteration with Shifts Rayleigh Methods
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes LU Decomposition Inverse Iteration Back Substitution Matrix Iteration with Shifts Rayleigh Methods Introduction to Inverse Iteration Inverse iteration is based on the fact that the symmetric stiffness matrix has a banded structure, that is a relatively large triangular portion of the matrix is composed by zeroes. The banded structure is due to the FEM model: in every equation of equilibrium the only non zero elastic force coefficients are due to the degrees of freedom of the few FE’s that contain the degree of freedom for which the equilibrium is written. Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes LU Decomposition Inverse Iteration Back Substitution Matrix Iteration with Shifts Rayleigh Methods Definition of LU decomposition Every symmetric, banded matrix can be subjected to a so called LU decomposition, that is, for K we write K = L U where L and U are, respectively, a lower- and an upper-banded matrix. If we denote with b the bandwidth of K , we have � i < j � � l ij L = with l ij ≡ 0 for j < i − b and � i > j � � U = u ij with u ij ≡ 0 for j > i + b Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes LU Decomposition Inverse Iteration Back Substitution Matrix Iteration with Shifts Rayleigh Methods Twice the equations? In this case, with w n = M x n , the recursion can be written L U x n +1 = w n or as a system of equations, U x n +1 = z n +1 L z n +1 = w n Apparently, we have doubled the number of unknowns, but the z j ’s can be easily computed by the procedure of back substitution . Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes LU Decomposition Inverse Iteration Back Substitution Matrix Iteration with Shifts Rayleigh Methods Back Substitution Temporarily dropping the n and n + 1 subscripts, we can write z 1 = ( w 1 ) /l 11 z 2 = ( w 2 − l 21 z 1 ) /l 22 z 3 = ( w 3 − l 31 z 1 − l 32 z 2 ) /l 33 · · · i − 1 � z i = ( w i − l ij z j ) /l ii j = i − b · · · Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes LU Decomposition Inverse Iteration Back Substitution Matrix Iteration with Shifts Rayleigh Methods Back Substitution Temporarily dropping the n and n + 1 subscripts, we can write z 1 = ( w 1 ) /l 11 z 2 = ( w 2 − l 21 z 1 ) /l 22 z 3 = ( w 3 − l 31 z 1 − l 32 z 2 ) /l 33 · · · i − 1 � z i = ( w i − l ij z j ) /l ii j = i − b · · · The x are then given by U x = z . Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes LU Decomposition Inverse Iteration Back Substitution Matrix Iteration with Shifts Rayleigh Methods Back Substitution We have computed z by back substitution, we must solve U x = z but U is upper triangular, so we have x N = ( z N ) /u NN x N − 1 = ( z N − 1 − u N − 1 ,N z N ) /u N − 1 ,N − 1 x N − 2 = ( z N − 2 − u N − 2 ,N z N − u N − 2 ,N − 1 z N − 1 ) /u N − 2 ,N − 2 · · · j − 1 � x N − j = ( z N − j − u N − j,N − k z N − k ) /u N − j,N − j , k =0 Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes LU Decomposition Inverse Iteration Back Substitution Matrix Iteration with Shifts Rayleigh Methods Back Substitution We have computed z by back substitution, we must solve U x = z but U is upper triangular, so we have x N = ( z N ) /u NN x N − 1 = ( z N − 1 − u N − 1 ,N z N ) /u N − 1 ,N − 1 x N − 2 = ( z N − 2 − u N − 2 ,N z N − u N − 2 ,N − 1 z N − 1 ) /u N − 2 ,N − 2 · · · j − 1 � x N − j = ( z N − j − u N − j,N − k z N − k ) /u N − j,N − j , k =0 For moderately large systems, the reduction in operations count given by back substitution with respect to matrix multiplication is so large that the additional cost of the LU decomposition is negligible. Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Inverse Iteration Matrix Iteration with Shifts Rayleigh Methods
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Inverse Iteration Matrix Iteration with Shifts Rayleigh Methods Introduction to Shifts Inverse iteration can be applied to each step of matrix iteration with sweeps, or to each step of a different procedure intended to compute all the eigenpairs, the matrix iteration with shifts . Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Inverse Iteration Matrix Iteration with Shifts Rayleigh Methods Matrix Iteration with Shifts, 1 If we write ω 2 i = µ + λ i , where µ is a shift and λ i is a shifted eigenvalue , the eigenvalue problem can be formulated as K ψ i = ( µ + λ i ) M ψ i or ( K − µ M ) ψ i = λ i M ψ i . If we introduce a modified stiffness matrix K = K − µ M , we recognize that we have a new problem, that has exactly the same eigenvectors and shifted eigenvalues, K φ i = λ i Mφ i , where λ i = ω 2 φ i = ψ i , i − µ. Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Inverse Iteration Matrix Iteration with Shifts Rayleigh Methods Matrix Iteration with Shifts, 2 The shifted eigenproblem can be solved, e.g., by matrix iteration and the procedure will converge to the smallest absolute value shifted eigenvalue and to the associated eigenvector. After convergence is reached, ω 2 ψ i = φ i , i = λ i + µ. Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Inverse Iteration Matrix Iteration with Shifts Rayleigh Methods Matrix Iteration with Shifts, 2 The shifted eigenproblem can be solved, e.g., by matrix iteration and the procedure will converge to the smallest absolute value shifted eigenvalue and to the associated eigenvector. After convergence is reached, ω 2 ψ i = φ i , i = λ i + µ. The convergence of the method can be greatly enhanced if the shift µ is updated every few steps during the iterative procedure using the current best estimate of λ i , ˆ x n +1 M x n λ i,n +1 = x n +1 , x n +1 M ˆ ˆ to improve the modified stiffness matrix to be used in the following iterations, K = K − λ i,n +1 M Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Inverse Iteration Matrix Iteration with Shifts Rayleigh Methods Matrix Iteration with Shifts, 2 The shifted eigenproblem can be solved, e.g., by matrix iteration and the procedure will converge to the smallest absolute value shifted eigenvalue and to the associated eigenvector. After convergence is reached, ω 2 ψ i = φ i , i = λ i + µ. The convergence of the method can be greatly enhanced if the shift µ is updated every few steps during the iterative procedure using the current best estimate of λ i , ˆ x n +1 M x n λ i,n +1 = x n +1 , x n +1 M ˆ ˆ to improve the modified stiffness matrix to be used in the following iterations, K = K − λ i,n +1 M Much thought was spent on the problem of choosing the initial shifts, so that all the eigenvectors can be computed in sequence without missing any of them. Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Inverse Iteration Matrix Iteration with Shifts Rayleigh Methods Rayleigh-Ritz Method Rayleigh-Ritz Example Subspace iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Rayleigh-Ritz Method Inverse Iteration Rayleigh-Ritz Example Matrix Iteration with Shifts Subspace iteration Rayleigh Methods Rayleigh Quotient for Discrete Systems The matrix iteration procedures are usually used in conjunction with methods derived from the Rayleigh Quotient method. Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Rayleigh-Ritz Method Inverse Iteration Rayleigh-Ritz Example Matrix Iteration with Shifts Subspace iteration Rayleigh Methods Rayleigh Quotient for Discrete Systems The matrix iteration procedures are usually used in conjunction with methods derived from the Rayleigh Quotient method. The Rayleigh Quotient method was introduced using distributed flexibilty systems and an assumed shape function, but we have seen also an example where the Rayleigh Quotient was computed for a discrete system using an assumed shape vector. Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Rayleigh-Ritz Method Inverse Iteration Rayleigh-Ritz Example Matrix Iteration with Shifts Subspace iteration Rayleigh Methods Rayleigh Quotient for Discrete Systems The matrix iteration procedures are usually used in conjunction with methods derived from the Rayleigh Quotient method. The Rayleigh Quotient method was introduced using distributed flexibilty systems and an assumed shape function, but we have seen also an example where the Rayleigh Quotient was computed for a discrete system using an assumed shape vector. The procedure to be used for discrete systems can be summarized as x ( t ) = φ Z 0 sin ωt, x ( t ) = ω φ Z 0 cos ωt, ˙ 2 T max = ω 2 φ T M φ , 2 V max = φ T K φ , Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Rayleigh-Ritz Method Inverse Iteration Rayleigh-Ritz Example Matrix Iteration with Shifts Subspace iteration Rayleigh Methods Rayleigh Quotient for Discrete Systems The matrix iteration procedures are usually used in conjunction with methods derived from the Rayleigh Quotient method. The Rayleigh Quotient method was introduced using distributed flexibilty systems and an assumed shape function, but we have seen also an example where the Rayleigh Quotient was computed for a discrete system using an assumed shape vector. The procedure to be used for discrete systems can be summarized as x ( t ) = φ Z 0 sin ωt, x ( t ) = ω φ Z 0 cos ωt, ˙ 2 T max = ω 2 φ T M φ , 2 V max = φ T K φ , equating the maxima, we have ω 2 = φ T K φ φ T M φ = k ⋆ m ⋆ , where φ is an assumed shape vector, not an eigenvector. Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Rayleigh-Ritz Method Inverse Iteration Rayleigh-Ritz Example Matrix Iteration with Shifts Subspace iteration Rayleigh Methods Ritz Coordinates For a N DOF system, an approximation to a displacement vector x can be written in terms of a set of M < N assumed shape, linearly independent vectors, φ i , i = 1 , . . . , M < N and a set of Ritz coordinates z i , i − 1 , . . . , M < N : � x = φ i z i = Φ z . i We say approximation because a linear combination of M < N vectors cannot describe every point in a N -space. Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Rayleigh-Ritz Method Inverse Iteration Rayleigh-Ritz Example Matrix Iteration with Shifts Subspace iteration Rayleigh Methods Rayleigh Quotient in Ritz Coordinates We can write the Rayleigh quotient as a function of the Ritz coordinates, ω 2 ( z ) = z T Φ T K Φ z z T φ T M φz = k ( z ) m ( z ) , but this is not an explicit function for any modal frequency... Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Rayleigh-Ritz Method Inverse Iteration Rayleigh-Ritz Example Matrix Iteration with Shifts Subspace iteration Rayleigh Methods Rayleigh Quotient in Ritz Coordinates We can write the Rayleigh quotient as a function of the Ritz coordinates, ω 2 ( z ) = z T Φ T K Φ z z T φ T M φz = k ( z ) m ( z ) , but this is not an explicit function for any modal frequency... On the other hand, we have seen that frequency estimates are always greater than true frequencies, so our best estimates are the the local minima of ω 2 ( z ) , or the points where all the derivatives of ω 2 ( z ) with respect to z i are zero: m ( z ) ∂k ( z ) − k ( z ) ∂m ( z ) ∂ω 2 ( z ) ∂z i ∂z i = = 0 , for i = 1 , . . . , M < N ∂z j ( m ( z )) 2 Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Rayleigh-Ritz Method Inverse Iteration Rayleigh-Ritz Example Matrix Iteration with Shifts Subspace iteration Rayleigh Methods Rayleigh Quotient in Ritz Coordinates Observing that k ( z ) = ω 2 ( z ) m ( z ) we can substitute into and simplify the preceding equation, ∂k ( z ) − ω 2 ( z ) ∂m ( z ) = 0 , for i = 1 , . . . , M < N ∂z i ∂z i Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Rayleigh-Ritz Method Inverse Iteration Rayleigh-Ritz Example Matrix Iteration with Shifts Subspace iteration Rayleigh Methods Rayleigh Quotient in Ritz Coordinates With the positions Φ T K Φ = K Φ T M Φ = M and we have � � k ( z ) = z T Kz = k rs z r z s , r s hence �� � � ∂k ( z ) � � = k is z s + k ri z r . ∂z i s r Due to symmetry, k ri = k ir and consequently � � � ∂k ( z ) � � = 2 k is z s = 2 Kz . ∂z i s Analogously � ∂m ( z ) � = 2 Mz . ∂z i Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Rayleigh-Ritz Method Inverse Iteration Rayleigh-Ritz Example Matrix Iteration with Shifts Subspace iteration Rayleigh Methods Reduced Eigenproblem Substituting these results in ∂k ( z ) ∂z i − ω 2 ( z ) ∂m ( z ) = 0 we can write a new ∂z i eigenvector problem , in the M DOF Ritz coordinates space, with reduced M × M matrices: K z − ω 2 M z = 0 . Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Rayleigh-Ritz Method Inverse Iteration Rayleigh-Ritz Example Matrix Iteration with Shifts Subspace iteration Rayleigh Methods Modal Superposition? After solving the reduced eigenproblem, we have a set of M eigenvalues ω 2 i and a corresponding set of M eigenvectors z i . What is the relation between these results and the eigenpairs of the original problem? The ω 2 i clearly are approximations from above to the real eigenvalues, and if we write ψ i = Φ z i we see that, being T i Φ T M Φ i Mψ j = z T z j = M i δ ij , ψ � �� � M the approximated eigenvectors ψ i are orthogonal with respect to the structural matrices and can be used in ordinary modal superposition techniques. Giacomo Boffi Truncated Sums, Matrix Iteration
Introduction Fundamental Mode Analysis Second Mode Analysis Higher Modes Rayleigh-Ritz Method Inverse Iteration Rayleigh-Ritz Example Matrix Iteration with Shifts Subspace iteration Rayleigh Methods A Last Question One last question: how many ω 2 i and ψ i are effective approximations to the true eigenpairs? Experience tells that an effective approximation is to be expected for the first M/ 2 eigenthings. Giacomo Boffi Truncated Sums, Matrix Iteration
RR Example m x 5 k m x 4 k m x 3 k m x 2 k m k x 1
RR Example m The structural matrices +2 − 1 0 0 0 1 0 0 0 0 x 5 k − 1 +2 − 1 0 0 0 1 0 0 0 m K = k 0 − 1 +2 − 1 0 M = m 0 0 1 0 0 x 4 k 0 0 − 1 +2 − 1 0 0 0 1 0 m 0 0 0 − 1 +1 0 0 0 0 1 x 3 k m x 2 k m k x 1
RR Example m The structural matrices +2 − 1 0 0 0 1 0 0 0 0 x 5 k − 1 +2 − 1 0 0 0 1 0 0 0 m K = k 0 − 1 +2 − 1 0 M = m 0 0 1 0 0 x 4 k 0 0 − 1 +2 − 1 0 0 0 1 0 m 0 0 0 − 1 +1 0 0 0 0 1 x 3 k The Ritz base vectors and the reduced matrices, m 0 . 2 − 0 . 5 � 0 . 2 � 0 . 2 x 2 ¯ k 0 . 4 − 1 . 0 K = k 0 . 2 2 . 0 m Φ = 0 . 6 − 0 . 5 � 2 . 2 � 0 . 2 ¯ 0 . 8 +0 . 0 M = m k x 1 0 . 2 2 . 5 1 . 0 1 . 0
RR Example m The structural matrices +2 − 1 0 0 0 1 0 0 0 0 x 5 k − 1 +2 − 1 0 0 0 1 0 0 0 m K = k 0 − 1 +2 − 1 0 M = m 0 0 1 0 0 x 4 k 0 0 − 1 +2 − 1 0 0 0 1 0 m 0 0 0 − 1 +1 0 0 0 0 1 x 3 k The Ritz base vectors and the reduced matrices, m 0 . 2 − 0 . 5 � 0 . 2 � 0 . 2 x 2 ¯ k 0 . 4 − 1 . 0 K = k 0 . 2 2 . 0 m Φ = 0 . 6 − 0 . 5 � 2 . 2 � 0 . 2 ¯ 0 . 8 +0 . 0 M = m k x 1 0 . 2 2 . 5 1 . 0 1 . 0 � 2 − 22 ρ � � z 1 � � 0 � 2 − 2 ρ Red. eigenproblem ( ρ = ω 2 m/k ): = 2 − 2 ρ 20 − 25 ρ z 2 0
RR Example m The structural matrices +2 − 1 0 0 0 1 0 0 0 0 x 5 k − 1 +2 − 1 0 0 0 1 0 0 0 m K = k 0 − 1 +2 − 1 0 M = m 0 0 1 0 0 x 4 k 0 0 − 1 +2 − 1 0 0 0 1 0 m 0 0 0 − 1 +1 0 0 0 0 1 x 3 k The Ritz base vectors and the reduced matrices, m 0 . 2 − 0 . 5 � 0 . 2 � 0 . 2 x 2 ¯ k 0 . 4 − 1 . 0 K = k 0 . 2 2 . 0 m Φ = 0 . 6 − 0 . 5 � 2 . 2 � 0 . 2 ¯ 0 . 8 +0 . 0 M = m k x 1 0 . 2 2 . 5 1 . 0 1 . 0 � 2 − 22 ρ � � z 1 � � 0 � 2 − 2 ρ Red. eigenproblem ( ρ = ω 2 m/k ): = 2 − 2 ρ 20 − 25 ρ z 2 0 The roots are ρ 1 = 0 . 0824 , ρ 2 = 0 . 800 , the frequencies are � � ω 1 = 0 . 287 k/m [ = 0 . 285] , ω 2 = 0 . 850 k/m [ = 0 . 831] , while the k/m normalized exact eigenvalues are [0 . 08101405 , 0 . 69027853] . The first eigenvalue is estimated with good approximation.
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