The pitfalls of Ritz vector analysis and the development of its replacement Mante Zemaityte † Fran¸coise Tisseur Ramaseshan Kannan ‡ †
†
The University of Manchester ‡
Arup
January 30, 2018
mante.zemaityte@manchester.ac.uk
K φ = λMφ
January 30, 2018
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Overview
1
Introduction to the modelling and analysis of structures
2
Current methods and their drawbacks
3
A novel approach to finding frequencies and modes of interest
4
An example and some numerical results
mante.zemaityte@manchester.ac.uk
K φ = ΝMφ
January 30, 2018
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Structural Dynamics Problem
Equation of dynamics for a finite element model M u¨ + Ku = f , where M 0, n × n mass matrix, K 0, n × n stiffness matrix, u ∈ Rn – displacement vector, f ∈ Rn – dynamic load vector, f = Mr , r ∈ Rn – rigid body vector.
mante.zemaityte@manchester.ac.uk
K φ = λMφ
January 30, 2018
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The usual approach - projection methods Projecting the matrix ODE onto a subspace spanned by the columns of some projection matrix X ∈ Rn×k , where k n, results in the following: M ∗ v¨ + K ∗ v = f ∗ , where M ∗ = X T MX ∈ Rk×k , K ∗ = X T KX ∈ Rk×k , f ∗ = XTf , u ≈ Xv , for some v ∈ Rk . It is crucial to determine the right subspace and its dimension k for subsequent analysis.
mante.zemaityte@manchester.ac.uk
K φ = λMφ
January 30, 2018
4 / 17
Choosing a projection space Choices for a basis {x1 , . . . , xk } of a projection space include: 1. Eigenvectors φj of the generalised eigenvalue problem K φ = λMφ, corresponding to the smallest k eigenvalues λj , j = 1, . . . , k. I
Eigenvalues λ correspond to natural frequencies of the structure.
I
Eigenvectors φ correspond to the modes of vibration.
2. Ritz vectors {y1 , . . . , yk } (good and bad approximations to φ). 3. Lanczos vectors {q1 , . . . , qk }. We focus on the first two in this talk.
mante.zemaityte@manchester.ac.uk
K φ = λMφ
January 30, 2018
5 / 17
Lanczos algorithm and Ritz vectors Given starting vector q ∈ Rn , k steps of the Lanczos algorithm applied to the transformed problem K φ = λMφ
=⇒
K −1 Mφ = θφ,
λ=
1 , θ
computes the following: matrix Qk ∈ Rn×k of Lanczos vectors, the M-orthonormal basis of the Krylov subspace Kk (K −1 M, q), matrix Sk ∈ Rk×k of eigenvectors of QkT MK −1 MQk , matrix Yk = Qk Sk ∈ Rn×k of Ritz vectors, some of which are good approximations to the eigenvectors φj of (K , M).
mante.zemaityte@manchester.ac.uk
K φ = λMφ
January 30, 2018
6 / 17
Response spectrum analysis
Standard method for deriving an elastic response to a design earthquake. Imposes a criterion on the mass captured in a modal analysis. mante.zemaityte@manchester.ac.uk
K φ = ΝMφ
January 30, 2018
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Mass participation Define the (normalised) mass participation factor (MPF) of the jth column xj of a projection matrix X to be the following m(xj ) = where
(xjT Mr )2 , r T Mr
( 1 xiT Mxj = 0
for i = j, for i = 6 j.
The dimension k of a projection space is deemed large enough when the following MPF condition is satisfied: k X m(xj ) ≥ 0.9. j=1
mante.zemaityte@manchester.ac.uk
K φ = λMφ
January 30, 2018
8 / 17
Eigenvectors vs Ritz vectors Problem
DOF
φi
yi (K −1 Mr )
ccnb x
57,152
119
5
ccnb y
12
2
ccnb z
27
4
Table: Projection space size k required to satisfy the MPF condition.
1 MPFs m(φj ) of eigenvectors MPFs m(yj ) of Ritz vectors
0.8 0.6 0.4 0.2 0
0
1
2
3
4 λi
mante.zemaityte@manchester.ac.uk
K φ = λMφ
5
6
7
8 ·104
January 30, 2018
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Pitfalls The following are the issues that we face at present: 1. Subspace iteration often employed to compute eigenvectors for the analysis is not an adaptive algorithm - projection space k has to be specified in advance. 2. Altough adaptive, methods employing Ritz vectors in analysis of structures are not proved to be accurate. 3. For more problematic cases (large k) other known algorithms struggle with: I
numerical instabilities,
I
time and memory constraints,
I
uneven distribution of eigenvalues on the spectrum,
I
large number of eigenvectors of negligible mass participation are computed.
We wish to develop an efficient, adaptive and stable algorithm that computes eigenvectors of high mass participation.
mante.zemaityte@manchester.ac.uk
K φ = ΝMφ
January 30, 2018
10 / 17
Shift-and-invert Lanczos
Shift-and-invert Lanczos algorithm refers to applying the Lanczos algorithm to the shifted GEP 1 (K − σM)−1 Mφ = θφ, λ = σ + , θ to force the convergence of eigenvalues of (K , M) close to some chosen shift σ. This overcomes the time and memory constraints, however: Choosing shifts without knowing the distribution of the eigenvalues is a difficult and unpredictable problem, sometimes leading to the failure of the algorithm. Lots of eigenvectors of negligible mass participation are still computed.
mante.zemaityte@manchester.ac.uk
K φ = λMφ
January 30, 2018
11 / 17
Predicting frequency intervals of interest Define the cummulative mass participation factor sum to be ( n X 1, λ ≥ 0, m(φi )H(λ − λi ), H(λ) = ω(λ) = 0, λ < 0. i=1 If we start the Lanczos iteration with the rigid body vector r , we can then derive from k steps of the Lanczos algorithm an approximation τ (λ) to ω(λ) at a negligible cost, where k X τ (λ) = τi2 H(λ − θi ), i=1
where the θi is the Ritz value of (K , M). In particular, we have that
Theorem (Karlin and Shapley) If the difference function ω(λ) − τ (λ) is not identically zero, it has exactly 2k − 1 sign changes.
mante.zemaityte@manchester.ac.uk
K φ = λMφ
January 30, 2018
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Example (I) Horizontal and vertical steps of τ (λ) intersect ω(λ) exactly 2k − 1 times. To satisfy the MPF condition: 122 eigenvectors were required with the Lanczos algorithm, 36 with pSIL. 1 ω(λ) τ (λ) m(φj ), (Lanczos) m(φj ), (pSIL)
0.8 0.6 0.4 0.2 0
0
100
200
300
400
500
600
λi mante.zemaityte@manchester.ac.uk
K φ = λMφ
January 30, 2018
13 / 17
Example (II) After removing the eigenvectors of negligible mass participation, only 7 of them are required to satisfy the MPF condition (pSIL x). 1 ω(λ) τ (λ) m(φj ), (Lanczos) m(φj ), (pSIL) m(φj ), (pSIL x)
0.8 0.6 0.4 0.2 0
0
100
200
300
400
500
600
λi mante.zemaityte@manchester.ac.uk
K φ = λMφ
January 30, 2018
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Numerical Results
Problem
SIL
pSIL
Reduction
pSIL x
Reduction
lm z
987
372
62%
189
81%
TT y
493
406
18%
135
73%
TT z
1184
1077
9%
318
73%
Table: The number of eigenvectors required to satisfy the MPF condition for the lm problem (DOF 51,348) and TT problem (DOF 131,835).
mante.zemaityte@manchester.ac.uk
K φ = λMφ
January 30, 2018
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Summary
Ritz vectors may not be a good substitute for eigenvectors - response spectrum analysis is not guaranteed to be accurate. Shift-and-invert Lanczos algorithm proves to be a good choice of an algorithm for an adaptive search for eigenvectors. The novel approach of estimating the frequency intervals of interest reduces the computational cost and provides an option for further reducing the number of eigenvectors required to satisfy the MPF condition.
M. Zemaityte, F. Tisseur and R. Kannan. A shift-and-invert Lanczos algorithm for the dynamic analysis of structures. (In preparation)
mante.zemaityte@manchester.ac.uk
K φ = λMφ
January 30, 2018
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Wilson, Edward L. and Yuan, Ming-Wu and Dickens, John M. Dynamic analysis by direct superposition of Ritz vectors. Earthquake Engineering and Structural Dynamics, 10(6):813–821, 1982. Bahram Nour-Omid and Ray W. Clough. Dynamic analysis of structures using Lanczos co-ordinates. Earthquake Engineering and Structural Dynamics, 12(4):565–577, 1984. Chen, Harn C. and Taylor, Robert L. Using Lanczos vectors and Ritz vectors for computing dynamic responses. Engineering computations, 6(2):151–157,1989. Fischer, Bernd, and Roland W. Freund. On adaptive weighted polynomial preconditioning for Hermitian positive definite matrices. SIAM Journal on Scientific Computing 15.2 (1994): 408–426.