Enhancing Subspace Iteration by AMLS for Huge Eigenvalue Problems Heinrich Voss
[email protected] Joint work with Pu Chen and Jiacong Yin (Peking University) Hamburg University of Technology Institute of Mathematics
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Heinrich Voss
AMLS&SubspaceIteration
Cambridge, June 13, 2013
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Outline
1
Introduction
2
Automated Multi-Level Sub-Structuring (AMLS)
3
Typical behavior of AMLS: A Numerical Example
4
Combining Subspace iteration and AMLS
5
Back to the Numerical Examples
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Heinrich Voss
AMLS&SubspaceIteration
Cambridge, June 13, 2013
2 / 25
Introduction
Problem
Determine all eigenvalues in an interval [0, λmax ] and corresponding eigenvectors of the huge eigenproblem Kx = λMx
(1)
where K and M are symmetric and positive definite (typically the stiffness and mass matrices of a FE model of a structure). A efficient and robust method is the subspace iteration method (SIM) which was developed about 40 years ago by Bathe (1972). A typical task at that time was to determine a small number of eigenmodes (10 or 20, e.g.) at the lower end of the spectrum, but today often hundreds of eigenpairs are needed for huge problems with millions of degrees of freedom.
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Heinrich Voss
AMLS&SubspaceIteration
Cambridge, June 13, 2013
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Introduction
Subspace Iteration
Given a matrix V0 containing approximations to the wanted eigenvectors the basic task in the k th step of SIM is to solve the block linear system ˆk = MVk −1 KV ˆk and to M-orthonormalize the columns of V ˆk to obtain the next matrix Vk . for V Crucial for the success of SIM is to establish effective starting vectors V0 and (in particular for huge problems) to solve the linear systems efficiently. We propose to take advantage of automated multi-level sub-structuring (AMLS) for both tasks which combines block Gaussian elimination and modal reduction of sub-structures.
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Heinrich Voss
AMLS&SubspaceIteration
Cambridge, June 13, 2013
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Automated Multi-Level Sub-Structuring (AMLS)
Automated Multi-Level Substructring AMLS was introduced by Bennighof (1998) and was applied to huge problems of frequency response analysis. The large finite element model is recursively divided into very many substructures on several levels based on the sparsity structure of the system matrices. Assuming that the interior degrees of freedom of substructures depend quasistatically on the interface degrees of freedom, and modeling the deviation from quasistatic dependence in terms of a small number of selected substructure eigenmodes the size of the finite element model is reduced substantially yet yielding satisfactory accuracy over a wide frequency range of interest. Recent studies in vibro-acoustic analysis of passenger car bodies where very large FE models with more than six million degrees of freedom appear and several hundreds of eigenfrequencies and eigenmodes are needed have shown that AMLS is considerably faster than Lanczos type approaches. TUHH
Heinrich Voss
AMLS&SubspaceIteration
Cambridge, June 13, 2013
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Automated Multi-Level Sub-Structuring (AMLS)
Automated Multi-Level Substructuring Consider Kx = λMx
(2)
where K and M are symmetric and positive definite. Similarly as in the component mode synthesis method (CMS) the structure is partitioned into a small number of substructures based on the sparsity pattern of the system matrices, but more generally than in CMS these substructures in turn are sub-structured on a number of levels yielding a tree topology for the substructures. AMLS consists of two ingredients. 1
First, based on the substructuring the stiffness matrix K is transformed to block diagonal form by Gaussian elimination,
2
and secondly, the dimension is reduced substantially by modal condensation of the substructures.
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Heinrich Voss
AMLS&SubspaceIteration
Cambridge, June 13, 2013
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Automated Multi-Level Sub-Structuring (AMLS)
1 2 3 4 5 6 7 8
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Heinrich Voss
AMLS&SubspaceIteration
Cambridge, June 13, 2013
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Automated Multi-Level Sub-Structuring (AMLS)
Automated Multi-Level Substructring If Kss is a sub-matrix of K corresponding to a particular substructure, then after reordering rows and columns in (2) the pencil obtains the form Kss Ksr Mss Msr , . Krs Krr Mrs Mrr With block Gaussian elimination, i.e. post- and premultiplying this pencil with −1 I −Kss Ksr Us = O I and UsT , respectively, Kss is decoupled, and the pencil obtains the following form ˜ sr Kss O Mss M UsT KUs , UsT MUs = , . ˜rr ˜T ˜ rr O K M M sr Repeating the block elimination for all substructures 1, . . . , m we get ˜ = U T KU, M ˜ = U T MU with U = U1 U2 . . . Um K ˜ has block diagonal form. where the transformed stiffness matrix K TUHH
Heinrich Voss
AMLS&SubspaceIteration
Cambridge, June 13, 2013
10 / 25
Automated Multi-Level Sub-Structuring (AMLS)
Automated Multi-Level Substructring To reduce the dimension of the eigenproblem we determine for every substructure (after decoupling it from the remaining degrees of freedom in the stiffness matrix as above, and neglecting connections to other substructures in the mass matrix) all eigenvalues λsj not exceeding a cut off frequency λcutoff and corresponding eigenvectors zsj , j = 1, . . . , ms . Then with Zs = [zs1 , . . . , zsms ] and the global block diagonal projection matrix Z = diag{Z1 , . . . , Zm } we finally get the reduced eigenvalue problem Kc xc = λMc xc
(3)
˜ Z = Z T U T KUZ is a diagonal matrix and where Kc = Z T K T ˜ Mc = Z MZ = Z T U T MUZ has generalized block arrowhead form. Important: In an implementation the block Gaussian eliminations and the condensations are performed in an interleaving way to avoid the storage of large dense sub-matrices of the transformed mass matrix which would occur in the course of the block elimination: as soon as a sub-matrix pencil ˜ss , M ˜ ss ) has been formed, the eigenproblem K ˜ss Zs = M ˜ ss Zs Λs is solved and (K the corresponding projection is executed. TUHH
Heinrich Voss
AMLS&SubspaceIteration
Cambridge, June 13, 2013
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Automated Multi-Level Sub-Structuring (AMLS)
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Heinrich Voss
AMLS&SubspaceIteration
Cambridge, June 13, 2013
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Typical behavior of AMLS: A Numerical Example
Numerical example Finite element model of a blade of a 1.5 MW wind turbine. 117990 DoF rel. error AMLS: blade of wind turbine
0
10
−1
rel. err. of eigenvalues
10
λc=1e6
−2
λc=1e7
10
−3
10
−4
10
−5
10
−6
10
0
1
2
3 eigenvalue
4
5
6 5
x 10
Figure: Blade of 1.5 MW wind turbine TUHH
Heinrich Voss
AMLS&SubspaceIteration
Cambridge, June 13, 2013
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Typical behavior of AMLS: A Numerical Example
Numerical example Finite element model of a blade of a 1.5 MW wind turbine. 117990 DoF modal errors AMLS: blade of wind turbine
2
10
modal error
λc=1e6
1
10
λ =1e7 c
0
10
0
1
2
3 eigenvalue
4
5
6 5
x 10
Figure: Blade of 1.5 MW wind turbine TUHH
Heinrich Voss
AMLS&SubspaceIteration
Cambridge, June 13, 2013
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Typical behavior of AMLS: A Numerical Example
Applications of AMLS
The applications of AMLS are mainly limited to areas where eigenvalues are needed with low accuracy or to frequency response analysis, in which the accuracy of eigenvectors is of lower importance. However, in structural analysis its accuracy is not satisfactory because the precision of extracted eigenvector approximations are too low to meet the requirements in strain and stress computations. In structural analysis, the modal errors are required to be as low as 10−3 . Otherwise, no sufficiently accurate strain or stress can be derived.
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Heinrich Voss
AMLS&SubspaceIteration
Cambridge, June 13, 2013
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Combining Subspace iteration and AMLS
Eigenvector approximation by AMLS
AMLS is a one shot projection method, i.e. after having chosen a cut-off frequency the method produces a fixed subspace V := span{V }, V := UZ and the corresponding projected eigenproblem. Differently from iterative projections methods such as Krylov subspace or Jacobi–Davidson methods there is no way to expand the subspace V further reusing the projected problem if the computed approximate eigenpairs turn out to be not accurate enough. One has to repeat the reduction with a higher cut-off frequency. Alternatively, one can improve the subspace V obtained with AMLS by subspace iteration.
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Heinrich Voss
AMLS&SubspaceIteration
Cambridge, June 13, 2013
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Combining Subspace iteration and AMLS
Subspace iteration Let the columns of V0 ∈ R n×p form an approximate basis of the invariant subspace of the pencil (K , M) corresponding to the wanted eigenvalues. Then one step of subspace iteration requires to solve a linear system (K − σM)V1 = MV0 for V1 where σ is some shift close to the wanted eigenvalues. However, for huge matrices K and M a factorization of K − σM and a solution of this system is very costly. Alternatively, we may apply subspace iteration to the transformed problem ˜ z := U T KUz = λU T MUz =: λMz, ˜ K where U = U1 , . . . , Um is the matrix constructed in the AMLS process that transforms K to block diagonal form. ˜ and M ˜ are Due to the interleaving implementation of AMLS the matrices K usually not stored when computing the reduced model, but in principle this ˜ then obtains block diagonal form with could be easily done. The matrix K ˜ will moderate block sizes, but owing to fill in during the elimination process M contain many dense sub-matrices requiring a huge amount of storage. So, this approach is also not efficient. TUHH
Heinrich Voss
AMLS&SubspaceIteration
Cambridge, June 13, 2013
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Combining Subspace iteration and AMLS
Subspace iteration
The way out is to combine the benefits of both approaches, i.e. to apply ˜V ˜ taking subspace iteration to the transformed system, but to evaluate M advantage of the transformation matrix U and the sparse structure of the original mass matrix M.
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Heinrich Voss
AMLS&SubspaceIteration
Cambridge, June 13, 2013
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Combining Subspace iteration and AMLS
Subspace iteration with AMLS ˜ , the transformed stiffness matrix K ˜, Require: Transformed eigenvectors V and the transformation matrix U from AMLS ˜ (0) = V ˜ 1: initialize the iteration matrices Q 2: for k = 1, 2, . . . , nk do ˜ (k −1) 3: transform backward Q (k −1) = U Q (k −1) 4: compute R = MQ ˜ = UT R 5: transform forward R (k ) ˜ ˜ (k ) ˜ ˜ 6: solve for Q : K Q = R 7: end for ˜ (nk ) ˜c = R ˜TQ 8: project transformed stiffness matrix K (nk ) (nk ) ˜ 9: transform backward Q = UQ ˜ c = (Q (nk ) )T MQ (nk ) 10: project transformed mass matrix M ˜ ˜ 11: solve projected problem Kc Xc = Mc Xc Λ 12: compute eigenvector approximations V (nk ) = Q (nk ) Xc .
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Heinrich Voss
AMLS&SubspaceIteration
Cambridge, June 13, 2013
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Back to the Numerical Examples
Numerical Example Computations were performed on a 64-bit Linux platform with an Intel Pentium D CPU (3.64 GHz, 2 Cores) and 7.7 GB memory. FE model of a blade of a 1.5 MW wind turbine with 117990 DoF nz (K ) = 11243248, nz (M) = 5590256
Table: Computation time of NormalSIM and AMLS-SIM
AMLS reduction eigenvectors AMLS-SIM normal SIM
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Heinrich Voss
100.2s 7.6s 65.6s 167.0s
AMLS&SubspaceIteration
Cambridge, June 13, 2013
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Back to the Numerical Examples
Blade of 1.5 MW wind turbine relative errors after k subspace iteration steps
−1
10
AMLS −2
10
−3
10
k=1
−4
rel. error
10
−5
k=3
10
−6
10
−7
10
−8
10
−9
10
0
1
2
3 eigenvalue
4
5
6 5
x 10
Figure: Relative errors of eigenvalue approximations TUHH
Heinrich Voss
AMLS&SubspaceIteration
Cambridge, June 13, 2013
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Back to the Numerical Examples
Blade of 1.5 MW wind turbine modal errors after k subspace iteration steps
2
10
AMLS 0
10
−2
modal error
10
k=1 −4
10
k=3 −6
10
−8
10
−10
10
0
2
4
6 eigenvalue
8
10 4
x 10
Figure: Modal errors TUHH
Heinrich Voss
AMLS&SubspaceIteration
Cambridge, June 13, 2013
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