enhancing eigenvector approximations of huge gyroscopic eigenproblems from amls … ·...

Enhancing Eigenvector Approximations of HugeGyroscopic Eigenproblems from AMLS with

Subspace Iteration

Heinrich [email protected]

Joint work with Pu Chen and Jiacong Yin (Peking University)

Hamburg University of TechnologyInstitute of Mathematics

TUHH Heinrich Voss AMLS&SubspaceIteration Prague, July 01, 2013 1 / 44

Outline

1 Introduction

2 Automated Multi-Level Substructuring

3 Typical behavior of AMLS: A Numerical Example

4 Subspace iteration

5 Numerical Examples – Revisited

6 Gyroscopic eigenvalue problem

7 Numerical Example

8 Conclusions

Introduction

Outline

1 Introduction

7 Numerical Example

8 Conclusions

Introduction

Problem

Figure: Acoustic finite element model of a tire.

Simulation of acoustic properties has gained increasing importance in theengineering design process, in particular in automobile industries.

Sound radiation from the rolling tires has been identified as a major source oftraffic noise generated by vehicles moving at speeds above 40 km/h forpassenger cars, and above 60 km/h for trucks.

Introduction

Problem

Figure: Acoustic finite element model of a tire.

Simulation of acoustic properties has gained increasing importance in theengineering design process, in particular in automobile industries.

Sound radiation from the rolling tires has been identified as a major source oftraffic noise generated by vehicles moving at speeds above 40 km/h forpassenger cars, and above 60 km/h for trucks.

Introduction

The simulation of the tire noise is performed in three steps.

First, the nonlinear tire deflections under steady state conditions arecomputed using an Arbitrary Lagrangian Eulerian (ALE) approach.

Next, the transient vibrations governed by the eigenpairs of a gyroscopiceigenvalue problem

Q(ω)x := Kx + iωGx − ω2Mx = 0. (1)

are assumed to be superimposed onto the nonlinear deflections.

Finally, the acoustic analysis is carried out solving Helmholtz’s equation wherethe normal velocities at the wheel surface, extracted from the vibrationanalysis, are taken as boundary conditions.

Introduction

Q(ω)x := Kx + iωGx − ω2Mx = 0. (1)

Introduction

Q(ω)x := Kx + iωGx − ω2Mx = 0. (1)

Introduction

Q(ω)x := Kx + iωGx − ω2Mx = 0. (1)

Introduction

In this presentation we consider only the second step, i.e. the numericalsolution of the gyroscopic eigenproblem (1) where K is the stiffness matrixmodified by the presence of centripetal forces, M is the mass matrix, and G isthe gyroscopic matrix stemming from the Coriolis force.

Clearly, K and M are symmetric and positive definite, and G isskew–symmetric.

Due to the complicated interior structure of a belted tire the matrices K , Mand G of a sufficiently accurate FE model are very large and sparse.

Moreover, for the acoustic analysis many eigenpairs not necessarily at theend of the spectrum are needed.Therefore, well-established sparse eigensolvers of Arnoldi type with shift andinvert techniques for a linearization of problem (1), methods which are basedon structure preserving linearizations like SHIRA, and iterative projectionmethods for nonlinear eigenproblems are very costly since LU factorizationsof complex valued matrices Q(ωj ) for several parameters ωj are required.

Introduction

Moreover, for the acoustic analysis many eigenpairs not necessarily at theend of the spectrum are needed.

Therefore, well-established sparse eigensolvers of Arnoldi type with shift andinvert techniques for a linearization of problem (1), methods which are basedon structure preserving linearizations like SHIRA, and iterative projectionmethods for nonlinear eigenproblems are very costly since LU factorizationsof complex valued matrices Q(ωj ) for several parameters ωj are required.

Introduction

Moreover, for the acoustic analysis many eigenpairs not necessarily at theend of the spectrum are needed.Therefore, well-established sparse eigensolvers of Arnoldi type with shift andinvert techniques for a linearization of problem (1),

methods which are basedon structure preserving linearizations like SHIRA, and iterative projectionmethods for nonlinear eigenproblems are very costly since LU factorizationsof complex valued matrices Q(ωj ) for several parameters ωj are required.

Introduction

Moreover, for the acoustic analysis many eigenpairs not necessarily at theend of the spectrum are needed.Therefore, well-established sparse eigensolvers of Arnoldi type with shift andinvert techniques for a linearization of problem (1), methods which are basedon structure preserving linearizations like SHIRA,

and iterative projectionmethods for nonlinear eigenproblems are very costly since LU factorizationsof complex valued matrices Q(ωj ) for several parameters ωj are required.

Introduction

Automated Multi-Level Substructuring

Outline

1 Introduction

7 Numerical Example

8 Conclusions

Automated Multi-Level SubstructringA method which is capable to determine a large number of eigenpairs of ahuge problem with moderate accuracy is the Automated Multi-LevelSubstructuring (AMLS).

AMLS was introduced by Bennighof (1998) and was applied to huge problemsof frequency response analysis.

The large finite element model is recursively divided into very manysubstructures on several levels based on the sparsity structure of the systemmatrices.

Assuming that the interior degrees of freedom of substructures dependquasistatically on the interface degrees of freedom, and modeling thedeviation from quasistatic dependence in terms of a small number of selectedsubstructure eigenmodes the size of the finite element model is reducedsubstantially yet yielding satisfactory accuracy over a wide frequency range ofinterest.

Recent studies in vibro-acoustic analysis of passenger car bodies where verylarge FE models with more than six million degrees of freedom appear andseveral hundreds of eigenfrequencies and eigenmodes are needed haveshown that AMLS is considerably faster than Lanczos type approaches.

Automated Multi-Level Substructring

ConsiderKx = λMx (2)

where K and M are symmetric and positive definite (typically the stiffness andmass matrices of a FE model of a structure).

Similarly as in the component mode synthesis method (CMS) the structure ispartitioned into a small number of substructures based on the sparsity patternof the system matrices, but more generally than in CMS these substructuresin turn are sub-structured on a number of levels yielding a tree topology forthe 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.

Sub-structure tree

Decoupling sub-structuresAfter reordering the degrees of freedom the current reduced problem obtainsthe following form Kp O O

O Kc KcrO Krc Kr

xpxcxr

Mp Mpc MprMcp Mc McrMrp Mrc Mr

xpxcxr

where xp denotes the degrees of freedom which where already obtained inprevious reduction steps, xc are the degrees of freedom which are treated inthe current step of AMLS, and xr collects the ones to be handled in remainingsteps of the algorithm.

Applying the variable transformation xpxcxr

I O OO I −K−1

c KcrO O I

xpxcxr

=: Uc x

and multiplying with UTc from the left to retain the symmetry of the eigenvalue

problem, the degrees of freedom of the current sub-structure are decoupledfrom the remaining ones in the stiffness matrix.

Decoupling sub-structuresAfter reordering the degrees of freedom the current reduced problem obtainsthe following form Kp O O

O Kc KcrO Krc Kr

xpxcxr

Mp Mpc MprMcp Mc McrMrp Mrc Mr

xpxcxr

where xp denotes the degrees of freedom which where already obtained inprevious reduction steps, xc are the degrees of freedom which are treated inthe current step of AMLS, and xr collects the ones to be handled in remainingsteps of the algorithm.

Applying the variable transformation xpxcxr

I O OO I −K−1

c KcrO O I

xpxcxr

=: Uc x

and multiplying with UTc from the left to retain the symmetry of the eigenvalue

problem, the degrees of freedom of the current sub-structure are decoupledfrom the remaining ones in the stiffness matrix.

Decoupling sub-structuresThus, the eigenvalue problem obtains the following form Kp O O

O Kc OO O Kr

xpxcxr

Mp Mpc Mpr

Mcp Mc Mcr

Mrp Mrc Mr

xpxcxr

,where

Kr = Kr − KrcK−1c Kcr ,

Mr = Mr −MrcK−1c Kcr − KrcK−1

c Mcr + KrcK−1c McK−1

Mpr = Mpr −MpcK−1c Kcr = MT

Mcr = Mcr −McK−1c Kcr = MT

Recall that most of the columns of Kcr are null vectors. Only those columns ofKcr contain non-zero components the corresponding degree of freedom ofwhich lies on the boundary of the current sub-structure. Hence, K−1

c Kcr canbe computed very efficiently since only a very small number of small linearsystems has to be solved.

Decoupling sub-structuresThus, the eigenvalue problem obtains the following form Kp O O

O Kc OO O Kr

xpxcxr

Mp Mpc Mpr

Mcp Mc Mcr

Mrp Mrc Mr

xpxcxr

,where

Kr = Kr − KrcK−1c Kcr ,

Mr = Mr −MrcK−1c Kcr − KrcK−1

c Mcr + KrcK−1c McK−1

Mpr = Mpr −MpcK−1c Kcr = MT

Mcr = Mcr −McK−1c Kcr = MT

Recall that most of the columns of Kcr are null vectors. Only those columns ofKcr contain non-zero components the corresponding degree of freedom ofwhich lies on the boundary of the current sub-structure. Hence, K−1

c Kcr canbe computed very efficiently since only a very small number of small linearsystems has to be solved.

Order reduction

To reduce the dimension of the eigenproblem we determine for everysubstructure (after decoupling it from the remaining degrees of freedom in thestiffness matrix as above, and neglecting connections to other substructuresin the mass matrix) all eigenvalues λcj of Kcz = λcjMcz not exceeding a cut offfrequency λcutoff and corresponding eigenvectors zsj , j = 1, . . . ,ms.

Then with Zs = [zs1, . . . , zsms ] and the global block diagonal projection matrixZ = diag{Z1, . . . ,Zm} we finally get the reduced eigenvalue problem

Kcxc = λMcxc (3)

where Kc = Z T K Z = Z T UT KUZ is a diagonal matrix andMc = Z T MZ = Z T UT MUZ has generalized block arrowhead form.

Important: In an implementation the block Gaussian eliminations and thecondensations are performed in an interleaving way to avoid the storage oflarge dense sub-matrices of the transformed mass matrix which would occurin the course of the block elimination: as soon as a sub-matrix pencil (Kc ,Mc)has been formed, the eigenproblem KcZc = McZcΛc is solved and thecorresponding projection is executed.

Order reduction

Kcxc = λMcxc (3)

Order reduction

Kcxc = λMcxc (3)

Reduced mass matrix

Typical behavior of AMLS: A Numerical Example

Outline

1 Introduction

7 Numerical Example

8 Conclusions

Numerical exampleFinite element model of a blade of a 1.5 MW wind turbine. 117990 DoF

0 1 2 3 4 5 6

10−6

10−5

10−4

10−3

10−2

10−1

eigenvalue

rel. error AMLS: blade of wind turbine

es λc=1e7

λc=1e6

Figure: Blade of 1.5 MW wind turbine

Numerical exampleFinite element model of a blade of a 1.5 MW wind turbine. 117990 DoF

0 1 2 3 4 5 6

modal errors AMLS: blade of wind turbine

eigenvalue

λc=1e7

λc=1e6

Figure: Blade of 1.5 MW wind turbine

Applications of AMLS

The applications of AMLS are mainly limited to areas where eigenvalues areneeded with low accuracy or to frequency response analysis, in which theaccuracy of eigenvectors is of lower importance (only the projection of anexcitation to an eigenspace is needed).

However, in structural analysis its accuracy is not satisfactory because theprecision of extracted eigenvector approximations are too low to meet therequirements 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.

Subspace iteration

Outline

1 Introduction

7 Numerical Example

8 Conclusions

Subspace iteration

Eigenvector approximation by AMLS

AMLS is a one shot projection method, i.e. after having chosen a cut-offfrequency the method produces a fixed subspace V := span{V}, V := UZand the corresponding projected eigenproblem.

Differently from iterative projections methods such as Krylov subspace orJacobi–Davidson methods there is no way to expand the subspace V furtherreusing the projected problem if the computed approximate eigenpairs turnout to be not accurate enough. One has to repeat the reduction with a highercut-off frequency.

Alternatively, one can improve the subspace V obtained with AMLS bysubspace iteration.

Subspace iteration is an efficient and robust method for computingeigenvalues and eigenvectors which was developed about 40 years ago byBathe (1972). A typical task at that time was to determine a small number oeigenpairs at the lower end of the spectrum, but today it is used to determinehundreds of eigenmodes of large problems (Bathe (2013)).

Subspace iteration

Subspace iterationLet the columns of V0 ∈ Rn×p form an approximate basis of the invariantsubspace 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 wantedeigenvalues. However, for huge matrices K and M a factorization of K − σMand a solution of this system is very costly.

Alternatively, we may apply subspace iteration to the transformed problem

K z := UT KUz = λUT MUz =: λMz,

where U = U1, . . . ,Um is the matrix constructed in the AMLS process thattransforms K to block diagonal form.

Due to the interleaving implementation of AMLS the matrices K and M areusually not stored when computing the reduced model, but in principle thiscould be easily done. The matrix K then obtains block diagonal form withmoderate block sizes, but owing to fill in during the elimination process M willcontain many dense sub-matrices requiring a huge amount of storage. So,this approach is also not efficient.

Subspace iteration

The way out is to combine the benefits of both approaches, i.e. to applysubspace iteration to the transformed system, but to evaluate MV takingadvantage of the transformation matrix U and the sparse structure of theoriginal mass matrix M.

Subspace iteration

Subspace iteration with AMLS

Require: Transformed eigenvectors V , the transformed stiffness matrix K ,and the transformation matrix U from AMLS

1: initialize the iteration matrices Q(0) = V2: for k = 1,2, . . . ,nk do3: transform backward Q(k−1) = UQ(k−1)

4: compute R = MQ(k−1)

5: transform forward R = UT R6: solve for Q(k): K Q(k) = R7: end for8: project transformed stiffness matrix Kc = RT Q(nk )

9: transform backward Q(nk ) = UQ(nk )

10: project transformed mass matrix Mc = (Q(nk ))T MQ(nk )

11: solve projected problem KcXc = McXcΛ12: compute eigenvector approximations V (nk ) = Q(nk )Xc .

Numerical Examples – Revisited

Outline

1 Introduction

7 Numerical Example

8 Conclusions

Numerical Example

Computations were performed on a 64-bit Linux platform with an Intel PentiumD 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 DoFnz(K ) = 11243248, nz(M) = 5590256

Table: Computation time of NormalSIM and AMLS-SIM

AMLS reduction 100.2seigenvectors 7.6sAMLS-SIM 65.6snormal SIM 167.0s

Numerical Example

Blade of 1.5 MW wind turbine

0 1 2 3 4 5 6

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

eigenvalue

relative errors after k subspace iteration steps

Figure: Relative errors of eigenvalue approximations

Blade of 1.5 MW wind turbine

0 2 4 6 8 10

10−10

10−8

10−6

10−4

10−2

modal errors after k subspace iteration steps

eigenvalue

Figure: Modal errors

Gyroscopic eigenvalue problem

Outline

1 Introduction

7 Numerical Example

8 Conclusions

Consider the gyroscopic eigenproblem

Kx + iωGx − ω2Mx = 0

with K = K T > 0, M = MT > 0, and G = −GT describing the eigenproblem ofa rotating structure.

Hermitian linearization[iG KK O

[M OO K

], y = ωx

AMLS does not apply directly since the matrix on the left is not positivedefinite, and for the transformed problem with µ = ω−1 it yieldsapproximations to eigenpairs corresponding to the largest eigenvalues inmodulus which are everything else of note.

[M OO K

], y = ωx

[M OO K

], y = ωx

Gyroscopic eigenvalue problemSince the influence of the gyroscopic matrix G on the eigenvectors is usuallynot very high compared to the mass and stiffness matrices, it is reasonable toneglect the linear term when constructing the ansatz space for theRayleigh–Ritz projection, i.e. when defining the sub–structuring, computingthe transformations of K to block diagonal form, and employing the modalreductions corresponding to the sub–structures.

Hence, the AMLS reduction is applied to the pencil (K ,M) as for a vibrationproblem in frequency response analysis, and the resulting congruencetransformations and modal reductions are also applied to the skew-symmetricmatrix G.

Thus, one obtains a reduced model

K x + iωGx − ω2Mx = 0,

where the reduced stiffness matrix K = Z T UT KUZ and mass matrix M aredetermined in the AMLS reduction, and G = Z T UT GUZ is the projectedgyroscopic matrix which can be evaluated along with K and M within theAMLS process for (K ,M).

The gyroscopic eigenvalue problem where K ∈ Cn×n and M ∈ Cn×n areHermitian and positive definite and G ∈ Cn×n is skew–Hermitian has 2n realeigenvalues, n of which are negative and n are positive.

For real matrices we even have that −ωj are the negative eigenvalues if ωj ,j = 1, . . . ,n denote its positive eigenvalues.

For the general complex case positive and negative eigenvalues are of thesame magnitude in modulus if G is small compared to K and M.

The eigenvectors corresponding to the positive eigenvalues form a basis ofCn, and the same holds true for the eigenvectors corresponding to thenegative eigenvalues (Duffin 1960).

If the subspace iteration is applied to the linearized eigenvalue problem withshift θ = 0 (this is the only way to take advantage of the transformed blockdiagonal matrix K ) with an initial basis X ∈ Rn×m, then one obtainsconvergence to eigenvalues, m/2 of which are negative and m/2 are positive.This suggests to apply the subspace iteration in the following way:

Let V ∈ Rn×m be the matrix of eigenvector approximations obtained fromAMLS and Λ ∈ Rm×m be the diagonal matrix containing the approximations ofthe m smallest positive eigenvalues then we apply the subspace iteration withthe initial basis [

V Λ1/2 −V Λ1/2

Taking advantage of the special structure of the matrices the k th step ofsubspace iteration reads

P(k) = Q(k−1), KQ(k) = MP(k−1) − iGQ(k−1).

V Λ1/2 −V Λ1/2

P(k) = Q(k−1), KQ(k) = MP(k−1) − iGQ(k−1).

V Λ1/2 −V Λ1/2

P(k) = Q(k−1), KQ(k) = MP(k−1) − iGQ(k−1).

Subspace iteration

Require: Diagonal matrix Λ containing eigenvalue approximations fromAMLS, transformed eigenvectors V , the transformed stiffness matrix K thetransformation matrix U from AMLS

1: initialize the iteration matrices Q(0) = [V , V ] and P(0) = [V Λ1/2,−V Λ1/2]2: transform backward P(0) = UP(0)

3: for k = 1,2, . . . ,nk do4: transform backward Q(k−1) = UQ(k−1)

5: compute R = MP(k−1) − iGQ(k−1)

6: transform forward R = UT R7: P(k) = Q(k−1)

8: solve for Q(k): K Q(k) = R9: end for

10: T = RHQ(nk )

11: projected mass matrix M = (P(nk ))HMP(nk ) + T12: reload R and compute S = RHP(nk )

13: projected stiffness matrix K = i(P(nk ))HGP(nk ) + S + SH .14: solve projected problem K Z = MZ Λ15: sort out positive eigenvalue Λ+ and corresponding eigenvectors Z+

16: compute improved eigenvectors V (nk ) = QZ+.TUHH Heinrich Voss AMLS&SubspaceIteration Prague, July 01, 2013 33 / 44

Numerical Example

Outline

1 Introduction

7 Numerical Example

8 Conclusions

Numerical Example

Numerical exampleConsider a rotating tire the FE model of which consists of 39204 brickelements with 124992 degrees of freedom and accounts for 20 differentmaterial groups. The speed is assumed to be 60 km/h. Our aim is todetermine approximations to the smallest 200 eigenvalues and correspondingeigenvectors.

Figure: Continental AG 205/55R16-91H tire

Numerical Example

Numerical example

The numerical tests were performed on a 64-bit HP workstation with an IntelXeon CPU (3.20 GHz, 2 cores) and 24GB memory. AMLS and the twosubspace iteration algorithms were implemented with Matlab R2009a.

The AMLS method addressing the linear eigenvalue problem Kx = λMx costs881.2 seconds for the AMLS projection and 270.4 seconds for solving theprojected linear eigenvalue problem of dimension 2263 by eig.

Numerical Example

Numerical example

The numerical tests were performed on a 64-bit HP workstation with an IntelXeon CPU (3.20 GHz, 2 cores) and 24GB memory. AMLS and the twosubspace iteration algorithms were implemented with Matlab R2009a.

The AMLS method addressing the linear eigenvalue problem Kx = λMx costs881.2 seconds for the AMLS projection and 270.4 seconds for solving theprojected linear eigenvalue problem of dimension 2263 by eig.

Numerical Example

Numerical example

Table: Computation time of NormalSIM and AMLS-SIM with 208 iteration vectors

Computational Steps NormalSIM(s) AMLS-SIM(s)Compute initial eigenvectors from AMLS 9.2 2.41 Iteration 2401.0 148.92 Iterations Not computed 304.93 Iterations Not computed 462.14 Iterations Not computed 621.8Compute K and M if nk = 1 10.0 9.7Compute K and M if nk > 1 Not computed 19.6Solve K Z = MV Λ by eig 2.6 2.6Compute final eigenvectors 0.9 25.6

Numerical Example

Numerical example

0 20 40 60 80 100 120 140 160 180 20010-11

Mode Number

Linear AMLS1 Iteration2 Iterations3 Iterations4 Iterations

Figure: Relative errors of eigenvalues computed with subspace iteration with AMLS

Numerical Example

Numerical example

0 20 40 60 80 100 120 140 160 180 20010-7

Mode Number

Linear AMLS1 Iteration2 Iterations3 Iterations4 Iterations

Figure: Modal errors computed with subspace iteration with AMLS

Numerical Example

Taking advantage of symmetry

Our approach does not use the special structure to its full capacity.

If for some k the bases satisfy the following symmetry properties:

P(k−1) =[Pk−1,−Pk−1

], Q(k−1) =

[Qk−1,Qk−1

thenP(k) =

[Pk ,Pk

], Q(k) =

[Qk ,−Qk

andP(k+1) =

[Pk+1,−Pk+1

], Q(k+1) =

[Qk+1,Qk+1

Hence the symmetry is preserved in every other step.

Numerical Example

P(k−1) =[Pk−1,−Pk−1

], Q(k−1) =

[Qk−1,Qk−1

thenP(k) =

[Pk ,Pk

], Q(k) =

[Qk ,−Qk

andP(k+1) =

[Pk+1,−Pk+1

], Q(k+1) =

[Qk+1,Qk+1

Numerical Example

P(k−1) =[Pk−1,−Pk−1

], Q(k−1) =

[Qk−1,Qk−1

thenP(k) =

[Pk ,Pk

], Q(k) =

[Qk ,−Qk

andP(k+1) =

[Pk+1,−Pk+1

], Q(k+1) =

[Qk+1,Qk+1

Numerical Example

P(k−1) =[Pk−1,−Pk−1

], Q(k−1) =

[Qk−1,Qk−1

thenP(k) =

[Pk ,Pk

], Q(k) =

[Qk ,−Qk

andP(k+1) =

[Pk+1,−Pk+1

], Q(k+1) =

[Qk+1,Qk+1

Numerical Example

P(k−1) =[Pk−1,−Pk−1

], Q(k−1) =

[Qk−1,Qk−1

thenP(k) =

[Pk ,Pk

], Q(k) =

[Qk ,−Qk

andP(k+1) =

[Pk+1,−Pk+1

], Q(k+1) =

[Qk+1,Qk+1

Numerical Example

Subspace iteration using symmetry

Require: Diagonal matrix Λ with eigenv. approx. from AMLS, transf.eigenvec. V , transf. stiffness matrix K , transformation matrix U

1: initialize the iteration matrices Q0 = V and P0 = V Λ1/2

2: transform backward P0 = UP03: for k = 1,2, . . . ,nk do4: transform backward Qk−1 = UQk−15: compute R = MPk−1 − iGQk−16: transform forward R = UT R7: Pk = Qk−1

8: solve for Qk : K Qk = R9: end for

10: T = [R,−R]H [Qnk ,−Qnk ]

11: projected mass matrix M = [Pnk ,Pnk ]HM[Pnk ,Pnk ] + T12: reload R and compute S = [R,−R]H [Pnk ,Pnk ]

13: projected stiffness matrix K = i[Pnk ,Pnk ]HG[Pnk ,Pnk ] + S + SH .14: solve projected problem K Z = MZ Λ15: sort out positive eigenvalue Λ+ and corresponding eigenvectors Z+

16: compute improved eigenvectors V (nk ) = [Qnk ,−Qnk ]Z+.

Numerical Example

Improvement using symmetry

Table: Comparison of symmetric and non-symmetric approach

Computational Steps symmetric (s) non-symmetric (s)Initial eigenvectors from AMLS 2.4 2.41 Iteration 67.8 148.92 Iterations 141.3 304.93 Iterations 216.8 462.14 Iterations 289.4 621.8Compute K and M if nk = 1 6.2 9.7Compute K and M if nk > 1 8.7 19.6Solve K Z = MZ Λ by eig 2.6 2.6Compute final eigenvectors 25.6 25.6

Conclusions

Outline

1 Introduction

7 Numerical Example

8 Conclusions

Conclusions

AMLS is an efficient condensation method for computing a huge number ofeigenmodes and frequency responses for large complex structures.

It usually provides approximate solutions which are less accurate than theones obtained with standard Krylov type methods.

However, in many applications the underlying algebraic eigenproblem is a FEmodel of a continuous structure, and so the required level of accuracy is nomore than what is furnished by the FE model.

Numerical examples demonstrate that the approximations to eigenvaluescomputed with AMLS are often of this limited but sufficient accuracy, whereasthe modal errors of eigenvectors are usually still quite large.

Combining AMLS with subspace iteration taking advantage of the blockstructure of the transformed stiffness matrix, but avoiding the use of the highlypopulated transformed mass matrix, the accuracy can be improved efficiently.

We demonstrated its efficiency for a huge gyroscopic eigenvalue problem.

Conclusions

enhancing eigenvector approximations of huge gyroscopic eigenproblems from amls … ·...

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