ITERATIVE PROJECTION METHODS FOR SPARSE .ITERATIVE PROJECTION METHODS FOR SPARSE LINEAR SYSTEMS AND

  • View
    213

  • Download
    0

Embed Size (px)

Text of ITERATIVE PROJECTION METHODS FOR SPARSE .ITERATIVE PROJECTION METHODS FOR SPARSE LINEAR SYSTEMS AND

  • ITERATIVE PROJECTION METHODS FORSPARSE LINEAR SYSTEMS AND

    EIGENPROBLEMSCHAPTER 12 : AMLS METHOD

    Heinrich Vossvoss@tu-harburg.de

    Hamburg University of TechnologyInstitute of Numerical Simulation

    TUHH Heinrich Voss AMLS Summer School 2006 1 / 45

    http://www.tu-harburg.de/mat/hp/voss/

  • Automated Multi-Level Substructuring

    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.

    TUHH Heinrich Voss AMLS Summer School 2006 2 / 45

  • Condensation

    Given (a finite element model of a structure, e.g.)

    Kx = Mx (1)

    where K Rnn and M Rnn are symmetric and M is positive definite.

    Aim: Reduce the number of unknowns by some sort of elimination.

    TUHH Heinrich Voss AMLS Summer School 2006 3 / 45

  • Exact condensation

    Partition degrees of freedom into variables xi to be kept (for substructurings:interface DoF) and variables x` to be droped (local DoF). After reorderingproblem (1) obtains the following form

    (K`` K`iKi` Kii

    ) (x`xi

    )=

    (M`` M`iMi` Mii

    ) (x`xi

    )(2)

    Solving the first equation for x` yields

    x` = (K`` M``)1(K`i M`i)xi

    and substituting in the second equation one gets the exactly condensedeigenproblem

    T ()xi = Kiixi + Miixi + (Ki` Mi`)(K`` M``)1(K`i M`i)xi

    TUHH Heinrich Voss AMLS Summer School 2006 4 / 45

  • Static condensation

    Linearizing the exactly condensed problem at = 0 yields the staticallycondensed eigenproblem (introduced independently by Irons (1965) andGuyan (1965))

    Kiixi = Miixi (3)

    where

    Kii = Kii Ki`K1`` K`iMii = Mii Ki`K1`` M`i M`iK

    1`` K`i + Ki`K

    1`` M``K

    1`` K`i

    For vibrating structures this means that the local degrees of freedom areassumed to depend quasistatically on the interface degrees of freedom, andthe inertia forces of the substructures are neglected.

    TUHH Heinrich Voss AMLS Summer School 2006 5 / 45

  • Substructuring

    Consider the vibrations of a structure which is partitioned into r substructuresconnecting to each other through the variables on the interfaces only.

    Then ordering the unknowns appropriately the stiffness matrix obtains thefollowing block form

    K =

    K``1 O . . . O K`i1O K``2 . . . O K`i2...

    .... . .

    ......

    O O . . . Kssr KsmrKi`1 Ki`2 . . . Kmsr Kii

    and M has the same block form.

    TUHH Heinrich Voss AMLS Summer School 2006 6 / 45

  • Substructuring ct.

    For the statically condensed problem we obtain

    Kii = Kii r

    j=1

    KmsjK1ssj Ksmj

    Mii = Mii r

    j=1

    Mmmj ,

    where

    Mmmj = KmsjK1ssj Msmj + MmsjK1ssj Ksmj KmsjK

    1ssj MssjK

    1ssj Ksmj .

    The submatrices corresponding to the individual substructures can bedetermined independently from smaller subproblems and in parallel.

    TUHH Heinrich Voss AMLS Summer School 2006 7 / 45

  • ExampleFEM model of a container ship: 35262 DoF, bandwidth: 1072

    0

    50

    100

    150

    200

    100

    10

    0

    20

    40

    TUHH Heinrich Voss AMLS Summer School 2006 8 / 45

  • Example ct.

    10 substructures; condensation to 1960 interface DoF

    TUHH Heinrich Voss AMLS Summer School 2006 9 / 45

  • Example ct.

    Container ship: relative errors of static condensation

    # eigenvalue nodal cond.1 1.2555112888e-01 5.02e-052 1.4842667377e-01 2.36e-053 1.8859647898e-01 6.32e-054 8.2710672903e-01 1.06e-045 1.4571047916e+00 3.98e-046 1.8843144791e+00 6.16e-047 2.4004294125e+01 5.47e-038 5.2973437588e+01 2.11e-029 5.6869743387e+01 2.49e-02

    10 1.7501327597e+02 8.41e-0211 2.0806150033e+02 1.08e-0112 2.8210662009e+02 1.25e-01

    TUHH Heinrich Voss AMLS Summer School 2006 10 / 45

  • A projection approach

    We transform the matrix K to block diagonal form using block Gaussianelimination, i.e. we apply the congruence transformation with

    P =(

    I K1`` K`i0 I

    )to the pencil (K , M) obtaining the equivalent pencil

    (PT KP, PT MP) =((

    K`` 00 Kii

    ),

    (M`` M`iMi` Mii

    )). (4)

    Here K`` and M`` stay unchanged, and

    Kii = Kii Ki`K1`` K`i is the Schur complement of K``M`i = M`i M``K1`` K`i = M

    Ti`

    Mii = Mii Mi`K1`` K`i Ki`K1`` M`i + Ki`K

    1`` M``K

    1`` K`i .

    TUHH Heinrich Voss AMLS Summer School 2006 11 / 45

  • static condensation revisited

    Neglecting in (4) all rows and columns corresponding to local degrees offreedom, i.e. projecting problem (1) to the subspace spanned by columns of(K1`` K`i

    I

    )one obtains the method of static condensation

    Kiiy = Miiy

    To model the deviation from quasistatic behavior thereby improving theapproximation properties of static condensation we consider the eigenvalueproblem

    K`` = M``, T M`` = I, (5)

    where is a diagonal matrix containing the eigenvalues.

    TUHH Heinrich Voss AMLS Summer School 2006 12 / 45

  • CraighBampton form

    Changing the basis for the local degrees of freedom to a modal one, i.e.applying the further congruence transformation diag{, I} to problem (4) onegets ((

    00 Kii

    ),

    (I T M`i

    Mi` Mii

    )). (6)

    In structural dynamics (6) is called CraighBampton form of the eigenvalueproblem (1) corresponding to the partitioning (2).

    In terms of linear algebra it results from block Gaussian elimination to reduceK to block diagonal form, and diagonalization of the block K`` using a spectralbasis.

    TUHH Heinrich Voss AMLS Summer School 2006 13 / 45

  • Component Mode Synthesis (CMS)

    Selecting some eigenmodes of problem (5), and dropping the rows andcolumns in (6) corresponding to the other modes one arrives at thecomponent mode synthesis method (CMS) introduced by Hurty (1965) andCraigh & Bampton (1968).

    If the diagonal matrix 1 contains in its diagonal the eigenvalues to drop and1 the corresponding eigenvectors, and if 2 and 2 contain the eigenvaluesand eigenvectors to keep, respectively, then the eigenproblem (6) can berewritten as1 0 00 2 0

    0 0 Kii

    x1x2x3

    = I 0 M`i10 I M`i2

    Mi`1 Mi`2 Mii

    x1x2x3

    (7)with

    Msmj = Tj (M`i M``K1`` K`i) = MTmsj , j = 1, 2,

    TUHH Heinrich Voss AMLS Summer School 2006 14 / 45

  • CMS ct.

    and the CMS approximations to the eigenpairs of (1) are obtained from thereduced eigenvalue problem(

    2 00 Kii

    )y =

    (I M`i2

    Mi`2 Mii

    )y (8)

    Usually the eigenvectors according to eigenvalues which do not exceed a cutoff threshold are kept. In vibration analysis of a structure this choice ismotivated by the fact that the high frequencies of a substructure do notinfluence the wanted low frequencies of the entire substructure very much.

    Notice however that in a recent paper Bai and Lia (2006) suggested adifferent choice based on a momentmatching analysis.

    TUHH Heinrich Voss AMLS Summer School 2006 15 / 45

  • Container ship

    We consider the structural deformation caused by a harmonic excitation at afrequency of 4 Hz which is a typical forcing frequency stemming from theengine and the propeller.

    Since the deformation is small the assumptions of the linear theory apply, andthe structural response can be determined by the mode superposition methodtaking into account eigenfrequencies in the range between 0 and 7.5 Hz(which corresponds to the 50 smallest eigenvalues for the ship underconsideration).

    To apply the CMS method we partitioned the FEM model into 10 substructuresas shown before. This substructuring by hand yielded a much smaller numberof interface degrees of freedom than automatic graph partitioners which try toconstruct a partition where the substructures have nearly equal size.

    For instance, our model ends up with 1960 degrees of freedom on theinterfaces, whereas Chaco ends up with a substructuring into 10substructures with 4985 interface degrees of freedom.

    TUHH Heinrich Voss AMLS Summer School 2006 16 / 45

  • Container ship ct.We solved the eigenproblem by the CMS method using a cut-off bound of20,000 (about 10 times the largest wanted eigenvalue 50 2183). 329eigenvalues of the substructure problems were less than our threshold, andthe dimension of the resulting projected problem was 2289.

    0 10 20 30 40 5010

    8

    107

    106

    105

    104

    103

    102

    CMS: cut off frequency 20000

    number of eigenvalue

    rela

    tive

    erro

    r

    TUHH Heinrich Voss AMLS Summer School 2006 17 / 45

  • Reducing interface DoF

    The number of interface degrees of freedom may still be very large, andtherefore the dimension of the reduced problem (8) may be very high. It canbe reduced further by modal reduction of the interface degrees of freedom inthe following way:

    Considering the eigenvalue problem

    Kii