AMLS METHOD Automated Multi

Automated Multi-Level Substructuring
AMLS was introduced by Bennighof (1998) and was applied to huge problems
of frequency response analysis.
CHAPTER 4 : AMLS METHOD
The large finite element model is recursively divided into very many
substructures on several levels based on the sparsity structure of the system
matrices.
Heinrich Voss
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.
[email protected]
Hamburg University of Technology
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.
Heinrich Voss (Hamburg University of Technology)
AMLS
Eigenvalue problems 2012
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Heinrich Voss (Hamburg University of Technology)
Condensation
AMLS
Eigenvalue problems 2012
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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 reordering
problem (1) obtains the following form
Given (a finite element model of a structure, e.g.)
Kx = λMx
(1)
K``
Ki`
K`i
Kii
x`
M``
=λ
xi
Mi`
M`i
Mii
x`
xi
(2)
Solving the first equation for x` yields
where K ∈ Rn×n and M ∈ Rn×n are symmetric and M is positive definite.
x` = −(K`` − λM`` )−1 (K`i − λM`i )xi
Aim: Reduce the number of unknowns by some sort of elimination.
and substituting in the second equation one gets the exactly condensed
eigenproblem
T (λ)xi = −Kii xi + λMii xi + (Ki` − λMi` )(K`` − λM`` )−1 (K`i − λM`i )xi
Heinrich Voss (Hamburg University of Technology)
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Eigenvalue problems 2012
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Heinrich Voss (Hamburg University of Technology)
AMLS
Eigenvalue problems 2012
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Static condensation
Substructuring
Linearizing the exactly condensed problem at ω = 0 yields the statically
condensed eigenproblem (introduced independently by Irons (1965) and
Guyan (1965))
˜ii xi = λM
˜ ii xi
K
(3)
Consider the vibrations of a structure which is partitioned into r substructures
connecting to each other through the variables on the interfaces only.
Then ordering the unknowns appropriately the stiffness matrix obtains the
following block form


K``1
O
...
O
K`i1
 O
K``2 . . .
O
K`i2 


 ..
..
..
.. 
..
K = .
.
.
.
. 


 O
O
. . . Kssr Ksmr 
Ki`1 Ki`2 . . . Kmsr
Kii
where
˜ii
K
˜ ii
M
−1
= Kii − Ki` K``
K`i
−1
−1
−1
−1
= Mii − Ki` K``
M`i − M`i K``
K`i + Ki` K``
M`` K``
K`i
For vibrating structures this means that the local degrees of freedom are
assumed to depend quasistatically on the interface degrees of freedom, and
the inertia forces of the substructures are neglected.
Heinrich Voss (Hamburg University of Technology)
AMLS
Eigenvalue problems 2012
and M has the same block form.
5 / 45
Substructuring ct.
Heinrich Voss (Hamburg University of Technology)
AMLS
Eigenvalue problems 2012
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Example
FEM model of a container ship: 35262 DoF, bandwidth: 1072
For the statically condensed problem we obtain
˜ii = Kii −
K
r
X
j=1
˜ ii = Mii −
M
where
−1
Kmsj Kssj
Ksmj
r
X
Mmmj ,
j=1
200
40
Mmmj =
−1
Kmsj Kssj
Msmj
+
−1
Mmsj Kssj
Ksmj
−
−1
−1
Kmsj Kssj
Mssj Kssj
Ksmj .
100
The submatrices corresponding to the individual substructures can be
determined independently from smaller subproblems and in parallel.
Heinrich Voss (Hamburg University of Technology)
AMLS
Eigenvalue problems 2012
150
20
0
10
7 / 45
50
0
−10
0
Heinrich Voss (Hamburg University of Technology)
AMLS
Eigenvalue problems 2012
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Example ct.
Example ct.
Container ship: relative errors of static condensation
10 substructures; condensation to 1960 interface DoF
#
1
2
3
4
5
6
7
8
9
10
11
12
Heinrich Voss (Hamburg University of Technology)
AMLS
Eigenvalue problems 2012
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A projection approach
˜ `i
M
˜ ii
M
=
˜ ii
M
=
=
−1
Kii − Ki` K``
K`i
Heinrich Voss (Hamburg University of Technology)
AMLS
Eigenvalue problems 2012
10 / 45
˜ii y = λM
˜ ii y
K
.
(4)
To model the deviation from quasistatic behavior thereby improving the
approximation properties of static condensation we consider the eigenvalue
problem
K`` Φ = M`` ΦΩ, ΦT M`` Φ = I,
(5)
is the Schur complement of K``
−1
˜T
M`i − M`` K``
K`i = M
i`
−1
−1
Mii − Mi` K``
K`i − Ki` K``
M`i
AMLS
Neglecting in (4) all rows and columns corresponding to local degrees of
freedom,
i.e.
projecting problem (1) to the subspace spanned by columns of
−1
−K`` K`i
one obtains the method of static condensation
I
Here K`` and M`` stay unchanged, and
˜ii
K
˜ `i
M
nodal cond.
5.02e-05
2.36e-05
6.32e-05
1.06e-04
3.98e-04
6.16e-04
5.47e-03
2.11e-02
2.49e-02
8.41e-02
1.08e-01
1.25e-01
static condensation revisited
We transform the matrix K to block diagonal form using block Gaussian
elimination, i.e. we apply the congruence transformation with
−1
I −K``
K`i
P=
0
I
to the pencil (K , M) obtaining the equivalent pencil
K`` 0
M``
(P T KP, P T MP) =
˜ii , M
˜ i`
0 K
Heinrich Voss (Hamburg University of Technology)
eigenvalue
1.2555112888e-01
1.4842667377e-01
1.8859647898e-01
8.2710672903e-01
1.4571047916e+00
1.8843144791e+00
2.4004294125e+01
5.2973437588e+01
5.6869743387e+01
1.7501327597e+02
2.0806150033e+02
2.8210662009e+02
where Ω is a diagonal matrix containing the eigenvalues.
−1
−1
+ Ki` K``
M`` K``
K`i .
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Heinrich Voss (Hamburg University of Technology)
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Eigenvalue problems 2012
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Craig–Bampton form
Component Mode Synthesis (CMS)
Changing the basis for the local degrees of freedom to a modal one, i.e.
applying the further congruence transformation diag{Φ, I} to problem (4) one
gets
˜ `i
Ω 0
I
ΦT M
.
(6)
,
˜ii
˜ i` Φ
˜ ii
0 K
M
M
In structural dynamics (6) is called Craig–Bampton form of the eigenvalue
problem (1) corresponding to the partitioning (2).
In terms of linear algebra it results from block Gaussian elimination to reduce
K to block diagonal form, and diagonalization of the block K`` using a spectral
basis.
Heinrich Voss (Hamburg University of Technology)
AMLS
Eigenvalue problems 2012
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Selecting some eigenmodes of problem (5), and dropping the rows and
columns in (6) corresponding to the other modes one arrives at the
component mode synthesis method (CMS) introduced by Hurty (1965) and
Craig & Bampton (1968).
If the diagonal matrix Ω1 contains in its diagonal the eigenvalues to drop and
Φ1 the corresponding eigenvectors, and if Ω2 and Φ2 contain the eigenvalues
and eigenvectors to keep, respectively, then the eigenproblem (6) can be
rewritten as
 


˜ `i1  x1 
Ω1 0
0
I
0
M
x1
˜ `i2  x2 
 0 Ω2 0  x2  = λ  0
(7)
I
M
˜ii
˜
˜
˜
x3
x
0
0 K
3
Mi`1 Mi`2 Mii
with
˜ smj = ΦT (M`i − M`` K −1 K`i ) = M
˜ T , j = 1, 2,
M
j
msj
``
Heinrich Voss (Hamburg University of Technology)
CMS ct.
14 / 45
We consider the structural deformation caused by a harmonic excitation at a
frequency of 4 Hz which is a typical forcing frequency stemming from the
engine and the propeller.
Usually the eigenvectors according to eigenvalues which do not exceed a cut
off threshold are kept. In vibration analysis of a structure this choice is
motivated by the fact that the high frequencies of a substructure do not
influence the wanted low frequencies of the entire substructure very much.
Notice however that in a recent paper Bai and Liao (2006) suggested a
different choice based on a moment–matching analysis.
AMLS
Eigenvalue problems 2012
Container ship
and the CMS approximations to the eigenpairs of (1) are obtained from the
reduced eigenvalue problem
˜ `i2
Ω2 0
I
M
y
=
λ
(8)
˜ii
˜ i`2 M
˜ ii y
0 K
M
Heinrich Voss (Hamburg University of Technology)
AMLS
Eigenvalue problems 2012
Since the deformation is small the assumptions of the linear theory apply, and
the structural response can be determined by the mode superposition method
taking into account eigenfrequencies in the range between 0 and 7.5 Hz
(which corresponds to the 50 smallest eigenvalues for the ship under
consideration).
To apply the CMS method we partitioned the FEM model into 10 substructures
as shown before. This substructuring by hand yielded a much smaller number
of interface degrees of freedom than automatic graph partitioners which try to
construct a partition where the substructures have nearly equal size.
For instance, our model ends up with 1960 degrees of freedom on the
interfaces, whereas Chaco ends up with a substructuring into 10
substructures with 4985 interface degrees of freedom.
15 / 45
Heinrich Voss (Hamburg University of Technology)
AMLS
Eigenvalue problems 2012
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Container ship ct.
Reducing interface DoF
We solved the eigenproblem by the CMS method using a cut-off bound of
20,000 (about 10 times the largest wanted eigenvalue λ50 ≈ 2183). 329
eigenvalues of the substructure problems were less than our threshold, and
the dimension of the resulting projected problem was 2289.
The number of interface degrees of freedom may still be very large, and
therefore the dimension of the reduced problem (8) may be very high. It can
be reduced further by modal reduction of the interface degrees of freedom in
the following way:
CMS: cut off frequency 20000
−2
10
Considering the eigenvalue problem
−3
10
˜ii Ψ = M
˜ ii ΨΓ, ΨT K
˜ii Ψ = Γ, ΨT M
˜ ii Ψ = I,
K
−4
relative error
10
and applying the congruence transformation to the pencil in (6) with
˜ = diag{I, Ψ}, we obtain the equivalent pencil
P
−5
10
−6
10
−7
10
with
−8
10
0
10
Heinrich Voss (Hamburg University of Technology)
20
30
number of eigenvalue
40
AMLS
50
Eigenvalue problems 2012
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Reducing interface DoF ct.
Selecting eigenmodes of (5) and of (10) and neglecting rows and columns in
(11) which correspond to the other modes one gets a reduced problem which
is the one level version of the automated multilevel substructuring method,
introduced by Bennighof (1992).
0
0
Ω2
0
Heinrich Voss (Hamburg University of Technology)
 
I
0
ˆ


0  M21
,
0  0
ˆ 41
Γ2
M
ˆ 12
M
I
ˆ 32
M
0
AMLS
0
ˆ
M23
I
ˆ 43
M
ˆ 14 
M

0 


ˆ
M34 
I
Heinrich Voss (Hamburg University of Technology)
. . . where
ˆ `i
M
I
(11)
(12)
AMLS
Eigenvalue problems 2012
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Reducing interface DoF ct.
ˆ 12
M
ˆ 14
M
=
=
=
=
−1
ˆT
ΦT1 (M`i − M`` K``
K`i )Ψ1 = M
21
−1
T
ˆ
Φ (M`i − M`` K K`i )Ψ2 = M T
1
ΦT2 (M`i
ΦT2 (M`i
−
−
``
−1
M`` K``
K`i )Ψ1
−1
M`` K`` K`i )Ψ2
41
ˆT
=M
23
ˆ
= MT .
43
For the container ship we reduced the interface degrees of freedom as well
with the same cut-off bound 20,000. This reduced the dimension of the
projected eigenproblem further from 2289 to 436.
(13)
Eigenvalue problems 2012
O
I
, ˆT
Γ
M`i
Then the single level approximations of AMLS to eigenpairs are obtained from
ˆ 34
Ω2 0
I
M
y =λ ˆ
y.
(14)
0 Γ2
M43
I
and rearranging the rows and columns beginning with the modes
corresponding to Φ1 and Ψ1 to be dropped followed by the ones
corresponding to Φ2 and Ψ2 problem (11) obtains the form
0
Γ1
0
0
Ω
O
ˆ `i = ΦT (M`i − M`` K −1 K`i )Ψ = M
ˆ T.
M
i`
``
ˆ 32
M
ˆ 34
M
Similarly as for the CMS method we partition the matrices Γ and Ψ into
Γ1 0
Γ=
and Ψ = (Ψ1 , Ψ2 )
0 Γ2

Ω1
 0

 0
0
(10)
The next picture shows the relative errors of CMS and the single level version
of AMLS.
19 / 45
Heinrich Voss (Hamburg University of Technology)
AMLS
Eigenvalue problems 2012
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Relative errors CMS and AMLS(1)
−1
Multi-Level Substructuring: Level 0
CMS and AMLS(1): cut off frequency 20000
10
−2
10
−3
relative error
10
−4
10
−5
10
−6
10
−7
10
−8
10
0
Heinrich Voss (Hamburg University of Technology)
10
20
30
number of eigenvalue
AMLS
40
50
Eigenvalue problems 2012
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Multi-Level Substructuring: Level 1
Heinrich Voss (Hamburg University of Technology)
AMLS
Eigenvalue problems 2012
Heinrich Voss (Hamburg University of Technology)
AMLS
Eigenvalue problems 2012
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Multi-Level Substructuring: Level 2
23 / 45
Heinrich Voss (Hamburg University of Technology)
AMLS
Eigenvalue problems 2012
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Multi-Level Substructuring: Level 3
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AMLS
Eigenvalue problems 2012
Multi-Level Substructuring: Level 4
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Heinrich Voss (Hamburg University of Technology)
Multi-Level Substructuring: Level 5
AMLS
Eigenvalue problems 2012
26 / 45
AMLS - Algorithm (Kx = λMx)
Reorder System (using Graph Partitioner):

Ks
T
Ksm
KsrT
Ksm
Km
T
Kmr

Ksr
Kmr 
Kr
with


Ks = 
Ks1
..

.
Ksn


1
2
3
Kr
4
5
Heinrich Voss (Hamburg University of Technology)
AMLS
Eigenvalue problems 2012
27 / 45
6
Km
7
Ks
Heinrich Voss (Hamburg University of Technology)
AMLS
Eigenvalue problems 2012
28 / 45
AMLS - Algorithm ct.
Congruence transformation with

I −Ks−1 Ksr
U = O
I
O
O
yields

Ks
0
0
0
ˆm
K
ˆT
K
mr

0
ˆmr  ,
K
ˆr
K
AMLS - Algorithm ct.
Solving of substructure EVPs

−Ks−1 Kmr

O
I

Ms
ˆT
M
sm
ˆT
M
sr
ˆ sm
M
ˆm
M
ˆ
MT
mr
Ks Φs = Ms Φs Ωs ,
and projecting on a subset of Φs (usually corresponding to eigenvalues not
exceeding a cut-off frequency) yields


˜

˜
˜
ˆ sm M
ˆ sr
Is
M
Ωs
0
0
˜


ˆm K
ˆmr  , M
0 K
T
ˆ
ˆ
ˆ
M
M
m
mr
 sm

T
ˆ
ˆ
˜
0 Kmr Kr
ˆT M
ˆT
ˆr
M
M
ˆ sr 
M
ˆ mr 
M
ˆr
M
sr
Notice that Ks is block-diagonal, and determining Ks−1 Ksr means that a large
number of linear system of small dimension have to solved. Moreover, the
congruence transformation consists of block matrix multiplications for blocks
of small dimension.
Heinrich Voss (Hamburg University of Technology)
AMLS
Eigenvalue problems 2012
ΦTs Ms Φs = I
29 / 45
This first step of AMLS was introduced already by Hurty (1965) and by Craig
and Bampton (1968), and it is called Component Mode Synthesis (CMS).
Heinrich Voss (Hamburg University of Technology)
AMLS - Algorithm ct.
Once substructures on the lowest level have been transformed and reduced
by modal projection they are assembled to parent substructures on the next
level.
mr
AMLS
Eigenvalue problems 2012
30 / 45
AMLS - Algorithm ct.
Interface and local degrees of freedom are identified, and the substructure
models are transformed similarly as on the lowest level.
˜
Ω1
O

O
O
O
˜2
Ω
O
O
O
O
˜ii
K
˜H
K
ir
 

O
I
z1
z2 
O
O
 

˜ir  z3  = λ  M
˜H
K
1i
˜rr
˜H
z4
K
M
1r
O
I
˜H
M
2i
˜H
M
2r
˜ 1i
M
˜ 2i
M
˜ ii
M
˜H
M
˜ 1r  z 
M
1
˜ 2r  z2 
M
 
˜ ir  z3  ,
M
˜ rr
z4
M
O
I
˜H
M
2i
ˆH
M
2r
˜ 1i
M
˜ 2i
M
˜ ii
M
ˆ
MirH
ˆ 1r  w 
M
1
ˆ 2r  w2 
M
 ,
ˆ ir  w3 
M
ˆ
w4
Mrr
ir
˜jr yields
Block-elimination of K
˜
Ω1
O

O
O
Heinrich Voss (Hamburg University of Technology)
AMLS
Eigenvalue problems 2012
31 / 45
O
˜2
Ω
O
O
O
O
˜ii
K
O
 

I
O
w1
w2 
O
O
  = λ
M
˜H
O  w3 
1i
H
ˆ
ˆ
w4
Krr
M1r
Heinrich Voss (Hamburg University of Technology)
AMLS
Eigenvalue problems 2012
32 / 45
AMLS - Algorithm ct.
AMLS - Algorithm ct.
Treating coarser levels one after the other in the same way one gets a
projected eigenvalue problem of significantly lower dimension
To perform the modal reduction of the interior degrees of freedom of the
current substructure one would have to solve the eigenvalue problem
˜
Ω1
O
O
 

O
I
w1




w2 = ω O
O
˜ii
˜H
w3
K
M
1i
O
˜
Ω2
O
O
I
˜H
M
2i
Kc x = λMc x
˜ 1i  w1 
M
˜ 2i  w2  .
M
˜ ii
w3
M
with Kc spd and diagonal and Mc spd in generalized arrowhead structure.
Massmatrix of AMLS
However, since the number of interior degrees of freedom of substructures
grows too large in the course of the algorithm, we reduce the dimension only
taking advantage of the eigenvalue problem corresponding to the right lower
diagonal block, i.e.
˜ii Φi = M
˜ ii Φi Ωi , ΦH M
˜ ii Φi = I.
K
i
Applying the congruence transformation with T = diag{I, I, Φi , I} and dropping
all rows and columns in the third block if the corresponding eigenvalue
exceeds the cut-off frequency we further reduce the dimension of the
eigenproblem.
Heinrich Voss (Hamburg University of Technology)
AMLS
Eigenvalue problems 2012
33 / 45
Heinrich Voss (Hamburg University of Technology)
AMLS
Container ship
We substructured the FE model of the container ship by Metis with 4 levels of
substructuring. Neglecting eigenvalues exceeding 20,000 and 40,000 on all
levels AMLS produced a projected eigenvalue problem of dimension 451 and
911, respectively.
Eigenvalue problems 2012
34 / 45
Example
FEM model of 2D problem in vibrational analysis with linear Lagrangean
elements.
n = 68 862 degrees of freedom.
Method
Arnoldi
Jacobi-Davidson
0
10
10 eigvals
10.7
42.2
50 eigvals
37.8
148.3
200 eigvals
221.1
901.9
secs
secs
−1
10
AMLS
−2
10
ωc
10 ∗ λ50
40 ∗ λ50
50 ∗ λ50
65 ∗ λ50
−3
10
−4
10
−5
10
−7
0
5
Heinrich Voss (Hamburg University of Technology)
10
15
20
25
AMLS
30
35
40
45
50
Eigenvalue problems 2012
tsolve
1.7
7.5
10.3
14.7
nc
418
1407
1720
2166
max.err. 10
0.14%
0.014%
0.0097%
0.0068%
max.err. 50
3.63%
0.37%
0.24%
0.15%
max.err. 200
25.2%
2.67%
1.75%
1.05%
Even this quite small sized eigenvalue problems demonstrates that AMLS
becomes competitive if a large number of eigenvalues is wanted the accuracy
of which need not be too high.
−6
10
10
tred
205.7
209.1
209.2
211.3
35 / 45
Heinrich Voss (Hamburg University of Technology)
AMLS
Eigenvalue problems 2012
36 / 45
Connected beams
Connected beams ct.
We report on the performance of AMLS for a FE model of a structure of
connected beams
For the following analysis a discretization with linear Lagrangean elements
with n = 517161 DoFs is used.
The AMLS method is applied with cut-off frequency ωc = 4 · 109 .
Due to the linear elements the matrices are relatively sparse resulting in small
interfaces over all levels. Consequently, the eigenvalue problems are small as
well, which can be seen in the average size of the eigenvalue problems on
each level.
The computer used is a 32-bit workstation with a 3.0 GHz Pentium and 1.5
GByte memory. AMLS is implemented (by Kolja Elssel) in C using METIS for
computing graph partitions and LAPACK.
Heinrich Voss (Hamburg University of Technology)
AMLS
Eigenvalue problems 2012
37 / 45
Connected beams ct.
The distribution of component normal modes over the levels is typical for large
scale problems. The average number of component normal modes (CNM) for
the interface and substructure eigenvalues problems that are below the cut-off
frequency decreases on lower levels. Quite commonly no CNMs are used for
the lowest level substructures.
Heinrich Voss (Hamburg University of Technology)
AMLS
Eigenvalue problems 2012
38 / 45
Connected beams ct.
The following table contains substructuring information for AMLS
level
1
2
3
4
5
6
7
8
9
10
11
12
nsub
1
2
4
9
27
55
110
220
440
882
1060
953
Heinrich Voss (Hamburg University of Technology)
Avg.EVP size
681
537
376
548
425
309
268
160
87
126
112
153
AMLS
The limiting factors for the applicability of the algorithm are the computational
time and the memory requirements. For the computations discussed external
storage was used to store contemporary data and data needed for
subsequent calculations such as the computation of Ritz vectors. The
following figure shows the memory consumption and the temporary storage
needed by the algorithm.
Avg. # CNM (Σ)
39.00 (39)
11.50 (23)
7.00 (28)
9.67 (87)
4.70 (127)
2.80 (154)
1.04 (114)
0.19 (41)
0 (0)
0 (0)
0 (0)
0 (0)
Eigenvalue problems 2012
The large peak at the beginning of the calculation and in the middle are due
the graph partitioner which computes partitions for the graph corresponding to
the system matrices. For larger systems this becomes a limiting factor.
For systems which have a denser structure the size of the interface problems
become larger and cause problems with memory consumption and the
solution of the interface eigenvalue problems.
39 / 45
Heinrich Voss (Hamburg University of Technology)
AMLS
Eigenvalue problems 2012
40 / 45
Connected beams ct.
Connected beams ct.
Memory allocation profile of AMLS method
The computational time for this problem can be roughly divided into three
parts. With 60% the largest part of the computational time is spent on matrix
multiplications resulting from the variable transformations in step 2 of the
AMLS algorithm.
700
Memory
Harddisk
Memory Allocation [MByte]
600
500
The second largest part is with 20% due to the eigenvalue solver, followed by
the solution of linear systems of equations with 15%.
400
The remaining five percent consist of matrix partitioning (about 3%), matrix
substructuring and algorithmic overhead.
300
200
Note, the matrix multiplications originating from the eigensolver and the linear
system solver are included into their respective percentages and are not
included in the percentage of the matrix multiplications.
100
0
0
50
100
Heinrich Voss (Hamburg University of Technology)
150
200
250
300
Time [seconds]
350
400
AMLS
450
500
Eigenvalue problems 2012
41 / 45
Connected beams ct.
One of the largest systems which has been reduced with the AMLS method
has been discretized with linear Lagrange elements and has n = 1 951 170
degrees of freedom (about 78 million non-zeros in the stiffness matrix and 26
million in the mass matrix).
The computational time for this discretization is tred = 5 431 seconds
(approximately 1.5 hours).
Eigenvalue problems 2012
42 / 45
Eigenvalue problems 2012
Another discretization has been computed with quadratic Lagrange elements
and has n = 1 270 947 degrees of freedom.
Here, the interface problems are larger than for the linear Lagrange element
system. For instance the highest level has 2 202 degrees of freedom and the
average size of eigenvalue problems on the fourth level is 1 019.
The reduction over 13 levels and nsub = 8 223 substructures takes tred = 4 161
seconds.
Significantly raising the cut-off frequency to ωc = 1011 results in a nc = 10 240
dimensional system.
Bisections are used for the substructuring which results in nsub = 13 694
substructures over nlevel = 14 levels.
AMLS
AMLS
Connected beams ct.
To compare the scalability other discretization of the same model have been
computed.
Heinrich Voss (Hamburg University of Technology)
Heinrich Voss (Hamburg University of Technology)
Notice that the computational time increases by less than 3%.
43 / 45
Heinrich Voss (Hamburg University of Technology)
AMLS
Eigenvalue problems 2012
44 / 45
Connected beams ct.
Results of AMLS applied to large linear eigenvalue problems
Elements
Linear
Linear
Quadratic
Quadratic
Linear
Linear
n
517 161
517 161
1 270 947
1 270 947
1 951 170
2 297 175
Heinrich Voss (Hamburg University of Technology)
ωc
4·108
4·109
4·109
1·1011
4·109
4·109
nc
218
613
653
10 240
648
651
AMLS
nsub
3 763
3 763
8 223
8 223
13 694
15 283
nlevel
12
12
13
13
14
14
tred
499 sec
502 sec
4 161 sec
4 232 sec
5 431 sec
7 928 sec
Eigenvalue problems 2012
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