Alternative Model Order Reduction in Elastic Multibody Systems
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Transcript of Alternative Model Order Reduction in Elastic Multibody Systems
Institute of Engineering and Computational Mechanics University of Stuttgart, Germany Prof. Dr.-Ing. Prof. E.h. Peter Eberhard
6th European ATC
Turin, April 22-24, 2013
Alternative Model Order Reduction in
Elastic Multibody Systems
Philip Holzwarth, Peter Eberhard
Institute of Engineering and Computational Mechanics University of Stuttgart, Germany Prof. Dr.-Ing. Prof. E.h. Peter Eberhard
Example: FE Structure
FEM-model of a structure
11873 nodes 4986 elements
about 35000 elastic degrees of freedom
goal is control
lower plate assumed to be rigid
modelled as point mass
connected with CERIG command to rods
hole in upper plate is interface to remaining part
of the structure
modelled with spider web of beams
diameter 1 mm
Young's modulus 1018 N/m2
density 100 kg/m3
Institute of Engineering and Computational Mechanics University of Stuttgart, Germany Prof. Dr.-Ing. Prof. E.h. Peter Eberhard
Comparison with Modal
Reduction F
FE
F
F
)(
)(
)(
)()()(
H
H
H
HH
21
2
1
21
2
1 F
d)(
d)(
Q2
2
E
H
H
Q
Krylov (173) 2.06 10-7
Krylov+gram
(35/173)
5.16 10-6
POD (36/24) 2.64 10-5
modal (40) 7.27 10-3
alternative reduction methods
show better results
Institute of Engineering and Computational Mechanics University of Stuttgart, Germany Prof. Dr.-Ing. Prof. E.h. Peter Eberhard
Outline
motivation
model order reduction in elastic MBS – Why is this an
important step to obtain a good model?
different methods to obtain reduced flexible bodies
examples
large systems (industrial application)
software package Morembs
summary
Institute of Engineering and Computational Mechanics University of Stuttgart, Germany Prof. Dr.-Ing. Prof. E.h. Peter Eberhard
Basis of Elastic
Multibody Systems multibody system
elastic body
discretization
finite element,
finite difference,
...
continuum
elastic multibody system
rigid body
bearings and
coupling elements p bodies
f degrees of freedom
q reaction force
C
reduction of the
elastic degrees
of freedom
models are getting larger
and more detailed
many degrees of freedom
FE-models have to be reduced
with the floating frame of reference
formulation linear model order
reduction is possible
Institute of Engineering and Computational Mechanics University of Stuttgart, Germany Prof. Dr.-Ing. Prof. E.h. Peter Eberhard
industrial practice
fine discretization
many dof (e.g.
10 million)
Modeling Elasticity with
the FEM continuum formulation
PDE
spatial discretization
ODE
finite element method
linear model order reduction
reduced FE equation of motion
with
with
𝐌 e ∙ 𝐪 e + 𝐃 e ∙ 𝐪 e + 𝐊 e ∙ 𝐪 e = 𝐡 e
𝐌 e = 𝐕T ∙ 𝐌e ∙ 𝐕, … 𝐡 e = 𝐕T ∙ 𝐁e ∙ 𝐮e
projection matrix
dim 𝐪 e ≪ dim 𝐪e , 𝐪e ≈ 𝐕 ∙ 𝐪 e
𝐕 ∈ ℝn×m
finite element method
FE equation of motion
input/output aspect
define input or control matrix
define output/observation matrix
𝐌e ∙ 𝐪 e + 𝐃e ∙ 𝐪 e + 𝐊e ∙ 𝐪e = 𝐡e
𝐌e ∙ 𝐪 e + 𝐃e ∙ 𝐪 e + 𝐊e ∙ 𝐪e = 𝐁e ∙ 𝐮e 𝐲 = 𝐂e ∙ 𝐪e
𝐁e 𝐂e
Institute of Engineering and Computational Mechanics University of Stuttgart, Germany Prof. Dr.-Ing. Prof. E.h. Peter Eberhard
Model Reduction
Techniques model reduction techniques
used for elastic bodies
condensation
static condensation (Guyan, constraint modes)
dynamic condensation
modal truncation
free-free modes
fixed-interface modes
component mode synthesis
Hurty/Craig-Bampton method
Craig-Chang method
...
Component Mode Synthesis
static moment-matching via
Padé-type approximation
moment-matching with
Krylov subspaces
Arnoldi, Lanczos
iterative methods (IRKA, MIRIAM),
adaptive methods (SOAGA)
...
tangential interpolation
interpolation methods
balanced truncation
second order balancing
Lyapunov balancing
stochastic balancing
bounded real balancing
…
frequency weighted balanced
truncation
Proper Orthogonal Decomposition
(POD)
Gramian-based methods
Block-Krylov vectors as component
modes
(Extended) Singular Value
Decomposition Model Order
Reduction (E)SVDMOR
Laguerre-based model reduction
RK-ICOP
…
hybrid methods
Institute of Engineering and Computational Mechanics University of Stuttgart, Germany Prof. Dr.-Ing. Prof. E.h. Peter Eberhard
approach
approximation of the nodal displace-
ment vector by a linear combination of
the dominant eigenvectors (normal modes)
quadratic eigenvalue problem
projection matrices
problem: how to select important normal modes?
standard: sorting by eigenfrequency
and experience of the user
Modal Approximation/
Truncation
modal
1
e~
qΦ
Φq
r
i
tie
i
0ΦKM ieei )(2
TΦ3rd
6th
30th
, ...
, ...
𝐪𝐞 ≈ 𝛗𝑖 ∙𝐫
𝐢=𝟏𝐪 e,𝑖
= 𝚽 ∙ 𝐪 e 𝚽T =
−λ𝑖2𝐌e + 𝐊e ∙ 𝛗𝑖 = 0
Institute of Engineering and Computational Mechanics University of Stuttgart, Germany Prof. Dr.-Ing. Prof. E.h. Peter Eberhard
approach
definition of boundary nodes, where
further components are connected
constraint modes for all boundary coordinates (as in Guyan condensation)
unit displacement of one boundary while others held fixed
additional fixed-interface normal modes
for the inner part of the system
projection matrices
𝐪e,i𝐪e,b
= 𝚽k 𝚿c ∙𝐪 e,i𝐪e,b
CMS/ Craig-Bampton
big improvements to modal truncation
inter-component compatibility
exact static response
movement of boundary dofs is
explicitly available
normal modes for internal
dynamics are selected
by their frequency
err
or 𝜀 [−]
frequency f [Hz] normal modes for internal
dynamics are selected
by their frequency
Institute of Engineering and Computational Mechanics University of Stuttgart, Germany Prof. Dr.-Ing. Prof. E.h. Peter Eberhard
approach
partitioning into boundary and internal dof
second order LTI system: definition of input and output, transfer matrix
transfer matrix at s = 0, using Schur complement
static matching of the transfer function for reduced and original system if
Guyan-Condensation
𝐌ii 𝐌ib
𝐌bi 𝐌bb⋅𝐪 i𝐪 b
+𝐊ii 𝐊ib
𝐊bi 𝐊bb⋅𝐪i𝐪b
=𝐡i𝐡b
𝐡i𝐡b
=𝐁i𝐁b
⋅ 𝐮, 𝐲 = 𝐂i 𝐂b ⋅𝐪i𝐪b
𝐇 s = 𝐂e ∙ s2𝐌e + 𝐊e−1 ∙ 𝐁e
𝐇 0 = 𝐂i ⋅ 𝐊ii−1 ⋅ 𝐁i + 𝐂 b ⋅ 𝐊 bb
−1 ⋅ 𝐁 b with 𝐊 bb = 𝐊bb − 𝐊bi ⋅ 𝐊ii−1 ⋅ 𝐊ib
𝐂 b = 𝐂b − 𝐂i ⋅ 𝐊ii−1 ⋅ 𝐊ib
𝐁 b = 𝐁b − 𝐊bi ⋅ 𝐊ii−1 ⋅ 𝐁i
= 𝟎
𝐕 = 𝐈ii
−𝐊ii−1 ⋅ 𝐊ib
Krylov subspace methods:
numerically robust extension of this concept to
• arbitrary combinations of matching frequencies
• derivatives of 𝐇(s)
Institute of Engineering and Computational Mechanics University of Stuttgart, Germany Prof. Dr.-Ing. Prof. E.h. Peter Eberhard
transfer matrix
series expansion with expansion point
moment matching of original and
reduced system is achieved implicitly
when using the Krylov subspace
with
Moment Matching via
Krylov-Subspaces
e1
eee2
e )ss()s( BKDMCH
𝐇 s = 𝐂e ∙ s2𝐌e + s𝐃e + 𝐊e−1 ∙ 𝐁e
)s()s(
s~s
10
j
0j
j
TTT
TH
𝐇 s = 𝐓0σ + 𝐓1
σ s − σ +⋯+ 𝐓∞σ s − σ ∞
𝐓j
σ: moments of the transfer function
σ }{},,,{),( 1VRMRMRRΜ colspspan r
rK 𝒦r 𝐌,𝐑 = span 𝐑,𝐌 ∙ 𝐑,⋯ ,𝐌r−1 ∙ 𝐑 ⊆ colsp{𝐕}
𝐌 = 𝐊e−1 ∙ 𝐌e
𝐑 = 𝐊e−1 ∙ 𝐁e
σ
‖𝐇f‖F
frequency f [Hz]
Institute of Engineering and Computational Mechanics University of Stuttgart, Germany Prof. Dr.-Ing. Prof. E.h. Peter Eberhard
advantages
a priori error
bound exists
weighting of a certain
frequency range
disadvantage
only efficiently possible
for small models
MOR with Frequency
Weighted Gramian Matrices approach
Gramian matrices provide an energy
interpretation of the system’s states
controllability Gramian matrix
observability Gramian matrix
balanced representation
Hankel singular values
truncation of states represented by
small singular values
Balanced Truncation
frequency weighted reduction
usage of frequency weighted Gramian
matrices
very good approximation of a specific
frequency range
advantages
a priori error
bound exists
weighting of a certain
frequency range
disadvantage
only efficiently possible
for small models 2-step approach
POD
𝐏 𝐐
σi = λi(𝐏 ∙ 𝐐)
err
or 𝜀 [−]
frequency f [Hz]
35 30
20
13
err
or 𝜀 [−]
frequency f [Hz]
Institute of Engineering and Computational Mechanics University of Stuttgart, Germany Prof. Dr.-Ing. Prof. E.h. Peter Eberhard
Numerical Examples –
Frequency Domain crankshaft
exhaust
err
or 𝜀 [−]
frequency f [Hz] err
or 𝜀 [−]
frequency f [Hz]
Institute of Engineering and Computational Mechanics University of Stuttgart, Germany Prof. Dr.-Ing. Prof. E.h. Peter Eberhard
Numerical Examples –
Time Domain
time t [s] acce
lera
tion a [mm/s
2]
me
an
err
or
ca
lcu
lation
tim
e [s]
err
or
time t [s]
large improvement on mean error
with shorter calculation time
Krylov (50)
POD (50) Gram (50)
CMS (110)
Institute of Engineering and Computational Mechanics University of Stuttgart, Germany Prof. Dr.-Ing. Prof. E.h. Peter Eberhard
- Preprocessor
to reduce elastic bodies
MOREMBS contains implementations
of converters for a wide variety of FE-
programs and numerous reduction
methods. The software is available as
a Matlab-based version (MatMorembs)
as well as one in C++ (Morembs++).
MOREMBS (Model Order Reduction of
Elastic Multibody Systems)
- users in
research & industry
Departement
Werktuigkunde
LUT Metal Technology,
Faculty of Technology
VDLAB (Vehicle
Dynamics
Laboratory) Universität
Kassel
Institut für
Mechanik
cooperation
and project
partners
Institute of Engineering and Computational Mechanics University of Stuttgart, Germany Prof. Dr.-Ing. Prof. E.h. Peter Eberhard
FE-Software
workflow in
import from Ansys, Abaqus,
Permas, …
reduction with traditional and
modern methods
export to Neweul-M2, RecurDyn,
Simpack, VL.Motion, …
advantages
usage of
standard FEM programs
standard MKS programs
modern reduction methods
instead of only modal methods
preserving the familiar process
chain
MKS-Software
Institute of Engineering and Computational Mechanics University of Stuttgart, Germany Prof. Dr.-Ing. Prof. E.h. Peter Eberhard
Morembs in the
HyperWorks Process
Chain
process
control
full body
reduced
body
FE Solver
RADIOSS
ABAQUS
LS-DYNA
NASTRAN
ANSYS
…
MBS
Motion Solve
Adams
SIMPACK
…
making use of HyperWorks’
various interfaces
(sketch, current project with Altair)
Institute of Engineering and Computational Mechanics University of Stuttgart, Germany Prof. Dr.-Ing. Prof. E.h. Peter Eberhard
Summary advantages of alternative MOR
better, more reliable results
(guaranteed error bounds)
no mode selection by hand
necessary
shorter computation times
automated algorithms available
many examples with industrial
relevance
challenges and current topics
many inputs
coupled bodies
moving loads
break squeal
uncertainties
industrial applicability,
interfaces
…
advanced MOR techniques
• improve results if computational
effort is the same
• speed up calculations for the
same quality of results
Thank you for your attention!