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


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


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


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


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


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


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


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


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


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)


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]


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]


Numerical Examples –

Frequency Domain crankshaft

exhaust

err

or 𝜀 [−]

frequency f [Hz] err

or 𝜀 [−]

frequency f [Hz]


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)


- 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


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


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)


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!