1-s2.0-S0377042703010045-main

Journal of Computational and Applied Mathematics 168 (2004) 383–391www.elsevier.com/locate/cam

A dual Craig–Bampton method for dynamic substructuring

Daniel J. Rixen∗;1

T.U. Delft, Faculty of Design, Engineering and Production, Engineering Mechanics—Dynamics,Mekelweg 2, Delft 2628 CD, The Netherlands

Received 19 July 2002

Abstract

A novel component mode synthesis method for dynamic analysis of structures is presented. It is basedon free interface vibration modes and residual ,exibility components. Although the ingredients are the sameas in previously published procedures (e.g. MacNeal or Rubin), our method is fundamentally di/erent inthat it assembles the substructures using interface forces (dual assembly) and enforces only weak interfacecompatibility. The new formulation is based on a fully consistent reduction approach and the reduced matricesso-obtained are exactly dual to the Craig–Bampton reduced matrices. The new free interface substructuringmethod proposed here is thus more natural then classical free interface synthesis procedures and leads tosimpler reduced matrices. We illustrate the e4ciency of the dual Craig–Bampton approach for reducing thedynamical representation on a three-dimensional frame.c© 2003 Elsevier B.V. All rights reserved.

Keywords: Component mode synthesis; Dynamic substructuring; FETI; Model reduction; Free interface mode

1. Introduction

The power and storage capability of modern computing hardware as well as remarkable advancesin numerical methods and in compilers make it possible nowadays to solve very large linear systems(typically of the order of millions of degrees of freedom). Nevertheless, since dynamic analysis (e.g.inverse iteration based eigensolvers or time-integration schemes) requires solving many linear systemsand because the complexity and re:nement of :nite element models is increasing at least as fast asthe computing capabilities, dynamic substructuring remains an essential tool for engineers. Buildingreduced models of subparts (known as super- or macro-elements in :nite elements) facilitates sharing

∗ Tel.: 3115-278-1523; fax: 3115-278-2150.E-mail address: [email protected] (D.J. Rixen).1 Supported by the Koiter Institute, Delft University of Technology.

0377-0427/$ - see front matter c© 2003 Elsevier B.V. All rights reserved.doi:10.1016/j.cam.2003.12.014

mailto:[email protected]

384 D.J. Rixen / Journal of Computational and Applied Mathematics 168 (2004) 383–391

models between design groups. Reducing the representation of the dynamical behavior of subpartsis also very relevant for building reduced order systems and designing high performance controllers.For an overview of substructuring methods see for instance [1].Basically, there are two classes of mode components synthesis methods: those where the substruc-

ture dynamics is represented by :xed interface vibration modes and those where free interface modesare used. The method commonly used is the Craig–Bampton procedure [2]. Nearly all substructuringmethods reduce the subparts in sub-domains with displacement connectors on the interface so thatthey can be used in :nite element codes as macro-elements.In this paper, we will present a method that is based on free interface modes and where the

substructures are assembled through interface forces. We will see that this combination is fullynatural and leads to an attractive dual Craig–Bampton formulation.In Section 2, we discuss the di/erence between primal and dual assembly. In Section 3, we then

develop the dual Craig–Bampton method. A numerical example is given in Section 4.

2. Primal and dual assembly of substructures

Let us assume that a :nite element modal de:ned on a domain � is subdivided into a numberN (s) of substructures called �(s) such that every node belongs to one and only one substructureexcept for those on the interface boundaries. The linear dynamic behavior of each substructure �(s)

is governed by the local equilibrium equations

M (s) Gu(s) + K (s)u(s) = f (s) + g(s); s= 1; : : : ; Ns; (1)

where M (s) and K (s) are the substructure mass and sti/ness matrices, u(s) are the local degreesof freedom, f (s) the external loads applied to the substructure and g(s) the internal forces on theinterfaces between substructures that ensure compatibility.

2.1. Primal assembly of substructures

Every substructure can be looked at as a macro-element: the local degrees of freedom u(s) arerelated to a global set of assembled degrees of freedom by

u(s) =

[u(s)b

u(s)i

]=

[L(s)b 0

0 I

] [ub

u(s)i

]; (2)

where the subscripts i and b refer, respectively, to the internal and boundary degrees of freedom andwhere L(s)b is a Boolean localization matrix relating assembled degrees of freedom ub on the interfaceto u(s)b . Using the compatibility condition (2), the local equilibrium Eq. (1) can be assembled as

Ma Gua + Kaua = fa(t); (3)

D.J. Rixen / Journal of Computational and Applied Mathematics 168 (2004) 383–391 385

where

Ma =

Ns∑s=1

L(s)T

b M (s)bb L

(s)b L(1)

T

b M (1)bi · · · L(Ns)

T

b M (Ns)bi

M (1)ib L

(1)b M (1)

ii 0

.... . .

M (Ns)ib L(Ns)b 0 M (Ns)

ii

; ua =

ub

u(1)i...

u(Ns)i

;

Ka =

Ns∑s=1

L(s)T

b K (s)bb L

(s)b L(1)

T

b K (1)bi · · · L(Ns)

T

b K (Ns)bi

K (1)ib L

(1)b K (1)

ii 0

.... . .

K (Ns)ib L(Ns)b 0 K (Ns)

ii

(4)

and where Gua is the second-order time derivative of ua. The interface forces g(s) cancel out whenassembled on the interface.

2.2. Dual assembly of substructures

Another way to enforce the compatibility between substructures consists in explicitly expressingthe compatibility constraints, namely the equality between corresponding degrees of freedom on theinterface

M (s) Gu(s) + K (s)u(s) +

[b(s)

T

0

]= f (s);

Ns∑s=1

b(s)u(s)b = 0;

(5)

where b(s) are signed Boolean matrices. Comparing with (1), b(s)T represent the interconnecting

forces between substructures. We introduce the block-diagonal notations

M =

M (1) 0

. . .

0 M (Ns)

; K =

K (1) 0

. . .

0 K (Ns)

;


u =

u(1)

...

u(Ns)

; f =

f (1)

...

f (Ns)

;

B = [B(1) · · · B(Ns) ]; B(s)T=

[b(s)

T

0

]: (6)

The set of Eqs. (5) can then be written in block form[M 0

0 0

] [Gu

]+

[K BT

B 0

] [u

]=

[f

0

]: (7)

The dual assembled system (7) is equivalent to the assembled system (3) since they express thesame local equilibrium and enforce the same interface compatibility.

3. The dual Craig–Bampton method

3.1. Craig–Bampton

According to the primal assembled system (3), every substructure can be looked at as beingexcited through its interface degrees of freedom, namely

M (s)ii Gu

(s)i + K (s)

ii u(s)i = f (s)i − K (s)

ib u(s)b −M (s)

ib Gu(s)b ; (8)

which indicates that one can de:ne an approximation for u(s)i in every subdomain as a superpositionof a static response and of eigenmodes associated to M (s)

ii and K (s)ii . Hence, one can de:ne a

transformation matrix for reducing the primal assembled system as

ua =

ub

u(1)i...

u(Ns)i

� TCB

ub

�(1)

...

�(Ns)

=

I 0 · · · 0

(1)L(1)b �(1) 0

.... . .

(Ns)L(Ns)b 0 �(Ns)

ub

�(1)

...

�(Ns)

; (9)

where

(s) =−K (s)ib K

(s)−1

ii (10)

are the static response modes and where �(s) are n(s)i × n(s)� matrices which columns contain the

:rst n(s)� free vibration modes of the substructure clamped on its interface. Applying the reductionprocedure (9), the standard Craig–Bampton reduced matrices LKCB=TTCBKaTCB and LMCB=TTCBMaTCBare found [2].


3.2. Free interface modes as reduction basis

Let us now consider the dual assembled problem (5) or (7). In that case, every substructure canbe seen as being excited through interface connection forces. The local dynamical behavior cantherefore be expressed in terms of eigenmodes of the entire local matrices, hence in terms of freevibration modes of the substructures with free interfaces, and in terms of local static solutions

u(s) = u(s)stat +n(s)−m(s)∑r=1

�(s)r �(s)r ; (11)

where

u(s)stat =−K (s)+B(s)T +

m(s)∑i=1

R(s)i �(s)i : (12)

K (s)+ is the inverse of K (s) when there are enough boundary conditions to prevent the substructurefrom ,oating when its interface with neighboring domains is free. If a substructure is ,oating, K (s)+

is a generalized inverse of K (s) and R(s) is the matrix having as column the m(s) corresponding rigidbody modes. �(s)i are amplitudes of the local rigid body modes and �(s)r are the amplitude of thelocal eigenmodes.An approximation is then constructed by considering only the :rst n(s)� free modes �(s) in the

expansion. Calling �(s) the matrix containing these n(s)� modes,

u(s) � −K (s)+B(s)T + R(s)�(s) +�(s)�(s): (13)

Let us observe that, in this approximation, part of the solution in the subspace of �(s) is also includedin K (s)+B(s)

T since the generalized inverse has as spectral expansion [4,6]

K (s)+ =n(s)−m(s)∑r=1

�(s)r �(s)T

r

!(s)2

r

: (14)

Hence, the approximation (13) can be equivalently written as

u(s) =−G (s)resB

(s)T + R(s)�(s) +�(s)�(s); (15)

where

G (s)res =

n(s)−m(s)∑r=n(s)� +1

�(s)r �(s)T

r

!(s)2

r

= K (s)+ −n(s)�∑r=1

�(s)r �(s)T

r

!(s)2

r

(16)

which is computed using the second equality in (16). K (s)+ is computed from any generalizedinverse K (s)† by projecting out the rigid body modes [4,6]. G (s)

res is commonly called the residuallocal ,exibility matrix and has the properties

G (s)res = G

(s)Tres ; �(s)TK (s)G (s)

res = 0;

G (s)Tres K

(s)G (s)res = G

(s)res ; �(s)TM (s)G (s)

res = 0;

R(s)TM (s)G (s)

res = 0:

(17)


In summary, the local degrees of freedom and the Lagrange multipliers (interconnecting forces) canbe approximated by

[u

]=Tdual

�(1)

�(1)

...

�(Ns)

�(Ns)

=

R(1) �(1) 0 −G (1)res B

(1)T

. . . . . ....

0 R(Ns) �(Ns) −G (Ns)res B

(Ns)T

0 · · · 0 I

�(1)

�(1)

...

�(Ns)

�(Ns)

: (18)

3.3. Reduced matrices and dual assembly

Approximation (18) corresponds to the representation used in substructuring methods such asthe MacNeal method and the Rubin method [3,7,9]. In those procedures, the interface forces areeliminated locally in order to obtain reduced matrices in terms of the substructure interface displace-ments only. Hence these procedure go back to a primal assembly expression which simpli:es theimplementation of the procedure in standard :nite element codes. However, the reduced matrices soobtained have a cumbersome expression and are not as sparse as the Craig–Bampton matrices.In this work, we will go the “fully dual assembly” way: since the approximation (18) has been

shown in the previous section to be related to the dual assembly process, we will reduce the dualassembled form. Using approximation (18) in the dual formulation (7) and assuming that the rigidand elastic modes are orthonormalized with respect to M (s), one obtains the hybrid system

M̃

G�(s)

G�(s)

...G

+ K̃

�(s)

�(s)

...

= T

Tdual f (19)

with the reduced hybrid matrices

M̃ = TTdual

[M 0

0 0

]Tdual =

[I 0

0 Mres

](20)


K̃ =

[0 0

0 �(1)2

]0

[R(1)

TB(1)

T

�(1)TB(1)T

]

. . ....

0

[0 0

0 �(Ns)2

] [R(Ns)

TB(Ns)

T

�(Ns)TB(Ns)T

]

[B(1)R(1) B(1)�(1) ] · · · [B(Ns)R(Ns) B(Ns)�(Ns) ] −Fres

; (21)

where

Fres =Ns∑s=1

B(s)G (s)resB

(s)T Mres =Ns∑s=1

B(s)G (s)resM

(s)G (s)resB

(s)T : (22)

Remark. The following remarks on the reduced matrices (20) and(21) are noteworthy:

• The reduced matrices (20) and (21) of the Dual Craig–Bampton method are similar but dualto the classical Craig–Bampton matrices. The cost for building those reduced matrices is alsocomparable.

• Compared to the Mac Neal and the Rubin methods, the procedure described here enforces only aweak compatibility between the substructures. Indeed, the last equation in (19) is obtained from

[ · · · −B(s)G (s)res · · · I ]

...

M (s) Gu(s) + K (s)u(s) = f (s)

...Ns∑s=1

B(s)u(s) = 0

: (23)

This observation can be interpreted as follows: call Pf (s) the residual forces in the substructuresdue to the reduction. Call Pu(s) =G (s)

resPf (s) the displacements these residual would create locallyaccording to the residual ,exibility. The weak compatibility condition then expresses that aninterface displacement “jump” equal to the incompatibility of the Pu(s) is permitted.

• It is easy to show that when the eigensolutions associated are computed using inverse iterationschemes, the associated static problem leads to an interface problem equivalent to the dual interfaceproblem found in the Finite Element Tearing and Interconnecting (FETI) method [5,8]. Hence thesolution techniques developed for the FETI method can be applied in a straightforward manner tosolve the hybrid problems associated to the reduced system (19).

4. Application examples

Let us consider the example problem (Fig. 1) of a frame made of steel beams. It is divided into5 substructures and clamped at one end. Each cell in the frame has a height of 0:35 m and a width


red fullDual C.-B.MacNealCraig-Bampton

4 modes/substr.

eigenfrequency number

SUBSTRUCTURE 1

SUBSTRUCTURE 4

SUBSTRUCTURE 3

SUBSTRUCTURE 5

SUBSTRUCTURE 2

1 8765432

100

10-5

10-4

10-3

10-2

10-1

10-6

2ω 2ω_

full2ω

Fig. 1. Craig–Bampton and dual Craig–Bampton reduction of a beam frame.

and depth of 0:5 m. All outer beams have a hollow circular cross-section, the outside and insidediameters being 0.02 and 0:018 m. The diagonal members inside the cells have plain circular sectionof diameter 0:008 m.In Fig. 1 we plot the relative error on the :rst eigenfrequencies computed by the Craig–Bampton,

the MacNeal and the Dual Craig–Bampton methods, compared to the frequencies obtained for thenon-reduced system. We used four non-rigid modes for the substructures of the frame. The resultsshow that whereas the Craig–Bampton and the MacNeal methods yield similar accuracy, the DualCraig–Bampton reduction technique leads to an accuracy nearly two orders of magnitude better inthe low frequency range. The MacNeal and the Dual Craig–Bampton approaches have the samereduction basis. However, strong interface compatibility is enforced in the MacNeal approach whilethe global eigenmodes computed with the Dual Craig–Bampton satisfy only a weak form of theinterface compatibility. One can thus speculate that the Dual Craig–Bampton method is remarkablye4cient because it enforces interface compatibility consistent with the reduction basis.

5. Conclusions

We propose a novel substructuring technique based on free interface modes and dual assembly.Unlike other well-known substructuring techniques using free interface modes, only a weak compat-ibility is enforced between substructures. This method exhibits strong similarities with the classicalCraig–Bampton method and can be considered as its dual. The reduced matrices associated to theDual Craig–Bampton method presented here are very similar to the Craig–Bampton matrices and lesscumbersome then reduced matrices obtained by other mode synthesis procedures based on free inter-face modes. Numerical results indicate that the Dual Craig–Bampton exhibits remarkable e4ciency.

References

[1] R.R. Craig, Coupling of substructures for dynamic analyses: an overview, in: Structures, Structural Dynamics andMaterial Conference, 41st AIAA/ASME/ASCE/AHS/ASC, Atlanta, 2000, AIAA-2000-1573.

[2] R. Craig, M. Bampton, Coupling of substructures for dynamic analysis, Amer. Inst. Aero. Astro. J. 6 (7) (1968)1313–1319.


[3] R.R. Craig, C.J. Chang, On the use of attachment modes in substructure coupling for dynamics analysis, in: Proceedingsof the 18th Structures, Structural Dynamics and Material Conference, AIAA/ASME, San Diego, AIAA 77-405, 1977,pp. 89–99.

[4] C. Farhat, D. Rixen, Encyclopedia of Vibration, Academic Press, New York, 2002, pp. 710–720, ISBN 0-12-227085-1(Chapter Linear Algebra).

[5] C. Farhat, F.X. Roux, Implicit parallel processing in structural mechanics, Comput. Mech. Adv. 2 (1) (1994) 1–124.[6] M. GVeradin, D. Rixen, Mechanical vibrations. Theory and Application to Structural Dynamics, 2nd Edition, Wiley,

Chichester, 1997.[7] R. MacNeal, A hybrid method of component mode synthesis, Comput. Structures 1 (4) (1971) 581–601.[8] D. Rixen, Encyclopedia of Vibration, Academic Press, New York, 2002, pp. 990–1001, ISBN 0-12-227085-1 (Chapter

Parallel Computation).[9] S. Rubin, Improved component-mode representation for structural dynamic analysis, Amer. Inst. Aero. Astro. J. 13

(8) (1975) 995–1006.

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