Efficient Component Mode Synthesis with a New Interface Reduction Method

Geng Zhang, Matthew P. Castanier, and Christophe PierreDepartment of Mechanical Engineering, The University of Michigan, Ann Arbor, MI 48109-2125

ABSTRACT

When the finite element model of a complex structure is partitioned into substructures in order to enable componentmode synthesis (CMS), there may be a large number of degrees of freedom (DOF) on the interface betweencomponents. In such a case, the constraint-mode-related coordinate transformation at the substructure analysisstage becomes computationally expensive, and the CMS model may be cumbersome due to the interface DOFthat dominate the model size. A more compact model can be obtained using a previously introduced method forreducing the interface DOF. This method involves an eigenanalysis performed on the interface partitions of the CMSmatrices, followed by an interface mode selection. However, because the interface matrix partitions usually havehigh densities, this reduction process may demand a high computational cost. In this work, several alternativeinterface reduction methods from the literature are reviewed and compared, and a new method is proposed in whichinterface modes are obtained based on approximate constraint modes. With only a slight sacrifice of accuracy, theproposed method exhibits greatly improved efficiency for both the constraint-mode-related coordinate transformationrequired to generate the CMS model as well as any subsequent interface eigenanalysis. For an application toa large-scale finite element model of an automotive vehicle, numerical results are provided to demonstrate theperformance of the method.

Nomenclature

K stiffness matrixk stiffness matrix partitionM mass matrixm mass matrix partitionni number of selected normal modes for substructures inint number of selected interface modesnss number of substructuresv velocity vectorβ Boolean matrixε small value for filtration operationΘ set of interface modesθ interface modeΘ set of approximate interface modesλ eigenvalueΦ set of normal modesφ normal modeΨ set of constraint modesψ constraint modeψ filtered constraint mode

Subscripts

a index for substructure eigenvalue and eigenvectorb index for interface eigenvalue and eigenvectori index for substructures index for approximate interface eigenvalue and eigenvector

Superscripts

C system constraint-mode DOF (global interface DOF)c substructure constraint-mode DOF (local interface DOF)CMS component mode synthesis modelL DOF of local interface modelN normal-mode DOFROM reduced-order modelΓ interface DOFΩ interior DOFΘ interface-mode DOFΘ approximate interface-mode DOF

1 Introduction

Component mode synthesis (CMS) methods [1–3] are well established for constructing efficient models to analyzethe dynamics of complex structures that are often described by separate substructure (or component) models.These methods allow the accurate representation of the dynamic characteristics using a relatively small number ofDOF. Among various CMS methods, the fixed-interface Craig-Bampton method [2] is well known for its numericalstability. It is also found to be suitable for finite-element-based power flow analysis [4–6], because it retains thecomplete interface DOF in the CMS model.

However, when a complex model with a fine mesh is partitioned into many substructures, there may be a largenumber of DOF on the interface between components. This could then cause two problems for the fixed-interfaceCraig-Bampton CMS method. First, at the substructure analysis stage, a large number of interface DOF leads to alarge number of constraint modes, and consequently the constraint-mode-related transformation may be very timeconsuming. Second, a Craig-Bampton CMS model may be cumbersome due to the dominant interface DOF, andany further dynamic analysis based on this model may still be very expensive.

The first problem has not been discussed much in the literature. More attention has been placed on the cost ofsolving for constraint modes rather than the cost of the constraint-mode-related transformation, because the formerinvolves the matrix inversion and the latter only involves matrix multiplication. However, when the CMS procedure isapplied to a complex model with a large interface, such as the vehicle model described in this paper, the constraint-mode-related transformation may be much more expensive than solving for the constraint modes. In fact, the totalcost of the Craig-Bampton CMS procedure may be dominated by this transformation.

The second problem, that of a large number of constraint-mode DOF in the CMS model, could be solved by an in-terface reduction technique. More compact reduced-order models (ROMs) can be obtained by reducing the originalinterface DOF with a set of interface modes. The idea of interface reduction was first suggested by Craig and Changin a report to NASA [7]. Three reduction methods were suggested, namely Guyan reduction, Ritz reduction, andmodal reduction (MR). However, they did not publish a paper on the application of this interface reduction technique,

possibly because the smaller maximum model size allowed by computers’ capability at that time did not necessitatethe use of this technique. Unaware of Craig and Chang’s NASA report, the MR method had also been derivedindependently by Brahmi et al. [8], Balmes [9], and Tan and co-workers [6, 10].

The MR method is a straightforward, accurate, and effective method for interface reduction. In the MR method,the interface DOF are reduced by a set of interface modes obtained through a direct eigenanalysis of the interfacepartitions of the CMS mass and stiffness matrices. The reduced order model (ROM) from the MR method ishighly compact and accurate. However, the interface eigenanalysis in the MR method may be expensive, becausethe interface matrices usually have high densities. Therefore, several alternative methods [8, 11–14] have beenproposed for implementing interface reduction without using direct eigenanalysis.

In this paper, a matrix filtration technique is introduced to improve the standard Craig-Bampton CMS procedure bymaking it more efficient for the dynamic analysis of complex models with large interfaces. The filtration is appliedto both the constraint mode matrices and the interface matrices, so that the computational savings are achievedfor both the constraint-mode-related transformation and the interface eigenanalysis. Therefore, the filtration methodenables a more efficient CMS procedure to produce highly compact and accurate ROMs.

This paper is organized as follows. In section 2, the standard Craig-Bampton CMS procedure is summarized. Insection 3, improvements in the efficiency of CMS by the filtration of constraint modes and by interface reduction arediscussed. Then, as an example application, the finite element model of a vehicle structure is described in section4, and numerical results are provided in section 5. Finally, conclusions are drawn in section 6.

2 Background

In this section, the Craig-Bampton CMS method [2] is summarized. For a substructure labeled by index i, theunknown velocities vi can be partitioned as

vi =

vΓi

vΩi

(1)

where vΓi are the velocities of the interface DOF, and vΩ

i are the velocities of the interior DOF. Correspondingly,substructure mass and stiffness matrices can be partitioned as

Mi =[

mΓΓi mΓΩ

i

mΩΓi mΩΩ

i

]Ki =

[kΓΓi kΓΩ

i

kΩΓi kΩΩ

i

](2)

The fixed-interface Craig-Bampton method [2] of component mode synthesis utilizes two sets of substructure modes:normal modes, Φi, and constraint modes, Ψi. The normal modes of substructure i are calculated with all DOF ofthe interface held fixed. Thus, they are the eigenvectors of the eigenvalue problem

kΩΩi φia = λamΩΩ

i φia for a = 1, 2, 3, ...ni (3)

where ni is the number of normal modes selected for substructure i. A constraint mode is defined as the staticdeflection induced in the substructure by applying a unit displacement to one interface DOF while all other interfaceDOF are held fixed. The constraint modes for substructure i, Ψi, can be readily calculated by [2]

Ψi = −kΩΩ−1

i kΩΓi (4)

The constraint modes provide a complete and mathematically convenient set of deformation shapes associated withthe motion of the interface DOF. The original physical DOF, vΓ

i and vΩi , thus can be represented by the constraint

mode DOF, vci , and the normal mode DOF, vNi , by the following formvΓi

vΩi

=[

I 0Ψi Φi

]vcivNi

(5)

With this coordinate change, the substructure mass and stiffness matrices are

MCMSi =

[mci mcN

i

mcNT

i mNi

]KCMSi =

[kci 00 kNi

](6)

where the matrix partitions are

mci = mΓΓ

i + mΓΩi Ψi + ΨT

i mΓΩT

i + ΨTi mΩΩ

i Ψi (7)

mcNi = mΓΩ

i Φi + ΨTi mΩΩ

i Φi (8)

mNi = ΦT

i mΩΩi Φi (9)

kci = kΓΓi + ΨT

i kΩΓi (10)

kNi = ΦTi kΩΩ

i Φi (11)

Matrices from substructures are then assembled into a full system CMS model, with substructures coupled atinterfaces by enforcing displacement compatibility and traction force cancellation. This synthesis yields a modalvelocity vector, vCMS , of the form

vCMS =[

vCT

vNT

1 vNT

2 · · · vNT

nss

]T(12)

where nss is the number of substructures contained in the global structure. A boolean matrix βi is introduced tomap the global (system) interface DOF, vC , back to the local (subsystem i) interface DOF, vci as

vci = βivC (13)

The corresponding synthesized CMS mass and stiffness matrices have the following forms

MCMS =

mC mCN

1 mCN2 · · · mCN

nss

mCNT

1 mN1 0 · · · 0

mCNT

2 0 mN2 0

......

. . ....

mCNT

nss 0 0 · · · mNnss

(14)

KCMS =

kC 0 0 · · · 00 kN1 0 · · · 00 0 kN2 0...

.... . .

...0 0 0 · · · kNnss

(15)

where

mC =nss∑i=1

βiTmc

iβi, kC =nss∑i=1

βiTkciβi (16)

andmCNi = βi

TmcNi (17)

3 Reduction of CMS Computational Costs

3.1 Filtration of Constraint Modes

Among the coordinate transformations in Eqs. (7)-(11), the constraint-mode-related transformation of the interfacemass matrix in Eq. (7) is the most computationally demanding. When the size of substructure interface is large, the

mass transformation, in particular the triple-product that appears as the last term of Eq. (7), may be very expensive.Furthermore, this cost remains the same no matter what frequency range is selected. The computational cost of theconstraint-mode-related transformation in Eq. (7) is due to both the large size and the high density of Ψi. The largesize of Ψi is determined by the finite element model (FEM) and the partitioning scheme, while the high density ofΨi comes from the definition of the constraint modes. When a unit displacement is applied at one interface DOF, itwill induce displacement throughout most of the interior field. Therefore, the constraint mode matrix, Ψi, is usuallya matrix with very high density.

However, as seen in Fig. 1, which shows a typical constraint mode for one component of a two-component platemodel, significant displacements for a constraint mode are usually limited to the area close to the displaced interfaceDOF, since all the other interface DOF are held fixed. If the very small displacements far away from the interfacewere set to be zero, it would dramatically reduce the density of the constraint modes and the computational cost ofthe constraint-mode-related transformation, while the sacrifice of accuracy is expected to be slight.

Fig. 1: Two-component plate model and a typical constraint mode for one component

The above idea can be implemented numerically as a filtration of the original constraint modes. For each constraintmode, ψq, the filtration process can be expressed as

|ψpq| < ε ∗maxp|ψpq| ⇒ ψpq = 0 (18)

where ψpq is the pth element of the qth constraint mode. An appropriate value of ε can be chosen by monitoring themodal assurance criterion (MAC) values between the original constraint modes and the filtered constraint modes,which is defined as

MAC =(ψTq ψq)

2

(ψTq ψq)(ψT

q ψq)(19)

where ψq is an original constraint mode and ψq is the corresponding filtered constraint mode. It is noted thatiteration may be necessary to find an appropriate value of ε. However, the iteration procedure is performed beforethe mass transformation, and it includes only the filtration of constraint modes and the calculation of MAC values.Therefore, the computational cost is far less than the cost of the mass transformation using full constraint modes.

Note that the transformation of the interface stiffness matrix in Eq. (10) is not very expensive, and also that theaccuracy of the interface stiffness matrix is critical to the overall modeling accuracy in the low-frequency range.Therefore, it is recommended to use the exact constraint modes for the transformation of the interface stiffnessmatrix and the filtered constraint modes for the transformation of the interface mass matrix.

The filtration criterion can be adjusted slightly to handle any special elements in the model. For example, in thevehicle model described in the next section, the doors are modeled with concentrated masses and rigid bars. Theconstraint-mode DOF corresponding to these large concentrated masses should be kept without filtration. In thiscase, this adjustment greatly improves the accuracy with trivial increase of computational cost, because there areonly dozens of concentrated mass elements but the vehicle model has more than 500,000 DOF.

3.2 Review of Interface Reduction Methods

In a Craig-Bampton CMS model, the substructure normal mode matrices mNi and kNi are diagonal and their sizes

depend on the number of modes selected for the frequency range of interest. However, the number of constraint-mode DOF, or the size of matrices mC and kC , is equal to the number of interface DOF in the original finite elementmodel, which is determined by the finite element mesh. If the mesh is fine in the interface regions, or if there aremany substructures, then the constraint-mode partitions of the CMS matrices may be relatively large. In such acase, an interface reduction technique can be adopted to obtain more compact ROMs.

In the MR method, the interface modes Θ = [θ1 θ2 · · · θnint ] are obtained through a direct eigenanalysis:

kCθb = λbmCθb for b = 1, 2, 3, ..., nint (20)

where nint is the number of selected interface modes, which is relatively small compared to the total number ofinterface DOF. The frequencies associated with λb are defined as interface natural frequencies. The velocities ofthe interface DOF, vC , can then be represented as

vC = ΘTvΘ (21)

where vΘ are velocities of interface-mode DOF. A more compact ROM can be obtained by applying this coordinatetransformation to the Craig-Bampton CMS model. Now, the velocity vector vROM is partitioned as

vROM =[

vΘT vNT

1 vNT

2 · · · vNT

nss

]T(22)

The mass matrix, MROM , and the stiffness matrix, KROM can be explicitly written as

MROM =

mΘ mΘN

1 mΘN2 · · · mΘN

nss

mΘNT

1 mN1 0 · · · 0

mΘNT

2 0 mN2 0

......

. . ....

mΘNT

nss 0 0 · · · mNnss

(23)

KROM =

kΘ 0 0 · · · 00 kN1 0 · · · 00 0 kN2 0...

.... . .

...0 0 0 · · · kNnss

(24)

wheremΘ = ΘTmCΘ, kΘ = ΘTkCΘ (25)

andmΘNi = ΘT mCN

i (26)

Although the system ROM from the MR method is highly compact and accurate [15–18], the interface eigenanalysisin Eq. (20) is expensive because kC and mC are usually dense matrices. Therefore, several alternative methodshave been proposed for performing interface reduction without using direct eigenanalysis.

Brahmi et al. proposed a “dynamic condensation” method [8] and a “before assembly” reduction method [11]. Inthe first method, the interface DOF were partitioned into master DOF and slave DOF. With the slave DOF beingdynamically condensed, the final ROM contained only the master DOF. In the second method, a set of substructureinterface modes were obtained by performing an eigenanalysis on the individual substructure interface matrices(mc

i and kci ) before assembling the system matrices. Substructure interface modes were mapped into the globalinterface coordinates, and then a singular value decomposition (SVD) was performed to get the global interfacemodes.

Rixen [12] proposed a “force modes” method. The name came from the explicit use of interface traction forces asLagrange multipliers in the equations of motion and the projection of traction forces onto interface mode coordinates.The interface modes in Rixen’s method were obtained through an eigenanalysis of approximate interface flexibilityand unit interface mobility matrices.

Aoyama [13] and Yagawa proposed a “local interface modes” method. For each local interface, a local modelconsisting of all substructures attached to this interface was constructed. An eigenanalysis was then performed toget the local interface modes. These local interface modes were mapped into the global interface coordinates toyield the global interface modes.

Balmes [14] proposed a “local model” method. A local model was constructed by extracting several layers of ele-ments surrounding the interface, resulting in local stiffness and mass matrices, KL and ML. A direct eigenanalysiswas performed on KL and ML to obtain the eigenvalues and eigenvectors of the local model. Approximate interfacemodes Θ can be obtained from these eigenvectors by extracting rows corresponding to the interface DOF.

The cost-reduction strategies adopted by these alternative methods can be broken into three general approaches:

1. Avoid the cost of an interface eigensolution by performing alternative operations

2. Break the global interface eigenanalysis into a set of local analyses

3. Use approximate interface matrices to reduce the densities

All of these alternative methods improve efficiency with some sacrifice of accuracy. When tested on the vehiclemodel described in the next section, the MR method was the most accurate method, while Balmes’ method was themost efficient method due to the sparsity of KL and ML.

3.3 Filtration of Interface Matrices

Since the high densities of interface matrices is the primary factor for the computational cost of interface eigenanal-ysis, a new method is proposed for directly reducing matrix density. The high densities of interface matrices are dueto the constraint-mode-related transformation, which makes non-coupled interface DOF in the original FEM coupledin the Craig-Bampton CMS model. From Eq. (10), two interface DOF are stiffness-coupled in the Craig-BamptonCMS model if one constraint mode has any non-zero displacement at any interior DOF that is stiffness-coupled tothe other interface DOF. From Eq. (7), two interface DOF are mass-coupled in the Craig-Bampton CMS model ifone constraint mode has any non-zero displacement at any interior DOF that is mass-coupled to the other interfaceDOF, or if two constraint modes share any non-zero displacement at any interior DOF. Therefore, two interface DOFare usually mass- and stiffness-coupled as long as they belong to the same substructure. This is the fundamentalreason for the high densities of the interface mass and stiffness matrices.

Since most of the coupling terms of interface matrices come from the constraint-mode-related transformation, someof the coupling terms can be neglected to reduce the densities of the interface matrices. Again, this is basedon the observation that significant displacements of constraint modes are usually limited to the area close to theinterface. If two interface DOF are physically close to each other in the original FEM, the constraint-mode-relatedtransformation will create a large coupling term, because the two constraint modes share large displacements atthe interior DOF. On the contrary, the coupling term for two interface DOF that are far away from each other will besmall. The new interface matrices with small coupling terms being neglected should still allow reasonable accuracy,while the densities of the interface matrices and the associated computational cost of the interface eigenanalysiswill be greatly reduced.

The above idea can be implemented numerically as a simple filtration of the interface matrices. The correspondinginterface reduction method is called the filtered MR method to differentiate it from the original MR method. Using

the interface mass matrix as an example, the filtration criterion can be expressed as

|mpq|√mpp ∗mqq

< ε⇒ mpq = 0 (27)

The filtration of the interface stiffness matrix can be carried out in the same way. Different values of ε can be chosenfor the filtration of mass and stiffness matrices.

Selection guidelines for appropriate values of ε are still under investigation. In this study, they are selected basedon the authors’ knowledge of the finite element model and the interface. For the coupling terms in the massand stiffness matrices, only a portion of them are considered as strong coupling terms, which are critical for theaccuracy and should be kept in the filtered matrices. The expected densities of matrices can be estimated once thepercentage of strong coupling terms is decided. The matrices are then filtered iteratively with different ε values, untilthe matrices approach the expected densities. The filtration of matrices is handled efficiently by NASTRAN, and thecomputational cost is trivial compared to the cost of interface eigenanalysis.

It is noted that a Sturm sequence check may be a more objective approach to the selection of ε. The number ofmodes within the selected frequency range is first counted using the original matrices, then the number is countedagain using the filtered matrices. If the numbers are close to each other, it means the filtration values are ade-quate, and the filtered matrices have not lost too much information as far as the eigenvalues and eigenvectors areconcerned. However, the cost of Sturm sequence check is not trivial, especially for an iterative process.

Following the filtration procedure, a set of approximate interface modes Θ can be obtained through an eigenanalysison the filtered interface mass and stiffness matrices. These modes are then substituted for the exact interface modesΘ in Eqs. (25) and (26), to obtain the condensed interface matrices and mass coupling as follows

mΘ = ΘTmCΘ, kΘ = ΘTkCΘ (28)

mΘNi = ΘT mCN

i (29)

Matrices from above equations can be assembled with normal mode matrices to yield the system ROM. Note thatunlike mΘ and kΘ, mΘ and kΘ are not diagonal. As an accuracy check, a special modal analysis can be carriedout as

kΘφs = λsmΘφs for s = 1, 2, 3, ..., nint (30)

Frequencies associated with λs are considered as approximate interface natural frequencies. The accuracy of thefiltered MR method can be evaluated by comparing the approximate interface eigenvalues from Eq. (30) with theoriginal interface eigenvalues from Eq. (20). Note that Eqs. (28)-(30) are also valid for the approximate interfacemodes obtained from Balmes’ method or other interface reduction methods.

4 Finite Element Model of the Vehicle Structure

The finite element model of a vehicle (body) structure is shown in Fig. 2, which is comprised of almost 600,000 DOF.In order to apply CMS, the vehicle structure has been partitioned into 12 substructures as shown in Fig. 3. Somenomenclature and size information for these substructures is summarized in Table 1.

With this partitioning scheme, there are a total of 29,430 interface DOF, which is large enough to necessitate theuse of the interface reduction technique. The size of each substructure interface is also very large, varying roughlyfrom 3000 to 7000 DOF, which causes the constraint-mode-related transformation performed at the substructureanalysis stage to be very time consuming.

In order to apply Balmes’ method [14], a local model is constructed as shown in Fig 4. For this local model, fourlayers of elements surrounding the interface are collected. The size of this local model is about 220,000 DOF, whichis about 37 percent of the full model size, and 7 times the interface size.

Fig. 2: Finite element model of the vehicle struc-ture

Fig. 3: Partitioning of the vehicle finite elementmodel

Table 1: Substructures of the vehicle modelSubstructure Substructure Description Interface Interior Substructure normal

number name nodes nodes modes under 400 Hz1 front1 front - lower 1,086 8,146 502 front2 front - upper 491 7,162 803 left1 left side - front 1,161 12,141 504 left2 left side - rear 1,053 7,196 1305 right1 right side - front 1,198 12,220 506 right2 right side - rear 824 7,016 1507 back1 back - lower 821 10,460 1008 back2 back - upper 560 4,739 259 roof roof 670 3,421 150

10 floor1 floor pan - front 752 8,909 7511 floor2 floor pan - middle 584 8,999 5512 floor3 floor pan - rear 624 6,526 25

5 Numerical Results

Due to the large size of the vehicle model, all the calculations were carried out using NASTRAN software (version70.5). In particular, because the filtration operation is not provided by any standard NASTRAN solver, it has beenimplemented using NASTRAN Direct Matrix Abstraction Program (DMAP) routines. Therefore, all the numericalresults for accuracy and computational cost shown in this section are based on NASTRAN DMAP calculations.

5.1 Filtration Method Applied to Constraint Modes

Here the filtration method is applied to constraint modes in order to improve the efficiency of generating a CMSmodel of the vehicle structure shown in Fig. 2. The computational cost of the constraint-mode-related transformationis summarized in Table 2. The computational times given are the totals for all 12 substructures. Considering thatonly 3.5 hours were required for the normal modes solution and 0.45 hours were required for the constraint modessolution, the computational cost for generating the Craig-Bampton CMS model is in fact dominated by the 35.1hours spent on the constraint-mode-related transformation. Of the 35.1 hours, 34.9 hours were spent on calculatingthe triple-product for the mass transformation.

Fig. 4: Local model of the interface region

The filtration method with ε = 0.01 is applied to the constraint modes to reduce the cost. With the density of theconstraint modes being reduced by 94.8 percent, the computational cost (user CPU time) of the constraint-mode-related transformation is reduced by 95.2 percent.

Table 2: Computational cost of constraint-mode-related transformationDensity of Ψi CPU time for triple-product Total CPU time

Original constraint modes 84.7% 34.9 hr 35.1 hrFiltered constraint modes 4.4% 1.6 hr 1.7 hrPercentage savings 94.8% 95.4% 95.2%

Figure 5 shows the sorted MAC values between the original constraint modes and the filtered constraint modes(Eq. (19)) of the “right1” substructure, which is the largest substructure. For the MAC values of the 7188 constraintmodes, all of them are above 0.97 and 7156 of them are above 0.99. Therefore, the computational savings of theconstraint-mode-related transformation is achieved with very slight sacrifice of accuracy.

0 1000 2000 3000 4000 5000 6000 7000 80000.97

0.975

0.98

0.985

0.99

0.995

1

1.005Sorted Model Assurance Criterion (MAC) Values

number of modes

MA

C

Fig. 5: Sorted MAC values between the original and filtered constraint modes of the “right1” substructure

5.2 Filtration Method Applied to Interface Reduction

The filtration method is applied to the vehicle structure together with the most accurate interface reduction method,the MR method [7, 10], and the most efficient interface reduction method, the Balmes’ method [14]. The resultsunder examination here are the system natural frequencies calculated up to 200 Hz. For all substructures, thenormal modes up to 400 Hz have been calculated, with the numbers of modes listed in the last column of Table 1.

After selecting the substructure normal modes up to 400 Hz, the Craig-Bampton CMS model contains 29,430interface DOF and 940 substructure normal mode DOF. The model is cumbersome due to the dominant interfaceDOF, which constitute 96.9 percent of the total DOF. The interface reduction methods are then applied to obtain amore compact ROM. According to a previous convergence study of the MR method [18], the interface modes up toabout 2.5f0 should be retained when the frequency range of interest is 0–f0 Hz. Therefore, 640 interface modes(up to 500 Hz) are obtained with the MR method, yielding an ROM with 1580 DOF. The reduction ratios achievedby the MR method are 46:1 for the interface and 19:1 for the system ROM.

Although the model size reduction is impressive, the computational cost of the interface eigenanalysis is significant,as shown in Table 3. The filtration method is then applied to the interface matrices (ε = 10−2 for mass matrixfiltration, and ε = 10−6 for stiffness matrix filtration). With the densities of the stiffness and mass matrices beingreduced by 77.1 percent and 86.1 percent respectively, the cost (user CPU time) of solving for the same numberof interface modes has been reduced by 44.4 percent. The computational savings of the filtration method comesfrom the reduction of the matrix densities. Therefore, besides saving CPU time, the filtration method also reducescomputer memory usage and I/O time.

Table 3: Computational cost of interface eigenanalysis (Lanczos method)Density Density Average Time of Average time of Average time Total

of of forward/backward matrix-vector of CPUkC mC substitution (FBS) multiply shift and factor time

Original matrices 25.4% 25.4% 71.1 s 54.5 s 8287.1 s 12.6 hrFiltered matrices 5.8% 3.5% 69.5 s 13.1 s 5293.6 s 7.0 hrPercentage savings 77.1% 86.1% 2.9% 75.9% 36.1% 44.4%

In Figs. 6 and 7, the accuracy of the interface natural frequencies (Eq. (30)) are compared for the filtered MR methodand Balmes’ method, using the results from the MR method (Eq. (20)) as reference. Balmes’ method shows pooraccuracy in the low-frequency range, which comes mostly from the incomplete information of the interface stiffnessmatrix. Accuracy of the filtered MR method is excellent, with no more than 1 percent error for the 0–500 Hz range.

In Fig. 8, the accuracy of the system natural frequencies are compared for the original and filtered MR method,using the CMS results as reference. The largest error for the 0–200 Hz range is 1 percent for the MR method and1.2 percent for the filtered MR method. The accuracy of the filtered MR method is comparable with that of theoriginal MR method, despite the large savings in computational cost provided by the filtration technique.

6 Conclusions

An efficient CMS procedure based on a matrix filtration technique was introduced in this paper for the dynamicanalysis of complex models with large interfaces. The density of constraint modes and the cost of constraint-mode-related transformation are greatly reduced by the filtration of the constraint mode matrix. In addition, the densityof interface matrices and the cost of interface eigenanalysis are reduced by the filtration of interface matrices.Therefore, the proposed method can be used to greatly reduce the CPU time required to generate a CMS modelas well as highly compact and accurate ROMs. The excellent performance of the proposed method has beendemonstrated by its application to a realistic vehicle model.

0 50 100 150 200 250 300 350 400 450 5002

3

4

5

6

7

8

9

10

Percentage Error of Interface Natural Frequencies(Balmes’ Method versus the Original MR Method)

Natural Frequencies (Hz)

Per

cent

age

Err

or (

%)

Fig. 6: Percentage error of interface natural fre-quencies from Balm es’ method, relative to re-sults from the original MR method

0 50 100 150 200 250 300 350 400 450 5000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Percentage Error of Interface Natural Frequencies(The Filtered MR Method versus the Original MR Method)

Natural Frequencies (Hz)

Per

cent

age

Err

or (

%)

Fig. 7: Percentage error of interface natural fre-quencies from the filtered MR method, relativeto results from the original MR method

0 20 40 60 80 100 120 140 160 180 2000

0.2

0.4

0.6

0.8

1

1.2

1.4

Natural Frequencies (Hz)

Per

cent

age

Err

or (

%)

Percentage Error of System Natural Frequencies(Using Craig−Bampton Method as Reference)

The Original MR MethodThe Filtered MR Method

Fig. 8: Percentage error of system natural frequencies from the original and filtered MR method, relative toresults from Craig-Bampton CMS method

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3 Craig, R. R., Structural Dynamics: An Introduction to Computer Methods, chap. 19, John Wiley & Sons, NewYork, NY, 1981.

4 Mace, B. R. and Shorter, P. J., “Energy Flow Models from Finite Element Analysis,” Journal of Sound and Vibra-tion, Vol. 233, No. 3, 2000, pp. 369–389.

5 Tan, Y.-C., Castanier, M. P., and Pierre, C., “Approximation of Power Flow Between Two Coupled Beams UsingStatistical Energy Methods,” Journal of Vibration and Acoustics, Vol. 123, No. 4, 2001, pp. 510–523.

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