Next-generation parametric reduced-order models

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Mechanical Systems and Signal Processing

Mechanical Systems and Signal Processing 37 (2013) 403–421

0888-32

http://d

n Corr

E-m

matt.ca

journal homepage: www.elsevier.com/locate/ymssp

Next-generation parametric reduced-order models

Sung-Kwon Hong a, Bogdan I. Epureanu a,n, Matthew P. Castanier b

a Department of Mechanical Engineering, University of Michigan, 2350 Hayward Street, Ann Arbor, MI 48109-2125, USAb US Army Tank Automotive Research, Development, and Engineering Center, Warren, MI 48397-5000, USA

a r t i c l e i n f o

Article history:

Received 8 December 2011

Received in revised form

4 December 2012

Accepted 26 December 2012Available online 31 January 2013

Keywords:

Parametric reduced order models

Thickness variation

Transformation matrix

Parameterization techniques

Interface reduction

70/$ - see front matter & 2013 Elsevier Ltd.

x.doi.org/10.1016/j.ymssp.2012.12.012

esponding author. Tel.: þ1 734 647 6391;

ail addresses: [email protected], hsung

[email protected] (M.P. Castanier).

a b s t r a c t

Novel parametric reduced-order models are proposed for fast reanalysis to predict the

dynamic response of complex structures, which suffered thickness variations caused by

design changes or damage in one or more substructures. Parametric reduced-order

models developed previously have two important challenges to overcome to improve

accuracy and performance: (a) the transformation matrix is not mathematically stable

and (b) the Taylor series parameterization techniques do not capture thickness variations

of the structure modeled with solid-type elements due to the highly nonlinear

dependence on thickness changes. Thus, herein, a new transformation matrix and novel

parameterization techniques are proposed. Usual reduced-order models have an addi-

tional challenge, namely the difficulty in reducing the interface degrees of freedom. Thus

a way of reducing the interface degrees of freedom is also proposed. The predicted

vibration responses of complex structures are shown to agree very well with results

obtained using a much more computationally expensive commercial tool.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Structural dynamic analyses often require high computational costs because of repetitive calculations needed in theprocess of optimization, stochastic, and statistical analysis. Even though the computing power increases, the repetitivecalculations are very time consuming when conventional complex and large models are used. In the field of structuraldynamic analysis, component mode synthesis (CMS) [1–7] is well established as an alternative to conventional finiteelement models (FEMs) with large numbers of degrees of freedom (DOFs). CMS belongs to a wide class of domaindecomposition techniques. CMS divides the global structure into several substructures, and the DOFs of each individualsubstructure are reduced significantly. Then, the substructures are reconnected, and the dynamic response of the system ispredicted very efficiently and accurately. Applications of CMS include the work of Wang and Kirkhope [8,9] who developeda free-interface CMS approach. Their method uses truncated component real modes, complemented with residualflexibility, inertial-relief modes and a residual dissipative effect to assemble a damped rotor system with flexible interfacesusing any number of components. In addition, Liu and Zheng [10] proposed an improved CMS for nonclassically dampedsystems. They established a new criterion for the selection of the component vibration modes in the frequency domain,and proposed state-level synthesis procedures in the formulation. Morgan et al. [11] developed a new concept of CMS fornonproportionally damped systems. They developed two new methods based on (a) constraint mode CMS (CB-CMS) andon (b) residual flexibility CMS. The constraint mode method can be used when existing finite element matrices can beemployed for analytical implementation. The residual flexibility method is used within an experimentally basedimplementation with test data. Maess and Gaul [12] applied CMS to fluid-filled pipe components interacting with

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[email protected] (S.-K. Hong), [email protected] (B.I. Epureanu),

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S.-K. Hong et al. / Mechanical Systems and Signal Processing 37 (2013) 403–421404

acoustic fluids. Both the linear acoustical domain and the structural domain were modeled by finite elements, and theywere fully coupled by a fluid-structure interface. Butland and Avitabile [13] proposed a test-based CMS approach to obtainmore accurate predictions of system-level response. Their goal was to evaluate the accuracy of the models of certaincomponents of a system separately, and then combine them analytically to ultimately determine the dynamiccharacteristics of the system. However, static constraint modes of each component are used in CB-CMS, and they arenot easily evaluated from modal testing. Thus, Butland and Avitabile [13] proposed an alternate CMS approach whichutilizes frequency and shape information for each component acquired from modal testing. They used an exact analyticaltechnique to improve their CMS-based ROMs. Tran [14] developed a new CMS method to reduce the number of interfacecoordinates for cyclically symmetric structures by using interface modes. The interface modes are a truncated set of modesobtained by solving an eigenvalue problem with system-level mass and stiffness matrices projected on global constraintmodes. These interface modes replace the static constraint modes otherwise used in CB-CMS. Also, Tran et al. [15]developed a robust CMS technique for stochastic damped vibroacoustics. That is necessary because the classical CMSmethod is not always efficient to predict the dynamical behavior of structures including visco-elastic and/or poro-elasticpatches. Thus, they used a Ritz basis to complete static residuals associated to viscoelastic and poro-elastic behavior.

In general, CMS has become a very popular numerical tool in aerospace and automotive engineering because itusually meets high standards of computational efficiency. Computational efficiency is illustrated by significant costsaving when remeshing is needed, since this task can be done locally, i.e., on each substructure separately. However,the remeshing process might also be time consuming computationally and manually for design purposes such asstructural optimization, and for damage modeling, for structural health monitoring. Therefore, reduced-ordermodeling techniques for design and damage modeling purposes are needed.

For that purpose, Kirsch [16] developed an approximate structural reanalysis method based on series expansion foroptimal design of large systems. Specifically, they used a modified nonpolynomial series to improve a quality of theapproximations of polynomial series expansion. However, the method was not focused on reducing DOFs, but ondeveloping a parameterization technique for reanalyses. Thus, ROMs for design and damage modeling were introducedalmost 15 yr ago by Balme�s et al. [17,18] to avoid the relatively expensive process of reanalysis of complex structures. Inaddition, several other ROMs referred to as parametric reduced-order models (PROMs) have been developed [19–22]. Inparticular, multi-component PROMs (MC-PROMs) have been developed recently by Hong et al. [22]. For robustsubstructure (re)analysis, MC-PROMs are advantageous because they allow several substructures to have parametricvariability in characteristics such as geometric parameters (e.g., thickness), or material properties (e.g., Young’s modulus).MC-PROMs are perfectly suited for predicting the vibration response of structures modeled with shell-type finite elementswhich can have thickness variations. However, if the structure is modeled with brick-type finite elements, and if the brick-type elements require local volume changes during reanalysis, then MC-PROMs cannot be effectively used to predict thedynamic response. This is because MC-PROMs use third-order Taylor series for parameterization. These Taylor series donot capture accurately the variation of the mass and stiffness matrices for brick-type finite elements because the volume oflocal finite elements can change during the reanalysis. Consequently, some entries of the mass and stiffness matrices forbrick-type finite elements vary highly nonlinearly with respect to geometric variations in the structure. Herein, a novelparameterization technique is proposed to capture these element-level nonlinearities.

Another challenge for MC-PROMs is that they can be numerically unstable due to the transformation matrix theyemploy. Specifically, the transformation matrix consists of static constraint modes and fixed interface normal modescomputed for a set of nominal parameters, and a few sets of perturbed parameter values (typically up to three sets perparameter) [22]. If all static constraint modes are kept and many normal modes are included, then the system-level mass iskept and many normal modes are included, then the system-level mass and stiffness matrices can be nearly singular (andcan even be larger than that of the full-order models). This is because the transformation matrix can contain vectors whichare nearly linearly dependent. These vectors are usually normal modes for substructures where the parametric variation(e.g., modulus of elasticity) does not affect the component-level normal modes. Moreover, the transformation matrix usedin MC-PROMs was designed for small parameter variations which ensure that the space spanned by the basis vectors at acomponent-level does not depend nonlinearly on parameter variations. That approach can break down because of thevolume variations which can occur when brick elements are used.

Another challenge of CMS methods and MC-PROMs is that they often require an excessively large number of interfaceDOFs because (often) these DOFs are hard (or impossible) to reduce. To address this issue, Castanier et al. [6] proposed thatthe physical interface DOFs be replaced by global interface modes, which were also called characteristic constraint (CC)modes. However, this concept is not optimal for substructural-based techniques because CC modes are system-levelinterface modes, not substructural-level interface modes. Thus, a new technique to reduce the interface DOFs locally isproposed herein and referred to as local-interface reduction.

This paper is organized as follows. In Section 2, the element-level nonlinearity due to the volume variations offinite elements of brick or other type is evaluated, and a novel parameterization technique is proposed to capturethis nonlinearity. Next, in Section 3, CMS is briefly reviewed, and next-generation parametric reduced-order models(NX-PROMs) are proposed. In Section 4, to locally reduce the interface DOFs, a local-interface reduction technique ispresented. Section 5 discusses the procedure to assemble substructural mass and stiffness matrices (with andwithout implementing the local-interface reduction technique). In Section 6, numerical examples such as a platestructure, an L-shaped structure, and a realistic vehicle model (a high mobility multipurpose wheeled vehicle,

S.-K. Hong et al. / Mechanical Systems and Signal Processing 37 (2013) 403–421 405

HMMWV) modeled with brick-type finite elements are used to demonstrate the proposed methods. Finally,conclusions are summarized in Section 7.

2. Robust parameterization techniques for element-level nonlinearity

For structural design and damage modeling purposes, the parameterization of the mass and stiffness matrices can bethe most important step. This is because the parameterization techniques enable capturing mass and stiffness variationsdue to design changes or damage in the structure. Thus, the reanalysis time can be significantly reduced because the finiteelement mesh does not need to be modified and remodeled. The parameterization technique has to be adapted for thedifferent characteristics of each type of finite element used. For example, the thickness, Young’s modulus, and materialdensity variations of shell-type elements can be captured well by third-order Taylor series [17,18,21,22]. However, wefound that the thickness variations for a brick and other types of finite elements such as hexagonal and tetrahedronelements cannot be captured well by Taylor series of low order due to element-level nonlinear characteristics caused byvolume variations. For elements which have volume, local thickness variations induce volume variations in the elements.In contrast, Taylor series works well for parameterizing shell-type elements because these do not have actual volume. Inthis section, a parameterization technique that captures thickness variations of brick and other types of finite elements isthe focus.

Fig. 1 shows an 8-node brick-type element which uses first-order (linear) shape functions. Coordinates x, y, and z areglobal, and coordinates x, Z, and z are local. As a conceptual example, consider that the four nodes on the top surface inFig. 1 move by a distance Dt. The brick-type element has a volume, so when each node moves, the volume of the brick-typeelement varies. Thus, the parameterization technique has to account for these volume variations. To that aim, let us firstrevisit the formulation used to derive stiffness matrices for brick-type elements [23,24]. The equation used to obtain thestiffness matrix can be expressed as

K¼

ZV

BT DB dV

¼

Z z ¼ 1

z ¼ �1

Z Z ¼ 1

Z ¼ �1

Z x ¼ 1

x ¼ �1BT DB detðJðxi,Zj,zkÞÞ dx dZ dz

¼X8

i ¼ 1

X8

j ¼ 1

X8

k ¼ 1

WiWjWkBTðxi,Zj,zkÞDBðxi,Zj,zkÞ detðJðxi,Zj,zkÞÞ, ð1Þ

where B is a strain matrix (which contains derivatives of the linear shape functions in global coordinates) and D is anelasticity matrix (which contains Poisson’s ratio n and the elastic modulus E). The determinant of the Jacobian $\bf{J}$ J inEq. (1) is obtained from the coordinate transformation of the strain matrix B. The determinant contains in its denominatora cubic polynomial of x, Z, and z, which reflects volume variations. Thus, the parameterization should also contain a cubicpolynomial in the denominator. To establish the coefficients of this cubic polynomial, the volume variations of brick-typeelements are considered. As shown in Fig. 1, one or several nodes on the top surface move to capture thickness variations.The resulting volume can be expressed as

V ¼ V0 1þdp�p0

p0

� �¼ V0 1þd

Dp

p0

� �,

where V and V0 are the final and the initial volume of the brick-type element, respectively, and p and p0 are the targetparameter value (thickness) and the initial parameter value, respectively. When only one node on the top surface moves,the coefficient d is 1/3. Similarly, when two nodes move, d¼ 1=2. Also, when three nodes move, d¼1. Finally, when four

31

2

4

75

6

4

ξ

ζη

1: (-1,-1,-1)2: (1,-1,-1)3: (1,1,-1)4: (-1,1,-1)5: (-1,-1,1)6: (1,-1,1)7: (1,1,1)8: (-1,1,1)

31

2

4

75

6

4

x

y

z

1: (0,0,0)2: (10,0,0)3: (0,0,-10)4: (10,0,-10)5: (0,10,0)6: (10,10,0)7: (0,10,-10)8: (10,10,-10)

Global Coordinates Local Coordinates

Fig. 1. Sample 8-node brick element with global and local coordinates.


nodes move, the volume variation is proportional to Dp, i.e., V ¼ V0ðDp=pÞ. This last type of variation, proportional to Dp, isvery well captured by a regular interpolation. The other three cases, however, are not. To address this issue, a cubicpolynomial which considers the volume variation of brick-type elements with a target parameter variation Dp is defined as

DðDpÞ ¼ 1þDp

p0

� �1þ

1

2

Dp

p0

� �1þ

1

3

Dp

p0

� �: ð2Þ

The new parameterization equation consists of a fourth-order interpolation in the numerator and the cubic polynomial inEq. (2) in the denominator, which yields the new parameterization as

Kðp0þDpÞ �K0þK1DpþK2ðDpÞ2þK3ðDpÞ3þK4ðDpÞ4

DðDpÞ: ð3Þ

To calculate the matrices K0, K1, K2, K3, and K4 in Eq. (3), five equations are needed. For that, stiffness matrices for fiveparameter values are computed. First, consider the case where Dp¼ 0. One obtains

Kðp0Þ �K0: ð4Þ

Next, consider Dp¼ idp (with i¼ 1,2,3,4), one obtains

Kðp0þ idpÞ �K0þK1ðidpÞþK2ðidpÞ2þK3ðidpÞ3þK4ðidpÞ4

DðdpÞ: ð5Þ

Rearranging Eqs. (4) and (5) into matrix form, for each entry e, q of the matrices K1, K2, K3, and K4, one obtains

C

K0,eq

K1,eq

K2,eq

K3,eq

K4,eq

26666664

37777775¼

Kðp0Þeq

Kðp0þdpÞeq

Kðp0þ2dpÞeq

Kðp0þ3dpÞeq

Kðp0þ4dpÞeq

26666664

37777775

, ð6Þ

where

C¼

1 0 0 0 01

DðdpÞdp

DðdpÞðdpÞ2

DðdpÞðdpÞ3

DðdpÞðdpÞ4

DðdpÞ

1Dð2dpÞ

ð2dpÞDð2dpÞ

ð2dpÞ2

Dð2dpÞð2dpÞ3

Dð2dpÞð2dpÞ4

Dð2dpÞ

1Dð3dpÞ

ð3dpÞDð3dpÞ

ð3dpÞ2

Dð3dpÞð3dpÞ3

Dð3dpÞð3dpÞ4

Dð3dpÞ

1Dð4dpÞ

ð4dpÞDð4dpÞ

ð4dpÞ2

Dð4dpÞð4dpÞ3

Dð4dpÞð4dpÞ4

Dð4dpÞ

26666666664

37777777775:

Eq. (6) can be easily solved by simply inverting the 5�5 matrix on the left hand side. This matrix is nonsingular andvery well behaved for inversion. Also, note that this inversion has to be done only once (for a given dp). Let us denoteby A this inverse matrix, i.e.,

A¼

1 0 0 0 01

DðdpÞdp

DðdpÞðdpÞ2

DðdpÞðdpÞ3

DðdpÞðdpÞ4

DðdpÞ

1Dð2dpÞ

ð2dpÞDð2dpÞ

ð2dpÞ2

Dð2dpÞð2dpÞ3

Dð2dpÞð2dpÞ4

Dð2dpÞ

1Dð3dpÞ

ð3dpÞDð3dpÞ

ð3dpÞ2

Dð3dpÞð3dpÞ3

Dð3dpÞð3dpÞ4

Dð3dpÞ

1Dð4dpÞ

ð4dpÞDð4dpÞ

ð4dpÞ2

Dð4dpÞð4dpÞ3

Dð4dpÞð4dpÞ4

Dð4dpÞ

26666666664

37777777775

�1

¼

A11 A12 A13 A14 A15

A21 A22 A23 A24 A25

A31 A32 A33 A34 A35

A41 A42 A43 A44 A45

A51 A52 A53 A54 A55

26666664

37777775:

Rearranging Eq. (6) using the entries in A, one obtains

Kðp0þDpÞ � b0Kðp0Þþb1Kðp0þdpÞþb2Kðp0þ2dpÞ

þb3Kðp0þ3dpÞþb4Kðp0þ4dpÞ, ð7Þ

where

b0 ¼ ðA11þA21DpþA31Dp2þA41Dp3þA51Dp4Þ,





b4 ¼ ðA15þA25DpþA35Dp2þA45Dp3þA55Dp4Þ:

Eq. (7) shows that Kðp0þDpÞ is simply a linear combination of five (pre-computed) matrices. The coefficients in thelinear combination depend very nonlinearly on Dp. That is the key factor which ensures the high accuracy of the newparameterization. Note that the computational cost of the new parameterization is the same as that for a regular fourth-order interpolation. The accuracy, however, is higher (as shown in Section 6.1).

3. Reduced-order models

The parameterization techniques proposed in Section 2 are for the full-order finite element model. However, the mainobjective of this work is to predict vibration responses using ROMs (as opposed to full-order models) to reduce thecalculation time. To detail the construction of ROMs, the fixed-interface Craig-Bampton component mode synthesis(CB-CMS) [5] is reviewed briefly. Next, a new transformation matrix is presented and used in conjunction with the newparameterization technique discussed in Section 2. Finally, NX-PROMs are constructed.

3.1. Brief review of Craig-Bampton component mode synthesis

In this section, the fixed-interface CB-CMS [2] method is reviewed. This modeling approach is broadly used because ofits simplicity and computational stability. To apply the CB-CMS, the complex structure of interest is partitioned intosubstructures. The DOFs of each substructure are further partitioned into active DOFs on the interface (indicated by thesuperscript A), and omitted DOFs in the interior (indicated by the superscript O). The mass and stiffness matrices for acomponent i can then be partitioned to obtain

Mi ¼mAA

i mAOi

mOAi mOO

i

" #and Ki ¼

kAAi kAO

i

kOAi kOO

i

24

35:

Next, the physical coordinates are changed to a set of coordinates representing the amplitudes of a selected set of fixed-interface component-level normal modes UN

i (indicated by the superscript N), and the amplitudes of the full set of staticconstraint modes UC

i ¼�kOO�1i kOA

i (indicated by the superscript C). The transformed mass and stiffness matrices forcomponent i can be expressed as

MCBCMSi ¼

m̂Ci m̂

CNi

m̂NCi m̂

Ni

24

35 and KCBCMS

i ¼k̂

C

i k̂CN

i

k̂NC

i k̂N

i

24

35:

In this work, the CB-CMS method is used only for the substructures which do not have any parameter variation ordamage.

3.2. Next-generation parametric reduced-order models

In this section, MC-PROMs are improved to be more robust and mathematically stable. The resulting models arereferred to as NX-PROMs.

3.2.1. Transformation matrix for NX-PROMs

The transformation matrix for NX-PROMs is constructed somewhat similar to MC-PROMs (which was constructed byusing the idea behind CB-CMS). It also has a set of static constraint modes WC and a set of fixed-interface normal modesUN . However, the transformation matrix for NX-PROMs has a different set of static constraint modes and a different set offixed-interface normal modes compared to CB-CMS and MC-PROM. This transformation matrix can be written forcomponent i as

Ti ¼I 0

WCaug,i UN

aug,i

" #,

where WCaug is referred to as the matrix of augmented constraint modes

WCaug,i ¼ ½W

C0,i WC

1,i WC2,i WC

3,i WC4,i�

and UNaug is referred to as the matrix of augmented fixed-interface normal modes

UNaug,i ¼ ½U

N0,i UN

1,i UN2,i UN

3,i UN4,i�:

Matrices WC0,i and UN

0,i correspond to the nominal parameter values, whereas matrices WCl,i and UN

l,i ðl¼ 1,2,3,4Þ correspondto four other parameter values.


In general, the columns of UNaug are not orthogonal. Therefore, for numerical stability, an orthogonal basis for the space

spanned by these modes is computed. This basis is obtained by a truncated set of left singular vectors UN of UNaug [22].

Thus, the new transformation matrix can be expressed as

~Ti ¼I 0

WCaug,i UN

i

" #: ð8Þ

An orthogonal basis for the space spanned by the fixed interface mode set is used for numerical stability. This basis ðUNÞ is

orthogonal, so numerical conditioning problems are avoided. In addition, the left singular vectors in Eq. (8) are truncatedto lower the size of the ROM. The choice of the cutoff point can affect accuracy if it is chosen too high, but it does not affectnumerical stability. Thus, the cutoff value can be estimated by a standard convergence study where calculations are doneat ever lower cutoff values until convergence is obtained. For the calculations we performed, the cutoff was 0.01% of themaximum singular value.

Using linear interpolation for constructing (system-level) bases has important drawbacks. That is why we limited thatapproach to only the static constraint modes. The other modes/basis vectors are not linearly interpolated. Also, the linearinterpolation is done only at a component level (where the dynamics and geometry are simpler) rather than at a systemlevel. That makes the NX-PROMs approach even more robust. Such component level modeling is possible in structuraldynamics, but has not yet been done for fluids. Evaluating the effectiveness of the NX-PROMs in aeroelasticity (wherestructures and fluids interact) is an intriguing idea for future work. Amsallem and Farhat [25] have applied an interpolationmethod based on the Grassmann manifold and its tangent space at a point to construct bases of reduced-order models inaeroelasticity. Their method overcomes difficulties faced by direct interpolation when applied in aerodynamics. Theirinterpolation method is a better approach for aeroelasticity compared to NX-PROMs which are designed for structuraldynamics.

The transformation matrix in Eq. (8) can be used to project the physical domain onto the NX-PROM domain. Thestiffness matrix KNXPROM

i ¼ ~TT

i KFEMi

~Ti for component i (where KFEMi represents the stiffness matrix of the full-order model of

component i) can be partitioned to obtain

KNXPROMi ¼

KC00,i KC

01,i KC02,i KC

03,i KC04,i KCN

00,i

KC10,i KC

11,i KC12,i KC

13,i KC14,i KCN

11,i

KC20,i KC

21,i KC22,i KC

23,i KC24,i KCN

22,i

KC30,i KC

31,i KC32,i KC

33,i KC34,i KCN

33,i

KC40,i KC

41,i KC42,i KC

43,i KC44,i KCN

44,i

KNC00,i KNC

11,i KNC22,i KNC

33,i KNC44,i KN

i

2666666666664

3777777777775:

A similar relation is obtained for the mass matrix of component i. Here, the DOFs corresponding to the constraint part(superscript C) are repeated for the five parameter values (denoted by subscript 0, 1, 2, 3, and 4). Note that in such anapproach, the size of the mass and stiffness matrices can be quite large. Also, these matrices may be ill-conditionedbecause the columns of WC

aug,i are not necessarily linearly independent.To address this issue, a new method to account for the static constraint modes is developed. This new method avoids

duplicating the interface DOFs (C) and captures the interface effects more accurately. The approach reduces the number ofsets of static constraint modes (used to obtain MNXPROM

i and KNXPROMi ) from five sets to just one set. Consider that the actual

value p of the parameter where the reanalysis is needed exists between the lth and ðlþ1Þth parameter values (l¼ 0,1,2,3,or 4) which were used to construct MNXPROM

i . In that case (described in Fig. 2), a new static constraint mode can begenerated by linearly interpolating between the static constraint modes for the lth and the ðlþ1Þth parameter values toobtain

~WC

i ¼plþ1�p

plþ1�pl

� �WC

l,iþp�pl

plþ1�pl

� �WC

lþ1,i ¼ aiWCl,iþbiW

Clþ1,i: ð9Þ

This new static constraint mode ~WC

i replaces all five static constraint modes used to construct ~T i in Eq. (8).Note that this reduction can be implemented without re-constructing ~T i for each case of parameter variation. Instead, only

simple linear combinations of partitions of the matrices MNXPROMi and KNXPROM

i , are needed. Details are given in Section 3.2.2

p0 p0 + lδp p0 + (l+1)δp

p

Ψ ......C0 Ψ

Cl Ψ

Cl+1 ΨC

4

p0 + 4δp

Fig. 2. The case where the value of parameter p is between p0þ ldp and p0þðlþ1Þdp for some l value between 0 and 3.


below. In the end, the final NX-PROM mass and stiffness matrices have only a single set of constraint modes ~WC

i which alwayshas linearly independent columns.

3.2.2. Parameterization for NX-PROMs

The new interpolation presented in Section 2 is applied to NX-PROMs. Five mass and five stiffness matrices areconstructed for each component i (for l¼ 0,1,2,3,4) as follows:

MNXl,i ¼

~TT

i Mðp0þ ldpÞ ~Ti and KNXl,i ¼

~TT

i Kðp0þ ldpÞ ~T i:

These matrices are not all used independently to form NX-PROMs. Instead, they are linearly combined to implement thesingle set of static constraint modes ~W

C

i in Eq. (9). Thus, conceptually, ~Ti is replaced by T̂i, given by

T̂i ¼I 0

WCaug,i UN

i

" #ðaiRl,iþbiRlþ1,iÞ ¼

~T iðaiRl,iþbiRlþ1,iÞ,

where Rl,i and Rlþ1,i are masking matrices of zeros and ones. The matrix Rl,i is a 6�2 block matrix where the blocks ðl,1Þand (6,2) are unit matrices, while all other blocks are zero. The first five rows correspond to WC

l,i ðl¼ 0,1,2,3,4Þ and the lastrow corresponds to UN

i . This new transformation matrix T̂i is applied to the mass and stiffness matrices of component i toconstruct NX-PROMs. First, five mass and stiffness matrices are constructed for each parameter variation sdp (fors¼ 0,1,2,3,4) as follows:

M̂NX

s,i ¼ ðaiRTl,iþbiR

Tlþ1,iÞM

NXs,i ðaiRl,iþbiRlþ1,iÞ,

K̂NX

s,i ¼ ðaiRTl,iþbiR

Tlþ1,iÞK

NXs,i ðaiRl,iþbiRlþ1,iÞ:

Next, the new interpolation discussed in Section 2 is applied using M̂NX

s,i and K̂NX

s,i . Eq. (7) is used to obtain

Mðp0þDpÞNXi � b0M̂

NX

0,i þb1M̂NX

1,i þb2MNX2,i þb3M̂

NX

3,i þb4M̂NX

4,i ,

Kðp0þDpÞNXi � b0K̂

NX

0,i þb1K̂NX

1,i þb2KNX2,i þb3K̂

NX

3,i þb4K̂NX

4,i ,

where bs ðs¼ 0,1,2,3,4Þ are computed for each Dp by using the matrix A and the expression in Eq. (7). Note that the fivecoefficients bs which depend on the actual Dp are easily calculated because they are just five scalars that depend only onDp and dp irrespective of the size of the finite element mesh.

Finally, the mass and stiffness matrices for the ith component of the NX-PROMs can be partitioned as follows:

Mðp0þDpÞNXi ¼

MCDp,i MCN

Dp,i

MNCDp,i MN

Dp,i

24

35,

Kðp0þDpÞNXi ¼

KCDp,i KCN

Dp,i

KNCDp,i KN

Dp,i

24

35, ð10Þ

where superscript C indicates a constraint partition and superscript N indicates a nominal mode partition.

4. Local-interface reduction

If the finite element mesh used is very fine, the size of the reduced system-level matrices is dominated by the constraintDOFs corresponding to the C partition in Eq. (10). The constraint DOFs of matrices constructed by CMS-based techniquesare difficult to reduce. This is an important issue because, if the constraint DOFs cannot be reduced, then the overallstructure cannot be efficiently divided into many substructures. To address this issue, Castanier et al. [6] suggested the useof characteristic constraint (CC) modes. This technique is based on performing a secondary eigenanalysis of the constraintpartition (C) of the system-level mass and stiffness matrices constructed by CB-CMS. This technique is applied after thesystem-level matrices are constructed. However, the core idea of NX-PROMs is that all analyses are accomplished at thesubstructure-level, and not at the system-level. Thus, an alternate interface reduction technique is proposed next. The newapproach is applied at the substructure-level, and it is referred to as local-interface reduction (LIR).

The local-interface reduction technique is also based on a secondary eigenanalysis of the constraint partition. However,the secondary eigenanalysis is executed on the constraint partitions (C) of the substructural matrices, not the system-levelmatrices. The secondary eigenanalyses on constraint DOFs (C) of the mass and stiffness matrices of component i

constructed by either CB-CMS or the NX-PROM approach are given by

KCDp,iU

CCDp,i�MC

Dp,iUCCDp,iLDp,i ¼ 0,

where LDp,i is a diagonal matrix which contains the eigenvalues, and UCCDp,i are the characteristic constraint (CC) modes of

the ith substructure. They are truncated for the frequency range of interest by using the eigenvalues in LDp,i. The rows ofthe CC modes indicate the constraint DOFs of the substructure, and the columns represent the set of truncated CC modes.The CC modes for each substructure are used to reduce the interface DOFs for each boundary locally. Note that joining all CC


modes for each interface between different components may lead to vectors which are not necessarily linearlyindependent. However, they span the adequate space. Thus, the left singular vectors of the CC interface modes may haveto be used for certain interfaces. To demonstrate the LIR procedure, a simple plate model is used in Section 6.2.

The set of orthogonal basis vectors used for all interfaces that a component i has with other components are grouped ina block diagonal matrix which contains the entire interface component i has. The number of blocks is equal to the numberof components that are connected to component i. These matrices are denoted by UDp,i. Of course, if component i connectsto only one other component, then there is only one block in UDp,i. Next, the mass and stiffness matrices in Eq. (10) areprojected using UDp,i as follows:

MCCDp,i ¼UT

Dp,iMCDp,iUDp,i, MCCN

Dp,i ¼UTDp,iM

CNDp,i, MNCC

Dp,i ¼MCCNDp,iT,

KCCDp,i ¼UT

Dp,iKCDp,iUDp,i, KCCN

Dp,i ¼UTDp,iK

CNDp,i, KNCC

Dp,i ¼KCCNDp,iT: ð11Þ

Thus, the final mass and stiffness matrices with reduced constraint DOFs are given for component i by

MLIRDp,i ¼

MCCDp,i MCCN

Dp,i

MNCCDp,i MN

Dp,i

24

35, KLIR

Dp,i ¼KCCDp,i KCCN

Dp,i

KNCCDp,i KN

Dp,i

24

35,

where superscript LIR indicates that the matrices are constructed using local-interface reduction. An example is providedin Section 6.2.

In some cases it is faster to perform an exact analysis using sparse matrices than to perform an approximate analysisusing full matrices obtained when there are many interface DOFs. However, NX-PROMs can be enhanced to address thatissue through a local interface reduction (LIR) method. Also, the number of interface DOFs can be adjusted by a judiciousdefinition of the components (and their interfaces). In the cases we analyzed, even though the matrices for the exactanalysis are sparse, the size of the matrices constructed by NX-PROMs is 1000–10,000 times smaller. Hence, for the caseswe studies, NX-PROMs significantly enhance the computational speed compared to the full-order model.

5. Assembly

To predict the system-level dynamics, the mass and stiffness matrices obtained in Sections 3 and 4 for eachsubstructure have to be assembled. To do that, geometric compatibility conditions must be enforced. In the following,we discuss separately the case where LIR is not used and the case where it is used.

Let us consider the case where the geometric compatibility conditions are used for models without LIR. In this case, theconstraint partitions (C) of component-level matrices keep the meaning of the physical interface DOFs of matricesobtained from FEMs. This means that the geometric compatibility conditions between interface DOFs (constraint DOFs)can be applied directly to construct the system-level matrices. The complete component-level equations of motion forcomponent i based on CB-CMS or the NX-PROM approach can be expressed as

MROMi

€qROMi þKROM

i qROMi ¼ FROM

i , ð12Þ

where ROM indicates component-level matrices obtained using either CB-CMS or the NX-PROM approach. The stiffnessmatrices in Eq. (12) obtained for components without parameter variability can be expressed as

KROMi ¼KCBCMS

i ¼KC

i KCNi

KNCi KN

i

" #CBCMS

: ð13Þ

For component with parameter variability, the stiffness matrices in Eq. (12) can be expressed as

KROMi ¼KNX

i ¼KCDp,i KCN

Dp,i

KNCDp,i KN

Dp,i

24

35

NXPROM

: ð14Þ

The formulas for the mass matrices are similar to those for the system matrices in Eqs. (13) and (14) (and are omitted herefor the sake of brevity).

Next, the matrices in Eq. (12) are grouped for all i to obtain

M̂ ¼ Bdiag½MROM1 � � � MROM

n �,

K̂ ¼ Bdiag½KROM1 � � � KROM

n �,

F̂ ¼ ½FROM1 T � � � FROM

n T�T, ð15Þ

where n is the number of components and Bdiag½�� denotes a block-diagonal matrix.The geometric compatibility condition for the ROM is expressed as

qCi ¼ qC

j , ð16Þ


where qi and qj are the generalized coordinates for the constraint partitions (C for CB-CMS or NX-PROMs) that correspondto the interface between components i and j. Of course, there is no compatibility condition to be enforced for twocomponents which do not have a common interface.

Eq. (16) is used to transform the matrices in Eq. (15) similar to the assembly process in all finite element modelingmethods. The final assembled, reduced-order, system-level matrices are given by

MROMsys ¼

MC MCN

MNC MN

" #, KROM

sys ¼KC KCN

KNC KN

" #,

FROMsys ¼

FC

FN

" #,

where

KCN¼ ½KCN

1 KCN2 � � � KCN

n �, KNC¼KCNT

,

KN¼

KN1 0 � � � 0

0 KN2 � � � 0

^ ^ & 0

0 0 � � � KNn

266664

377775

and KC is a matrix which is obtained by the assembly of each interface. In general, KC is a matrix which has a smaller sizethan the C partitions in K̂. The same process is applied to obtain FC . Also, similar relations are obtained for the mass matrixMROM

sys (and are omitted here for the sake of brevity).Finally, the compatibility conditions for models constructed using LIR can be expressed almost identically to those for

models without LIR. The only difference is that the generalized coordinates in Eq. (16) represent amplitudes ofcharacteristic constraint modes or amplitudes of the basis vectors used to capture the space spanned by the characteristicconstraint modes.

Fig. 3 shows the procedure used to construct a NX-PROM. The process may be summarized as follows: (a) the system isdivided into components to accommodate the various types of parameter variation, (b) CB-CMS is used to constructmodels for each substructure without parameter variations, (c) NX-PROM is used to construct models for eachsubstructure with parameter variations, (d) LIR is applied at interfaces between substructures (one interface at a time)to reduce the number of physical interface DOFs, and (e) all substructures are assembled, and geometric compatibilityconditions are applied using generalized coordinates in the LIR domain.

Parameter variations

No variations

(a) Divide system into components

PROM

CB-CMS

PROM

CB-CMS

(b) Apply CB-CMS to each substructure without variations

(c) Apply NX-PROM to each substructure with variations

LIC

(d) Apply LIR to each interfaces of each substructure

(e) Assembled togetherFinite Element Model NX-PROMs

Fig. 3. Conceptual diagram showing the process of constructing NX-PROMs.


6. Numerical results

6.1. Element-level nonlinearity of brick and other types of finite elements

Fig. 4 shows a simple plate structure modeled with a shell-type finite elements, where t is the thickness of the plateðt¼ 0:2 mmÞ. To examine the variation in the entries of the finite element mass and stiffness matrices for this substructure,the thickness t is varied by increments of Dt¼ 0:01 mm.

The size of the mass and stiffness matrices of the simple structure shown in Fig. 4 is 15,582� 15,582. As an example,the 32nd diagonal entries of the mass and stiffness matrices, M32,32 and K32,32, are shown in Fig. 5 as the thickness varies(approximated matrices are shown by a dash-dot line, and the actual matrices are shown by a solid-line for various valuesof Dp). One may observe that those entries of the mass and stiffness matrices vary almost linearly. To capture thesevariations very accurately, a cubic interpolation is used as follows:

Kðp0þDpÞ � ~K0þ~K1Dpþ ~K2Dp2þ ~K3Dp3: ð17Þ

The matrices ~K0, ~K1, ~K2, and ~K3 are computed by using the stiffness matrices evaluated at three parameter values. Thesevalues are the reference value p0, the first perturbed value p1 ¼ p0þdp, the second perturbed value p2 ¼ p0þ2dp, and thethird perturbed value p3 ¼ p0þ3dp. The procedure is standard and similar to the one used in Section 2, so its details areomitted here for the sake of brevity. A similar interpolation is used for the mass matrix. Note that, in contrast to Taylorseries, the cubic interpolation does not require the calculation of derivative terms.

Next, to examine the variations in the entries of the mass and stiffness matrices for brick-type elements, the same platestructure is modeled with brick-type elements. The nominal thickness of the plate is (the same) 0.2 mm and is varied by(the same) increments of Dt¼ 0:01 mm, as shown in Fig. 6. The thickness is varied in a region near the center of the plate.The entries of the mass and stiffness matrices near the DOFs where the thickness is varied are affected. A sample DOFaffected is the 645th. The 645th diagonal entry of the mass matrix varies linearly (and is omitted for the sake of brevity).The same entry of the stiffness matrix, however, does not vary linearly, as shown in Fig. 7 (left), where exact values areshown by a solid-line. To capture this nonlinear variation, the cubic interpolation function in Eq. (17) is used. Theapproximate values obtained are shown by a dash-dot line. These results show that Eq. (17) is not good enough to capturethe highly nonlinear variation of the stiffness matrix. Therefore, a fourth-order interpolation is used as follows:

Kðp0þDpÞ � ~K0þ~K1Dpþ ~K2Dp2þ ~K3Dp3þ ~K4Dp4: ð18Þ

This fourth-order interpolation captures well the nonlinear variation in the entries of the stiffness matrix as shown in Fig. 7(right).

Based on the results in Fig. 7, one may assume that the errors obtained based on Eq. (18) are negligible. However, that isnot correct, as demonstrated by the forced response of the plate calculated using exact and approximated matrices. Fig. 8shows the forced response at one of the DOFs on the plate for excitation frequencies near the first resonance. The solid-line

t

Fig. 4. Simple plate structure modeled with shell-type elements.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.42

3

4

5

6

7

8

9

10x 10−11

Thickness (mm)

M32

,32

ExactClassic Interpolation

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.43

4

5

6

7

8

9

10

11

12x 104

Thickness (mm)

K32

,32

ExactClassic Interpolation

Fig. 5. The 32nd diagonal entries of the exact and the parametrized mass and stiffness matrices obtained by using a classic cubic interpolation for

shell-type elements.

t

Fig. 6. Simple plate structure modeled with brick-type elements.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.43.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

4x 104

Thickness (mm)

K64

5,64

5

Exact

Classic Interpolation

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.43.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

4x 104

Thickness (mm)

K64

5,64

5

Exact

Fourth Order Interpolation

Fig. 7. The 645th diagonal entry of the exact and the parametrized stiffness matrices obtained by using a classic cubic interpolation (left) and an classic

fourth-order interpolation (right) for brick-type elements.

4250 4300 4350200

300

400

500

600

700

800

900

Frequency (Hz)

Dis

plac

emen

t (m

m)

ExactFourth Order Interpolation

Fig. 8. Forced responses calculated for Dp¼ 0:37 mm using the exact and parametrized mass and stiffness matrices obtained based on a fourth-order

interpolation for brick-type elements.


represents the response computed by the actual mass and stiffness matrices, and the dash-dot line indicates the responsecomputed by the mass and stiffness matrices parametrized using Eq. (18) for the case of Dt¼ 0:37 mm. Clearly, the forcedresponse computed by the parametrized mass and stiffness matrices does not agree with that computed by the actualmatrices. This means that the errors in the entries of the stiffness matrix from the fourth-order interpolation are not smallenough to capture accurately the dynamic response of the structure with a modified thickness. Note that, the errors in thedynamic response are induced by inaccuracies in the stiffness matrix, not in the mass matrix. These results demonstratedthat a new parameterization technique focused on capturing element-level nonlinear variations in the stiffness is neededfor brick-type finite elements.

The parametrized stiffness matrix was calculated using Eq. (7) for Dp¼ 0:37 mm. As a sample of results, Fig. 9 showsthat the 645th diagonal entry of the exact and the approximated stiffness matrices match extremely well. Similar matchesare observed for all entries of the matrices. Next, forced responses were calculated using these matrices. The results areshown in Fig. 10. The solid-line indicates the response computed using the exact matrices and the dash-dot line indicatesthe response computed by using the new parametrized matrices. The results closely match. Moreover, the naturalfrequencies for the exact matrices and the new parametrized matrices, also match, as shown in Table 1.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.43.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

4x 104

Thickness (mm)

K64

5,64

5

ExactNew Parameterization

Fig. 9. The 645th diagonal entry of the exact and the parametrized stiffness matrices obtained by using the new interpolation for brick-type elements.

4250 4300 4350200

300

400

500

600

700

800

900

Frequency (Hz)

Dis

plac

emen

t (m

m)

ExactNew Parameterization

Fig. 10. Forced response calculated for Dp¼ 0:37 mm using the exact and the parametrized mass and stiffness matrices obtained based on the new

interpolation for brick-type elements.


6.2. Example of local-interface reduction

Consider a structure composed of five substructures and three boundaries as shown in Fig. 11. Gj represents the jthboundary. First, using the constraint partitions (C) of the reduced mass and stiffness matrices, the CC modes are computed.These are UCC

Dp,1, UCCDp,2, UCC

Dp,3, UCCDp,4, and UCC

Dp,5. UCCDp,1 has interface DOFs for boundary G1 and G2, while UCC

Dp,2 has interfaceDOFs for G2 and G3. Substructures 3, 4, and 5 each have one boundary G1, G2, and G3, respectively. The mathematicalrepresentation for these partitions for each CC mode can be expressed as

UCCDp,1 ¼

ðUCCDp,1Þ

G1

ðUCCDp,1Þ

G2

24

35, UCC

Dp,2 ¼ðUCC

Dp,2ÞG2

ðUCCDp,2Þ

G3

24

35,

UCCDp,3 ¼ ðU

CCDp,3Þ

G1 , UCCDp,4 ¼ ðU

CCDp,4Þ

G2 , UCCDp,5 ¼ ðU

CCDp,5Þ

G3 :

By using each boundary partition of the CC modes, the augmented set of CC modes for each boundary is constructed as

UG1

Dp,aug ¼ ½ðUCCDp,1Þ

G1 ðUCCDp,3Þ

G1 �,

UG2


G2 ðUCCDp,2Þ

G2 ðUCCDp,4Þ

G2 �,

UG3


G3 ðUCCDp,5Þ

G3 �: ð19Þ

Table 1About 10 lowest natural frequencies for exact and parametrized matrices with volume variations.

Mode Exact (Hz) Approximated (Hz)

1 4278.24 4278.24

2 9024.39 9011.52

3 9068.45 9053.21

4 14,323.12 14,323.11

5 24,952.70 24,952.34

6 25,024.01 25,023.94

7 27,463.11 27,460.10

8 27,656.15 27,654.70

9 36,098.49 36,098.51

10 41,275.58 41,262.95

1

23

45

Fig. 11. A simple structure used to demonstrate the local-interface reduction.


Eq. (19) describes the augmented CC bases for boundaries G1, G2, and G3. However, the augmented bases UG1

Dp,aug, UG2

Dp,aug,and UG3

Dp,aug are not guaranteed to be linearly independent. Thus, they cannot be directly used to reduce the constraintDOFs because of lack of numerical stability. Thus, an orthogonal basis for the space spanned by the augmented CC basis isused. Specifically, the left singular vectors ~U

G1

Dp, ~UG2

Dp, and ~UG3

Dp of the three augmented CC bases UG1

Dp,aug, UG2

Dp,aug, and UG3

Dp,aug

in Eq. (19) are computed for each substructure, and the left singular vector corresponding to singular values larger than0.01% of the maximum singular value is selected for each boundary. The rows of the orthogonal bases ~U

G1

Dp, ~UG2

Dp, and ~UG3

Dp

are (re)sorted for each substructure to match the interface DOFs for each boundary. The resorted matrices are denoted byUG1

Dp, UG2

Dp, and UG3

Dp. The (re)sorted matrices are grouped for each component i to obtain matrices UDp,i which UDp,i are usedto project the interface DOFs onto the secondary generalized coordinates (CC domain). For example, substructure 1includes the G1 and G2 boundaries. To reduce the interface DOFs of substructure 1, the orthogonal bases UG1

Dp and UG2

Dp aregrouped in a matrix UDp,1 given by

UDp,1 ¼

UG1

Dp 0G1

0G2 UG2

Dp

24

35: ð20Þ

The orthogonal basis for substructure 2 is constructed in the same way, to obtain

UDp,2 ¼

UG2

Dp 0G2

0G3 UG3

Dp

24

35: ð21Þ

For some of the substructures, the grouping of UGDp matrices is not necessary. For example, the bases used for

substructures 3, 4, and 5 are simply given by

UDp,3 ¼UG1

Dp, UDp,4 ¼UG2

Dp and UDp,5 ¼UG3

Dp ð22Þ

Next, the orthogonal basis for each substructure is constructed using Eqs. (20)–(22), and the interface DOFs (C) of theNX-PROMs generalized coordinates are projected into the secondary generalized CC coordinates as shown in Eq. (11).

6.3. L-shaped plate modeled with brick-type finite elements

To demonstrate the proposed NX-PROM and LIR methodologies, an L-shaped structure modeled with brick-typeelements (shown in Fig. 12) and containing thickness variations is investigated numerically. Fig. 12 shows the pristinestructure and the structure with thickness variations. The structure consists of eight substructures. Substructures 7 and 8have three cases of thickness variations, as given in Table 2. The nominal thickness of the structure is 1 mm and the

12

3

4

5

6

78

F

F

1

2

3

4

5

6

78

thickness variations

F

F

Г2

Г1Г3

Г4

Г5 Г6

Г7

Г8 Г9

Г10

Fig. 12. An L-shaped plate shown without thickness variation (left) and with thickness variation (right); interfaces between components are denoted by Gi.

Table 2Thickness variations in substructures 7 and 8 of the L-shaped plate.

Substructure Thickness, case 1 Thickness, case 2 Thickness, case 3

7 1:00 mm-1:22 mm 1:00 mm-1:42 mm 1:00 mm-1:81 mm

8 1:00 mm-1:22 mm 1:00 mm-1:42 mm 1:00 mm-1:81 mm


elemental thickness is 0.2 mm. The thickness variations applied are very large compared to the nominal elementalthickness considered. This causes the entries of the mass and stiffness matrices to vary nonlinearly. CB-CMS is applied forthe 1st to the 6th substructure, and the NX-PROM approach is applied for the 7th and the 8th substructures.

Fig. 13 shows the system-level forced responses of the nominal structure and the three cases of thickness variation. Thedotted lines represent the vibration response of the nominal structure. The crosses and circles represent the responses ofthe structure with thickness variations predicted using NX-PROMs and full-order models, respectively. To reveal theenhanced performance of the NX-PROMs, the vibration response predicted by MC-PROMs is also shown by the stars inFig. 13. For all three cases, the forced response predicted by the full-order models and the NX-PROMs show excellentmatching. However, the forced responses predicted by MC-PROMs are not accurate compared to the full-order models.This is due to the fact that MC-PROMs cannot capture the volume variation of brick-type finite elements, which leads topoor predictions of the vibration response. Note also that the thickness variations affect significantly the structure, asshown by the significant differences between the response of the nominal structure and the other responses.

The number of DOFs of the full-order model and the NX-PROMs is 18,300 and 3502, respectively. The system-levelDOFs of the NX-PROMs include 3000 interface DOFs and 502 generalized internal DOFs. The number of generalizedinternal DOFs is small. However, the number of interface DOFs is large, and should be reduced. Thus, the LIR techniquedescribed in Section 4 was applied. Fig. 12 shows the interfaces which are reduced, where Gm indicates the mth interface.Fig. 14 shows the forced responses for the nominal structure and the three cases of thickness variations. The dotted linesrepresent the response of the nominal structure. The crosses and circles indicate the responses of the structure withthickness variations predicted using NX-PROMs with full interface DOFs and full-order models, respectively. The squaresrepresent the responses predicted using NX-PROMs with LIR. These latter models use only 533 interface DOFs, reduced byLIR from 2498. The response predicted by NX-PROMs with LIR agrees very well with the responses of the full-order model.

6.4. Results for a high mobility multipurpose wheeled vehicle model

In this section, NX-PROMs are used to predict the dynamic response of a realistic vehicle model. We consider the baseframe of a high mobility multipurpose wheeled vehicle (HMMWV). The finite element model for the HMMWV is aconventional model used to examine its dynamic response [19–22,26].

Fig. 15 shows the system-level and substructure-level finite element models of the HMMWV frame. The cross-barstructure is composed of substructures CL and CR, which are modeled with solid-type finite elements, as shown in Fig. 16.The marked region in Fig. 16 indicates the nodes which move due to thickness variation. Table 3 indicates two cases ofthickness variation for CL and CR. The thickness variations applied are much larger than those used in the L-shapedexample. We chose these large variations to demonstrate that the proposed methods are very accurate even when thethickness variations are very large. Such large variations are encountered in practice especially when components arere-designed. The structural and the elemental thicknesses of the CL and CR substructures are 5 mm and 2.5 mm,respectively. NX-PROMs were created for the CL and CR substructures, and CB-CMS was applied to the remainingsubstructures. Next, forces and moments were applied to the engine cradle, and the resulting forced response wascomputed. Fig. 17 shows the response of the HMMWV frame for cases 1 and 2. The measured point is shown in Fig. 15. Thedotted line shows the forced response of the nominal HMMWV structure. The circles and crosses indicate the responses of

240 245 250 255 260 265 270 275 2801

1.5

2

2.5x 10−3

Frequency (Hz)

Dis

plac

emen

t (m

m)

NominalCase 1 for FEMCase 1 for NX−PROMCase 1 for MC−PROM

240 250 260 270 2801

1.5

2

2.5x 10−3

Frequency (Hz)

Dis

plac

emen

t (m

m)


250 260 270 280 290 3001

1.21.41.61.8

22.22.42.62.8

3 x 10−3

Frequency (Hz)

Dis

plac

emen

t (m

m)


Fig. 13. Forced response predictions of the L-shaped plate for the nominal structure and for cases 1–3 computed using full-order models, MC-PROMs and

the novel NX-PROMs. (a) Case 1. (b) Case 2. (c) Case 3.


the re-designed HMMWV frame predicted by full-order models and NX-PROMs, respectively. The stars show the responsespredicted by MC-PROMs. Results obtained using the full-order models and the NX-PROMs show excellent agreement forboth cases 1 and 2, but the results predicted by MC-PROMs do not agree well with the response obtained using the full-order models. Also, note that the re-design has important effects on the structure, as demonstrated by the significantdifference between the responses of the nominal and the re-designed structures. Table 4 shows the computational timerequired for obtaining essentially the same accuracy as the full order model. The difference between the frequenciespredicted by NX-PROMs and full-order analyses is less than 0.01% for all these calculations. The reanalysis based on NX-PROMs is over 1000 times faster in all calculations.

The full-order model of the HMMWV has 123,201 DOFs. The NX-PROMs have 2683 DOFs, of which 1473 are constraintDOFs and 1210 are fixed-interface generalized DOFs. To reduce the number of constraint DOFs, LIR was applied. Fig. 15shows the interfaces of the substructures in the HMMWV frame. Table 5 shows what are the interface DOFs for eachsubstructure. Fig. 18 shows the response of the HMMWV model predicted by NX-PROMs for cases 1 and 2 using differentlevels of reduction of the overall interface DOFs. Magnified plots near the resonant frequencies are included also. Note thatthe accuracy of the response predicted by LIR depends on the number of remaining interface DOFs. In these two cases, anacceptable accuracy is obtained when the remaining interface DOFs are not fewer than approximately 1000.

There are two possible routes to enhancing accuracy. As expected, both routes lead to models of somewhat larger size.The first route is to increase the number of normal modes and local characteristic constraint modes which are kept in thetransformation matrix. Typically, these are the modes from a frequency range which is roughly twice as broad as thefrequency range of interest. If the global accuracy obtained is not sufficient, then these modes are selected from an evenbroader range. The second route is to choose a distinct substructuring. That may involve choosing the same number ofcomponents or more, depending on the nature of the parametric variations considered. If many components are defined,then it is likely that the number of modes kept in the transformation matrix for each component can be smaller. If fewcomponents are defined, then it is likely that the number of modes kept in the transformation matrix has to be larger.Hence, there is a tradeoff between number of components and number of modes for the same level of accuracy. Thistradeoff affects the size of the resulting NX-PROM and depends on the nature of the parametric variations considered.A method like that proposed by Bennighof and Lehoucq [27] may provide an optimal tradeoff. They developed anautomated multilevel substructuring method to achieve a high level of dimensional reduction, locally and inexpensively,while balancing the errors associated with truncation and the finite element discretization. Such balancing is a route forautomatically defining components and selecting modes.

35 40 450

0.050.1

0.150.2

0.250.3

0.350.4

0.450.5

Frequency (Hz)

Dis

plac

emen

t (m

m)

NominalCase 1 for full−orderCase 1 for NX−PROM without LIRCase 1 for NX−PROM with LIR

35 40 450

0.050.1

0.150.2

0.250.3

0.350.4

0.450.5

Frequency (Hz)

Dis

plac

emen

t (m

m)


35 40 450

0.050.1

0.150.2

0.250.3

0.350.4

0.450.5

Frequency (Hz)

Dis

plac

emen

t (m

m)


Fig. 14. Forced response predictions of the L-shaped plate for the nominal structure and for cases 1–3 computed using full-order models, NX-PROM, and

NX-PROM with LIR. (a) Case 1, (b) Case 2 and (c) Case 3.

Fig. 15. System-level and substructure-level finite element models of the frame of a high mobility multipurpose wheeled vehicle.


7. Conclusions and discussion

The key contributions of this paper are as follows. The proposed next-generation parametric reduced-order models(NX-PROMs) were developed by using a new parameterization technique to capture the element-level nonlinearity due tovolume variations of finite elements of brick or other types. In addition, to establish a mathematically stable formulationfor NX-PROMs, a new transformation matrix was developed using a novel interpolation of static constraint modes. Finally,a local-interface reduction (LIR) technique was proposed for further enhancing the computational efficiency of theNX-PROMs.

∆t

CR

CL

Fig. 16. Nominal and re-designed cross-bar composed of CL and CR.

Table 3Thickness variations in substructures CL and CR of the HMMWV frame.

Substructure Thickness, case 1 Thickness, case 2

CL 5 mm-20 mm 5 mm-32 mm

CR 5 mm-20 mm 5 mm-32 mm

145 150 1550

1

2

3

4

5

6

7

8x 10−4

Frequency (Hz)

Dis

plac

emen

t (m

m)

NominalFEMNX−PROMMC−PROM

145 150 1550

1

2

3

4

5

6

7

8x 10−4

Frequency (Hz)

Dis

plac

emen

t (m

m)

NominalFEMNX−PROMMC−PROM

Fig. 17. Forced response predictions for the HMMWV frame for the nominal structure and for cases 1 (a) and 2 (b).

Table 4Computational time for full-order and NX-PROM calculations of natural frequencies of a structure with two cases of parameter variations.

Case of variations Mode Full-order (Hz) Time (s) NX-PROM (Hz) Time (s) Error (%)

Case 1 1 24.353 17,650 24.353 16.7 0.0

2 45.491 45.493 0.003

3 52.141 52.146 0.008

4 54.804 54.809 0.007

5 56.140 56.144 0.006

Case 2 1 24.268 17,710 24.267 17.2 0.0

2 45.492 45.491 0.0004

3 52.143 52.139 0.003

4 54.793 54.789 0.007

5 56.047 56.044 0.006


Novel, next-generation parametric reduced order models (NX-PROMs) for predicting the vibration response of complexstructures have been presented. These models address two main drawbacks of MC-PROMs. The first is that theparameterization techniques used in MC-PROMs cannot capture the thickness variation of brick and other types of finite

Table 5Interfaces between substructures in the HMMWV model.

Interface Substructure

1 2 3 4 5 6 7 8 9 10

G1 O O

G2 O O

G3 O O

G4 O O

G5 O O

G6 O O

G7 O O

G8 O O

G9 O O

G10 O O

G11 O O

G12 O O

G13 O O

G14 O O

145 150 1550

1

2

3

4

5

6

7

8x 10−4

Frequency (Hz)

Dis

plac

emen

t (m

m)

FEM: 1,473NX−PROM w/ LIR: 612NX−PROM w/ LIR: 814NX−PROM w/ LIR: 939NX−PROM w/ LIR: 1,047



Case 1

145 150 1550

1

2

3

4

5

6

7

8 x 10−4

Frequency (Hz)

Dis

plac

emen

t (m

m)

Case 2

Frequency (Hz)15

0.5 151

151.5 15

215

2.5 153

153.5 15

415

4.5

3

4

5

6

78 x 10−4

Dis

plac

emen

t (m

m)

Case 1

151

151.5 15

215

2.5 153

153.5 15

415

4.5

2.5

3

3.5

4

4.5

5

5.5

6

6.5x 10−4

Frequency (Hz)

Dis

plac

emen

t (m

m)

Case 2


Fig. 18. Forced response predictions for the HMMWV frame obtained using NX-PROMs with LIR for different levels of reduction; the total number of

DOFs obtained for each model using LIR are indicated for case 1 (left) and 2 (right).


elements due to element-level nonlinearity of the stiffness matrix. The second drawback is that the transformation matrixfor MC-PROMs is not numerically stable. Thus, a new parameterization technique was developed to capture thenonlinearity of the stiffness matrix, and a new transformation matrix was proposed to make the NX-PROMs more stablenumerically and more accurate compared to MC-PROMs.

To reduce the interface DOFs, a new method called local-interface reduction (LIR) was developed. NX-PROMs weredeveloped for realistic substructural analysis. In such cases, the interface DOFs should be reduced before the system-levelmatrices are constructed. The LIR technique uses characteristic constraint modes computed for each substructure by usingthe constraint partition of the reduced mass and stiffness matrices constructed by CB-CMS or the NX-PROM approach.


By using these characteristic constraint modes, orthogonal bases were defined to reduce the interface DOFs of eachsubstructure. That is a key advantage of this reduction technique, and it is very useful for substructural analysis.

Similar to MC-PROMs, the novel NX-PROMs also provide smaller system matrices and shorter analysis and reanalysistime to predict the vibration response of complex structures. This means that NX-PROMs are especially useful for therepetitive analyses needed in optimization problems where geometric changes are applied in the design cycle forstructures modeled with brick and other types of finite elements.

Acknowledgment

The authors gratefully acknowledge the financial support of the Automotive Research Center, a US Army Center ofExcellence for Modeling and Simulation of Ground Vehicles led by the University of Michigan.

Reference herein to any specific commercial company, product, process, or service by trade name, trademark,manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring bythe United States Government or the Department of the Army (DoA). The opinions of the authors expressed herein do notnecessarily state or reflect those of the United States Government or the DoA, and shall not be used for advertising orproduct endorsement purposes.

UNCLASSIFIED: Dist A. Approved for public release.

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Next-generation parametric reduced-order models