Component-centric reduced order models (ROMs) are introduced here as small-size ROMs providing an accurate prediction of the linear response of part of a structure (the β component) without focusing on the rest of it (the α component). Craig–Bampton (CB) substructuring methods are first considered. In one method, the β component response is modeled with its fixed interface modes while the other adopts singular value eigenvectors of the β component deflections of the linear modes of the entire structure. The deflections in the α component induced by harmonic motions of these β component modes are processed by a proper orthogonal decomposition (POD) to model the α component response. A third approach starts from the linear modes of the entire structure which are dominant in the β component response. Then, the contributions of other modes in this part of the structure are approximated in terms of those of the dominant modes with close natural frequencies and similar mode shapes in the β component, i.e., these nondominant modal contributions are “lumped” onto dominant ones. This lumping permits to increase the accuracy in the β component at a fixed number of modes. The three approaches are assessed on a structural finite element model of a nine-bay panel with the modal lumping-based method yielding the most “compact” ROMs. Finally, good robustness of the ROM to changes in the β component properties (e.g., for design optimization) is demonstrated and a similar sensitivity analysis is carried out with respect to the loading under which the ROM is constructed.

Introduction

Modal analysis is a very efficient tool to predict the linear response of complex structures in the low-frequency band. The LF range is characterized by a typically small modal density and relatively well isolated resonances for which modal reduction methods [14] are particularly well suited. The modes on which they are based are typically extended to the entire structure so that a global approximation of the response is constructed. This property of the modal approach can be very desirable as it permits the analysis of the entire structure in one pass and fully accounts for the coupling between the substructures it may involve. Yet, this global approach may also: (i) lead to a large number of modes and (ii) not be efficient when focusing on specific parts of the structure. The design of aircraft frames for example is not accomplished monolithically but rather by considering separate components (e.g., panels) and modeling their interactions.

To support such a localized design process, it is desirable to develop compact (i.e., with a small number of basis functions) reduced order models (ROMs), referred to as “component-centric” here, that lead to a close prediction of the response in one component (referred to as “the β component”) of the structure without necessarily providing an equally good prediction in the rest of it (referred to as “the α component”). The focus of this paper is then on the construction of such component-centric ROMs for linear structures, especially those made of stiffened panels. Note that the ROMs corresponding to different parts of the structure could be run in parallel and thus, could provide a prediction of the response of the entire structure faster than by using a single monolithic model of it.

The definition of the problem seems to call for the use of component mode synthesis (CMS) techniques and indeed two strategies based on the Craig–Bampton (CB) approach [5] will be presented. Because of the large number of boundary degrees-of-freedom that they typically induce, even after appropriate reduction, current CMS strategies appear to be ill-suited for the present setting. A third alternative will also be formulated which relies directly on the linear modes of the structure and optimizes their use for the representation of the response of the β component without consideration of its α counterpart.

While the formulation of the component-centric ROMs is described and validated here only in the context of linear structures, the low number of basis functions found necessary makes them particularly attractive for nonlinear geometric ROMs ([68] and references therein). This issue is thus briefly discussed and the potential of component-centric ROMs in the nonlinear geometric case confirmed.

Validation Model: Multibay Structure

The nine-bay fuselage sidewall panel of Refs. [9] and [10], was considered for the validation of the component-centric ROMs. It is a representative example of the class of structures for which a component-centric ROM would be desirable as it exhibits well delineated components, i.e., the nine different bays. As described in Ref. [9], the overall dimensions of the panel are 58.1 in by 25.06 in and it is subdivided into nine geometrically identical bays riveted by a frame and longeron substructure. Each bay measures 18.75 in by 7.5 in between rivet lines and is identified by its number, 1–9, as shown in Fig. 1. The Nastran finite element model consists of quadrilateral plate elements (referred to as CQUAD4) and beam elements (referred to as CBEAM) and has a total of 96,156 degrees-of-freedom. The selected material properties are: Young's modulus of 10.5 × 106 psi, Poisson's ratio equal to 0.33, and a density of 2.614 × 10−4 lbf s2/in4. Finally, the edges of the skin are simply supported. Further information regarding this model can be found in Refs. [9] and [10]. In these references, the bays have equal thickness of 0.05 in and thus, the overall model is nearly symmetrical with the dissymmetry arising only from the frame and the longeron configuration. This property implies that the response of the panel to a uniform pressure is dominated by only a limited set of modes versus the entire set. To promote richer dynamics, the nine bays were taken here to have different thicknesses selected as (in order of bay, from 1 to 9) 0.0554, 0.0683, 0.0274, 0.0586, 0.0532, 0.0369, 0.0457, 0.0534, and 0.0858 (unit: inch). The thickness of the rest of the skin and the frame substructure was selected as in the original model, i.e., 0.05 in and 0.04 in. The structure was assumed to be subjected to a uniform pressure of unit magnitude on its skin only and, a Rayleigh damping model was adopted, i.e.,

 
C=r1M+r2K
(1)

where C denotes the damping matrix and r1 = 7.55/s and r2  = 5.6 × 10−6 s. This model provides classical damping with damping ratios ranging from 0.65% to 1% of critical in the frequency band of interest.

In most of the ensuing discussion, the β component was selected as bay 4, which was found to be quite representative of all bays, the conclusions derived from its analysis hold for all bays unless explicitly stated. The rest of the structure was then assigned as the α component. Figure 2 shows the magnitude of the transverse displacement frequency response function of the middle point of bay 4. This curve was obtained using the first 85 linear modes of natural frequencies ranging from 68 to 500 Hz; it is nearly indistinguishable from the full finite element solution. The in-plane deflections were found much smaller than their transverse counterparts and thus, the latter were selected as the primary quantities of interest. The curve of Fig. 2 will be referred to as the “baseline” against which the component-centric ROM predictions of Secs. 3 and 4 will be compared.

Several observations can be drawn from Fig. 2 and the list of natural frequencies of Table 1. First, while some natural frequencies are well separated from the others, there are several zones densely populated with frequencies. This occurrence is reflected by either tightly packed sharp peaks or broad peaks in Fig. 2. Second, there is a sharp drop of the response around 350 Hz and thus, the analysis will focus solely on the band [0,350] Hz.

Craig–Bampton-Based Approaches

Craig–Bampton Method.

Since the structure will be decomposed into two parts, α and β, a component mode synthesis framework seems appropriate as a mean to emphasize the latter part over the former one. More specifically, the CB method is adopted here and, for completeness and notational convenience, is briefly reviewed below.

The equations of motion for the undamped substructure s (s representing either α or β) can be written as 
Msüs+Ksus=Fs
(2)
where Ms, Ks, us, and Fs are, respectively, the mass and stiffness matrices and the displacement and force vectors of the substructure in the physical coordinates. The mass and stiffness matrices can be partitioned as 
Ms=[MiiMibMbiMbb]
(3)
and 
Ks=[KiiKibKbiKbb]
(4)
where the subscripts i and b correspond to the interior and boundary degrees-of-freedom of the substructure s. In the CB method, the internal (uis) and boundary (ub ) displacements are represented as 
uis=Φsqis+ΞsY
(5)
 
ub =Y
(6)
where Φs=[φ1sφ2sφns] denotes a matrix of n selected fixed interface modes φk, i.e., with the boundary displacements set to zero. They are derived from the eigenvalue problem 
Kiisφjs=λjsMiisφjs
(7)
with λjs representing the corresponding eigenvalue. Further, in Eq. (5), Ξs denotes the matrix of constraint modes in the substructure s which are determined as the static responses inside the substructure s corresponding to all boundary degrees-of-freedom set to zero except one of them in turn which is then set to a unit value. That is 
Ξs=(Kiis)1Kibs
(8)
Finally, the vector qis of Eq. (5) contains the generalized coordinates of the fixed interface modes. Note in the above discussion that the boundary degrees-of-freedom are common to both substructures. The reduction of variables, from (uis,  ub ) to (qis, Y), is accompanied by the transformation matrix 
Ts=[ΦsΞs0I]
(9)
where I denotes the identity matrix of appropriate dimensions. Then, the reduced CB mass and stiffness matrices can be obtained as 
M̃CBs=(Ts)TMsTs=[M̃qqM̃qYM̃YqM̃YY]
(10)
 
K̃CBs=(Ts)TKsTs=[K̃qq00K̃YY]
(11)

where the superscript T denotes vector/matrix transposition. Note further that both K̃qq and M̃qq are diagonal and the ratios of their components are the eigenvalues of the fixed-interface problem.

Considering the two substructures, the overall equations of motion of the undamped structure are 
[M̃qqαM̃qYα0M̃YqαM̃YY M̃Yqβ0M̃qYβM̃qqβ][q̈iαŸq̈iβ]+[K̃qqα000K̃YY 000K̃qqβ][qiαYqiβ]=[FiαFb Fiβ]
(12)
A known aspect of the original CB method, i.e., Eqs. (5) and (6), is the much larger size of the reduced model as compared to the linear modes, primarily due to the often large number of boundary degrees-of-freedom. As discussed in Ref. [13] for the nine-bay panel with identical bays, the number of degrees-of-freedom in the CB model is very much larger than the 50 linear modes of the structure in the band of interest, [0,350] Hz. This situation may be improved by the secondary reduction of the boundary degrees-of-freedom using characteristic constraint modes [11,12] resulting [13] in 232 degrees-of-freedom for the nine-bay panel with identical bays. Two different secondary reductions are discussed below in Secs. 3.2 and 3.3.

Component-Centric Reduced Order Model Based on β-Fixed Interface and α-Proper Orthogonal Decomposition Modes.

A potential improvement of the large size ROM issue may result here from the condition that only the β component needs to be modeled accurately. Then, the mode reduction should extract only the modes in α that have significant effects on the response in the β component. Given reciprocity, these modes can be obtained as the responses in α due to imposed β motions. With that perspective, the modal basis was determined as follows:

  • (i)

    the fixed interface modes of the β component were assumed to vary harmonically one at a time and for all frequencies in the range of interest,

  • (ii)
    the damped response of the α component and the boundary were determined from Eq. (12) without additional forces on them, i.e., 
    [M̃qqαM̃qYαM̃YqαM̃YY ][q̈iαŸ]+[C̃qqαC̃qYαC̃YqαC̃YY ][q˙iαY˙]+[K̃qqα00K̃YY ][qiαY]=[0M̃Yqβq̈iβ]
    (13)
  • (iii)

    a proper orthogonal decomposition (POD) analysis of the responses obtained in (ii) was carried out to extract the dominant components (see Ref. [14] for a recent review), and finally

  • (iv)

    the component-centric ROM was built with the fixed interface modes of the β component (referred to as “β-fixed”) and the POD modes for α and the boundary (referred to as “α-POD”).

Note that the singular value decomposition (SVD) was used for all POD analyses to avoid the loss of precision when forming the covariance matrix [14]. The assessment of the above procedure was carried out by first comparing the magnitude of the frequency response of the β component to its baseline. Figure 3 provides this comparison at the middle point of the β component with six β-fixed and 30 α-POD modes. The excellent matching with the baseline curve indicates that this method does provide a good prediction of the response in the β component with enough α-POD modes.

To obtain a more global representation of the matching over the two components, a representation error was defined as 
representationerror=normofresponseerrornormofresponse×maximumresponse
(14)

which was computed in both components (α and β) and at each frequency. Shown in Fig. 4 are the representation errors in α and β components with the six β-fixed and 30 α-POD modes component-centric ROM. They confirm the good match seen in Fig. 3 and demonstrate that the modeling of the β component is better than the one for the α component as could be expected. However, 36 modes are necessary while a similar prediction can already be achieved with as few as 21 linear modes (see Sec. 4.1). Moreover, reducing the number of α-POD modes degrades rapidly the accuracy: the ROM built with six β-fixed and 20 α-POD modes fails to capture some of the peaks, see Fig. 5. It is concluded that this method provides a good matching of the response in the β component but does not appear to lead to a very compact ROM.

Component-Centric Reduced Order Model Based on β-POD and α-POD Modes.

The large ROM size required in Sec. 3.2 could be attributed in part to the use of the fixed interface modes of the β component which are not representative of the actual motions in that component. Further, to each mode in the β component is associated a series of α-POD modes forcing the model size to be larger than necessary. To try to mitigate this situation, a different representation of the β component response was sought. Specifically, the deflections of the β component of the linear modes in the [0,350] Hz band were first extracted and then, a POD analysis of the snapshot matrix composed of these responses was carried out. These computed POD modes are referred to as “β-POD” modes in the sequel and were used to generate an associated basis in the β component following a process similar to the one of Sec. 3.2.

A notable difference between β-POD and β-fixed interface modes is however that the degrees-of-freedom of the interface are included in the β-POD modes versus separate degrees-of-freedom in the previous formulation. Then, the imposition of the motions in β to obtain the α-POD modes is actually achieved by prescribing the motion of the entire boundary, i.e., the degrees-of-freedom Y and computing the corresponding response in α according to Eq. (13) or 
M̃qqαq̈iα+C̃qqαq˙iα+K̃qqαqiα=M̃qYαŸ
(15)

Shown in Fig. 6 is the comparison of the magnitudes of the frequency responses at the middle point of the β component obtained with this 30 α-POD and ten β-POD basis component-centric ROM and the baseline model. The excellent matching indicates the convergence of the prediction of the response in the β component with enough modes. The representation errors in α and β components shown in Fig. 7 confirm the finding and indicate that a better modeling of the β component than of the α one is achieved. These observations support the suitability of this approach as a component-centric ROM. However, the method suffers from the same large size issue as the previous one: 40 modes are used for Figs. 6 and 7. Reducing the number of either α-POD or β-POD modes degrades the accuracy of the predictions as seen in Figs. 8 and 9.

Linear Modes Selection and Lumping

Analyzing the results obtained by the two previous methods suggests that the large ROM size issue originates from the large number of α-POD modes needed, i.e., from the difficulty in modeling the coupling between the α and β components. This modeling is however achieved naturally with the linear modes suggesting that they could be the foundation for a compact component-centric ROM. Seeking a reduction of the number of modes from the 50 existing in [0,350] Hz, it is first observed in Fig. 2 that there are only 16 separate peaks versus the 50 expected. There are two reasons for the lower number of peaks. First, some of the linear modes have very small modal amplitudes resulting from the uniform pressure loading. Second, some peaks have merged with each other to form single broader peaks. Based on these observations, the third approach to construct a compact component-centric ROM starts from the linear modes in the band (all 50) and proceeds with two successive reductions of the modal basis: the linear mode selection and modal lumping which are described in Secs. 4.1, 4.2, and 4.3, respectively.

Linear Mode Selection.

The first step of the modal basis reduction process is the careful selection of the linear modes based on their contribution to the response in the β component. This process is performed recursively as follows starting with the baseline model (including all modes) and its frequency response in the band:

  • (a)

    Compute the response in the β component when one of the linear modes from the current model is removed and evaluate the corresponding norm error with respect to the predictions of the current model in the β component,

  • (b)

    repeat the computations of (a) for each mode of the current model in turn and determine which of the modes induced the smallest error,

  • (c)

    remove the mode with the smallest error from the current model to obtain a new, smaller size current model, and

  • (d)

    repeat steps (a)–(c) until only one mode is left in the model.

Note that the “importance” of the linear modes should be opposite to the elimination order (e.g., the linear mode left at the end should be the most important one). The above recursive method provides a rigorous way to find the modes that contribute most to the response in the β component, but it requires the computation of a series of full solution. However, a much more practical strategy to determine these dominant modes can be based on the response estimate Ri derived from mode i maximum response in the β component, i.e., 
Ri=|Fi2ωi2ζi|max(|Uβi|)
(16)

where Fi, ωi, ζi are the modal force, natural frequency, and damping ratio associated with the linear mode i. This mode has values Uβi in the β component and their maximum (in absolute value) is max(|Uβi|). Shown in Tables 2 and 3 are the 21 most important linear modes predicted, respectively, from the recursive algorithm described above and from Eq. (16), i.e., those with the largest values of Ri. Comparing them, it is seen that the 14 most dominant modes are identical, i.e., correctly predicted by Eq. (16), with some switches of order occurring on the following ones due, most likely, to the lack in Eq. (16) of coupling effects between two modes of close natural frequencies. The frequency response functions of the middle point of bay 4 predicted by the two 21 modes models of Tables 2 and 3 are compared to each other and to the baseline in Fig. 10. An excellent match between the three curves is observed demonstrating that: (i) the number of modes necessary to achieve a close match of the frequency response in the β component is much smaller than the total number of modes in the band for a given loading (see Sec. 5.2 for further discussion) and (ii) the two selections algorithms, the recursive one and the one based on Eq. (16), are effectively equivalent in identifying the modes that matter.

Linear Mode (Direct) Lumping Method.

The presence of stiffeners leads to two particular properties. First, it induces a partial localization of higher modes and to the appearance of groups of modes with close natural frequencies, as discussed in Sec. 2, see Table 1 and Fig. 2. Second, the displacement field of a particular bay is often similar in many of these modes, especially those with close natural frequencies. For illustration, the five linear modes 8–12 with close frequencies (between 127 Hz and 136 Hz) are plotted in Figs. 11 and 12. It is clearly seen that the five mode shapes in each bay are most of the time similar. This phenomenon provides a potential to further reduce the number of modes for the ROM by “lumping” these linear modes together—that is, by approximating the contributions of these five close frequencies modes in the β component only by that of a single one of them or two if needed.

Formally, modal lumping will refer here to as approximating in the β component the sum of two or more modal contributions by that of a single mode. For the approximation to be accurate, the two or more modes must have: (i) close frequencies and (ii) similar modal deflections in the β component. Stated differently, modes that can be lumped would be those which would be difficult to differentiate/identify if a modal testing of the structure was carried out and only the response in the β component was captured. To clarify the lumping process, let the displacement field in the β component be represented as 
uβ(t)=q1(t)Uβ1+q2(t)Uβ2++qi(t)Uβi+qj(t)Uβj++qn(t)Uβn
(17)
where Uβ=[Uβ1 Uβ2  Uβi Uβj  Uβn] are the displacements in the β component of the mass normalized linear modes (e.g., those selected in Sec. 4.1) and q1,q2, , qn are the corresponding generalized coordinates. Next, assume that Uβi and Uβj are the modes that can potentially be lumped, i.e., their mode shapes in the β component are very similar so that 
UβjaUβi
(18)

where the scalar a may be estimated as a=(Uβj)TUβi/Uβi2.

Introducing the approximation of Eq. (18) in Eq. (17), it is found that 
uβ(t)=q1(t)Uβ1+q2(t)Uβ2++q¯i(t)Uβi++qn(t)Uβn
(19)
where 
q¯i(t)=qi(t)+aqj(t)
(20)
Consider next the equations of motion for the generalized coordinates  qi(t) and qj(t), i.e., 
q̈i+2ζiωiq˙i+ωi2qi=Fi
(21)
and 
q̈j+2ζjωjq˙j+ωj2qj=Fj
(22)
where Fi and Fj are the corresponding modal forces. Assuming that the mode j has both natural frequency and damping ratio close to those of mode i, Eq. (22) can be approximated as 
q̈j+2ζiωiq˙j+ωi2qj=Fj
(23)
which can be combined with Eq. (21) to yield 
q¯̈i+2ζiωiq¯˙i+ωi2q¯i=Fi+aFj
(24)

Thus, the sum of the contributions of the two modes i and j has been approximated by that of a single one (mode i) in both time, Eq. (24), and space, Eq. (19).

The lumping process, and the resulting equations (19) and (24), can also be recognized as originating from a rotation of the two modes Ui and Uj, defined in the entire structure (i.e., in both α and β components) as 
Ui=[UαiUβi]andUj=[UαjUβj][UαjaUβi]
(25)
Since the natural frequencies of these two modes are assumed to be identical, the structure has repeated frequencies (assumed two here) and thus any linear combination of these two modes is also a mode. On this basis, introduce the two new modes 
Ũi=(Ui+aUj)/1+a2Ũj=(aUiUj)/1+a2
(26)
which are readily shown to exhibit the same orthonormality properties as the original modes Ui and Uj, i.e., 
(Ũi)TM Ũi=1;(Ũj)TM Ũj=1;(Ũi)TM Ũj=0;(Ũi)TM Ul=0;(Ũj)TM Ul=0
(27)
where M denotes the mass matrix of the entire structure and the index l of the mode Ul is such that l ≠ i or j. Introducing Eq. (25) in Eq. (26), it is found that 
Ũi=[Uαi+aUαj(1+a)Uβi]/1+a2Ũj=[aUαiUαj0]/1+a2
(28)

Note that the elements in Ũj corresponding to the β component are all zero. Thus, the displacement in that component is not a function of the generalized coordinate q̃j associated with the mode Ũj. Moreover, this generalized coordinate is uncoupled to q̃i and the remaining generalized coordinates ql, l ≠ i or j, because of the orthogonality properties of Eq. (27). Thus, the value/time history of q̃j does not appear anywhere in the prediction of the response of the β component (it will however appear in the displacement of the α component which is not of concern here) and mode j has indeed been lumped into mode i. Note finally that the governing equation for the generalized coordinate q̃i can be shown to reduce to Eq. (26) with q¯i=q̃i1+a2 so that the above two perspectives on lumping are consistent. Figure 13 illustrates the rotation of the mode shapes for the two linear modes 10 and 11, see Figs. 11(c), 11(d), and 12, comparing the original linear modes, Ui and Uj to the rotated ones, Ũi and Ũj. As shown in the two top right figures, the mode after rotation, Ũj, has nearly zero deformation in β.

To optimize the accuracy of the approximation, the lumping of two modes should be done on the mode of largest modal coordinate and/or the one with the most representative deflection in the β component of the modes to be lumped. Moreover, the lumping of multiple modes on one of them proceeds as described above with two modes under the same assumptions of close natural frequencies and similar modal deflection in the β component for all modes involved.

As a first example of application of the lumping process, consider bay 4 and note from Table 2 that modes 8–12 are all listed in the 21 most important modes. They exhibit close natural frequencies, between 127 and 136 Hz, and very similar modal deflection in the β component, see Fig. 12. They are thus prime candidates for modal lumping. Of these five modes, mode 11 is the largest contributor to the response in the β component and thus, was selected as the mode on which the other four were lumped. Shown in Fig. 14 are the magnitudes of the frequency responses of the middle point of bay 4 computed using (a) the 21 selected modes, (b) the 17 modes remaining after the elimination of modes 8, 9, 10, and 12 from the 21 selected ones, and (c) the 17 modes model of (b) with modes 8, 9, 10, and 12 lumped onto mode 11. The magnitude of the response obtained with the latter 17 modes model clearly matches much better the 21 selected modes predictions than the 17 modes model of (b). In fact, the matching of the lumped modes model frequency response function with its counterpart from the 21 selected modes is so close that it can be substituted for it leading to a reduction of the number of modes in the ROM from 21 to 17.

To further demonstrate the applicability and benefits of the lumping method, shown in Fig. 15 are the frequency responses of the middle point of bay 1 (assumed as the β component) with the largest 20 selected linear modes and a 20-mode model including these modes as well as the lumping of six other modes (modes 9, 10, 12, 15, 22, and 42) of frequencies 129.01, 130.43, 135.43, 143.17, 195.42, and 296.87 Hz. The contributions of the first three modes were lumped on the one of mode 11 of frequency 132.00 Hz while those of the last three modes were lumped with those of modes 14, 21, and 40 of frequencies equal to 142.23, 193.86, and 292.14 Hz. As shown, the lumping method improves notably the prediction of the response.

Modal Lumping Optimization.

In the lumping approach described above, referred to as “direct lumping” in the sequel, the resulting natural frequency and damping ratio are those of the dominant mode, see Eq. (24). Further improvements of the matching of the resulting frequency response function with its baseline can be sought by optimizing these two parameters as well as the modal force. Two optimization options have been investigated.

  • (A) A frequency domain optimization in which the natural frequency, damping ratio, and modal force are sought to minimize the L2 norm difference of the frequency responses of the model with the lumped modes and the baseline model over the frequency band of the lumped modes natural frequencies. The frequency response corresponding to each linear mode is obtained by solving Eq. (21) and the summation of these solutions forms the approximation of the actual structural response, i.e., for two modes 
    H(ω)=Fp(ωp2ω2)+2iζpωpω+a Fq(ωq2ω2)+2iζqωqω
    (29)
    where p and q represent the indices of the linear modes being lumped, a=(Uβq)TUβp/Uβp2, and i is the imaginary unit (i2=1). Then, the natural frequency ωˇ, damping ratio ζˇ, and modal force Fˇ of the optimum lumped mode will be selected to minimize 
    ωpδωq+δ||H(ω)||Fˇ(ωˇ2ω2)+2iζˇωˇω||dω
    (30)
    where δ  > 0 specifies the range of frequencies to be considered outside of the domain of the natural frequencies of the two modes to be lumped. Initial guesses for ωˇ, ζˇ, and Fˇ were selected as 
    ω0=(ωp+ωq)/2;ζˇ0=(ζp+ζq)/2;Fˇ0=Fp+aFq
    (31)
    with the initial guess on the modal force consistent with the direct lumping approach. The selection of ωˇ0 as the average of the natural frequencies stems from the expectation that ωˇ would lie between ωp and ωq.

    The above procedure was applied first to bay 4, to the lumping of the contributions of modes 8, 9, 10, and 12 onto the one of mode 11. The optimization was carried out numerically using the optimizer fminsearch in matlab. The average of the natural frequencies of the five modes is 131 Hz and the damping ratios of these modes range from 0.0068 to 0.0069. Moreover, the parameter δ was chosen equal to 5 Hz. The convergence result shows that the optimal natural frequency is 132 Hz and the damping ratio is 0.0095. The frequency response function of the middle point of bay 4 corresponding to this approximation is shown in Fig. 14. Comparing it to the direct lumping and the 21 selected modes curves, it is seen that the optimization does provide an improved matching of the baseline frequency response over the direct lumping method.

  • (B) An identification-based approach in the time domain was considered next. In this strategy, the responses of the modes to be lumped are combined in the time domain to simulate “measured data” (e.g., impulse response) to which a single degree-of-freedom (“1DOF”) model is fit using an autoregressive (AR) modeling, Fig. 16 illustrates the process.

The simulated measured data x(n) is assumed here to be the impulse response of a representative point of the β component constructed only from the set of linear modes to be lumped, i.e., assuming the two modes p and q as before 
x(n)=FpωdpeζpωpnΔtsin(ωdpnΔt)+a FqωdqeζqωqnΔtsin(ωdqnΔt)
(32)
where a=(Uβq)TUβp/Uβp2 provides the scaling of mode q in comparison to mode p as in Eq. (29). Further,  ωdp and  ωdq are the damped circular frequencies of modes p and q defined as 
ωdp=ωp1ζp2;ωdq=ωq1ζq2
(33)

Figures 17 and 18 show the simulation data constructed for bay 4 with the linear modes 8–12. As expected, this curve is more complex than the response of a 1DOF system and thus, the time window over which this data is modeled will affect the estimates of the natural frequency and especially of the damping ratio. For the simulated measured data to appear as originating from a 1DOF system, the observation time must be small enough that the difference in frequencies between the various modes is not perceived. Then, the modal contributions will effectively be lumped. This window must however be long enough to get an accurate estimate of the frequency and damping ratio. These conflicting requirements led to the use of 1.5 period of the average frequency of the modes to be lumped. Finally, the window was started at the peak of the data so that the decay of the response be visible in the short analysis window. For the simulated measured data of Figs. 17 and 18, the analysis window was thus selected from 0.0668 s to 0.0782 s.

Modeling the free response of Fig. 18 by that of a 1DOF system was carried out through an AR fit of the data. That is, the data was modeled according to the recursion [1517] 
x(n)=iMaix(ni)+ε(n)
(34)

where x(n) is the current value of the time series, ai are predictor (weighting) coefficients, M is the model order, indicating the number of the past values used to predict the current value, and ε(n) represents a one-step prediction error, i.e., the difference between the predicted value and the current value at this point.

Since the free response of a 1DOF data is exactly representable in the form of Eq. (34) with M = 2 [15,16], this value was adopted; leading to the model 
x(n)+a1x(n1)+a2x(n2)=ε(n)
(35)
The coefficients a1 and a2 were obtained by the correlation approach [15] seeking the minimum of n[ε(n)]2. The natural frequency  ωˇ and damping ratio ζˇ (assumed less than 1) could then be estimated by equating the free responses of the AR and 1DOF models. The latter is of the form eλt, sampled at t=nΔt, where 
λ=ζωˇ±iωˇ1ζ2
(36)
The free response of the AR model of Eq. (35) is of the form ρn, where ρ is any of the two solutions of the quadratic equation 
ρ2+a1ρ+a2ρ=0
(37)
Then, matching the two free responses implies that ρ=eλΔt from which it is found that [16,17] 
ωˇ=(ln(|ρ|)Δt)2+(arg(ρ)Δt)2
(38)
 
ζˇ=ln(|ρ|)ωˇΔt
(39)
Finally, the equivalent modal force Fˇ is determined by matching the signal energy of the simulated measured data of Eq. (32) and its 1DOF approximation Fˇx(n), where, 
x(n)=1ωdeζωnΔtsin(ωdnΔt)
(40)
where ωd=ω1ζ2. That is 
Fˇ=sign(Fb)xxˇ
(41)

The above procedure was first applied to bay 4 and the lumping of modes 8–12. The time step was chosen as Δt=π/(30ω¯), where ω¯ is the average of the natural frequencies of the modes to be lumped. Shown in Fig. 14 is the resulting approximation of the the magnitude of the frequency response of the middle point of bay 4. It is seen that the AR identification approach of (B) yields very similar results as the frequency optimization of (A).

Another assessment of the two lumping methods (A) and (B) was carried out on bay 1 where the contributions of modes 9, 10, 12, 15, 22, and 42 were lumped onto those of modes 11, 14, 21, and 40 as performed earlier with the direct lumping method. Figure 15 is the corresponding comparison of the frequency responses. As in Fig. 14, it is seen from Fig. 15 that the full optimization approach (A) provides an improved matching of the baseline frequency response over the direct lumping method. Moreover, the AR identification approach of (B) again yields very similar results to those obtained with the frequency optimization method (A) except in the neighborhood of 325 Hz. A similar, although clearer, perspective on the benefits of the lumping optimization methods can finally be obtained from the mean representation error, see Table 4. Given its lower computational cost, the AR identification approach (B) appears to be the best tradeoff between accuracy of the resulting model and computational effort.

Component-Centric Reduced Order Model Benefits and Robustness

Computational Benefits.

As discussed in Sec. 3, the Craig–Bampton-based approaches of Secs. 3.2 and 3.3 only provide a marginal reduction of the number of modes necessary, say 30–35 for bay 4 versus the consideration of all 45 active (i.e., with a non-negligible modal force) linear modes in the band. On the contrary, the modal lumping-based approaches lead to much more compact ROMs: less than 20 modes are sufficient to obtain excellent approximations of the frequency response functions of the middle points of bay 4, see Fig. 14, and bay 1, see Fig. 15. In fact, this observation is true for all bays as can be seen from Table 4 (results for bays 2, 3, 5–9 not shown here for brevity but available in Ref. [18]).

The reduction of the number of modes observed in Table 5, by 58–67% depending on the bay analyzed, is potentially quite valuable for the linear structures considered here but is likely to have a dramatic impact when considering the nonlinear geometric ROMs developed within the last decade (see Refs. [68] and references therein). The computational cost associated with such ROMs can be split into the costs of running the ROM and of identifying its linear, quadratic, and cubic stiffness coefficients. Denoting by N the number of modes, the former cost is proportional to the number of coefficients, i.e., approximately N4/6, while the latter cost is as small as N2/6 when using the tangent stiffness approach of Ref. [8]. These costs are shown in the last two columns of Table 5 with N = 2 M since the representation of the full displacement field in the nonlinear case requires more modes than in the linear one, often of the order of twice as many (see the dual modes discussion of Refs. [68]). These numbers clearly demonstrate the advantages of reducing the number of modes present in the model, even at the cost of some preliminary computations, i.e., the mode selection and the modal lumping processes, especially when parallel computations can be carried out to identify and run the component-centric nonlinear geometric ROMs. More details on these nonlinear ROMs is presented in Refs. [18] and [19].

Robustness Assessment.

The construction of component-centric ROMs by the modal lumping approaches, see Sec. 4, relied on a known loading and on the occurrence of close frequencies of a particular set of modes (those being lumped). These conditions may change if the environment varies and/or the structure is modified, e.g., as would happen in the design process. Accordingly, it is desirable to assess the sensitivity of the response predictions accuracy to variations in both the loading and/or the structural properties of the β component.

These assessments were carried out with the 17 mode ROM of bay 4 obtained with the AR lumping approach, see Sec. 4.3(B) and Fig. 14 or 19(a). First, the uniform loading was retained but the thickness of bay 4 was reduced (see Fig. 19(b)) or increased (see Fig. 19(c)) by 20%. The modal basis used for these computations, including the lumping details, was the same as for the original thickness but the ROM mass and stiffness matrices were computed for each thickness by projecting the corresponding finite element mass and stiffness matrices on the modes. Accordingly, they were full for the latter modified cases versus diagonal for the original model. While the accuracy of the predictions is somewhat degraded, the ROM still captures well the β component dynamics suggesting that this component-centric ROM could be used in an optimization effort without having to reselect the modal basis.

To assess the sensitivity to the loading, a triangular pressure load varying from zero to full value along the x-axis (long side of the panel) was applied to the ROM of bay 4 constructed with the uniform loading. Then, shown in Fig. 19(d) is the resulting prediction of the frequency response of the middle point of bay 4 as compared to a full modal solution. With the exception of the peak around 101 Hz, the matching is still excellent. The large increase of the response at that frequency results from the symmetry breaking of the triangular loading which excites particularly that mode (mode 3). This observation suggests that the loading used to develop the component-centric ROM should either closely approximate the one that the structure will be subjected to, if known, or otherwise not exhibit a particular symmetry that may induce a zero modal force in some important modes. If mode 3 had been included in the original lumping process to form an 18 mode model based on a uniform loading, the prediction would have been excellent for the triangular load as is shown on Fig. 19(e).

Summary

This paper focused on developing compact ROMs that provide an accurate prediction of the response of part of the structure considered without focusing on the rest of it. Three strategies were presented and discussed to construct the basis of these ROMs, referred to as component-centric ROMs. The first two approaches are both based on the Craig–Bampton substructuring technique and start with a set of modes for the component of interest (referred to as the β component). The response in the rest of the structure (referred to as the α component) induced by these modes is then determined and optimally represented using a POD strategy. The main difference between the two approaches is the set of modes used for the β component. The first approach relies as usual on the fixed interface modes of the β component while the second one adopts POD modes computed from the deflections in the β component of the linear modes in the frequency band of interest. These first two methods are effectively basis reductions techniques of the CB basis. On the test structure considered, a sizable nine-bay panel model, these methods were found to lead to accurate representations of the responses of the various bays but with ROM sizes only slightly, 20–25%, smaller than the number of linear modes present in the frequency band of interest.

This observation led to the consideration of a third approach which starts from the linear modes of the structure, reducing first this set to those dominant in the β component response (the “selected modes”) and then lumping the contributions of other modes to the ones of the dominant modes of close frequencies and similar mode shapes in the β component. This process leads to an increased accuracy for a given ROM number of modes and/or permits a decrease of this number for a given accuracy. Three variations of this modal lumping strategy were discussed and compared with the last one, the autoregressive modeling-based method, providing the best compromise between computational cost and accuracy of predictions. The application of the modal lumping approaches to the various bays of the nine-bay panel led to ROM sizes equal at most to 40% of the number of linear modes of the structure in the frequency band, a potentially worthwhile saving. This reduction in ROM size is a benefit even if the response of the entire structure is required. Indeed, by modeling each critical component of the structure in turn, a single large ROM for the entire structure can be replaced by multiple smaller component-centric ROMs, which are perfectly suited to run in parallel. The computational benefits can thus be measured by the computational effort required for the most complex component. It was noted that the reduced ROM sizes would be particularly important for the nonlinear geometric ROMs developed within the last decade.

Finally, a good robustness of the ROM to changes in the β component properties (e.g., for design optimization) was demonstrated and a similar sensitivity analysis was carried out with respect to the loading under which the ROM is constructed—all of which support the usefulness of the proposed component-centric ROMs.

Acknowledgment

The authors gratefully acknowledge the support of this work by the AFRL-University Collaborative Center in Structural Sciences (Cooperative Agreement No. FA8650-13-2-2347), with Dr. Ben Smarslok as program manager. In addition, the authors wish to express their appreciation to Dr. S.A. Rizzi and Dr. A. Przekop for the use of the nine-bay finite element model.

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