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2 Chapter 7 Vbraton and Optmzaton Analyss of Large-Scale Structures usng Reduced-Order Models and Reanalyss Methods Zssmos P. Mourelatos, Dmtrs Angels and John Skaraks Addtonal nformaton s avalable at the end of the chapter 1. Introducton Fnte element analyss (FEA) s a well-establshed numercal smulaton method for structural dynamcs. It serves as the man computatonal tool for Nose, Vbraton and Harshness (NVH) analyss n the low-frequency range. Because of developments n numercal methods and advances n computer software and hardware, FEA can now handle much more complex models far more effcently than even a few years ago. However, the demand for computatonal capabltes ncreases n step wth or even beyond the pace of these mprovements. For example, automotve companes are constructng more detaled models wth mllons of degrees of freedom (DOFs) to study vbro-acoustc problems n hgher frequency ranges. Although these tasks can be performed wth FEA, the computatonal cost can be prohbtve even for hgh-end workstatons wth the most advanced software. For large fnte element (FE) models, a modal reducton s commonly used to obtan the system response. An egenanalyss s performed usng the system stffness and mass matrces and a smaller n sze modal model s formed whch s solved more effcently for the response. The computatonal cost s also reduced usng substructurng (superelement analyss). Modal reducton s appled to each substructure to obtan the component modes and the system level response s obtaned usng Component Mode Synthess (CMS). When desgn changes are nvolved, the FEA analyss must be repeated many tmes n order to obtan the optmum desgn. Furthermore n probablstc analyss where parameter uncertantes are present, the FEA analyss must be repeated for a large number of sample ponts. In such cases, the computatonal cost s even hgher, f not prohbtve. Reanalyss methods 2012 Mourelatos et al.; lcensee InTech. Ths s an open access artcle dstrbuted under the terms of the Creatve Commons Attrbuton Lcense ( whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted.

3 134 Advances n Vbraton Engneerng and Structural Dynamcs are ntended to analyze effcently structures that are modfed due to varous changes. They estmate the structural response after such changes wthout solvng the complete set of modfed analyss equatons. Several revews have been publshed on reanalyss methods [1-3] whch are usually based on local and global approxmatons. Local approxmatons are very effcent but they are effectve only for small structural changes. Global approxmatons are preferable for large changes, but they are usually computatonally expensve especally for cases wth many desgn parameters. The well-known Raylegh-Rtz reanalyss procedure [4, 5] belongs to the category of local approxmaton methods. The mode shapes of a nomnal desgn are used to form a Rtz bass for predctng the response n a small parametrc zone around the nomnal desgn pont. However, t s ncapable of capturng relatvely large desgn changes. A parametrc reduced-order modelng (PROM) method, developed by Balmes [6, 7], expands on the Raylegh-Rtz method by usng the mode shapes from a few selected desgn ponts to predct the response throughout the desgn space. The PROM method belongs to the category of local approxmaton methods and can handle relatvely larger parameter changes because t uses multple desgn ponts. An mproved component-based PROM method has been ntroduced by Zhang et al. [8, 9] for desgn changes at the component level. The PROM method usng a parametrc approach has been successfully appled to desgn optmzaton and probablstc analyss of vehcle structures. However, the parametrc approach s only applcable to problems where the mass and stffness matrces can be approxmated by a polynomal functon of the desgn parameters and ther powers. A new parametrc approach usng Krgng nterpolaton [10] has been recently proposed [11]. It mproves effcency by nterpolatng the reduced system matrces wthout needng to assume a polynomal relatonshp of the system matrces wth respect to the desgn parameters as n [6, 7]. The Combned Approxmatons (CA) method [12-14] combnes the strengths of both local and global approxmatons and can be accurate even for large desgn changes. It uses a combnaton of bnomal seres (local) approxmatons, called Neumann expanson approxmatons, and reduced bass (global) approxmatons. The CA method s developed for lnear statc reanalyss and egen-problem reanalyss [15-19]. Accurate results and sgnfcant computatonal savngs have been reported. All reported studes on the CA method [12-19] use relatvely smple frame or truss systems for statc or dynamcs analyss wth a relatvely small number of DOF and/or modes. For these problems, the computatonal effcency was mproved by a factor of 5 to 10. Such an mprovement s benefcal for a sngle desgn change evaluaton or even for gradent-based desgn optmzaton where only a lmted number of reanalyses (e.g. less than 50) s performed. However, the computatonal effcency of the CA method may not be adequate n smulaton-based (e.g. Monte-Carlo) probablstc dynamc analyss of large fnte-element models where reanalyss must be performed hundreds or thousands of tmes n order to estmate accurately the relablty of a desgn. A large number of modes must be calculated and used n a dynamc analyss of a large fnte-element model wth a hgh modal densty, even f a reduced-order modelng approach (Secton 2) s used. In such a case, the mplct assumpton of the CA method that the cost of

4 Vbraton and Optmzaton Analyss of Large-Scale Structures usng Reduced-Order Models and Reanalyss Methods solvng a lnear system s domnated by the cost of matrx decomposton way no be longer vald (see Secton 3.4) and the computatonal savngs from usng the CA method may not be substantal. For ths reason, we developed a modfed combned approxmaton (MCA) and ntegrated t wth the PROM method to mprove accuracy and computatonal effcency. The computatonal savngs can be substantal for problems wth a large number of desgn parameters. Examples n ths Chapter demonstrate the benefts of ths reanalyss methodology. The Chapter presents methodologes 1. for accurate and effcent vbraton analyss methods of large-scale, fnte-element models, 2. for effcent and yet accurate reanalyss methods for dynamc response and optmzaton, and 3. for effcent desgn optmzaton methods to optmze structures for best vbratory response. The optmzaton s able to handle a large number of desgn varables and dentfy local and global optma. It s organzed as follows. Secton 2 presents an overvew of reduced-order modelng and substructurng methods ncludng modal reducton and component mode synthess (CMS). Improvements to the CMS method are presented usng nterface modes and fltraton of constrant modes. The secton also overvews two Frequency Response Functon (FRF) substructurng methods where two substructures, represented by FRFs or FE models, are assembled to form an effcent reduced-order model to calculate the dynamc response. Secton 3 presents four reanalyss methods: the CDH/VAO method, the Parametrc Reduced Order Modelng (PROM) method, the Combned Approxmaton (CA) method, and the Modfed Combned Approxmaton (MCA) method. It also ponts out ther strong and weak ponts n terms of effcency and accuracy. Secton 4 demonstrates how the reanalyss methods are used n vbraton and optmzaton of large-scale structures. It also presents a new reanalyss method n Crag-Bampton substructurng wth nterface modes whch s very useful for problems wth many nterface DOFs where the FRF substructurng methods cannot be used. Secton 5 presents a vbraton and optmzaton case study of a large-scale vehcle model demonstratng the value of reduced-order modelng and reanalyss methods n practce. Fnally, Secton 6 summarzes and concludes. 2. Reduced-Order Modelng for Dynamc Analyss Computatonal effcency s of paramount mportance n vbraton analyss of large-scale, fnte-element models. Reduced-order modelng (or substructurng) s a common approach to reduce the computatonal effort. Substructurng methods can be classfed n mathematcal and physcal methods. The mathematcal substructurng methods nclude the Automatc Mult-level Substructurng (AMLS) and the Automatc Component Mode Synthess (ACMS) n NASTRAN. The physcal substructurng methods nclude the well known fxed-nterface Crag-Bampton method. Both the AMLS and ACMS methods use graph theo

5 136 Advances n Vbraton Engneerng and Structural Dynamcs ry to obtan an abstract (mathematcal) substructurng usng matrx parttonng of the entre fnte-element model. The computatonal savngs from the mathematcal and physcal methods can be comparable dependng on the problem at hand Modal Reducton For an undamped structure wth stffness and mass matrces K and M respectvely, under the exctaton force vectorf, the equatons of moton (EOM) for frequency response are 2 [ ] K - w M d = F (1) where the dsplacement d s calculated at the forcng frequencyω. If the response s requred at multple frequences, the repeated drect soluton of Equaton (1) s computatonally very expensve and therefore, mpractcal for large scale fnte-element models. A reduced order model (ROM) s a subspace projecton method. Instead of solvng the orgnal response equatons, t s assumed that the soluton can be approxmated n a subspace spanned by the domnant mode shapes. A modal response approach can be used to calculate the response more effcently. A set of egen-frequences ω and correspondng egenvectors (mode shapes) φ are frst obtaned. Then, the dsplacement d s approxmated n the reduced space formed by the frst n domnant modes as (2) where s the modal bass and U s the vector of prncpal coordnates or modal degrees of freedom (DOF). Usng the approxmaton of Equaton (2), the EOM of Equaton (1) can be transformed from the orgnal physcal to the modal degrees of freedom as (3) The response d can be recovered by solvng Equaton (3) for the modal response U and projectng t back to the physcal coordnates usng Equaton (2). If ω max s the maxmum exctaton frequency, t s common practce to retan the mode shapes n the frequency range of 0 2ω max. The system modes n the hgh frequency range can be safely truncated wth mnmal loss of accuracy. Due to the modal truncaton, the sze of the ROM s reduced consderably, compared to the orgnal model. However, the sze ncreases wth the maxmum exctaton frequency. An added beneft of the ROM s that Equatons (3) are decoupled because of the orthogonalty

6 Vbraton and Optmzaton Analyss of Large-Scale Structures usng Reduced-Order Models and Reanalyss Methods of the mode shapes and can be therefore, solved separately reducng further the overall computatonal effort. Note that for a damped structure wth a dampng matrxc, Equaton (1) becomes K + jωc ω 2 M d = F and Equaton (3) s modfed as by addng the modal dampng term. For proportonal (structural or Raylegh) dampng, C s a lnear combnaton of M and K ;.e. C =α M + β K where αand βare constants. In ths case, the reduced Equatons (3) of the modal model are decoupled. Otherwse, they are not. In ths Chapter for smplcty, we present all theoretcal concepts for undamped systems. However for forced vbratons of damped systems, the addton of dampng s straghtforward Substructurng wth Component Mode Synthess (CMS) To model the dynamcs of a complex structure, a fnte-element analyss of the entre structure can be very expensve, and sometmes nfeasble, due to computer hardware and/or software constrants. Ths s especally true n the md-frequency range, where a fne fnte element mesh must be used n order to capture the shorter wavelengths of vbraton. Component mode synthess (CMS) was developed as a practcal and effcent approach to modelng and analyzng the dynamcs of a structure n such crcumstances [20 23]. The structure s parttoned n component structures and the dynamcs are descrbed by the normal modes of the ndvdual components and a set of modes that couple all component. Besdes the sgnfcant computatonal savngs, ths component-based approach also facltates dstrbuted desgn. Components may be modfed or redesgned ndvdually wthout re-dong the entre analyss. One of the most accurate and wdely-used CMS methods s the Crag-Bampton method [22] where the component normal modes are calculated wth the nterface between connected component structures held fxed. Attachment at the nterface s acheved by a set of constrant modes. A constrant mode shape s the statc deflecton nduced n the structure by applyng a unt dsplacement to one nterface DOF whle all other nterface DOF are held fxed. The moton at the nterface s thus completely descrbed by the constrant modes. The Crag-Bampton reduced-order model (CBROM) results n great model sze reducton by ncludng only component normal modes wthn a certan frequency range. However, there s no sze reducton for constrant modes because CBROM must have one DOF for each nterface DOF. If the fnte element mesh s suffcently fne, the constrant-mode DOFs wll domnate the CBROM mass and stffness matrces, and ncrease the computatonal cost. We address ths problem by usng nterface modes (also called characterstc constrant CC- modes) n order to reduce the number of retaned nterface DOFs of the Crag-Bampton approach. For that, a secondary egenvalue analyss s performed usng the constrantmode parttons of the CMS mass and stffness matrces. The CC modes are the resultant egenvectors. The basc formulaton s descrbed n Sectons and The nterface modes represent more natural physcal moton at the nterface. Because they capture the

7 138 Advances n Vbraton Engneerng and Structural Dynamcs characterstc moton of the nterface, they may be truncated as f they were tradtonal modes of vbraton, leadng to a hghly compact CC-mode-based reduced order model (CCROM). In addton, the CC modes provde a sgnfcant nsght nto the physcal mechansms of vbraton transmsson between the component structures. Ths nformaton could be used, for example, to determne the desgn parameters that have a crtcal mpact on power flow. Fgure 1 compares a conventonal constrant mode used n Crag-Bampton analyss wth an nterface mode, for a smple cantlever plate whch s subdvded n two substructures. It should be noted that the calculaton of the CC modes s essentally a secondary modal analyss. Therefore, the benefts are the same as those of a tradtonal modal analyss. For nstance, refnng the fnte-element mesh ncreases the accuracy of a CCROM wthout ntroducng any addtonal degrees of freedom. The ablty of the CC mode approach to produce CCROM whose sze does not depend on the orgnal level of dscretzaton makes t especally well suted for fnte-element based analyss of md-frequency vbraton. Fgure 1. Illustraton of nterface modes Crag-Bampton Fxed Interface CMS Ths secton provdes the bascs of Crag-Bampton method usng the fxed-nterface assumpton. The method s commonly used n CMS algorthms. The fnte-element model of the entre structure s parttoned nto a group of substructures. The DOFs n each substructure are dvded nto nterface (Γ) DOF and nteror (Ω) DOF. The equatons of moton for the th substructure are then expressed as

8 Vbraton and Optmzaton Analyss of Large-Scale Structures usng Reduced-Order Models and Reanalyss Methods ém ê ëm GG WG m m GW WW ùìu&& úí ûîu&& G W ü ék ý + ê þ ëk GG WG k k GW WW ùìu úí ûîu G W ü ìf ý = í þ îf G W ü ý þ (4) The fxed-nterface Crag-Bampton CMS method utlzes two sets of modes to represent the substructure moton; substructure normal (N) modesφ N, and constrant (C) modesφ C, where denotes the th substructure. The sze reducton of the Crag-Bampton method comes from the truncaton of the normal modes Φ N = φ 1 φ 2 φ n whch are calculated by fxng all nterface DOFs and solvng the followng egenvalue problem k ΩΩ 1 Φ C =Λ N m ΩΩ Φ C k ΩΩ {φ n }=λ n m ΩΩ {φ n } for n =1, 2,... (5) The statc constrant modesφ C are calculated by enforcng a set of statc unt constrants to the nterface DOFs as C WW -1 WG = - é ëφ ù û é ëk ù û é ëk ù û (6) The orgnal physcal DOFs DOFs G u and C u and the normal-mode DOFs W u can be thus represented by the constrant-mode N u as G C G ìu ü é I 0 ù ìu º u ü í W ý = ê C N ú í N ý îu þ ëφ Φ û î u þ (7) Usng the above Crag-Bampton transformaton, the orgnal EOM of Equaton (7), can be expressed as CC CN C CC CN C C ém m ù ì u&& ü é k k ù ì u ü ì f ü ê NC NN ú í + = N ý ê NC NN ú í N ý í N ý ëm m û îu&& þ ëk k û îu þ îf þ (8) where the superscrpts C and N are used to ndcate partton related to statc constrant mode DOFs and fxed-nterface normal mode DOFs, respectvely. The matrx parttons of Equaton (8) are (9)

9 140 Advances n Vbraton Engneerng and Structural Dynamcs (10) (11) (12) CN NC ( ) T k = k = 0 (13) (14) (15) (16) The matrces of each substructure are then assembled by applyng dsplacement contnuty and force balance along the nterface to obtan the EOM of the reduced system. A secondary modal analyss s fnally carred out usng the mass and stffness matrces of the reduced system to obtan the egenvalues and egenvectors. Note that constrant mode matrx C Φ s usually a full matrx. Therefore Equaton (9) can be computatonally expensve due to the trple-product ( C T ) WW C Φ m Φ nvolvng constrant modes. The computatonal cost of the Crag-Bamtpon method s mostly related to 1. solvng for the normal modes, 2. solvng for the constrant modes, and 3. the transformaton calculaton n Equaton (9) Crag-Bampton CMS wth Interface Modes In Crag-Bampton CMS, the matrces from all substructures are assembled nto a global CBROM wth substructures coupled at nterfaces by enforcng dsplacement compatblty. Ths synthess yelds the modal dsplacement vector parttoned as CMS d of the syntheszed system to be

10 Vbraton and Optmzaton Analyss of Large-Scale Structures usng Reduced-Order Models and Reanalyss Methods CMS CT NT N T N T d = é ë d d1 d2 L d ss ù n û (17) T where n ss s the number of substructures n the global structure. The correspondng syntheszed CMS mass and stffness matrces are as follows M K CMS C CN CN CN é m m1 m2 L m SS ù n ê CN T N ú êm1 m1 0 L 0 ú ê CN T N = m ú 2 0 m2 L 0 ê ú ê M M M O M ú ê CN T N ú SS SS ë m 0 0 L m n n û é L ê 0 k 0 L 0 C k N ê 1 ú CMS N = ê 0 0 k ú 2 L 0 ê ú ê M M M O M ê ë L k N SS n ù ú ú ú û (18) where the component modal matrces N m and N k are dagonalzed and ther szes depend on the number of selected modes for the frequency range of nterest. However, the number of constrant-mode DOFs, or the sze of matrces C m andk C, s equal to the number of DOFs of the nterfaces between components and s therefore, determned by the fnte-element mesh. If the mesh s fne n the nterface regons, or f there are many substructures, the constrant-mode parttons of the CMS matrces may be relatvely large. For ths reason, we further reduce the CMS matrces by performng a modal analyss on the constrant-mode DOFs as follows k C ψ n =Λ n m Cψ n for n =1, 2, 3,... (19) The egenvectors ψ n are transformed nto the fnte-element DOFs for the th component structure usng the followng transformaton Φ = Φ β Ψ (20) CC C C where Ψ = ψ 1 ψ 2 ψ n CC (21)

11 142 Advances n Vbraton Engneerng and Structural Dynamcs s a selected set of n CC nterface egenvectors whch are few compared to the number of the constrant-mode DOFs. The matrx the local (subsystem ) DOFd. The vectors n C C β maps the global (system) nterface DOFs C d back to CC Φ are referred to as the nterface modes or characterstc constrant (CC) modes, because they represent the characterstc physcal moton assocated wth the constrant modes. Relatvely few CC-mode DOFs are used compared to the number of nterface DOFs. Fnally, the CMS matrces are transformed usng the CC modes and the reduced-order CMS matrces are obtaned smlarly to Equatons (18). Now, the unknown dsplacement vector d ROM s parttoned as ROM CCT NT N T N T d = é ë d d1 d2 L d ss ù n û (22) T where superscrpt CC ndcates the partton assocated wth the CC modes. The equatons of moton of the reduced order CMS model (ROM) are expressed by 2 ROM ROM ROM ROM é ë- w M + K ù û d = f (23) The mass matrx M ROM, the stffness matrx K ROM, and the appled force vector f ROM, are explctly wrtten as M ROM CC CN CN CN é m m1 m2 L m SS ù n ê CN T N ú êm1 m1 0 L 0 ú ê CN T N = m ú 2 0 m2 L 0 ê ú ê M M M O M ú ê CN T N ú SS SS ë m 0 0 L m n n û (24) K ROM é ê CC k N ê 1 ú N = ê 0 0 k ú 2 L 0 ê ú ê ê ë L 0 k 0 L 0 M M M O M L k N SS n ù ú ú ú û (25) ROM CCT NT N T N T f = é ë f f1 f2 L f SS ù n û (26) T

12 Vbraton and Optmzaton Analyss of Large-Scale Structures usng Reduced-Order Models and Reanalyss Methods where (27) and m CN =Ψ T β C T m CN (28) Fltraton of Constrant Modes Fgure 2 shows a typcal constrant mode for a plate substructure. The non-zero dsplacement feld (ndcated by red color) s usually lmted to a very small regon close to the perturbed nterface DOF. Fgure 2. Illustraton of a fltered constrant mode. If the small-dsplacement part of the constrant mode shape s explctly replaced by zero, the densty of the resultng fltered constrant mode wll be sgnfcantly reduced. Consequently, the computatonal cost n Equaton (9) wll be consderably reduced. To flter the constrant modes, the followng crteron s used φ C pq =0 f φ pq C <ε * max p φ C pq (29) where φ C pq s the p th element of the q th constrant modeφ C. If the rato of an element of the constrant mode vector to the maxmum value n the vector s smaller than a defned smallε, the element of the constrant mode s truncated to zero. For the constrant mode of Fgure 4, the constrant mode densty reduces from 100% to 16% fε =0.03.

13 144 Advances n Vbraton Engneerng and Structural Dynamcs 2.3. Frequency Response Functon (FRF) Substructurng and Assemblng If the number of nterface nodes (or DOFs) between connected substructures s small, a reduced-order model of small sze can be developed usng an FRF representaton of each substructure. Ths s known as FRF substructurng. The FRF representaton can be easly obtaned from a fnte element (FE) model or even expermentally. If the FE model of one substructure s very small (e.g. a vehcle suspenson model), t can be easly coupled drectly to another substructure whch s represented usng FRFs. Ths secton provdes the fundamentals of FRF substructurng for both FRF-FE and FRF-FRF couplng Algorthm for FRF/FE Couplng The numercal algorthm s explaned usng the two-substructure example of Fgure 3. Sub1 s an FRF type substructure, meanng that ts dynamc behavor s descrbed usng FRF matrces whch are denoted by H (see Equaton 30 for notaton). The elements of H are frequency dependent and complex f dampng s present. A bold letter ndcates a matrx or vector. Accordng to Equaton (30), H AC for example, represents the dsplacement X A of DOF A due to a unt force F C on DOF C. Sub2 s a fnte element (FE) type substructure. Its dynamc behavor s descrbed usng the stffness K, mass M and dampng B matrces. Fgure 3. Two-substructure example and notaton. The FRF matrx of Sub1 can be calculated by ether a drect frequency response method or a modal response method. In the former case, the orgnal FE equatons are used n the frequency doman whle n the latter a modal model s frst developed and then used to calculate the FRF matrx. The sze of the FRF matrx s small and depends on the number of DOFs of the exctaton, response and nterface DOFs. Usually FRFs are calculated between exctaton and response DOFs. However n order to assemble two substructures, FRFs are also calculated between nterface DOFs and exctaton/response DOFs. The Sub1 FRF matrx H n Equaton (30) s thus parttoned nto nteror (A) DOFs and nterface/couplng (C) DOFs. The nteror DOFs nclude all exctaton and all response DOFs (Fgure 3). The second substructure Sub2 s expressed n FE format. The system FE matrces K, M, and B form the frequency dependent dynamc matrx Z=K + ωb ω 2 M whch s then parttoned accordng to the nteror (B) and nterface (C) DOFs. The nterface DOFs for Sub1 and Sub2 have the same node IDs so that they can be assembled to obtan the system FRF matrx.

14 Vbraton and Optmzaton Analyss of Large-Scale Structures usng Reduced-Order Models and Reanalyss Methods The procedure to assemble the H matrx of Sub1 wth the Z matrx of Sub2 and calculate (solve for) the system matrx H s descrbed below. The equatons of moton for Sub1 are expressed as éx A ù éh AA H AC ù éfa ù ê ú = ê ú ê 1 ú ëxc û ëhca HCC û ëfc û (30) where subscrpt A ndcates the nteror (exctaton plus response) DOFs of Sub1, and subscrpt C ndcates the connecton/common/couplng DOFs between Sub1 and Sub2. The equatons of moton for Sub2 are expressed as éfb ù ézbb ZBC ù éxb ù ê = 2 ú ê ú ê ú ëfc û ëzcb ZCC û ëxc û (31) where subscrpt B ndcates the nteror DOFs of Sub2, and subscrpt C ndcates the connecton/common/couplng DOFs between Sub2 and Sub1. Because of dsplacement compatblty at the nterface, X C appears on the left-hand sde of Equaton (30) for Sub1 and on the rght-hand sde of Equaton (31) for Sub2. Superscrpts 1 and 2 are used to dfferentate the nterface forces F C at Sub1 and Sub2. To couple Sub1 and Sub2, compatblty of forces at the nterface s appled as F = F + F where the force vector wth -1 CC CA 1 2 C C C Φ = H H s obtaned from the second row of Equaton (30) and the second row of Equaton (31) provdes the force vec 2 tor F = Z X + Z X. We thus have C CC C CB B (32) From Equaton (31), (33) where. Substtuton of Equaton (33) n Equaton (32) yelds (34)

15 146 Advances n Vbraton Engneerng and Structural Dynamcs where. From Equaton (34), X C can be expressed n terms of F A, F B and F C as (35) Substtuton of Equaton (35) n Equaton (33) gves X B n terms of F A, F B and F C as (36) Solvng Equaton (30) for and substtutng yelds (37) We can now express X A n terms of F A, F B and F C by substtutng Equaton (35) n Equaton (37), as (38) Based on Equatons (38), (36) and (35), X A, X B and X C are expressed n terms of F A, F B and F A as follows (39) resultng n the followng FRF of the assembled system

16 Vbraton and Optmzaton Analyss of Large-Scale Structures usng Reduced-Order Models and Reanalyss Methods (40) where S = [Φ Θ I] and Algorthm for FRF/FRF Couplng Fgure 4 shows the couplng of two FRF type substructures. The equatons of moton for Sub1 and Sub2 and are expressed as éx A ù éh AA H AC ù éfa ù ê ú = ê 1 ú ê 1 ú ëxc û ëhca HCC û ëfc û (41) and éxb ù éh BB H BC ù éfb ù ê ú = ê 2 ú ê 2 ú ëxc û ëhcb HCC û ëfc û (42) Fgure 4. Two FRF type substructures example and notaton To couple Sub1 and Sub2, we enforce dsplacement compatblty at the nterface and apply the nterface force relatonshp. In ths case, the assembled system equatons can be re-arranged n matrx form as

17 148 Advances n Vbraton Engneerng and Structural Dynamcs T éx A ù æ éh AA H AC ù éh AC ù éh AC ù ö éfa ù ê ú ç ê 1 ú ê 1 ú é ù ê ú ê ú ê XB ú = ç ê HCA HCC ú + ê HCC ú ëhcc + HCC û ê HCC ú ê FC ú ê ú ç ê ú ê ú ê ú ëx ê ú C û è ë H BB û ëh BC û ëh BC û ø ëfb û (43) The FRF/FRF couplng s a specal case of the FRF/FE couplng. 3. Reanalyss Methods for Dynamc Analyss 3.1. CDH/VAO Method The CDH/VAO method, developed by CDH AG for vbro-acoustc analyss, s a Raylegh- Rtz type of approxmaton. If the stffness and mass matrces of the baselne desgn structure are K 0 and M 0, the exact mode shapes n Φ 0 are obtaned by solvng the egen-problem (44) where Λ 0 s the dagonal matrx of the baselne egenvalues. A new desgn (subscrpt p) has the followng stffness and mass matrces K = K + D K M = M + DM (45) p 0 p 0 For a modest desgn change where ΔK and ΔM are small, t s assumed that the change n mode shapes s small and the new response can be therefore, captured n the sub-space spanned by the mode shapes Φ 0 of the ntal desgn. The new stffness and mass matrces are then condensed ask R =Φ T 0 K p Φ 0 and M R =Φ T 0 M p Φ 0 and the followng reduced egenvalue problem s solved to calculate the egen-vector Θ K p Θ =M p ΘΛ p (46) The approxmate egenvalues of the new desgn are gven by Λ p and the approxmate egenvectors Φ p are Φ p = RΘ (47) wherer =Φ 0. Thus, the modal response of the modfed structure can be easly obtaned and the actual response can be recovered usng the egenvectorsφ p.

18 Vbraton and Optmzaton Analyss of Large-Scale Structures usng Reduced-Order Models and Reanalyss Methods Parametrc Reduced-Order Modelng (PROM) Method The PROM method approxmates the mode shapes of a new desgn n the subspace spanned by the domnant mode shapes of some representatve desgns, whch are selected such that the formed bass captures the dynamc characterstcs n each dmenson of the parameter space. Balmes et al. [6, 7] suggested that these representatve desgns should correspond to the mddle ponts on the faces of a box n the parameter space representng the range of desgn parameters. For a structure wth m desgn varables, Zhang [9] suggested that the representatve desgns nclude a baselne desgn for whch all parameters are at ther lower lmts plus m desgns obtaned by perturbng the desgn varables from ther lower lmts to ther upper lmts, one at a tme. The ponts representng these desgns n the space of the desgn varables are called corner ponts (see Fgure 5). Ths selecton of representatve desgns results n a more accurate PROM algorthm. Fgure 5. Desgn space of three parameters. The mode shapes of a new desgn are approxmated n the space of the mode shapes of the corner ponts as Φ Φ p = PΘ (48) where the modal matrx P ncludes the bass vectors as n Equaton (49) and Θ represents the partcpaton factors of these vectors. The columns of P are the domnant mode shapes of the above (m + 1) desgns, P = Φ 0 Φ 1 Φ m (49)

19 150 Advances n Vbraton Engneerng and Structural Dynamcs whereφ 0 s the modal matrx composed of the domnant mode shapes of the baselne desgn, and Φ s the modal matrx of the th corner pont. The mode shapes of the new desgn satsfy the followng egenvalue problem, K Φ p =M Φ pλ KPΘ =MPΘ Λ (50) where Λ s a dagonal matrx of the frst s egenvalues. A reduced egenvalue problem s obtaned by pre-multplyng both sdes of Equaton (50) by P T as K R Θ =M R ΘΛ (51) where the reduced stffness and mass matrces are K R = P T KP and M R = P T MP (52) Thus, the matrx Θ n Equaton (48) conssts of the egenvectors of the reduced stffness and mass matrces K R andm R. For m desgn varables, (m + 1)egenvalue problems must be solved n order to form the bass P of Equaton (49). Therefore, both the cost of obtanng the modal matrces Φ and the sze of matrx P ncrease lnearly wth m. The PROM approach uses the followng algorthm to compute the mode shapes of a new desgn: 1. Calculate the mode shapes of the baselne desgn and the desgns correspondng to the m corner ponts n the desgn space, and form subspace bassp. 2. Calculate the reduced stffness and mass matrces K R and M R from Equaton (52). 3. Solve egenproblem (51) for matrxθ. 4. Reconstruct the approxmated egenvectors n Φ p usng Equaton (48). Step 1 s performed only once. A reanalyss requres only steps 2 to 4. For a small number of mode shapes and a small number of desgn varables, the cost of steps 2 to 4 s much smaller than the cost of a full analyss. The computatonal cost of PROM conssts of 1. the cost of performng (m + 1) full egen-analyses to form subspace bass Pn Equaton (49), and 2. the cost of reanalyss of each new desgn n steps 2 to 4. The former s the fxed cost of PROM because t does not depend on the numbers of reanalyses and the latter s the varable cost of PROM because t s proportonal to the number of reanalyses. The fxed cost s not attrbuted to the calculaton of the response for a partcular

20 Vbraton and Optmzaton Analyss of Large-Scale Structures usng Reduced-Order Models and Reanalyss Methods desgn. It s smply requred to obtan the nformaton needed to apply PROM. The varable cost (cost of reanalyss of a new desgn n part b) s small compared to the fxed cost. The fxed cost of PROM s proportonal to the number of desgn varables m because t conssts of the domnant egenvectors Φ 0 of the baselne desgn and the domnant egenvectors Φ, =1,..., m of the m corner desgn ponts (see Equaton 49). When the sze of bass P ncreases so does the fxed cost because more egenvalue problems and mode shapes must be calculated to form bassp. The PROM method results n sgnfcant cost savngs when appled to problems that nvolve few desgn varables (less than 10) and requre many analyses (e.g. Monte Carlo smulaton or gradent-free optmzaton usng genetc algorthms) Combned Approxmatons (CA) Method The PROM method requres an egenvalue analyss for multple desgns (corner ponts) to form a bass for approxmatng the egenvectors at other desgns. It s effcent only when the number of desgn parameters s relatvely small. On the contrary, the CA method of ths secton does not have such a restrcton because the reanalyss cost s not proportonal to the number of desgn parameters. The CA method s thus more sutable than the PROM method, when the number of reanalyses s less than or comparable to the number of desgn parameters, such as n gradent-based desgn optmzaton. The fundamentals of the combned approxmatons (CA) method [15-19] are gven below. A subspace bass s formed through a recursve process for calculatng the natural frequences and mode shapes of a system. If K 0 and M 0 are the stffness and mass matrces of the orgnal (baselne) desgn, the exact mode shapes Φ 0 are obtaned by solvng the egen-problem K 0 Φ 0 =λ 0 M 0 Φ 0. We want to approxmate the mode shapes of a modfed desgn (subscrpt p) wth stffness and mass matrces K p = K 0 + ΔK and M p =M 0 + ΔM where ΔK and ΔM represent large perturbatons. The CA method estmates the new egenvalues λ p and egenvectors Φ p wthout performng an exact egenvalue analyss. The egen-problem for the modfed desgn can be expressed as Φ p =λ p K 0 1 M p Φ p K 0 1 ΔK Φ p (53) leadng to the followng recursve equaton Φ p, j =λ p K 0 1 M p Φ p, j 1 K 0 1 ΔK Φ p, j 1 (54) whch produces a sequence of approxmatons of the mode shapesφ p, j, j =1, 2,, s. The CA method uses the changes R j =Φ p, j Φ p, j 1 to form a subspace bass to approxmate the modes of the new desgn. In order to smplfy the calculatons, λ p K 1 0 M p Φ p, j 1 n Equaton (53) s replaced wth λ p K 1 0 M p Φ 0 and Equaton (54) becomes

21 152 Advances n Vbraton Engneerng and Structural Dynamcs Φ p, j =λ p K 0 1 M p Φ 0 K 0 1 ΔK Φ p, j 1 showng that the bass vectors satsfy the followng recursve equaton R = -K D KR = K (55) -1 j 0 j-1 j 2,, s where the frst bass vector s assumed to ber 1 = K 1 0 M p Φ 0. The CA method forms a subspace bass [ ] R = R R L R (56) 1 2 s where s s usually between 3 and 6 [16-18, 23] and the mode shapes of the new (K p,m p ) desgn are then approxmated n the subspace spanned by R usng the followng algorthm: Condense the stffness and mass matrces as K = R K R M = R M R (57) T T R p R p Solve the reduced egen-problem (usng matrces K R and M R ) to calculate the egenvector matrxθ. Reconstruct the approxmate egenvectors of the new desgnφ p as Φ p = RΘ (58) The egenvalues of the new desgn are approxmated by the egenvalues λ p of the reduced egen-problem. The CA method has three man advantages: 1. t only requres a sngle matrx decomposton of stffness matrx K 0 n Equaton (55) to calculate the subspace bassr, 2. t s accurate because the bass s updated for every new desgn, and 3. the egenvectors of a new desgn are effcently approxmated by Equaton (58) where the egenvectors Θ correspond to a much smaller reduced egen-problem. However for a large number of reanalyses, the computatonal cost can ncrease substantally because a new bass and the condensed mass and stffness matrces n Equaton (57) must be calculated for every reanalyss. Examples where many analyses are needed are optmzaton problems n whch a Genetc Algorthm s employed to search for the optmum, and probablstc analyss problems usng Monte-Carlo smulaton. The PROM method can be more

22 Vbraton and Optmzaton Analyss of Large-Scale Structures usng Reduced-Order Models and Reanalyss Methods sutable for these problems because the subspace bass R does not change for every new desgn pont. Note that steps 1 and 3 (Equatons 57and 58) are smlar to steps 2 and 4 of PROM. CA uses bass R and PROM uses bassp. The CA method s more effcent than PROM for desgn problems where few reanalyses are requred for two reasons. Frst, t does not requre calculaton of the egenvectors Φ, =1,, m, of the m corner desgn ponts, and second the cost of matrx condensaton of Equaton (57) s much lower than that of Equaton (52), because the sze (number of columns) of bass R s not proportonal to the number of parameters m as n bass P. For problems wth a large number of desgn parameters, the PROM approach s effcent only when a parametrc relatonshp s establshed [7] because a large overhead cost, proportonal to the number of desgn parameters, s requred. In contrast, the CA method does not requre such an overhead cost because the reanalyss cost s not proportonal to the number of desgn parameters. The CA method s therefore, more sutable than PROM, f the number of reanalyses s less than or comparable to the number of desgn parameters Modfed Combned Approxmatons (MCA) Method In the lterature, the accuracy and effcency of the CA method has been mostly tested on problems nvolvng structures wth up to few thousands of DOFs, such as frames or trusses [12-19]. We have tested the CA method usng, among others, the structural dynamcs response of a medum sze (65,000 DOFs) fnte-element model (Fgure 7 of Secton 4.3). Due to ts hgh modal densty, there were more than two hundred domnant modes n the frequency range of zero to 50 Hz. It was observed that the computatonal savngs of the CA method, usng the recursve process of Equaton (55), were not substantal. For ths reason, we developed a modfed combned approxmatons method (MCA) by modfyng the recursve process of Equaton (55) whch s much more effcent than the orgnal CA method for large sze models. The cost of calculatng the subspace bass n Equaton (55) conssts of one matrx decomposton (DCMP) and one forward-backward substtuton (FBS). The DCMP cost s only related to the sze and densty of the symmetrc stffness matrx, whle the FBS cost depends on both the sze and densty of the stffness matrx and the number of columns ofφ 0. As the frequency range of nterest ncreases, more modes are needed to predct the structural response accurately. In such a case, although a sngle DCMP s needed n Equaton (55), the number of columns n Φ 0 may ncrease consderably, thereby ncreasng the cost of the repeated FBS. When the number of domnant modes becomes very large, the cost of performng the calculatons n Equaton (55) can be domnated by the FBS cost. For example, the vehcle model of Secton 4.3 (Fgure 7) has 65,000 degrees of freedom and 1050 modes n the frequency range of Hz. The cost of one DCMP s 1.1 seconds (usng a SUN ULTRA workstaton and NASTRAN v2001) and the cost of one FBS s less than 0.1 seconds f Φ 0 has only one mode. In ths case, the total cost s domnated by the DCMP, and the CA method reduces the cost of one reanalyss consderably. However, f Φ 0 has 1050 modes, the cost of FBS ncreases to 29 seconds domnatng the cost of the DCMP. The CA method can therefore, mprove the

23 154 Advances n Vbraton Engneerng and Structural Dynamcs effcency only when the number of retaned modes s small. Otherwse, the computatonal savngs do not compensate for the loss of accuracy from usng K 0 (stffness matrx of baselne desgn) nstead of K p (stffness matrx of new desgn). The modfed combned approxmatons (MCA) method of ths secton addresses ths ssue. The MCA method uses a subspace bass T whose columns are constructed usng the recursve process T 1 = K p 1 (M p Φ 0 ) T = K p 1 (M p T 1 ) =2, 3,, s (59) nstead of that of Equaton (55). The selecton of the approprate number of bass vectors s s dscussed later n ths secton. The only dfference between Equatons (55) and (59) s that matrx K 0 s nverted n the former whle matrx K p s nverted n the latter. The DCMP of K p must be repeated for every new desgn. However, the cost of the repeated DCMP does not sgnfcantly ncrease the overall cost n Equaton (59) because the latter s domnated by the FBS cost. The teratve process of Equaton (59) provdes a contnuous mode shape updatng of the new desgn. If the process converges n s teratons, the mode shapes T s wll be the exact mode shapesφ p. Equaton (55) does not have the same property. The vectors T provde therefore, a more accurate approxmaton of the exact mode shapes Φ p than the R vectors of the orgnal CA method. Ths s an mportant advantage of MCA. Because the mode shapes T n Equaton (59) can quckly converge to the exact mode shapes Φ p, for many practcal problems only one teraton (.e. s = 1) may be needed, resultng n T = T (60) 1 If the convergence s slow, multple sets of updated mode shapes can be used so that T = Φ 0 T 1 T 2 T s (61) For better accuracy, the above bass also ncludes the mode shapes Φ 0 of the baselne desgn. Because the approxmate modes T are more accurate than the CA vectors R n approxmatng the exact mode shapesφ p, MCA can acheve smlar accuracy to the CA method usng fewer modes. The example of Secton 4.3 demonstrates that the MCA method acheves good accuracy wth only 1 bass vector whereas the CA method requres 3 to 6 bass vectors [13-17]. In summary, the proposed MCA method nvolves four steps n calculatng the approxmate egenvectors Φ pas follows

24 Vbraton and Optmzaton Analyss of Large-Scale Structures usng Reduced-Order Models and Reanalyss Methods Calculate bass T usng Equaton (60) or Equaton (61). Calculate the condensed stffness and mass matrces K R and M R as K = T K T M = T M T (62) T T R p R p Solve the followng reduced egen-problem to calculate the egenvalues and the projectons of the modes n the reduced space spanned by T (K R λm R )Θ =0 (63) Reconstruct the approxmate egenvectors Φ p as Φ p =TΘ (64) The slghtly ncreased cost of usng Equaton (61) nstead of Equaton (60) s usually smaller than the realzed savngs n steps 2 through 4 of Equatons (62) through (64) due to the smaller sze of the reduced basst. The bases of Equatons (60) and (61) are smaller n sze than the CA bass of Equaton (56) for comparable accuracy. The MCA method requres therefore, less computatonal effort for steps 2 through 4. The computatonal savngs compensate for the ncreased cost of DCMP for dynamc reanalyss of large fnte-element models wth a large number of domnant modes. All mode shapes n Equaton (63) must be calculated smultaneously n order to ensure that the approxmate mode shapes Φ p are orthogonal wth respect to the mass and stffness matrces. However, the cost of estmatng the mode shapes Φ pusng Equatons (62) to (64) may ncrease quckly (quadratcally) wth the number of modes, and as a result, the MCA method may become more expensve than a drect egen-soluton when the number of domnant modes exceeds a certan lmt. One way to crcumvent ths problem s to dvde the frequency response nto smaller frequency bands and calculate the frequency response n each band separately nstead of solvng for the frequency response n one step. The modal bass T n Equaton (61) s dvded nto k groups as T ét T L T ù (65) 1 2 k = ë û where T = Φ 0 T 1 T 2 T s (66)

25 156 Advances n Vbraton Engneerng and Structural Dynamcs Each group T contans roughly n / k orgnal modes Φ 0 fromφ 0, and ther correspondng mproved modes. The egenvectors of the new desgn are calculated usng T nstead of T n Equatons (62) to (64). The process s repeated k tmes usng a modal bass that s 1 / k of the sze of the orgnal modal bass. All k groups of egenvectors are then collected together to calculate the frequency response of the new desgn. As demonstrated n Secton 4.3.2, ths reduces the cost consderably wth mnmal loss of accuracy Comparson of CA/MCA and PROM Methods As we have dscussed, a large overhead cost whch s proportonal to the number of desgn parameters s requred before the PROM reanalyss s carred out. However, the CA/MCA method does not requre ths overhead cost because t does not need the bass P of Equaton (49) (see Secton 3.2). Therefore, the CA/MCA method s more sutable, when the number of reanalyses s comparable to the number of desgn parameters. Ths s usually true n gradent-based desgn optmzaton. The CA and MCA methods can become expensve however, when many reanalyses are needed because, for each reanalyss, they requre a new bass R or T (see Equatons 56and 61, respectvely) and new condensed mass and stffness matrces n Equatons (57) and (62). Ths s the case n gradent-free optmzaton problems employng a Genetc Algorthm for example, and n smulaton-based probablstc analyss problems employng the Monte-Carlo method. For these problems, the PROM method s more sutable because the subspace bass P does not change for every new desgn pont. Table 1 summarzes the man characterstcs, advantages and applcaton areas of the CA/MCA and PROM methods. CA/MCA Method PROM Method Overhead Cost None Requred cost to construct P. Cost proportonal to the number of desgn parameters m. Bass Vector Varable bass R/T. Sze proportonal to n and s. Reanalyss Cost Moderate Relatvely small sze of R/T. Must recalculate R/T at every new desgn. Best Applcaton Small number of reanalyses compared to the number of desgn parameters. Evaluaton of few desgn alternatves and gradent-based optmzaton. Constant bass P. Sze proportonal to n and m. Hgh f no parametrc relatonshp exsts due to the condensaton of large sze and dense P. Low f a parametrc relatonshp exsts. Very large number of reanalyses. Gradent-free optmzaton (e.g. genetc algorthms) and probablstc analyss. Table 1. Comparson of the CA/MCA and PROM methods.

26 Vbraton and Optmzaton Analyss of Large-Scale Structures usng Reduced-Order Models and Reanalyss Methods Reanalyss Methods n Dynamc Analyss and Optmzaton The reanalyss methods of Secton 3 can be used n dfferent dynamc analyses such as modal or drect frequency response and free or forced vbraton n tme doman. Dependng on the problem and the type of analyss, a partcular reanalyss method may be preferred consderng how many tmes t wll be performed and how many desgn parameters wll be allowed to change. Ths secton demonstrates the computatonal effcency and accuracy of reanalyss methods n dynamc analyss and optmzaton. It also ntroduces a new reanalyss method n Crag-Bampton substructurng wth nterface modes whch s very useful for problems wth many nterface DOFs where FRF substructurng s not practcally applcable Integraton of MCA Method n Optmzaton We have mentoned that the MCA method provdes a good balance between accuracy and effcency for problems that requre a moderate number of reanalyses, as n gradent-based optmzaton. For problems where a large number of reanalyses s necessary, as n probablstc analyss and gradent-free (e.g. genetc algorthms) optmzaton, a combnaton of the MCA and PROM methods s more sutable. Fgure 6 shows a flowchart of the optmzaton process for modal frequency response problems. The DMAP (Drect Matrx Abstracton Program) capabltes n NASTRAN have been used to ntegrate the MCA method and the NASTRAN modal dynamc response and optmzaton (SOL 200). The hghlghted boxes ndcate modfcatons to the NASTRAN optmzer. Startng from the orgnal desgn, the code frst calculates the desgn parameter senstvtes n order to establsh a local search drecton and determne an mproved desgn along the local drecton. At the mproved desgn, an egen-soluton s obtaned to calculate a modal model and the correspondng modal response. The dynamc response at certan physcal DOFs s then recovered from the modal response. At ths pont, a convergence check s performed to decde f the optmal desgn s obtaned. If not, further teratons are needed and the above procedure s repeated. Many teratons are usually needed for practcal problems to obtan the fnal optmal desgn. Secton demonstrates how ths process was used to optmze the vbro-acoustc behavor of a 65,000 DOF, fnte-element model of a truck. Usng the MCA method, the computatonal cost of the entre optmzaton process was reduced n half compared wth the exstng NASTRAN approach. As for a stand alone modal frequency response, the egen-soluton accounts for a large part of the overall optmzaton cost for vbratory problems where a modal model s used. A reanalyss method can be nserted nto the procedure as shown n Fgure 6 to provde an approxmate egen-soluton savng therefore, substantal computatonal cost. Other reanalyss methods such as the CDH/VAO, CA or PROM can also be used dependng on the number of desgn varables and the number of expected teratons.

27 158 Advances n Vbraton Engneerng and Structural Dynamcs Fgure 6. Flowchart for mca-enhanced nastran optmzaton Integraton of MCA and PROM Methods The PROM method requres exact calculaton of the mode shapes for all desgns correspondng to the corner ponts of the parameter space n order to calculate the subspace bass Pof Equaton (49). The requred computatonal effort can be prohbtve for a large number of parameters (optmzaton desgn varables). Ths effort can be reduced substantally f the modes of each corner pont are approxmated by the MCA method. In ths case, an exact egen-soluton s requred only for the baselne desgn. The followng steps descrbe an algorthm to ntegrate the MCA and PROM methods: Perform an exact egen-analyss at the baselne desgn pont p 0 all parameters are at ther lower lmt, to obtan the baselne mode shapesφ 0. Use the MCA method at desgn pont p wth all parameters at ther low lmt except the th parameter whch s set at ts upper lmt. Obtan approxmate mode shapes for the th corner pont usng the followng recursve process T,1 = K 1 (M Φ o ) T, j+1 = K 1 (M T, j ) j =2, 3,, s (67) Form the subspace bass T as

28 Vbraton and Optmzaton Analyss of Large-Scale Structures usng Reduced-Order Models and Reanalyss Methods T = Φ o T 0,s T 1,s T m,s (68) where m s the total number of parameters. Obtan the approxmate mode shapes Φ p usng the subspace projecton procedure of Equatons (50) through (52) where T s used nstead ofp. The modal bass Φ p can be subsequently used n a modal dynamc response soluton. Only step 4 s repeated n reanalyss. The computatonal savngs can be substantal especally for problems where many reanalyses are needed Combned MCA and PROM Methods: Vbro-Acoustc Response of a Vehcle The pckup truck vehcle model wth 65,000 DOFs of Fgure 7 s used n ths secton to demonstrate the advantages of the combned MCA and PROM method n optmzng the vbroacoustc response of a vehcle. The model has 78 components and roughly 11,000 nodes and elements. The example s performed on a SUN ULTRA workstaton usng NASTRAN v2001. The MCA and PROM methods have been mplemented n NASTRAN DMAP. Fgure 7. FE model of a pckup truck. The sound pressure level at the drver s ear locaton s calculated usng a vbro-acoustc analyss. The structural forced vbraton response due to unt harmonc forces n x, y, and z drectons at the engne mount locatons, s coupled wth an nteror acoustc analyss. The frst and second egen-frequences of the acoustc volume nsde the cabn are 95.9 Hz and Hz. The sound pressure level s calculated n the 80 to 140 Hz frequency range. The structure and flud domans are coupled through boundary condtons ensurng contnuty

29 160 Advances n Vbraton Engneerng and Structural Dynamcs of vbratory dsplacement and acoustc pressure. A fnte-element formulaton of the coupled undamped problem yelds the followng system equatons of moton [24]. æ ék S -HSF ù 0 2 é MS ù ö éds ù éfb ù ç ê jw 2 T 0 ú - ê = ê ú F r 0c ú ê ú è ë K û ë 0HSF M F û ø ëpf û ëfq û (69) where the vbratory dsplacement d S and the acoustc pressure p F are the prmary varables. The fnte-element representaton of the two domans conssts of stffness and mass matrx pars K S, M S and K F, M F, respectvely. The ar densty and wave speed are ρ 0 andc 0. The rght hand sde of Equaton (69) denotes the external forces. The spatal couplng matrx H SF ndcates couplng between the two domans whch s usually referred to as two-way couplng. Due to ths couplng, the combned structural-acoustc system of equatons s not symmetrc. If the acoustc effect on the structural response s small, the couplng term can be omtted, resultng n the so-called one-way couplng, where the structural response s frst calculated and then used as nput ( f q n Equaton 69) to solve for the acoustc response. The coupled structure-acoustc system can be solved ether by a drect method, or more effcently by a modal response method whch can be appled to both the structural and acoustc domans Combned MCA and PROM Methods To demonstrate the computatonal effectveness and accuracy of ntegratng MCA n PROM, a reanalyss was performed for a modfed desgn where fve plate thckness parameters vary; chasss and ts cross lnks, cabn, truck bed, left door, and rght door. All parameters were ncreased by 100% from ther baselne values. The sound pressure at the drver s ear was calculated usng two-way couplng. A structural modal frequency response was used. The acoustc response was calculated usng a drect method because the sze of the acoustc model s relatvely small. For the structural analyss, 1050 modes were retaned n the 0 to 300 Hz frequency range. The combned MCA and PROM approach was compared aganst the NASTRAN drect soluton for a modfed desgn where all fve parameters were at ther upper lmts. Only one teraton was used n Equaton (59) n order to get the set of once-updated mode shapes for each corner desgn pont. The subspace bass, whch ncludes nformaton for all fve desgn parameters, s therefore, represented by T = Φ o T 0,1 T 1,1 T 5,1 (70) The maxmum error n natural frequences as predcted by the combned MCA and PROM method and NASTRAN, s less than 0.45% n the entre frequency range. Fgure 8 ndcates that the sound pressures calculated by both methods are almost dentcal. The computatonal effort for the MCA method to obtan approxmate mode shapes at each corner desgn pont s about 30 seconds. In contrast, t takes about 180 seconds for an exact egen-soluton

30 Vbraton and Optmzaton Analyss of Large-Scale Structures usng Reduced-Order Models and Reanalyss Methods usng NASTRAN. The computatonal cost to construct the reduced bass (Pn PROM and T n PROM+MCA) s compared n Table 2. The total cost was reduced from 1080 seconds to 330 seconds. The computatonal savng s more sgnfcant f the number of desgn parameter ncreases. Fgure 8. Comparson of sound pressure at drver s ear between combned MCA and PROM method and NASTRAN. Method Solvng for mode Solvng for mode Total Cost shape Φ 0 at baselne shapes at 5 corner desgn desgn ponts PROM 180 sec 180*5=900 sec 1080 sec PROM+MCA 180 sec 30*5=150 sec 330 sec Table 2. CPU tme to construct reduced bass Optmzaton usng MCA Method The goal here s to mnmze the sound pressure at the drver s ear. A total of 41 desgn parameters are used representng the thckness of all vehcle components modeled wth plate elements. All thcknesses are allowed to change by 100% from ther baselne values. Table 3 descrbes all desgn parameters. At the ntal pont of the optmzaton process, all parameters are at ther low bound.

31 162 Advances n Vbraton Engneerng and Structural Dynamcs Prm. Descrpton Prm. Descrpton Prm. Descrpton # (thckness of) # (thckness of) # (thckness of) 1 Bumper 15 Radator mtg. 29 Tre, front rght 2 Rals 16 Radator mtg., md. 30 Tre, rear left 3 A-arm, low left 17 Fan cover, low 31 Tre, rear rght 4 A-arm, low rght 18 Fan cover, up 32 Engne outer 5 A-arm, up left 19 Cabn 33 A-arm conn., up left 6 A-arm, up rght 20 Cabn mtg. renf. 34 A-arm conn., up rght 7 Tre rm 21 Door, left 35 A-arm conn., low left 8 Engne Ol-box 22 Door, rght 36 A-arm conn., low rght 9 Fan 23 Bed 37 Glass, left 10 Hood 24 Brake, front left 38 Glass, rght 11 Fender, left 25 Brake, front rght 39 Glass, rear 12 Fender, rght 26 Ral conn., rear 40 Glass, front 13 Wheel house, left 27 Ral mount 41 Ral conn., front 14 Wheel house, rght 28 Tre, front left Table 3. Descrpton of desgn parameters. Because of the large number of desgn parameters, the combned MCA and PROM approach Secton s not computatonally effcent because the sze of the PROM bass s very large (see Equaton 70). For ths reason, we use the MCA reanalyss method and demonstrate ts capablty to handle a large number of parameters. It approxmates the mode shapes at ntermedate desgn ponts usng only T 1 n Equaton (59). The subspace bass at each optmzaton step s thust = Φ o T 1. Because 1050 modes exst n the frequency range of 0 to 300 Hz of the ntal desgn, the sze of the MCA modal bass s 2*1050 = k=1 k=21 Eq. (59) 31 sec 31 sec Eq. (62) 258 sec 50 sec Eq. (63) 48 sec 6 sec Eq. (64) 67 sec 10 sec Total Cost 404 sec 97 sec Table 4. CPU tme of the MCA method. The cost of solvng for 1050 modes drectly from NASTRAN s 180 seconds (see Table 2). In the MCA method, the cost of solvng the lnear system of equatons n Equaton (59) s 31

32 Vbraton and Optmzaton Analyss of Large-Scale Structures usng Reduced-Order Models and Reanalyss Methods seconds, and the addtonal combned cost of Equatons (62) to (64) s 373 seconds, resultng n a total cost of 404 seconds (see Table 4). To reduce ths cost, the 1050 modes are dvded nto 21 groups and the modes n each group are obtaned separately as explaned n the last paragraph of Secton 3.4. Ths reduces the cost of Equatons (62) to (64) to 66 seconds for a total cost of 97 seconds, whch s about half the cost of the drect NASTRAN method. Fgure 9. Comparson of sound pressure at drver s ear between ntal and optmal desgns. Fgure 10. Percent ncrease of optmal desgn parameters relatve to baselne desgn parameters. The gradent-based optmzer n NASTRAN (SOL 200) usng the optmzaton process of Fgure 6 needed three teratons to calculate the optmal desgn. Fgure 9 compares the

33 164 Advances n Vbraton Engneerng and Structural Dynamcs sound pressure at the drver s ear between the optmal and ntal desgns. Fgure 10 shows the percentage ncrease of optmal values relatve to the ntal values for all 41 desgn parameters. In the frequency range of Hz, the maxmum sound pressure s slghtly reduced from 7.9E-7 to 7.2E-7 Pascal. Most parameters are mnmally changed. The largest ncrease s 20% for the ral mount thckness (parameter #27). Fgure 11. Comparson of sound pressure at drver s ear between baselne and optmal desgns wth 20 ntal populatons and 4 generatons. Fgure 12. Comparson of sound pressure at drver s ear between baselne and optmal desgns wth 100 ntal populatons and 6 generatons.

34 Vbraton and Optmzaton Analyss of Large-Scale Structures usng Reduced-Order Models and Reanalyss Methods The Sequental Quadratc Programmng (SQP) algorthm of NASTRAN can only fnd a local optmum. To obtan a more sgnfcant desgn mprovement, two addtonal studes were performed usng a Genetc Algorthm wth the MCA method. The frst study used 20 ntal populatons and 4 generatons, and the second study used 100 ntal populatons and 6 generatons. Fgures 11and 12 show that the number of ntal populatons and the number of generatons, affect the optmzaton results. Whle a hgher number of ntal populatons and generatons results n a slghtly better result, both studes produced a much better optmum than the SQP algorthm. In the case of 100 ntal populatons and 6 generatons, the sound pressure s reduced from 7.9E-7 Pascal to 2.0E-7 Pascal, whch s equvalent to about 15 db n sound pressure level (SPL). To verfy the accuracy of the MCA approxmaton, the sound pressure response at the optmal desgn from MCA+GA wth 100 ntal populatons and 6 generatons was evaluated by both drect NASTRAN and MCA. Fgure 13 shows that the MCA method s very accurate. For a smlar to MCA accuracy, the orgnal CA method needed three sets of mode shapes to form the subspace bass, requrng 90 seconds to solve the lnear equatons. The much larger mode bass R n CA ncreases the computatonal cost to calculate the trple matrx products of Equaton (57). Therefore for large scale, fnte-element models wth a hgh modal densty, the proposed MCA method can be more effcent compared to ether a complete NASTRAN analyss or the orgnal CA method. Fgure 13. Comparson of sound pressure at drver s ear between drect nastran and mca Reanalyss n Crag-Bampton Substructurng wth Interface Modes The FRF substructurng of Secton 2.3 couples two structures usng FRF nformaton between the couplng (nterface) DOFs, and the exctaton and/or response DOFs. Although ths approach s very effcent, t s practcal only f we have a few couplng DOFs; e.g. connecton of a vehcle suspenson to chasss or connecton of the exhaust system to body

35 166 Advances n Vbraton Engneerng and Structural Dynamcs through a few hangers. If the physcal substructures have nterfaces wth many DOFs, a dfferent reduced-order modelng (ROM) approach must be used such as the Crag-Bampton ROM of Secton The Crag-Bampton ROM can be large however, f the number of retaned nterface DOFs s large. We address ths problem by performng a secondary egenvalue analyss whch yelds the so-called nterface modes (see Secton 2.2.2). The followng secton descrbes a reanalyss methodology for physcal substructurng wth Crag-Bampton ROMs usng nterface modes. We show that ts accuracy s very good and the computatonal savngs are substantal Crag-Bampton wth Interface Modes and Reanalyss In the Crag-Bampton CMS method (Crag-Bampton reduced-order model or CBROM), the mass and stffness matrces of each substructure are parttoned nto nterface sub-matrces, nteror (omtted DOF) sub-matrces, and ther couplng sub-matrces. The dynamcs of a structure are then descrbed by the normal modes of ts ndvdual components, plus a set of modes called constrant modes that couple the components. In CBROM, there s no sze reducton for constrant modes snce all of them are kept n the reduced equatons. If the fnte element mesh s suffcently fne, the constrant-mode DOFs wll domnate the sze of CBROM mass and stffness matrces and result n a large computatonal cost. Ths ssue s addressed by usng nterface modes (also called characterstc constrant CC- modes). For that, a secondary egenvalue analyss s performed usng the constrant-mode parttons of the CMS mass and stffness matrces. The CC modes are the resultant egenvectors. Detals are provded n Sectons and The number of constrant modes n c equals to the number of nterface DOF. For many FE models of large structures, the number of nterface DOF can be rather large. The calculaton of constrant modes n Equaton (6) nvolves a decomposton step and a FBS step. The cost of FBS s proportonal ton c. For any matrx multplcaton that nvolvesφ C, the cost s proportonal ton c. For any trple-product that nvolves Φ C the cost s proportonal ton c 2. The matrces from all substructures are assembled nto a global CBROM wth substructures coupled at nterfaces by enforcng dsplacement compatblty. If k C and m C are the component (substructure) matrces, the global matrces K C and M C are assembled as å K C C C C = k, M = m (71a) å and a secondary egenvalue analyss s performed to calculate the nterface modes Φ CC as K C λ CC M C Φ CC =0 (71b) The matrces n Equatons (9), (10) and (12) are then reduced as

36 Vbraton and Optmzaton Analyss of Large-Scale Structures usng Reduced-Order Models and Reanalyss Methods m CC =Φ CCT T m C Φ CC m CCN =Φ CCT T m CN (72) k CC =Φ CCT T k C Φ CC where the matrces m CC, m CCN and k CC are of much smaller sze than matrces m C, m CN and k C. The nterface modes reduce the nterface sze producng a smaller reduced order model (ROM) compared wth the tradtonal Crag-Bampton ROM (CBROM). However, they are calculated from the assembled nterface K and M matrces. Thus, the calculaton of constrant modes and all matrx multplcatons related to constrant modes are stll necessary. The nterface mode method reduces the sze of ROM but t does not reduce the computatonal cost related to the constrant modes. If the nterface modes Φ CC were known before hand, the calculatons n Equatons (6), (9), (10) and (12) and Equaton (72) could be performed much more effcently as follows 55 Φ^ C =Φ C Φ CC = k ΩΩ 1 (k ΩΓ Φ CC ) (73) m CC =( Φ CC) T m ΓΓ Φ CC + ( Φ CC) T m ΓΩ Φ^ C + (Φ^ C ) T m ΩΓ Φ CC + (Φ^ C ) T ΩΩ m (Φ^ C ) T (74) m CCN =( Φ CC) T m ΓΩ Φ N + (Φ^ C ) T m ΩΩ Φ N (75) k CC =( Φ CC) T k ΓΓ Φ CC ( Φ CC) T k ΓΩ Φ^ CC (76) The followng observatons can be made: 1. In Equatons (74) to (76), the computaton nvolves Φ CC and Φ^ C and does not nvolve Φ C. Therefore, the calculaton of orgnal constrant modes Φ C s no longer needed. 2. In Equaton (73), the number of columns of matrx (k ΩΓ Φ CC ) s equal to the number of nterface modes n cc whch s usually smaller thann c. Therefore, the FBS cost of solvng for Φ^ C s proportonal ton cc and t s much smaller than the FBS cost of solvng forφ C. 3. Because both Φ CC and Φ^ C are of sze n cc the cost of matrx multplcaton and trpleproduct n Equatons (74) to (76) are now proportonal to n cc andn cc 2. Therefore, the cost s much smaller than the correspondng cost n Equatons (9), (10) and (12). In ths CCROM method whch s based on CBROM, the nterface modes Φ CC are obtaned usng the assembled nterface parttons of the CBROM formulaton. Thus, t s mpossble to know Φ CC before hand for a new desgn. For ths reason, Equatons (73) to (76) can not be

37 168 Advances n Vbraton Engneerng and Structural Dynamcs theoretcally mplemented to mprove effcency. For ths reason, we propose a reanalyss approach where the calculated nterface modes Φ CC for orgnal (baselne) desgn can be used as an approxmaton of the new nterface modes at any modfed desgn. In ths case, Equatons (73) to (76) are appled to mprove the computatonal effcency A Car Door Example The car door model of Fgures 14 and 15 s used to demonstrate the proposed reanalyss method for substructurng wth Crag-Bampton method usng nterface modes. It has 25,800 nodes and 25,300 elements and s dvded nto two substructures. The frst substructure ncludes the outer door shell and a bar attached to t. The second substructure ncludes the rest of the door. There are 293 nodes (1758 DOFs) on the nterface. Therefore, the CBROM or CCROM method must calculate 1758 constrant modes accordng to Equaton (6) for both substructures. The 1758 constrant modes are nvolved n matrx multplcaton or trpleproducts n Equatons (9), (10) and (12). Fgure 16 shows the nterface nodes. For the ntal desgn usng the CCROM method, 52 nterface modes are calculated below 600 Hz. A modfed desgn s created where the shell thcknesses for the outer door (substructure 1) and nner door (substructure 2) are doubled. To provde baselne numbers, the CCROM method s used on the new desgn to solve for the system natural frequences. The new reanalyss approach s used on the new desgn to calculate approxmate natural frequences whch are then compared wth the baselne numbers. The nterface modes calculated at the orgnal desgn are used as an ntal guess for the nterface modes of the new desgn. Fgure 14. Outsde and nsde vews of car door model.

38 Vbraton and Optmzaton Analyss of Large-Scale Structures usng Reduced-Order Models and Reanalyss Methods Fgure 15. Substructure 1 (outer door shell) and substructure 2 (rest of door). Fgure 16. Interface nodes ndcated by whte dots.

39 170 Advances n Vbraton Engneerng and Structural Dynamcs Fgure 17. Comparson of natural frequences between orgnal CCROM method and CCROM wth reanalyss for the car door example. Fgure 17 compares the natural frequences of the new (modfed) desgn between the orgnal CCROM (Crag-Bampton wth Interface modes) method and the new approach where reanalyss s used n CCROM to approxmate the nterface modes. We observe that the natural frequences of the modfed desgn are very dfferent from those of the orgnal desgn. Also, the accuracy of the proposed reanalyss method s excellent. The frequences for the modfed desgn calculated by the orgnal CCROM and the proposed new approach are almost dentcal. The percentage error of the new approach versus the orgnal CCROM approach s less than 1% on average. The computaton cost s summarzed n Table 5. Substructure 1: CPU Tme Normal Constrant Multplcaton Other Cost Total Cost Modes Modes CCROM 8 sec 61 sec 65 sec 3 sec 137 sec New Approach 7 sec 2 sec 0.3 sec 2 sec 11 sec Substructure 2: CPU Tme Normal Constrant Multplcaton Other Cost Total Cost Modes Modes CCROM 108 sec 282 sec 927 sec 10 sec 1327 sec New Approach 110 sec 16 sec 3 sec 10 sec 139 sec Table 5. Summary of computatonal cost for the car door example.

40 Vbraton and Optmzaton Analyss of Large-Scale Structures usng Reduced-Order Models and Reanalyss Methods In the new approach to reduce the cost related to constrant modes, the total remanng cost s domnated by the cost of calculatng the normal modes for each substructure. For example, the calculaton of the normal modes for Substructure 2 took 110 seconds out of a total of 139 seconds (see Table 5). It should be noted that the normal modes cost can be further reduced by applyng another reanalyss method such as CDH/VAO, CA or MCA to approxmate the normal modes. Therefore, the overall cost of substructurng based on Crag- Bampton wth nterface modes, can be drastcally reduced by usng the proposed reanalyss to approxmate the constrant modes and a CDH/VAO or MCA reanalyss to approxmate the normal modes at a new desgn. 5. Optmzaton of a Vehcle Model A detaled optmzaton study s presented usng a large-scale FE model of a vehcle. For smplcty, we call t BETA car model. It s composed of approxmately 7.1 mllon DOFs and 1.1 mllon elements. Fgure 18 shows all modelng detals. Fgure 18. Detals of BETA car model.

41 172 Advances n Vbraton Engneerng and Structural Dynamcs Fgure 19. Ten response locatons on two front doors. We form an optmzaton problem n terms of the maxmum vbratory dsplacement at any locaton of the outer shell of the two front doors by mnmzng the maxmum dsplacement among ten locatons of the two front doors (Fgure 19) due to a hypothetcal engne exctaton n the vertcal (up-down) drecton. The engne s represented by a lumped mass connected rgdly to the engne mounts (Fgure 20). The powertran-exhaust model has about 1.3 mllon DOFs and s composed of 29 PSHELL components and 12 PSOLID components. There are also some RBE2 and PBUSH elements whch are used as connectors. The maxmum dsplacement at each of the ten door locatons s observed n the y drecton (lateral drecton perpendcular to the door plane). Fgure 20. Descrpton of the ffteen desgn varables.

42 Vbraton and Optmzaton Analyss of Large-Scale Structures usng Reduced-Order Models and Reanalyss Methods Fgure 21. Descrpton of the fve desgn varables on the doors. Ffteen desgn varables are chosen; fve structural elements of each door (thckness of door shell, front frame, rear frame, top panel, mddle ppe), vertcal stffness of each of the four engne mounts, and vertcal stffness of each of the sx exhaust system supports. All desgn varables are schematcally ndcated n Fgures 20 and 21. The optmzaton problem s stated as follows: The optmal value of each of the ffteen desgn varables s calculated n order to mnmze the maxmum response among the ten locatons on the doors whle the mass of the vehcle remans less or equal to the mass of the ntal (nomnal) vehcle. The response s calculated n the 100 Hz to 200 Hz frequency range and a 3% structural dampng s used. The optmzaton problem s numercally very challengng because of 1. the many local optma and 2. the computatonal cost of each dynamc analyss. The former was handled by usng a hybrd optmzaton algorthm whch frst explores the entre desgn space usng a Nchng Genetc Algorthm (GA) [25] and then swtches to a gra

43 174 Advances n Vbraton Engneerng and Structural Dynamcs dent-based optmzer (fmncon n MATLAB) usng the best estmate of the optmal pont from the GA as ntal pont. Ths ensures a rapd convergence to the fnal optmum because although all GA optmzers can move quckly to the vcnty of the fnal optmum, they have a very slow convergence rate n pnpontng the fnal optmum. FRF substructurng s used to assemble all components of the vehcle (body, doors, and engne-exhaust) nto a small reduced-order model. Ths keeps the computatonal cost of each dynamc analyss low (4 mnutes per analyss). A modal model s created only once for the body subsystem and then used to generate an FRF representaton. Ths modal model does not change durng the optmzaton because the chosen desgn varables are not assocated wth the body. However, the modal models of the doors change durng the optmzaton. The fnal model for the entre vehcle s created by assemblng the FRF models of each component. The FRF assembly operaton s repeated durng optmzaton because the FRF models of the two doors keep changng. The Nchng GA optmzer maxmzes a ftness functon by modfyng all desgn varables. A proper ftness functon whch mnmzes the maxmum response among the ten door locatons whle satsfyng the vehcle mass constrant s chosen as follows Ftness = 10 max(res p ) Nomnal =1 10 max(res p ) =1 1 + p mn(c, 0) The rato of the nomnal maxmum response over the actual maxmum response s used so that the ftness value ncreases when the actual response s reduced. Ths rato s multpled by 1 + p * mn(c, 0) where p = 10 s a penalty value andc =1 Mass Mass Nomnal. Thus, c s postve f Mass s less than Mass Nomnal satsfyng the constrant and the value of 1 + p * mn(c, 0) s equal to one. Otherwse, c becomes negatve f Mass s greater thanmass Nomnal and the term 1 + p * mn(c, 0) assumes a large negatve value whch reduces the ftness value consderably. As a result, the GA optmzer always satsfes the mass constrant whle maxmzng the value of the ftness functon. Fgure 22 summarzes the optmzaton results by comparng the maxmum door response between the optmal and ntal desgns. The optmzer determned that the maxmum response occurs at locaton 9 (center of left front door of Fgure 19) at approxmately 105 Hz. Fgure 23 shows that ths represents a vehcle local mode nvolvng moton of the doors only. At the optmal desgn the maxmum response was reduced from the ntal 10-3 m to 0.47*10-3 m (Table 6).

44 Vbraton and Optmzaton Analyss of Large-Scale Structures usng Reduced-Order Models and Reanalyss Methods Fgure 22. Comparson of optmal and ntal desgns. Fgure 23. Vehcle local mode at 105 Hz ndcatng door deformaton. Table 6 compares the value of each desgn varable between the ntal (nomnal) and fnal optmal desgns. It also ndcates that all desgned varables were allowed to vary wthn a lower and upper bound. The values of the fve door desgn varables changed consderably between the ntal and optmal desgns. Ths s expected because the optmzer tred to suppress the local door mode. The stffness of the four engne mounts and the sx exhaust supports also changed. Although we ntutvely expect the stffness of the engne mounts to

45 176 Advances n Vbraton Engneerng and Structural Dynamcs change but not the stffness of the exhaust supports, ths s not the case n ths example. Table 6 also ndcates that at the optmum we not only reduced the maxmum response from 10-3 m to 0.47*10-3 m but the vehcle mass was also reduced from the ntal unts to the fnal unts. Desgn Thckness Lower Upper Nomnal Optmal Varables Bound Bound Desgn Desgn X 1 Door Shell X 2 Front Frame X 3 Rear Frame X 4 Top Panel X 5 Mddel Ppe X 6 Engne mount X 7 Engne mount X 8 Engne mount X 9 Engne mount X 10 Exhaust support X 11 Exhaust support X 12 Exhaust support X 13 Exhaust support X 14 Exhaust support X 15 Exhaust support Max Resp. 1* *10-3 Door Mass Table 6. Summary of optmal desgn. Fgure 24 shows the actual functon evaluatons (desgn ponts where the vehcle dynamc response was calculated) n the X 1 -X 2 -X 3 space and ndcates the vcnty of the optmal desgn pont. The GA optmzer needed only 359 functon evaluatons and used a populaton sze of 5*(15+1) = 80 and a maxmum of 10 generatons. The populaton sze and the number of allowed generatons were kept at a mnmum n order to locate the vcnty of the optmum quckly wthout wastng valuable computatonal effort.

46 Vbraton and Optmzaton Analyss of Large-Scale Structures usng Reduced-Order Models and Reanalyss Methods Fgure 24. Functon evaluatons of the Nchng GA n the X 1 -X 2 -X 3 space. Consderng that the computatonal cost for each functon evaluaton was 4 mnutes, the total computatonal tme was (359 functon evaluatons) * (4 mnutes per evaluaton) = 1436 mnutes or 23.9 hours. Ths s acceptable consderng the sze and type of performed analyss. The computatonal cost, n terms of number of functon evaluatons, was kept low by couplng the Nchng GA wth a Lazy Learnng metamodelng technque [26, 27]. The latter estmates the value of the ftness functon from exstng values at close by desgns wthout calculatng the actual response. It uses an error measure to fgure out f the estmaton s accurate. The error s small f enough prevous desgns, for whch the ftness value was evaluated, are close to the new desgn. In ths case, the metamodel estmates the current ftness value wthout runnng an actual dynamc response. If the error s large, an actual response s calculated and the ftness value of ths new desgn s added to the pool of prevous desgns the Lazy Learnng metamodelng technque wll use downstream. 6. Conclusons and Future Work Reduced-order models and reanalyss methodologes were presented for accurate and effcent vbraton analyss of large-scale, fnte element models, and for effcent desgn optmzaton of structures for best vbratory response. The optmzaton s able to handle a large number of desgn varables and dentfy local and global optma. For large FE models, t s common to solve for the system response through modal reducton n order to mprove computatonal effcency. An egenanalyss s performed usng the system stffness and mass matrces and a modal model s formed whch s then solved for the response. The computatonal cost can be also reduced usng substructurng (or reduced-order modelng) methods. A modal reducton s appled to each substructure to obtan the component modes and the system level response s then obtaned usng component mode synthess. In optmzaton of dynamc systems nvolvng desgn changes (e.g. thcknesses,