Discrete Adjoint Method for the Sensitivity Analysis of Flexible Multibody Systems

Transcription

1 Dscrete Adjont Method for the Senstvty Analyss of Flexble Multbody Systems Alfonso Callejo, Valentn Sonnevlle, and Olver A. Bauchau Department of Aerospace Engneerng, Unversty of Maryland College Park, Maryland 2742 Abstract The gradent-based desgn optmzaton of mechancal systems requres robust and effcent senstvty analyss tools. The adjont method s regarded as the most effcent sem-analytcal method to evaluate senstvty dervatves for problems nvolvng numerous desgn parameters and relatvely few objectve functons. Ths paper presents a dscrete verson of the adjont method based on the generalzed-alpha tme ntegraton scheme, whch s appled to the dynamc smulaton of flexble multbody systems. Rather than usng backward ntegraton, the proposed approach leads to a straghtforward algebrac procedure that provdes desgn senstvtes evaluated to machne accuracy. The approach s based on an ntrnsc representaton of moton that does not requre a parameterzaton of rotaton. Desgn parameters assocated wth rgd bodes, knematc jonts, and beam sectonal propertes are consdered. Rgd and flexble mechancal systems are nvestgated to valdate the proposed approach and demonstrate ts accuracy, effcency, and robustness. 1 Introducton Senstvty analyss s an essental ngredent of gradent-based desgn optmzaton. Indeed, gradentbased optmzaton methods refne the soluton near an optmal desgn based on senstvty dervatves at that pont. It s possble to derve senstvty dervatves analytcally for very smple models only; n most practcal applcatons, numercal, algorthmc, or sem-analytcal dfferentaton methods are requred. Flexble multbody systems consst of rgd and flexble bodes nterconnected through knematc jonts; typcally, they are modeled through hghly-nonlnear dfferental-algebrac equatons [1 3]. The analyss of these systems faces numerous challenges, ncludng the modelng of rgd-body moton, the choce of coordnates, the tme ntegraton scheme and the enforcement of constrants. The desgn optmzaton of flexble multbody systems also calls for the development of accurate and effcent desgn senstvty evaluaton tools. When t comes to numercal approaches, the most accurate and effcent method s complex-valued dfferentaton, whch provdes the senstvty dervatves wth respect to a sngle desgn parameter To appear n ASME Journal of Computatonal and Nonlnear Dynamcs,

2 at the cost of performng one smulaton [4, 5]. Ths approach s mpractcal for systems nvolvng a large number of desgn parameters, partcularly when runnng a sngle smulaton s tself costly [6,7]. Sem-analytcal methods, on the other hand, can handle numerous desgn parameters and objectve functons wth equal or hgher accuracy and greater performance. For nstance, drect dfferentaton methods excel when the number of parameters s smaller than the number of objectve functons. In contrast, adjont varable methods are recommended when the number of desgn parameters s larger than the number of objectve functons. Although sem-analytcal methods are the most effcent way to compute senstvty dervatves, the mplementaton of these approaches s complex and requres consderable development effort. Consequently, most commercal software packages for flexble multbody systems lack dedcated senstvty analyss tools. Optmzaton s often performed va gradent-free desgn algorthms; whenever gradent-based desgn optmzaton tools are mplemented, senstvty dervatves are commonly evaluated usng numercal dfferentaton approaches. Sem-analytcal senstvty analyss tools frst appeared n the area of optmal control and were soon extended to elastc structures, multbody systems, and many other felds of engneerng. Drect dfferentaton methods arse naturally from the dfferentaton of the equatons of moton; for multbody systems, such approaches were presented by Krshnaswam and Bhatt [8] and Chang and Nkravesh [9]. These methods, however, requre the computaton of the state senstvtes over tme for each desgn parameter,.e., roughly one smulaton of the system s requred to evaluate the senstvty dervatve wth respect to each desgn parameter. Although less ntutve, adjont varable methods have been used for years n computatonal mechancs. Haug and Arora [1, 11] adapted the approach to multbody systems; a more recent treatment s provded by Cao et al. [12]. The effcency of adjont methods along wth ncreased computatonal resources enable the treatment of ncreasngly complex desgn optmzaton problems. Consequently, the multbody dynamcs communty has often used ths approach, especally the contnuous verson (see Sonnevlle et al. [13], Dopco et al. [14], or Nachbagauer et al. [15]). Lately, the dscrete adjont method has been embraced by several authors such as Boopathy and Kennedy [16] or Lauß et al. [17,18]. When the adjont varable method s used for tme-dependent problems, a backward soluton process that requres the storage of the complete forward soluton s necessary; for large systems, ths becomes cumbersome. Furthermore, the backward process n the contnuous case requres tme ntegraton of the adjont equatons; problems arse when usng varable tme-step-sze ntegrators, leadng to accumulaton of numercal errors. The overall goal of ths work s the multdscplnary desgn optmzaton of helcopter rotors that requre the combned use of computatonal flud and structural dynamcs tools. The flud dynamcs problem nvolves a large number of desgn parameters such as arfol twst, thckness, camber, and shape at numerous locatons along the blade s span. Consequently, dscrete adjont methods have been developed for Naver-Stokes solvers (see, for nstance, Nelsen and Dskn [19] and Wang et al. [7]). Because rotor dynamcs s often formulated as a flexble multbody dynamcs problem [2], the present paper focuses on the desgn optmzaton of flexble multbody systems, where the dscrete adjont paradgm arses naturally to complement the approach used n the flud dynamcs module. The dscrete adjont method developed here presents unque features that ease soluton challenges assocated wth the backward tme ntegraton of the adjont equatons and the global parameterzaton of rgd-body moton. By dfferentatng both the governng equatons and the tme ntegraton scheme, an algebrac backward method that bypasses the need for backward tme ntegraton s 2

3 obtaned. Consequently, the soluton s the dscrete adjont of the orgnal dscrete forward problem. The analyss of multbody systems dynamcs used here s based on the moton formalsm proposed by Sonnevlle et al. [21, 22]. In the moton formalsm, poston and rotaton varables are coupled and treated as a unt referred to as a frame or a moton, whch streamlnes the treatment of rgd-body motons. Ths work makes systematc use of materal dervatves, flterng out the geometrc nonlneartes from the equlbrum equatons. 2 Flexble multbody formulaton Ths work uses a fnte element based formulaton of flexble multbody dynamcs as descrbed n the textbooks of Géradn and Cardona [2] or Bauchau [3]. The formulaton of structural elements such as rgd bodes, beams, plates, and shells are geometrcally exact,.e., fnte moton s treated exactly, although deformatons are assumed to reman small. These structural elements are connected through knematc jonts that mpose restrctons on ther relatve moton; here agan, fnte relatve motons at the jonts are treated exactly. The subsectons below summarze the formulaton of geometrcally exact beams that consttute the workhorse of many flexble multbody dynamcs codes. Ths formulaton was presented frst by Smo and Vu-Quoc [23] and further developed by Borr and Merln [24] and Hodges [25]. The present formulaton s cast wthn the framework of the moton formalsm, whch leads to mproved numercal performance, as demonstrated by Sonnevlle et al. [21, 22]. 2.1 Knematc descrpton Because flexble multbody systems undergo arbtrarly large moton, proper tools must be used to descrbe ther knematcs. Earler formulatons treat the dsplacement and rotaton felds as separate and rely on specfc parameterzatons of rotaton (see the revew paper of Géradn and Rxen [26]). More recently, the moton formalsm has attracted consderable attenton, startng wth the work of Borr et al. [27] and many other authors [28 3]. In the moton formalsm, the poston and orentaton felds are treated as a sngle unt, represented by the moton tensor of sze 6 6. Fgure 1 depcts a beam n ts reference and deformed confguratons; the nertal frame s 1 3 O F I α 1 2 C (α 1 ) C(α 1 ) b 3 F B b 2 B 3 t B b 1 F C (α 1 ) r Fgure 1: Geometrcally exact beam element F I = [O, I = (ī 1, ī 2, ī 3 )]. In the reference and deformed confguratons, the plane of the cross-secton s defned by frames F = [B, B = ( b 1, b 2, b 3 )] and F = [B, B = ( B 1, B 2, B 3 )], respectvely. The B 2 B 1 3

4 moton tensors [3] that brng frame F I to F and frame F to F are defned as [ ] R ũ R C (α 1 ) =, (1a) R [ ] Rr ũ r R C r (α 1 ) = r, (1b) R r where curvlnear varable α 1 measures length along the beam s reference lne. Rotaton tensors R and R r, both of sze 3 3, brng bass I to B and bass B to B, respectvely; the relatve poston vectors of pont B wth respect to pont O and of pont B wth respect to pont B are denoted u and u r, respectvely; notaton ã ndcates the skew-symmetrc matrx of sze 3 3 constructed from the components of vector a. If all tensor components are resolved n nertal frame F I, composton of the two motons leads to C = C r C, where moton tensor C brngs frame F I to F. The vector and skew-symmetrc matrx forms of the reference lne curvature are expressed as K(α 1 ) = { T K K(α 1 ) = C 1 C = }, (2a) [ ] K T, (2b) K where K and T are the sectonal curvature and stran vectors, respectvely. The components of curvature vector K are resolved n materal frame F. The components of the curvature vector of the reference lne, resolved n frame F and denoted K (α 1 ), are defned n a smlar manner. The sectonal strans now become { } ɛ E(α 1 ) = = K K κ. (3) Smlarly, the vector and skew-symmetrc matrx forms of the components of the sectonal velocty vector resolved n materal frame F are { } v V(α 1 ) =, (4a) ω ] [ ω ṽ Ṽ(α 1 ) = C 1 Ċ =, (4b) ω where v and ω are the translatonal and angular velocty vector of the secton. Fnally, the vrtual moton vector s defned n an analogous manner as δu = C 1 δc. (5) Because the moton tensor s a contnuous functon of ts varables, compatblty condtons mply that (Ċ) = (C ), leadng to the transpostonal equatons, V Ė = ṼK. 2.2 Equlbrum equatons The sectonal consttutve laws of the beam relate the sectonal stress resultants, stored n 6- dmensonal array F, to the sectonal stran measures defned by eq. (3) as F = D E, (6) 4

5 where matrx D, of sze 6 6, s the sectonal stffness matrx. In general, ths matrx s fully populated and accounts for couplng between the varous deformaton modes. Typcally, the sectonal stffness and mass matrces are evaluated through beam analyss tools that compute the soluton to Sant-Venant s problem. For nstance, Han and Bauchau [31] have developed SectonBulder, a fnte element based tool for the analyss of cross-sectons of beams of arbtrary confguraton made of ansotropc materals. The partal dfferental equatons that descrbe the dynamc behavor of the beam are F I + F E = F A, F I = M V ṼT M V, F E = F K T F, (7a) (7b) (7c) where vectors F I, F E, and F A are the sectonal nertal, elastc, and appled force vectors, respectvely, and matrx M, of sze 6 6, s the sectonal mass tensor. Typcally, these equatons are dscretzed n both space and tme. The system s dscretzed n the spatal doman by usng n nodes wth sx degrees of freedom at each node for a total of 6n degrees of freedom. The dscretzed confguraton feld s represented by a set u = {C 1,..., C n } that gathers the confguratons (moton tensors) of all nodes; the varatons of ths set form a lnear space δu T = {δu T 1,..., δu T n}, of sze 6n. Smlarly, the velocty and acceleraton vectors at all nodes are stored n arrays v T = {V T 1,..., V T n} and v T = { V T 1,..., V T n}, both of sze 6n. Wthn a fnte element, the stran, velocty, and acceleraton felds are nterpolated as δe = B(α 1 ) δu, V = H(α 1 ) v, V = H(α 1 ) v, (8a) (8b) (8c) where matrces H and B are the dsplacement and stran nterpolaton matrces, respectvely, both of sze 6 6n. The spatally dscretzed equatons of moton now become f I + f E = f A, [ L ] f I = H T M H dα 1 v f E = L B T F dα 1. L H T ( H v ) T M H v dα1, To formulate the tme ntegraton process, t s convenent to lnearze the nertal and elastc forces to obtan the mass, gyroscopc, and stffness matrces, defned as M f I G f I v = K f E v = u = L L L H T M H dα 1, (ṼT H T M + P ) H dα 1, ( ) B T D + H T F B dα 1, 5 (9a) (9b) (9c) (1a) (1b) (1c)

6 where P = M V s the 6-dmensonal momentum, and ( ) s an operator that, gven two generc vectors x and y, satsfes x T y = yx. The lnearzed equatons of moton now read M δ v + G δv + K δu = δf A. (11) The equatons of moton for other structural components such as rgd bodes, plates, or shells can be obtaned n a smlar manner. The matrces defned n eqs. (1) wll also be useful to compute the adjont of the dynamc equatons. 2.3 Multbody equatons In flexble multbody problems, the varous elastc components of the system are connected by mechancal jonts that are modeled as knematc constrants. Because jonts restrct the relatve moton between adjacent elastc bodes, the constrants they mpose depend on ths relatve moton only and hence, are nvarant under the superposton of a rgd-body moton. The m holonomc (but possbly rheonomc) constrant equatons to be enforced are collected n an array of sze m that s a functon of system confguraton, C(u, t) =. (12) The varaton of the constrant vector gves δc = B(u, t)δu, (13) where B s the Jacoban matrx that can be evaluated easly for varous types of jonts [22]. The constrant forces are then formulated through the augmented Lagrangan approach [32, 33] as f C = B T (λ + pc), (14) where array λ, of sze m, stores the Lagrange multplers and p s a user-defned penalty factor. The varaton of the constrant forces becomes δf C = ( X + pb T B ) δu + B T δλ, (15) where matrx X = X(λ + pc) stores the second dervatves of the constrants postmultpled by a partcular value of λ + pc. The fnal expresson of the equatons of moton becomes f I (v, v) + f E (u) + f C (u, λ) = f A (u, v), Ċ k = C k Ṽ k, k = 1,..., n, (16a) C(u, t) =. (16b) (16c) Equaton (16a) s known as the knematc compatblty equaton because t expresses the compatblty of the confguraton and velocty felds. Equatons (16) form a system of nonlnear dfferental-algebrac equatons that represent the dynamcs of the system n ntrnsc form. 6

7 2.4 Tme dscretzaton The generalzed-α scheme [34] s used to dscretze the governng equatons (16) n tme. A typcal tme step extends from tme t 1 to tme t, and h = t t 1 denotes the tme step sze. Subscrpt ( ) ndcates quanttes evaluated at tme t. The ntegrator s characterzed by the followng equatons, q = hv 1 + h 2 [ ( 1 2 β)x 1 + βx ], v = v 1 + h [ (1 γ)x 1 + γx ], x = c 6 x 1 + c 5 v 1 + c 9 v, (17a) (17b) (17c) where vector x denotes algorthmc acceleraton varables that are related to the physcal acceleratons through recurrence relatonshp (17c). Vector q = {Q T,..., 1 QT } stores the nodal ncremental n motons of all nodes between tme t 1 and tme t. The followng auxlary ntegraton parameters are defned, c 1 = βα ( f h 2, c 4 = 1 γ ) h, c 7 = γ(1 α f) h, (18a) 1 α m 1 α m 1 α m c 2 = c 3 = ( 1 2 β ) h 2, c 5 = α f 1 α m, c 8 = β(1 α f) 1 α m h 2, (18b) 1 α m γα f h, c 6 = α m, 1 α m 1 α m c 9 = 1 α f, 1 α m (18c) where α m (ρ), α f (ρ) are closed-form functons of the level of numercal dsspaton nherent to the ntegraton scheme, ρ, as are γ(α m, α f ) and β(γ). More detals can be found n [34]. Let C 1 and C be the moton tensors of a partcular node at tmes t 1 and t, respectvely. The soluton of the generalzed-α equatons provdes the ncremental moton vector at tme t, denoted Q. The confguraton of the node at tme t s now obtaned from the followng composton of moton operaton C = C 1 C nc, (Q ), (19) where C nc, s the ncremental moton tensor assocated wth ncremental moton vector Q. To evaluate ths ncremental moton tensor, the ncremental moton vector s assumed to form the Cayley-Gbbs-Rodrgues moton parameter vector [3], leadng to C nc, = I + Z( ζ 1, ζ 1 ) Q + Z( ζ 2, ζ 2 ) Q Q. (2) The parameters are evaluated as follows: Q T = {q T 1, qt 2 }, r = 1/(1 + q T 2 q 2 /4), ζ 1 = r, ζ 2 = r /2, ρ = q T 2 q 1, ζ 1 = ρr 2 /2, ζ 2 = ρr 2 /4, and fnally, Z(α, β) = [ ] βi αi. (21) βi Note that the ncremental moton vector can be assumed to form any moton parameter vector because all converge to the nfntesmal moton vector at the same rate [3]. Because the moton tensor s of sze 6 6, eqs. (19) form a hghly-redundant set of equatons: ndeed, only sx equatons are ndependent. To overcome ths problem, the knematc update equaton 7

8 s recast as C 1 C = C 1 C 1 C nc, = I and the sx ndependent equatons are selected to be the axal part of the tensor, ( ) ( A C, C 1, Q axal C 1 C 1 C nc, ) =, (22) because axal(i) =. The knematc update equaton s now expressed as mplct functon A for a sngle node. Collectng these condtons at all nodes provdes mplct functon A = for the complete system. To solve the nonlnear equatons teratvely, the ntegrator equatons can be recast nto a predctor-corrector scheme where the ncremental correctons at each teraton are computed as [M + c 7 G + c 8 ( K + X + pb T B ) c 8 B T c 8 B ] } { v = λ { f A f I f E C f C }, (23) untl convergence. More nformaton on ths soluton scheme can be found n Sonnevlle et al. [21]. 2.5 Intal condtons To enable the soluton of the problem, boundary condtons must be provded. For typcal ntal value problems, the confguraton and velocty of all nodes are provded at tme t =. Two mplct functons, denoted P (u, v ) = and S(v, v ) =, are used to mpose the ntal condtons on the confguraton and velocty felds, respectvely. The ntal acceleratons are obtaned by solvng the equatons of moton at the ntal tme. Fnally, the ntal value of the Lagrange multplers s obtaned from the constrant equatons by means of one addtonal mplct functon denoted C (u, v, v ) =. To obtan detaled expressons of these functons, consder the confguraton of a partcular node at tme t =, denoted C n ; the ntal condton at ths node now becomes C = C n. Ths condton s recast as C 1 C = I and the ntal condtons are extracted as the axal part of ths tensor, n P axal(c 1 C ) =. (24) n Collectng these condtons at all nodes provdes mplct functon P = for the complete system. Next, the condton on the ntal veloctes, denoted v n, smply states that S v v n =. (25) Fnally, takng a second dervatve of eq. (16c) wth respect to tme provdes an addtonal m equatons that can be used to calculate the ntal values of the Lagrange multplers C B v + Ḃ v + Ċt =, (26) where C t s the partal dervatve of the constrant equatons wth respect to tme at t =. Ths equaton represents the acceleraton level constrant equatons. 3 Dscrete adjont method The dscrete adjont method derves the adjont equatons of the fully dscretzed system. Hence, the dscrete adjont equatons depend on both spatal and temporal dscretzaton schemes. The strategy descrbed below apples to both rgd and flexble multbody systems. It can be adapted easly to deal wth dfferent tme ntegrators and can handle arbtrary ntal condtons. 8

9 3.1 Forward equatons To facltate the development of the senstvty analyss, the dscrete equatons of moton used n the forward analyss are recast as mplct functons of the unknowns of the problem. Array b gathers the desgn parameters. Intal condtons: = P (u, v ) =, S(v, v ) =, X (x ) x x n =, F (u, v, v, λ, b) f I (v, v ) + f E (u ) + f C (u, λ ) f A =, C (u, v, v, b) =, (27a) (27b) (27c) (27d) (27e) Tme steppng: < N F (u, v, v, λ, b) f I (v, v ) + f E (u ) + f C (u, λ ) f A C (u, b) =, U (q, v 1, x, x 1 ) q hv 1 h 2 [ ( 1 2 β)x 1 + βx ] =, V (v, v 1, x, x 1 ) v v 1 h [ (1 γ)x 1 + γx ] =, X (x, x 1, v, v 1 ) x c 6 x 1 c 5 v 1 c 9 v =, A (u, u 1, q ) =. =, (28a) (28b) (28c) (28d) (28e) All mplct functons appearng n eqs. (27) and (28) are of sze 6n 1, except for mplct functons C and C, whch are of sze m 1. Implct functons (27a) and (27b) mpose the ntal condtons on confguraton and velocty whle mplct functon (27c) provdes ntal condtons for the algorthmc acceleratons. Typcally, algorthmc acceleraton x are set to vansh t =. Fnally, mplct functon (27e) s used to compute the ntal values of the Lagrange multplers. Implct functons (27d) and (28a) enforce the governng equatons of moton at all tmes whle mplct functon (28b) mposes the knematc constrants. The tme ntegrator equatons are mposed by mplct functons (28c) (28e), and mplct functon (28f) corresponds to the knematc update equaton. All state varables depend on the desgn parameters mplctly; potentally, the governng and constrant equatons depend on the desgn varables explctly. The followng varables are defned only at tme step = : u, v, v, x, and λ. The remanng varables, u, v, v, x, λ, and q are defned at tme steps < N. 3.2 Adjont equatons A generc objectve functon that depends on the dscrete states and on the desgn parameters s defned frst, ψ = ψ(u,..., u N, v,..., v N, v,..., v N, λ,..., λ N, b). (29) Note that the algorthmc acceleratons and moton ncrements are defned only for algorthmc purposes and, hence, are not part of the objectve functon. Next, the Lagrangan of the system s 9 (28f)

10 defned as L = ψ + Λ T P P + Λ T SS + Λ T XX + Λ T F F + Λ T CC N ( ) (3) + Λ T F F + Λ T CC + Λ T UU + Λ T V V + Λ T XX + Λ T AA. =1 Felds of Lagrange multplers, denoted Λ and referred to as adjont varables, have been defned. Each ntal condton (27) and each mplct governng equaton (28) s multpled by a feld of adjont varables. Because all ntal condtons and governng equatons wll be satsfed, the adjont varables can be chosen arbtrarly and the Lagrangan reduces to the objectve functon. Note that for > N, the adjont varables are not defned and are assumed to vansh, Λ F = Λ C = Λ U = Λ V = Λ X = Λ A =. The varaton of the Lagrangan s δl = δψ = ψ N b δb + ψ N ψ N ψ N ψ δu u + δv = v + δ v = v + δλ = λ = ( P + Λ T P δu u + P ) ( S δv v + Λ T S δv v + S ) δ v v + Λ T X X δx x ( F + Λ T F b δb + F δu u + F δv v + F δ v v + F ) δλ λ ( + Λ T C C b δb + C δu u + C δv v + C ) δ v v N ( F + Λ T F b δb + F δu u + F δv =1 v + F δ v v + F ) δλ λ N ( + Λ T C C b δb + C ) δu u =1 ( ) N + Λ T U U δq q + U δv =1 v 1 + U δx 1 x + U δx x 1 1 N ( V + Λ T V δv v + V δv =1 v 1 + V δx 1 x + V ) δx x 1 1 N ( + Λ T X X δx x + X δx =1 x 1 + X δ v 1 v + X ) δ v v 1 1 ( ) N + Λ T A A δu u + A δu u 1 + A δq 1 q, =1 (31) where the partal dervatve wth respect to the confguraton, denoted / u, represents the drectonal dervatve of the functon along the drecton of vrtual moton δu. Equatons (31) nvolve varatons of state varables δu, δv, δx, δλ and δq, whch should be nterpreted as δu = (du /db)δb, wth smlar expressons for the other state varables. Ths observaton stems from the fact that the response of the system depends on the desgn varables; 1

11 the term state senstvtes s used to refer to total dervatves of state varables wth respect to the desgn varables,.e., terms such as du /db, dv /db, etc.. Typcally, state senstvtes cannot be evaluated easly wthout carryng out a costly drect senstvty analyss. The goal of the adjont method s to bypass the need to compute state senstvtes. To acheve ths goal, the followng strategy s mplemented: senstvty dervatves are factored out n eq. (31), and the terms multplyng them are set to vansh. Because the adjont varables can be chosen arbtrarly, they are selected to satsfy the equatons correspondng to the vanshng of the terms multplyng the state senstvtes; the resultng equatons are called the adjont equatons. Once the adjont varables are known, the remanng terms n eq. (31) can be evaluated easly. Implementaton of the process descrbed above leads to two sets of adjont equatons. Tme steppng: < N ψ + Λ T F F + Λ T C C + Λ T A A + Λ T A +1 A,+1 =, (32a) u u u u u ψ + Λ T F F + Λ T U +1 U,+1 v v v ψ v + Λ T F Λ T U + Λ T V V v F v + Λ T X X v U q + Λ T A A =, (32b) q + Λ T V +1 V,+1 + Λ T X +1 X,+1 Λ T U U + Λ T U +1 U,+1 + Λ T V V + Λ T V +1 V,+1 + Λ T X X + Λ T X +1 X,+1 x x x x x Intal condtons: = ψ λ + Λ T F v =, (32c) v =, (32d) =, (32e) x F =, (32f) λ ψ + Λ T P P + Λ T F F + Λ T C C + Λ T A 1 A1 =, u u u u u (33a) ψ + Λ T P P + Λ T S S + Λ T F F + Λ T C C + Λ T U 1 U1 + Λ T V 1 V 1 =, v v v v v v v (33b) ψ + Λ T S S + Λ T F F + Λ T C C + Λ T X 1 X1 =, v v v v v (33c) Λ T X X + Λ T U 1 U1 + Λ T V 1 V 1 + Λ T X 1 X1 x x x x =, (33d) ψ + Λ T F F =. λ λ (33e) The soluton of the adjont equatons s a backward process that goes from = N to =. The soluton of the adjont equatons at = N makes use of the fact that, > N, Λ F = Λ C = Λ U = Λ V = Λ X = Λ A =. 11

12 3.3 Partal dervatves The soluton of the adjont equatons requres the evaluaton of a number of partal dervatves, such as the partal dervatves of the knematc update equaton (22). Partal dervatves A / U, A / U 1, and A / Q wll be evaluated for a sngle node frst. It s convenent to start from the knematc update equaton wrtten n tensor form as A C 1 leads to δa = δc 1 δa = δu + ( C 1 C 1 C nc, + C 1 C 1 C nc, = I, whose varaton δ(c 1 C nc, ). Equaton (5) and further manpulaton now yeld δu nc, 1) + ( T (Q )δq ), where T s the tangent operator correspondng to a sngle node. Fnally, the vector form of the equaton s δa axal(δa ) = δu + C 1 nc, δu 1 + T (Q )δq. (34) For the complete system, the desred drectonal dervatves become A = I, (35a) u A ( ) = dag C 1 u,..., nc,1 C 1 C 1, (35b) nc,n nc, 1 A ( = dag T (Q q 1 ),..., T (Q n )) T (q ), (35c) where C 1 nc, collects the nverse ncremental moton of all n nodes at tme level, and T (q ) collects the ncrement tangent operators of all n nodes at tme level. The drectonal dervatve of constrant equatons (28b) are obtaned from eq. (13) as C u = B. (36) Partal dervatves of the system elastc an nertal forces are obtaned easly from the lnearzaton of the governng equatons of moton (11). Fnally, the partal dervatves of the constrant forces wth respect to the confguraton and the Lagrange multplers stem from eq. (15) as 3.4 Adjont soluton process f C = ( X + pb T B ) u, (37a) f C = B T λ. (37b) The frst step n the soluton of the adjont problem, s to solve eqs. (32) for < N. Introducng partal dervatves (1), (35), (36) and (37) nto eq. (32) and rearrangng the equatons leads to the followng backward soluton process [ M T + c 7 G T ( + c 8 T T nc, K T + X T + pb T B ) c 8 T T nc, BT c 8 B 12 ] {ΛF } Λ C { } BF =. (38) B C

13 The rght-hand sde term s ( ) T ( ) T ( ) T ψ ψ ψ B F = c 8 T T c nc, 7 c 8 T T u v v nc, C T Λ nc,+1 A,+1 c ( 8 β γ 1 ) Λ β 2 U,+1 + c 8 hβ Λ V,+1 + (c 5 + c 9 c 6 ) Λ X,+1, B C = c 8 ( ψ λ (39a) ) T, (39b) where T nc, = T (q ). At the frst level of the process ( = N), the terms wth + 1 subscrpts vansh. The leadng matrx n eq. (38) s the transpose of the leadng matrx n the forward teratve problem (see eq. (23)), whch tself s the lnearzed verson of the governng equaton n eq. (16b). Note that T nc, was omtted n eq. (23) as an approxmaton that typcally does not affect convergence n the forward problem; on the other hand, t must be ncluded n eq. (38) to obtan accurate senstvtes. The remanng adjont varables can be computed n the followng sequence, ( ) T ψ Λ V = G T v Λ F + hλ U,+1 + Λ V,+1, (4a) [ ( Λ X = 1 ) ] T ψ + M T c 9 v Λ F c 5 Λ X,+1, (4b) Λ U = 1 [ h 2 (β 1 h 2 2 β )Λ ] U,+1 hγλ V + h(γ 1)Λ V,+1 + Λ X c 6 Λ X,+1, (4c) Λ A = ( T T nc,) 1ΛU. (4d) These equatons provde suffcent nformaton to solve for the adjont varables n a backward manner from = N to = 1. Note that no tme ntegraton s requred, but merely an algebrac procedure that s carred out sequentally. 3.5 Intal adjont varables The last step of the adjont soluton conssts of computng the adjont varables at =. Substtutng the partal dervatves n eq. (33) and rearrangng the equatons leads to (K T + X T + pb T B) ( C u ) T ( P u ) T Λ ( ψ B F u ) T C T Λ nc,1 A1 Λ M T ( C ) T ( S C ( ψ ) T λ = ) T Λ v v P ( ψ ) ( C ) T ( P ) T ( S ) T Λ S T, (41a) + c v 5 Λ X1 ( ψ ) T + hλ v v v v U1 + Λ V 1 G T Λ X = h 2 (β 1 2 )Λ U1 h(γ 1)Λ V 1 + c 6 Λ X1. (41b) The partal dervatves appearng n these equatons depend on the type of ntal condtons mposed on the problem. As an example, these quanttes wll be evaluated for ntal value problems. For a sngle node, the ntal condton on the confguraton states that P C 1 C = I, and the varaton n 13

14 of ths equaton yelds δp = δc 1 C = δu n C 1 C = δu n. The vector form of ths statement s δp = axal(δp) = δu. For the complete system, the followng partal dervatves result, P = I, u P =, v S = I, v S =, v C = Ḃ, v (42a) C = B v. (42b) Equaton (41) now reduces to 4 Desgn senstvtes ( ) [ M T B T ] { } T ψ ΛF v + c5 Λ = X1 ( ) B Λ C T ψ, (43) λ Λ P = ( K T + X T + pb T B ) ( ) T ψ Λ F + + C T u Λ nc,1 A1, (44) ( ) T ψ Λ S = + hλ v U1 + Λ V 1 G T Λ F. (45) The desgn senstvty expressons assocated wth dfferent types of desgn parameters are presented here. The dscrete adjont method elmnates state senstvtes from eq. (31), and the varaton of the Lagrangan now reduces to δψ = ψ N b δb + F ΛT F b δb + C ΛT C b δb + ( ) Λ T F F b δb + C ΛT C b δb, (46) =1 where the adjont varables are known. 4.1 Jont parameters Certan multbody systems nclude sprng and damper forces that act on the knematc jonts, e.g. vehcle suspenson bushngs or elastc manpulator jonts. If the relatve moton and velocty at the jont are denoted u r and v r, respectvely, the force n the jont s then fr A = cv r ku r, where c and k are the dampng and stffness coeffcents, respectvely. If those dampng and stffness coeffcents are consdered to be desgn parameters, b = {c, k} T, the constrant equatons reman ndependent of these parameters, that s, C / b = and C / b =. The varaton of the Lagrangan then reduces to N dψ db = F ΛT F b + ( ) Λ T F F. (47) b =1 The partal dervatves n eq. (47) become F r b = f A r b = {v r, u r }, (48) 14

15 where =,..., N. Fnally, the desgn senstvtes can be wrtten as 4.2 Cross-sectonal parameters N dψ db = Λ F r {v r, u r }. (49) = In multbody systems that nvolve beam elements, the sectonal stffness and mass propertes of the beam characterze the behavor of the system, as seen n eq. (6). Sectonal propertes, however, do not consttute a relevant set of desgn parameters. Indeed, gven a set of sectonal stffness and mass propertes, t mght be dffcult and often mpossble to manufacture a secton that presents those propertes. Ths s often dsregarded by many authors, who select sectonal stffness and mass propertes as desgn parameters. In aerospace applcatons, wngs or rotor blades often present complex geometrc shapes and are made of advanced, hghly ansotropc composte materals. Sectonal propertes must then be evaluated based on a detaled descrpton of the secton ncludng geometry, ply thcknesses, fber orentaton angles, materal constants, etc. In mechancal engneerng applcatons, complex geometres and advanced materals are often used to desgn hgh-precson mechansms; here agan, the accurate evaluaton of sectonal propertes s requred. In many applcatons, the sectonal stffness matrx becomes fully populated and ts evaluaton becomes a complex task that requres the use of fnte element models of the cross-secton. Callejo et al. [35] presented an adjont varable method for the computaton of cross-sectonal senstvtes. By couplng ther sectonal senstvty analyss and the dynamc senstvty analyss presented n ths paper, t becomes possble to address a realstc desgn problem: the desgn optmzaton of dynamcal systems wth respect to the local propertes of flexble elements. Let the 6 6 mass and stffness propertes of the cross-secton be functons of sectonal desgn parameters,.e., M = M(b) and D = D(b). In ths case, the constrant equatons are ndependent of the desgn parameters and eq. (47) stll provdes desgn senstvtes. The unknown partal dervatves can be obtaned by usng the chan rule of dfferentaton for eq. (1), F b = f I M ( L f I M = f E D = L M b + f E D D b, H T M M Hdα 1 B T D D E dα 1, ) v L H T ( H v ) T M M H v dα 1, (5a) (5b) (5c) where M/ M and D/ D are fourth-order tensor denttes that are defned as follows: a jkl = 1 f = k and j = l; a jkl = otherwse. The partal dervatves of the sectonal mass and stffness matrces wth respect to the desgn parameters can be obtaned from the adjont method appled to the cross-sectonal analyss problem [35] or, alternatvely, can be computed through real- or complex-valued dfferentaton. 15

16 5 Results Two mechancal systems are analyzed to valdate the proposed method. In order to nvestgate the effect of drastcally dfferent structural propertes, a fully rgd system and a flexble one are selected. The sem-analytcal desgn senstvtes predcted by the proposed adjont method are compared to those obtaned wth real- and complex-step numercal dfferentaton approaches. The real-step approach s mplemented va the central fnte dfference scheme wth a relatve perturbaton of b/b = 1 7 ; two evaluatons of the objectve functon per desgn parameter are necessary. The complex-step approach, on the other hand, uses a relatve perturbaton sze of b/b = 1 15, and requres only one evaluaton per desgn parameter [4, 5]. 5.1 Quarter-car suspenson The frst example conssts of a quarter-car suspenson system n whch the wheel s connected 8 to the fxed element through a MacPherson strut, Strut a lower control arm, and a steerng mechansm. 9 Knuckle The system topology s shown n fg. 2 and the Te rod coordnates of the ponts are lsted n table 1. 1 The reference frame s located under the front 5 4 axle at ground level, wth the x axs pontng forward and the z axs pontng up. 6 Revolute jonts are located at ponts 1, 2, and 1 3 6; sphercal jonts at ponts 3, 5, and 8; prsmatc Lower control arm jonts at ponts 9 and 11; and a unversal jont at pont 4. Ponts 1, 2, 8, and 11 are attached 7 2 to the vehcle body, whch s consdered to be fxed. The system has 3 degrees of freedom and a total of 246 dependent coordnates. The lower Fgure 2: Quarter-car suspenson control arm s attached to the fxed element va two revolute jonts; the redundant constrant equatons are dscarded by the algorthm. Table 1: Quarter-car model pont coordnates # Coordnates (m) # Coordnates (m) x y z x y z Steerng All bodes are consdered to be rgd. The bars have a unform densty of ρ = 7, 8 kg/m 3 and a dameter D = 3 mm. The knuckle has a mass m k = 3 kg and an nerta tensor J k = dag (1, 1, 16 11

17 1) kg m 2 wth respect to pont 5, whereas the wheel has a mass m w = 23 kg and an nerta tensor J w = dag (.633, 1.15,.633) kg m 2 wth respect to ts attachment pont. The prsmatc jont located at pont 9 holds a sprng-damper set of stffness constant k = 1 kn/m and dampng constant c = 1 kn s/m. The system s ntally at rest, and no gravtatonal feld exsts. The angular velocty of the wheel s set to ω = 2π rad/s, whereas the degree of freedom assocated wth the steerng rack s prescrbed as d = 5 sn(πt) mm. To excte the degree of freedom of the control arm, a vertcal force f z = 2(1 cos 2πt) kn s appled to pont 6. The system s smulated for t f = 2π/ω s wth a tme-step sze of t = 2 ms. The effect of the suspenson parameters on the vertcal response of the wheel s nvestgated. The stffness and dampng constants of the strut are the desgn parameters, whereas the objectve functon s defned as ψ tf t v 2 6z(t) dt, (51) where v 6z s the vertcal component of the velocty of the wheel attachment pont. The dscrete adjont method was used to obtan senstvty dervatves dψ/dk and dψ/dc. Table 2: Accuracy of quarter-car senstvty dervatves Method dψ/dk [m 3 /(N s)] Rel. error Adjont Complex step Real step Method dψ/dc [m 3 /(N s 2 )] Rel. error Adjont Complex step Real step Table 2 lsts the desgn senstvty dervatves obtaned through the three dfferentaton methods; the last column of the table lsts the relatve error n the senstvtes wth respect to those obtaned wth the adjont method. The senstvty dervatves obtaned by the three approaches are found to be n good agreement. The predctons of the adjont and complex-step methods are the most accurate. The real-step dfferentaton approach s very senstve to the perturbaton sze, as shown n fg. 3, whch depcts senstvty dervatve dψ/dk versus the relatve perturbaton sze: relable predctons are obtaned only wth relatve perturbaton szes b/b [1 8, 1 2 ]. On the other hand, the complex-step method exhbts lower senstvty to the perturbaton sze: accurate predctons are obtaned wth relatve perturbaton szes b/b < 1 2. Of course, because the adjont method s sem-analytcal, t does not rely on an assumed perturbaton sze. 5.2 Rotatng beam 17

18 Absolute Value Perturbaton Adjont method Complex step Real step Value Absolute Adjont method Complex step Real step Perturbaton Fgure 3: Quarter-car senstvty dervatve dψ/dk versus relatve perturbaton sze k/k Fgure 4: Rotatng beam senstvty dervatve dψ/dd 55 versus relatve perturbaton sze D 55 /D 55 The second example conssts of a beam of length L = 2 m that s ntally located along the x axs and at rest. No gravtatonal feld exsts. The root of the beam s constraned to rotate at a constant angular velocty ω = 1 rad/s about the z axs. A constant vertcal load, f z,tp = 1 kn, s appled to the tp of the beam. The system s smulated for t f = 2π/ω s wth a tme-step of t = 2 ms. The sectonal stffness matrx of the beam s D = 1 5 dag(1 3, 1 2, 1 2, 1 1, 1, 1), (52) where the unts of the axal and shearng components are N and the unts of the torson and Fgure 5: Rotatng beam bendng components are N m 2. The sectonal mass matrx of the beam s M = 1 1 dag(1 2, 1 2, 1 2, 1, 1, 1), (53) where the unts for the translatonal and rotatonal components are kg/m and kg m 2 /m, respectvely. The beam s dscretzed nto 1 four-node fnte elements. A sequence of the dynamc response s shown n fg. 5. In ths problem, the dagonal components of the sectonal stffness matrx D are selected as the desgn parameters, and the objectve functon s ψ tf t ω 2 y,tp(t) dt, (54) where ω y,tp s the y component of the angular velocty of the tp node n the materal frame. Table 3 lsts the desgn senstvty dervatves obtaned through the three dfferentaton methods; the last column of the table lsts the relatve error n the senstvtes wth respect to those obtaned wth the adjont method. Here agan, the senstvty dervatves obtaned by the three approaches are found to be n good agreement. As expected, the real-step dfferentaton approach s very senstve 18

19 Table 3: Accuracy of the rotatng beam senstvty dervatves Method dψ/dd 11 [rad 2 /(N s)] Rel. error Adjont Complex step Real step Method dψ/dd 22 [rad 2 /(N s)] Rel. error Adjont Complex step Real step Method dψ/dd 33 [rad 2 /(N s)] Rel. error Adjont Complex step Real step Method dψ/dd 44 [rad 2 /(N m 2 s)] Rel. error Adjont Complex step Real step Method dψ/dd 55 [rad 2 /(N m 2 s)] Rel. error Adjont Complex step Real step Method dψ/dd 66 [rad 2 /(N m 2 s)] Rel. error Adjont Complex step Real step to the perturbaton sze, as shown n fg. 4, whch depcts senstvty dervatve dψ/dd 55 versus the relatve perturbaton sze. Relable predctons are obtaned only wth relatve perturbaton szes b/b [1 9, 1 3 ]. On the other hand, the complex-step method exhbts lower senstvty to the perturbaton sze: accurate predctons are obtaned wth relatve perturbaton szes b/b < 1 3. Of course, because the adjont method s sem-analytcal, t does not use perturbatons. The proposed dscrete adjont method took 11 s of CPU tme to compute the senstvty dervatves wth respect to sx desgn parameters wthn a MATLAB mplementaton. In contrast, the complex- and real-step methods took 567 s and 797 s, respectvely. Ths mples that the complexand real-step methods are 5.63 and 7.92 tmes slower that the dscrete adjont method on the bass of ths prelmnary mplementaton. 19

20 6 Conclusons Ths paper has presented a dscrete adjont varable method for the senstvty analyss of rgd and flexble multbody systems. In contrast wth the more commonly used contnuous adjont method, the dscrete approach computes the exact adjont of the dscrete forward soluton. By dfferentatng both the governng equatons and the tme ntegraton scheme, the dscrete adjont approach leads to a backward algebrac procedure that bypasses the need for backward tme ntegraton. Moreover, the proposed approach s based on the moton formalsm, whch provdes a unfed representaton of the dsplacement and rotaton felds, resultng n a forward method that s computatonally effcent and devod of sngulartes. The dervatons of the governng, adjont and desgn senstvty equatons were presented n detal. The proposed sem-analytcal approach has been appled to the analyss of a quarter-car suspenson system and to a rotatng beam. The real- and complex-step dfferentaton methods have been used to valdate the predctons of the dscrete adjont method. The senstvty dervatves obtaned by the three approaches are found to be n good agreement. The results of the real-step dfferentaton approach are very senstve to the relatve perturbaton sze; relable predctons are obtaned only for a range of perturbaton szes. A far lower senstvty to the relatve perturbaton sze was observed for the complex-step method. Overall, the dscrete adjont method has proven to be sgnfcantly more accurate and effcent than the numercal approaches. Ongong work ams at the mplementaton of the dscrete adjont method n a general-purpose flexble multbody dynamcs code, openng the door to optmzaton procedures that would be mpractcal wth less effcent and accurate senstvty analyss tools. Acknowledgments Ths work was supported by NASA s Revolutonary Vertcal Lft Technology Project contract NNL15AB93T. References [1] P. E. Nkravesh. Computer-Aded Analyss of Mechancal Systems. Prentce-Hall, Englewood Clffs, New Jersey, [2] M. Géradn and A. Cardona. Flexble Multbody System: A Fnte Element Approach. John Wley & Sons, New York, 21. [3] O. A. Bauchau. Flexble Multbody Dynamcs. Sprnger, Dordrecht, Hedelberg, London, New-York, 211. [4] J. C. Newman, W. K. Anderson, and D. L. Whtfeld. Multdscplnary senstvty dervatves usng complex varables. Techncal Report MSSU-EIRS-ERC-98-8, Msssspp State Unversty, Msssspp State, MS, [5] J. R. R. A. Martns, P. Sturdza, and J. J. Alonso. The complex-step dervatve approxmaton. Transactons on Mathematcal Software (TOMS), 29(3): , 23. 2

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