Adjoint Methods of Sensitivity Analysis for Lyapunov Equation. Boping Wang 1, Kun Yan 2. University of Technology, Dalian , P. R.

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1 th World Congress on Structural and Multdscplnary Optmsaton 7 th - th, June 5, Sydney Australa Adjont Methods of Senstvty Analyss for Lyapunov Equaton Bopng Wang, Kun Yan Department of Mechancal and Aerospace Engneerng, Unversty of exas at Arlngton, P.O. Box 93, Arlngton, X 769, USA, bpwang@uta.edu State Key Laboratory of Structural Analyss for Industral Equpment, Faculty of Vehcle Engneerng and Mechancs, Dalan Unversty of echnology, Dalan 63, P. R. Chna. Abstract he exstng drect senstvty analyss of optmal structural vbraton control based on Lyapunov s second method s computatonally expensve when appled to fnte element models wth a large number of degree-of-freedom and desgn varables. A new adjont senstvty analyss method s proposed n ths paper. Usng the new method the senstvty of the performance ndex, a tme ntegral of a quadratc functon of state varables, wth respect to all desgn varables s calculated by solvng two Lyapunov matrx equatons. wo numercal examples demonstrate the accuracy and effcency of the proposed method. Fnally, we use the adjont senstvty analyss scheme to solve a topology optmzaton problem.. Keywords: Adjont method, senstvty analyss, topology optmzaton 3. Introducton In tme doman, there s a classc problem formulaton of passve structural vbraton control that deals wth the dynamc system dsturbed by ntal condtons. he objectve s to fnd desgn parameters of the damped vbraton system that mnmze the performance ndex n the form of tme ntegral of the quadratc functon of state varables (dsplacement and veloctes, e.g. see Eq.(5)). hs performance ndex can be evaluated by Lyapunov s second method []. Based on the Lyapunov equaton, the evaluaton of performance ndces are smplfed nto matrx quadratc forms and do not requre the tme doman ntegraton. Parameter optmzaton problems wth a quadratc performance ndex have been solved by ths method [~8]. Wang et al. [9] appled the Lyapunov equaton to solve the transent response optmzaton problem of lnear vbratng systems excted by ntal condtons. In ther wor, the Lyapunov equaton was expanded to a set of lnear equaton and drect senstvty was carred out by use of the same system of lnear equaton. he computatonal effectveness of the method s llustrated by applyng t to the classcal vbraton absorber and to a cantlever beam carryng an absorber at ts mdpont. Du [] appled the Lyapunov equaton to obtan the optmum confguraton of dynamc vbraton absorber (.e., DVA) attached to an undamped or damped prmary structure. he Lyapunov equaton s also used n other felds of optmal desgn. In ths paper, we consder one case of passve control optmzaton problem, that s, to mnmze an ntegrated quadratc performance measure for damped vbratng structures subjected to ntal condtons. he goal of ths paper s to present an adjont senstvty analyss method consderng the above mentoned objectve functon based on Lyapunov s second functon. he results ndcate the potental of applcaton of the proposed method to topology optmzaton under the specal tme doman crteron. 4. Applcaton of Lyapunov s second method to optmze transent response of mechancal systems Consder a vscously damped lnear vbraton system governed by the equaton: M u& + Cu& + Ku () where M(N N) s the mass matrx, C(N N) s the dampng matrx, K(N N) s the stffness matrx, and u(n ) s dsplacement vector. N s the structural degree of freedoms. Assume the system s excted by ntal dsplacements or veloctes. he desgn problem s to fnd n M, K and C matrces to mnmze a performance matrx n the form & where, q( u, u) u Q u + u Q u (,u) J q u & dt () & & s a quadratc functon of u and u&. ransent dynamc responses have to be u &u performed to evaluate the objectve functon. Drect or adjont methods can be appled to evaluate the response senstvty requred for evaluaton senstvty of the performance. Alternatve, f we replace the upper bound of ntegraton to nfnte, we can use Lyapunov s second method to evaluate the performance wthout performng transent dynamc response analyss. o apply Lyapunov s second method to ths system, t s necessary to rewrte Eq.() n the state space form

2 Where X & AX (3) O I u X M K M C u & A he matrx A s (N N). he vector X s (N ). Structural desgn parameters such as mass densty, dampng rato and sprng stffness are contaned n the matrx A. he optmzaton problem s to choose these parameters to mnmze the performance measure J defned by J X QX dt (5) for a gven ntal exctaton X(). In Eq.(5), Q(N N) s a postve sem-defnte symmetrc weghtng matrx and t denotes tme. Accordng to Lyapunov theory of stablty, for an asymptotcally stable system, there exst a symmetrc postve sem-defnte matrx P(N N) satsfyng (6) A P + PA Q Eq.(6) s the well-nown Lyapunov equaton. Based on the Lyapunov s second equaton, the Eq.(5) can be further smplfed as J X PX (7) ( ) ( ) hat s to say, to mnmze J n Eq.(5) s equvalent to mnmze X ( ) PX( ), where X( ) vector and the unnown symmetrc matrx P can be obtaned by solvng Eq.(6). (4) s the ntal state 5. Senstvty analyss scheme o apply gradent-based optmzaton method to solve the above optmzaton problem, senstvty (gradent) of the objectve functons wth respect to the desgn varables s needed. he adjont method wll be developed n ths paper. he new method just needs to solve the Lyapunov functon twce to obtan the senstvtes wth respect to all the desgn varables. For ease of presentaton of the new senstvty analyss scheme, we adopt Du s approach of usng Kronecer product and column expanson to expand the Lyapunov equaton as a system of lnear equaton. he column expanson of matrx V s defned as a vector that stacs all columns of ths V matrx. For example, for the 3 3 matrx V V V V3 V V V V3 (8) V3 V3 V33 the column expanson cs(v) of V s V cs V V V V V V V V V (9) ( ) [ ] V33 Note that cs(v) s a 9 vector. he operator cs(*) refers to the expanson operaton. For an N-dof system, usng the Kronecer product, (6) can be wrtten as GP Q () where P cs( P), Q cs( Q) and the matrx G can be obtaned from the matrx A by Kronecer product. hat s G ( A E + E A ) () and E(N N) s an dentty matrx wth the same sze of A. Now, by drect calculaton, the objectve functon n Eq.(7) can be wrtten as where S J S P () cs( S), X( ) X( ) S (3) S also s a postve sem-defnte symmetrc matrx as matrx Q. From the Eq.(), the term by P can be obtaned

3 hus senstvty can be expressed as X he rght hand of Eq.(5) can be rewrtten as where P G P G Q P + G Q ( ) X( ) S G P + (4) (5) P X D ( ) X( ) λ D (6) λ S G (7) Q G + P Note that λ and D are the column expanson of matrces λ and D, respectvely. Aλ + λa + S (9) λ, the adjont matrx, can be obtaned by solvng the above Lyapunov matrx equaton. D can be also computed by D Q A + A P + P Fnally, the senstvty of the objectve functon wth respect to the desgn varable can be expressed as X() PX() + X() X() P + For the case X() ndependent of desgn varables, the Eq.() can smplfed as N N j N N j λ D (8) () () λ D () 6. Numercal example wo examples are presented n ths secton. he frst example s used to demonstrate the accuracy and effcency of the proposed methods. he optmal support locaton s solved as a topology optmzaton problem n the second example. 6.. Example In ths example, we consder a clamped-free beam (3m.m.m) attached wth several dentcal damped sprngs (along Y drecton). he beam materal s lnear elastc wth the elastc modulus. Pa and mass densty 785Kg/m 3. he sprng stffness s s to be determned (N/m), and the dampng coeffcent s 3 N s/m. Fgure shows the beam model used n ths example. Specally, the beam s unformly meshed nto 5 -node beam elements. Each node has DOFs (lateral dsplacement and rotaton about Z-axs). Fve equally spaced damped sprng supports are consdered. he ntal dsplacements and veloctes of all nodes are zero and m/s respectvely. he stffness of each sprng s chosen as the desgn varable. hus, there are 5 desgn varables. Frstly, we compare the senstvty results from three methods, central dfference method, adjont method and drect method to valdate the proposed adjont method. 3m Y X.6m Fgure : he beam model wth 5 damped sprngs 3

4 he objectve functon s J R y dt where y s the Y-drecton dsplacement of the th node of the beam, R s the total number of the free nodes of the beam. o study the effect of step sze n central dfference analyss, we calculate the approxmaton of senstvty of sprng stffness of the sprng at rght hand of beam at s 5 N/m for three dfferent step szes. he results are shown n table. We chooseδ s N/m for further study n ths example. able : Senstvty results at s. 5 N/m from dfferent step szes Step sze 4 3 Senstvty result he senstvty results of the objectve functon wth respect to s of each sprng at s. 5 N/m from central dfference method, adjont method and drect method are shown n fgure and are represented by the blac crosses, red squares and blue rounds, respectvely. he results show that the adjont method obtans dentcal results wth the central dfference method and drect method. (3).4 x + Fnte dfference method Drect method Adjont method Senstvty Sprng s number Fgure : Senstvty results of the stffness of each sprng from three methods In ths paper, the CPU tme results are the average values of CPU tme of repeated analyses. he computer used n ths paper s GHz, Wndows 7. Now, we compare the CPU tme of drect method and adjont method. he CPU tme results of these two methods are summarzed n table. he results show that total CPU tme of the senstvty analyss process of adjont method s less than drect method, especally when the problem has large number of DOFs. A s the CPU tme of the senstvty analyss process of adjont method, and D s the CPU tme of the senstvty analyss process of drect method. able : CPU tme of two methods vs. number of DOFs n the model Number of CPU tme (s) DOFs A (adjont) D (drect) A / D % % 6.. Example opology optmzaton problems always have large numbers of desgn parameters. We construct a topology optmzaton problem to test the new senstvty analyss methods. In ths example, we consder a m m.m plate attached wth several dentcal damped sprngs( s 6 N/m, c s N s/m). One edge of the plate s clamped 4

5 and other three edges are free. he materal of the plate s lnear elastc wth elastc modulus. Pa, Posson rato.3, and mass densty 785Kg/m 3. he ntal velocty of Z drecton of all the free nodes of plate s m/s. he desgn problem s to decde the optmzed locaton of H damped sprngs to mnmze a crteron defned below. We formulate ths problem as a topology optmzaton problem. hs s acheved by ntroducng an artfcal densty varable to descrbe the spatal dstrbuton of the damped sprngs and use nterpolaton model of SIMP to obtan - desgn. Specfcally, an dentcal potental damped sprng (along Z drecton) s placed between every free nodes and the ground. Set a vrtual densty ρ to every sprng as the desgn varable. ρ s a contnuous varable, and ρ [ ρ mn,] We ntroduce an artfcal relaton between densty ( ρ ) and the parameters of the damped sprngs. l ρ K. l K, C ρ C (4) where l s the penalty parameter. In ths example, l s chosen as.. he analyss model s shown n Fgure 3, where the purple lnes are the dumped sprngs and the blue square elements are the 4-node square plate elements (shell63 n Ansys). Each node of the element has 3 DOFs, u, θ and θ. he element sze of the plate s.m (there are z x free nodes). he analyss model has 33 DOFs and desgn parameters. he topology optmzaton problem can be expressed as mn J M y z dt const. ρ H (5) < ρ mn ρ where H specfes the materal volume avalable for the damped sprngs. Here we assume each sprng, f any, uses materal volume, H wll be the number of damped sprngs n the fnal optmum desgn. z s the Z-drecton dsplacement of the th node of the plate. he model has free nodes, so the objectve functon concerns the dsplacements of all the free nodes. It should be noted ths example manly serves to compare dfferent methods of senstvty analyss through solvng the topology optmzaton problem descrbed by Eq.(5). Z Y X Fgure 3: he analyss model n topology optmzaton he CPU tme of solvng processes of adjont method and drect method s summarzed n table 3. he computng tme of adjont method s much shorter than the computng tme of drect method. able 3: CPU tme of senstvty analyss of two methods Adjont Drect CPU tme (s) Fnally, we use above mentoned four senstvty analyss schemes to solve the topology optmzaton descrbed n Eq.(5). H s set to. Fgure 4 shows that optmzaton usng dfferent senstvty analyss methods have dentcal teraton hstores and obtan same optmzed desgns. he CPU tme of solvng processes of topology optmzaton problem usng dfferent senstvty analyss methods s summarzed n table 4. he optmzaton process usng adjont method just taes about mnutes whch s far less than the CPU tme of other one. 5

6 Objectve..5 Objectve Iteraton (a) Iteraton (b) Fgure 4: Iteraton hstores of the objectve functon of optmzaton process: (a) Ajont method; (b) Drect method able 4: CPU tme of optmzaton processes usng dfferent senstvty analyss methods AVMF DVMF CPU tme (s) Concluson A new adjont senstvty analyss method for the ntegral square performance ndex s proposed n ths paper. he new approach requres the solutons of two Lyapunov equatons only, one for the performance ndex and one for the adjont vector. In contrast, drect senstvty analyss requres the soluton of a Lyapunov equaton for each desgn varable. hs mprovement n computatonal effcency maes the approach applcable to optmal desgn problem wth a large number desgn varable. he accuracy and effcency of the proposed method are demonstrated by two numercal examples. 8. Acnowledgement he frst author s grateful to the support of ths wor from State Key Laboratory of Structural Analyss for Industral Equpment (Project No. GZ3) 9. References [] Kalman RE, Bertram, JE,Control System Analyss and Desgn Va the Second Method of Lyapunov: I Contnuous-me Systems. Journal of Fluds Engneerng, 8(), , 96. [] Ogata K, Yang Y, Modern control engneerng, 97. [3] ruhar N, Veselc K. An effcent method for estmatng the optmal dampers' vscosty for lnear vbratng systems usng Lyapunov equaton. SIAM Journal on Matrx Analyss and Applcatons, 3(): 8-39, 9. [4] Kuzmanovć I, omljanovć Z, ruhar N. Optmzaton of materal wth modal dampng. Appled mathematcs and computaton, 8(3): ,. [5] Cox S J, Nać I, Rttmann A, et al. Lyapunov optmzaton of a damped system. Systems & control letters, 53(3): 87-94, 4. [6] Marano GC, Greco R, Chaa B, A comparson between dfferent optmzaton crtera for tuned mass dampers desgn. Journal of Sound and Vbraton, 39(3), ,. [7] Marano GC, Greco R, rentadue F, Chaa B, Constraned relablty-based optmzaton of lnear tuned mass dampers for sesmc control. Internatonal Journal of Solds and Structures, 44(), , 7. [8] Chang W, Gopnathan S V, Varadan V V, et al. Desgn of robust vbraton controller for a smart panel usng fnte element model. Journal of vbraton and acoustcs, 4(): 65-76,. [9] Wang BP, Kts L, Pley WD, ransent Response Optmzaton of Vbratng Structures by Lapunov s Second Method. J. Sound Vb., 96, pp. 55 5, 984. [] Du D, Analytcal solutons for DVA optmzaton based on the Lyapunov equaton. Journal of Vbraton and Acoustcs, 3(5), 545, 8. 6