Optimal Control of An Os illating BodyUsing Adjoint Equation MethodHisaki SAWANOBORIDepartment of Civil Engineering, Chuo University,E-mail : kawa ivil. huo-u.a .jpMutsuto KAWAHARADepartment of Civil Engineering, Chuo University,E-mail : kawa ivil. huo-u.a .jpAbstra tThe purpose of this study is to determine an angle of a wing whi h is atta hed to an os illating bodylo ated in transient in ompressible vis ous ows, using the Arbitrary Lagrangian Eulerian (ALE) nite elementmethod and the optimal ontrol theory, in whi h a performan e fun tion is expressed by the displa ement of thebody. At present, there are some bridges with wings to prevent os illation by wind ows. When the angle of thewing hanges, the state of os illation is also hanged. Therefore, the angle of the wing is very important so as tominimize the os illation of bridge. In this resear h, we solve this problem based on the optimal ontrol theory.In order to minimize the os illation of a body, the performan e fun tion is introdu ed as the minimizationindex. The performan e fun tion is dened by the square sum of displa ements of a body. This problem anbe transformed into the non-restri ted minimization problem by the Lagrange multiplier method. The adjointequations an be obtained by the stationary ondition of the extended performan e fun tion. We an derive thegradient to update the angle of the wing from solving the adjoint and the state equations. As a minimizationte hnique, the weighted gradient method is applied. In this study, the determination of the angle whi h theos illation of the body is made to minimize is presented by this theory. To express the motion of uids around abody, the Navier-Stokes equations des ribed in the ALE form is employed as the state equation. The motion ofthe body is expressed by the motion equations using displa ements and rotational angle of the body supportedby springs. As a numeri al study, the optimal ontrol of the angle of the wing is shown at low Reynolds number ows. By the numeri al omputation, the angle of the wing with whi h the os illation of the body be omesminimum an be shown.Key words : ALE Finite Element Method, Fluid-Stru ture Intera tion, Optimal ControlTheory, Performan e Fun tion, First Order Adjoint Equation, Weighted Gradient MethodMar h, 2012.1

1 INTRODUCTIONIn re ent years, ompli ated and large-s ale omputation an be performed be ause of rapid development of thenite element method and omputer hardwares. It is very important to obtain the physi al quantity of uid bynumeri al analysis, for example, to analyze uid ows around a body. In ase that a body is lo ated in uid ows,it is well-known that the Karman vortex o urs behind the body. The Karman vortex has various in uen e on thebody. A ollapse of the Ta oma Narrows bridge is a typi al example. Re ently, the weight and rigidity of re entbridges are redu ed be ause of the use of high quality material, and are resulted in that the unexpe ted os illationof those bridges will o ur with low velo ity wind. The os illation of bridge may be ome to in uen e on the traÆ son the bridge.To prevent the os illation by the wind, there are some bridges equipped with wings as shown in Figure 1.These wings are alled as wind-resistant wing. Optimal angles of these wings are determined by the wind tunnelexperiment. A lot of data of physi al quantity an be olle ted with the wind tunnel tests. Contrary to this,the experiment requires a high ost and it takes a lot of times to prepare models and to perform experiments.Therefore, it is ne essary to determine the method to obtain the optimal angle by numeri al analysis.The analysis of os illating body by uids for e has a moving boundary problem, therefore, the ALE method(e.g.[1; [2; [6) an ee tively be applied to this problem. The ALE method is a te hnique ombining the Lagrangedes ription with the Euler des ription based on the moving velo ity of nodal points. It is possible to use independentmesh velo ity from uid partial velo ity, to prevent ex essive distortion of the nite element mesh and to give movingmesh in a suitable manner. An os illating body is a phenomenon that is the oupled behavior of the uid and thestru ture. This phenomenon is alled the uid-stru ture intera tion problem. It is ne essary for the uid-stru tureintera tion problem to nd the solution satisfying the momentum equations on erning with the uid and the bodysupported by elasti springs.An optimal ontrol of an os illating bridge with a wing is presented in this study. The bridge is assumedto be a body with a wing supported by the elasti springs. The Navier-Stokes equation is employed to expressthe motion of uid around the body. The angle of wing is determined based on the optimal ontrol theory. Theperforman e fun tion in the optimal ontrol theory is assumed to be the square sum of displa ements of a body. Theverti al displa ement of the body is omputed so as to minimize the performan e fun tion under the onstraintof the equation of motion. The os illation of the body an be minimized by the ontrol variable, whi h is theangle of wing. The extended performan e fun tion is dened by using the Lagrange multiplier method. The rstorder adjoint equations an be obtained by the stationary ondition of the extended performan e fun tion. We anderive the gradient from solving both the adjoint and state equations. As the minimization te hnique, the weightedgradient method is applied. The performan e fun tion is minimized by al ulation of the angle in the weightedgradient method. The present approa h is the extended formulation presented by [10.In the numeri al study, an optimal ontrol of an os illating bridge with wings is arried out.2

Figure 1: Metropolitan Expressway Route 11 Daiba line2 STATE EQUATION2.1 ALE des riptionIndi ial notation and summation onvention with repeated indi es are used to express equations in this paper. Amotion of a body and uid motion around a body are expressed by the Lagrangian, Eulerian or referen e des ription.The referen e des ription is independent of the rst two and is possible to hoose an arbitrary oordinate system.A ording to these des ription, the Lagrangian, Eulerian and referen e oordinate systems are expressed by Xi,xi, or i, respe tively. If a fun tion f is arbitrary physi al quantity des ribed in the Euler des ription, relationbetween the real time derivative and the referen e time derivative is written as follows:DfDt = f(i; t)t i+ f(i; t)j j(Xi; t)t Xi : (1)If f is a position ve tor xi at the present time, eq.(1) an be written in the following form.xi(Xi; t)t Xi = xi(i; t)t i+xi(i; t)j j(Xi; t)t Xi (2)= xi(i; t)t i+wj xi(i; t)j Xi : (3)3

where wj = j(Xi; t)t Xi ; (4)Introdu ing relative velo ity bi of the material point for the referen e oordinate system.bi = ui ~ui = wj xi(Xi; t)j Xi ; (5)where ui = xi(Xi; t)t Xi ; (6)~ui = xi(i; t)t i : (7)Substituting the velo ity relation eq.(5) into the referen e time derivative equation, the referen e time derivativein the Euler domain is obtained as follows:DfDt = f(i; t)t i+bj f(i; t)xj : (8)Eq.(8) is the time derivative equation of a fun tion f in the ALE method.2.2 Governing equation of uidLet denote the omputational domain with boundary , and suppose that an transient in ompressible vis ous ow o upies . The Navier-Stokes equations are employed and an be expressed by the non-dimensional form inthe ALE method as follows: _ui + bjui;j + p;i (ui;j + uj;i);j = 0 in ; (9)ui;i = 0 in ; (10)where ui, p, and are velo ity, pressure and kinemati vis osity oeÆ ient, respe tively. The inverse of Reynoldsnumber is denoted by , and bj is onve tive velo ity using the referen e oordinate system in eq.(5). The quantity~uj in eq.(5) is alled as mesh velo ity.The boundary onsists of separated in ow, edge, out ow and body boundaries, U , S , D and B , respe -tively. The boundary onditions are given as follows:ui = U on U ; (11)4

Ω

UΓ

SΓ

DΓ

SΓ

BΓ

U

Body

Figure 2: S hemati view of the problem Figure 3: Coordinate systemu2 = 0; t1 = 0 on S ; (12)ti = ti = 0 on D; (13)u1 = 0; b2 = 0 on B ; (14)where ti is tra tion, whi h is expressed as;ti = fpÆij + (ui;j + uj;i)gnj ; (15)where nj is unit ve tor of outward normal to . Those are written in Figure 2. The initial onditions for velo ityand pressure are ui = u0i at t = 0 in ; (16)p = p0 at t = 0 in ; (17)where u0i means the onstant initial velo ity.2.3 Governing equation of a bodyA rigid body is assumed to have three degrees of freedom in x-y plane as shown in Figure 3. The equation ofmotion is expressed as follows: m V + _V + kV = F; (18)V = (X;Y;)T ; (19)F = (Fx; Fy;M)T ; (20)5

where X and Y denote displa ements of the body in the x and y dire tions and is the rotational angle of thebody around bary enter. Here, m, and k are mass, damping and rigidity, respe tively.3 DISCRETIZATION3.1 Spatial dis retizationFor the spatial dis retization, the te hnique presented by Kawahara's group([3; [4; [5) is used. This methodis the nite element method based on the mix interpolation with the bubble fun tion and linear interpolation.The velo ity eld is interpolated by the bubble fun tion element as shown in Figure 4, and the pressure eld isinterpolated by the linear fun tion element as shown in Figure 5.The interpolation relation is as follows:1. bubble fun tion interpolation ui = 1ui1 +2ui2 +3ui3 +4~ui4; (21)~ui4 = ui4 13(ui1 + ui2 + ui3); (22)1 = L1; 2 = L2; 3 = L3; 4 = 27L1L2L3; (23)2. linear interpolation p = 1p1 +2p2 +3p3; (24)1 = L1; 2 = L2; 3 = L3; (25)L1 + L2 + L3 = 1; (26)

2 3

4

1

Figure 4: Bubble fun tion element 2 3

1

Figure 5: Linear elementwhere L1 L3 are the area oordinate. The nite element equations of the Navier-Stokes equations are expressed6

as follows: Mij _uj +A jbju i Hip +Dijuj = fi; (27)Hiui = 0; (28)where Mij = Z()Æijd; A j = Z( ;j)d;Dij = Z(;k;k)Æijd+ Z(;j;i)d; Hi = Z(;i)d;fi = Z(ti)d; bj = uj ~uj :Variables in eqs.(27) and (28) an be separated into two omponents, those on the moving boundary B and othersare expressed as: uj = Uj + vj ; (29)~uj = ~Uj + vj : (30)where vj is the variable in the moving boundary, and Uj and ~Uj are those on the other omputational area.The ompatibility and equilibrium onditions are given asCompatibility onditions: vj = Tj _V on B : (31)_vj = Tj V on B : (32)Equilibrium ondition: Tifi + F = 0 on B ; (33)where Ti shows the geometri al relation as expressed in the next se tion.The nite element equations for the uid-stru ture intera tion problem an be obtained by substituting these onditions to eqs.(18) ,(27) and (28) and rearranging the terms:Mij _Uj +A jBjU i Hip +DijUj + Si = 0; (34)7

m V + _V + kV +N = 0; (35)HiUi +HiTi _V = 0; (36)where m = m + TiMijTj ; (37) = + Ti(A jBjT i +DijTj); (38)Si = MijTj V +A jBjT i _V +DijTj _V; (39)N = Ti(Mij _Uj +A jBjU i Hip +DijUj); (40)Eqs.(37) and (38) mean the mass and damping whi h in lude the intera tion ee ts between body and the uidaround it. The motion of uid is expressed by eq.(34), and the motion of body is expressed by eq.(35). Theseequations have to be solved satisfying the ondition of in ompressible ow des ribed in eq.(36).3.2 Transformation equationReferring to Figure 6, the ompatibility equation of displa ements at nodal points on the surfa e of the rigidbody (Ux; Uy) Ui and the displa ements at the bary enter (X;Y;) V an be written as eq.(41).xU 2

yU 2

xU1

yU1

xU β

yU β

11, yx LL

22 , yx LL

X

Y

Θ

Figure 6: Compatibility equation of displa ement

8

2666666666666666664U1xU1yU2xU2y...UnxUny

3777777777777777775 =2666666666666666664

1 0 L1y0 1 L1x1 0 L2y0 1 L2x...1 0 Lny0 1 Lnx3777777777777777775266664 XY 377775 : (41)

From eq.(41), eqs.(31) and (32) an be derived onsidering the oeÆ ient matrix in eq.(41) as Tj .Let (fx; fy) fi be tra tion for e at the nodal points on the surfa e of the rigid body and (Fx; Fy;M) Fbe the tra tion for e at the bary enter. Referring to Figure 7, the equilibrium equation of for e and moment anbe written as follows:xf2

yf2

xf1

yf1

xfα

yfα22 , yx LL

11, yx LL

yF

xFM

Figure 7: Equilibrium equation of for e266664 1 0 1 0 1 00 1 0 1 : : : 0 1L1y L1x L1y L1x L1y L1x 377775

2666666666666666664f1xf1yf2xf2y...fnxfny

3777777777777777775 = 266664 FxFyM 377775 : (42)Eq.(42) is written as eq.(33) onsidering the oeÆ ient matrix as Ti.

9

3.3 Temporal dis retizationFor the spatial dis retization, the Newmark- method is applied to the motion equation, and the predi tor- orre tor method is applied to the nite element equation of the Navier-Stoke equation. The Newmark- methodis a temporal dis retization te hnique whi h is used in the eld of stru tural dynami s widely. By the predi tor- orre tor method(e.g.[7; [8; [9), the solution at the new time point is al ulated using the predi tors and orre torsin ea h time iteration. The non-linear equation an be solved by the method exa tly. The al ulation algorithm onsists of three steps; i.e. the predi tor, in remental al ulation of a eleration and pressure, and the orre tor.In this study, the following al ulation algorithm is employed.Step.1 Predi t variables _U (0)j(n+1), U (0)j(n+1), P (0)(n+1), V (0)(n+1), _V (0)(n+1), V (0)(n+1) at next time pointStep.2 Cal ulate in rements _U (l)j ;p(l) ;V (l)Step.(a) Cal ulate residuals of the momentum equations R(l)i and r(l)Step.(b) Obtain the temporary in rements of a eleration _U(l)j and V (l)Step.( ) Obtain the in rement of pressure p(l)Step.(d) Cal ulate orre ted in rements of a eleration _U (l)j and V (l)Step.3 Corre t update variables _U (l+1)j(n+1), U (l+1)j(n+1), P (l+1)(n+1), V (l+1)(n+1), _V (l+1)(n+1), V (l+1)(n+1) at next time pointStep.4 Convergent test; if jU (l+1)i U (l)i j < " and jV (l+1) V (l) j < " then go to next iteration,else go to step.2where R(l)i and r(l) are expressed as follows:R(l)i = Mij _U (l)j(n+1) A jB(l)j(n+1)U (l) i(n+1) +Hip(l)(n+1) DijU (l)j(n+1) S(l)i(n+1); (43)rl = m V (l)(n+1) _V (l)(n+1) kV (l)(n+1) N (l)(n+1): (44)4 FORMULATION OF OPTIMAL CONTROL4.1 Performan e fun tionThe purpose of this study is to minimize the os illation of the body. In order to minimize the os illation ofbody, the performan e fun tion J , whi h is dened by the square sum of displa ements of the body is introdu ed.J = 12 Z tft0 (VQV) dt; (45)10

where V and Q are the displa ement of the body and the weighting diagonal matrix, respe tively. The dis-pla ement V is governed by eq.(35). On the optimal ontrol theory, the optimal ondition an be derived if theperforman e fun tion is minimized.4.2 Extended performan e fun tionThe performan e fun tion should be minimized satisfying the onstraint onditions whi h are the state equations(18), (27) and (28). The Lagrange multiplier method is suitable for minimization problems with the onstraint onditions. The Lagrange multipliers for the state equations (27), (28) and (18) are dened as the adjoint parameterui, V and p, respe tively. The performan e fun tion is extended by the adding inner produ ts between theadjoint parameters and the state equations (27), (28) and (18). The extended performan e fun tion J is expressedas follows: J = 12 Z tft0 (VQV) dt Z tft0 ui(Mij _uj +A jbju i Hip +Dijuj fi)dt+ Z tft0 p(Hjuj)dt Z tft0 V (m V + _V + kV F)dt (46)4.3 First order adjoint equationThe minimization problem with onstraint onditions results in satisfying the stationary onditions of theextended performan e fun tion. Eq.(46) an be derived from the rst variation of the extended performan efun tion. ÆJ = Z tft0 Æui(Mij _uj +A jbju i Hip +Dijuj fi)dt+ Z tft0 Æp(Hjuj)dt Z tft0 ÆV (m V + _V + kV F)dt Z tft0 Æuj(Mij _ui +Bijui Hjp +Dijui)dt+ Z tft0 Æp(Hiui)dt Z tft0 ÆV(m V _V + kV QV)dt11

+ Z tft0 (ui + TiV )Æfidt Mijui(tf )Æuj(tf ) mV (tf )Æ _V(tf )+ fm _V (tf ) V (tf )gÆV(tf )+ G X' Æ' (47)where Bij = A ju i +A b Æij (48)The stationary ondition means that the rst variation of the extended performan e fun tion vanishes.ÆJ = 0 (49)Considering eq.(49), the ea h term of eq.(47) equals zero. Therefore, the rst-order adjoint equations, terminal onditions, and boundary ondition are obtained as follows:Mij _ui + Bijui Hjp +Dijui = 0 in (50)Hiui = 0 in (51)m V _V + kV QV = 0 in (52)Mijui(tf ) = 0 in (53)mV (tf ) = 0 in (54)m _V (tf ) V (tf ) = 0 in (55)ui + TiV = 0 on B (56)Eqs.(50) (52) are the adjoint equations, and these equations should be solved ba kward from tf to t0 with theterminal onditions (53) (55) and boundary onditions of the variables satisfy eqs.(50) (56). we an obtain:ÆJ = G X' d'; (57)where G in eq.(57) is expressed asG = Z tft0 ui(MijX _uj + A jX bju i HiX p + DijX uj)dt+ Z tft0 p(HjX uj)dt: (58)12

From eq.(57), the gradient to update the angle of the wing ' an be derived as follows:grad(J) = G X' (59)5 MINIMIZATION5.1 Weighted gradient methodAs the minimization te hnique, the weighted gradient method, whi h seems to be s ar ely dependent on theinitial value, is applied. In this method, a modied performan e fun tion K(l) is introdu ed by adding a penaltyterm to the extended performan e fun tion. The modied performan e fun tion is expressed as follows:K(l) = J(l) + 12('(l+1) '(l))W ('(l+1) '(l)): (60)where l and W are iteration number and weighting parameter, respe tively. In ase that the modied performan efun tion onverges to zero, the penalty term an also be zero. The minimization of the modied performan efun tion is equal to the minimization of the extended performan e fun tion. Applying the following stationary ondition is applied to the modied performan e fun tion,ÆK(l) = 0; (61)the updated angle of the wing is al ulated at ea h iteration y le by the following equation:W'(l+1) =W'(l) G X' : (62)5.2 AlgorithmThe following al ulative algorithm is employed for the omputation.Step1. Sele t an initial angle of the wing '(0).Step2. Solve u(0)j ,p(0) ,V (0) by eqs.(18), (27), and (28) in .Step3. Solve u(l)i ,p(l) , V (l) by eqs.(50) (52) in .13

Step4. Compute '(l+1) by eq.(62)Step5. Update the mesh around the wing.Step6. Solve u(l+1)j ,p(l+1) ,V (l+1) by eqs.(18), (27), and (28) in .Step7. If jJ (l+1) J (l)j < then stop, else go to step.8Step8. Update a weighting parameter W (l+1);If J (l+1) J (l) < 0, then set W (l+1) = 0:9W (l) and go to step.3else W (l+1) = 2:0W (l) and go to step.4.6 MESH UPDATEIn ase that the body is moved, and then the mesh around the body should be moved to orre t distorted onguration of mesh. we assume that there are two areas, one of whi h we apply the remeshing and the othernon-remeshing. There are illustrated in Figure 17. Following the movement, all of the internal nodes should beremeshed. For the method of the internal node remeshing, the following rule is introdu ed, i.e. the oordinate ofea h nodal point obeys the Lapla e equation, ;ii = 0 in : (63)where means x or y oordinate. As the boundary ondition for , the moved oordinates of the body and thenon-moved oordinates on the internal boundary are used to obtain the uniform distribution of the oordinate ofnodal points.

14

7 CASE STUDY7.1 Preliminary studyAs a preliminary study, an optimal angle of a body lo ated in steady in ompressible vis ous ows is arried outto verify the present optimal ontrol s heme. An angle of a body is determined so as to minimize the lift for esubje ted to the body by the steady ows. The performan e fun tion is dened by the square sum of uid for es:J = 12 Z tft0 (qijFiFj) dt; (64)and the formulation is derived by the method deseribed in se tions 4 and 5. The omputational domain andboundary ondition are shown in Figure 8. A quadrilateral prism with slenderness ratio of 2:1 is lo ated in the enter of the omputational domain. Low Reynolds number Re = 1:0 is assumed to onsider the steady ow. Thenite element mesh is represented in Figure 9, The mesh has 4523 nodes and 9583 elements. The initial angle of al ulation is set '(0) = 45:0Æ. we an guess the performan e fun tion is onverged to zero in ase that the prismangle will be 0:0Æ or 90:0Æ, be ause the ow around the prism is symmetry.The omputed results are shown in Figures 10 14. The variation of the performan e fun tion and of theangle of body are plotted in Figures 10 and 11, respe tively. The performan e fun tion is onverged to zero atthe last iteration y le, and the angle of body is obtained as 90:0Æ. The omparison of the lift for es between theinitial angle of body and the optimal angle of body are shown in Figure 12. We an see the lift for e is zero at theoptimal angle. The omputed pressure distributions at initial angle and at optimal angle are illustrated in Figures13 and 14. We an see the pressure ontour around the prism is symmetry with respe t to the horizontal axis atthe optimal angle. This fa t veries the present method is useful for the determination of the optimal angle of abody.

15

ϕ0.0

0.1

==

v

u

0.0=v

0.0=v

0.0

0.0

==

v

u

)0.10,0.15(−

)0.10,0.15( −−

)0.10,0.15(

)0.10,0.15( −

x

y

D

D0.2

Figure 8: Computational domain of preliminary study

Figure 9: Finite element mesh of preliminary study16

Iteration number

Per

form

ance

func

tion

0 5 10 15 20 25 30 350

0.2

0.4

0.6

0.8

1

Figure 10: Variation of performan e fun tion Iteration number

Ang

leo

fbod

y[D

egre

e]

0 5 10 15 20 25 3040

50

60

70

80

90

100

Figure 11: Variation of angle of body

Time step

Lift

forc

e

20 40 60 80-5

-4

-3

-2

-1

0

1

Initial angleOptimal angle

Figure 12: Comparison of lift for e17

Figure 13: Pressure distribution at initial angle

Figure 14: Pressure distribution at optimal angle18

7.2 Vortex os illationIn the ase study No.1, an optimal angle of the wing similar to the one equipped to the bridge at Route 11Daiba line on Metropolitan Express Way in Tokyo as shown in Figure 1 is determined. The bridge is three span ontinuous bridge of whi h one span is 230 meter long. The shape of the bridge se tion is modelled referring tothe a tual one, however, the wind speed is assumed as slow ows (Re = 250) be ause the ow pattern around thebridge an be expressed learly by the numeri al omputation. The mass is estimated as 157:0. The stru turaldamping in eq.(18) is assumed zero. Thus, the damping is assumed to be indu ed only by the vis ous ow. Namely,the rst term in eq.(38) is zero, and the se ond term is used for damping. This is be ause the stru tural dampingis far smaller than the damping indu ed by the ow. The rigidity of the bridge is assumed as kx =1, ky = 34:138,k =1. This value is far smaller than that obtained by the simple beam assumption. The verti al rigidity an beestimated by the simple beam assumption, whi h is 1582:2. This value is assigned referring to the vortex os illationof the bridge. The omputational domain is shown in Figure 15.The nite element mesh is represented in Figure 16, whi h onsists of 6887 nodes and 13381 elements. Thismesh shown in the ase that the angle of wing is zero. The mesh is separated into two parts, one is the movingbut non-remeshing part whi h in ludes the bridge body, and the other is the moving and remeshing part whi hin ludes the wing. The omputation is arried out determining the angle of wing by eq.(63). The time in rementt = 0:01.The omputed results are shown in Figures 18 26. The omputed velo ity, pressure and vorti ity at time400 without the wing are illustrated in Figures 21; 23 and 25. The variation of the displa ement of the bary enterof the bridge se tion is represented in Figure 20. It is shown that the amplitude of the os illation of the bridge isgradually in reased. Starting from the angle of the wing ' = 40:0Æ[deg:, the optimal angle of the wing is obtainedas ' = 28:6[deg:. The variation of the performan e fun tion and of the angle of wing are plotted in Figures 18and 19, respe tively. The displa ement of the bary enter of the bridge is represented in Figure 20 by the red line.It is learly shown that the displa ement is onverged to zero. The omputed velo ity, pressure and vorti ity attime 400:0, with the optimal wing are represented in Figures 22; 23 and 26. Looking at the vorti ity in Figure 26,anti-symmetri al vorti ity is observed on the top and bottom of the bridge body. Moreover, the onne ting pointof the top and bottom ow is far from the bridge body. This fa t shows that the lift for e is redu ed by the wingoptimally lo ated. The displa ement with the optimal wing is in Figure 20, whi h is 98:0% redu ed ompared withthat of the non-equipped bridge se tion.19

D

D5.9

D9.28 D9.28

0.0

0.1

==

v

u

0.0

0.1

==

v

u

0.0=v

0.0=v10.0) (-30.0, 10.0) (30.0,

10.0)- (-30.0, 10.0) (30.0,Figure 15: Computational domain

Figure 16: Finite element mesh

Figure 17: Remeshing and non-remeshing parts20

Iteration number

Per

form

ance

func

tion

0 5 10 15

0.8

0.85

0.9

0.95

1

Figure 18: The variation of performan e fun tion Iteration number

Ang

leof

win

g[d

egre

e]

0 5 10 1528

29

30

31

32

33

34

35

36

37

38

39

40

41

Figure 19: The variation of angle of wings

Time

Dis

pla

cem

ent

0 100 200 300 400-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Without wingsWith wings of optimal angle

Figure 20: The omparison by the existen e of wings21

-4

-2

0

2

4

Figure 21: Velo ity around the bridge without wings

-4

-2

0

2

4

Figure 22: Velo ity around bridge with optimal wings22

-4

-2

0

2

4

Figure 23: Pressure around bridge without wings

-4

-2

0

2

4

Figure 24: Pressure around bridge with optimal wings23

-4

-2

0

2

4

Figure 25: Vorti ity around bridge without wings

-4

-2

0

2

4

Figure 26: Vorti ity around bridge with optimal wings24

7.3 Rotational os illationIn the ase of study No.2, an optimal angle of the wing similar to the one in Figure 1 is determined. In this ase, Optimal ontrol of verti al and rotational os illation is arried out. The omputational domain and some onditions are shown in Figure 27. This omputational domain is lose to the same ex ept a torsional elasti springis equipped to the rotational dire tion of the bridge. The Reynolds number is set to Re = 250 to express the ow pattern around brdige leary. The physi al properties of bridge are set; The mass is estimatied as 157.0, Thestru tual damping is assumed zero, The rigidity is assumed as kx = 1, ky = 34:138, k = 68:276. The niteelement mesh is represented in Figure 28, whi h onsists of 6887 nodes and 13381 elements. The time in rement isset to t = 0:01.The omputed results are shown in Figure 29 38. The variation of the performan e fun tion and angle ofbody are plotted in Figures 29 and 30, respe tively. The performan e fun tion is onverged to 0:817 at the lastiteration y le, and the angle of the body is obtained as 33:32Æ. The omparison of the verti al and rotationaldispla ements between the initial angle of wings and the optimal angle of wings are shown in Figures 31 and 32.We an see the state of os illations be omes small. Espe ially, it is seen that the verti al displa ement is suppressedby the optimal ontrol. From the results, it is on luded that the wind-resistant wings is ee tive for the verti alos illation of the bridge.The omputed velo ity, pressure and vorti ity at time 100:0 with the initial wing are illustrated in Figures 33; 35and 37, and the omputed velo ity, pressure and vorti ity at time 100:0, with the optimal wing are represented inFigures 34; 36 and 38.

25

D

D6.10

D32 D32

0.0

0.1

==

v

u

0.0

0.1

==

v

u

0.0=v

0.0=v11.1) (-33.3, 11.1) (33.3,

11.1)- (-33.3, 11.1) (33.3,Figure 27: Computational domain

Figure 28: Finite element mesh

26

Iteration number

Per

form

ance

func

tion

0 5 10 15

0.85

0.9

0.95

1

Figure 29: The variation of performan e fun tion

Iteration number

An

gle

ofw

ings

[Deg

ree]

0 5 10 15

33

34

35

36

37

38

39

40

Figure 30: The variation of angle of wings27

Time

Dis

pla

cem

ent

0 20 40 60 80

-0.02

0

0.02

0.04

Optimal angleInitial angle

Figure 31: The omparison of verti al displa ement

Time

Dis

plac

emen

t[D

egre

e]

0 20 40 60 80-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Optimal angleInitial angle

Figure 32: The omparison of rotational displa ement28

-4

-2

0

2

4

Figure 33: Velo ity around bridge with initial wings

-4

-2

0

2

4

Figure 34: Velo ity around bridge with optimal wings29

-4

-2

0

2

4

Figure 35: Pressure around bridge with initial wings

-4

-2

0

2

4

Figure 36: Pressure around bridge with optimal wings30

-4

-2

0

2

4

Figure 37: Vorti ity around bridge with initial wings

-4

-2

0

2

4

Figure 38: Vorti ity around bridge with optimal wings31

8 CONCLUSIONIn this study, an optimal ontrol of an os illating bridge with a wing in an in ompressible ow is presented. Anos illating bridge lo ated in the transient in ompressible vis ous ows an be analyzed by the ALE nite elementmethod. As numeri al studies, an optimal ontrol of an os illating bridge with a wing in an in ompressible ows are arried out. The angle with whi h the os illation is minimized is obtained. It is seen that the ex essive os illation an be drasti ally redu ed by using the optimal ontrol.Referen es[1 T.Okamoto and M.Kawahara,\Two-Dimensional Sloshing Analysis by the Arbitrary Lagrangian-Eulerian Finite ElementMethod", pro . Jap. So . Civil Engr., No.44I-18, pp207-216, 1992[2 A.Anju, A.Maruoka and M.Kawahara,"2-D Stru tural Intera tion Ploblems by an Arbitrary Lagrangian-Eulerian Finite ElementMethod", Int. J. Comp. Fluid Dyn., Vol.8, No.1, pp1-9, 1996[3 J.Matsumoto and M.Kawahara,\Shape Identi ation for Fluid Stru ture Intera tion Problem Using Bubble Element", Int. Jour.Comp. Fluid Dyn.,Vol.15, pp33-45, 2001[4 J.Matsumoto and M.Kawahara,\Shape Identi ation for Intera tion Problem Using Finite Element Method with MINI Element",Comp. Meth. Cont. Appl.,Vol 16, pp293-312, 2001[5 H.Okumura and M.Kawahara,\A New Stable Bubble Element for In ompressible Fluid Flow Based on a Mixed Petrov-GalerkinFinite Element Formulation", Int. J. Comp. Fluid Dyn., Vol.17, pp275-282, 2003[6 T.J.Hughes, W.K.Liu and T.K.Zimmerman,\Lagrangian-Eulerian nite element formulation for in ompressible vis ous ows",Comp. Meth. Appl. Me h. Engrg.,29, pp.329-349,1981[7 T.Nomura,\Appli ation of predi tor- orre tor method to ALE nite element analysis of ow-stru ture intera tion problems andasso iated omputational te hniques", Journal of Hydrauli , Coastal and Environmental Engineering, No.455/I-21, pp.55-63,1992.10[8 T.Nomura and T.J.Hughes,\An arbitrary Lagrangian-Eulerian nite element method for intera tion of uid and a rigid body",Comp. Meth. Appl. Me h. Engrg.,95, pp.411-430, 1992[9 Z.Qun and T.Hisada,\Investigations of the oupling method for FSI Analysis by FEM ",Trans Japan So . Me h. Eng., A67(662)pp,1555-1562 2001.[10 T.Kurahashi and M.Kawahara,\Water Quality Control by Bank Pla ement Based on Optimal Control and Finite ElementMethod", Int. J. Num. Meth. Fluids., Vol.41, pp319-338, 200332