TEAM 2007, Sept. 10-13, 2007,Yokohama, Japan Hydoelastic Analysis of a 1900 TEU Containe Ship Using Finite Element and Bounday Element Methods Ahmet Egin 1)*, Levent Kaydıhan 2) and Bahadı Uğulu 3) 1) Pofesso, Faculty of Naval Achitectue and Ocean Engineeing, Istanbul Technical Univesity, Maslak, 34469, Istanbul, Tukey e-mail: egina@itu.edu.t 2) Depatment Manage, Delta Maine, Okul Sk., Altunizade Sitesi, E/20 Altunizade, 34662, Üsküda, Istanbul, Tukey e-mail: l.kaydihan@mesh-e.com 3) D., Faculty of Naval Achitectue and Ocean Engineeing, Istanbul Technical Univesity, Maslak, 34469, Istanbul, Tukey e-mail: bugulu@itu.edu.t Abstact This pape pesents a hydoelastic analysis of a 1900 TEU containe ship using finite element and bounday elements. The method of analysis is sepaated into two pats. In the fist pat, the in vacuo dynamic popeties of the containe ship wee obtained by using a standad finite element method. In the second pat of the analysis, the ship stuctue was assumed vibating in its in vacuo modes when it is in contact with fluid, and the pessue distibution on the wetted suface was calculated sepaately fo each mode. The fluid-stuctue inteaction effects wee calculated in tems of the genealized added mass tems. The wetted suface of the containe ship was idealized by using appopiate bounday elements, efeed to as hydodynamic panels. A highe ode panel method (linea distibution) was adopted fo the calculations. In a futhe analysis, the wet calculations wee epeated using a finite element softwae (ABAQUS). The wet fequencies calculated fom the both analysis wee compaed, and a vey good compaison was obtained between the esults. Keywod: Hydoelasticity, bounday element method, finite element method, containe ship, vibation. 1. INTRODUCTION This pape pesents the dynamic chaacteistics (e.g., natual fequencies and mode shapes) of a 1900 TEU containe ship in full load and ballast conditions. The method pesented is based on a bounday integal equation method togethe with the method of images in ode to impose appopiate bounday condition on the fluid s fee suface. The method poposed in this study has aleady been successfully applied to stuctues patially filled with o patially submeged in quiescent and flowing fluids (see, fo instance, [1] [4]). In this investigation, it is assumed that the fluid is ideal, i.e., inviscid, incompessible and its motion is iotational. It is also assumed that the ship hull vibates in its in vacuo eigenmodes when it is in contact with fluid, and that each in vacuo mode gives ise to a coesponding suface pessue distibution on the wetted suface of the ship stuctue. The in vacuo dynamic analysis entails the vibation of the ship hull in the absence of any extenal foce and stuctual damping, and the coesponding dynamic chaacteistics of the containe ship ae obtained using a standad finite element softwae. At the fluid-stuctue inteface, continuity consideations equie that the nomal velocity of the fluid is equal to that of the stuctue. The nomal velocities on the wetted suface ae expessed in tems of the modal stuctual displacements. By using a bounday integal equation method the fluid pessue is eliminated fom the poblem, and that the fluid-stuctue inteaction foces ae calculated in tems of the genealized hydodynamic added mass coefficients. Duing this analysis, the wetted suface is idealized by using appopiate bounday elements, efeed to as hydodynamic panels. To assess the influence of the fluid on the dynamic esponse behavio of the containe ship, the wet fequencies and associated mode shapes ae calculated.
In a futhe analysis, a standad finite element softwae (ABAQUS [5]) was used fo modeling both the stuctual and fluid domains. The fluid domain suounding the ship hull was idealized by using acoustic fluid elements. The cago loads, ballast weight, etc. applied as inetia mass elements ove the cago aea, etc. The wet fequencies and mode shapes wee calculated by solving the fluid-stuctue inteaction poblem. The wet fequency values obtained fom the bounday element method (BEM) compae vey well with those calculated by ABAQUS. 2. MATHEMATICAL MODEL 2.1 Fluid stuctue inteaction poblem The fluid is assumed ideal, i.e., inviscid and incompessible, and its motion is iotational and thee exists a fluid velocity vecto, v, which can be defined as the gadient of the velocity potential function φ as v ( x, y, z, t) = φ ( x, y, z, t). (1) 2 whee φ satisfies the Laplace s equation, φ = 0, thoughout the fluid domain. Befoe descibing the esponses of the flexible stuctue, it is necessay to assign coodinates to the deflections at vaious degees of feedom. One paticula set of genealized coodinates having the advantage of being unambiguous and easily commended is the pincipal coodinates of the dy stuctue (see, fo example, Egin et al. [6]). Fo a stuctue vibating in an ideal fluid, with fequency ω, the pinciple coodinate, descibing the esponse of the stuctue in the th modal vibation, may be expessed by p ( t) = p e 0 iω t (2) The velocity potential function due to the distotion of the stuctue in the th in vacuo vibational mode may be witten as follows (,,, ) i (,, ) e i ω φ x y z t = ωφ x y z p t 0 = 1, 2,, M (3) whee M epesents the numbe of modes of inteest, and p 0 is an unknown amplitude fo the th pincipal coodinate. On the wetted suface of the vibating stuctue the nomal fluid velocity must equal to the nomal velocity on the stuctue and this condition fo the th modal vibation of the elastic stuctue submeged in fluid can be expessed as φ = u n, n whee n is the unit nomal vecto on the wetted suface and points into the egion of inteest. The vecto u denotes the displacement esponse of the stuctue in the th pincipal coodinate and it may be witten as (4) (,,, ) (,, ) e i ω u x y z t = u x y z p t 0 (5) whee u (x, y, z) is the th modal displacement vecto of the median suface of the stuctue, and it is obtained fom the in vacuo analysis. It is assumed that the elastic stuctue vibates at elatively high fequencies so that the effect of suface waves can be neglected. Theefoe, the fee suface condition (infinite fequency limit condition) (see, fo instance, [3]) fo the petubation potential can be appoximated by φ = 0, on the fee suface. (6) The method of images may be used to satisfy this bounday condition. By adding an imaginay bounday egion, the condition given by equation (6) at the hoizontal suface can be omitted; thus the poblem is educed to a classical Neumann case. The details of the method of images may be found in Egin and Temael [4].
2.2 Numeical evaluation of petubation potential φ The bounday value poblem fo the petubation potential,φ, may be expessed in the following fom: c s q s s q s ds * * ( ξ) φ ( ξ) = ( φ (, ξ) ( ) φ ( ) (, ξ)) S W (7) * whee ξ and s denote, espectively, the evaluation and field points on the wetted suface. φ is the fundamental solution and expessed as follows, φ ( s, ξ ) = 4π * 1 q = φ / n is the flux, and the distance between the evaluation and field points. The fee tem c (ξ ) is defined as * the faction of φ (ξ ) that lies inside the domain of inteest. Moeove, q ( s, ξ) can be witten as (8) q ( s, ξ ) = ( / n) / 4π * 3 (9) Fo the solution of equation (7) with bounday condition (4), the wetted suface can be idealized by using bounday elements, efeed to as hydodynamic panels, and the distibution of the potential function and its flux ove each hydodynamic panel may be descibed in tems of the shape functions and nodal values as ne ne φ = N φ q = N q e e e, e e e = 1 = 1 (10) Hee, n e epesents the numbe of nodal points assigned to each hydodynamic panel, and N e the shape function adopted fo the distibution of the potential function. e and indicates the numbes of the hydodynamic panels and nodal points, espectively. In the case of a linea distibution adopted in this study, the shape functions fo a quadilateal panel may be expessed as (see Wobel [7]) N e1 = ((1 ς )(1 η)) / 4 N e 2 = ((1 + ς )(1 η)) / 4 (11) N e 3 = ((1 + ς )(1 + η)) / 4 N e 4 = ((1 ς )(1 + η)) / 4 Afte substituting equations (10) and (11) into equation (7) and applying the bounday condition given in equation (4), the unknown potential function values can be detemined fom the following set of algebaic equations m nm m nm * * k φ k + ( φ i = ( i φ ), i= 1 = 1 S i= 1 = 1 i Si c N q ds u n N ds k =1, 2,, m (12) whee m denotes the numbe of nodal points used in the discetization of the stuctue. φ i and u i epesent, espectively, the potential and displacement vecto fo the th nodal point of the ith hydodynamic panel. 2.3 Calculation of genealized fluid-stuctue inteaction foces Using the Benoulli s equation and neglecting the second ode tems, the dynamic fluid pessue on the elastic stuctue due to the th modal vibation becomes
P ( x, y, z, t) = ρ φ, t (13) whee ρ is the fluid density. Substituting equation (3) into (13), the following expession fo the pessue is obtained, P x y z t = ρ ω φ (14) 2 iω t (,,, ) ( x, y, z) p0 e. The kth component of the genealized fluid-stuctue inteaction foce due to the th modal in-vacuo vibation of the elastic stuctue can be expessed in tems of the pessue acting on the wetted suface of the stuctue as Z = P ( x, y, z, t) u n ds k k SW = p e ρ ω φ u n ds iω t 2 0 k SW (15) The genealized added mass, A k tem can be defined as A = ρ φ u n ds, k k SW (16) Theefoe, the genealized fluid-stuctue inteaction foce component, Z k, can be witten as Z t = A ω p = A && p t 2 iωt k ( ) k 0 e k ( ) (17) 2.4 Calculation of wet fequencies and mode shapes It should be noted that, in the case when a body oscillates in o nea a fee suface, the hydodynamic coefficients exhibit fequency dependence in the low fequency egion, but show a tendency towads a constant value in the high fequency egion. In this study, it is assumed that the stuctue vibates in a elatively high fequency egion so that the genealized added mass values ae constants and evaluated by use of equation (16). Hence the genealized equation of motion fo the dynamic fluid stuctue inteaction system (see, e.g., Egin et al. [6]), assuming fee vibations with no stuctual damping, is ω 2 ( a + A ) + c p = 0, (18) whee a and c denote the genealized stuctual mass and stiffness matices, espectively. The matix A epesents the infinite fequency genealized added mass coefficients. Solving the eigenvalue poblem, expessed by (18), yields the wet fequencies and associated wet mode shapes of the stuctue in contact with fluid. To each wet fequency ω, thee is a coesponding wet eigenvecto p = p, p, K, p. The coesponding uncoupled wet mode shapes fo the stuctue patially and totally in { } 0 1 2 m contact with fluid ae obtained as whee u ( x, y, z) = { u, v, w} = { } = u ( x, y, z) u, v, w u ( x, y, z) p, (19) = 1 M denote the in vacuo mode shapes of the elastic stuctue and M the numbe of mode shapes included in the analysis. It should be noted that the fluid stuctue inteaction foces associated with the inetial effect of the fluid do not have the same spatial distibution as those of the in vacuo modal foms. Consequently, this poduces hydodynamic coupling between the in vacuo modes of the stuctu. This coupling effect is intoduced into equation (18) though the genealized added mass matix A.
3. NUMERICAL RESULTS A containe ship is adopted fo the numeical calculations, and its main paticulas ae given in Table 1. Two diffeent loading conditions ae consideed: full load and ballast conditions. The cago loads ae distibuted as inetia mass elements ove the cago aea inne bottom plating. Ballast weight, heavy fuel oil and othe tank weights ae also applied as inetia mass elements. The light weight of the ship (LWT) is 9000 tons. The load goups ae given in Table 2 fo the full load and ballast conditions. The stuctual finite element model of the containe ship is shown in Figue 1. Table 1. The main paticulas of the containe ship Length (L bp ) 171 m Beadth (B) 28 m Depth (D) 16.10 m Daught (T) 10.00 m Speed (V) 19.50 knot The in vacuo dynamic chaacteistics of the containe ship wee obtained using the ABAQUS finite element softwae [5]. This poduces infomation on natual fequencies and the nomal mode shapes of the dy containe ship in vacuum. In these calculations, the ship hull was discetized by using 176 800 stuctual elements. The in vacuo dynamic chaacteistics of the ship stuctue ae scaled to a genealized stuctual mass of 1 ton m 2. Table 2. Load goups in full load and ballast conditions (tons). Full Load Condition Ballast Condition Cago loads 17150 - Ballast weight 3021 11250 Heavy Fuel Oil 1886 305 Maine Diesel Oil 165 36.8 Fesh wate 206 26.7 Othe weights 165 165 Fig.1 The stuctual finite element model of the containe ship.
It should be noted that the finite element calculations wee caied out by the Delta Maine of Tukey. The dy natual fequencies of the containe ship ae given in Tables 3 and 4 fo the full load and ballast conditions. In the wet pat of the calculations, the fluid is assumed ideal, i.e., inviscid and incompessible, and its motion is iotational. The wetted suface of the containe ship was idealized by using bounday elements, efeed to as hydodynamic panels. 10772 and 6739 hydodynamic panels wee distibuted ove the wetted suface of the ship hull, espectively, fo the full load and ballast conditions. A linea distibution of unknown potentials was assumed ove the each hydodynamic panel. These unknown potentials wee calculated using the kinematic bounday conditions imposed at each nodal point. The discitized wetted suface is shown in Figue 2 fo the full load condition. The kinematic bounday conditions wee obtained fom the in vacuo finite element calculations. By using Eq.(18), the eigenvalues and eigenvectos of the fluid-stuctue inteaction system wee calculated, and the wet fequencies ae pesented in Tables 3 and 4 fo the fist five global ship hull vibations togethe with the dy esults, espectively, fo the full load and ballast conditions. A numbe of 12 in vacuo modes was adopted fo the wet calculations. In a futhe analysis, a standad finite element pogam (ABAQUS) was adopted fo the wet calculations. The ship stuctue and fluid suounding the ship hull wee discetized by finite elements. Fo the fluid medium, 60 300 acoustic finite elements wee employed fo the full load condition. Fig.2 The discetized wetted suface of the containe ship fo the full load condition. Table 3. Calculated fequencies fo the full load condition. In vacuo Wet Wet Diffeence Mode FEM FEM BEM % 1. Tosion 1.119 0.998 0.999 0.1 1. Bending 1.331 1.015 1.041 2.5 Bend.&Tos. 1.515 1.328 1.349 1.6 2. Tosion 2.547 2.325 2.344 0.8 2. Bending 2.676 2.059 2.072 0.6 Table 4. Calculated fequencies fo the ballast condition. In vacuo Wet Wet Diffeence Mode FEM FEM BEM % 1. Tosion 1.340 1.221 1.217 0.3 1. Bending 1.608 1.290 1.210 6.2 Bend.&Tos. 1.981 1.787 1.776 0.6 2. Tosion 3.337 2.504 2.386 4.7 2. Bending 3.455-3.129 -
It can be seen fom Tables 3 and 4 that thee ae vey good ageement between the esults obtained fom the finite element analysis (ABAQUS) and bounday element analysis poposed in this study. The lagest diffeences obtained ae 2.5% and 6.2%, espectively, fo the full load and ballast conditions. On the othe hand, it can be obseved fom the tables that the fequencies decease with inceasing aea of contact with the fluid. The ship hull has the lagest aea of contact fo the full load condition, and theefoe, the lowest fequencies wee obtained fo this case. The mode shapes indicated in Tables 3 and 4 ae the global vibation modes of the ship hull, and the fist fou global mode shapes ae shown in Figue 3. (a) (b) (c) (d) Fig.3 The fist fou global modes of the containe ship: (a) 1 st tosion, (b) 1 st bending, tosion, (d) 2 nd bending. (c) hoizontal bending & CONCLUSIONS The dynamic chaacteistics (wet fequencies and associated mode shapes) of a containe ship fo two diffeent load conditions wee calculated by an appoach based on the bounday integal equation method, and the fequencies obtained wee compaed with the esults of the finite element calculations. A vey good compaison was obtained between the pedictions of the bounday element and finite element methods. It can be concluded fom the esults that the bounday integal equation method poposed in this study is suitable fo the dynamic analysis of ship hull stuctues and also fo othe elastic stuctues fully o patially in contact with fluid. ACKNOWLEDGEMENTS The authos acknowledge the suppot of a eseach gant fom the Scientific and Technological Reseach Council of Tukey (TUBITAK); Poect No: 105M041.
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