Fast computation for vibration study of partially submerged structures using low resolution hydrodynamic model

Journal of Fluids and Structures 91 (2019) 102756

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Fast computation for vibration study of partially submerged structures using low resolution hydrodynamic model ∗

Shahrokh Sepehrirahnama , Eng Teo Ong, Heow Pueh Lee, Kian-Meng Lim 3 Engineering Drive 2, National University of Singapore, Singapore 117578, Singapore

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Article history: Received 24 April 2019 Received in revised form 5 September 2019 Accepted 30 September 2019 Available online xxxx Keywords: Fluid–structure interaction Fluid added mass Hydroelasticity Ship vibration

a b s t r a c t A fast computation scheme is presented for simulating fluid–structure interaction of partially submerged structures. Fluid added mass is simulated using the boundary element method, which only requires a mesh at the fluid–structure interface while satisfying the far field boundary conditions in a semi-infinite domain. The boundary element solver is coupled to a finite element solver for structural vibration analysis. To speed up the boundary element simulation, a low resolution and geometrically simplified surface mesh was used. A projection scheme is introduced to map fluid elements to structural elements, allowing proper transfer of displacement and fluid loading between the two solvers. A series of low-resolution meshes for a realistic container ship model were used to study the efficiency of the proposed scheme, comparing the errors incurred in calculating the wet natural frequencies against the speed-up in computation. With a 20 times decrease in the number of boundary elements, the relative error in wet natural frequency calculation was found to be less than 10%, while computation speed-up was found to be almost 500 folds. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Additional loading from surrounding fluid affects the vibration response of any structure that is fully or partially immersed (Newman, 1979; Korotkin, 2008; Brennen, 1982). This fluid loading is normally represented as a system of mass and damper, which are called added mass and added damping, respectively (Newman, 1979; Korotkin, 2008; Yang, 1990). Added mass and damping are functions of fluid density and the displacement profile at the fluid–structure interface. In addition, added damping depends on the fluid viscosity and free-surface condition (Yang, 1990; Salvesen, 1978). For special case of large structures such as containership in contact with seawater, the added damping can be neglected since fluid viscosity is relatively small (Brennen, 1982; Yang, 1990). The main challenge of Fluid–Structure Interaction (FSI) modeling for vibrating structures is to find the fluid loading for the desired mode of vibration (Ergin and Price, 1992; Ergin and Temarel, 2002; Everstine and Henderson, 1990; Liang et al., 2007; Zhou et al., 2005; Fossen and Smogeli, 2004; Haddara and Cao, 1996). For rigid body oscillation, such as heave or sway motions of ships, added mass calculation is straightforward since the displacement amplitude is the same for the entire structure (Yeung, 1981; Lin and Liao, 2011; Rahman and Bhatta, 1993). However, the effect of the surrounding fluid on the flexural vibration of the structure requires numerical computations, especially for complex structures such as ships and off-shore marine structures, since the displacement profile varies across the surface area of the wetted part ∗ Corresponding author. E-mail address: [email protected] (S. Sepehrirahnama). https://doi.org/10.1016/j.jfluidstructs.2019.102756 0889-9746/© 2019 Elsevier Ltd. All rights reserved.

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of structure (Ergin and Price, 1992; Ergin and Temarel, 2002; Haddara and Cao, 1996; Han and Xu, 1996; Yadykin et al., 2003; Sinha et al., 2003; Liu et al., 2014; Ge et al., 2010). Generally, the hydroelastic analysis of ship structures are conducted to obtain the wave loads, restoring stiffness, whipping and sloshing phenomena, etc. Malenica and Derbanne (2014). Due to their complex nature, investigating these effects requires numerical simulations (Malenica and Derbanne, 2014; Senjanović et al., 2014b; Im et al., 2017). In many studies, ship hull was modeled as a beam to obtain its dry natural frequencies and mode shapes (Malenica and Derbanne, 2014; Senjanović et al., 2014b; Im et al., 2017; Senjanović et al., 2014a; Verma et al., 2019; Thekinen and Datta, 2019; Datta and Thekinen, 2016). The advantage of using beam theory is to solve the free vibration of the dry structure analytically and only solve the hydrodynamic part numerically. However, the shortcomings of beam modeling of the ship hull in terms of capturing coupling between bending and torsion and warping mode shapes as well as its integration with 3D BEM simulation led to the full-scale 3D modeling of the ship with all geometry details for numerical hydroelasticity analysis (Malenica and Derbanne, 2014; Senjanović et al., 2014b; Im et al., 2017; Senjanović et al., 2014a; Verma et al., 2019; Thekinen and Datta, 2019; Datta and Thekinen, 2016; Sepehrirahnama et al., 2019; Kim and Kim, 2017). Among the reported numerical methods of simulating the added fluid mass effect on flexural vibration, as a part of hydroelasticity analysis, finite element method (FEM) and boundary element method (BEM) are commonly used (Malenica and Derbanne, 2014; Senjanović et al., 2014b; Im et al., 2017; Senjanović et al., 2014a; Sepehrirahnama et al., 2019; Kim and Kim, 2017). Compared to other CFD methods, BEM is preferred to accommodate the semi-infinite domain of the fluid without loss of accuracy. Also, FEM is commonly used for the structure vibration (Everstine and Henderson, 1990; Liang et al., 2007; Lin and Liao, 2011; Cabos and Ihlenburg, 2003; Sepehrirahnama et al., 2019). Coupling FEM and BEM at the formulation level is one of the techniques to solve such problems (Liang et al., 2007; Wang and Meylan, 2004; Förster et al., 2007; Malenica and Derbanne, 2014; Datta and Thekinen, 2016); however, the augmented system matrix, which contains both BEM and FEM matrices, no longer keeps the sparse and symmetric properties of FEM matrices. The order of computational complexity of BEM scales by O(N 3 ), where N is the number of boundary elements, and this would contribute a major part of the computational time for the entire simulation. Another popular algorithm is based on modal superposition by using the dry mode shapes (Ge et al., 2010; Ergin and Uğurlu, 2003; Sepehrirahnama et al., 2019). Partial fluid loading is calculated for each of the dry mode shapes by using BEM. The total fluid loading is obtained from the partial ones by applying the modal superposition (Ge et al., 2010; Ergin and Uğurlu, 2003). In the calculation of the partial fluid loading, the interface displacements, extracted from the vibration mode shapes, are passed to the fluid panels or elements of BEM. These fluid panels are obtained from FEM model at the fluid–structure interface, so that the transfer of displacement information is simple and straightforward (Cabos and Ihlenburg, 2003). The calculated fluid loading is passed back to the structural FEM model in a similar manner. However, for a one-to-one mapping of displacement and pressure fields on both FEM and BEM discretizations of a full 3D model, the number of fluid panels are constrained to be equal to the FEM elements (Sepehrirahnama et al., 2019; Kim and Kim, 2017). This is not desirable since BEM is computationally expensive due to the O(N 3 ) computational complexity. To reduce the computational effort while maintaining reasonable accuracy, low-order BEM panels or constant field elements, were used for complex geometries (Sahin et al., 1993, 1997). It is also possible to speed up the BEM simulation by using Fast Multipole Method (Lin and Liao, 2011). Despite the promising speed-up in the BEM computation, the reported studies were limited to the rigid body oscillation of the structures. The proposed way to circumvent the issue of high computation cost of the BEM simulation is to reduce the number of fluid panels. This will give rise to a mismatch between the FEM elements and the BEM panels at the interface; thus requiring extra steps to properly transfer the displacement and force information between the two meshes. In this work, a projection method is proposed to map displacement and fluid pressure fields from the BEM fluid panels to FEM elements and vice versa, at the interface. A separate surface, not necessarily conforming to the interface, with an arbitrary geometry is used to generate the BEM fluid panels. To reduce the number of fluid panels, details of the interface geometry (such as curvature, indented regions or extruded sections) can be removed to create a low-resolution representation of the interface. Coarse elements on surfaces with simple geometry are also used. In this study, several low resolution surfaces were proposed and used to calculate the wet natural frequency of a vibrating structure that are partially submerged in water. A real-life container ship structure was used to demonstrate the accuracy and efficiency of the low resolution models used. 2. Methodology The additional fluid loading against structure displacement is incorporated in the vibration equation as an external force, denoted by {f}f , as follows,

( 2 ) −ω M + K {X} = {f}f

(1)

where {X} is the amplitude of displacement, M and K are the mass and stiffness matrices, and ω is the circular frequency of the excitation force. The damping in the system and any other external excitation forces were neglected for the present analysis to calculate the wet natural frequencies. To solve the above system, a modal superposition method is applied by

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Fig. 1. Schematic illustration of floating structure in the fluid domain with the free surface. Image-method is applied to automatically impose the zero-pressure condition (pf = 0) on the free-surface of the fluid. n and nI are the unit normal vector to the immersed part of the structure and its image, respectively.

first obtaining the dry mode shapes when the fluid loading is not present, that is {f}f = 0. A set of dry mode shapes are obtained from the FEM vibration analysis of the dry structure and they are put in the modal matrix D, D = [X(1) , X(2) , . . . , X(j) , . . . , X(m) ].

(2)

Here, only m dry mode shapes are used, where m is typically much smaller than the total number of degrees of freedom N in the FEM model. (j) For each of the dry mode shape X(j) , the partial fluid loading ff corresponding to that mode shape is calculated using the BEM. Fig. 1 shows a schematic illustration of the partially immersed structure in a semi-infinite fluid domain, and the use of image-method to impose zero pressure condition on the free-surface. For the details of BEM modeling and simulation, please refer to reference (Sepehrirahnama et al., 2019). The unknown wet mode shape and fluid load in Eq. (1) are then written as linear combinations of the dry mode shapes and partial fluid loading,

{X} = [X(1) , X(2) , . . . , X(j) , . . . , X(m) ]{L} {f}f = ω [ 2

(1) ff

,

(2) ff

,...,

(j) ff

,...,

(m) ff

(3)

]{L}

(4) (j)

where {L} is the vector of participation factors to be solved. The expressions for fluid loading ff have ω2 factored out, so that they represent the fluid added effect that acts at the fluid–structure interface. The original system of equations then gives the following eigenvalue problem (Ergin and Price, 1992; Ergin and Temarel, 2002; Ergin and Uğurlu, 2003)

(

[ ] ) ˆ +M ˆ a + Kˆ {L} = {0}, −ω2 M

(5)

where

ˆ = DT KD, K

ˆ = DT MD, M

(6)

and (1)

(2)

(m)

ˆ a = DT [f , f , . . . , f ] M f f f

(7)

are the reduced structural stiffness, structural mass, and fluid added mass matrices, respectively. The eigenvalues would give the wet natural frequencies ωj of the fluid–structure interaction problem, and the wet mode shapes {X}j are reconstructed from the dry mode shapes:

{X}j = D{L}j ,

j = 1, 2, . . . , m.

(8)

If the BEM mesh is obtained directly from the FEM mesh at the fluid–structure interface, the transfer of displacement (j) from the dry modes X(j) and the calculation of the partial fluid loading ff would be straightforward. However, to reduce the computational cost of hydrodynamic simulation, a low resolution mesh for BEM is sought. First, the interface mesh, which is referred to as original mesh, is extracted from 3D FEM mesh. Then, a low-resolution BEM mesh, called Reference mesh hereinafter, is constructed by removing geometry details of the original mesh. The panels on Reference mesh are always coarser than FEM elements at the interface; hence, a projection method is required to transfer information from one mesh to the other (Fig. 2). The coarser elements of Reference mesh results in speed-up in BEM computations. The proposed projection method in this study is based on the normal unit vector to the elements of the original mesh. As shown in Fig. 2, a line is drawn from the centroid of an FE element, point p, in the direction of the unit normal vector n. If the line intersects with any fluid panel of the reference mesh, the distance between the centroid p and the intersection

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Fig. 2. Schematic illustration of projection method from three elements of original mesh to a coarser hydro panel from the reference mesh. The line from the centroid point p to the projection point h is along the unit normal vector to the smaller element. Table 1 Particulars of the container ship. Mode shape

Dry ship

Length overall Length between perpendicular Beam molded Depth molded Design draft Deadweight at design draft (8 m)

204 m 194 m 28 m 17 m 10 m 31 kt

point h is recorded. Subsequently, the fluid panel that corresponds to the smallest projection distance is considered for mapping of displacement from the structural element to the reference mesh. The component of displacement at point p that is normal to the reference element is then extracted and assigned to point h. For fluid panels that contain more than one projected points, the displacement at its centroid q is calculated by averaging the displacements of the projected points. If a fluid panel contains no projected point, its centroid displacement would be set to zero. The fluid panels are constant field surface elements with three nodes at the vertices of triangle. The advantage of using such fluid panels is the uniqueness of the unit normal across the entire surface of the fluid interface (Sahin et al., 1997). Main steps of the projection technique are summarized as follow: 1. 2. 3. 4.

Find centroid of each FEM element, denoted by p, Draw a line from the centroid in the direction of element normal vector Find projection of centroid p on one of the panels, denoted by h, Repeat for all FEM elements

Once the pressure at each fluid panel is obtained from the BEM simulation, it is transferred back to the original structural elements that are mapped to this fluid panel. The forces acting at the nodes of the structural element are calculated by the FEM element shape function. Structural elements that are not in contact with the fluid will experience zero fluid loading. Using this projection scheme between the original FEM mesh and coarse reference BEM mesh helps (j) to construct partial fluid loading vectors ff that have the same dimension as the number of degrees of freedom in the original FEM mesh. 3. Container ship model To demonstrate the proposed scheme, the wet natural frequencies of the first six modes of vibration of a container ship are calculated. A sample container ship model (from the American Bureau of Shipping (ABS)) with particulars listed in Table 1 is used for the case studies in the following section. As shown in Fig. 3, the model includes the ship hull, the deckhouse and machinery propulsion system. Plates and beam elements were used to achieve better structural stiffness representation. In addition, mass elements were applied to represent cargo, water ballast in tanks, fuel oil in tanks and non-structural masses. The element size is equal to the representative spacing of the girders for most part of the model. The global coordinate (right-hand) system of the model is defined with x-axis (longitudinal) positive from aft to fore, y-axis (vertical) positive upward, and z-axis (transverse, athwart-ship) positive towards starboard. The x-origin is located

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Fig. 3. Finite Element model of container ship with internal partitions. The average mesh size is less than 2 m which gives approximately 12 000 elements with 4000 of them on the wet part of the ship hull.

at the aftermost of the ship, while the y-origin and z-origin are located at the intersection of the baseline and centerline planes. The result of mesh convergence study showed that element size of less than 2 m is appropriate for extracting the first 50 mode shapes accurately. In this study, only the first six mode shapes are investigated; hence, the mesh used is sufficient to ensure convergence and good accuracy of the Finite Element analysis. The free vibration of the dry ship was performed by using NASTRAN that is the most commonly used commercial FEM software package in the shipping industry. For draft of 8 m, the number of elements on the interface of the fluid and the ship structure is 4004. The wet natural frequencies of the first six flexural mode shapes, given in Table 2, were calculated by using the original mesh, with the proper meshsize obtained from convergence study (Sepehrirahnama et al., 2019), directly for the BEM simulation. As

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S. Sepehrirahnama, E.T. Ong, H.P. Lee et al. / Journal of Fluids and Structures 91 (2019) 102756 Table 2 Natural frequencies of the container ship model obtained by using the original mesh of the ship hull (Sepehrirahnama et al., 2019). The frequency values are in Hz. Mode shape

Dry ship

Wet ship

% reduction

1-node 2-node 2-node 3-node 3-node 2-node

0.686 0.850 1.191 1.827 2.272 2.519

0.624 0.631 1.038 1.332 1.987 2.250

9.037 25.764 12.846 27.093 12.544 10.678

torsional bending vertical bending horizontal bending vertical bending horizontal bending torsion bending

expected, the natural frequency decreases for all the mode shapes, showing a positive fluid added mass. The downshift of the frequencies depends on the mode of vibration. Only global torsion, vertical bending, and horizontal bending of the ship are discussed in this work. Due to the large number of BEM fluid panels, the number of operations for the BEM computation is of order of O(1011 ). The proposed projection scheme helps to drastically reduce the computational time of the BEM calculations. 4. Results Theoretically, there is a trade-off between the speed-up of the BEM computations and the accuracy. By finding an appropriate low-resolution BEM mesh, one can achieve acceptable accuracy within a reasonable computation time. Two indices were introduced to correlate low-resolution mesh to the one extracted from FE discretization with all geometry details. The first index, denoted by α , is the ratio between the surface area of the low-resolution reference mesh and the original one,

∫

Γ α= ∫ R

ΓO

dΓ dΓ

,

(9)

where subscripts R and O refer to the low-resolution reference mesh with coarse elements and the original mesh, respectively. This index indicates how much the area of structure–fluid interface changes after simplifying the geometry or removing the details such as curvature, indented or extruded sections. The second index, denoted by β , is the root mean square (RMS) of the projection distance, which is the distance between the two points p and h in Fig. 2,

√ β=

∑N

j=1

|pj − hj |2 N

(10)

where pj is the centroid of the jth element of the original mesh, hj is the projection of pj on the reference surface with coarser elements and N is the number of elements of the original mesh. This index, which is of the unit of length, indicates how close the reference mesh is to the original discretization of the fluid–structure interface. Ten low-resolution reference meshes were considered, as shown in Fig. 4. Four of these meshes, Fig. 4(a)–(d), are generated from a surface with geometry conforming to the ship hull by using coarser elements of sizes 6 m and larger than 8 m. The first three, as shown in Fig. 4(a)–(c), include the visible details of the original mesh while the elements were coarsened. Further simplification was done to create the fourth reference mesh, Fig. 4(d), with the element size of larger than 8 m. These four reference meshes were used to investigate the influence of element size on the accuracy of natural frequency estimation. The next three reference meshes, as shown in Fig. 4(e)–(g), are based on a cylindrical surface with the same draft level but with varying width at the rim. Three width values were chosen such that they are larger, equal, and smaller than the width of the original mesh. The radii of these cylindrical surfaces were calculated based on their width and draft. Cylindrical surface was chosen to replace the original mesh that is a surface of continuously varying curvature. The geometrical details at the stern and bow were removed to further simplify the low resolution mesh. The last three meshes, as shown in Fig. 4(h)–(j), are three rectangular boxes that have the same dimensions as the three cylindrical surfaces. The rectangular box geometry is inspired by the middle section of the ship, from l ≈ 40 to 150 m. The lower resolution, compared to the original mesh, was achieved by extending the rectangular surface to both ends of the ship hull. The draft level was maintained during the discretization of the low-resolution reference meshes while the width was varied to find the appropriate choice of reference mesh that gives good accuracy. 4.1. Conforming reference mesh Increasing the size of linear elements on a given surface changes the curvature within the occupied area by coarser elements. Mesh coarsening is one way to keep extruded features and indented regions of the surface while reducing the curvature by using lesser number of elements. Fig. 5(a)–(c) show the wet part of the ship hull with coarser elements than

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Fig. 4. Ten reference meshes for replacing the original finite element mesh of the hull for the BEM simulation. Panels (a)–(d) are conforming meshes, (e)–(g) cylindrical reference meshes, and (h)–(j) rectangular box reference meshes. Table 3 Wet natural frequencies of the container ship model obtained by using the conforming reference meshes, as shown in Fig. 4(a)–(d). Mode shape

Orig. mesh

Ref. mesh 1

Ref. mesh 2

Ref. mesh 3

Ref. mesh 4

1-node 2-node 2-node 3-node 3-node 2-node

0.624 0.631 1.038 1.332 1.987 2.250

0.624 0.636 1.031 1.341 1.980 2.247

0.628 0.639 1.035 1.353 1.990 2.256

0.626 0.635 1.033 1.345 1.964 2.223

0.630 0.632 1.040 1.341 1.984 2.263

torsional bending vertical bending horizontal bending vertical bending horizontal bending torsion bending

the original mesh. The propeller shaft and skeg at the stern and the indentations at the bow were preserved in these three reference meshes. The fourth reference mesh, Fig. 4(d), has coarse elements of size larger than 8 m, while the local details at bow and stern sides were removed. These conforming surfaces are shown in Fig. 5 while projected points are illustrated on the coarse fluid panels. The estimated wet frequencies obtained by using these four reference meshes are reported in Table 3.

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Fig. 5. Four conforming reference meshes with the projected points, shown across all their fluid panels.

Fig. 6. Relative error of the wet frequencies, calculated for the four conforming reference meshes, with respect to (a) the area ratio index and (b) RMS of the projection distance for six mode shapes. Torsional, Vertical and Horizontal bending modes are denoted by T, V and H, respectively.

The frequency results for all the mode shapes show that the frequency downshift due to the added fluid mass was achieved accurately, since the computed wet natural frequencies from Reference meshes are close to those that were obtained by using the original mesh. The modal added masses for all cases are positive, in line with the theory of hydroelasticity (Ergin and Price, 1992). The relative accuracy for the various meshes can be compared using a relative error indicator, denoted by ϵ , that is

ϵ=

|ωˆ − ω| , ω

(11)

where ω and ω ˆ are the wet natural frequency calculated by using the original mesh and low resolution reference mesh, respectively.

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Fig. 7. Three cylindrical reference meshes with the projected points, shown across all their fluid panels. Table 4 Wet natural frequencies of the container ship model obtained by using the cylindrical reference meshes, as shown in Fig. 7. Mode shape

Orig. mesh

Ref. mesh 5

Ref. mesh 6

Ref. mesh 7

1-node 2-node 2-node 3-node 3-node 2-node

0.624 0.631 1.038 1.332 1.987 2.250

0.651 0.668 1.032 1.438 2.004 2.363

0.633 0.635 1.009 1.365 1.984 2.377

0.614 0.625 0.998 1.328 1.973 2.369

torsional bending vertical bending horizontal bending vertical bending horizontal bending torsion bending

Fig. 6 shows the accuracy of the wet natural frequencies, obtained by using conforming reference meshes, with respect to area ratio α and RMS projection distance β . It was observed that the estimation accuracy varies for different mode shapes. The relative error falls within 0.02% to 1.5%, for all the mode shapes. This accuracy range is very promising, considering the aggressive mesh coarsening process (more than 4 times of the original element size), removal of geometry details, such as the propeller shaft and indented parts of the bulbous bow, and the simplifications in the projection technique. It is concluded that conforming reference mesh, which can be generated easily by coarsening the existing elements on the FE model, is suitable for speeding up the fluid simulation by using BEM. Good accuracy is achieved for the first six natural frequencies because these are global modes of vibration that are unaffected by the removal of small features and details in the ship structure. Comparing different cases with respect to area ratio α shows that the accuracy of the computed frequencies increases as the surface area of the reference mesh approaches to the one of the original mesh i.e. α → 1. The two cases that give error of less than 1% are the Ref. Mesh 1 (with element size of 6 m) and Ref. Mesh 4 (with element size of larger than 8 m and without bow and stern details). Between these two cases, the surface area ratio α of Ref. Mesh 4 is very close to one. This implies that an acceptable accuracy is achievable for heavily simplified meshes as long as the surface area is kept close to the original mesh. Looking at the RMS projection distance, Ref. Mesh 2 (with full details and 8 m elements) has a slightly lower measure of β = 0.447 than Ref. Mesh 1 (with 6 m elements) with β = 0.541. This implies that the fluid panels in Ref Mesh 2 are generally closer than Ref Mesh 1 to the original mesh. However, this Ref Mesh 2 gives less accurate results compared to the Ref. Mesh 1. This indicates that β is not a good measure to indicate the level of inaccuracy due to using the conforming reference meshes. 4.2. Cylindrical reference mesh To investigate other acceptable choices of the reference mesh, the geometry is further simplified into a half cylinder with two flat ends. Since the curvature of the cylinder geometry is constant, it is considered as a potential choice for generating low-resolution reference meshes. In addition, the geometry complexity at bow and stern were removed from this set of reference meshes. Fig. 7 shows the three cylindrical reference meshes that were used to replace the original mesh of the fluid–structure interface. The length and draft of the cylindrical surfaces are equal to the original mesh. By varying the width at the rim, three different surface meshes were constructed. Table 4 shows the estimated wet frequencies that were obtained by using the cylindrical reference meshes. It is observed that, even though the cylindrical reference meshes are much simplified, the wet natural frequencies for all the six mode shapes are quite close to those from the original mesh. The frequency downshift or added fluid mass is effectively captured by the cylindrical reference meshes. It implies that the current projection scheme onto a surface with simple geometry may yield acceptable estimation of the wet frequencies corresponding to the global modes of flexural vibration of the ship, as long as the length and the draft are the same as the original mesh. To study the influence of the cylinder width on the accuracy of the results, the relative errors of the reported frequencies are shown in Fig. 8. Considering all the three cases, the range of the relative errors is from around 0.1 to

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Fig. 8. Relative error of the wet frequencies, calculated for the three cylindrical reference meshes, with respect to (a) the area ratio index and (b) RMS of the projection distance for the six mode shapes. Torsional, Vertical and Horizontal bending modes are denoted by T, V and H, respectively.

10%. The smallest error was found for the wet frequency corresponding to the 3-node horizontal bending mode shape, obtained by using Ref. Mesh 6 (28.4 m width), which is the same as the original mesh. Using Ref. Mesh 5 (22 m width) gave the largest error for the frequency of the 3-node vertical bending mode. Comparing the overall error, the case of 22 m width gives the least accurate results. It is concluded that equal or larger width of the reference mesh leads to better accuracy. As expected, Ref. Mesh 6 (28.4 m width) has the closest surface area to the original mesh (α = 1.056). Its RMS projection distance β is 3.165 m which is higher than those reported for conforming reference meshes. Similarly, the other two cases have relatively large β values, which is an indication of the shape difference between the original mesh and the low-resolution cylindrical meshes. Considering the two indices α and β , the most accurate estimation of the wet frequencies was achieved for Ref. Mesh 7 (32 m width), since the errors of half of the estimated frequencies are less than 1%. The area ratio α = 1.152 of Ref. Mesh 7 (32 m width) is around 15% larger than the original mesh; however, the achieved accuracy is better than Ref. Mesh 6 (28.4 m width) with α = 1.056. Ref Mesh 7 (32 m width) with a RMS projection distance β = 2.939, the smallest among the three cases, also gives the least error overall. In this case of highly simplified cylindrical reference meshes, the index β takes on larger values (ranging from 3 to 4) compared to the small values of 0.4 to 1.2 in the previous case of conforming reference meshes. Thus, a larger value of β seems to give a better indicator of accuracy for non-conforming meshes. 4.3. Rectangular reference mesh To simplify the reference mesh further, rectangular boxes with larger disparity from the original mesh were considered. Compared to the cylindrical ones, these reference meshes has the lowest resolution of geometry details since the curvature is zero everywhere. Fig. 9 shows the three choices of the rectangular reference meshes with the projected points being shown on the hydro panels. This geometry of the reference meshes was inspired by the middle part of the ship hull, between l ≈ 40 to 150 m, that is very similar to a rectangular box. The selected reference meshes have the same length and draft as the original mesh. Similar to the cylindrical meshes, the width was varied to construct three reference meshes with element size of larger than 8 m. Using these rectangular reference meshes, the estimated wet frequencies are reported in Table 5. Similar to the previous case studies, the values in Table 5 show that the wet natural frequencies from the reference meshes are comparable to those from the original mesh. The added fluid mass effect is still captured by the frequency downshift from the dry natural frequencies. It also implies that the proposed projection technique works well regardless of the geometry of the low-resolution reference mesh. Fig. 10 show the relative error ϵ of the estimations of the wet frequencies with respect to area ratio α and RMS projection distance β . It is observed that the error range is much higher compared to the previous cases, typically between 1% to 10%. An exception (with relatively good accuracy) was obtained for 2-node vertical and 2-node horizontal bending modes, using Ref. Mesh 10 with the largest width (32 m). Compared to the cylindrical reference meshes, the lower bond

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Fig. 9. Three rectangular reference meshes with the projected points, shown across all their fluid panels.

Fig. 10. Relative error of the wet frequencies, calculated for the three reference meshes with rectangular cross-section, with respect to (a) the area ratio index and (b) RMS of the projection distance for the six mode shapes. Torsional, Vertical and Horizontal bending modes are denoted by T, V and H, respectively. Table 5 Wet natural frequencies of the container ship model obtained by using the rectangular box meshes, as shown in Fig. 9. Mode shape

Orig. mesh

Ref. mesh 8

Ref. mesh 9

Ref. mesh 10

1-node 2-node 2-node 3-node 3-node 2-node

0.624 0.631 1.038 1.332 1.987 2.250

0.636 0.663 1.076 1.381 2.054 2.368

0.591 0.642 1.060 1.289 2.046 2.352

0.572 0.631 1.048 1.259 2.039 2.361

torsional bending vertical bending horizontal bending vertical bending horizontal bending torsion bending

of the error range increased by one order of magnitude. This is due to further reduction of the curvature by replacing the cylinder geometry with flat faces. The overall accuracy achieved for the Ref. Mesh 10 (32 m width) is better than the other two cases, even though the area ratio α and RMS projection distance β of the 32 m case are the largest. This suggests that α or β are no longer good measures of accuracy of the reference mesh when the deviation between the reference mesh and original mesh is too large. In this case, the parameter values of α = 1.403 and α = 1.519 for Ref. Meshes 9 and 10 indicate that there is a big mismatch in area. However, it is found that better accuracy is expected when the width of either cylindrical or rectangular reference meshes is equal to or larger than the one of the original mesh. If the width is smaller, the overall relative error increases irrespective of the geometry of the reference mesh. It is concluded that a reference mesh that encapsulates the original mesh is more desirable, although the error may be high (between 1% to 10%).

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Fig. 11. Wet natural frequencies of the container ship corresponding to the first six flexural mode shapes, obtained by using the original mesh and 10 Reference meshes.

Fig. 12. Total simulation time, plotted with respect to the number of elements of the hydrodynamic model.

4.4. Accuracy and computation speed-up Three sets of reference meshes with different geometries and increasing level of mismatch with the original mesh at the fluid structure interface were used to study the accuracy and error introduced by a low resolution BEM mesh. Based on the wet natural frequencies calculated for the first six global modes (as shown in Fig. 11), the relative error increases with greater mismatch between the reference and original meshes. Black dotted lines showed the wet natural frequencies obtained by using the original mesh with N = 4004 elements. The relative errors were within 10% for the cases investigated, implying that low-resolution meshes with coarse elements may be acceptable if one is ready to trade off accuracy for speed in the BEM simulations. When the element size on the reference mesh is set to larger than 8 m, the number of panels reduces to around N = 180. Compared to the original mesh, there is approximately 20 times reduction in the number of degrees of freedom. Since the computational complexity for solving the fully-populated BEM matrix system is of the order O(N 3 ), a reduction of computational time by about 8000 times would be expected. However, this is not the case as there are additional computations for the projection scheme and the coupling with the original structural FEM solver (albeit using a computationally less intensive modal superposition scheme). Fig. 12 shows the total simulation time with respect to the number of elements for the original mesh and the first three conforming reference meshes that were generated by coarsening the elements of the original mesh. The reduction in computational time is about 500 times (from 500 s to about

S. Sepehrirahnama, E.T. Ong, H.P. Lee et al. / Journal of Fluids and Structures 91 (2019) 102756

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1 s), after reducing the number of degrees of freedom in the BEM by 20 times. The anticipated speed-up by an order of O(N 3 ) was not achieved; rather, a speed-up of O(N 1.92 ) was observed due to the additional computations in the projection scheme. This is still very promising and effective for real-life design simulations. For the present case study of a container ship model with realistic dimensions, the simulation time was reduced from 500 s or about 8 min to approximately 1 s, which allows running more number of simulations for parametric studies and design modifications within a given period of time. All the simulations were run on a Dell Workstation (model Precision T7810) with Intel-Xeon CPU running at 2 GHz (2 Processors), 128 GB of RAM and Windows operating system. 5. Conclusions In this study, a low resolution BEM mesh was used together with a FEM structure mesh to solve a fluid–structure interaction problem. The method was demonstrated using a realistic container ship model to calculate its wet natural frequencies when the ship is partially submerged. A projection scheme was proposed, enabling a fine (original) mesh on the structural model to be mapped to a coarse (reference) mesh of the fluid model with proper transfer of displacement and fluid loading information between the two solvers. The low-resolution discretization of the interface was achieved by coarsening the original elements to reduce the geometry curvature and removing the complexities at the bow and stern such as propeller shaft, skeg and bulbous bow. A total of ten low-resolution reference meshes based on three geometries (conforming surface, cylinder, and rectangular box) were investigated. The first six flexural mode shapes, which are of global vertical bending, horizontal bending, and torsion were considered. Form the case studies, the following conclusions were drawn:

• It was found that the error of estimating wet frequency is less than 10% for all the case studies. • The best accuracy was achieved for the conforming reference meshes with coarse elements (Reference meshes 3 and 4).

• It was shown that removing the geometry details of the bow and stern has little influence on the calculated frequencies, as they correspond to global modes and elimination of fine features and details do not affect the accuracy. • The computation speed-up in using this projection scheme was found to be approximately second order, O(N 1.95 ). The presented coupled FEM–BEM algorithm with the projection methodology was implemented as a software package that is useful for actual design simulations of partially submerged structures. Acknowledgments This study is supported by the research project ‘‘Advanced Computational Tools for Fast Analysis and Design of Ships against Vibration and Noise’’ funded by the Singapore Maritime Institute (Project ID: SMI-2015-MA-08). The sample container ship model used in this study was kindly provided by our collaborator, American Bureau of Shipping (ABS). References Brennen, C.E., 1982. A Review of Added Mass and Fluid Inertial Forces. Department of the Navy. Cabos, C., Ihlenburg, F., 2003. Vibrational analysis of ships with coupled finite and boundary elements. J. Comput. Acoust. 11 (01), 91–114. http://dx.doi.org/10.1142/S0218396X03001821. Datta, N., Thekinen, J., 2016. A rayleigh–ritz based approach to characterize the vertical vibration of non-uniform hull girder. Ocean Eng. 125, 113–123. Ergin, A., Price, W.G., 1992. Dynamic characteristics of a submerged, flexible cylinder vibrating in finite water depths. J. Ship Res. 36, 154–167. Ergin, A., Temarel, P., 2002. Free vibration of a partially liquid-filled and submerged, horizontal cylindrical shell. J. Sound Vib. 254 (5), 951–965. Ergin, A., Uğurlu, B., 2003. Linear vibration analysis of cantilever plates partially submerged in fluid. J. Fluids Struct. 17 (7), 927–939. Everstine, G.C., Henderson, F.M., 1990. Coupled finite element/boundary element approach for fluid–structure interaction. J. Acoust. Soc. Am. 87 (5), 1938–1947. Förster, C., Wall, W.A., Ramm, E., 2007. Artificial added mass instabilities in sequential staggered coupling of nonlinear structures and incompressible viscous flows. Comput. Methods Appl. Mech. Engrg. 196 (7), 1278–1293. Fossen, T.I., Smogeli, Ø.N., 2004. Nonlinear time-domain strip theory formulation for low-speed manoeuvering and station-keeping. Model. Identif. Control 25 (4), 201. Ge, F., Lu, W., Wu, X., Hong, Y., 2010. Fluid-structure interaction of submerged floating tunnel in wave field. Procedia Eng. 4, 263–271. Haddara, M.R., Cao, S., 1996. A study of the dynamic response of submerged rectangular flat plates. Mar. Struct. 9 (10), 913–933. Han, R.P.S., Xu, H., 1996. A simple and accurate added mass model for hydrodynamic fluid–structure interaction analysis. J. Franklin Inst. B 333 (6), 929–945. Im, H.-I., Vladimir, N., Malenica, Š., Cho, D.-S., 2017. Hydroelastic response of 19,000 TEU class ultra large container ship with novel mobile deckhouse for maximizing cargo capacity. Int. J. Nav. Archit. Ocean Eng. 9 (3), 339–349. Kim, J.-H., Kim, Y., 2017. Numerical computation of motions and structural loads for large containership using 3d rankine panel method. J. Mar. Sci. Appl. 16 (4), 417–426. Korotkin, A.I., 2008. Added Masses of Ship Structures, Vol. 88. Springer Science & Business Media. Liang, Q.W., Rodriguez, C.G., Egusquiza, E., Escaler, X., Farhat, M., Avellan, F., 2007. Numerical simulation of fluid added mass effect on a francis turbine runner. Comput. & Fluids 36 (6), 1106–1118. Lin, Z., Liao, S., 2011. Calculation of added mass coefficients of 3d complicated underwater bodies by fmbem. Commun. Nonlinear Sci. Numer. Simul. 16 (1), 187–194.

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