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Advances in Marine Structures – Guedes Soares & Fricke (eds)© 2011 Taylor & Francis Group, London, ISBN 978-0-415-67771-4

Efficient calculation of fluid structure interaction in ship vibration

M. WilkenGermanischer Lloyd SE, Hamburg, Germany

A. MenkRobert Bosch GmbH, CR/APJ3, Stuttgart-Schwieberdingen, Germany

H. VossHamburg University of Technology, Hamburg, Germany

C. CabosGermanischer Lloyd SE, Hamburg, Germany

ABSTRACT: Simulating global ship vibration can be split into three steps: firstly, the computation of the dry elastic vibration of the ship structure, secondly determination of the hydrodynamic pressures caused by a given time harmonic velocity distribution on the outer shell and thirdly, the solution of the coupled vibration problem by considering the interaction of fluid and structure. In this paper various approaches for the solution of the third problem for large models are compared and discussed. They are based on reduction methods for the hydrodynamic mass matrix and make use of fast solution methods for the exterior fluid problem for given velocity distributions of the shell. A numerical example is used to assess the accuracy and the speed of the solution procedures.

1.2 Equations describing the hydrodynamic mass effect

Since the structural displacements in ship vibration are small compared to the dimensions of the ship, the ship and the surrounding fluid can be modeled by a set of linear PDEs. Moreover the flow of the water around the ship’s hull is assumed to be invis-cid and irrotational.

Hence, the velocity field of the fluid is the gradient of a velocity potential—which due to mass conservation satisfies the Laplace equation

∆p = 0 (1)

1 INTRODUCTION

Forces induced by engines and propellers excite ship vibrations which despite of their small amplitudes can affect human comfort and may cause fatigue damages. In order to predict ship vibrations it is indispensable to account for the effect of the sur-rounding water because the hydrodynamic forces acting on the ship’s hull can considerably reduce the natural frequencies of the dry ship and there-fore can significantly affect the vibration response.

1.1 The effect of water on a vibrating structure

The hydrodynamic influence of the water on the vibrations of a ship can be modeled as an addi-tional mass distribution on the outer shell. The acceleration of the structure causes the fluid near the interface to accelerate which in turn exerts an opposing force on the ship’s hull (see Figure 1). The additional force which is needed to accelerate the surrounding fluid can be interpreted by Newton’s law as an additional mass distributed on the ship’s hull. That mass is often called hydrodynamic mass or added mass.

Figure 1. Illustration of the effect of water on the vibrating outer shell.

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Assuming an infinitely wide and deep fluid domain, boundary conditions have to be speci-fied at the free water surface and at the submerged ship’s hull (Figure 2). The exact boundary con-dition at the free surface is nonlinear (Newman 1977), but for frequencies above 1 Hz and small displacements it can be linearized to yield a pres-sure release condition which takes the form p = 0 on Γf . At the submerged ship surface the fluid velocity normal to this surface must be equal to the normal velocity of the structure,

∂∂

ρωp

nu nTT k=

2 r rT on Γ (2)

where p denotes the pressure at the outer shell, ρ the fluid density, ω the excitation frequency,

ru

the displacement of the structure and rn is the out-

ward normal.

2 CONSIDERATION OF HYDRODYNAMIC MASS EFFECTS IN GLOBAL SHIP VIBRATION ANALYSIS

2.1 Standard procedures

2.1.1 Full hydrodynamic mass matrix methodTo account for the surrounding water, an FE model of the ship can be complemented by an FE discretization of the water to solve the Laplace equation with coupling boundary condition (2) (Arman et al., 1979). This causes considerable additional cost since only a bounded region of the fluid domain can be modeled this way and suitable boundary conditions on the outer boundary have to be specified or the remaining unbounded region of the water has to be discretized by semi infinite elements. If Boundary Element (BE) methods are used in combination with a special fundamental solution, an unbounded fluid domain can be mod-eled, but only the submerged ship hull has to be dis-cretized. Thus the problems previously mentioned are avoided. The mesh can simply be generated from the FE mesh of the ship’s hull. Today this is a standard approach if three dimensional effects have to be included in the analysis (Cabos et al., 2003).

An advantage of this method is the fact that the discretization of the coupled fluid-solid problem has the same dimension as the FE model of the dry ship alone in case the free surface boundary condition is handled using the method of images (Wilken et al., 2009). However, as a drawback the part of the mass matrix corresponding to the wet hull of the ship is fully populated.

To determine the eigenfrequencies of the coupled problem, the following eigenvalue prob-lem must be solved:

K u uS wK u et− wet ( )M MS HMMM =Ω 2 0 (3)

where KS, MS are the stiffness and structural mass matrix of a FE model of the ship which are large-scaled and sparse. The hydrodynamic mass matrix MH models the impact of the surrounding water on the ship. Only rows and columns of MH corresponding to wet degrees of freedom contain entries which are different from zero. However, the total number of non-zero entries in the coupled system is still increased considerably. Computing the complete hydrodynamic mass matrix in this way leads to a cubic scaling of required computa-tion time and a quadratic scaling of memory usage with the number of fluid panels.

2.1.2 Lewis approachConsideration of the effect of the surrounding water for the computation of global ship vibra-tions dates back to the first half of the 20th cen-tury. Regarding a ship as a slender body, Lewis (Lewis 1929) showed that the inertia of the water can be approximately be accounted for by analyz-ing the two-dimensional flow around ship cross sections.

The hydrodynamic mass of a cylindrical cross section was generalized to more complex shapes by introducing reduction coefficients. Assuming that a ship is a slender body, Lewis succeeded in determining the hydrodynamic mass affecting vertical bending vibrations of a ship (Figure 3). Since the hydrodynamic mass derived with this method depends on the particular bending mode of the ship, it is typically valid only for a specific range of frequencies around the corresponding eigenfrequency of this mode. Wendel (Wendel 1950) and Landweber (Landweber 1957) extended Lewis’ work by considering also horizontal and rotational acceleration of ship cross sections. Grim (Grim 1953, 1960) examined the reduction coef-ficients for higher modes. The Lewis method is most appropriate for a Finite Element analysis, if the ship is modeled by several beam elements. For the three dimensional analysis of ship vibrations based on a Finite Element model, the use of the

Γf Γf

Γk ΩΓk

Ω

outer shell

Figure 2. Problem description.

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so called Lewis Method leads to the problem that only the total force onto a ship cross section (or better its total hydrodynamic mass) can be com-puted. The actual distribution of the hydrodynamic mass over the contour of the section is determined with a heuristic approach. In a three dimensional vibration analysis this can lead to hydrodynamic forces having components tangential to the shell surface. Despite these approximations and short-comings, the Lewis approach has proven to yield good results in the low frequency range having a very good performance in terms of CPU time and memory consumption.

2.2 Advanced approaches using fast BEM

2.2.1 Fast multipole methodThe standard boundary element method com-putes the pressure on a fluid panel caused by each other vibrating panel. This is done in an exact manner with no restriction on the shape of the immersed hull. The so called fast multipole method takes advantage of the concrete hull shape of the vibrating structure based on the following advisement:

Solving the Laplace equation (1) for vibrat-ing point sources close to each other, the far field result is a pressure field that decreases with one to the square of the distance to these points. It can therefore be idealized in the far field as a pressure field caused by a single vibrating point source. This effect is exploited by the fast multipole method yielding a fast and memory saving procedure for computing the resulting pressure field of a vibrat-ing structure (Wilken et al., 2009).

Considering a typical hull form, it is obvious that the far field approximation above could be applied for the majority of pairs of panels since only a small fraction of them is close to each other. For this reason the fast multipole bound-ary element method seems to be well suited for its

application to vibrating ship structures as shown in the next sections.

2.2.2 Projection approachNeglecting the influence of the surrounding water one obtains the so called dry eigenvalue problem for the ship structure

K u M uS dK u rydd SM =M ud M− Ω2 0 (4)

that can be solved efficiently due to the sparse structure of the matrices.

While the eigenvectors of (3) and (4) exhibit similar characteristics, it is found that the eigen-modes of these problems are quite close to each other. This observation suggests to project the eigenproblem (3) onto a space spanned by a selected subset of nsel eigenvectors Udry of (4). This is done by assuming the wet eigenvectors as linear combination of the dry eigenvectors and multiplying the resulting equation from the left side with UTdry yielding a new eigenvalue problem of a smaller dimension nsel:

U K U wdrU yrT

S wK et drU yrwet− ( )M MS HMMM( ) =Ω 2 0 (5)

The required subset of dry eigenvectors may be choosen as eigenvectors with eigenfrequencies smaller than an appropriate multiple of the upper frequency bound of interest. Instead of explic-itly calculating the hydrodynamic mass matrix it suffices to evaluate the matrix vector product. This can be done by a standard boundary ele-ment method but also more efficiently by the fast multipole boundary element method.

Solving this eigenvalue problem yields eigen-vectors w defining linear combination factors to be used for approximating the wet eigenvectors from the dry eigenvectors. This approach can be improved by projecting onto approximated wet modes (“semi wet modes”) resulting from an eigen-value problem with an approximate hydrodynamic mass matrix:

usu emi( )K M MS sK emi S H sM emi+MSM(( ) = 0 (6)

Using these semi wet modes for projection yields

U K U wseU miT

S wK et seU mi− wet ( )M MS HMMM( ) =Ω 2 0 (7)

respectively a new eigenvalue problem

Ω Ωsemi wet semiT

semi

I Us

U wsemi

2 2Ω

0

(

( )H H semiM MH H ) =× (8)

x

λ

x

λ

Figure 3. Lewis assumption of deformation of the ship structure over the ship length.

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Depending on the number of semi wet modes used for projection, the precision of the wet modes can be influenced.

In the present work the semi wet modes are calculated using the Lewis approach described in section 2.1.2. The solution of the forced vibration problem is then easily obtained by modal superpo-sition using the derived wet modes from (8).

2.2.3 Modal hydrodynamic mass matrix approach

The eigenvalue problem (3) can be solved by the shift-and invert Lanczos method (Bai et al., 2000). Using this method requires the solution of the lin-ear system

KSx − σ(MS + MH)x = b (9)

for x in every iteration step. This can be done by a direct method which requires the explicit form of the matrix MH (i.e., nH solves of the BE sys-tem) and which, due to the structure of the sys-tem matrix A: = KS − σ(MS + MH), is very time and memory consuming. Solving (9) by an iterative solver like MINRES (Saad 2003), every Lanczos step requires a suitable Krylov subspace. Hence, one has to apply the system matrix A to a couple of vectors, and each of these multiplications demands the solution of one BE system. In the following we derive a reduction method which is much more efficient from a computational point of view. The hydrodynamic mass matrix MH is symmetric and positive definite. We take advantage of the spectral decomposition

M x xHM i i ix xT

i

hHh

∑λ (10)

of MH, where the eigenvalues λ λ λi iλ λλ nλ H≥λiλ ≥… are ordered by magnitude. Then the best approxima-tion to MH by a matrix of rank ɶn with respect to the spectral norm is the truncation of (10)

ɶɶ

M x xHM i i ix xT

i

n

∑λ (11)

The Lanczos method is favorable for computing an approximation to ɶMHM since it does not employ the explicit form of the matrix MH but only matrix-vector products and according to the Kaniel-Paige theorem (Golub et al., 1996) it converges first to extreme eigenvalues and in particular to the largest ones which are better separated than the smallest ones. Notice that the Lanczos process for com-puting ɶMHM can even be accelerated by replacing the solution of the BE approximation by the fast multipole approach (Wilken et al., 2009).

Having determined the approximation ɶMHM to MH, we have to solve the reduced eigenvalue problem

K u uS wK u et= wet ( )M MS HMMMɶΩ 2 (12)

Tackling it by the shift-and-invert Lanczos method one has solve a linear system

K x x bSK x − ( )M MS HMMSM (13)

in every iteration step where σ is a preselected shift. Since KS and MS are sparse the LU factorization of KS − σMS can be determined efficiently and since the rank of ɶMHM is quite small compared to the dimension of the problem it is inexpensive to employ the Sherman-Morrison-Woodbury formula (Golub et al., 1996) for solving problem (13).With this approach the shift-and-invert Lanczos method for the reduced problem (12) essentially requires the following work: To initialize one provides those vectors and matrices which are independent of the right hand side b when solving (13) by the Sher-man-Morrison-Woodbury formula. To this end a slightly larger number than ɶn solutions of the BE system are necessary in a Lanczos process for com-puting ɶMHM . Additionally ɶn solutions of linear sys-tems (Ks − σMS)wj = xj and ɶn

2 scalar products of length n are required in this preprocessing phase. Thereafter, every iteration step requires one solve of a linear system of dimension n, and ɶn scalar products of length n.

3 NUMERICAL EXAMPLE

3.1 Model

An FE model of a typical container vessel of 250 m length and 32 m breadth having 35262 degrees of freedom was investigated to assess the accuracy and the speed of the described techniques. This model is capable to compute global vibration responses

Figure 4. Container Ship of 252.2 m length and 32.2 m breadth.

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to a frequency up to 20 Hz. A draft of 10.8 m was used in the calculations leading to 1490 wetted elements of the outer shell with 4554 degrees of freedom.

3.2 Computations

For the assessment of the various methods, the eigenvalues and the forced vibration response caused by a typical propeller induced excitation were calculated. The following approaches for the solution of the vibrational fluid-structure interac-tion problem were performed

1. Usage of the explicit hydrodynamic mass matrix computed by a standard BEM method herein-after referred as “FULL_HYM” method.

2. Usage of a diagonal hydrodynamic mass matrix approximation hereinafter referred as “LEWIS_HYM” approach.

3. Projection of vibration equation (3) onto a set of semi wet modes hereinafter referred as “PROJECTION” approach. The semi wet modes were taken from an eigenvalue computation approximating the hydrodynamic mass matrix according to Lewis. This Lewis approximation yields 338 eigenvectors below 20 Hz (88 modes more than the reference FULL_HYM method) which all were used for the projection.

4. Usage of a modal approximation of the hydro-dynamic mass matrix hereinafter referred as “MODAL_HYM” approach. Due to com-parability reason with the PROJECTION approach the same number of (Fast-) BEM applications were performed leading to a modal hydrodynamic mass matrix approximation of rank 338.

3.3 Comparison

In the following, the modes, i.e. the pairs of eigenvalue and eigenvector, and forced vibration results (i.e. velocity at dedicated locations) of the FULL_HYM method serves as reference values for the comparison of the various approaches.

The FULL_HYM method yields 250 modes below 20 Hz.

3.3.1 PrecisionFor comparing the eigenvectors of the different approaches the modal assurance criterion (MAC), see e.g. Allemang (1980), will be used

MACijCC =( )v M vi refT H jM v appr

( )v M vi refT H iM v referef H j

ref H i

2

( )v M vjv apprT H jM v app, ,appr H j rr (14)

where vi,ref denotes the i-th eigenvector computed by the FULL_HYM method, MH denotes the full hydrodynamic mass matrix and vj,appr denotes the j-th mode computed by the approaches 2–4. A MAC value near to 1 indicates that the approached eigenvector vj,appr is quite similar to the reference eigenvector vi,ref .

This criterion can be subsequently used to com-pare the eigenvalues of the most similar eigenvec-tors computed by the FULL_HYM method and the particular approach:

relErrEE irri appr

j ref

= { }j ( )MACMM i jCCΩΩ, (15)

As it can seen from Figure 7 the eigenvectors computed according to LEWIS matching the reference eigenvectors only for the first 100 modes. The differences in eigenfrequencies is below 5% in the lower frequency range and goes up to 20% in the higher frequency range to 20 Hz.

Relative differences in eigenvalues of the PRO-JECTION and MODAL_HYM approach are of the same magnitude, i.e. below 5% over the total frequency range from 0 Hz to 20 Hz. Also the eigenvectors of these 2 different approaches are of the same similarity compared to the reference eigenvectors.

Figure 5. BEM mesh with 1490 boundary elements.

Figure 6. Forced vibration configuration.

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Figure 7. Similarity of LEWIS_HYM eigenvectors and reference eigenvectors according to MAC.

Figure 8. Relative error of LEWIS_HYM eigenvalues and reference eigenvalues ordered by MAC value.

Figure 9. Similarity of MODAL_HYM eigenvectors and reference eigenvectors according to MAC.

Figure 10. Relative error of MODAL_HYM eigenval-ues and reference eigenvalues ordered by MAC value.

Figure 11. Similarity of PROJECTION eigenvectors and reference eigenvectors according to MAC.

Figure 12. Relative error of PROJECTION eigenvalues and reference eigenvalues ordered by MAC value.

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In Figure 13 typical frequency response curves caused by the single and double propeller blade passage excitations computed by the different approaches are shown. The response curves com-puted according to LEWIS gives the qualitative characteristics in the low frequency range but devi-ates in the higher frequency range considerable from the reference curve. Application of the PRO-JECTION approach yields the best accordance with the reference velocity whereupon the velocity results of the MODAL_HYM approach are only a little less inaccurate.

3.3.2 Run timeAll computations were performed on a 64-bit linux computer Quad-Core AMD Opteron(tm) Proces-sor 2356 with 32 GB RAM and a clock rate of 2.3 GHz.

The FULL_HYM method as the most accu-rate procedure is also the most time consuming method with a total computation time of ∼1.8 h and the most inaccurate approach (LEWIS) is the fasted with ∼2 min CPU time. The PROJECTION and the MODAL_HYM approach having total

computation time of the same order of magnitude (∼15 min and ∼20 min) only the distribution between pre computation time and mode compu-tation differ considerably: The pre computation time of the PROJECTION approach, i.e. the semi wet mode computation time according to Lewis is much less than the pre computation time of the MODAL_HYM approach where the modal approximation of the hydrodynamic mass matrix has to be computed.

4 CONCLUSIONS

The presented PROJECTION approach com-bines the fast and robust LEWIS method with an advanced fast boundary element technique yielding very accurate eigenfrequencies and accu-rate forced vibration results within small com-putation times. The MODAL_HYM approach exhibits only slightly worse characteristics in preci-sion and run time. Both approaches require user experience: the PROJECTION approach in case of selecting the number of eigenvectors used for projection and the MODAL_HYM approach in case of number of modes needed for approxima-tion the hydrodynamic mass matrix.

A particular advantage of the proposed meth-ods is that they scale very well. The effort to com-pute the hydrodynamic mass effect is dominated by evaluations of the hydrodynamic mass opera-tor. Through application of the fast multipole method, the cost for this application grows approx-imately like N log2(N) for large numbers N of wet panels.

REFERENCES

Allemang, R.J. 1980. Investigation of Some Multiple Input/Output Frequency Response Function Experi-mental Modal Analysis Techniques. Doctor of Philoso-phy Dissertation, University of Cincinnati, Department of Mechanical Engineering, pp. 141–214.

Armand, J.-L. & Orsero, P. 1979. A method for evaluat-ing the hydrodynamic added mass in ship hull vibra-tions. SNAME Transactions, 87:99–120.

Bai, Z., Demmel, J., Dongarra, J., Ruhe, A. & van der Vorst, H.A. 2000. Templates for the Solution of Alge-braic Eigenvalue Problems: A Practical Guide. SIAM, Philadelphia.

Cabos, C. & Ihlenburg, F. 2003. Vibrational Analysis of Ships with Coupled Finite and Boundary Elements. Journal of Computational Acoustics, 11(1):91–114.

Golub, G.H. & Van Loan, C.F. 1996. Matrix Computa-tions. The John Hopkins University Press, Baltimore and London, 3rd edition.

Grim, O. 1953. Berechnung der durch Schwingungen eines Schiffskörpers erzeugten hydrodynamischen Kräfte. STG Jahrbuch.

Figure 13. Typical response curves of the different approaches.

870744126Lewis modesPROJECTION

Approach

Mode Computation

CPU Time [s]CPU Time [s]

787

2

6060

CPU Time [s]Item

6613553full hydro massFULL_HYM

TotalPre Computation

1196

128

409

126

Modal hydro massMODAL_HYM

Lewis hydro massLEWIS_HYM

Figure 14. Comparison of run time.

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Grim, O. 1960. Elastische Querschwingungen des Schiffskörpers. Schiffstechnik, 7(35):1–3.

Landweber, L. 1957. Mass of Lewis Forms Oscillating in a Free Surface. Proceedings: Symposium on the Behav-iour of Ships in a Seaway, Wageningen.

Lewis, F.M. 1929. The Inertia of the Water Surrounding a Vibrating Ship. Transactions of the SNAME, 37.

Newman, J.N. 1977. Marine Hydrodynamics. The MIT Press, Cambridge, Mass.

Saad, Y. 2003. Iterative Methods for Sparse Linear Sys-tems. SIAM, Philadelphia, 2nd edition.

Wendel, K. 1950. Hydrodynamische Massen und hydrodynamische Massenträgheitsmomente 14. Jahr-buch der STG, vol. 44, 207–255.

Wilken, M., Of, G., Cabos, C. & Steinbach, O. 2009. Effi-cient calculation of the effect of water on ship vibra-tion. C. Guedes Soares and P.K. Das, Analysis and Design of Marine Structures, pages 93–101, London, Taylor & Francis.