Boundary element simulation of large amplitude standing waves in vessels

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Boundary element simulation of large amplitude standing waves in vessels

Kenji Hamanoa,*, Sunao Murashigeb, Ken Hayamic

aDepartment of Mathematical Engineering and Information Physics, Graduate School of Engineering, The University of Tokyo, 7-3-1 Hongo,

Bunkyo-ku, Tokyo 113-0033, JapanbDepartment of C omplexity Science and Engineering, Graduate School of Frontier Sciences, The University of Tokyo, 7-3-1 Hongo,

Bunkyo-ku, Tokyo 113-0033, Japanc

Mathematical Informatics Research, Foundations of Informatics Research Division, National Institute of Informatics, 2-1-2 Hitotsubashi,Chiyoda-ku, Tokyo 101-8430, Japan

Received 3 September 2002; revised 4 February 2003; accepted 12 February 2003

Abstract

In this paper, a fluid in a vessel is considered and large amplitude standing waves (LASW) of the fluid are simulated directly using the

boundary element method (BEM).

In the simulation, two problems come out. The first problem is that the energy of the LASW increases gradually when using double nodes

at corners in the BEM. The second problem is that projection-like profiles appear near the point where the free surface meets the vessel wall

when regridding is not used at each time step. These projection-like profiles are not physical and indicate numerical error, and cause the

simulation to break down.

We found that the use of discontinuous elements solves the first problem, and the use of the half shift technique solves the second

problem.In addition, a method called RIG for highly accurate simulation using regridding is proposed and verified.

q 2003 Elsevier Science Ltd. All rights reserved.

Keywords: Boundary element method; Large amplitude standing waves; Direct simulation

1. Introduction

The study of large amplitude standing waves (LASW) is

required in various areas. For example, melted iron

containers are carried on rails in ironworks. The containers

may oscillate and melted iron may spill out of the

containers. Besides, for launching a rocket successfully, itis important to analyze how a liquid fuel moves inside the

tank due to the rockets oscillation and how in turn that

effects the rockets movement. Moreover, it is important to

study the motion of liquid in ships, when the ships oscillate

due to ocean waves. If the liquid in the ships starts to

resonate, the ships oscillation increases rapidly and in the

worst case, the ship will overturn.

Ref. [12] can be cited for the direct simulation of LASW

using boundary element method (BEM). They assume

periodic waves as in the formulation of Ref. [7]. They map

the free surface to a closed curve in the complex plane using

the assumption of periodicity. Then, they apply BEM to the

closed curve in the complex plane.

Most papers which simulate the dynamics of liquids in a

vessel using BEM have an interest in handling sloshing

analysis, and few papers simulate LASW in a vessel usingBEM. Moreover, those papers which address LASW

assume the periodicity in most cases, and in some cases, a

vortex-sheet method is used instead of BEM [15]. On the

other hand, in our paper, we directly apply BEM to a

boundary of a fluid in a vessel in order to simulate LASW,

and do not use the periodicity. Thus, BEM has a corner

problem in our simulation, and the corner points become the

peak of LASW. It has been pointed out that in some

applications, the intersection of the free surface and the wall

(the corner point) sometimes leads to inaccurate solutions

and wave instability after propagation in time [4,14]. In this

respect, our simulation is more difficult than Refs. [7,12].

The overview of this paper is as follows. In Section 2,

the settings of our simulation are explained. In Section 3,

0955-7997/03/$ - see front matter q 2003 Elsevier Science Ltd. All rights reserved.

doi:10.1016/S0955-7997(03)00041-9

Engineering Analysis with Boundary Elements 27 (2003) 565574www.elsevier.com/locate/enganabound

* Corresponding author. Present address: Second Research Center,

Technical Research and Development Institute, Japan Defense Agency,

1-2-24 Ikejiri, Setagaya-ku, Tokyo 154-8511, Japan.E-mail address: [email protected] (K. Hamano); [email protected]

tokyo.ac.jp (S. Murashige); [email protected] (K. Hayami).
http://www.elsevier.com/locate/enganaboundhttp://www.elsevier.com/locate/enganabound


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the basic formulation is given. In Section 4, simulationresults on the comparison of using double nodes and using

discontinuous elements, the simulation result using the

half shift technique, and the simulation result with the

vertical excitation are presented. In Section 5, RIG

algorithm for the highly accurate simulation is explained

and verified. The simulation result with the vertical

excitation using RIG is also shown. In Section 6, the

large amplitude cosine wave is simulated in order to show

that our method can also simulate transient standing

waves.

2. Settings

A perfect fluid contained in a rectangular vessel in two

dimensional space is considered (Fig. 1). We take the x-axis

in the horizontal direction and the y-axis in the vertical

direction. The fluid at surface in contact with air is called the

free surface. The center of the vessel is x 0 m; the widthof the vessel is L 2p [m], the distance between the bottomof the vessel and the still water surface y 0 m ish $ L=2: When the depth of the water is greater than or

equal to half the wavelength, the motion of water beneath

the surface is not affected by the bottom. (Waves under

these conditions are classified as deep-water waves.) Time isdenoted by t [s], the displacement of the free surface from

the still water surface is h hx; t [m]. The bottom of thevessel is at y 2h [m], and the height of the vessel is1 [m]. The gravitational acceleration is g 9.8 [m/s2].The initial wave profile of our simulation is as shown in

Fig. 1.

3. The numerical simulation method

3.1. The basic formulation

We consider an incompressible irrotational flow.The velocity potential f can be defined. The velocity

potential satisfies the Laplaces equation,

Df 0: 1The velocity of fluid particles u; vT satisfies the followingequations:

u fx

; 2

v fy

: 3

u; vT is also given by the following equations

u DxDt

; 4

v DyDt

; 5

where D=Dt is the material derivative.

Positions of fluid particles on the free surface x;yTsatisfy the following equations:

Dx

Dt f

x; 6

Dy

Dt f

y: 7

From Bernoullis equation, the velocity potential satisfies

the following equation at the free surface,

Df

Dt 1

2

f

x

2 f

y

2( )2 gh: 8

Using the above evolution Eqs. (6)(8), the time evolution

of the free surface is simulated. f=x; f=y can be

calculated using BEM as the velocity potential f satisfies

the Laplaces equation. In this paper, the time evolution

scheme based on the following Taylor expansion [9] was

used:

xtDt

< xt DtDx

Dt Dt

2

2

D2x

Dt2 Dt

3

3!

D3x

Dt3

Dt4

4!

D4x

Dt4: 9

ytDt and ftDt are expanded similarly. The second-orderLagrangian derivative is expressed as

D2x

Dt2 ft

x u u

x v u

y; 10

where ft f=t:ft is also computed using BEM,since ft satisfies the Laplaces equation: Dft 0; and theboundary values that ft Df=Dt2 u2 v2 on the freesurface and ft=n 0 on the bottom and the wall of thevessel can be obtained. Higher order derivatives of fwith

respect to t are computed using BEM similarly. The reasonfor usi ng t he ti me evolution scheme based on

Fig. 1. Schematic illustration of the simulation.

K. Hamano et al. / Engineering Analysis with Boundary Elements 27 (2003) 565574566


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the Taylor expansion is as follows. When the fourth-order

Runge Kutta method is used for the time integration,

one must compute the coefficient matrices of the system of

equations in the BEM four times in each time step. On the

other hand, if the Taylor expansion is used for the time

integration, the computation of the coefficient matrices is

required only once in each time step, so that the computation

time is significantly reduced.

3.2. BEM

In this part, we briefly explain the BEM [3] formulation.

Let Gf be the free surface, Gw be the boundary where the

fluid contacts with the vessel. G Gf < Gw is a closedcurve. Let Vbe the domain inside G:

The boundary integral equation (BIE)

u

2pf

Gffp

ndG

G

f

nfp dG; 11

holds, where uis the angle subtending Vat the point on the

boundary G; and fp is the fundamental solution.

Then, G and f; f=n on G are discretized using linear

elements G


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Namely, the case using double nodes, the case using

discontinuous elements, and the case using the method of

Ref. [7] (250 nodes were placed onGf in this case). Note that

when using the method of Ref. [7], there is no corner

problem since it assumes periodicity. The effect of the

corner nodes can be observed by comparing the three cases.

It is observed that there is a little declination between the

profile using double nodes and using the method of Ref. [7],

whereas, the profile using discontinuous elements matches

well with that using the method of Ref. [7].

As a result, we recommend that discontinuous elementsshould be used when simulating LASW with BEM.

4.2. The half shift technique

Next, we consider the use of methods which can simulate

LASW without using regridding. The reason considering

this is that when the amplitude of the standing waves

becomes large, the accuracy of spline interpolation used in

regridding may deteriorate, so that, if we want to simulate

the limit amplitude standing waves, the simulation without

using regridding may be required.

When the initial wave profile was set as Fig. 1, that is,

when the center displacement of the free surface from the

still water surface at t 0 : h0; 0 is at the highest andhp; 0;h2p; 0 are at the lowest, and LASW is simulatedwithout using regridding, projection-like profiles appear at

the end of the free surface as shown in Figs. 4 and 5.

These projection-like profiles are not physical and

indicate numerical error, and that causes the simulation to

break down.

Then, we consider two initial wave profiles. For the

first profile, h0; 0 is at the highest and hp; 0;h2p; 0are at the lowest (no_shift), for the second, h0; 0 is atthe lowest and hp; 0;h2p; 0 are at the highest(half_shift) (Fig. 6). In both cases, 63 nodes were placed

on the free surface.

When the initial wave profile is set to no_shift of Fig. 6,

the evolution of the free surface is as shown in Fig. 7.

This simulation fails before half the period of oscillation,

and nodes are found to concentrate towards the end of the

free surface.

Fig. 4. Three wave profiles when the simulation without using regridding

fails. Time progresses as the index number increases.

Fig. 3. Wave profiles for three cases: using double nodes and regridding

(double node, regrid), using discontinuous elements and regridding

(discontinuous, regrid) and using the method of Ref. [7] and withoutusing regridding with 250 nodes on the free surface (L-H, C, noregrid,

250points).

Fig. 2. Evolution of the total energy of the standing waves. Two cases are

shown. Namely, the case using double nodes and regridding (double node,

regrid) and the case using discontinuous elements and regridding

(discontinuous, regrid).

Fig. 5. Magnification of Fig. 4.

K. Hamano et al. / Engineering Analysis with Boundary Elements 27 (2003) 565574568


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When the initial wave profile is set to half_shift ofFig. 6,the evolution of the free surface is as shown in Fig. 8.

Different from the simulation of no_shift, this simulation

does not fail, and the nodes do not concentrate towards the

end of the free surface.

Therefore, shifting the initial wave profile by half the

wave length prevents the concentration of nodes at the

end of the free surface. This, in turn, enables us to avoid

the formation of projection-like profiles, and renders the

simulation without regridding feasible.

4.3. Simulation of excited standing waves

In this part, we excite the LASW with Ac 21=gDfy=Dt 0:85 in the vertical direction and make theLASWs amplitude larger and larger. The initial wave

profile was set to the half_shift given in Fig. 6, and

discontinuous elements were used. The oscillating

amplitude was d 0:01 m: Since the depth of thevessel h is sufficiently large, the angular frequency v of

the free surface obtained from linear analysis isffiffig

p 1=s:According to Mathieus theory [1,8,15], when the angular

frequency of the forced vertical oscillation is set to twice

of v; the amplitude of the standing waves becomes

larger and larger. The oscillation is represented as

follows:

gt g d2v2 sin2vt m=s2: 16

Fig. 9 shows the evolution of Ac; of the acceleration of

h0; t normalized by g; using regridding. Two cases areshown. Namely, the case when 127 and 251 nodes were

placed on the free surface, respectively. The results for

both cases are clearly different and there are some

disturbances.Fig. 10 shows the evolution of Ac without using

regridding. Two cases are shown. Namely, the case when

127 and 251 nodes were placed on the free surface,

respectively. The results for both cases are close, and the

disturbances seen in Fig. 9 have diminished.

As a result, in order to simulate the limit LASW, we

recommend that regridding should not be used. Splineinterpolation in the regridding procedure might be causing

the disturbances.

Fig. 9. Evolution of the amplitude of the crest acceleration normalized by

the gravitational acceleration Ac: The case with 127 nodes and 251 nodeson the free surface are shown, respectively. In each case, regridding wasused.

Fig. 8. Evolution of the wave profile. The initial amplitude of the free

surface is set to half_shift of Fig. 6.

Fig. 7. Evolution of the wave profile. The initial amplitude of the free

surface is set to no_shift of Fig. 6.

Fig. 6. The initial amplitude of the free surface hx; 0 Ac 0:85:

K. Hamano et al. / Engineering Analysis with Boundary Elements 27 (2003) 565574 569


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5. The RIG method

If we want to simulate not only standing waves but more

general waves like transient waves, regridding may be

required. But regridding with simple spline interpolation

gives rise to numerical error.

In this part, we propose a method called RIG in order to

make the wave simulation more accurate. The idea is based

on Refs. [10,13]. The basic idea is as follows. First, the

velocity potential is expanded using the basic functionswhich satisfy the Laplaces equation. Then, these

coefficients are corrected to satisfy Bernoullis equation

rigorously. Note that this proposed method can solve

the problem which has symmetrical geometry and uniform

depth.

5.1. The RIG algorithm

The position of the node i on the free surface is xi;yiT;and the velocity potential of the node i is fi: The number of

nodes on the free surface is f: The following algorithm gives

the time evolution from t to t

Dt:

Assume that at time t; xi;yi;fi i 1; 2; ;f satisfyBernoullis equation (8) rigorously.

1. Using a time evolution scheme such as in Section 3.1,

compute xi;yi;fi i 1; 2; ;f at t Dt:2. Regrid the free surface so that xi remains the same

for each node i i 1; 2; ;f: Update yi and fii 1; 2; ;f using interpolation.

3. Using the expansion of the velocity potential [5,13]

fx;y Xkmax21k0

Akcosh ky h

coshkh coskx; 17

where kmax is the truncation number of the summationand Ak k 0; 1; ; kmax21 are the coefficients of

the expansion, determine Ak k 0; 1; ; kmax21 fromxi;yi;fi; i 1; 2; ;f using the least square method.

The appropriate value for kmax requires trial and

error. Here, we set it to bf 1=2c: The expansion(17) is derived from the Laplaces equation: Df 0and the bottom and wall boundary conditions: f=

n 0:4. Execute the convergence computation with respect to

yi i 1; 2; ;f using the Newton method andBernoullis equation as follows.

for i 1 to fwhile yi converges do

yi yi 2ft 12 f2x f2y gyfty fxfxy 2fyfxx g

:

In order to calculate ft; the time derivative of Akk 0; 1; ; kmax21 are required. To calculate thetime derivative of Ak; we must store the Ak k0; 1; ; kmax21 of the previous time steps, andapproximate as follows,

dAkdt

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Boundary element simulation of large amplitude standing waves in vessels