The Interaction of an Underwater Explosion Bubble and an Elastic-plastic Structure

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    Applied Ocean Research 30 (2008) 159171

    Contents lists available at ScienceDirect

    Applied Ocean Research

    journal homepage: www.elsevier.com/locate/apor

    Review

    The interaction of an underwater explosion bubble and an elasticplasticstructure

    A.M. Zhang , X.L. Yao, J. LiCollege of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001, China

    a r t i c l e i n f o

    Article history:

    Received 27 February 2008Received in revised form

    22 October 2008

    Accepted 9 November 2008

    Available online 20 December 2008

    Keywords:

    Underwater explosion

    Bubble

    Ring

    Jet

    Elastic-plasticity

    Structure

    Surface ship

    a b s t r a c t

    Based on the potential flow theory, the boundary element method (BEM) is applied to calculate the

    dynamics of an underwater explosion bubblenear boundaries, and in conjunction with thefinite elementmethod (FEM) it is employed to compute the interaction between a bubble and an elasticplasticstructure. A complete 3D underwater explosion bubble dynamics code is developed;the simulatedresultscompare well with an underwater explosion experiment. With this code, the interactions between an

    underwater explosion bubble andelasticplasticstructuressuch as a flatplate, a cylinder andother simplestructures arecalculated andanalyzed. Besides, the damages caused by theafter flow, pulsating pressure,

    and jetting load on thestructures arealso calculated, with or without a free surface. From thetime historyof the pressure and stress of the structure, it can be observed that the stress reaches its maximum value

    when the bubble collapses, which proves that the pressure and jet impact induced by the collapse ofthe bubble can result in severe damage to the structure. In particular, the 3D analysis code is applied to

    some engineering problems, for example it is used on a surface ship to study the interaction between abubble and a complex elasticplastic structure. Under the bubble load, the low-order eigenfrequency of

    the ship is aroused usually, leading to the so-called whipping effect, because the pulsating frequency ofthe bubble matches the low-order eigenfrequency of the ship. The ship moves up and down with the

    expansion and collapse of the bubble respectively. Meanwhile, the power of the bubble generated by anear-field underwater explosion in short range is discussed, and some important conclusions which canbe applied to project application field are drawn.

    2008 Elsevier Ltd. All rights reserved.

    Contents

    1. Introduction........................................................................................................................................................................................................................1602. Theory and numerical model ............................................................................................................................................................................................160

    2.1. Introduction ...........................................................................................................................................................................................................1602.2. Boundary-element method (BEM) for the fluid part...........................................................................................................................................160

    2.3. Toroidal bubble model ..........................................................................................................................................................................................1612.4. Time-step size control and numerical procedures ..............................................................................................................................................162

    2.5. Finite-element method (FEM) solver for the structural part and coupling with fluid part ..............................................................................1623. Results and discussions .....................................................................................................................................................................................................163

    3.1. Circular plate model ..............................................................................................................................................................................................1643.2. Cylinder model.......................................................................................................................................................................................................165

    3.2.1. Comparison of the simulated results and experimental data..............................................................................................................165

    3.2.2. The case with a free surface ...................................................................................................................................................................1653.2.3. The case with bubble under cylinder ....................................................................................................................................................166

    3.3. The interaction between a bubble and a complex elasticplastic structure (e.g. a surface ship) ....................................................................1673.3.1. Explosion in middle and far field...........................................................................................................................................................167

    3.3.2. Explosion in near field............................................................................................................................................................................1694. Conclusions.........................................................................................................................................................................................................................169

    Acknowledgments .............................................................................................................................................................................................................170References...........................................................................................................................................................................................................................170

    Corresponding author. Fax: +86 0451 82518296.E-mail address: [email protected](A.M. Zhang).

    0141-1187/$ see front matter 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.apor.2008.11.003

    http://www.elsevier.com/locate/aporhttp://www.elsevier.com/locate/apormailto:[email protected]://dx.doi.org/10.1016/j.apor.2008.11.003http://dx.doi.org/10.1016/j.apor.2008.11.003mailto:[email protected]://www.elsevier.com/locate/aporhttp://www.elsevier.com/locate/apor
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    1. Introduction

    During an underwater explosion, there will be an initial shockwave propagating radially outwards to be followed by a high-pressure bubble containing hot gaseous products of the explosion.Under the effects of hydrostatic pressure, gravity and inertia, theso-called after flow, jet and pulsating pressure are developedowing to the motion of the bubble. Not only the shock wave butalso the bubble load can have great damages on the underwaterstructures. For instance, the pulsating pressure and after flow cancause global damage on the structure (e.g. a surface ship), whilethe high-speed re-entrant water jet will cause local damage tothe structure. Nowadays, researches are mainly focused on theinteraction between a bubble and a rigid wall (e.g. [17]), butfewer published literatures are about the interaction betweenan underwater explosion bubble and an elasticplastic structure.Kalumuck andChahine et al.[8] calculated the interaction betweena bubble and an elasticplastic structure with the combinationof the finite element method (FEM) and boundary elementmethod (BEM), and also developed the 2DYNAFS, 3DYNAFS andother codes. Klaseboerk [9] studied the underwater explosionbubble dynamics and the interaction between a bubble and a

    simple flat plate numerically and experimentally. Based on theirachievements, this paper discusses the interaction between abubble and a complex elasticplastic structure (e.g. a surfaceship), taking the free surface into account simultaneously. Thedamages caused by the after flow, pulsating pressure, jetting loadare investigated, aiming at revealing the power of the bubble load.

    2. Theory and numerical model

    2.1. Introduction

    The numerical calculations can be split into two parts: the fluidpart and the structural part. The fluid part is carried out usingthe boundary integral method. Special care must be taken after

    the jet impact induced by the collapse of the bubble, since thefluid domain then becomes doubly connected; a vortex ring isplaced inside the bubble to account for this phenomenon. Thestructural part is solved using the finite-element solver ABAQUS. Afull coupling between the two codes has been made; informationconcerning displacements of the structure is passed on from thestructural code to the fluid code and forces (deduced from thepressure loading on thestructure) arepassed on from thefluidcodeto the structural code. A controlling interface code decides whichcode has to be called to do the appropriate calculations, making therelevant results available as input to the other code. Time steps andmeshes can be different for the two codes.

    2.2. Boundary-element method (BEM) for the fluid part

    To investigate the dynamics of the underwater explosionbubble, it is assumed that the surrounding fluid field is fullof ideal fluid which is inviscid, irrotational and incompressible.Therefore the velocity potential is introduced and the velocityvector can be derived from this potential as: u = . When itis coupled with the continuity equation u = 0, the velocitypotential is governed by the Laplace equation:

    2 = 0. (1)

    As the Laplace equation (1) is an elliptic equation, the solution canalways be computed everywhere in the fluid domain,provided thateither the potential (Dirichlet condition) or the normal velocity/ n (Neumann condition) is given on the boundaries of the

    problem. Here /n = n is the normal inward derivative ofthe boundary S and n is directed out of the fluid. According to

    Greens formula, the velocity potential of any point in the fluidcan be obtained by the velocity potential on the boundary and itsnormal derivative, in other words, the function in the fluid field

    can be described by laying distributing sources on the boundaryand distributing dipoles along the normal direction. The boundarycondition at infinity is of the form:

    r = x2 + y2 + z2 , 0 (2)where r = (x,y,z) is the position vector. Then Eq. (1) can bewritten as:

    (p) =

    S

    (q)

    nG(p, q) (q)

    nG(p, q)

    dS. (3)

    Eq. (3) is Greens integral formula, where S is the boundary of thefluid domain including bubble surface Sb, free surface Sf and wallsurface Sw; p and q are a fixed point and the integration variablesituated on S, respectively; is the solid angle viewed from thepoint p : = 4 for an interior fluid point, c(p) = 2 for point pon a smooth surface and c(p) < 4 for point p at the corner.

    The solid angle subtended at the control point p by surface Scanbe obtained through integral as follows:

    =

    S

    G

    n(p, q) dSq, p S. (4)

    The three-dimensional field Green function is givens:

    G(p, q) = |p q|1 . (5)

    Ignoring the motion of the gas inside the bubble and itscorresponding influence on gas pressure, we suppose that thegas pressure is only relative to the initial state and volume ofthe bubble, and then the gas pressure inside the bubble may beexpressed in term of its volume V as:

    pb = pc + p0

    V0

    V

    (6)

    where Pc is saturated vapor pressure of noncondensable gas; P0 andV0 are the initial pressure and volume of the bubble respectively,referring to literature [6]; and is the ratio of specific heat ofthe gas, which is equal to 1.25 for the gaseous explosion productsresulting from an TNT explosion [10] and 1.4 for ideal gas [11].

    To generalize our research, a system of non-dimensionalisationis adopted here in which all the parameters are scaled: Rm forlength, P = P Pc for pressure, Rm(/P)

    1/2 for time,(P/)1/2 for velocity and Rm(P/)

    1/2 for velocity potential.Here P = gH + Patm is defined as the hydrostatic pressure atinfinity on the Oxy coordinate plane located at the initial bubblecenter, where is the density of the fluid; g is the gravityacceleration; H is the initial depth of the bubble center andPatm is the normal atmospheric pressure; Rm is the maximum

    radius that the bubble would attain in an infinite fluid domainunder the pressure of P.The surface tension is neglected inthis investigation, since it is insufficient to cause appreciableeffect for most of the lifetime of a cavitation or an underwaterexplosion bubble [1113]. Based on the potential flow theory andpublished literature, the non-dimensional Bernoulli equation onthe boundaries of the bubble, structure and free surface are asfollows (all variables are non-dimensionalized in the equations):

    d

    dt= 1

    V0

    V

    2z +

    1

    2|u|2 (On a bubble surface), (7a)

    d

    dt= 1 P 2z +

    1

    2|u|2 (On a structural surface), (7b)

    ddt

    = 12

    |u|2 2z f

    (On the free surface), (7c)

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    where and u are the velocity potential and velocity vector on

    the boundary respectively; z is the displacement along the gravitydirection on the boundary, P is the pressure on the structuresurface, = P0/Pis the dimensionless initial pressure parameter

    inside the bubble and = (gRm/P)1/2 is the dimensionless

    buoyancy parameter, f = H/Rm is the dimensionless initial depthparameter. Providing that r is the spatial vector coordinate of a

    fluid particle on the boundary, and then the motion equation ofthe fluid particle on the boundary could be given as:

    dr

    dt= . (8)

    If the initial conditions are given, Eqs. (3), (7) and (8) forms a

    completed set of equations for the motion of bubble and otherboundaries. In order to attain theinitial conditions, a high-pressurespherical bubble with radius R0, whose initial expanding velocity is

    zero, is assumed to be formed at the initial stage of an underwaterexplosion. During the early phase of the bubble motion, the effectsof the buoyancy and boundary can be ignored due to the bubbles

    small size and the high-pressure gas inside, therefore the motionof the bubble can be described by the Rayleigh equation [14]:

    RR +3

    2R2 =

    R0

    R

    3

    1. (9)

    As long as an arbitrary initial velocity is given, the new initial

    radiusand pressure of thebubblecan be obtained by integrating (9)backwards in time. The method to determine the initial pressureand radial velocity is presented in [9].

    2.3. Toroidal bubble model

    The simulation of the dynamics of the bubble consists of two

    main stages, namely the pre-jet stage (singly-connected region)and post-jet stage (doubly-connected region, Toroidal Bubble). Forthe former stage, it can be solved with the method described

    in Section 2.2; while for the latter stage, the bubble evolvesinto a toroidal shape after jetting and the fluid becomes doubly-connected, at which time the velocity potential on the surface of

    the bubble may be a multiform function. So far, there have beenseveral axis-symmetrical models that can simulate the dynamicsof a toroidal bubble. Lundgren and Mansour [15] divided the

    bubbles collapse into two phases. They dealt with the first phasewhich lasted from the initiation to the moment of jetting witha simple boundary element method. In the other phase, they

    introduced a vortex line into their work. As a result, it not onlymade the calculation of the bubble collapse to be prolonged, butalso the numerical simulation of the phenomenon that vortex was

    produced in the anaphase of bubble collapse to be carried out;while this method is limitedto the simulation of thebubble having

    a constant volume. Best [16] introduced a branch cut approachand formulated a boundary integral equation valid for both on thebubble surface and the branch cut. However, the deficiency of thismethod is that the cuttingbubble surface needs special disposal,so

    it is hard to be popularized. In the process of simulation, Zhang &Duncan et al. [17,18] defined a layer to separate the water jet andthe surrounding domain during the toroidal bubble phase, which

    acts as a vorticity sheet and moves with the flow, and employedmodified boundary element method to calculate the whole process

    of bubble collapse. But the deformation of the layer cant exceedthe bubble surface, which makes the tracking very challenging,especially to the simulation of three-dimensional models. To solve

    this problem, Wang et al. [11] employed a so-called surgical-cutto convert the originally singly-connected bubble to a multiply-

    connected toroidal bubble after jet impact. Instead of adding avortex sheet at theimpact area, a vortex ring was placed inside the

    Fig. 1. A toroidal bubble with the technique of surgical-cut: the identification ofthe jet top node I and the impact node J; I and J will be joined when the distance d

    between them is less than some constant.

    Fig. 2. Motion of a typical toroidal bubble: (a) shows the state when the bubble

    begins to collapseand a jetjust appears; (b)shows thestate whenthe jetpenetrates

    the bubble and a doubly-connected bubble has been formed; (c) shows the state

    when bubble rebounds and goes into the second period; the brown bar in the (b)

    and (c) represents the vortex ring.

    bubble to account for the double connectivity of the bubble. There

    is no longer the need for meticulously tracking the vorticity sheet

    asin [17]; we just ensurethat thevortexring stays insidethe toroid

    as the bubble evolves. The models above are all axis-symmetrical.

    As for three-dimensional models, Zhang et al. [6] extended the

    research of Wang et al. [11], putting the vortex ring into the

    simulation of the three-dimensional toroidal bubble; based on this

    model, the whole process of expansion, collapsing, jet formation

    and rebounding of a three-dimensional bubble can be simulated.

    Until now, most of the numerical simulations of 3D toroidal

    bubble [19,13] are mainlybased on the vortex-ring model of Zhang

    et al. [6]. The 3D surgical-cut procedure is showed in Fig. 1.This paper adopts the vortex-ring model posed by Zhang

    et al. [6] too, in which the total potential is decomposed into

    two parts: one is the potential associated with the circulation

    generated by the impact, termed the ring potential; the other is

    the remnant energy which is uniformly distributed in the entire

    fluid domain. Then:

    (r, t) = (r) + (r, t). (10)

    Now the total potential consists of the ring potential and remnant

    potential, and Eqs. (7) and (8) should be transformed into

    corresponding forms, referring to Zhang et al. [6]. Because of

    the severe instability of the bubble after instantaneous impact,

    should be smoothed every several time steps to guarantee theconvergence of the model. Furthermore, the location of the vortex

    ring should be updated continuously with the change of the bubble

    shape so that the vortex ring can be inside the bubble all the time.

    Fig. 2 shows the typical dynamic behavior of a toroidal bubble,

    i.e. the transition of an underwater explosion bubble from a singly-

    connected to multiply-connected domain.

    Whats worse, the distortion of the mesh may occur in the

    simulation of the three-dimensional bubble dynamics, therefore

    a smoothing scheme is needed on the bubble surface and other

    boundaries. Especially, after jet impact most elements concentrate

    in the region where the jet is formed, which makes the meshes

    close to it too dense, resulting in the abortion of the calculation. In

    order to avoid this problem, an elastic mesh technology (EMT) [20]

    is employed in this paper.

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    Fig. 3. Flow chart of the numerical algorithm (E. and B.C. are short for equation and boundary condition respectively in the chart).

    2.4. Time-step size control and numerical procedures

    To update the position and velocity potential of the boundaries,the time-advancing scheme is adopted in current study. If the

    velocity potential (t) and position r(t) of the boundaries fortime t (step i) are given, then the normal velocity component/ t(t) and material velocity u(t) can be obtained through

    integral equation (3) and the finite difference method respectively.Then the velocity potential (t + t) on the boundaries for nexttime step (step i + 1) can be attained by Bernoulli equation (7).

    Combining the finite difference method, the update of the velocitypotential is prescribed as:

    i+1 = i + (i + i+1)t/2. (11)

    The position vector r(t + t) of the boundaries for next time step(step i + 1) can be acquired by Eq. (8), and the update position ofthe bubble boundaries would thus be of the form:

    ri+1 = ri + (ri + ri+1)t/2. (12)

    In Eqs. (11) and (12) t is the time-step size, which must becontrolled carefully to maintain the stability of the solution. In the

    present paper the time-step size t is chosen as:

    t = min {t1, t2} (13a)

    t1 =

    max 1 + 12

    |

    |2

    2z V0

    V

    (13b)

    t2 =

    max 1

    2||2 2(z f)

    (13c)where is some constant andthe changeof thevelocity potentialon every node in each stepshould be not morethan . A value of

    0.02 is set for in the computation of current study, with whichthe calculation is stable throughout. The code is implemented asfollows:

    (a) Initialize the routine, i.e. read the initial information of thenodes and elements on the structure and initial parameters ofthe bubble such as R0, , ;

    (b) Begin time stepping (assume the total time Ttotal = 0,constant = 0.02, step counter i = 1,and get

    0and t

    0through

    Eq. (13) with the initial data);

    (c) Solve the boundary integral equation (3); obtain the normal

    velocity (/ n)i on the node of the bubble and free surface

    and also the distribution of the velocity potential mi on thestructural surface; combine the infinite difference method to

    solve the velocity vector ui on the node of the boundaries.

    (d) Calculate the pressure Pmi on the fluid-structural interface(theso-called wetted surface) with unstable Bernoulli equation

    (7b),and impose the pressure load Pmi on the structure through

    current solver ABAQUS with an interface routine;

    (e) Solve the displacement rmi and velocity rmi of the structure

    under new pressure load Pmi with finite element method insolver ABAQUS;

    (f) Update the boundary condition(B.C.) with the new velocity rmi

    and displacement rmi and apply new B.C. to the fluid boundaryintegral code, perform step (c) to (f) until (/ n)i = r

    mi n

    is satisfied on the wetted surface, then obtain the new normal

    velocity (/n)i and new position ri of the boundaries;

    (g) Get the stress, strain and other dynamic variables on the

    structure to check the damages of bubble load (if necessary);

    (h) Add ti to Ttotal, update the location ri and velocity potential

    i of the bubble surface with Eqs. (7) and (8) and acquire next

    time step ti+1 with Eq. (13);

    (i) Return to (c) and carry out next step i + 1 until Ttotal is more

    than certain time T, and then the whole calculation ends.

    To illustrate the process more clearly, the flow chart is shown as

    Fig. 3:

    2.5. Finite-element method (FEM) solver for the structural part and

    coupling with fluid part

    The fluidstructure interaction is usually calculated with twomethods [9,21]. One is the so-called full coupling method and the

    other method is termed the loose coupling method. Full coupling

    method solves all the unknowns simultaneously: (i) computethe hydrodynamic loads applied on the structure from the fluid

    and calculate the structural response; (ii) update the boundary

    conditions based on the calculated displacement and velocity;(iii) calculate the response of the fluid and the structure until the

    velocity and displacement on the structural wetted surface match

    the fluid motion. The loose coupling method is similar to the fullcoupling one: the fluid and the structure are solved in stagger

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    iteration process, but the iteration calculations are completed in

    two adjacent time steps instead of the same step. Both these two

    methods are available in solving the fluidstructure interaction.

    However, the full coupling method gets the full coupling solution

    with huge calculation work; the loose coupling method is actually

    an approximate method with incomplete solution, the solution of

    which is usually not convergent. So it is not suitable for the long-

    time numerical simulation.The full coupling method is used in this paper incorporating the

    BEM with FEM. The BEM is used to solve the bubble dynamics and

    the pressure caused by thebubble motion, while theFEM is used to

    solve the response of the structure (e.g. ship) under the pressure.

    Several nonlinear FEM solvers such as PAM-CRASH, ABAQUS, LS-

    DYNA, and MSC-DYTRAN can be used to calculate the structural

    part, which have good precision and utility routines available. In

    this paper, the nonlinear FEM solver ABAQUS is adopted to solve

    the dynamic response of the structure according to the following

    Eq. (14):V

    : dV +

    V

    rm rmdV +

    V

    rm rmdV

    +

    Sfs

    Prm ndS

    Sf

    rm fdS = 0 (14)

    where and n are the stress and unit outward normal of the

    structure respectively, the density of the material, the mass

    proportional damping factor, rm the acceleration of the structure,

    rm the velocity of the structure, f the surface pressure applied to

    the structure, rm a variational displacement field, the strain

    variation that is compatible with rm, Sfs the structural wetted

    surface, Sf the surface which f acting on, and P the pressure acting

    on the wetted surface which can be calculated through Eqs. (7b) or

    (15),

    P(t) = ((t) (t t))/t 2z(t) +1

    2

    |u(t)|2. (15)

    As described above, the response of structure under bubble load

    can be solved through combining Eqs. (14) and (15), and the

    calculationscan be carried outin theway described in thestep (f)in

    last section, which indicates that the velocity on the wetted surface

    coincides with that on the fluid boundary Eq. (16). Details can refer

    to flow chart (3) in the Section 2.4.

    /n = rmt n. (16)

    An interface routine alternately calls the BEM and FEM solvers.

    The time steps of the two codes can be, and are usually, different.

    Therefore, a leapfrog time-advancing scheme is selected in the

    present work to cope with the different time steps of the fluid and

    solid codes.

    3. Results and discussions

    Underwater explosion can be divided into two stages, namely

    the stage of the shock wave and bubble pulsation. Generally the

    load caused by the shock wave is very high but the corresponding

    duration is very short (several milliseconds); while the load caused

    by bubble is low but the duration is longer (several seconds) [10],

    showed as Fig. 4. Although both the loads will inflict severe

    damages on the adjacent structures, their damage mechanisms

    are different: the shock wave will result in great damages on the

    structure whose natural period is in the order of milliseconds, and

    this kind of structures is generally local structures on the ship such

    as hull platesand grillages, therefore theshockwave usually causeslocal damages on ships;

    Fig. 4. Phenomenon of the underwater explosion:shock wave and a high-pressure

    bubble appear after explosion; the shock wave generates high-pressure load but

    sustains short while pulsating bubble induces low-pressure load but sustains long

    and the pressure caused by bubble collapse is getting lower along with time.

    Fig. 5. The damage of an underwater explosion bubble on a ship: (a) shock wave

    moves at a very high speed and generates the first shock on the ship; (b) gas

    bubble is expanding and the ship is raising; (c) bubble is collapsing with a jet being

    developed and the ship is pulled downward; (d) jet penetrates the bubble and

    impacts on the ship, breaking the ship off.

    However, the bubble pulsation drives the fluid around moving

    in a big area, which induces the after flow thus, and the collapse

    of the bubble results in pulsating pressure. Both the after flow

    and pulsating pressure will cause global damage on ships. If the

    frequency of the bubble pulsation approaches to eigenfrequency

    of the ship, it will cause whipping motion of the ship which

    endangers the total longitudinal strength and even breaks off the

    ship from its middle part. At the same time, the high-speed jet

    formed in bubble collapse aggravates the damages on ships so

    that the destroyed ship may submerge. The whole process of the

    dynamic response of a surface ship under the effect of bubble

    pulsation is showed as Fig. 5.

    The underwater explosion shock wave and its damage on

    structures have been the core of underwater explosion problems

    studied before the middle 1990s. There have been many literatures

    and numerical algorithms (DAA2, ALE and so on) to simulate the

    explosion shock wave from non-contact explosion in far field

    to contact explosion in near field. For example, the commercial

    solvers like ABAQUS/UNDEX, LS-DYNA/USA can simulate the

    damage of the underwater explosion shock wave on structures in

    good precision. Therefore ABAQUS/UNDEX is adopted in present

    paper to analyze the underwater explosion, and the shock waveload refers to the model of Geers and Hunter [22]. The pressure of

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    164 A.M. Zhang et al. / Applied Ocean Research 30 (2008) 159171

    the shock wave versus time is:

    P(t) =1

    R

    f

    4

    acR

    AV(t) (17)

    where R is the distance from arbitrary measuring point to thecharge center, f is the mass density of the fluid (i.e. water), ac isthe radius of the explosive charge,A is a constant concerning aboutthe material property of the charge (for TNT A = 0.18), V(t) can

    be obtained through Eq. (18):

    V(t) =4 ac

    fPc [0.8251 exp (1.338t/Tc)

    + 0.1749 exp (0.1805t/Tc)] (18)

    where Pc = K

    m

    1/3c /ac

    1+A, Tc = km

    1/3c

    m

    1/3c /ac

    B, mc is the

    mass of the explosive charge. K, k, and B are also constants forcharge material (for TNT K, k, and B are taken as 5.21 107,9.0 105and 0.185 respectively).

    Although current FEM solvers have good precision whensimulating the damage of the shock wave on underwaterstructures, there are many limitations to simulate the interactionbetween the underwater explosion bubble and structures, which isbecause that the bubble pulsating period is usually very long andthe meshes of the bubble will distort during evolution, resultingin huge calculation for explicit finite element scheme. Thereforebased on the potential flow theory, boundary element code isexploited in this paper to study the interaction between theunderwater explosion bubble and structures.

    3.1. Circular plate model

    The interaction between a bubble and an elasticplastic circularplate is studied in this section; the experimental data are fromReference [9]. The experiment was carried out in a pool: anexplosive charge of 55 g exploded under the circular steel plateof thickness 2 mm. The steel plate was fixed on the surface, one

    side encountered the blasting load and the other side was exposedin the air. The mass density of the plate is 7800 kg/m3; the yieldstressis 240MPa;the shear modulus is 80.7 GPa while the Youngsmodulus is 210 GPa; and the Poissons ratio is 0.3. Assumed themodel was ideal elasticplastic and the dimensionless dampingratio was taken as 0.05. The steel plate had been fixed at itsboundaries for all six degrees of freedom and it was thick enoughto allow relatively large deformation of the plate to occur. Thestandoff distance had been chosen to be 1.2 m below the plate(or 2.2 times of the maximum bubble radius). The distancebetween the explosive charge and the floor of the pond was 3.5 m(the charge depth was also 3.5 m). In the experiment, severaldisplacement sensors were used to record the deformation of thesteel plate. Since the standoff distance was far larger than the

    bubble radius, the bubble behaved, to some extent, like a free-fieldbubble. The numerical results are shown in the Fig. 6; the verticaldisplacement of the plate at 89 ms is shown in Fig. 7.

    Fig. 6(a) shows the initial states of the charge and plate; ataround 46 ms, the centre of the plate moves towards the bubblebecauseof thenegative pressure,whilethe bubbleremains roughlyspherical (Fig. 6(b); at the time of 89 ms, it can be seen that thecentre of plate moves away from the bubble resulting from thecollapse of the bubble, and the bubble evolves into a toroidal form(Fig. 6(c)). The time history of the displacement of the plate centeris given in Fig. 8, both the experimental and numerical results areshown.

    As can be seen from Fig. 8, both the experiment and simulationhave a deformation of zero initially. After the explosion, the

    displacement of the plate center approaches nearly 30 mm inseveral milliseconds. Then, the plate center changes its direction

    (a) t = 0 ms. (b) t = 46 ms. (c) t = 89 ms.

    Fig.6. Thecouplingeffectbetween bubbleand plate at t = 0,46 and89ms; bubble

    first expands and then collapses to minimum; a toroidal bubble forms finally (the

    brown bar represents thevortex ring).The centreof plate moves upand down with

    the expansion and collapse of the bubble.

    Fig. 7. Deformation of a circular plate at t = 89 ms, the color contour represents

    the magnitude of the vertical displacement. (For interpretation of the references to

    colourin this figurelegend,the readeris referred to theweb version of this article.)

    Fig. 8. Comparison of the displacement of the plate center along z-axis: a positive

    displacement means that the plate is moving away from the bubble.

    and moves towards the bubble rapidly. Bubble pressure turns tobe lower than its ambient pressure, so a suction effect exerts

    on the plate. The plate then stays a relatively long time, nearly

    80 ms, when the plate is pulled below the water surface. Whenfinally the bubble collapses, the pressure caused by the ensuing jetand the compressed gas inside the bubble eventually leads to the

    permanent (and plastic) deformation of the plate. The deformation

    tested in the experiment is a little larger than the numerical result;the ultimate simulated plastic deformation of the plate is 37 mm

    while the experimental result is 40 mm, with an error of about8%. There are many reasons accounting for the difference such

    as the potential flow assumption and also error of the numericalsimulation. To illustrate the power of bubble load further, the time

    history of the pressure and stress of the heading-on element (this

    element is the nearest to the charge center) on theplate are shownin Fig. 9.

    Fig. 9 shows that when the bubble collapses, the plate suffers

    the maximum pressure load and its stress reaches its maximumvalue correspondingly, which indicates that the pressure resulted

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    (a) Pressure. (b) Stress.

    Fig. 9. Time history of the pressure and stress of the heading-on element (this element is the nearest one to the charge center) at the plate center.

    Fig. 10. Comparison of the displacement of the bubble center: the positive

    displacement means the bubble is rising.

    Fig. 11. The sketch map of a cylinder model: blast direction is showed in arrow.

    from bubble collapse and ensuing jet can cause severe damages onthe structure. During the whole dynamic process, the time historyof the displacement of the bubble center is shown in Fig. 10. Thebubble rises slowly in the expansion phase while it rises fast inthe collapse phase. The numerical results compare well with theexperimental results.

    3.2. Cylinder model

    The interaction between a bubble and an elasticplastic

    cylinder is studied in this section; the experimental data are fromReference [23]). The model is shown in Fig. 11. The outer diameterof the cylinder is 1 m; the thickness of the shell is 5 mm and thelength is 1.8 m. The experiment was carried out in a pool, with anexplosive charge of 1 kg. The cylinder was made in plain steel withthe density7800 kg/m3, yielding stress 230 MPa, Youngs modulus210 GPa, and Poissons ratio 0.3. The elasticplastic model wasisotropic hardening and the non-dimensional damping ratio wastaken as 0.05.

    3.2.1. Comparison of the simulated results and experimental data

    The experimental case is simulated with the developed codefirst. The center of the cylinder is located 5 m below the watersurface; the charge is parallel to the cylinder, with the blast

    direction is shown in Fig. 11. The charge center is located5 m awayfrom the cylinder in the horizontal direction,and the distance from

    Fig. 12. Time history of the displacement of the heading-on node (this node

    encounters the shock wave first): the positive displacement means the cylinder is

    moving away from the bubble.

    thechargecenterto the free surface is farlarger than the maximumradius of the bubble. Thereby the free surface effects on the freely

    constrained cylinder can be ignored. However, the effects inducedby the bubble pulsation pressure, after flow and jet are taken into

    account. Fig. 12 shows the time history of the displacement of the

    heading-on node (this node encounters the shock wave first) onthe cylinder.

    Fig. 12 shows that the cylinder moves against and towards the

    bubble with the bubbles expansion and collapse under the effectsof pulsating pressure and after flow. The maximum displacements

    of the experimental and numerical results are 13.51 mm and12.3 mm respectively, with an error of about 10%. Generally,

    the numerical simulation compares well to the experiment, and

    the dynamic process of the interaction between the bubble andcylinder is shown in Fig. 13.

    Fig. 13(a) shows the states ofthe bubble and cylinderat thetime

    of 1 ms, when the pressure inside the bubble is very high and thebubble expands rapidly, driving the ambient flow to move, which

    forms the after flow and impels the cylinder to move. The motiondirection of the cylinder is shown in the figure. At around 112 ms,

    the bubble reaches its maximum volume and the pressure insidethe bubble is lower than its ambient pressure, when the bubble

    begins to collapse. The cylinders moving direction is shown in

    Fig. 13(b). At the time of 218 ms (Fig. 13(c)), the bubble collapsesto its minimum volume and a toroidal bubble has been formed.

    The brown bar in the figure represents the vortex ring. Then thebubble begins to rebound and the motion direction of the cylinder

    is shown in the figure.

    3.2.2. The case with a free surface

    Based on the model in last section, the case with a free surface

    is studied by changing the boundary conditions. An explosivepackage charge of 1 kg is located 1.5 m below the free surface and

    the cylinder is placed on the right of the charge, 3 m away in thesame horizontal plane. The dynamic coupling process among the

    bubble, free surface and cylinder is shown in Fig. 14.

    In Fig. 14 the color contour represents the magnitude of thevelocity potential of the bubble,free surface and cylinder. Fig. 14(a)

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    (a) t = 1 ms. (b) t = 112 ms.

    (c) t = 218 ms.

    Fig. 13. The dynamic process of the interaction between bubble and cylinder at t = 1, 112 and 218 ms.

    (a) t = 0 ms. (b) t = 18 ms.

    (c) t = 96 ms. (d) t = 184 ms.

    (e) t = 204 ms.

    Fig. 14. The interaction process among bubble, free surface and cylinder at t = 0, 18, 96, 184 and 204 ms, the color contour represents the magnitude of the velocity

    potential. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

    shows the boundary states at the time of 0 ms, when the pressureinside the bubble is very high and the bubble expands fast. Ataround 18 ms (Fig. 14(b)), the bubble expands and drives theambient flow to move, forming the after flow and impelling thecylinder to move towards left. Besides, the free surface is raised alittle as well. At the time of 96 ms (Fig. 14 (c)), the bubble reachesits maximum volume and the pressure inside it is lower than itsambient pressure. The free surface is raised further and the bubblewill begin to collapse. At around 184 ms (Fig. 14(d)), the bubblebegins to collapse and the free surface keeps on rising. The cylindermoves towards the bubble overall. And finally, at the time of 204ms, the bubble has collapsed and the jet penetrates the oppositeside of the bubble, so the bubble evolves into a toroidal shape

    (Fig. 14(e)). The cylinder moves towards the bubble even more.From theprocess above it canbe seen that the cylinder moves away

    from and towards the bubble with the expansion and collapse ofthe bubble respectively.

    3.2.3. The case with bubble under cylinder

    This section focuses on the case that the charge bursts rightunder the cylinder. Similar to the cases above, a charge weighted1 kg blasts 5 m below the water surface. The cylinder is freelysuspended above the charge with its center 3 m above the initialbubble center. The mass density of the cylinder is 7800 kg/m3; theyielding stressis 230MPa; the Youngs modulus is 210 GPa and thePoissons ratio is 0.3. Here the free surface effects arent taken intoaccount. The dynamic process of the bubble and cylinder is shownin Fig. 15.

    Fig. 15(a) shows the states at the time of 1 ms, when thepressure inside the bubble is very high and the bubble expands

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    (a) t = 1 ms. (b) t = 112 ms. (c) t = 218 ms.

    Fig. 15. The dynamic process of the interaction between bubble and cylinder att = 1112 and 218 ms.

    Fig. 16. Time history of the vertical displacement of cylinder center: the positive

    displacement means the cylinder is moving away from the bubble.

    (a) t = 1 ms. (b) t = 112 ms. (c) t = 218 ms.

    Fig. 17. The dynamic process of the interaction between bubble and rigid fixed

    cylinder at t = 1112 and 218 ms.

    rapidly, driving the ambient flow to move and forming an afterflow. The after flow impels the cylinder to move upwards fast.At around 112 ms (Fig. 15(b)), the bubble reaches its maximumvolume and the pressure inside the bubble is lower than itsambient pressure and then it begins to collapse. The cylinderis raised greatly because of its inertia. At the time of 218 ms

    (Fig. 15(c)), the bubble has collapsed and the jet is formedwhile the bubble becomes toroidal, with the brown bar in thefigure representing the vortex ring. Then the bubble reboundsand the cylinder is pulled back. The time history of the verticaldisplacement of the cylinder center is shown in Fig. 16, from whichit can be seen that the cylinder moves away from and towardsthe bubble along with the expansion and collapse of the bubble.This coincides with the conclusion which has been drawn in theReference [23].

    Change the boundary conditions of the cylinder by fixing itsbottom rigidly, and then calculate the cases above. The dynamiccoupling process of the bubble and cylinder is shown in Fig. 17.

    Fig. 17(a) shows the states of the bubble and cylinder at thetime of 1 ms, when the pressure inside the bubble is very high

    andthe bubble expands outwards quickly but the cylinder deformsa little. At the time of 112 ms (Fig. 17(b)), the bubble reaches its

    Fig. 18. Permanent deformation of the middle part of the cylinder: the

    color contour represents the magnitude of the equivalent plastic strain. (For

    interpretation of the references to colour in thisfigure legend, the reader is referredto the web version of this article.)

    maximum volume. A concave appears at the bottom of the cylinder

    since its edges are fixed rigidly. At around 218 ms (Fig. 17(c)), thebubble collapses and develops a jet. The bubble becomes toroidal

    (the brown bar in the figure still represents the vortex ring) andthen begins to rebound. The middle part of the cylinder deforms

    greater and a clear concave is formed at last. The contour of its

    plastic deformation is shown in Fig. 18.The pressure caused by the ensuing jet acts upon the cylinder,generating the permanent deformation (plastic deformation) on it.

    The cylinder deforms badly, which indicates the great power ofthe bubble load. The time history of the pressure and stress of the

    heading-on element (this elementencounters theshockwave first)on the cylinder is shown in Fig. 19.

    Fig. 19 shows that when the bubble collapses, the pressurereaches the maximum value and then the structure suffers the

    maximum stress, which indicates that the pressure caused bybubble collapse and jet can inflict severe damage upon structure.

    3.3. The interaction between a bubble and a complex elasticplastic

    structure (e.g. a surface ship)

    The interaction between a bubble and a simple elasticplastic

    structure is studied above. In this section the three-dimensionalcode is converted to settle complex elasticplastic structure

    problem in project application field. Finally, a full couplingdynamic analysis code is exploited, which is fit for solving theinteraction between a three-dimensional bubble and a complex

    elasticplastic structure.

    3.3.1. Explosion in middle and far field

    The length of the ship adopted in the simulation is L, the widthis B and the draft is T. The mass density of the ship is 7800 kg /m3,the yielding stress is 350 MPa, the Youngs modulus is 210 GPa

    and the Poissons ratio is 0.3. The ideal elasticplastic model is

    isotropic hardening and the dimensionless damping factor is takenas 0.05. The charge weights N kg and is located 0.24 L under themidship section. The maximum diameter of the bubble generated

    by the explosion is about 0.08 L. The dimensionless distance fromthe bubble center to the free surface is greater than 3, so the free

    surface effects arent taken into account. The interaction betweenthe underwater explosion bubble and surface ship is simulated by

    the three-dimensional code in this paper and the dynamic processis shown in Fig. 20.

    The color contour represents the magnitude of the stress ofthe surface ship. Fig. 20(a) shows the initial size and mesh of

    the bubble and surface ship, when the bubble expands outwardsrapidly under the effect of the high-pressure gas inside. At around

    0.049 s, the bubble moves and drives the ambient flow to move,

    forming an after flow. The after flow generates a low-frequencypushing force on the ship and raises it (Fig. 20(b)). At the time of

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    (a) Pressure. (b) Stress.

    Fig. 19. Time history of the pressure and stress of the heading-on element (this element encounters the shock wave first) on the cylinder.

    (a) t = 0. (b) t = 0.049.

    (c) t = 0.48. (d) t = 0.84.

    (e) t = 0.95. (f) t = 1.12.

    Fig.20. The interaction between underwater explosionbubbleand shipat dimensionless t = 0, 0.049, 0.48, 0.84, 0.95and 1.12; the colorcontourrepresents the magnitude

    of the Mises stress. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

    0.48 s, the bubble reaches its maximum volume, and the pressure

    inside is less than the surrounding pressure, so the ship is pulleddownwards by the bubble. The bubble basically keeps spherical,

    and then begins to collapse (Fig. 20(c)). As can be seen from

    Fig. 20(d), at the time of 0.84 s, the bubble begins to collapseand the ship moves downwards wholly. The bubble is no longer

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    Fig. 21. Whipping deflection of hull: the time changes from 0.1 s to 1.1 s.

    Fig.22. Distribution of thestress along theship: thestress in middleship is bigger

    than that in the stem and stern.

    Fig. 23. Time history of the displacement of the typical part of hull: a positive

    displacement means that the ship is moving away from the bubble.

    spherical. In Fig. 20(e) the bubble collapses and the jet penetrates

    the bubble, impacting on the opposite side of the bubble. And thenthe bubble becomes toroidal, continuing collapsing. The brown barin the figure represents the vortex ring. The ship is in hoggingcondition because of the bubble collapse and jet impact. If themidship structure isnt strong enough, the ship will probably be

    broken off. Fig. 20(f) shows the states at the time of 1.12 s, whenthe bubble rebounds and gets into the next period.

    It can be clearly observed that the ship takes on its first-ordervertical mode and does whipping motion, because the pulsatingfrequency of the bubble approaches the low-order eigenfrequencyof the ship. The movement of the ship is shown in Fig. 21 and thestress distribution along its length direction is shown in Fig. 22.As Fig. 22 shows, the stress in the middle part of the ship is muchgreater than that in the stem and stern. The vertical displacementof the heading-on node (this node encounters the shock wave first)in the midship section is shown in Fig. 23, from which we can seethat the ship moves up and down with the expansion and collapseof the bubble.

    Along the length direction of the ship, three points are taken in

    the middle sheer plate to check the pressure on the ships outsideplate. The three points labeled with P1, P2 and P3 are 0, 0.5 L

    Fig.24. Contrastof the pressurein differentpositions: pressureat P1 is the biggestand pressures at P2 and P3 equals to each other.

    Fig. 25. Time history of the stress of the typical parts(outside plate, transverse

    bulkhead and deck) near the midship section.

    and 0.5 L from the midship section respectively. Fig. 24 shows thepressure on these points: the peak pressure of P1 is the biggest;

    the pressures ofP2 and P3 are equal, much lower than that of P1.This also indicates that the pressure along the motion directionof the bubble and attack direction of the jet is the highest. The

    time history of the stress of the typical parts amidships is shownin Fig. 25. The pressure reaches its maximum when the bubble

    collapses. This once again shows that the pressure induced by thebubble collapse can cause serious damage on the ship structures.

    3.3.2. Explosion in near field

    According to the studies, when the charge bursts under the

    bottom of the bull, the generated bubble load can have a greatpower. Moreover, the shorter the standoff distance is, the greater

    the power is. So the near-field explosion (the charge bursts nearthe structure) is studied in this section: The charge (TNT) weightsN kg and locates 0.04 L below the midship section. The dynamic

    response of the surface ship is shown in Fig. 26.

    Fig. 26 shows the whole dynamic response process of the ship.As Fig. 26(a) shows, the shock wave load only causes local damageon the ship. From Fig. 26(b)(e), it can be seen that the bubble load

    causes global damage and leads to a large range of plastic strain onthe ship. Time history of the equivalent plastic strain of an element

    selected from thedeck in midshipsection is shown in Fig.27, whichindicates that the plastic strain of the midship section is mainly

    caused by the bubble load (the time when the bubble collapses isabout 0.9 s).

    In the end, it has to be noted that cavitation is a significantphenomenon during an underwater explosion and also for the

    surviving environment of surface ships [24,25]. Unfortunately, theboundary-element method is poorly suited to the investigationof cavitation. So cavitation isnt taken into account in this paper;

    however, it should be paid much attention to in further study.

    4. Conclusions

    With the conjunction of BEM and FEM, the interaction between

    the underwater explosion bubble and elasticplastic structure isstudied in this paper. Firstly, a numerical algorithm to calculate

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    (a) t = 0.12. (b) t = 0.75.

    (c) t = 0.90. (d) t = 1.11.

    (e) t = 1.26.

    Fig. 26. The dynamic response of ship to near-field explosion at dimensionless t = 0.12, 0.75, 0.90, 1.11 and 1.26; the color contour represents the magnitude of the

    equivalent plastic strain. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

    Fig.27. Timehistoryof the equivalent plastic strain of the deckelementin midship

    section.

    the coupling dynamics of a bubble, an elasticplastic structure anda free surface is showed. Then, a complete 3D code based on thealgorithm is exploited, whose computational error is within 10%compared with the experimental results. Meanwhile, the code isused in the project application field, simulating the interactionbetween a surface warship and an underwater explosion bubble.Founded on the works above, some conclusions are got as follows:

    (1) From the time history of the pressure and stress on thestructure, it can be seen that both of them reach theirmaximum valueswhen the bubble collapses, which shows thatthepressure caused by the bubble collapse and ensuing jet cancause severe damages on the ship.

    (2) Because the bubble pulsating frequency approaches to ships

    low-order eigenfrequency, the low-order vertical mode of the

    warship is aroused and the ship presents whipping motion.

    (3) In an underwater explosion, the surface ships and underwater

    structures (e.g. cylinder model in this paper) all move up anddown with the expansion and collapse of the bubble.

    (4) The explosion bubble load has a great power when the charge

    bursts under the bottom of the hull. Besides, the shorter the

    standoff distance is, the greater the power is. Generally, the

    shock wave only cause local damage on the ship, while the

    bubble load will inflict total damage on it, endangering its totallongitudinal strength and even breaking it off.

    Acknowledgments

    Thanks to Prof. K.S. Yeo and Prof. B.C. Khoo at National

    University of Singapore and Prof. E.Klaseboer at Institute of High

    Performance Computing, who supported and assisted in many

    valuable ways during the course of this research. This study

    is supported by a Grant (50779007) from the National Science

    Foundation of China and also a Grant (50809018) from the National

    Science Foundation for Young Scientists of China.

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