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Simulation of dynamic behaviour of high-speedcatamaran craft subjected to underwater explosionJaeho Chunga & Young S. Shina

a Division of Ocean Systems Engineering, Korea Advanced Institute of Science andTechnology (KAIST), Daejeon, Republic of KoreaPublished online: 23 May 2013.

To cite this article: Jaeho Chung & Young S. Shin (2014) Simulation of dynamic behaviour of high-speed catamaran craftsubjected to underwater explosion, Ships and Offshore Structures, 9:4, 387-403, DOI: 10.1080/17445302.2013.793122

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Ships and Offshore Structures, 2014Vol. 9, No. 4, 387–403, http://dx.doi.org/10.1080/17445302.2013.793122

Simulation of dynamic behaviour of high-speed catamaran craft subjected to underwaterexplosion

Jaeho Chung and Young S. Shin∗

Division of Ocean Systems Engineering, Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Republic of Korea

(Received 15 August 2012; final version received 3 April 2013)

In recent years, shock trials for surface ships have been conducted in many countries for qualification of ship integrity, systemsand subsystems in response to shock. A ship trial identifies design and construction deficiencies that negatively impact shipand crew survivability. Such a trial also validates shock hardening and performance of shipboard equipments. However, live-fire ship shock trials and underwater explosion testing are both complex and expensive. As a possible alternative, numericalmodelling and simulation may provide viable information on the details of dynamic characteristics of ships, including at thecomponent and sub-component levels. Ship shock analyses were conducted using a finite-element-based coupled catamaran-type ship with a fluid model. This model is also applicable to underwater explosion (UNDEX) simulation for movements ofa high-speed catamaran-type ship. Catamaran-type ship shock modelling and simulation were performed and the simulationresults were compared with the empirical data. The high-speed catamaran-type ship shock analysis approach is presentedand the important parameters are discussed.

Keywords: underwater explosion (UNDEX); high-speed craft (HSC); catamaran; bubble motion; modelling and simulation

Introduction

The first high-speed passenger catamaran was built on theBlack Sea before World War II. The designers encounteredtwo problems: comfortable accommodation of passengersand reasonable seaworthiness. By that time it was knownthat a catamaran has better seaworthiness than a mono-hullof greater displacement. In oblique waves, motions of acatamaran are smaller because the hulls do not encounterwaves simultaneously. A twin-hull vessel with slender hullswas found to be stable and had good performance features(Dubrovsky and Lyakhovitsky 2001). Surface ship shocksimulation including an underwater explosion (UNDEX)is generally complicated by free-surface effects such assurface reflection waves resulting in bulk cavitation (Shin2004). In addition, there are phenomena such as hull cavita-tion, gas bubble oscillation and migration towards the freesurface and cavitation closure pulses (Shin 2004). Further-more, it is necessary to consider the complex fluid-structureinteraction phenomena and the dynamic behaviour of theship, shipboard systems and sub-systems (Shin 2004). Thebubble pulse loading can also cause critical transient motionof the ship. In current research, simulation of underwater ex-plosion situations has been applied only to stationary crafts.However, most UNDEX accidents occur on moving crafts.For that reason, applying an UNDEX situation to a mov-ing craft is the key focus area of this study. Furthermore,there is a growing trend in navies to build a catamaran-type. The response of a catamaran-type high-speed craft

∗Corresponding author. Email: yshin1234@kaist.edu

(HSC) is the main concern herein, including such aspectsas dynamic behaviour and structural response caused bythe bubble motions. If the catamaran moves at a planingspeed, the underlying physical phenomena have variationsin trim angle. In the planing stage, the weight of the ship issupported mainly by a hydro-dynamic pressure load. Thebuoyancy effect is no longer important. The variations inthe trim angle can cause variations in the global ship ge-ometry and can influence the dynamic response of the shipshock simulation. This paper shows that the variation of thetrim angle and the planing phenomenon can be new designparameters for the ship shock response in the UNDEX en-vironment. In this paper, shock analyses for a high-speedcatamaran-type ship that can travel at speeds greater than40 knots were conducted using finite element (FE)-basedcoupled ship and fluid models and an UNDEX situation inwhich 150 kg of trinitrotoluene (TNT) is detonated. As abenchmark step for the validation of the LS-DYNA (LSTC2007) and arbitrary Lagrangian–Eulerian (ALE)-couplingmethod in the UNDEX, several problems are analysed, suchas gas bubbles, explosive shock waves and cavitation, andthe results are compared with those obtained from analyti-cal solutions, simulation or experiments. The initial frame-work has been constructed and reported at the 81st Shockand Vibration Symposium (Chung and Shin 2010).

The special feature of a catamaranA catamaran is a type of multi-hulled ship consisting oftwo hulls, joined by a structure, the most basic of which

C© 2013 Taylor & Francis

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388 J. Chung and Y.S. Shin

Figure 1. Samples of the catamaran. (This figure is available in colour online.)

is a frame or deck, as shown in Figure 1. Catamarans canmaintain high stability in high-speed situations, owing tothe presence of the two hulls, and such ships have suitablehull forms for high-speed cruising. Thus, catamarans resultin less seasickness for crews. Moreover, catamarans aremore economical than mono-hulled vessels owing to theirlarge decks. They also offer several military advantagesinherent in their design to counter threats (Johnson andBurg 2000) as follows.

• High-speed catamaran hull form: The operator canavoid trouble and get out of trouble with the speedadvantage of the catamaran.

• Cushions: Numerous underwater shock tests by sev-eral navies have shown that an air-cushion vessel isless susceptible than traditional hull forms to thisdamage mechanism. The cushions also serve as anair void to block sound transmission into the water,thus, lowering the detectability of the vessel by sub-marines, torpedoes and mines.

• Low draft: The vessel can navigate to places thatdeeper draft vessels cannot. The operator can avoidtrouble and get out of trouble with the draft advan-tage of the catamaran. The shallower draft increasesstandoff distance from mines and torpedoes that maypass underneath the vessel.

• Excellent platform stability and ride characteristics:Other vessel motions are substantially reduced con-tributing to increased crew efficiency. Excellent sta-bility contributes to an expanded envelope for he-licopter operations in higher sea states. The stableplatform enhances weapon system accuracy, whichcontributes to one-shot kill capabilities.

UNDEX phenomena

High pressure originates in one section of the explosivecharge where detonation commences and propagates with

the detonation front throughout the remainder of the charge.As this detonation front propagates, it initiates the chem-ical reaction that creates more pressure. The detonationfront velocity steadily increases within the solid explo-sive until the front exceeds the speed of sound, creat-ing a shock wave. The detonation front then propagatesthough the solid at almost a constant velocity. As the det-onation front reaches the end of the explosive material,the shock wave propagates into the surrounding medium(Shin 2009). The high-pressure gas that results from theexplosion rapidly expands outward in a radial manner andimparts an outward velocity on the surrounding water aswell. Initially, the pressure is much greater than the at-mospheric and hydrostatic pressures that oppose it and istherefore compressive in nature (Smith 1999). At detona-tion, the pressure rise is discontinuous and decays expo-nentially with time. The pressure disturbance lasts only afew milliseconds. The detonation velocity in the explosivecharge is approximately 7620 m/s, near the charge, whichthen decreases rapidly as it travels radially outward in thewater.

Empirical equations have been determined to definethe profile of the shock wave (Shin 2009). These re-lations enable calculation of the pressure profile of theshock wave P(t), the peak magnitude of the pressureof the wave (Pmax), the shock wave decay constant (θ ),the shock wave impulse (I), the shock wave energy (E),the bubble period (T) and the maximum bubble radius(Amax). Table 1 shows shock wave parameters and Figure 2shows the surface phenomena.

P (t) = Pmaxe− t−t1

θ (Pa) , (1)

Pmax = 6894.7333 × K1

((0.4536 × W )

13

0.3048 × R

)A1

(Pa),

(2)

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Ships and Offshore Structures 389

Table 1. Shock wave parameters (Shin 2009).

HBX-1 TNT PETN NUKE

Pmax K1 22,347.6 22,505 24,589 4.38 × 106

A1 1144 1.18 1.194 1.18Decay constant K2 0.056 0.058 0.052 2.274

A2 −0.247 −0.185 −0.257 −0.22Impulse K3 1.786 1.798 1.674 11.760

A3 0.856 0.98 0.903 0.91Energy K4 3086.5 3034.9 3135.2 3.313 × 108

A4 2.039 2.155 2.094 2.04Bubble period K5 4.761 4.268 4.339 515Bubble radius K6 14.14 12.67 12.88 1500

θ = K2(0.4536 × W )13

((0.4536 × W )

13

0.3048 × R

)A2

(ms),

(3)

I = 6894.7333 × K3(0.4536 × W )13

×(

(0.4536 × W )13

0.3048 × R

)A3

(Pa − s), (4)

E = 703.08 × K4(0.4536 × W )13

×(

(0.4536 × W )13

0.3048 × R

)A4

(kg/m2), (5)

T = K5(0.4536 × W )

13

(0.3048 × D + 33)56

(s), (6)

Amax = 0.3048 × K6(0.4536 × W )

13

(0.3048 × D + 33)13

(m). (7)

Other variables in the equations are:

W = charge weight (kg): for nuclear (kton);R = standoff distance (m);

Table 2. Modelling of catamaran and component part ofconceptual catamaran.

Thickness WeightNo. of shellelements

Outer hull 1.5 cm 132 ton 11,106Bulkhead 1.5 cm 14 ton 1166Deck 1.5 cm 16 ton 1464Lumped masses – 35 ton –Total – 197 ton 13,736

D = charge depth (m);t1 = arrival time of shock wave (ms);t = time of interest (ms);K1, K2, K3, K4, K5, K6, A1, A2, A3, A4 = shock wave

parameters.

The maximum shock wave pressure, Pmax decreases byapproximately one-third after one decay constant time. Thesurface reflection wave is a tension wave, as opposed to acompressive wave, which is produced from the rarefactionof the shock wave off of the surface. Bottom reflection is thebouncing of the shock wave off of the bottom of the bodyof water; it is a compressive wave. Refraction causes theshock wave to travel through the bottom of the body of wa-ter before emerging again; this is also a compressive wave.In reasonably deep water, these two effects are not usu-ally a concern for surface vessels (Smith 1999). However,

Figure 2. Surface phenomena (Shin 2009).

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390 J. Chung and Y.S. Shin

Figure 3. Pulsations of the gas products from an UNDEX.

Figure 4. Displacements of the gas sphere from 136 kg TNTcharge fired 15.24 m below the surface.

free-surface reflection is an important effect. This rarefac-tion wave contributes to the creation of bulk cavitation nearthe free surface. Surface effects also occur as a result ofan UNDEX. A spray dome is caused by the incident shockwave. A bubble pressure pulse is produced when the gasbubble collapses during its oscillation and migration to thesurface.

The initial high pressure in the gas sphere decreasesconsiderably after the principal part of the shock wave hasbeen emitted, but it is still much higher than the equilib-rium hydrostatic pressure. The water in the immediate re-gion of the sphere or ‘bubble’, as it is usually called, hasa large outward velocity and the diameter of the bubbleincreases rapidly. This outward velocity is in excess of thatexpected from the magnitude of the pressure existing atthe time, owing to the after-flow characteristics of sphericalwaves, an effect that also has to be carefully considered inanalysis of the shock wave. The expansion continues fora relatively long time; the internal gas pressure decreasesgradually, but the motion persists because of the inertia ofthe outward flowing water. The gas pressure at later timesdecreases below the equilibrium value determined by at-mospheric and hydrostatic pressures; the pressure defectbrings the outward flow to a stop, and the boundary of thebubble begins to contract at an increasing rate. The inwardmotion continues until the compressibility of the gas, whichis insignificant in the phase of appreciable expansion, actsas a powerful check to abruptly reverse the motion. The in-ertia of the water, together with the elastic properties of thegas and water, thus provides the necessary conditions for anoscillating system, and the bubble does in fact undergo re-peated cycles of expansion and contraction. Ordinarily, theoriginal state of the bubble is approximately spherical andthe radial nature of the later flow results in an asymmetrical

Figure 5. FE models of the catamaran. (This figure is available in colour online.)

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Ships and Offshore Structures 391

Table 3. Material properties.

Density 7780 (kg/m3)

Shell elementsYoung’s modulus of elasticity 2.06e + 11 PaPoisson’s ratio 0.281Yield stress 3.55e + 08 PaUltimate stress 6.72e + 08 Pa

oscillation about the mean diameter, the bubble spendingmost of its time in an expanded condition. These phasesin bubble oscillation are shown schematically in Figure 3,which shows the bubble size as a function of time (Cole1948).

Oscillations of the gas sphere can persist for a number ofcycles, ten or more such oscillations having been detectedin some cases. The number of observable oscillations islimited by the sphere’s loss of energy due to radiation orturbulence, and by the disturbing effects of gravity andany intervening boundary surfaces. It is perfectly evidentthat the gaseous products must, because of their buoyancywhen in equilibrium with the surrounding pressure, eventu-ally rise to the surface. It is less evident that the gas spherewill in the course of its oscillation experience a net repul-sive force away from a free surface and will be attractedtowards a rigid boundary. The motion of the gas sphere isthus affected by its buoyancy and by the proximity of thesurface of the water, the seabed or other boundary surfaces(Cole 1948). Figure 4 shows the displacements of the gassphere from 136 kg of TNT detonated 15.24 m below thesurface.

Modelling and simulation

The conceptual catamaran model is adopted in this study.A three-dimensional (3D) FE model of the catamaran isshown in Figure 5. The catamaran lines are based on theDTMB 4667 series. The FE model includes ship structures,such as the bulkhead, deck and outer hull. Major stiffeners

Figure 6. Location of lumped masses. (This figure is available incolour online.)

Table 4. Input parameters for water and air.

Mass density(kg/m3)

Pressure cutoff(≤0.0)

Viscositycoefficient

Water 1000 −1.0e − 20 0.00113Air 1 −1.0e − 20 0.00113

are modelled in the shell elements. The structural materialproperties are assumed to remain plastic kinematic steelthroughout the simulation. Lumped masses were placed onthe upper deck, as shown in Figure 6. The total length is30 m, width is 15 m, depth is 3 m and shell thickness is0.015 m. There are 35 tons of lumped mass. Excluding thelumped mass, the weight of the model is 162 tons.

There are several bulkheads along the vessel and stiff-eners around the vessel. The space between two bulkheadsis 6 m. There is also a longitudinal stiffener in the middleof the catamaran.

The material model MAT PLASTIC KINEMATIC isused with shell elements to model the catamaran. It mod-els the isotropic and kinematic hardening plasticities andincludes a Cowper–Symonds strain-rate model. This mate-rial model is more cost effective than the linear plasticitymodel. Figure 7 shows the stress–strain curve of plastickinematic steel. Table 2 lists the thickness, weight and thenumber of shell elements of the component part of concep-tual catamaran. Table 3 lists the material properties.

Free field simulation

There are two types of UNDEX currently being studiedwith numerical methods. In the first, the explosive chargedetonates close to the ship hull. This is called a near-fieldUNDEX. In the second, the charge is further away from the

Figure 7. Stress strain curve of the material.

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392 J. Chung and Y.S. Shin

Figure 8. Eulerian model for free-surface UNDEX simulation. (This figure is available in colour online.)

ship hull. This is called a far-field UNDEX (Reid 1996). Inthis study, the near-field UNDEX was simulated.

Figure 8 shows the Eulerian model for the free-surfaceUNDEX simulation. Before the UNDEX simulation with ahigh-ship speed was performed, the free-field case UNDEXsimulation was checked. The simulation contained bothwater and air parts. As shown in Figure 8, the height is30 m, width is 60 m and length is 60 m for the water and airparts. An air height of 30 m is used to verify the water plumephenomenon. The simulation consists of 972,000 elements.

Figure 9. Simulation value and radius of bubble. (This figure isavailable in colour online.)

In water and air models, the MAT NULL keyword is usedfor the material. This material allows equations of state tobe considered without computing deviatoric stresses.

Pressure in null material is calculated using theGruneisen equation of state. Stress in the null material isthen calculated using Equation (8),

σij = −PEOS · Id + σdij , (8)

where Id is the identity matrix and PEOS is the pressurein the null material (LSTC 2007). Table 4 lists the inputparameters for water and air.

The TNT (150 kg) is located 10 m below the middleof the surface. For the TNT, the equation of state (EOS) iscalculated by Jones–Wilkins–Lee (JWL) of the EOS thatdefines pressure as a function of relative volume, V , and in-ternal energy per initial volume, E. The keyword EOS JWLis used for the explosion,

P = A

(1 − ω

R1V

)e−R1V

+B

(1 − ω

R2V

)e−R2V + ωE

V, (9)

where ω is the Gruneisen coefficient. A, B, R1, and R2 areinput parameters (LSTC 2007) as shown in Table 5.

Table 5. Input parameters for EOS JWL equation.

Equation ofstate label A B R1 R2

3 3.712e + 011 3.231e + 009 4.150 0.950

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Ships and Offshore Structures 393

Figure 10. The free-surface UNDEX simulation. (This figure is available in colour online.)

An EOS is a constitutive equation that provides a math-ematical relationship among the state variables, such aspressure, volume, internal energy and temperature.

An air density of 1.22 kg/m3 and water density of1025 kg/m3 were used. For boundary conditions, no reflec-tion of the shock wave and bubble pulse was applied. Thenon-reflecting boundary condition is applied to the exteriorboundaries of the fluid model where an infinite domain isrequired. The boundary condition prevents artificial stresswave reflections from reentering the model domain.

In Figure 8, there is a detonation zone in the middleof the Eulerian model. The mesh size of the detonationzone, which is a fine mesh, is 0.4 m (L) × 0.4 m (W) ×0.5 m (H). The out-of-detonation zone has a mesh sizeof 1.5 m (L) × 0.4 m (W) × 0.5 m (H). The empiri-cal values, obtained from empirical equations, were com-pared with the simulation values as shown in Table 6 andFigure 9.

Figure 10 shows the progress of the free-surface UN-DEX simulation using the ALE method. The total sim-

ulation time is 3 s. As seen in Figure 10, the gasbubble increases rapidly and oscillates until it creates awater plume.

The size of the plume formed by the UNDEX is afunction of explosive size, charge depth, charge type andfluid-shock-structure interaction effect. The variability ofthe plume size is relatively high owing to the chargecomposition. There are no particular rules on plume size

Table 6. A comparative table of radius of bubble and period ofbubble oscillation.

Bubble(empirical

value)Simulation

value

Amount of TNT (kg) 150.0 150.0Location of TNT; depth (m) 10.0 10.0Radius of bubble (m) 6.6146 6.6428Period of bubble oscillation (s) 0.9011 0.8314

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394 J. Chung and Y.S. Shin

Figure 11. Location of the TNT. (This figure is available in colour online.)

developed as a function of many variables. However, theALE method simulates plume effects well.

UNDEX simulation: structural response

Dynamic response with zero ship speed

Figure 11 shows the location of the TNT in the UNDEXsimulation with a ship speed of 0 knots. The TNT is located10 m below the centre line and has a mass of 150 kg. Thissimulation is generated by the ALE method, utilising 64CPUs with a computation time of 47 h. The total simulationtime is 5 s. In this simulation, the time step is 0.0001 until0.0078 s, 0.001 until 0.0108 s and 0.01 until 5.0 s. Thetrim angle of the catamaran is 0◦. The mesh size of thedetonation zone, which is a fine mesh, is 0.4 m (L) × 0.4 m(W) × 0.5 m (H). The out-of-detonation zone has a meshsize of 1.5 m (L) × 0.4 m (W) × 0.5 m (H). For boundaryconditions, no reflection of the shock wave and bubble pulsewas applied.

Figure 12 shows the process of the UNDEX simulationwith a ship speed of 0 knots. For initial conditions, a trimangle of 0◦ is applied. The 150 kg of TNT is detonatedat 0 s. After 0.001 s, the gas bubble expands gradationallyuntil 0.4308 s. Between 0.4308 and 0.8908 s, the gas bub-ble oscillates. After 1 s, the gas bubble generates a waterplume and the water plume impacts the catamaran. How-ever, after the water plume impacts the middle deck of thecatamaran, the water plume could not penetrate the deck ofthe catamaran.

Figure 13 shows five points of checked nodes at thekeel line for plots. Figure 14 shows the kick-off velocityin the UNDEX simulation with a ship speed of 0 knotsat the keel line. The vertical axis represents velocity andthe horizontal axis represents time. When the TNT is det-onated and the first bubble plume impacts, the verticalvelocity increases sharply. It subsequently decreases, and,the second bubble plume then impacts it, causing the ve-

locity to increase again. The resultant velocity is of al-most the same shape as the vertical velocity. This meansthat when the UNDEX occurs, the vertical response is themain response. The maximum vertical velocity is almost8 m/s.

Figure 15 shows the shock-induced response of the cata-maran: von-Mises stress at 0.008 s. When the shock impactsthe catamaran, stress spreads from the two hulls to the cen-tre deck. There is more stress at the side of the two hulls thanat the centre deck, which results in deformation of the sidesof the two hulls. The side hulls are particularly vulnerablepoints. However, the middle deck undergoes less deforma-tion. In Figure 15, the red parts indicate the sections understress.

Dynamic response with 40 knots ship speed

Euler model for dynamic response with 40 knotsship speed

Figure 16 shows the Euler model for a dynamic responsewith a ship speed of 40 knots. The total number of elementsis 1,545,156 and the length is 400 m. The height of theair part and the depth of the water part are 30 m each. Thewidth is 60 m. This simulation, using the ALE method, took251 h using 64 CPUs and computational dt is 6.10E-06. Thetotal simulation time is 25 s. In the detonation zone, thereis a TNT shell owing to detonation of the TNT at 18.9 s.Without the TNT shell, the TNT goes beneath the bottomof the Euler model. The TNT shell is generated to preventthis phenomenon. The thickness of the TNT shell is 0.1 cmand the radius is 0.288 m. The radius of the TNT shellis attributed to the amount of TNT (150 kg). The TNT islocated 300 m from the start line and 10 m below the waterline. The detonation zone has more fine meshes than anyother zone. The detonation zone is 14 m and its mesh sizeis 0.4 m (L) × 0.4 m (W) × 0.5 m (H). The convergedmesh is used in front and back of the detonation zone. For

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Ships and Offshore Structures 395

Figure 12. UNDEX simulation with a ship speed of 0 knots. (This figure is available in colour online.)

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396 J. Chung and Y.S. Shin

Figure 12. (Continued.)

boundary conditions, no reflection of the shock wave andbubble pulse was applied. The mesh size of the detonationzone is the same as the UNDEX simulation with a shipspeed of 0 knots.

Planing phenomena

The high-speed craft is characterised as having a relativelyhigh speed for its size and is classified by Froude number.The Froude number is defined as shown in Equation (10).

Figure 13. Position of checked nodes. (This figure is available in colour online.)

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Ships and Offshore Structures 397

Figure 14. Kick-off velocities in the UNDEX simulation with a ship speed of 0 knots. (This figure is available in colour online.)

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398 J. Chung and Y.S. Shin

Figure 15. von-Mises stress in the UNDEX with a ship speed of0 knots at 0.008 s (top view). (This figure is available in colouronline.)

In the equation, u is the speed of the vessel, g is thegravitational acceleration, and L is the waterline length ofthe vessel. When the Froude number is smaller than 0.5, it isa displacement vessel. In this case, buoyancy force carriesthe weight of the vessel. If the Froude number is largerthan 0.5, it can be classified as a high-speed vessel. Whenthe Froude number is between 0.5 and 1.0, it is a semi-displacement vessel. The buoyancy force is not dominantat the maximum operating speed. If the Froude number islarger than 1.0, it is a planing vessel. At this speed, theplaning phenomena will occur (Faltinsen 2005).

Figure 17 shows the planing phenomena. Fn is theFroude number. These relations enable the calculation ofthe speed of the vessel (U), water line length (L) and

gravity (g),

Fn = U√Lg

. (10)

The Froude number is 1.19 at 40 knots. In this simulation,the Froude number is almost 1.2 after 9 s. Figure 18shows the velocity of the catamaran. Until 3 s, the velocityof the catamaran is 0 m/s owing to initial equilibriumconditions. Between 3 and 9 s, the velocity of thecatamaran increases steadily until 21 m/s. After 9 s, thevelocity is almost 21 m/s, or 40 knots. Figure 19 shows theposition of the propulsion nodes. The keyword BOUND-ARY PRESCRIBED MOTION NODE is used forcatamaran propulsion.

Dynamic response with 40 knots ship speed

In the UNDEX simulation with high ship speed, there isalmost no impact on the catamaran due to the high speedand the trim angle, except the pitch angle. The high speedof the ship abates the water plume impact by moving theship out of the detonation zone rapidly. Figure 20 shows theentire process of the UNDEX simulation with the ship speedof 40 knots. The velocity of the catamaran is 0 m/s until3 s owing to the initial equilibrium conditions. Between 3to 9 s, the velocity of catamaran increases steadily until 40knots. After 9 s, the velocity of the catamaran remains at 40knots. The 150 kg of TNT was detonated at 18.9 s. The TNTwas located 300 m away from the start line and was placed10 m below the surface. The catamaran was positioned atthe middle of the detonation zone for a total of 19.6 s. As

Figure 16. Euler model for a dynamic response with a ship speed of 40 knots. (This figure is available in colour online.)

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Ships and Offshore Structures 399

Figure 17. Planing phenomena (Faltinsen, 2005). (This figure is available in colour online.)

seen in Figure 20, the changes in the trim angle are causedby the volumetric change of the bubble.

Figure 21 shows the shock-induced response of the cata-maran: von-Mises stress from UNDEX simulation with aship speed of 40 knots. When the gas bubble impacts thehigh-speed catamaran at 19.4 s, there is slight stress atthe sides of the two hulls. There is almost no impact on thecentre deck.

Figure 22 shows the pitch angle of the catamaran whenthe TNT is detonated. The vertical axis represents the pitchangle (degrees) and the horizontal axis represents time.The pitch angle is subject to a large change after the TNTis detonated. The first change is due to the volumetricchange of the bubble at approximately 19.2 s. The change

Figure 19. Position of the propulsion nodes. (This figure is avail-able in colour online.)

Figure 18. Velocity of the catamaran. (This figure is available in colour online.)

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400 J. Chung and Y.S. Shin

Figure 20. Simulation of the UNDEX with a ship speed of 40 knots. (This figure is available in colour online.)

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Figure 21. von-Mises stress distribution with a ship speed of 40 knots;19.4 s. (This figure is available in colour online.)

of pitch angle is 8◦ and subsequently decreases sharply.The second change is due to the first plume. This changeis largest at almost 20◦. There is a third change due tothe second plume. This third change is almost 10◦. After22 s, the catamaran has an initial pitch angle of 4◦ ow-

ing to its high speed. In this simulation, the velocity ofthe catamaran is almost 20 m/s, or 40 knots. The catama-ran moves out of the area of effect of the UNDEX after21.8 s. In this simulation, roll and yaw motions were ig-nored because the catamaran is symmetric in this simula-

Figure 22. Pitch angle in the UNDEX simulation with a ship speed of 40 knots.

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402 J. Chung and Y.S. Shin

Figure 23. Heave motion at zero speed – Case 1. (This figure is available in colour online.)

tion and the TNT is located under the middle point of thecatamaran.

Simulation results

The explosive charge is detonated at a distance of 10 m.The analysis results are used to compare heave motion at

zero speed (Case 1) with heave motion at 40 knots withthe trim angle (Case 2) at various locations in the keellines. The mesh sizes of the detonation zones are identical.Figures 23 and 24 show the heave motion at the keel lines.In these figures, the trends of the graphs are similar. In Case1, there are three changes, which are due to the volumet-ric change of the bubble and the first and second plumes.

Figure 24. Heave motion at 40 knots speed – Case 2. (This figure is available in colour online.)

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However, in Case 2, there are only two changes due tothe volumetric change of the bubble and the first plume.In Figure 23, Case 1, the maximum vertical displacementis almost 2 m. However, in Figure 24, Case 2, the maxi-mum vertical displacement is almost 1 m. It shows that thetrim angle and the planing phenomenon may be factors thatcan reduce damage due to UNDEX, including the verticaldisplacement. Moreover, there is no impact due to the sec-ond plume, because the catamaran moves out of the deto-nation zone after 21.8 s.

Conclusions

To the best of our knowledge, this is the first simulation ofa high-speed catamaran in an underwater explosion (UN-DEX) environment. It demonstrates the possibility of usingthe arbitrary Lagrangian–Eulerian (ALE) method. In pre-vious studies, only a stationary floating ship structure wassimulated.

For this study, the charge was detonated when the cata-maran was travelling at high speed with the planing phe-nomenon. Case 1 was the UNDEX simulation with a shipspeed of 0 knots, and Case 2 was the UNDEX simulationwith a ship speed of 40 knots. Prior to the UNDEX simu-lation with a ship speed of 40 knots (Case 2), the velocityof the catamaran, the Froude number of the catamaran andthe planing phenomena were confirmed. In Case 2, the rolland yaw motions were ignored because the catamaran issymmetric and the TNT is located under the centre point ofthe catamaran. This study shows that the trim angle and theplaning phenomenon can be new ship design parameters inUNDEX environments.

In this study, a comparative analysis between Case 1and Case 2 was performed. The same mesh layout of thedetonation zone was used for both cases. Owing to limi-tations of the computer operating systems, relatively largemeshes were used. However through the ALE code, simu-lation results were measured well. In Case 1, the sides ofthe two hulls have more stress (von-Mises stress) than thecentre deck in the UNDEX environment. In addition, thereis more impact in Case 1 than in Case 2.

UNDEX with high-speed catamaran-type ship shocksimulation is performed by modelling the 3D structures andsurrounding fluid volume using the ALE method. The shipshock simulation presented herein clearly demonstrates thatwhen a catamaran-type ship moves at high speed with thetrim angle in an UNDEX situation, it incurs less structuraldamage than it would in a zero speed situation.

Future work

It is recommended that additional studies be conducted toexamine in greater detail the effects of fluid mesh truncationin UNDEX simulations. First, such tests must vary the lo-cation of the charge to examine the diversity of the dynamicresponse of the catamaran. Second, such tests must vary thefluid element size to examine the effect fluid element pres-sure has on accuracy of the expected results. Third, suchtests must increase the geometric complexity of the cata-maran structure to study UNDEX effects on more complexmodels. Fourth, such tests must compare computed resultswith live-fire testing results. Finally, tests must apply fluidmesh truncation to more complex models, such as navywarships, to investigate the accuracy of the model responseto known live-fire testing data.

AcknowledgementsThis research was supported by WCU (World Class University)programme through the National Research Foundation of Koreafunded by the Ministry of Education, Science and Technology.(R31-2008-000-10045-0).

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