underwater shock response analysis of a ﬂoating vessel

53

Underwater shock response analysisof a floating vessel

J.E. van Aanhold∗, G.J. Meijer andP.P.M. LemmenTNO Building and Construction Research,Centre for Mechanical Engineering, P.O. Box 49,2600 AA Delft, The Netherlands

The response of a surface vessel to underwater shock hasbeen calculated using an explicit finite element analysis.The analysis model is two-dimensional and contains thefloating steel structure, a large surrounding water volumeand the free surface. The underwater shock is applied inthe form of a plane shock wave and cavitation is consideredin the analysis. Advanced computer graphics, in particu-lar video animations, provide a powerful and indispensablemeans for the presentation and evaluation of the analysisresults.

1. Introduction

The assessment of the vulnerability of naval ves-sels causes a continuous need for predicting the ef-fect of underwater shock. Essential items to be con-sidered in the analysis of ship structures subject tounderwater shock are shock wave propagation effectsduring the early time response stage, fluid cavita-tion, inertia dominated fluid behaviour during the latetime response stage and nonlinear structural response.In past analyses of structures subject to underwatershock, cavitation and nonlinear structural responsewere unfortunately too often neglected. Furthermore,the analysis of surface ships introduces an additionalchallenge.

This paper proposes a nonlinear finite elementanalysis based on a two-dimensional fluid/cavitationmodel for assessing the response of surface ship struc-tures. Explicit time integration is used and the incom-ing shock wave is considered plane. This approachhas an emphasis on early-time response, but the fluid

∗Corresponding author. E-mail: [email protected].

model is also capable of modelling the added hydro-dynamic mass for the late-time response stage. Themethod provides the following advantages over othersolutions:

• Suitable for the analysis of ship structures (pri-marily cross sectional deformations);• Better description and understanding of cavita-

tion;• Capability to model bulk and hull cavitation in

one and the same model;• Extension toward nonlinear analysis;• Improved assessment of the shock environment;• Inherent transition to late-time response;• Because of its 2D nature, acceptable in pre- and

postprocessing effort and computing times;• Possible extension toward 2 1

2 D analysis of a shipsection (strip theory) and/or simplified 3D anal-ysis of a ship section.

In order to demonstrate the feasibility of thismethod, first a number of simple one- and two-dimensional test problems was analyzed. This wassupported by theoretical studies regarding modellingtechniques, convergence and stability. This paper de-scribes the last subject of this study, the analysis ofa floating vessel, with an emphasis on the behaviourof the cavitating fluid. All analyses were done usingthe public domain version of DYNA3D (Whirley andHallquist [4]).

2. Underwater explosion analysis method

DYNA3D provides a fluid “material model” whichmay be used in combination with the standard 8-nodesolid brick element. This means that the hydrody-namic behaviour of the fluid following the acous-tic wave equation is modelled using a standard La-grangian displacement formulation. Although zero-energy deformation modes, also known as “hourglass-ing” are suppressed, the absence of shear stiffness in afluid still involves a serious risk that either excessive

Shock and Vibration 5 (1998) 53–59ISSN 1070-9622 / $8.00 1998, IOS Press. All rights reserved

54 J.E. van Aanhold et al. / Underwater shock response analysis of a floating vessel

hourglassing occurs or that significant shear stressesdevelop as a result of hourglassing control. Both phe-nomena can lead to unacceptable results. Hence, theavailable fluid analysis options had to be well testedfor this application. This displacement formulationhas the advantage that the fluid can be coupled di-rectly to the structure, but also increases the risk ofexcessive distortions of fluid elements near the fluid-structure interface. Because a fluid cannot transfershear stresses, coupling of fluid and structure in tan-gential direction results in a small mass increase ofthe structure.

The volumetric constitutive behaviour of the wateris described by the bilinear equation-of-state

p = max(−ρwc2wεv, 0), (1)

where p is the total pressure, εv the volumetric tensilestrain, cw the sonic velocity (1450 m/s) and ρw thedensity (1000 kg/m3). Cavitation is thus modelledby a zero fluid bulk stiffness for positive volumet-ric strains. The deviatoric constitutive behaviour ofthe water is described by means of a hydrodynamicviscosity coefficient.

With the exception of the free surface, the outerboundary of the water mesh is modelled as a nonre-flecting boundary. The plane step-exponential shockwave is modelled by means of pressure time historieswhich act on this boundary. These pressure time histo-ries are corrected for surface cutoff effects. The con-stitutive behaviour of the fluid also covers the addedhydrodynamic mass effects which are important in thelate time response stage. A necessary condition is thatthe nonreflecting model boundaries are at sufficientdistance from the structure, so that added mass ef-fects and cavitation are described properly by the fluidcontained within the model. This approach avoidsthe need for implementing approximations such as theDouble Asymptotic Approximation (DAA) at the non-reflecting boundary (Felippa and de Runtz [2]). Be-cause of its Lagrangian nature, this method cannotproperly describe fluid flow phenomena.

3. Test problems

The first test problem that was analyzed is the one-dimensional problem of a plate floating on the watersurface loaded by an underwater shock as describedby Bleich and Sandler [1]. This example, which isoften referred to in literature, was used as a firsttest. Various analyses using one- and two-dimensional

models were done and their results are in good ac-cordance with those by Bleich and Sandler [1]. Nohourglassing problems were encountered. The mainproblem appeared to be dispersion, i.e., a shock wavelooses its steepness during its propagation.

The second test problem was the two-dimensionalproblem described by Felippa and DeRuntz [2], amodified version of the problem originally analyzedby Newton [3]. This case consists of a deepwa-ter cylindrical shell of 10 m diameter in an infinitefluid domain loaded by a plane step-exponential shockwave. Also here the results agree well with thoseby Felippa and DeRuntz [2]. Again the solution suf-fered somewhat from dispersion, but it must be em-phasized that also the original results by Felippa andDeRuntz [2] exhibit significant dispersion effects.

In order to verify whether the fluid model correctlyaccounts for added hydrodynamic mass effects, an ex-tra analysis was done where the cylindrical shell ofthe previous test problem was fixed with relativelysoft node-to-ground springs. This analysis was runfor a long analysis time, i.e., a few seconds. Theeffective added mass as estimated from the resultingnatural frequency associated with the rigid body mo-tion showed that the added mass is described withsufficient accuracy.

The conclusions of these tests were sufficiently en-couraging to continue this study with a realistic struc-ture.

4. Analysis model of a floating structure

The structure considered in this study consists ofa cylindrical shell which is floating on the water sur-face. The shell has a radius 1.55 m and a thickness25 mm. A spring-supported rigid deck is mounted in-side the shell. The draught is 1.55 m. The structure isloaded by a heavy step-exponential shock wave whichtravels at an angle of 45 degrees with the horizontalplane. The decay time θ is approximately 1 ms. Onlylinear elastic structural behaviour was considered inthe current analysis. A number of analyses has beenperformed using different finite element models, in-cluding some for a fluid domain without the struc-ture.

The analysis presented here has a surrounding wa-ter domain of 15 m width and 10 m depth. Thesedimensions were based on the expected cavitation re-gion and the need for a proper description of addedhydrodynamic mass effects. The average element size

J.E. van Aanhold et al. / Underwater shock response analysis of a floating vessel 55

Fig. 1. Finite element model of fluid domain.

Fig. 2. Structural finite element model.

`element is 0.1 m, which corresponds to a dimensionlesselement length

α =`element

cwθ∼ 0.07, (2)

where θ is the decay time of the step-exponentialshock wave. Theoretical considerations showed thatthis dimensionless element size provides a reasonable

optimum between model size and minimization of dis-persion and spurious oscillations within the solution.A lesson learned from other analyses is that a regularmesh yields favourable results. Hence, the mesh con-tains mainly cube-shaped elements except for someinevitable slightly distorted elements near the struc-ture. The finite element model of the water is shownin Fig. 1, the model of the shell and spring-supporteddeck in Fig. 2.

The total mesh consists of 30 104 nodes, 14 694solid fluid elements, 96 shell elements for the shellstructure, 1 shell element for the rigid deck and 2× 4springs. A total of 350 nonreflecting boundary ele-ments have been applied for modelling the bottom,left and right side of the fluid mesh. The mesh hasa thickness 0.1 m in transverse direction. The shockwave enters the model at zero time at the left bot-tom corner nodes of the mesh and propagates at a 45◦

angle in right-upward direction.

5. Analysis

The analysis was done for two different durations:

(a) An early time response analysis until 15 mswith an output resolution of 0.05 ms. Outputare the initial coordinates, initial pressures, dis-placements and volumetric strains. These data


are output as binary datafiles and add up to314 Mb of data. The analysis used 1552 timesteps with an almost constant time step of ap-proximately 10 µs and almost 1 hour CPU timeon a Silicon Graphics work station.

(b) A cavitation closure analysis until 400 ms withan output resolution of 1 ms. Similar binaryoutput datasets add up to no less than 406 Mbof data. This analysis used 41 378 time stepswith an almost constant time step of approxi-mately 10 µs and approximately 21 hours CPUtime.

6. Presentation of results

The vast amount of output generated by DYNA3Dcreated the need to spend much effort on the presen-tation of results. Earlier analyses showed that the cal-culated fluid behaviour may be somewhat disturbedby the occurrence of spurious cavitation. Presenta-tion of results in the form of usual pressure contourplots does not allow to separate real and spuriouscavitation. It offers advantages to consider the vol-umetric tensile strain εv instead. It was discoveredthat spurious cavitation usually goes together withinsignificant volumetric tensile strains against muchlarger values for real cavitation. However, the vol-umetric compressive strains at the shock wave arevery low compared to the maximum volumetric ten-sile strain, which is 0.19 for the analysis presentedhere. The optimum solution is to use pressure/cavi-tation contour plots where pressures in non-cavitatingregions are plotted by real colours (57 colours rang-ing from dark blue for zero pressure through redfor peak explosion pressure), whilst in cavitating re-gions volumetric tensile strains are plotted using grayscales (7 gray scales ranging from dark for the cavi-tation threshold through light for maximum volumet-ric tensile strain using different contour increments).The threshold between spurious and real cavitationis set to 1/1000 times the maximum volumetric ten-sile strain, which makes spurious cavitation invis-ible. Such pressure/cavitation plots using the de-formed mesh are generated using MATLAB on a Sil-icon Graphics work station for every output time stepand saved as TIFF graphics format files. This re-quired approximately 30 hours CPU time and 127 Mbmore disk storage. Finally, Silicon Graphics anima-tion and video manipulation software was used to gen-erate a 400 frames video animation from these pic-tures.

7. Analysis results for water

The results were evaluated using a video animationwhich consists of a series of pressure/cavition contourplots on the deformed water mesh, where the defor-mations have been exaggerated by a factor 5. Figures3 through 6 show four examples of such plots.

Figure 3 shows the deformations and pressure/cavi-tation contours at 8 ms. The shock wave, which showsalready some dispersion, has reached the surface andthe structure, resulting in a large area of bulk cavita-tion. Note the thin layer of non-cavitating elementson top of the bulk cavitation. The water surface israising from the left.

Figures 4 and 5 show the deformations and pres-sure/cavitation contours at 22 and 60 ms, respectively,with large shell deformations, large deck deflectionsand an increasing bulk cavitation area. These figuresreveal minor spurious shear deformations in the fluidalong vertical lines aside the structure.

The bulk cavitation area remains increasing and thewater surface raising until approximately 96 ms. Atthe same time, the thickness of the water layer ontop of the bulk cavitation is increasing. The max-imum vertical displacement of the water surface is0.34 m against an average maximum vertical dis-placement 0.25 m. There is a continuous openingand closure of small cavitation regions near the hull.The water surface shows some small waves. Be-yond 96 ms, the thickness of the bulk cavitationlayer decreases while the thickness of the water layeron top of it increases. The bottom of the cavita-tion remains approximately at the same location. Af-ter approximately 150 ms the bulk cavitation layerstarts to show necking somewhat to the right under-neath the hull. The bulk cavitation starts to closeat 190 ms at the same location, which is shown inFig. 6.

From this moment onward, there are two bulkcavitation regions which close like a zipper towardthe nonreflecting boundaries of the model. Thiscavitation closure goes together with the genera-tion of shock waves which are cylindrical becauseof the 2D nature of the analysis. The resultingreloading pressures at the hull between 180 and220 ms are low compared to the primary shockwaves. After this moment some less significanthull cavitation continues until the end of the analy-sis (400 ms).

It may be discussed whether the observed cavitationclosure behaviour is realistic. In an additional analy-


Fig. 3. Pressure/cavitation contours and deformations for 8 ms. Fig. 4. Pressure/cavitation contours and deformations for 22 ms.

Fig. 5. Pressure/cavitation contours and deformations for 60 ms. Fig. 6. Pressure/cavitation contours and deformations for 190 ms,showing cavitation closure.

sis with a fully regular mesh of water only (withouta structure), the caviation closed as expected like azipper travelling from the left to the right at 210 ms,i.e., in the same direction as the shock wave passedby. On the other hand, a strong mesh dependency ofthe position and shape of the cavitation regions wasfound for more irregular meshes. It must neverthelessbe concluded that presence of the structure shortensthe duration of bulk cavitation.

Figure 7 shows pressure histories along the hull.Shock wave arrival is visible at approximately 8 ms.Bulk cavitation is shown as flat areas. The figurealso shows the reloading pressures resulting from bulkcavitation closure. Hull cavitation closure yields lo-

cal pressure peaks which are also of minor impor-tance.

Figure 8 shows the maximum hull pressures forshock wave arrival (7–10 ms) and reloading (180–220 ms). This shows that the maximum pressuresare found at the location of first shock wave arrival(φ = 45◦) and just before the location where the shockwave strikes the hull tangentially (φ = 135◦). Themaximum hull pressure is approximately 60% of thepeak explosion pressure. The maximum pressures inthe shadow zone beyond φ = 135◦ are considerablylower. Furthermore, the maximum hull pressures dur-ing reloading are low compared to those during shockwave arrival.


0

100

200

300

400 0

50

100

150

0

0.2

0.4

0.6

time [ms] element orientation [degrees]

pres

sure

/pea

k ex

plos

ion

pres

sure

Fig. 7. Hull pressures as a function of time and position angle φ.

0 20 40 60 80 100 120 140 160 1800

0.1

0.2

0.3

0.4

0.5

0.6

element orientation [degrees]

max

. pre

ssur

e/pe

ak e

xplo

sion

pre

ssur

e

− shock wave arrival −− reloading

Fig. 8. Maximum hull pressures during shock wave arrival (solidline) and reloading (dashed).

0 50 100 150 200 250 300 350 400−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

time [ms]

z−di

spla

cem

ent [

m]

Fig. 9. Vertical displacements of hull nodes as a function of time.

8. Analysis results for hull

The hull moves predominantly in vertical direction.Figure 9 shows the vertical displacements of the hullnodes. The maximum vertical displacement is approx-imately 0.25 m, which is mainly a rigid body trans-lation. This value equals the average maximum dis-placement of the water surface. The horizontal rigidbody motion is very small. The maximum deflectionsdue to vibrations are 55 mm. The maximum verticalspring deflection is 85 mm.

9. Conclusions

It is concluded that the analysis method as pre-sented in this paper is well suitable for this class ofproblems and produces results which are in reason-able agreement with solutions from literature. Thepresented demonstration analysis shows that it is fea-sible to calculate the underwater shock response of asurface vessel for the primary shock wave with bulkand hull cavitation. Advanced computer graphics, inparticular video animations, provide a powerful andindispensable means for the presentation and evalua-tion of the results. Although some questions about theresults remain, in particular about bulk cavitation clo-sure, the results are very instructive and encouragingfor future work. Most serious remaining problems aredispersion (a shock wave looses its steepness duringits propagation) and the present-day practical hard-ware and software limits, which unfortunately still ob-struct a full three-dimensional analysis of a completenaval ship structure. Further work needs to be done inaddressing cavitation closure, the effects of a pulsat-ing gas bubble and nonlinear elastic-plastic structuralresponse. Finally, validation by means of full scaleexperiments remains indispensable.

Acknowledgement

The authors express their gratitude to the RoyalNetherlands Navy for sponsoring the work presentedin this paper.

References

[1] H.H. Bleich and I.S. Sandler, Interaction between structuresand bilinear fluids, Int. J. Solids and Structures 6 (1970),617–639.


[2] C.A. Felippa and J.A. DeRuntz, Finite element analysis ofshock-induced hull cavitation, Computer Methods in AppliedMechanics and Engineering 44 (1984), 297–337.

[3] R.E. Newton, Effects of cavitation on underwater shock load-ing – Plane problem, final report, Report no. NPS-69-81-001A Naval Postgraduate School, Monterey, CA.

[4] R.G. Whirley and J.O. Hallquist, DYNA3D – A nonlin-ear, explicit, three-dimensional finite element code for solidand structural mechanics – User Manual, Report no. UCRL-MA-107254 Lawrence Livermore National Laboratory, Liv-ermore, CA.

Received 3 October 1997; Revised 4 February 1998

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttp://www.hindawi.com Volume 2010

RoboticsJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation http://www.hindawi.com

Journal ofEngineeringVolume 2014

Submit your manuscripts athttp://www.hindawi.com

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014

SensorsJournal of


Modelling & Simulation in EngineeringHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks


underwater shock response analysis of a ﬂoating vessel

Documents