draft: predictive simulation of underwater implosion ...€¦ · proceedings of the asme 2014 33rd...

Proceedings of the ASME 2014 33rd International Conference on Ocean, Offshore and Arctic EngineeringOMAE2014

June 8-13, 2014, San Francisco, California, USA

OMAE2014-23341

DRAFT: PREDICTIVE SIMULATION OF UNDERWATER IMPLOSION: COUPLINGMULTI-MATERIAL COMPRESSIBLE FLUIDS WITH CRACKING STRUCTURES

Kevin G. WangDepartment of Aerospace and

Ocean EngineeringVirginia Polytechnic Institute and

State UniversityBlacksburg, Virginia 24061

Patrick LeaDepartment of Aeronautics and

AstronauticsStanford University

Stanford, California 94305

Alex MainInstitute for Computational and

Mathematical EngineeringStanford University

Stanford, California 94305

Owen McGarityDepartment of Survivability,

Structures and MaterialsNaval Surface Warfare Center

West Bethesda, Maryland 20817

Charbel Farhat ∗Department of Aeronautics and Astronautics,Department of Mechanical Engineering, and

Institute for Computational andMathematical Engineering

Stanford UniversityStanford, California 94305

ABSTRACT

The implosive collapse of a gas-filled underwater structure canlead to strong pressure pulses and high-speed fragments thatform a potential threat to adjacent structures. In this work, a high-fidelity, fluid-structure coupled computational approach is devel-oped to simulate such an event. It allows quantitative predictionof the dynamics of acoustic and shock waves in water and theinitiation and propagation of cracks in the structure. This com-putational approach features an extended finite element method(XFEM) for the highly-nonlinear structural dynamics character-ized by large plastic deformation and fracture. It also featuresa finite volume method with exact two-phase Riemann solvers(FIVER) for the solution of the multi-material flow problem aris-ing from the contact of gas and water after the structure fractures.The Eulerian computational fluid dynamics (CFD) solver and theLagrangian computational structural dynamics (CSD) solver arecoupled by means of an embedded boundary method of second-order accuracy in space. The capabilities and performance ofthis computational approach are explored and discussed in the

∗Address all correspondence to this author.

full-scale simulations of a laboratory implosion experiment withhydrostatic loading and a three-dimensional manufactured im-plosion problem with explosion loading.

INTRODUCTION

The dynamic collapse of a submerged, gas-filled structure underhydrostatic or hydrodynamic pressure loading is a highly non-linear fluid-structure interaction (FSI) problem. Once the struc-ture starts collapsing under the effect of fluid pressure, the sur-rounding water follows with the same velocity. At the end, whenthe walls of the structure collide, this fast, inward traveling wa-ter flow is suddenly stopped; as a result a large amount of ki-netic energy associated to this water flow reflects in the form ofshock waves. This process is further complicated if the collaps-ing structure undergoes fracture or fragmentation, as the gas in-side the structure and the water outside come into direct contact.This type of problem, commonly referred to as underwater im-plosion, is a major area of concern in a variety of ocean engineer-ing applications, as the shock waves and structural fragments can

1 Copyright c© 2014 by ASME

cause damage to or failure of adjacent structures. Currently, pre-dictive simulation of underwater implosion is still a formidablechallenge. It requires high-fidelity physical models for the non-linear material behaviors and dynamics in both fluid and solidmedia, as well as accurate and robust computational methods forthe complex interaction among gas, water, and the implodablestructure. Recent efforts in this regard include [1–5], where fluid-structure coupled computational methods are developed and usedto simulate underwater implosions without fracture.

On the other hand, FSI with fracture is still an area of researchthat has yet to be fully explored, as methodologies for both FSIand fracture mechanics continue to evolve. Previous computa-tional efforts which address some of the aforementioned chal-lenges include [6–9]. More specifically, Cirak et al. [6] was ableto simulate the interaction of single-material fluid with fracturingshell, using a dilute interface representation in the fluid compu-tation and subdivision shell elements for fracture modeling. Thiswork was later extended by Deiterding et al. [7] in order to simu-late two-material fluid-structure interactions arising from under-water pipeline explosion. In [7], multiple immiscible fluid media(such as water and air) are modeled using a single generalizedequation of state with different values for certain parameters,while the material interface in between is blurred and not cap-tured. Rabczuk et al. [8] performed simulations on single-phaseFSI with cracking shells, using the immersed particle method,a Lagrangian meshfree method for both the fluid and the struc-ture. Another meshfree method, material point method (MPM),was used by Parker et al. [9] to model a fracturing structure as itinteracts with multi-material fluids with heat transfer.

In this work, a fluid-structure coupled computational frame-work is developed to simulate implosions of gas-filled under-water structures involving dynamic fracture. Both gas and wa-ter are modeled as compressible fluid. The implodable struc-ture is modeled using shells of elasto-plastic metal and is sub-ject to material failure in the form of cracking. The coupledfluid and structure governing equations are numerically solvedin a partitioned procedure. More specifically, the multi-materialcompressible fluid flow is solved on a fixed, non-body-fittedmesh using a finite volume method with exact two-phase Rie-mann solvers (FIVER) [2, 10, 11]. The nonlinear CSD prob-lem, characterized by large plastic deformation and fracture, issolved using an extended finite element method (XFEM) in thephantom node formulation [12, 13]. The main challenge in cou-pling FIVER and XFEM for cracking structures lies in interfacetracking. XFEM provides an explicit representation of the struc-tural geometry in general, except that the crack is tracked im-plicitly by means of signed distance functions (sometimes re-ferred to as local level set functions). In the context of FSI, thismixed geometry representation needs to be interpreted with spe-cial care in order to track the cracking fluid-structure interfacewith respect to the fixed, non-body-fitted fluid mesh. This work

is accomplished by extending the interface tracking algorithmspreviously prsented in [3] for FSI without fracture. Moreover,the fluid-induced loads on the fluid-structure interface are com-puted and transferred to the CSD solver using the conservativealgorithms presented in [2]. Finally, the semi-discretized fluidand structure governing equations are integrated in time usingsecond-order accurate, staggered fluid-structure time-integratorspresented in [1].

The capabilities and performance of this computational approachare explored and discussed in the full-scale simulations of twounderwater implosion problems. The first one is a laboratory ex-periment of Mode 3 collapse of an aluminum cylinder under hy-drostatic pressure loading. This example validates the proposedcomputational approach in the absence of fracture. The secondexample is a three-dimensional manufactured problem in whichan air-filled aluminum cylinder collapses due to hydrodynamicloading from an underwater explosion. This example is charac-terized by dynamic fracture and three different fluid materials,namely air, water, and the gaseous explosion product from TNT.

PHYSICAL MODELS

In this work, the multi-material fluid flow is formulated in an Eu-lerian setting, which is usually preferred over the alternative La-grangian and Arbitrary Lagrangian-Eulerian (ALE) approachesfor FSI problems involving large structural deformation and/orfracture. All the fluid materials are considered compressible andinviscid, therefore the dynamics is governed by the following Eu-ler equations

∂W∂ t

+~∇ · ~F (W ) = 0, in ΩF(t) (1)

where

W = (ρ, ρvx, ρvy, ρvz, E)T , (2)

~∇ =

(∂

∂x,

∂

∂y,

∂

∂ z

)T

, (3)

~F (W ) = (Fx(W ), Fy(W ), Fz(W ))T , (4)

Fx =(ρvx, p+ρv2

x , ρvxvy, ρvxvz, vx(E + p))T

, (5)

Fy =(ρvy, ρvxvy, p+ρv2

y , ρvyvz, vy(E + p))T

, (6)

Fz =(ρvz, ρvxvz, ρvyvz, p+ρv2

z , vz(E + p))T

. (7)

ΩF(t) denotes the time-dependent fluid domain, ρ denotes thefluid density, E denotes its total energy per unit volume, p de-notes the fluid pressure, and~v = (vx, vy, vz) is the velocity vec-tor.


The Euler equations are closed by an equation of state (EOS)which relates the thermodynamic variables ρ , p and e. In thiswork, the gas material inside the implodable structure is assumedto be air and modeled by the perfect gas EOS; the liquid water ismodeled by either the stiffened gas EOS:

p = (γ−1)ρe− γ pc (8)

with γ = 4.4 and pc = 87,000 psi, or the Tait EOS:

p = p0 +α

((ρ

ρ0

)β

−1

), (9)

where (p0,ρ0) is a given reference state, and α and β are empir-ical constants.

The nonlinear dynamics of the implodable structure is formu-lated in a Lagrangian setting:

ρsus−∇x ·σσσ s(us, us) = fexts in ΩS(0) (10)

σσσ snt = t on Γt

u = u on Γu

where ΩS(t) denotes the time-dependent structural domain, usdenotes the displacement of the structure with respect to the ref-erence configuration (ΩS(0)), ρs and σσσ s denote its density andCauchy stress tensor, respectively, and fext

s is the external forceacting on it. t is the applied traction on the Neumann boundaryΓt and us is the applied displacement on the Dirichlet boundaryΓu.

Both geometric and material nonlinearities are modeled, in orderto account for large deformation and fracture. The J2-plasticitymodel [14] is employed as the yield criterion for the ductilemetal materials encountered in the present work. A strain basedfracture criterion is employed to determine the propagation ofa crack. More specifically, when the maximum principal ten-sile strain at the tip of an existing crack reaches a pre-specifiedthreshold, the crack propagates from this tip. The direction ofpropagation is determined to be perpendicular to the directionof the principal tensile strain of a spatially averaged strain εavg

defined in [15].

A cohesive crack model is employed in order to ensure an ac-curate dissipation of fracture energy. It is assumed that acrossa crack, the normal component of the stress tensor satisfies thefollowing cohesive law:

σσσ+s ·n+

c = σσσ−s ·n−c = τ

c(JuK), (11)

in which nc is the unit normal to the crack surface, and τc is theapplied cohesive traction across it, expressed as a function of thejump of displacement across the crack in the normal direction(nc). Superscript plus and minus signs refer to the two sides ofthe discontinuity (crack). To ensure the accurate dissipation offracture energy, τc is designed to satisfy

G f =∫

∞

0τ

c(δ )dδ , (12)

where G f is the fracture energy of the material. In this work, apiecewise linear τc is used.

The fluid and structure are assumed to be impermeable, leadingto the following Dirichlet and Neumann transmission conditionsat the fluid-structure interface, namely ΩF(t)

⋂ΩS(t):

u ·n = v ·n (13)σσσ s ·n = −pn (14)

where n denotes the unit normal to the interface.

Equation (13) implies the continuity of the normal componentof velocity across the fluid-structure interface; Equation (14) im-plies the equilibrium of the interaction force between the fluidand the structure.

After fracture initiates, the gas inside the structure is directlyin contact with water (Fig. 1). In the context of underwaterimplosion, surface tension at the gas-water interface is negligi-ble compared to the pressure force. Therefore, in this work thegas-water interface is modeled as a free surface advected by thefluid flow. More specifically, let Ω

(1)F ⊂ ΩF and Ω

(2)F ⊂ ΩF be

the domains of two different fluid media. If they are in con-tact, i.e. ∂Ω

(1)F⋂

∂Ω(2)F 6= /0, the following conditions hold at

∂Ω(1)F⋂

∂Ω(2)F :

v(1) ·n = v(2) ·n (15)p(1) = p(2). (16)

where n denotes the normal direction of the gas-water interface.

In other words, the pressure field and the normal component(with respect to the interface) of the velocity field are continu-ous across the interface.

COMPUTATIONAL METHODS

The coupled fluid and structure governing equations presented inthe previous section are solved using a partitioned approach. Key


Fluid 2

Fluid 1

thin-walled structure

immiscible interface between fluid 1 and 2

FIGURE 1. Illustration of a multi-material fluid-structure interactionproblem involving fracture.

components of this computational approach include: (1) an Eule-rian multi-material CFD solver based on FIVER; (2) a nonlinearLagrangian CSD solver based on XFEM; (3) robust and efficientalgorithms for tracking the cracking shell structure with respectto the fixed, non-body-fitted CFD mesh; and (4) conservative al-gorithms for transferring fluid-induced loads onto the wetted sur-face of the structure.

A Finite Volume Method with Exact Two-Phase Rie-mann Solvers (FIVER)

The Euler equations governing the fluid flows are semi-discretized by a finite volume method equipped with exact two-phase Riemann solvers, also known as FIVER [11]. This methodoperates on fixed, non-body-fitted CFD meshes in which thefluid-structure interface is treated using a second-order accurateembedded boundary method [2] (see Figure 2 for an illustra-tion). Inside the domain of each fluid medium, the numerical fluxis evaluated based on approximate single-fluid Riemann solvers(e.g. Roe), same as the standard finite volume method. How-ever, at the fluid-structure and fluid-fluid interfaces, the numer-ical flux is evaluated based on the exact — or approximate, forcertain highly nonlinear EOS — solution of fluid-structure andfluid-fluid Riemann problems defined in [2, 10]. More specifi-cally, for the numerical flux associated to edge 〈i, j〉 in the fluidmesh (assuming a node-centered finite volume discretization),the choice of Riemann solver is completely determined basedon (1) the fluid medium occupying node i, denoted by Ii, (2) thefluid medium occupying node j, denoted by I j, and (3) whetheror not edge 〈i, j〉 intersects the fluid-structure interface, denotedby X(i, j). It is notable that the fluid-structure and fluid-fluidRiemann solvers also implicitly enforce the velocity conditions(Eqs. (13), (15)) at these interfaces.

FIVER is a particularly attractive method for the solution offluid-structure interaction problems with fracture. It does not re-

F

S

EnE

FIGURE 2. Domain setting of an embedded boundary method forfluid-structure interaction: fluid domain of interest ΩF , structure do-main of interest ΩS, embedded interface ΣE , and outward normal~nE toΣE .

quire any a priori knowledge of the crack initiation site or thecrack propagation path. However, in order to obtain Ii, ∀i in thisscenario, the dynamic gas-water interface needs to be captured.This is achieved in the present work by solving the level set equa-tion

∂ϕ

∂ t+v ·∇ϕ = 0, in ΩF(t) (17)

where ϕ(x, t) denotes the signed distance from x ∈ ΩF(t) to thegas-water interface at time t, and v denotes the fluid velocity.

Moreover, in the context of XFEM, special care is required ininterface tracking in order to obtain the intersection informationX(i, j), ∀〈i, j〉 for cracking structures. This aspect is to be dis-cussed later in this section.

XFEM and the Implicit Representation of Cracks

A crack in the structure medium corresponds to a strong discon-tinuity in its displacement field. It cannot be captured automat-ically in classical finite element analysis which relies on con-tinuous shape functions. In this work, cracking is modeled byXFEM [12, 17], following the phantom node formulation [13].The key idea of XFEM is to enrich the approximation basiswith shape functions that are discontinuous across the crack.The phantom node formulation [13] uses a transformation of thenodal variables which leads to a superposed element formula-tion. This formulation has been shown to be equivalent to XFEMthrough algebraic manipulations [18]. More specifically, withina cracked element the displacement field is expressed as

u(X, t) = ∑I∈Elem1

NIH (φ (X))u1I + ∑

J∈Elem2NJH (φ (X))u2

J (18)

where NI(X) and NJ(X) are the conventional continuous FEshape functions which, for example, can be the piecewise linear


P1 shape functions; φ(X) is a signed distance function used todefine the location of the crack; Elem1 and Elem2 are the indexsets of the nodes of superposed element 1 and 2, respectively;H(·) is the Heaviside step function defined by

H(α) =

1, α > 00, α ≤ 0 . (19)

In terms of implementation, once the failure criterion is reachedin one element, this element is replaced by two superposed anddisconnected phantom elements, each with a phantom region(Figure 3). These two phantom elements will move/deformindependently. The crack path within each phantom elementis assumed to be a straight line, and is tracked implicitly byφ(X). More specifically, for an arbitrary phantom element e,X ∈ e, φ(X) = 0, X ∈ e, φ(X)> 0, and X ∈ e, φ(X)< 0correspond to the cracking path in e, the real region of e, and thephantom region, respectively.

V2V1

V3V4

e

V1 V2

V5V6

V3V4

V8V7

real node

phantom node

f(x) > 0

f(x) < 0

f(x) < 0

f(x) > 0

e(1)

e(2)

f

f local distance function

crack

FIGURE 3. The phantom node formulation: each cracked element isreplaced by two phantom elements with additional phantom nodes.

Tracking an Embedded Fluid-Structure Interface withCracks and Phantom Elements

A Collision-Based Algorithm FIVER operates on fixed,non-body-fitted fluid meshes. It relies on intersection informa-tion X(i, j) (∀ edge 〈i, j〉 in the mesh) to track the location of thefluid-structure interface with respect to the fluid mesh. In [3],a geometric algorithm is presented which is capable of track-ing both closed and open interfaces with respect to arbitrary(i.e. structured and unstructured) fluid meshes. Referred to as

the collision-based approach, this algorithm is based on point-simplex collision detection with motivation from the computergraphics community. More specifically, it detects the intersec-tions between the edges of the fluid mesh and the triangle ele-ments1 of the embedded fluid-structure interface essentially bycasting rays from each node of fluid mesh to its neighbors andsearching for ray-triangle collisions. In other words, the inter-section between an edge 〈i, j〉 and a triangle is viewed as a colli-sion between the ray from node i towards node j (or from node jtowards node i) and the triangle.

Several efforts have been made to optimize computational effi-ciency, which include constructing bounding boxes for the edgesin the fluid mesh and the elements in the fluid-structure interface,and storing the bounding boxes in k-d trees (k = 3). With theseimprovements, it is shown in [3] that the computational cost isreduced to O(N logM), and in practice, 5% or less of the totalCPU time.

Treatment of cracking interfaces In the phantom nodeformulation, each cracked element is replaced by two super-posed elements. The crack path within each phantom elementis assumed to be a straight line and tracked implicitly by signeddistance function φ . In a fluid-structure coupled computation,information about newly generated phantom elements needs tobe transferred from the CSD solver to the CFD solver. Morespecifically, the node coordinates and connectivity of each newphantom element are sent to the CFD solver. The discrete em-bedded interface is then updated accordingly. The values of φ atthe nodes of phantom elements are also sent to the CFD solverand stored there. It is notable that in practice the computationaloverhead associated with these additional data transfers is trivial,as cracking usually occurs only locally in the structure medium.

The collision-based interface tracking algorithm is then modifiedsuch that a collision point located in a phantom element is reg-istered as a valid intersection if and only if it is located in thereal region of the phantom element. More specifically, let 〈i, j〉and e be an edge in the fluid mesh and a phantom element inthe XFEM mesh, respectively. An intersection is registered inbetween if and only if

〈i, j〉⋂

e 6= /0, (20)

and

φ(XI)> 0, (21)

1The elements of the discrete fluid-structure interface are assumed here to betriangles. This is a weak assumption as any n-noded surface element with n > 3can be easily divided into several triangles.


where XI := 〈i, j〉⋂

e is the intersection point.

Conservative Load Computation and Transfer UsingSurrogate Interface

In this work, the equilibrium condition (14) at fluid-structure in-terface is enforced by the computation of the flow-induced loadsat the fluid-structure interface and the transfer of these distributedloads to the structural solver. The conservative algorithm pro-posed in [2] is employed. The main ideas of this algorithm are

• first, integrate the flow-induced loads on a surrogate inter-face where the fluid pressure field is computed. The surro-gate interface serves as an approximation of the embeddedfluid-structure interface;

• transfer the distributed loads from the surrogate interface tothe embedded discrete interface in a conservative manner byapplying the virtual power principle.

Two options for the construction of surrogate interface in a non-body-fitted finite volume fluid mesh are proposed in [2]: (1) theunion of the control volume facets associated to edges intersect-ing the fluid-structure interface; and (2) the surface constructedby connecting all the intersection points.

APPLICATIONS

The FIVER method, the extended collision-based interface track-ing algorithm, and the conservative load computation and trans-fer algorithm are implemented in the massively parallel com-pressible flow solver AERO-F [19, 20]. XFEM and the phantomnode formulation are implemented in DYNA3D, a finite elementstructure/continuum solver developed by Lawrence LivermoreNational Laboratory (LLNL) [21]. The capabilities and perfor-mance of the presented computational approach are explored anddiscussed in the full-scale simulations of a laboratory implosionexperiment with hydrostatic loading and a three-dimensionalmanufactured implosion problem with explosion loading.

Mode 3 Collapse of An Aluminum Cylinder Under Hy-drostatic Pressure Loading

Experiment An underwater implosion experiment performedby S. Kyriakides and co-workers at the University of Texas atAustin is considered for the validation of the proposed compu-tational approach in the absence of fracture. The specimen is analuminum (Al6061-T6) cylinder of length L0 = 7.626 in, circularcross section with external diameter D= 1.5002 in, and thicknessτ = 0.0281 in (Figure 4). It is filled with air at the standard at-mospheric pressure and bonded at the ends to rigid steel plugs

which seal the cylinder. The unbonded region of the cylinder hasa length of 5.626 in. The specimen is maintained at the centerof a rigid water tank by a set of bars which connect the plugs tothe surface of the tank. It is surrounded by 5 pressure sensorswhich are distributed on the mid-plane (orthogonal to the axialdirection of the cylinder) with radial distance d = 1.6± 0.17 into the surface of the cylinder.

D = 1.5002 in

Aluminum Tube

(Al6061-T6, 2800 kg/m3)

t = 0.0281 in

L = 3.75D = 5.626 in

1.0 in

r = 7/64 in

Steel End Plug (7850 kg/m3)Pressure Sensor

FIGURE 4. Schematic drawing of the specimen used in an underwa-ter implosion experiment.

Initially, water outside the cylinder and air inside it are both atrest with the same pressure p0

w = p0a = 14.5 psi. Then, water

pressure is slowly increased at a constant rate until the cylin-der collapses. The final hydrostatic pressure, under which thecylinder collapses, is pco = 448.0 psi. The pressure time-historyrecorded by a sensor (Sensor 1 in Figure 7 – left) reveals first agradual pressure drop of about 100 psi before the cylinder self-contacts, followed by a sharp spike with a peak of 357 psi, then abroader pulse with a second peak of 230 psi. This pressure pulsedemonstrates the strong shock wave caused by the self-contact ofthe cylinder. The duration of implosion pulse is approximately0.4 ms. A photograph of the collapsed specimen is shown in Fig-ure 6. A cross section in the collapsed region shows three distinctlobes — this hydrostatic collapse pattern is referred to as Mode3.

Fluid-structure coupled simulation The geometric centerof the cylinder is chosen as the origin of the Cartesian coordi-nate system, and its axial direction is chosen as the x-axis. Halfof the cylinder (length-wise) is modeled. The aluminum mate-rial is represented as an isotropic elasto-plastic medium with J2-plasticity and linear hardening. The Young’s modulus is set toE = 10,037 ksi; the Poisson ratio is set to ν = 0.3, and the den-


sity is set to ρS = 2.52× 10−4 (lbf/in4).sec2. The yield stress isset to 38.2 ksi and the hardening modulus is set to 25.0 ksi.

The cylinder is discretized by a finite element mesh with 9,984four-noded shell elements. The steel plug to which the cylinder isbonded is represented by 6,656 four-noded rigid shell elements.A sliding boundary condition is imposed on the plug and the re-gion of the cylinder that is bonded to the plug, such that this partof the model can only slide in axial direction.

A small Mode 3 sinusoidal imperfection is imposed on the ge-ometry of the cylinder to trigger its collapse. More specifically,the circular cross section of the cylinder satisfies the followingparametric equation in the (r,θ) polar coordinate system:

r = r0(1−∆cos3θ).

The radius of the true circular cross section measured to the mid-surface of the cylinder is r0 = 0.736 in. The maximum imperfec-tion is set to ∆ = 0.005.

The fluid computational domain is a rectangular box: 0 ft≤ x≤10 ft, − 10 ft ≤ y ≤ 10 ft, − 10 ft ≤ z ≤ 10 ft. It is discretizedby an unstructured non-body-fitted CFD mesh with 3.3 M gridpoints and 20.0 M tetrahedra (Figure 5). Symmetry boundary

XXX TO BE IMPROVED XXXFIGURE 5. The finite element CSD model and the fixed, non-body-fitted CFD mesh.

conditions are applied to the fluid model at the transversal planepassing through the middle cross section of the cylinder, i.e. theplane determined by x = 0. Non-reflecting boundary conditionsare applied at the remaining boundaries of the fluid domain. Att = 0, the initial state of air inside the cylinder is set to ρ0

a , v0a,

and p0a. The initial density and velocity for water are set to ρ0

wand v0

w; the pressure is set to the collapse pressure pco.

The predicted pressure time-history at a sensor location is re-ported in Figure 7 – left together with the signal obtained fromthe experiment. The main features of the experimental result are

TABLE 1. The peak pressure and impulse at each sensor, showinggood agreement between the simulation and experimental results.

SensorPexp Psim

Diff.Iexp Isim

Diff.(psi) (psi) (psi-sec) (psi-sec)

1 357 383 7% 0.0537 0.0505 6%

2 348 328 6% 0.0493 0.0495 0%

3 360 334 7% 0.0545 0.0523 4%

4 351 361 3% 0.0569 0.0507 11%

5 331 369 11% 0.0570 0.0551 3%

Average 6.8% Average 4.8%

well replicated in the simulation. These include the gradual pres-sure drop, then a sudden pressure jump with two peaks, followednext by a sharp pressure drop. The magnitude and width of thefirst pressure drop are accurately captured. Also, the predictedfirst peak pressure matches the experimental data well with a rel-ative error around 7%. Comparison of peak pressure and impulsefor all five pressure sensors are reported in Table 1, showing goodagreement between the experiment and simulation results. Thepredicted deformation of the specimen at the end of the simula-tion is reported in Figure 6, together with the experimental result.Both exhibit a Mode 3 hydrostatic collapse showing three distinctlobes, and it is clear that the simulation result closely matches theexperimental result.

Experiment

Simulation

FIGURE 6. The geometry of the collapsed cylinder obtained fromsimulation (top) and experiment (bottom).


A B C

A

B

C

1

Gage 1, Experiment

Gage 1, Simulation

FIGURE 7. Comparison of experiment and simulation for the pres-sure time-history at a sensor location (left), and three snapshots ofthe fluid pressure showing the shock waves emanating from the fluid-structure interface.

Implosive Collapse of An Aluminum Cylinder UnderExplosion Loading

A challenging three-dimensional underwater implosion problemis manufactured to demonstrate the capabilities of the presentedcomputational approach. This problem is characterized by thecollapse and fracture of a shell structure due to hydrodynamicpressure loading from an underwater explosion. It involves theinteraction of three different fluid materials, namely air, water,and the gaseous explosion product from TNT.

In this problem, the structure specimen consists of an aluminumtube (the main tube) connected at each end to a wider tube (theend tube) through a tapering region. The main tube has radiusr1 = 3.94 in, length l1 = 55.12 in, and thickness τ1 = 0.25 in. Thetwo aluminum end tubes are identical, with radius r2 = 4.5 in,length l2 = 11.02 in, and thickness τ2 = 0.25 in. Each taperingregion, also aluminum, has length l3 = 3.94 in and thickness τ3 =0.25 in. Hence the total length of the specimen is 85.04 in. Thetube contains bulkheads of thickness τbulk = 0.25 in located atlb1 = 10.63 in, lb2 = 15.94 in, lb3 = 24.80 in, lb4 = 37.20 in,lb5 = 47.83 in, lb6 = 60.24 in, lb7 = 69.09 in, and lb8 = 74.41 in.The specimen is filled with air at atmospheric pressure, closedat either end by a solid end-cap (steel) with length lcap = 5.0 in,and submerged in water. Half of the specimen (length-wise) isshown in Fig. 8. Because of symmetry, the other half is simplya mirror image of the one shown in the figure. This fact will beexploited in the subsequent computations.

The collapse of the structure is induced by a near-field under-water explosion of 120.2 lb of TNT. The center of the charge is

FIGURE 8. The structure specimen (half model).

located on the plane of symmetry of the structure, at a distanceof 148.43 in to the geometric center of the structure.

Water Air

Explosive

Structure

148.43 in

y

z

x x

z

y

Structure

Water

Explosive

Air

FIGURE 9. Schematic drawing of the setup of an explosion-drivenunderwater implosion problem.

A multi-material fluid-structure coupled simulation is performedto predict the coupled fluid and structure dynamics both beforeand after the structure collapses. Half of the specimen (length-wise) is modeled in the simulation. The setup of computationaldomains and boundary conditions are shown in Fig. 9. Theheavy, solid lines identify the symmetry boundary planes of thefluid and structure domains. The dotted lines correspond to thefar-field boundaries of the fluid domain. Three fluid media areinvolved in the simulation, namely water, air, and the gaseousexplosion product from TNT. The water is modeled using theTait equation of state (Equation (9)). The air is modeled usingthe ideal gas equation of state. The explosion product is modeledusing the Jones-Wilkins-Lee (JWL) equation of state:

p =A1

(1− ωρ

R1ρ0

)exp[−R1ρ0

ρ

]+

A2

(1− ωρ

R2ρ0

)exp[−R2ρ0

ρ

]+ωρe,

with A1 = 5.38×107 psi, A2 = 4.68×105 psi, R1 = 4.15, R2 =0.95, ω = 0.3, and ρ0 = 1.83 slug/in3. The aluminum material of


the structure is represented as an isotropic elasto-plastic mediumwith J2-plasticity and linear hardening, same as in the previousexample.

The simulation is performed in two steps. First, the explosionof the TNT charge is simulated in one dimension (exploiting thespherical symmetry of the explosive) to the point in time justprior to when the blast wave reaches the structure (t = 2.0 ms)2

Then, the three-dimensional fluid-structure coupled simulationis started, using the one dimensional solution (mapped to threedimensions) as the initial condition. Three snapshots of the struc-tural deformation are provided in Fig. 10, showing the collapseof the structure: at t = 2.6 ms, as high mode buckling begins; att = 2.9 ms, as fracture begins; and at 3.05 ms as failure has pro-gressed further. The bulkheads cause each crack to be containedto its own compartment as would be expected. The majority offailure occurs near the center of tube which is also closest to theblast. Fig 11 is a zoom view of the fourth bay (one out fromcenter) showing fracture propagating along the tube. It is notablethat as cracks propagate the air inside the tubes comes into di-rect contact with water. In other words, the computation involvesa fracturing structure, the interaction of air and water with thestructure, an interface between air and water, and an interfacebetween water and the gaseous explosion product.

ACKNOWLEDGMENT

The Acknowledgment section will be completed in the final sub-mit.

REFERENCES

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[3] Wang K, Gretarsson J, Main A, and Farhat C. Computa-tional Algorithms for Tracking Dynamic Fluid-StructureInterfaces in Embedded Boundary Methods. InternationalJournal for Numerical Methods in Fluids. 2012; 70:515–535.

2The detonation itself is modeled using a programmed burn methodology.

FIGURE 10. Snapshots of the structural deformation, showing thecollapse and fracture of the tube.

[4] Turner S E, Ambrico J M. Underwater Implosion of Cylin-drical Metal Tubes. ASME Journal of Applied Mechanics.2013; 80:011013-1-11.

[5] Farhat C, Wang K, Main A, Kyriakides J S, Ravi-ChandarK, Lee L-H, and Belytschko T. Dynamic Implosion ofCylindrical Shells: Experiments and Computations. Inter-national Journal of Solids and Structures. 2013; 50:2943–2961.

[6] Cirak F, Deiterding R, Mauch S. Large-Scale Fluid-Structure Interaction Simulation of Viscoplastic and Frac-turing Thin-Shells subjected to Shocks and Detonations.Computers and Structures 2007; 85:1049–1065.


FIGURE 11. Zoom of fourth bay of the tube undergoing fracture.

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[11] Farhat C, Gerbeau J-F, and Rallu A. FIVER: A Finite Vol-ume Method Based On Exact Two-Phase Riemann Prob-lems and Sparse Grids for Multi-Material Flows with LargeDensity Jumps. Journal of Computational Physics 2012;231:6360–6379.

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[13] Song JH, Areias PMA, and Belytschko T. A Method forDynamic Crack and Shear Band Propagation with Phan-tom Nodes. International Journal for Numerical Methodsin Engineering 2006; 67:868–893.

[14] Lubliner J. Plasticity Theory. MacMillan, New York, NewYork, USA, 1990.

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[16] Osher S, Fedkiw R. Level Set Methods and Dynamic Im-plicit Surfaces. Springer-Verlag New York, Inc. New York,USA, 2003.

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