Advanced Computational Techniques for the Analysis of 3-D

Fluid-Structure Interaction

with Cavitation

by

Michael Alan Sprague

B.S., University of Wisconsin–Madison, 1997

M.S., University of Colorado at Boulder, 1999

A thesis submitted to the

Faculty of the Graduate School of the

University of Colorado in partial fulfillment

of the requirements for the degree of

Doctor of Philosophy

Department of Mechanical Engineering

2002

This thesis entitled:Advanced Computational Techniques for the Analysis of 3-D Fluid-Structure Interaction

with Cavitationwritten by Michael Alan Sprague

has been approved for the Department of Mechanical Engineering

Thomas L. Geers

Carlos A. Felippa

Date

The final copy of this thesis has been examined by the signatories, and we find that both the contentand the form meet acceptable presentation standards of scholarly work in the above mentioned

discipline.

iii

Sprague, Michael Alan (Ph.D., Mechanical Engineering)

Advanced Computational Techniques for the Analysis of 3-D Fluid-Structure Interaction

with Cavitation

Thesis directed by Prof. Thomas L. Geers

In an underwater-shock environment, cavitation, i.e., boiling, occurs as a result of reflection of

the shock wave from the free surface and/or wetted structure that causes the pressure in the water to

fall below its vapor pressure. If the explosion is sufficiently distant from the structure, the motion of

the fluid surrounding the structure may be assumed small, which allows linearization of the governing

fluid equations. Felippa and DeRuntz (1984) developed the cavitating acoustic finite element (CAFE)

method for modeling this phenomenon. While their approach is robust, it is too computationally

expensive for realistic 3-D simulations. In the work reported here, the efficiency and flexibility of the

CAFE approach has been greatly improved by: (i) separating the total field into equilibrium, incident,

and scattered components, (ii) replacing the bilinear CAFE basis functions with high-order Legendre-

polynomial basis functions, which produces a cavitating acoustic spectral element (CASE) formulation,

(iii) introducing a simple, non-conformal coupling method for the structure and fluid finite-element

models, and (iv) introducing structure-fluid time-step subcycling. Field separation provides flexibility,

as it allows the incorporation of non-acoustic incident fields, and propagates incident waves through

the mesh with total fidelity. The use of CASE admits a significant reduction in the number of fluid

degrees-of-freedom required to reach a given level of accuracy. The combined use of subcycling and non-

conformal coupling affords order-of-magnitude savings in computational effort. The benefits provided

by these improvements are illustrated with 1-D and 3-D canonical underwater-shock problems.

Dedication

To my wife, Hollie Sprague.

v

Acknowledgements

First and foremost, I thank my advisor, Prof. Thomas L. Geers. Under his guidance and demand

for excellence, I have become a better researcher and a better writer. He has shown amazing patience,

and has willingly suffered countless interuptions due to my visits to his office. Further, I have enjoyed

his friendship and our many discussions on politics, teaching, and the dangers of rock climbing. I

express my thanks to Prof. Carlos Felippa for sharing his knowledge of the finite-element method, and

to my other committee members: Prof. Martin Dunn, Prof. Charbel Farhat, and Prof. Bengt Fornberg.

I thank Tom Littlewood of ETC for his comments on my dissertation and for answering my questions

regarding finite elements. I also express my thanks to my undergraduate advisor, Prof. Roxann

Engelstad, whose early guidance gave me a huge advantage as a new graduate student.

This research was funded by the Office of Naval Research under Grant N00014-01-1-0154. I ex-

tend my appreciation to Dr. Luise Couchman and Mr. Stephen Schreppler of ONR and to Mr. Michael

Riley of NSWCCD for their interest and support.

I thank Kendall Hunter for his endless patience with my computer-based questions and our many

discussions regarding all facets of our lives during our stay at UCB, both academic and personal.

Finally, I extend my deepest appreciation to my wife, Hollie Sprague, and to my parents, Brian

and Sandra Sprague, for their faith, support, and encouragement throughout my undergraduate and

graduate career.

vi

Contents

Chapter

1 Introduction and Literature Review 1

1.1 Underwater Shock Problems and Cavitation . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Cavitating Acoustic Finite Elements (CAFE) . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 CAFE Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.2 Deficiencies in the Current CAFE Approach . . . . . . . . . . . . . . . . . . . . 3

1.3 Alternative Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3.1 Cavitation-Free Acoustic Domains . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3.2 Acoustic Domains Subject to Cavitation . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Improvements to the CAFE Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4.1 Cavitating Acoustic Spectral Elements . . . . . . . . . . . . . . . . . . . . . . . 6

1.4.2 Fluid-Structure Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4.3 Field Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4.4 Fluid-Structure Subcycling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4.5 Mesh Truncation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.5 Dissertation Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Formulation and Implementation 10

2.1 Continuum Fluid Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.1 Total-Field Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.2 Scattered-Field Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Spatial Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.1 Cavitating Acoustic Spectral Elements: Total-Field Model . . . . . . . . . . . . 14

2.2.2 Cavitating Acoustic Spectral Elements: Scattered-Field Model . . . . . . . . . 17

2.2.3 Cavitating Acoustic Finite Elements . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2.4 Structure FE Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

vii

2.2.5 Non-Reflecting Boundary (NRB) . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3 FE/Fluid-Volume/NRB Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3.1 Structure Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3.2 Fluid Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4 Temporal Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4.1 Staggered Integration with Subcycling . . . . . . . . . . . . . . . . . . . . . . . 23

2.4.2 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.5 Error Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 1-D Evaluation: 2-DOF Floating Spring-Mass Oscillator 27

3.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2 Discrete Model and Benchmark Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.3 CAFE and CASE Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.4 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4 3-D Evaluation: Submerged Spherical Shell 39

4.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2 Structure FE Model Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.3 Results and Discussion: No Cavitation . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.3.1 Benchmark FSI Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.3.2 CASE and CAFE with the TFM and SFM . . . . . . . . . . . . . . . . . . . . 44

4.3.3 CASE and CAFE and Time-Step Subcycling . . . . . . . . . . . . . . . . . . . 50

4.4 Results and Discussion: Cavitation at 150 m . . . . . . . . . . . . . . . . . . . . . . . 52

4.5 Results and Discussion: Cavitation at 50 m . . . . . . . . . . . . . . . . . . . . . . . . 55

4.5.1 Cavitation and Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.5.2 CAFE with Reduced Integration . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.6 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5 3-D Evaluation: Ship Structure 66

5.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.2 Discrete Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.4 Mesh Truncation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6 Conclusion 82

viii

Bibliography 84

Appendix

A CASE: Element-Level Matrices and Matrix-Vector Products 91

A.1 Explicit Matrix Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

A.2 Matrix-Vector Product: Tensor-Product Factorization . . . . . . . . . . . . . . . . . . 92

B Temporal Discretization 94

B.1 Total-Field Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

B.2 Scattered-Field Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

B.3 CWA Pressure Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

Chapter 1

Introduction and Literature Review

1.1 Underwater Shock Problems and Cavitation

In the context of fluid-structure interaction (FSI) and underwater shock, cavitation, i.e., boiling,

occurs as a result of the reflection of the shock wave from the free surface and/or wetted structure,

which causes the pressure in the water to fall below its vapor pressure [66]. The instantaneous bulk

modulus of cavitated water is orders of magnitude smaller than that of uncavitated water, which

produces nonlinear behavior. A simple but satisfactory constitutive model is a bilinear one, in which

the bulk modulus is that of water as an acoustic medium when the condensation (roughly, negative

volume strain) is positive, and zero when the condensation is negative [12]. Although pure water has

been found to sustain substantial negative pressures, even small amounts of dissolved gases nullify the

effect [32].

Near-free-surface underwater-shock problems are further complicated by the existence of three

greatly different spatial scales: the size of the ship, the decay length of the incident wave, and the spatial

extent of the cavitation. This is best illustrated by an example: a 45 kg charge of HBX-1 detonated

at a depth of 15 m yields a cavitation region with maximum dimensions of 489 m horizontally and 9

m in depth [25]. This region is shown with a representative ship (aft view) in Fig. 1.1. The associated

decay length of the shock wave when it reaches the ship is 0.9 m [22, 90].

-200 -100 0 100 200-20-10

045 kg HBX-1

Ship

Dep

th(m

)

Distance from Charge (m)

Figure 1.1: Cavitation-region envelope for a 45 kg charge of HBX-1 detonated at a 15 m depth. Alsoshown is the aft view of a box representation of a ship with 9.1 m beam, 4.6 m height, and 3.1 m draft.

2

1.2 Cavitating Acoustic Finite Elements (CAFE)

1.2.1 CAFE Approach

When an underwater explosion is sufficiently removed from the structure, the motion of the

fluid surrounding the structure is small, which allows the fluid to be treated as acoustic but subject to

cavitation. In 1984, Felippa and DeRuntz [36] developed a cavitating acoustic finite element (CAFE)

for FSI calculations based on the work of Newton [82, 83, 84, 85], in which the wave field in the fluid

is represented by a scalar displacement potential. Trilinear, isoparametric, eight-node brick elements

were formulated; a six-node wedge element was added later [37]. The CAFE semi-discrete equations

were integrated in time with a central-difference algorithm. Cavitation was treated node-by-node in

an on/off fashion in the time-update equations. Numerical damping was employed to suppress the

occurrence of spurious cavitation, which they called frothing ; the damping also has a beneficial effect

on structure responses in an acoustic fluid, as oscillatory numerical artifacts are smoothed. Their

choice of explicit time integration over unconditionally stable implicit integration seems appropriate

because the time scales associated with the incident wave and cavitation are typically very small [103];

an implicit scheme, which is inherently more costly for each time step, would still require very small

time increments for resolution of the physical effects.

The CAFE solution strategy for a ship-shock simulation consists of four steps (Fig. 1.2): (i)

construct a finite-element (FE) model of the structure, (ii) interface it with a CAFE model of the fluid

region in which cavitation is expected to occur, (iii) enclose the CAFE mesh with a non-reflecting

boundary (NRB), and (iv) start integrating the CAFE equations just before the incident wavefront

reaches either the free surface of the fluid or the wet surface of the structure [36, 99].

Non-Reflecting Boundary

Fluid CAFEModel

Incident Wavefront

Free Surface

Structure FE Model

Figure 1.2: FE/CAFE/NRB approach for ship-shock simulations; incident wavefront shown just beforethe onset of cavitation.

The CAFE elements were first implemented in the Cavitating Fluid Analysis (CFA) code [36],

3

with the structure equations being handled by the Structural Analysis of General Shells (STAGS)

[2] finite-element program. The fluid mesh was truncated by a first-order doubly asymptotic ap-

proximation (DAA) [42, 43], as implemented in the Underwater Shock Analysis (USA) code [27]. A

staggered-time-integration procedure [38] was used, in which the structure equations were integrated

with an implicit trapezoidal-rule scheme. A similar approach is used in the NASTRAN-CFA-DAA

code [80, 99]. The CAFE approach has been implemented in LS-DYNA/USA [27, 57] (the MAT90

element formulation), where the fluid and structure semi-discrete equations are integrated explicitly,

and the fluid mesh is bounded by a DAA or plane-wave-approximation (PWA) boundary. A similar

CAFE formulation is currently being incorporated into ABAQUS [59, 21], where the fluid domain is

truncated by a curved-wave-approximation (CWA) boundary.

For problems in which the underwater explosion is near or contacting the structure, the fluid

motion can no longer be considered small. In this case, a hydrocode [73] must be used, which, compared

to CAFE, is an extremely costly method as the nonlinear fluid-motion equations are solved.

1.2.2 Deficiencies in the Current CAFE Approach

Although the approach of Felippa and DeRuntz [36] is well developed, it has deficiencies that

make it too computationally expensive for accurate 3-D simulations. First, CAFE exhibit high nu-

merical dispersion due to their use of low-order basis functions [76]; high mesh refinement is typically

required. When cavitation, a phenomenon that is highly local in space and time, occurs, the refine-

ment requirement is extreme. This need for a “super-refined” mesh is demonstrated for 1-D and 2-D

cavitating problems by Sprague and Geers [103] and by Sprague [101].

As demonstrated in Sprague and Geers [103], integrating CAFE equations with a time increment

substantially smaller than the critical time increment for stable integration produces spurious regions

of cavitation. In an explicit-fluid/explicit-structure implementation with one-to-one fluid-structure

nodal coupling, the critical time increment for the structure equations is typically much smaller than

that required for the fluid equations as the sound speed in steel is more than three times that in

water. In a CAFE/FE implementation where the structure and fluid equations are integrated with

the same time increment [36, 57], integrating the fluid equations with a too-small time increment is

often unavoidable.

The CFA [36] and LS-DYNA [57] implementations feature one-to-one coupling between the

fluid and structural nodes on their interface. Thus, fluid-mesh refinement requires structure-mesh

refinement; if the structure is already adequately refined, this constitutes a waste of computer resources.

Furthermore, if the CAFE implementation employs explicit-fluid/explicit-structure time integration,

4

the structure-FE critical time increment is also reduced with mesh refinement. In the ABAQUS

implementation [59], the requirement for one-to-one structure-fluid nodal coupling has been removed

by a sophisticated membrane-coupling approach that allows an arbitrary CAFE mesh to be coupled

to an arbitrary structure mesh.

For accurate simulation, Felippa and DeRuntz [36] recommend that the CAFE mesh encompass

all regions of fluid that experience cavitation. However, as illustrated in Fig. 1.1, this is an impractical

recommendation for many near-free-surface shock problems because the size of the cavitation region

is extremely large relative to the decay length of the incident wave. Thus, a CAFE mesh that is

adequately refined to model the incident wave that also encompasses the entire cavitation region

would be enormous.

Finally, Felippa and DeRuntz [36] recommend that time integration begin before the incident

wavefront touches the structure or causes regions of cavitation resulting in the example initial con-

dition illustrated in Fig. 1.2. Here, the incident wavefront (which is usually discontinuous) must be

propagated through an appreciable portion of the dispersive mesh before reaching the structure; the

wave will therefore be deformed when it reaches the structure.

1.3 Alternative Methods

1.3.1 Cavitation-Free Acoustic Domains

There are several well-developed computational methods for treating underwater shock problems

in a deep-ocean environment where cavitation is absent and the fluid domain may be modeled as an

infinite homogeneous acoustic domain. Typically, the structure (and possibly the near-structure fluid)

in such a domain is modeled with finite elements and the effect of the infinite acoustic domain is

modeled with boundary elements (BEs) or infinite elements (IEs) that are coupled to the FE-model’s

wet-surface.

One BE formulation that has demonstrated much success is the doubly asymptotic approxima-

tion (DAA) [42, 43, 50, 51, 49]. The formulation is doubly asymptotic in that the solution approaches

exactness at early and late time. First-, second-, and third-order formulations have been developed;

the former two have seen extensive use by the underwater shock community [92, 5, 27, 69]. A funda-

mental limitation of the DAA is in convergence studies. Unlike the FE approach, the DAA converges

with both spatial refinement and increase in DAA order. However, implementation of a DAA with

order greater than two is exceedingly difficult [49]. Thus, the approach is not suitable for benchmark

solutions and convergence studies.

5

Other BE formulations have been developed based on one-sided asymptotic approximations: the

early-time approximations (ETA) [35, 45] and the late-time approximation (LTA) [20]. The ETA in

its first-order form is the plane-wave approximation [77], and in its second-order form is the curved-

wave approximation [58, 6]. When using such BEs for long-duration problems, the added-mass effects

[28] associated with low-frequency incompressible sloshing motion in the near-structure fluid must be

treated with a fluid FE model. The LTA is limited as it only treats added-mass effects.

An exact BE formulation may be developed with Kirchhoff’s retarded-potential formulation

(RPF). Several authors have implemented the RPF for scattering problems [67, 78, 98, 13] and FSI

problems [61, 33]. RPF carries the computational burden of being nonlocal in space and time, thus

requiring extensive data storage. In addition, the implementations for FSI problems have been plagued

by numerical instabilities [26, 33].

There has been extensive work done with infinite-element formulations [11]. These have seen

much success for steady-state problems [110, 4, 14, 16, 15] but have seen only limited application to

transient problems [3]. For transient problems, this approach requires that the structure be enclosed

in a fluid FE model bounded by a sphere or spheroid.

Finally, one may create a FE model of the structure and surrounding fluid that is bounded

by a sphere. A modal series solution for the external acoustic field may then be coupled to the FE

model, thereby creating what has become known as a Dirichlet-to-Neumann (DtN) boundary [65, 60].

This infinite-domain model is approximate in that only a finite number of modes may be employed,

whereas the true solution is an infinite series. This method is computationally very expensive when

one is modeling a long structure.

Besides the DAA, all of the above boundary methods will yield better structural results with

improvement of the near-structure-fluid FE model. Thus, while this dissertation is primarily focused

on the cavitation problem, any improvements to the wave-propagation capability of the fluid FE model

will benefit the numerical treatment of the acoustic shock problem as well.

1.3.2 Acoustic Domains Subject to Cavitation

As discussed earlier, underwater shock problems near a free surface are typically characterized by

the occurrence of cavitation. The methods discussed in Section 1.3.1 require a global representation of

the field, and are therefore inappropriate for treating cavitation, which is a highly local phenomenon.

The finite-element method, e.g. CAFE, is apparently the method best suited to this class of problems

because of its use of local basis functions.

As an alternative to a truncated CAFE mesh, various wet-surface approximations have been

6

proposed [31, 95, 93, 74]. These approximations apply a zero fluid pressure to the wet surface when

the fluid is cavitated and use the plane-wave or curved-wave radiation boundary otherwise. The

differences between methods lie in their determination of cavitation on/off times. Although wet-

surface approximations work well for heavy rigid structures floating on a 1-D fluid column, Sprague

and Geers [103] have revealed significant deficiencies in the approximations when the structure has

two degrees of freedom and the bulk of the mass resides in the dry portion of the structure. In

fact, for the 1-D cases studied, it was found that a single CAFE element with a plane-wave radiation

boundary yields results comparable in accuracy to those obtained with the best of the wet-surface

approximations studied.

Wet-surface approximations are deficient because they track data only at the structure’s wet sur-

face, whereas cavitation is a 3-D phenomenon. Therefore, current wet-surface approximations cannot

account for fluid accretion, i.e., a volume of uncavitated fluid separating the structure and a cavitated-

fluid volume. Sprague and Geers [103] have demonstrated that fluid accretion plays an important role

in structure response [103]. Further, the physical arguments for determining the cavitation-off times

are spurious at best. Thus, wet-surface-approximations cannot be used to check convergence, for which

a CAFE model must still be used.

1.4 Improvements to the CAFE Approach

In order to address the deficiencies of the CAFE approach, we incorporate several well-developed

methods that, taken together, promise to make the CAFE approach a viable treatment for 3-D near-

free-surface shock analysis. These improvements are:

• introduction of field separation,

• replacement of the trilinear basis functions (finite-element method) with Legendre-polynomial

spectral basis functions (spectral-element method),

• incorporation of a novel non-conformal fluid-structure coupling method,

• implementation of fluid-structure subcycling,

• exploitation of aggressive mesh truncation.

1.4.1 Cavitating Acoustic Spectral Elements

As discussed above, a fundamental limitation of low-order elements is the high dispersion seen

when applied to wave-propagation problems; this is exacerbated when the fields contain discontinuities,

7

which is typical of underwater-shock problems. Convergence is attained by brute-force refinement of

the mesh (h-refinement). With high-order schemes, convergence may be attained with a combination

of h-refinement and increasing the element order (p-refinement). High-order elements generate less

dispersion, but have traditionally been avoided for two reasons [23]: (i) they typically yield higher

maximum mesh eigenvalues, thus decreasing the critical time increment for explicit integration, and

(ii) they often produce more troublesome spurious oscillations when applied to wave-propagation

problems with solution discontinuities. However, the reduced dispersion can more than compensate

for the presence of the oscillations and reduced critical time increment [81]. This is clearly shown in

[53], where a numerical example is used to illustrate that information is contained in these oscillations

and high-order schemes retain more information than low-order schemes.

The most effective high-order elements appear to be Legendre-polynomial-based spectral ele-

ments [94, 71], which display impressive wave-propagation capabilities [68, 81]. The spectral-element

method (SEM) combines the accuracy of spectral methods [55, 86, 41, 17] with the geometric flexibility

of the finite-element method (FEM) [111]. For problems with regular domains and adequately smooth

solutions, the SEM enjoys exponential convergence; this is lost when the solution contains disconti-

nuities, which is the subject of several investigations [100, 53, 108, 54]. Finally, spectral methods are

well suited to parallel-CPU implementation [39].

The SEM was first introduced by Patera [88], who applied the method to the incompressible

Navier-Stokes equations in 1-D and 2-D. While most of the early work with SEM used spectral elements

(SEs) with Chebyshev-polynomial basis functions [52, 88, 87, 91, 97], Legendre polynomials [94, 71,

39, 72] have emerged as the preferred basis functions. This is because Legendre basis functions with

Gauss-Lobatto-Legendre quadrature

• allow fast matrix-vector-product evaluation with tensor-product factorization [71],

• produce diagonal mass matrices,

• yield matrices with better conditioning and sparsity when compared with those produced with

Chebyshev basis functions [68, 39],

• exhibit convergence rates comparable to those for Chebyshev polynomials.

In this dissertation, we apply the spectral-element method to two distinct applications: (i) wave

propagation in an acoustic medium subject to cavitation, and (ii) fluid-structure interaction. To

this end, we replace the trilinear basis functions of CAFE with Legendre-polynomial-based spectral

basis functions, thus yielding a cavitating acoustic spectral element (CASE). Further, we compare the

8

performance of CASE against CAFE in terms of required (i) CPU cost, (ii) memory storage, (iii) and

fluid-model degrees-of-freedom.

1.4.2 Fluid-Structure Coupling

The CAFE approach would be greatly improved if the fluid and structure meshes could be

refined separately. To this end, we propose a novel coupling method that is considerably simpler than

that used in the ABAQUS implementation [59]. With this method, the fluid may be refined to an

arbitrary level without requiring a change in the structure mesh.

While there are several methods for coupling two geometrically different meshes [34, 7], the

method used here, which may be classified as a consistent-interpolation coupling [34], is simple in

terms of implementation and usage. The gains associated with more complex coupling, which may be

“mathematically optimal”, are small and not worth the associated computational cost for the class of

problems studied here [34].

1.4.3 Field Separation

The far field produced by an acoustic shock wave in an unbounded fluid (the incident field) is

relatively easy to define [63] and is typically known in closed form. However, with CAFE, the incident

wave is propagated through the fluid mesh. This has two inherent problems:

• a discontinuous wave propagated through a spatially discrete mesh can become substantially

distorted,

• the incident wave cannot be propagated “quasi-acoustically”, as realistic shock waves do [47,

63].

In order to avoid propagating the incident wave through the mesh, we employ a technique used

frequently in linear scattering problems, which consists of separating the total field into a known

incident field and an unknown scattered field [79, 106]. The technique was applied to transient, FSI

FE calculations by Chan [18, 19] and is also implemented in ABAQUS [59]. It has also been applied

to transient, linear, finite-difference electrodynamics calculations [106]. With this technique, the fluid

mesh is used for scattered-field calculations only. To the author’s knowledge, field separation has never

been applied to a nonlinear field problem. We investigate the efficacy of field separation when applied

to FSI calculations in an acoustic fluid with and without the occurrence of cavitation.

9

1.4.4 Fluid-Structure Subcycling

Time-step subcycling [10, 8, 62] is a simple approach that is suitable for the CAFE/FE staggered

time-integration approach. It allows both the structure and fluid equations to be integrated with near-

optimal time increments. In this approach, the fluid equations are updated at some integer multiple

or division of the time increment for the structure equations. This is a valuable improvement, as it

allows gains in accuracy with considerable reduction in computational cost.

1.4.5 Mesh Truncation

Seeking to reduce CAFE overhead in 2-D near-free-surface simulations, van Aanhold et al.

[107] dramatically truncated the fluid mesh in the horizontal directions with considerable success.

In addition, Sprague and Geers [103] used 1-D models to show that the CAFE mesh may also be

truncated from below, provided that the mesh covers all regions that exhibit fluid accretion.

Recently, Malone and Shin [75] investigated the effect of mesh truncation on a CAFE mesh

surrounding a simple 3-D structural model of a barge excited by an underwater explosion. Their

results indicate that satisfactory structure-response results may be attained with the CAFE model

extending to at least half the maximum cavitation depth predicted by the methods of [25]. However,

there is no evidence that the fluid or structural model is adequately refined, nor is there any discussion

of fluid accretion.

An accurate 3-D study of mesh-truncation effects remains to be completed. This dissertation

addresses the issue with the optimized CASE approach discussed above.

1.5 Dissertation Outline

Chapter 2 discusses the formulation and implementation of the improvements discussed above.

In Chapter 3, a simple 1-D benchmark problem is used to evaluate the performance of CASE and

CAFE with the non-conformal fluid-structure coupling and field separation. Chapter 4 extends the

evaluation to the axisymmetric problem of a submerged spherical shell excited by spherical step-

exponential wave. Here, the performance of time-step subcycling is also evaluated. In Chapter 5,

the optimized approach is applied to the realistic problem of a surface ship excited by an underwater

explosion. The ship problem is also used to evaluate the efficacy of mesh truncation.

Chapter 2

Formulation and Implementation

In this chapter, we apply the improvements discussed in the previous chapter to the CAFE

approach of Felippa and DeRuntz [36] for underwater-shock FSI problems. The chapter begins with a

discussion of two continuum fluid models that are subsequently discretized with the spectral-element

and finite-element methods. Formulations for the structure FE model and non-reflecting boundary

are included for completeness. A novel fluid-structure-coupling approach is then introduced. The

chapter concludes with temporal discretization of the spatially discrete equations and discusses the

implementation of time-integration subcycling.

2.1 Continuum Fluid Models

We address the underwater shock problem illustrated in Fig. 1.2. We assume that fluid motion

is inviscid and irrotational, thus allowing a potential representation of the field. As discussed in the

Section 1.1, the fluid model uses a bilinear constitutive relation [12] to account for cavitation, which is

such that the fluid cannot transmit negative pressures and cavitation is treated as a macroscopically

homogeneous phenomenon.

In an underwater shock problem, the total field may be represented as the sum of three fields:

equilibrium, incident, and scattered. The equilibrium field is easily determined as that due to atmo-

spheric plus hydrostatic pressure, the incident field is less easily determined as that due to an acoustic

shock wave propagating in a homogeneous, unbounded fluid, and the complicated scattered field is

caused by the presence of both the structure and the free surface. This chapter introduces two models

for the field. We denote the first as the total-field model (TFM), as its dependent variables are com-

posed of incident and scattered quantities. We denote the second model as the scattered-field model

(SFM), as it treats only the scattered field on the assumption that the incident field is known at all

time throughout the domain.

11

2.1.1 Total-Field Model

In deriving the governing equations for the acoustic fluid subject to cavitation, we begin with

the bilinear equation of state [12],

p = pv +

Ba(ρ/ρ0 − 1),

0,

ρ ≥ ρ0,

ρ < ρ0,(2.1)

which is illustrated in Fig. 2.1. Here, Ba is the bulk modulus of the acoustic fluid in the absence of

cavitation, p( ~X, t) is the total pressure in the fluid, pv is the vapor pressure of water at temperature

T0, ρ0 is the corresponding density of the saturated liquid, and ρ( ~X, t) is the instantaneous density;

~X is the global-position vector of a fluid material point and t denotes time. We assume that T0, ρ0,

and pv are constant throughout the domain of interest. By definition, Ba = ρ0c2, where c is the sound

speed in the uncavitated acoustic fluid, and S = (ρ/ρ0−1) is the total condensation. These definitions

allow us to rewrite the equation of state as

p = pv +

ρ0c

2S,

0,

S ≥ 0,

S < 0.(2.2)

0(ρ− ρ0)/ρ0

p

Ba

pv

Figure 2.1: Bilinear equation of state for an acoustic fluid subject to cavitation.

The continuity equation may be written [1]

∂ρ

∂t+ ~∇ ·

(ρ∂~u

∂t

)= 0, (2.3)

where ~u( ~X, t) is the total fluid displacement. If |S| 1, (2.3) becomes

∂S

∂t+ ~∇ · ∂~u

∂t= 0. (2.4)

The Euler momentum equation [1], which is valid for inviscid flow, may be written

∂2~u

∂t2+

(∂~u

∂t· ~∇

)∂~u

∂t=

1ρ~∇p− ~∇Vg, (2.5)

12

where Vg( ~X) is the gravity-potential field. If |S| 1 and(

∂~u∂t · ~∇

)∂~u∂t ∂2~u

∂t2, (2.5) becomes

ρ0∂2~u

∂t2= −~∇p− ρ0

~∇Vg. (2.6)

Because fluid motion is assumed irrotational, we may write the total displacement field as

~u = −~∇Ψ + ~ueq, (2.7)

where Ψ( ~X, t) is the dynamic displacement potential and ~ueq( ~X) is the equilibrium displacement field.

We utilize (2.7) to write the continuity equation (2.4) as

∂S

∂t−∇2∂Ψ

∂t= 0, (2.8)

which may be integrated in time to yield

S = ∇2Ψ + Seq, (2.9)

where Seq( ~X) is the equilibrium condensation field. Similarly, the momentum equation (2.6) may be

written

~∇∂2Ψ∂t2

=1ρ0

~∇p+ ~∇Vg, (2.10)

which may be integrated in space to yield

∂2Ψ∂t2

=1ρ0p+ Vg + h, (2.11)

where h(t) is a constant of integration. If we consider the equilibrium problem, (2.11) becomes

∂2Ψ∂t2

=1ρ0

(p− peq), (2.12)

where peq = −ρ0(Vg + h) is the equilibrium pressure, i.e. the sum of atmospheric pressure and

hydrostatic pressure. The introduction of (2.2) into (2.12) then yields

ρ0∂2Ψ∂t2

= −peq + pv +

ρ0c

2S,

0,

S ≥ 0,

S < 0.(2.13)

Now, if we utilize peq = pv +ρ0c2Seq, define the dynamic densified condensation s = ρ0(S−Seq),

and the densified displacement potential ψ = ρ0Ψ, we may write from (2.9) and (2.13) the system of

governing equations

s = ∇2ψ, ψ =

c2s,

pv − peq,

c2s ≥ pv − peq,

c2s < pv − peq,(2.14)

where an overdot denotes a temporal partial derivative. The use of these scalar field quantities, as

opposed to vector quantities, minimizes the number of unknowns at each point in the fluid domain

13

and enforces irrotationality of the flow. The fluid domain governed by (2.14) has essential boundary

conditions at the free surface and natural boundary conditions along the interface with the structure

and the non-reflecting boundary.

The vapor pressure of the water is so much smaller than the equilibrium pressure that it may be

neglected. For example, at 10 C the vapor pressure of water is approximately 1.2 × 103 Pa, whereas

the minimum equilibrium pressure, i.e., atmospheric pressure, is 1.0 × 105 Pa.

In order to illustrate the numerical challenges associated with discrete representations of (2.14),

we eliminate ψ from those equations and rewrite them as

s = Ξ(s)∇2s, where Ξ(s) =

c2,

0,

c2s > −peq,

c2s ≤ −peq,(2.15)

where we have used used Laplace’s equation to eliminate peq, i.e., ∇2peq = 0. Equation (2.15) may

be classified as a quasilinear hyperbolic partial-differential equation [56]. As such, the domain is

susceptible to shock formation, a discontinuous phenomenon. For 1-D problems, (2.15) may be solved

with the method of characteristics [12]. However, for 2-D and 3-D domains, this approach becomes

intractable. We therefore discretize the fluid volume with finite elements. However, as discussed in

the introduction, solution discontinuities can have a strongly negative effect on the performance of

discrete numerical treatments.

2.1.2 Scattered-Field Model

Here, we separate the total field into three component fields: equilibrium, incident, and scattered.

Hence, we write s = sinc +ssc and ψ = ψinc +ψsc, where all of the variables constitute deviations from

equilibrium.

Introducing s = sinc + ssc and ψ = ψinc + ψsc into (2.14), we obtain the scattered-field model,

ssc = ∇2ψsc + ε1, ψsc =

c2ssc + ε2,

−(peq + pinc),

c2ssc > −(peq + pinc),

c2ssc ≤ −(peq + pinc),(2.16)

where pinc = ψinc is the incident pressure and

ε1 = ∇2ψinc − sinc, ε2 = c2sinc − ψinc. (2.17)

Because |sinc| 1 and the fluid motion is irrotational and inviscid, sinc = ∇2ψinc, which produces

ε1 = 0. If the incident field propagates acoustically, we may write from (2.14)

ψinc = c2sinc, (2.18)

14

which produces ε2 = 0. Note that the second of (2.16) involves a dynamic quantity on the right sides

of the inequalities, in contrast to the second of (2.14). In addition, the free-surface (FS) essential

boundary condition for the scattered-field equations is time dependent, i.e.,

psc|F S + pinc|F S = 0. (2.19)

An important advantage of the scattered-field model over the total-field model is the opportunity

to propagate any incident field through the fluid domain with total fidelity. For example, Hunter

and Geers [63] recently developed a realistic field model in which the shock wave propagates quasi-

acoustically, i.e., the wavefront propagates with the fluid sound speed, but the waveform slowly distorts

as it travels. Although incorporation of such a model violates the acoustic-field assumption in (2.18),

ε2 is negligible locally, and may be ignored.

2.2 Spatial Discretization

2.2.1 Cavitating Acoustic Spectral Elements: Total-Field Model

We seek to construct cavitating acoustic spectral elements (CASE) based on the irrotational-

motion formulation of Newton [82, 83, 84, 85]. A subparametric discretization is used; first-order

(trilinear) basis functions are used for geometry representation and higher-order basis functions are

used for field-variable representation. The finite fluid volume Ωfl is separated into ne hexagonal

elements defined by eight corner points. The geometry within each element is expressed as

X = ϕTX, Y = ϕTY, Z = ϕTZ, (2.20)

where X,Y,Z are column vectors of element-corner-point locations in global coordinates, ϕ is a column

vector of the standard trilinear shape functions (see, e.g., [23]), and a T superscript denotes vector

transposition.

The dependent field variables are represented within each element as

s(ξ, η, ζ, t) = φT (ξ, η, ζ)se(t) =N∑

i,j,k=0

φi(ξ)φj(η)φk(ζ)seijk(t),

ψ(ξ, η, ζ, t) = φT (ξ, η, ζ)ψe(t) =N∑

i,j,k=0

φi(ξ)φj(η)φk(ζ)ψeijk(t), (2.21)

where se and ψe are column vectors of (N + 1)3 time-dependent nodal values (seijk and ψe

ijk), and φ

is a column vector composed of 1-D, N th-order-polynomial basis functions φi(ξ)φj(η)φk(ζ); ξ, η, and

ζ are element natural coordinates (−1 ≤ ξ, η, ζ ≤ 1). The essence of the spectral-element method lies

15

in the choice of φi and the associated quadrature rule; here, we use Lagrangian interpolants given by

[94]

φi(ξ) = − (1 − ξ2)P ′N (ξ)

N(N + 1)PN (ξi)(ξ − ξi), (2.22)

where PN is the Legendre polynomial of degree N , the prime denotes differentiation with respect to

argument, and ξi is the ith Gauss-Lobatto-Legendre (GLL) quadrature point defined by the corre-

sponding root of

(1 − ξ2)P ′N(ξ) = 0. (2.23)

The expression (2.22) satisfies the relation

φi(ξj) = δij , (2.24)

where δij is the Kronecker delta. The 1-D Lagrangian interpolants (2.22) are shown in Fig. 2.2 for N

= 1, 4, 8, and 16. Element-node locations are coincident with the quadrature points, which are located

at the (N + 1) zeros of (2.23). Finally, the expressions for the derivative of (2.22) when evaluated at

GLL quadrature points may be written as [17]

∂φi

∂ξ

∣∣∣∣ξ=ξj

=

PN (ξj)PN (ξi)

1ξj−ξi

, i 6= j,

− (N+1)N4 , i = j = 0,

(N+1)N4 , i = j = N,

0, otherwise.

(2.25)

The governing equations for the TFM (2.14a) are discretized with a standard Galerkin approach

[111]: premultiplication of (2.14a) by φ, integration over the fluid volume, and application of Green’s

first identity. This yields ∫Ωe

φs dΩ +∫

Ωe

~∇φ · ~∇ψ dΩ =∫

Γe

φ~∇ψ · ~n dΓ, (2.26)

where Ωe is the element domain, Γe is its surface, and ~n( ~X) is the outward-normal vector to Γe.

Substitution of (2.21) into the dependent variables on the left side of (2.26) yields the element-level

algebraic equations

Qese + Heψe = be, (2.27)

where the capacitance matrix, reactance matrix, and boundary-interaction vector are given by

Qe =∫

Ωe

φφT dΩ, He =∫

Ωe

~∇φ · ~∇φT dΩ, be =∫

Γe

φ~∇ψ · ~n dΓ, (2.28)

respectively. Both Qe and He are symmetric. Note that in be, ~∇ψ is maintained in its continuum

form because it is provided by the displacements at the non-reflecting and structure boundaries.

16

-0

0

1

11ξ

(a) N = 1

-

0

0

1

11ξ

(b) N = 4

-

0

0

1

11ξ

(c) N = 8

-

0

0

1

11ξ

(d) N = 16

Figure 2.2: 1-D Lagrangian interpolants based on Legendre polynomials for N = 1, 4, 8, and 16.

17

The element-level systems may be assembled into a global system; the resulting semi-discrete

equations are

Qs + Hψ = b. (2.29)

where Q and H are the global matrices corresponding to Qe and He, respectively, and b, s, and ψ are

the global vectors corresponding to be, se, and ψe, respectively. The bilinear constitutive equations

(2.14b), when evaluated at node j, become

ψj =

c2sj ,

−peqj ,

c2sj > −peqj ,

c2sj ≤ −peqj ,(2.30)

where peq is a column vector of nodal equilibrium pressures.

Appendix A.1 contains a detailed account of the procedure for forming the element-level matrices

described in (2.28); the integrals of which are approximated with GLL quadrature. Because nodes

and quadrature points are coincident, and because of (2.24), Qe is diagonal, which facilitates the use

of the explicit time-integration scheme used here. For this study, the capacitance matrix is always

formed and stored in a global diagonal matrix. In the explicit scheme (discussed below), the vast

majority of computation time is devoted to evaluation of the matrix-vector product Hψ, which must be

evaluated at each time step. For Legendre-polynomial spectral elements, it is most efficient to evaluate

the matrix-vector product element-by-element using tensor-product factorizations [71, 39, 30, 40], of

which a detailed description is given in Appendix A.2. In addition to fast evaluation of the matrix-

vector product, tensor-product factorization removes the memory requirements associated with storing

H and/or He, which can be excessive due to the high inter-element nodal coupling. We postpone

discussing the formation of be until Section 2.3.2.

2.2.2 Cavitating Acoustic Spectral Elements: Scattered-Field Model

We spatially discretize (2.16a) with a standard Galerkin approach as described in the previous

section; the resulting scattered-field global CASE equations are

Qssc + Hψsc = bsc,

ψscj =

c2sscj ,

−peq + pincj ,

c2sscj > −peq + pincj ,

c2sscj ≤ −peq + pincj ,(2.31)

where Q and H are formed as discussed in Section 2.2.1, pinc is a column vector of known nodal

incident pressures, and the scattered boundary-interaction vector bsc is composed of element-level

vectors defined by

besc =

∫Γe

φ~∇ψsc · ~n dΓ. (2.32)

18

Numerical implementation of the above scheme is identical to that discussed in Section 2.2.1.

2.2.3 Cavitating Acoustic Finite Elements

We use the above CASE formulation to develop a CAFE formulation similar to that of Felippa

and DeRuntz [36]. This is achieved by replacing the Lagrangian interpolants (2.22) with a series of

bilinear functions, as illustrated in Fig. 2.3 for a 1-D element with endpoints at X0 and X1, and N = 4

CAFE refinement. This approach is equivalent to separating the original fluid element into multiple

CAFE. The basis functions and the natural coordinates for a single sub element are illustrated by

the solid lines in Fig. 2.3. Standard 8-point Gauss integration is used for evaluation of 3-D element-

level matrices (2.28). This approach produces a non-diagonal element-level capacitance matrix Qe.

Following the approach of Felippa and DeRuntz [36], we diagonalize Qe with row-sum lumping. The

semi-discrete equations appear identical to (2.29) and (2.31a) for the TFM and SFM, respectively.

0

1

X0 X1ξ = −1 ξ = 1

Figure 2.3: Basis functions for N = 4 CAFE refinement of a 1-D element with endpoints at X0 andX1. The natural coordinates (−1 ≤ ξ ≤ 1) pertain to the basis functions shown with the solid lines.This is the CAFE equivalent to Fig. 2.2b.

Henceforth, we use N as a spatial-refinement parameter. For CASE refinement, N denotes

the polynomial order of the basis functions; for CAFE refinement, N denotes the number of element

divisions applied to the base fluid element in each of the natural-coordinate directions. For either

method, N refinement fills a base fluid element with (N + 1)3 nodes.

As with the CASE model, the diagonal global capacitance matrix is formed and the most expen-

sive part of evaluating the time-update equations lies in the evaluation of Hψ. However, unlike CASE,

because of reduced nodal coupling and the lack of a tensor-product factorization, the most efficient

method for evaluation is achieved with a global matrix-vector product. The matrix H for CAFE is

created by forming the element-level matrices and storing them globally with a sparse-storage scheme

19

in which only nonzero entries are stored.

2.2.4 Structure FE Equations

For the purposes of this study, the structural finite-element model is limited to 4-node isopara-

metric quadrilateral shell elements and 2-node beam elements. The well-known semi-discrete FE

equations for linear, undamped structural motion (relative to equilibrium) at the element level may

be written as

Mexe + Kexe = f e, (2.33)

where Me and Ke are the symmetric consistent-mass and stiffness matrices, respectively, and xe and

f e are column vectors of nodal displacements and nodal forces, respectively.

For this study, the element-level matrices Me and Ke are generated with ABAQUS [59]. For

the shell elements, the fully integrated S4 quadrilateral element is used. This element has a consistent

mass matrix, six degrees of freedom at each node, and accounts for membrane, bending, and shear

stiffness, as well as rotary inertia. For the beam elements, the Timoshenko B31 element is used, which

uses a lumped-mass formulation. For the explicit time-integration scheme used here, evaluation of

(2.33) is simplified and accuracy is improved with the use of a diagonal lumped-mass matrix, Me. For

the shell elements, this is formed from Me with an HRZ lumping scheme [23].

The nodal-force vector f e may be decomposed as

f e = f edry + f e

wet (2.34)

where f edry and f e

wet are due to forcing on the dry and wet portions of the structure, respectively. In

this study, f edry = 0. Details regarding the evaluation of f e = f e

wet are discussed in Section 2.3.1.

Element-level matrices are incorporated into global matrices and are stored using a sparse-

storage scheme in which only non-zero terms are stored. The resulting semi-discrete global equations

are

Mx + Kx = f , (2.35)

where M and K are the global matrices corresponding to Me and Ke, respectively, and f and x are

the global vectors corresponding to f e and xe, respectively.

2.2.5 Non-Reflecting Boundary (NRB)

For this study, the non-reflecting boundary is the curved-wave approximation (CWA), which is

an early-time approximation [35] in that it approaches exactness as t→ 0. More advanced NRBs may

20

be used; Felippa and DeRuntz [36] truncated their CAFE mesh with a first-order DAA boundary.

While the DAA is much better the CWA because it treats added-mass effects, it is expensive in terms

of memory storage. In addition, the complications of the DAA are seen to be unnecessary, as a CAFE

or CASE mesh handles most of the added-mass effects in near-free-surface FSI calculations. In this

section, we introduce CWA formulations for both the TFM and SFM.

CWA: Total-Field Model

For the total-field model, we require the NRB to be transparent to incoming incident waves.

The boundary interaction vector (2.28c) is driven at the NRB by ~∇ψ · ~n. This dynamic quantity may

be separated into scattered and incident fields. Using the definition of ψ, we write

~∇ψ · ~n = −ρ0(u− ueq) = −ρ0(uinc + usc), (2.36)

where u = ~u · ~n, ueq = ~ueq · ~n, and uinc and usc denote the magnitudes of incident and scattered fluid

displacements, respectively, normal to the NRB surface (positive going out of Ωfl).

For the TFM, we restrict the incident field to an acoustic spherical wave with center ~Xc that is

expanding radially at the fluid’s sound speed. At t = 0, the wavefront has radius R0. For such waves,

an exact displacement-pressure relation at a fluid-boundary point ~X is [36]

uinc =(

1ρ0c

∗pinc +

1ρ0R

∗∗p inc

)~ξ · ~n (2.37)

where pinc is known, R = | ~X − ~Xc|, ~ξ = | ~X − ~Xc|/R, and an asterisk denotes temporal integration

from t = −R0/c through t. For the scattered field, the CWA may be written for the surface-normal

displacement at ~X as [58, 35]

usc =1ρ0c

∗psc +

κ

ρ0

∗∗p sc, (2.38)

where κ( ~X) is the mean curvature of the CWA surface, which is defined as positive for a convex

surface, and the constants of integration for∗psc and

∗∗p sc are determined from initial conditions. We

separate the total pressure into its three components, p = peq + pinc + psc, where peq and pinc are

known, and p is output from the fluid-volume mesh. We solve for psc and insert into (2.38). The result

may be combined with (2.36) and (2.37)to yield

u− ueq =(

1ρ0c

∗pinc +

1ρ0R

∗∗p inc

)~R · ~n+

1ρ0c

(∗p− ∗

peq −∗pinc

)+κ

ρ0

(∗∗p − ∗∗

p eq −∗∗p inc

)(2.39)

The above relation may be written in discrete form for nodes on the NRB as

u− ueq =(

1ρ0c

Θ∗pinc +

1ρ0

ΘR−1∗∗pinc

)+

1ρ0c

( ∗p− ∗

peq −∗pinc

)+

1ρ0κ

(∗∗p − ∗∗

peq −∗∗pinc

), (2.40)

21

where u, ueq, p are column vectors of nodal values corresponding to u, ueq, and p, respectively, Θ is

a diagonal matrix of the cosines of the angles between the incident-wave propagation direction ~ξ and

the surface normals at the boundary nodes, R is a diagonal matrix of distances between CWA surface

nodes and ~Xc, and κ is a diagonal matrix of NRB mean curvatures at surface nodes.

CWA: Scattered-Field Model

For the scattered-field model, the NRB is simplified as it only treats the scattered field; the

incident field is treated externally. Thus, the scattered-field boundary interaction vector along the

NRB is driven by ~∇ψsc · ~n = −ρ0~usc · ~n, where the scattered-displacement CWA, given in (2.38), may

be written for NRB nodes as

usc =1ρ0c

∗psc +

1ρ0κ∗∗psc. (2.41)

Here, psc and usc are column vectors of scattered pressures and scattered displacements normal to the

NRB surface, respectively. For the TFM, the center of the incident wave must lie outside the NRB

because (2.37) is singular at that point, but with the SFM, it can occur within the NRB.

2.3 FE/Fluid-Volume/NRB Coupling

With the above formulation, we have three separate semi-discrete models that interact along

their interfaces: structure, fluid-volume, and NRB. Displacements are output from the structure FE

model and NRB to the fluid-volume mesh, from which pressures are sent back [35]. This section

discusses the specifics of numerical implementation for the coupling of the three discrete models. Most

importantly, we introduce a novel technique for coupling the fluid to the structure.

As discussed in the introduction, we wish to couple the structure and fluid models in a manner

such that an arbitrary level of fluid refinement is allowed. To this end, we approach the problem

as follows: (i) create a “sufficiently refined” structure FE mesh composed of quadrilateral elements,

(ii) create a fluid-volume base mesh consisting of hexagonal elements whose quadrilateral faces are

coincident with structure-element faces along the fluid-structure interface. With this approach, a

coupling scheme may be constructed such that the base fluid mesh may be refined indefinitely with p

(CASE) or h (CAFE) refinement without changing the structure mesh. This is illustrated in Fig. 2.4

for N = 4 refinement, where we have maintained a one-to-one coupling between fluid and NRB nodes.

With this method, the fluid and structure models are separated by a membrane composed of

quadrilaterals. The fluid pressure may be numerically integrated over this surface and applied to the

structure nodes. Likewise, the structure displacement at any point on the quadrilateral may be found

by linear interpolation of the displacements at the corner points for output to the fluid mesh.

22

Structure FE

Base Finite Element(CAFE with N = 1)

Fluid-Mesh Refinement

h Refinement(CAFE with N = 4)

p Refinement(CASE with N = 4)

NRB

Structure FE NodesBase-Mesh Fluid NodesRefined-Mesh Fluid NodesNRB Nodes

Figure 2.4: Fluid-refinement approaches and fluid-structure coupling, shown for a single hexagonalfluid element.

2.3.1 Structure Forcing

Structure forcing due to the fluid action occurs as a pressure normal to the shell surface; there

is no forcing on structure rotational DOF. For the finite-element models used in this study, forcing

occurs only on the surfaces of wet quadrilateral shell elements. The element-level structure-FE forcing

vector f e is assembled from three 4-component (4-node) vectors f eX , f e

Y , and f eZ , each corresponding to

one of the three global Cartesian directions; again, the components of f e for rotational DOF are zero.

The vector for i-direction forcing is evaluated as

f ei = γe

i

∫Γwet

NpedyndΓ, i ∈ X,Y,Z, (2.42)

where N is a column vector of the standard bilinear shape functions [23], pedyn is the average of

the dynamic pressure (p − peq) over the element face, and γei is the cosine of the angle between

the structure-element normal (positive going out of the fluid) and the ith global Cartesian direction.

Numerical experiments comparing the use of pedyn instead of pe

dyn(ξ, η) have been performed, where ξ

and η are the natural coordinates over the shell surface. For the cases studied, the use of the constant

pressure yields more accurate structure responses due to reduced spurious ringing. A similar pressure-

averaging approach for fluid-structure coupling has been used effectively for over 20 years in the USA

code [29].

23

2.3.2 Fluid Forcing

The fluid CAFE or CASE model is coupled at every boundary point to either a known-pressure

boundary (free surface), a flexible structure, a non-reflecting boundary, or a symmetry plane. At a

known-pressure boundary, all dependent variables are known and these quantities are forced in the

time-update equations. For symmetry planes, the fluid displacements normal to the plane must vanish,

i.e., −ρ0~∇ψ ·~n = 0, where ~n is the normal vector of the plane. As this is a natural boundary condition,

ψ is left free along symmetry planes. Forcing at the structure-fluid and fluid-NRB interfaces occurs

through the boundary-interaction vector.

For the total-field fluid model, the boundary interaction vector (2.28c) is evaluated as

be = −ρ0

∫Γe

st

φxedΓ − ρ0

∫Γe

nrb

φφT (ue − ueeq)dΓ, (2.43)

where xe is the average normal structure displacement (positive going out of the fluid) at the center of

the wet-structure element, Γest is the quadrilateral area separating the wet-structure element from the

fluid element, and Γenrb is the quadrilateral area of the NRB. The first integral in (2.43) is evaluated in

closed form; the second is evaluated with GLL quadrature for CASE refinement and trapezoidal-rule

integration for CAFE refinement.

For the SFM, the boundary-interaction vector (2.32) is evaluated as

besc = −ρ0

∫Γst

φxescdΓ − ρ0

∫Γnrb

φφTuescdΓ, (2.44)

where xesc is the average scattered normal displacement (positive going into the fluid) at the center

of the wet-structure element; it is formed by subtracting the known incident displacement from each

node’s total displacement, projecting the resulting scattered displacement onto the normal of the

element, and finding the average over the element’s face.

2.4 Temporal Discretization

2.4.1 Staggered Integration with Subcycling

The semi-discrete fluid and structure equations are both integrated with a conditionally stable

explicit central-difference algorithm. Integration of the fluid and NRB equations follow the method of

[36], where the fluid equations are integrated with numerical damping proportional to s in order to

suppress frothing. Unlike previous implementations, we incorporate structure and fluid time-step sub-

cycling, which allows for efficient and accurate integration of the system equations. With subcycling,

we allow both the fluid and structure to be integrated with time increments close to those allowed

24

by their respective stability criteria. Below, we give a qualitative description of our time-integration

procedure. A detailed description is included in Appendix B. The implementation is the same for both

CASE and CAFE.

We begin by assuming ∆tfl

∆t = α ≥ 1, where ∆tfl and ∆t are the fluid and structure time

increments, respectively. As an example, the time-update procedure for α = 2 is illustrated in Fig. 2.5a.

Here, we assume that the state at all system nodes is known at time t (and at the previous time step)

and we wish to update the values at all nodes to time t+ α∆t.

The surface-normal displacements at all nodes on the wet structure and on the NRB are predicted

at time t+α∆t using a simple Euler scheme, i.e., f(t+∆t) = f(t)+∆tf(t) (step 1). These predicted

values are denoted by the circled × and circled square, respectively. With these predicted values, the

fluid equations are updated with central differences to t = α∆t = 2∆t (step 2). The updated fluid

values are then used to correct the normal displacements of the NRB nodes (step 3). With the fluid

state known at t + α∆t, the nodal pressures on the structure-fluid interface at t + ∆t are calculated

by interpolation and the structure equations are updated with central differences (step 4). This is

repeated until the state of all structure nodes is known at t+ α∆t (step 5).

For α < 1, the fluid equations are subcycled rather than those of the structure, as illustrated

in Fig. 2.5b for α = 12 . This procedure is very similar to that for α ≥ 1, however, the structure state

must be predicted for each fluid update.

2.4.2 Stability

Felippa and DeRuntz [36] performed a detailed stability analysis of the integration of the coupled

FE/CAFE/NRB approach. Because of the complexity of our system, we restrict ourselves to a stability

analysis of the uncoupled fluid equations. We begin by assuming that we have a conservative estimate

for the structure critical ∆t and we use the results of [36] for our stability analysis of the discrete fluid

equations. For the fluid equations, an upper bound for the the critical time increment is

∆tcr =2

c√λmax (1 + 2β)

, (2.45)

where β is the artificial-damping constant, and λmax is the maximum eigenvalue of the global gener-

alized eigenproblem

(H − λQ) z = 0, (2.46)

where z is the eigenvector associated with eigenvalue λ. The global λmax may be bounded by the

largest eigenvalue of the unassembled element-level matrices [23], i.e., λmax = max(λemax), where λe

max

25

1

1

1

22 2

2

2

3

2

4

4

3

44 4

4

55 5

5

5

Structure

Structure (predicted)

Fluid NRB

NRB (predicted)

(a)t

t+ ∆t

t+ 2∆t

4

5

55 5

5

5

112

222

2

1

2

3

3

4 4 6

67

7

77

7

t

t+ ∆t2

t+ ∆t

(b)

Figure 2.5: Information transfers for explicit-structure/explicit-fluid integration with (a) structuresubcycling (α = 2) and (b) fluid subcycling (α = 1/2). The circled numbers in (a) correspond to thestep numbers discussed in the text, and ∆t is the structure time increment.

is the maximum eigenvalue of the element-level eigenproblem

(He − λeQe) ze = 0. (2.47)

For the CAFE implementation, λmax may be bounded above by 4/∆x2min [9, 36] where ∆xmin

is the minimum distance between nodes in the fluid mesh. With this bound, the CAFE stability limit

may be written as

∆tcflcr =

∆tcfl√1 + 2β

, (2.48)

where ∆tcfl = ∆xmin/c is the Courant-Freidrichs-Lewy (CFL) [23] increment limit.

For CASE discretization, a simple upper-bound estimate on λmax is not available. However, we

may form the element-level matrices as discussed in Appendix A.1 and use Gerschgorin’s theorem to

obtain an upper bound on λmax. Given the generalized Eigenproblem (2.47), Gerschgorin’s theorem

26

[23, 64] states that

λemax ≤ λe,g

max = max

He

ii +(N+1)3∑j=1,j 6=i

∣∣Heij

∣∣ /Qe

ii

, ∀i ∈ 1, . . . , (N + 1)3, (2.49)

where Heij and Qe

ij are elements of He and Qe, respectively, and Qe is assumed diagonal. We define

the Gerschgorin critical time increment for CASE as

∆tgcr =2

c√λg

max (1 + 2β), (2.50)

where λgmax = max (λe,g

max), ∀e ∈ 1, . . . , nel.We note that ∆tcfl

cr does not require the formation of any matrices, whereas ∆tgcr requires the

formation of element-level matrices. However, if the procedure outlined in Appendix A.1 is used, this

formation requires little computational effort.

2.5 Error Factors

In this section, we discuss a modified form of the Geers comprehensive error factor [44, 46,

104], which may be used to quantify the error of transient response histories relative to an accepted

benchmark solution. The comprehensive error (C-error) factor is given by C =√M2 + P 2, in which

M =√ϑcc/ϑbb − 1 and P = 1

π arccos(ϑbc/√ϑbbϑcc), where

ϑbb = (t2 − t1)−1

∫ t2

t1

b2(t) dt, ϑcc = (t2 − t1)−1

∫ t2

t1

c2(t) dt,

ϑbc = (t2 − t1)−1

∫ t2

t1

b(t)c(t) dt. (2.51)

In these equations, c(t) is a candidate solution in the form of a response history, b(t) is the corre-

sponding benchmark history, and t1 ≤ t ≤ t2 is the time span of interest. M is the magnitude error

factor, which is insensitive to phase discrepancies, and P is the phase error factor, which is insensi-

tive to magnitude discrepancies. In its last published form [104], the phase error factor was given by

P = 1 − ϑbc/√ϑbbϑcc, which has been found to be too insensitive to phase errors.

Chapter 3

1-D Evaluation: 2-DOF Floating Spring-Mass Oscillator

3.1 Problem Description

For initial evaluation of the CASE approach, field separation, and the non-conformal fluid-

structure coupling, we use a benchmark model consisting of a semi-infinite, 1-D fluid column that

supports a 2-degree-of-freedom (2-DOF) mass-spring oscillator; the lower mass represents the ship’s

hull and the upper mass its internal structure and equipment. This model was productively employed

in a recent evaluation of CAFE, mesh truncation, and wet-surface-approximation methods [103].

The benchmark model mentioned above is shown in Fig. 3.1. The hull mass is m1 and the

total mass of internal structure and equipment is m2. The masses are separated by a linear spring

with stiffness k and the fluid column has cross-sectional area a. The displacements of m1 and m2 are

u1(t) and u2(t), respectively. Depth relative to the wet surface of m1 is denoted by X, and g is the

acceleration of gravity.

The incident field driving the hull mass is due to a plane, upward propagating, step-exponential

acoustic wave defined by

pinc(X, t) = p0 exp[− t+ (X − dinc)/c

τ

]H[t+ (X − dinc)/c], (3.1)

where p0 and τ are the peak pressure and decay time, respectively, and H() is the Heaviside step

function. At t = 0, the incident wavefront is located at a standoff dinc from the wet surface of m1.

The physical properties used in the 1-D calculations are ρ0 = 1026 kg/m3, c = 1500 m/s, patm

= 0.101 MPa, g = 9.81 m/s2, and a = (0.3 m)2. For the ship model, m2/m1 = 5 and m1 = 76.9 kg,

which produce an equilibrium draft for the corresponding 3-D structure of 5 m. The spring stiffness

k is such that the fixed-base natural frequency of m2 is 5 Hz. The incident wave corresponds to the

shock wave generated by a 45.4 kg charge of HBX-1 detonated at a 10 m standoff from the wet surface

of the hull mass, so that p0 = 16.2 MPa and τ = 0.42 ms [105].

28

u2

u1

m2

m1

k

dincX

patm

ρ0, c

Step-ExponentialPressure Wavefrontat t = 0

∞

g

Figure 3.1: Physical model used to represent a ship for the 1-D benchmark problem; the lower massrepresents the ship’s hull and the upper mass its internal structure and equipment.

29

3.2 Discrete Model and Benchmark Solution

The discrete representation of the benchmark model is shown in Fig. 3.2. The upper and lower

masses are represented by square 4-node structural plate finite-elements with cross-sectional area equal

to (0.3 m)2. The base fluid mesh (N = 1 refinement) is composed of 10 cube elements with 0.3 m sides.

The fluid-mesh is terminated by a plane-wave-approximation boundary element located at dpw = 3

m, which is exact for 1-D waves. Figure 3.2 also shows the data transfered between discrete models.

The CAFE and CASE models were integrated with β = 0.25, without subcycling (α = 1), and with

∆t = 0.75∆tcflcr and ∆t = ∆tgcr, respectively. At t = 0, the incident wavefront was located one base

fluid element from the wet surface of m1 (dinc = 0.3 m).

4-Node PlateElement

m2

m1

k/4k/4

dpw

pu1

up

PWA BE

8-Node Base

Fluid Elem.

Figure 3.2: Discrete model of the 1-D benchmark problem show in Fig. 3.1; data transfers betweenmodel components are denoted by arrows.

The benchmark results are the product of a 1-D CAFE research code, which has been validated

[103] with the benchmark problem of Bleich and Sandler [12]. The super-refined benchmark CAFE

mesh has 24,000 equal-length elements and employs a mesh with dpw = 3 m, which extends below the

2.5 m maximum depth of cavitation [103].

The benchmark velocity responses of m1 calculated with and without cavitation are shown in

Fig. 3.3a as a function of the delayed time t = t − dinc/c. Clearly, cavitation has a profound effect

30

on the structure response. Figure 3.3b shows the associated space-time cavitation zones; the gray

areas indicate the existence of cavitation, where and when absolute pressure is zero. Here, we see the

presence of fluid accretion during 2 < t < 46 ms and 117 < t < 126 ms. We also note that the abrupt

changes in velocity at t = 49 ms and t = 129 ms are due to the positive-pressure pulses caused by

closure of the initial two cavitation regions (see Fig. 3.3b). The closure of the third, and smallest,

cavitation region causes a small kink in the velocity response at t = 144 ms.

0 50 100 150-4

-2

0

2

4

6

u1

(m/s)

t (ms)

Benchmark

Benchmark (no cavitation)

(a)

0 50 100 150

0.0

0.5

1.0

1.5

2.0

2.5

3.0

t (ms)

slope cclosure points

X(m

)

(b)

Figure 3.3: (a) Benchmark velocity histories of m1 and (b) the associated space-time cavitation zones.CAFE results were calculated with a super-refined 1-D CAFE model (24,000 DOF) [103]. The dashedline in (b) denotes a fluid characteristic, which relates the pressure pulse caused by cavitation closureto the associated change in structure velocity.

When deriving the linearized governing equations (2.14), we assumed that the condensation was

small and that the convective term in the momentum equation was negligible. In order to test these

assumptions, we repeat the analysis of the above problem, but represent the fluid with a discrete

31

system of springs and masses. Hence, the model uses a Lagrangian treatment of motion, and makes

no small-motion assumptions. The dynamic pressure between two fluid masses mi and mj is given by

pij =

Ba

εε+1 ,

−peq,

pij ≥ peq,

pij < peq,(3.2)

where ε = (ui − uj)/∆X, ui is the dynamic displacement of mi, and ∆X is the equilibrium distances

between each fluid mass. As with the benchmark CAFE model, the system is bounded by a plane-wave

boundary at dpw = 3 m, and the semi-discrete equations are integrated in time with a central-difference

algorithm with damping proportional to s.

Figure 3.4 shows the m1 velocity histories calculated with the CAFE benchmark model and a

highly refined spring-mass model (20,000 DOF). The agreement is seen to be quite good (C = 0.040),

where the differences may be attributed to the neglected terms in the CAFE governing equations. The

difference between the response histories is too small to justify the expense associated with a hydrocode,

which treats the neglected terms. With cavitation effects neglected, the two models produced response

histories that were indistinguishable from one another.

0 50 100 150-4

-2

0

2

4

6

u1

(m/s)

t (ms)

Discrete Spring-Mass Fluid Model

Benchmark CAFE Model

Figure 3.4: Velocity histories of m1 produced with the benchmark CAFE model and a super-refinedLagrangian spring-mass fluid model.

3.3 CAFE and CASE Results

We first examine the performance of CASE and CAFE with the total-field model (TFM). Fig-

ure 3.5a shows the the velocity responses of m1 calculated with the benchmark model and the base-

fluid-element (N = 1) CAFE and CASE models, which have 44 fluid degrees of freedom (FDOF). In

the legend, the numbers in parentheses indicate the C-error of the response history relative to the

benchmark history calculated for 0 ≤ t ≤ 150 ms. We see that the CAFE model yields poor results

32

(C = 0.233) with the CASE model performing slightly better (C = 0.172). Figures 3.5b-c show results

produced by successively refined CAFE and CASE models; the number of FDOF for each model are

listed below the figures. Clearly, model refinement produces response histories that are converging to

the benchmark history, albeit slowly. Note that, because of the coupling method discussed in Section

2.3, fluid-model refinement does not require structure refinement. Although the response histories pro-

duced by the CASE models have more numerical noise than their CAFE counterparts, the histories

are substantially closer to the benchmark history; this is reflected in the C-error values.

Figure 3.6 shows u1 response histories calculated with four CASE-refined models (N = 1 − 4)

and either the TFM or the scattered-field model (SFM). Although the TFM results outperform the

SFM results for the two coarsest meshes (N = 1, 2), the TFM- and SFM-produced response histories

shown for N = 3, 4 exhibit good agreement.

In order to examine further the convergence rates of CASE and CAFE, Fig. 3.7 shows C-

error values as a function of fluid DOF for u1 histories calculated with CAFE-TFM, CASE-TFM,

and CASE-SFM. Clearly, CASE offers significant savings in the number of required FDOF when

compared to that required by CAFE. For example, if we set C = 0.15 as an error level for satisfactory

accuracy, CASE-TFM requires approximately 90 FDOF, whereas CAFE-TFM requires approximately

700 FDOF! Further, CASE-TFM and CASE-SFM models are seen to produce similar error levels for a

given refinement level for N > 2. Finally, we see that, for FDOF > 500, CASE-TFM and CASE-SFM

refinement have a slope (log(C)/ log(DOF)) that is 2.5 times that of CAFE-TFM refinement.

While the number of fluid DOF is a representative measure of computational cost, better mea-

sures are required operations, which is proportional to computation time, and required memory stor-

age, which can be a limiting factor if machine limits are exceeded and disk swapping is required. For

the explicit implementation of CAFE and CASE, most of the differences in required operations and

memory storage lie in the evaluation of the matrix-vector product Hψ, which must be performed at

each time step; other facets of the time-integration schemes are identical. The CAFE implementation

used here has been optimized for speed at the cost of memory required to store H. CASE, on the

other hand, has been optimized for speed at a minimal cost to required memory, which is due solely

to the storage of geometric information (see Appendix A.2).

Figure 3.8 shows the C-error values of Fig. 3.7 as a function of the total number of operations

required for the matrix-vector multiplication Hψ. Here, although CASE outperforms CAFE for

C < 0.15, the results are less impressive than those shown in Fig. 3.7. This is largely because, for

a given spatial-refinement level, CASE elements typically have a smaller critical time increment than

CAFE elements, i.e., more time steps must be taken with CASE.

33

0 50 100 150-4

-2

0

2

4

6

u1

(m/s)

t (ms)

Benchmark

CAFE (0.233)

CASE (0.172)

(a) N = 1, 44 FDOF

0 50 100 150-4

-2

0

2

4

6

u1

(m/s)

t (ms)

Benchmark

CAFE (0.180)

CASE (0.134)

(b) N = 2, 189 FDOF

0 50 100 150-4

-2

0

2

4

6

u1

(m/s)

t (ms)

Benchmark

CAFE (0.159)

CASE (0.111)

(c) N = 3, 496 FDOF

0 50 100 150-4

-2

0

2

4

6

u1

(m/s)

t (ms)

Benchmark

CAFE (0.143)

CASE (0.086)

(d) N = 4, 1025 FDOF

Figure 3.5: Velocity histories of m1 calculated with the total-field model, either CAFE or CASE,and four levels of spatial refinement. The numbers in parentheses are the C-error values for responsehistories relative to the benchmark history.

34

0 50 100 150-4

-2

0

2

4

6

u1

(m/s)

t (ms)

Benchmark

TFM (0.172)

SFM (0.274)

(a) N = 1, 44 FDOF

0 50 100 150-4

-2

0

2

4

6

u1

(m/s)

t (ms)

Benchmark

TFM (0.134)

SFM (0.173)

(b) N = 2, 189 FDOF

0 50 100 150-4

-2

0

2

4

6

u1

(m/s)

t (ms)

Benchmark

TFM (0.111)

SFM (0.108)

(c) N = 3, 496 FDOF

0 50 100 150-4

-2

0

2

4

6u1

(m/s)

t (ms)

Benchmark

TFM (0.086)

SFM (0.093)

(d) N = 4, 1025 FDOF

Figure 3.6: Velocity histories of m1 calculated with CASE, four levels of spatial refinement, and eitherthe TFM or the SFM. The numbers in parentheses are the C-error values for the response historiesrelative to the benchmark history.

100 1000 100000.01

0.11

23

45

6

810

1412

0.421

0.171

Unsatisfactory

Satisfactory

C-E

rror

Fluid DOF

CAFE-TFM

CASE-TFM

CASE-SFM

Figure 3.7: C-error of u1 response histories as a function of fluid-model DOF. The numbers beneaththe data indicate the order of refinement (N) for each DOF.

35

1e+06 1e+09 1e+120.01

0.1

Unsatisfactory

Satisfactory

C-E

rror

CAFE-TFM

CASE-TFM

CASE-SFM

Total Operations

Figure 3.8: C-error of the u1 response histories as a function of the total operations required forevaluation of the matrix-vector product, Hψ.

36

Figure 3.9 shows the C-error values of the previous two figures as a function of the required

memory storage for evaluation of the matrix-vector product. Here, we see huge gains in using CASE

instead of CAFE; CAFE-TFM requires 18 times more memory than that required by CASE-TFM to

produce a u1 history with C ≈ 0.15.

1e-02 1e-01 1e+00 1e+01 1e+020.01

0.1

Unsatisfactory

Satisfactory

C-E

rror

CAFE-TFM

CASE-TFM

CASE-SFM

Required Memory (Mb)

Figure 3.9: C-error of the u1 response histories as a function of the global memory storage requiredfor evaluation of the matrix-vector product, Hψ.

Except for the coarsest models, the results above indicate that CASE models with either TFM

or SFM produce results that are similar in accuracy. This was expected because dinc was quite

small. Here, we examine the performance of TFM and SFM when the incident wave must travel an

appreciable distance before striking the structure. Figure 3.10 shows C-error values for u1 response

histories relative to the benchmark history. Response histories were calculated with CASE models

with N = 3 refinement, either TFM or SFM, and several dinc values. The accuracy of the TFM-

produced solutions degrades with increasing dinc, whereas the accuracy of the SFM-produced solutions

is independent of the incident-wave travel distance. This degradation in TFM performance is due to

dispersion of the incident wave as it travels through the mesh. This is illustrated with the incident-

wave pressure-profile snapshots shown in Fig. 3.11, which were calculated with a CASE-TFM model

with N = 3, and dinc = 2.7 m; also shown are the exact solutions, which were produced with the SFM.

We see that the incident wave becomes quite distorted as it approaches the structure.

3.4 Summary and Conclusions

In this chapter, a simple 1-D fluid-structure-interaction problem was used to evaluate the per-

formance of several of the improvements proposed for the CAFE-FE approach. It was clearly demon-

strated that, for a given level of accuracy, CASE offers significant savings over CAFE in terms of

fluid DOF and required memory. It was also found that, while the scattered-field model degraded

37

0.0 0.5 1.0 1.5 2.0 2.5 3.00.10

0.12

0.14

C-E

rror

dinc (m)

CASE-TFM

CASE-SFM

Figure 3.10: C-error values of the u1 response histories as a function of incident-wave travel distance(dinc). Results are the product of a CASE model with N = 3 refinement and either the TFM or SFM.

0.0 0.5 1.0 1.5 2.0 2.5 3.00

5

10

15

20

t = 1.5 ms t = 1.0 ms t = 0.5 ms

t = 0 ms

Inci

den

tP

ress

ure

(MPa)

X (m)

CASE-SFM (Exact)

CASE-TFM

Figure 3.11: Incident-wave pressure-profile snapshots calculated with CASE-TFM, N = 3 refinement,and dinc = 2.7 m. Also shown is the exact solution, which was obtained with the SFM.

38

CASE performance for very coarse meshes, SFM produces solutions with accuracy similar to that

produced with the TFM for somewhat refined meshes. Finally, the non-conformal coupling approach

allowed significant savings in the structure FE model, as no refinement was required to accommodate

fluid-model refinement.

Chapter 4

3-D Evaluation: Submerged Spherical Shell

In the previous chapter, a 1-D problem was used to test the performance of CAFE-TFM, CASE-

TFM, CASE-SFM, and the fluid-structure coupling. In this chapter, we extend the analysis to 3-D by

investigating the axisymmetric response of an empty, submerged spherical shell excited by an incident

step-exponential spherical wave. We also examine the performance of time-step subcycling.

4.1 Problem Description

A schematic of the spherical shell and incident wave is shown in Fig. 4.1. The fluid is infinite

and homogeneous and gravity is neglected. The shell has thickness h, midsurface radius a, Young’s

Modulus E, density ρ0, and Poisson’s ratio ν. Meridional and radial displacements are denoted by v

and w, respectively. Cartesian (X,Y ) and spherical (r, θ) coordinate frames are shown. The incident-

wave center is located at ~Xc = (Xc, Yc, Zc), and is defined by

pinc( ~X, t) =R0

Rp0 exp

[− t− (R−R0)/c

τ

]H [t− (R−R0)/c] , (4.1)

where R = | ~X− ~Xc|, and the wave has radius R0 and maximum pressure p0 when t = 0. As in the 1-D

problem, dinc is the wavefront standoff from the structure when t = 0. The system properties used

for this analysis are listed in Table 4.1. The incident-wave peak pressure and decay time correspond

to a 60 kg charge of HBX-1 [22, 90] detonated at a 12 m standoff with dinc = 0.909 m. The uniform

hydrostatic pressure corresponds to a depth of 150 m.

With cavitation effects neglected, this problem has been solved with modal methods for plane-

step-wave excitation by Zhang and Geers [109]; their solution method has been incorporated into

SPHSHK/MODSUM [48], a public-domain FORTRAN program. The program has been extended by

Sprague and Geers [102] to include spherical and plane step-exponential waves.

40

Inc. Wavefront at t = 0p0, τ

a

c

dinc

vw

r

h

Infinite Fluidρ0, c, phyd

θ

E, ρs, ν

X

Y

(Xc, Yc, Zc)

R0

Figure 4.1: Schematic of the spherical-shell geometry and incident spherical wave with associatedvariables.

Table 4.1: Properties used for the spherical shell, water, and incident wave.

Young’s Modulus E 195 × 109 PaShell Density ρs 7700 kg/m3

Poisson’s Ratio ν 0.3Shell Radius a 5 m

Shell Thickness h 0.05 mWater Density ρ0 1026 Kg/m3

Water Sound Speed c 1500 m/sHydrostatic Pressure phyd 1.611 MPaInc.-Wave Peak Press. p0 16.26 MPaInc.-Wave Decay Time τ 0.464 ms

Inc.-Wave Center Xc, Yc, Zc (17, 0, 0) m

41

4.2 Structure FE Model Validation

Before investigating the performance of the fluid-modeling technique, we validate the perfor-

mance of the dry-structure FE model by applying the incident wave to the structure while neglecting

all fluid-structure-interaction effects. Although this is a physically unrealizable system, it is well suited

for structure-model validation. Figure 4.2 shows a representative quarter-symmetry FE mesh for the

structure which is composed of 150 quadrilateral shell elements.

Figure 4.2: Representative 150-element quarter-symmetry structure finite-element mesh used for theresponse calculations of the submerged spherical shell.

Figure 4.3 shows the nondimensional radial-velocity response histories at the front (θ = 0) and

back (θ = 180) of the shell as a function of nondimensional delayed time. The response histories were

produced with three structure FE models containing 96, 150, and 216 elements, respectively. The

structure was excited by the incident wave defined by (4.1). In the legend, the numbers in parentheses

are the C-error values of the response histories relative to those produced with the 216-element model

at the front and back of the shell, respectively. The FE equations were integrated with ∆t = 7.5×10−6

s, which was experimentally found to be 75% of the critical time increment for the 150-element model.

The agreement between the solutions produced by the three models is satisfactory. All subsequent

sphere calculations use the 150-element structure model.

42

-1

0

1

2

0 2 4 6 8 10-2

-1

0

1

2

tc/a

wρ0c/

p0

96-Element Model (0.029, 0.088)

150-Element Model (0.013, 0.037)

216-Element Model

θ = 0

θ = 180

Figure 4.3: Spherical-shell radial-velocity histories calculated with FSI effects neglected and FE modelswith three levels of refinement. The numbers in parentheses are the C-error values of the responsehistories relative to the 216-element histories at θ = 0 and θ = 180, respectively.

43

4.3 Results and Discussion: No Cavitation

4.3.1 Benchmark FSI Solution

With a satisfactorily refined structure model, we now focus on producing a benchmark solution

with a highly refined fluid model coupled to the 150-element structure model; the discrete models

are coupled with the method introduced in Section 2.3. Figure 4.4 shows the base fluid mesh that

surrounds the structure mesh shown in Fig. 4.2. The mesh is composed of 1,650 elements, has an

outside radius of 15 m, and is surrounded by CWA boundary elements. For subsequent calculations,

the structure is integrated with ∆t = 7.5 × 10−6 s and the incident wave is located a single base

element from the structure when t = 0 (dinc = 0.909 m).

Figure 4.4: Quarter-symmetry base fluid finite-element mesh used for the response calculations of thesubmerged spherical shell.

For the benchmark solution, subcycling and the scattered-field model were employed. Figure 4.5

shows the nondimensional structure radial-velocity response histories at θ = 0 and θ = 180 as a

function of delayed time. The responses were calculated with the 1,650-element base fluid model and

either N = 10 or N = 14 CASE refinement with α = 1/5 or α = 1/9, respectively (∆tfl = α∆t). Also

shown are the response histories produced by SPHSHK/MODSUM with the first 8 modes. Differences

between the two CASE solutions are virtually indiscernible, which indicates convergence in terms of

both spatial and temporal refinement. In addition, good agreement is seen between the FE and modal

solutions except at late time for θ = 180. The differences between the solutions may be attributed to

differences between the two structure formulations (the modal model neglects rotary inertia and shear

stiffness). Regardless, the agreement between the histories produced by the two methods is satisfactory

and the CASE N = 14 solution is used as the benchmark for all subsequent sphere calculations without

cavitation. Further, the results shown in Fig. 4.5 serve to validate both the fluid-structure coupling

44

approach and the use of time-step subcycling. We note that in order to achieve a fluid-refinement

level equivalent to the N = 14 model with the original one-to-one CAFE-FE coupling approach, the

structure model would require 150N2 = 29, 400 elements!

Close examination of the response histories for θ = 180 reveals that nonphysical oscillations

occur for 2 < tc/a < 2.2 due to the arrival of the incident wave. While these oscillations are noticeable

for the N = 10 solution, they are almost indiscernible for the N = 14 solutions. The oscillations may

be removed by applying a Savitsky-Golay filter [96, 89] to the response histories. The filtered θ = 180

responses are used as the benchmark for subsequent comparisons.

-0.8

-0.4

0

0 2 4 6 8 10-0.2

0

0.2

tc/a

wρ0c/

p0

SPHSHK/MODSUM (8 modes)

N = 10 CASE (1.7 × 106 Fluid DOF)

N = 14 CASE (4.6 × 106 Fluid DOF)

θ = 0

θ = 180

Figure 4.5: Nondimensional spherical-shell radial-velocity histories calculated with two super-refinedCASE fluid meshes coupled to the 150-element structure model. Cavitation effects were neglected.Also shown are the modal solutions produced by SPHSHK/MODSUM [48, 102]. The C-error valuesfor the N = 10 solutions relative to the N = 14 solutions are C = 0.003 and C = 0.006 for θ = 0 andθ = 180, respectively.

In the following sections, we compare the performance of CAFE and CASE models with many

fewer fluid DOF than the number in the benchmark model. Table 4.2 shows the fluid DOF and

subcycling ratios used for each refinement level. The subcycling ratios are based on ∆t = 7.5×10−6 s,

and were chosen such that α∆t was less than 75% ∆tcflcr for CAFE and less than ∆tgcr for CASE. We

see that, for the less-refined models, subcycling offers significant computational savings. The savings

are more pronounced for CAFE, which, for a given number of DOF, has a larger critical time increment

than CASE. For CASE models with N ≥ 5, subcycling removes the need to reduce the structure’s ∆t.

4.3.2 CASE and CAFE with the TFM and SFM

Figure 4.6 shows radial-velocity response histories produced at the front and back of the shell.

The histories were produced with the TFM and three refinement levels (N = 1−3) with either CAFE

45

or CASE. For the coarsest mesh (N = 1), CASE performs somewhat better at θ = 0, but CAFE

performs better at θ = 180. For further refinement, CASE outperforms CAFE at both shell positions,

e.g., for N = 3, CASE produces solutions with half the error of those produced with CAFE. For both

models, the histories quickly converge to the benchmark histories with increasing N . These results

serve as a validation for time-step subcycling and the non-conformal fluid-structure coupling. We note

that 150N2 = 1, 350 wet-structure elements would be required to accommodate the N = 3 fluid mesh

with the original CAFE-FE coupling method.

Figure 4.7 shows another set of radial-velocity response histories produced at the front and back

of the shell; this set was produced with three CASE-refinement levels (N = 1 − 3) and with either

the TFM or SFM. We see that the SFM-produced solutions for N = 1 are highly oscillatory and

yield unacceptably large C-error values. For N > 1 and at θ = 180, the oscillations begin when the

incident wave arrives at tc/a = 2. This poor performance may be attributed to the application of

the discontinuous incident wave to the structure; the coarse fluid mesh is insufficient for treating the

resulting discontinuous scattered field. Further, we see that these oscillations are mesh dependent;

mesh refinement produces oscillations with higher frequency and smaller amplitude. In the TFM, the

incident field is subject to an ameliorative smoothing due to numerical dispersion before striking the

structure. Thus, the scattered field is also smoothed. For N = 2, the SFM produces solutions with

acceptable error levels, and for N = 3 refinement, the SFM-produced solutions for θ = 0 are as good

as those produced with TFM. The solutions for θ = 180 are not as good due to the oscillations that

begin at tc/a = 2. These figures demonstrate that while TFM outperforms SFM at the coarsest mesh

levels, SFM quickly produces solutions with accuracy near that of the TFM as the model is refined.

As with the 1-D problem, we investigate the performance of the various methods in terms of

fluid DOF, total operations, and required memory. Figure 4.8 shows the C-error values as a function

of fluid DOF for solutions produced with CAFE-TFM, CASE-TFM, and CASE-SFM with many levels

of fluid refinement. We have again set C = 0.15 as an upper bound for satisfactory accuracy. Besides

the solutions produced by the coarsest model, those produced with CASE-TFM outperform those

produced with CAFE-TFM. Further, histories produced with CASE-SFM at θ = 0 yield accuracy

levels similar to those produced with CASE-TFM for N ≥ 3. The performance of CASE-SFM at

θ = 180 is less impressive; the oscillations, which occur due to incident wave’s arrival, increase the

error values. Regardless, all methods are producing solutions that are converging to the benchmark

solutions.

The C-error values of Fig. 4.8 are shown in Figs. 4.9 and 4.10 as a function of total operations

and memory required, respectively, for evaluation of the matrix-vector product Hψ. Here, we see

46

Table 4.2: Fluid-model properties for CAFE and CASE refinement of the 1650-element base fluidmesh shown in Fig 4.4.

N Fluid DOF CAFE α CASE α

1 2,052 20 142 14,743 10 43 47,974 6 24 111,645 5 25 215,656 4 1/26 369,907 3 1/28 868,729 2 1/310 1,687,311 2 1/5

-0.8

-0.4

0

0 2 4 6 8 10-0.2

0

0.2

wρ0c/

p0

tc/a

Benchmark

θ = 0

θ = 180

CAFE (0.158, 0.037)

CASE (0.127, 0.104)

(a) N = 1: 2,052 Fluid DOF

-0.8

-0.4

0

0 2 4 6 8 10-0.2

0

0.2

wρ0c/

p0

tc/a

Benchmark

θ = 0

θ = 180

CAFE (0.091, 0.032)

CASE (0.064, 0.020)

(b) N = 2: 14,743 Fluid DOF

-0.8

-0.4

0

0 2 4 6 8 10-0.2

0

0.2

wρ0c/

p0

tc/a

Benchmark

θ = 0

θ = 180

CAFE (0.057, 0.020)

CASE (0.022, 0.011)

(c) N = 3: 47,974 Fluid DOF

Figure 4.6: Nondimensional radial-velocity histories calculated with either CASE or CAFE and theTFM with cavitation effects neglected. The benchmark solutions are the product of a CASE-SFMmodel with N = 14. The numbers in parentheses are the C-error values associated with the responsehistories at θ = 0 and θ = 180, respectively.

47

-0.8

-0.4

0

0 2 4 6 8 10-0.2

0

0.2

wρ0c/

p0

tc/a

Benchmark

θ = 0

θ = 180

SFM (0.432, 0.248)

TFM (0.127, 0.104)

(a) N = 1: 2,052 Fluid DOF

-0.8

-0.4

0

0 2 4 6 8 10-0.2

0

0.2

wρ0c/

p0

tc/a

Benchmark

θ = 0

θ = 180

SFM (0.105, 0.061)

TFM (0.064, 0.020)

(b) N = 2: 14,743 Fluid DOF

-0.8

-0.4

0

0 2 4 6 8 10-0.2

0

0.2

wρ0c/

p0

tc/a

Benchmark

θ = 0

θ = 180

SFM (0.035, 0.024)

TFM (0.034, 0.008)

(c) N = 3: 47,974 Fluid DOF

Figure 4.7: Nondimensional radial-velocity histories calculated with the CASE and either the TFMor SFM with cavitation effects neglected. The benchmark solutions are the product of a CASE-SFMmodel with N = 14. The numbers in parentheses are the C-error values associated with the responsehistories at θ = 0 and θ = 180, respectively.

48

that, overall, CAFE slightly outperforms CASE in terms of required operations for a given accuracy

level. However, CASE admits huge gains in terms of required memory. For example, in order to reach

C ≈ 0.1 for θ = 0, CAFE requires 3.9 Mb of memory, whereas CASE only requires 0.9 Mb, a savings

of 77%.

1e+03 1e+04 1e+05 1e+06

0.01

0.1

C-E

rror

Fluid DOF

Unsatisfactory

Satisfactory

CAFE-TFM

CASE-TFM

CASE-SFM

(a) θ = 0

1e+03 1e+04 1e+05 1e+06

0.01

0.1

C-E

rror

Fluid DOF

UnsatisfactorySatisfactory

CAFE-TFM

CASE-TFM

CASE-SFM

(b) θ = 180

Figure 4.8: C-error values as a function of total fluid DOF for radial-velocity histories at the (a) frontand (b) back of the spherical shell. Benchmark results were produced with a N = 14 CASE-SFMmodel. Cavitation effects were neglected.

49

1e+07 1e+08 1e+09 1e+10 1e+11 1e+12

0.01

0.1

C-E

rror

Total Operations

Unsatisfactory

Satisfactory

CAFE-TFM

CASE-TFM

CASE-SFM

(a) θ = 0

1e+07 1e+08 1e+09 1e+10 1e+11 1e+12

0.01

0.1

C-E

rror

Total Operations

UnsatisfactorySatisfactory

CAFE-TFM

CASE-TFM

CASE-SFM

(b) θ = 180

Figure 4.9: C-error values as a function of total operations required for evaluation of the matrix-vectorproduct Hψ, for radial-velocity histories at the (a) front and (b) back of the spherical shell. Benchmarkresults were produced with a N = 14 CASE-SFM model. Cavitation effects were neglected.

1e+00 1e+01 1e+02

0.01

0.1

C-E

rror

Total Memory (Mb)

UnsatisfactorySatisfactory

CAFE-TFM

CASE-TFM

CASE-SFM

(a) θ = 0

1e+00 1e+01 1e+02

0.01

0.1

C-E

rror

Total Memory (Mb)

Unsatisfactory

Satisfactory

CAFE-TFM

CASE-TFM

CASE-SFM

(b) θ = 180

Figure 4.10: C-error values as a function of total memory storage required for evaluation of thematrix-vector product Hψ, for radial-velocity histories at the (a) front and (b) back of the sphericalshell. Benchmark results were produced with a N = 14 CASE-SFM model. Cavitation effects wereneglected.

50

4.3.3 CASE and CAFE and Time-Step Subcycling

In the previous section, time-step subcycling was used for all calculations. Here, we demonstrate

subcycling’s importance. Figure 4.11 shows the C-error values as a function of fluid DOF for response

solutions produced with or without subcycling and either CASE-TFM or CAFE-TFM. For θ = 0,

the use of subcycling actually improves the solution and at a lower computational cost. For θ = 180,

subcycling offers significant improvement in accuracy for CASE, whereas the accuracy of the CAFE

results is largely unaffected. The savings offered by subcycling are further illustrated in Fig. 4.12,

which shows the C-error values of Fig. 4.11, but as a function of required operations for evaluation of

the matrix-vector product.

1e+03 1e+04 1e+050.01

0.1

C-E

rror

Fluid DOF

UnsatisfactorySatisfactory

CAFE, Subcyc.

CAFE, No Subcyc.

CASE, Subcyc.

CASE, No Subcyc.

(a) θ = 0

1e+04 1e+05

0.01

0.1

C-E

rror

Fluid DOF

UnsatisfactorySatisfactory

CAFE, Subcyc.

CAFE, No Subcyc.

CASE, Subcyc.

CASE, No Subcyc.

(b) θ = 180

Figure 4.11: C-error values as a function of total fluid DOF for radial-velocity histories at the (a) frontand (b) back of the spherical shell. Demonstrates the importance of time-step subcycling. Benchmarkresults were produced with a N = 14 CASE-SFM model. Cavitation effects were neglected.

51

1e+07 1e+08 1e+09 1e+10 1e+11 1e+120.01

0.1

C-E

rror

Total Operations

UnsatisfactorySatisfactory

CAFE, Subcyc.

CAFE, No Subcyc.

CASE, Subcyc.

CASE, No Subcyc.

(a) θ = 0

1e+06 1e+07 1e+08 1e+09 1e+10 1e+11 1e+12

0.01

0.1

C-E

rror

Total Operations

UnsatisfactorySatisfactory

CAFE, Subcyc.

CAFE, No Subcyc.

CASE, Subcyc.

CASE, No Subcyc.

(b) θ = 180

Figure 4.12: C-error values as a function of total operations required for evaluation of the matrix-vector product Hψ, for radial-velocity histories at the (a) front and (b) back of the spherical shell.Demonstrates the importance of time-step subcycling. Benchmark results were produced with a N =14 CASE-SFM model. Cavitation effects were neglected.

52

4.4 Results and Discussion: Cavitation at 150 m

In this section, we investigate the system studied in the previous section, but we include cavi-

tation effects. Figure 4.13 shows the response histories for the front and back of the shell, which were

produced with CASE-SFM, subcycling, and either N = 10 or N = 14 refinement. Also shown are the

benchmark solutions for the noncavitating system. Again, the N = 10 and N = 14 response histories

are virtually indiscernible, thus indicating a converged solution. We see that cavitation has a minimal

effect on structure response with differences noticeable only for early time at θ = 0.

An investigation comparing the the performance of CAFE-TFM, CASE-TFM, and CASE-SFM

was again performed. The results regarding the performance of the methods are very similar to those

of the previous section: CASE and CAFE demonstrated similar performance in terms of required fluid

dof and operations. However, CASE offered significant savings in terms of required memory.

-0.8

-0.4

0

0 2 4 6 8 10-0.2

0

0.2

tc/a

wρ0c/

p0

N = 14 CASE (No Cavitation)

N = 10 CASE (1.7 × 106 Fluid DOF)

N = 14 CASE (4.6 × 106 Fluid DOF)

θ = 0

θ = 180

Figure 4.13: Nondimensional spherical-shell radial-velocity histories calculated with two super-refinedCASE-SFM fluid models coupled to the 150-element structure model. The hydrostatic pressure cor-responded to a 150 m depth, and cavitation effects were included. Also shown are the benchmarkresponses from Fig 4.5, which were calculated with cavitation effects neglected. The C-error valuesfor the N = 10 solutions relative to the N = 14 solutions are C = 0.006 for both θ = 0 and θ = 180.

53

Figure 4.14 shows axisymmetric cavitation snapshots at various nondimensional times. In these

figures, the light gray areas indicate location of the structure and the dark areas indicate the location

of the fluid model. White areas within the dark indicate the location of cavitation. The results are

the product of the benchmark CASE-SFM model with N = 14 refinement. We see that a region of

cavitation forms at the front of the shell at early time but quickly disperses. The cavitated volume

is responsible for the differences between the response histories for the cavitating and noncavitating

solutions at θ = 0. Later, we see the formation of cavitation region on the shell’s back side, which has

no discernible affect on the structure response shown in Fig. 4.13. There is no cavitation for tc/a > 4.

54

-10 0 100

5

10

15

X (m)

Y(m

)

(a) tc/a = 0.5

-10 0 100

5

10

15

X (m)

Y(m

)

(b) tc/a = 1.0

-10 0 100

5

10

15

X (m)

Y(m

)

(c) tc/a = 1.5

-10 0 100

5

10

15

X (m)

Y(m

)

(d) tc/a = 2.0

-10 0 100

5

10

15

X (m)

Y(m

)

(e) tc/a = 2.5

-10 0 100

5

10

15

X (m)

Y(m

)

(f) tc/a = 3.0

-10 0 100

5

10

15

X (m)

Y(m

)

(g) tc/a = 3.5

-10 0 100

5

10

15

X (m)

Y(m

)

(h) tc/a = 4.0

Figure 4.14: Axisymmetric cavitation snapshots for a spherical shell at a depth of 150 m for 0.5 ≤tc/a ≤ 4.0. Results are a product of a CASE-SFM model with N = 14.

55

4.5 Results and Discussion: Cavitation at 50 m

In this section, we again examine the response of the spherical shell, but we set the uniform

hydrostatic pressure to 0.604 MPa, which corresponds to a depth of 50 m. Because of this lower

pressure, we expect to see more cavitation than that seen for a depth of 150 m. Figure 4.15 shows the

response histories for the front and back of the shell, which were produced with CASE-SFM, subcycling,

and eitherN = 10 orN = 14 refinement. Also shown are the benchmark solutions for the noncavitating

system. Again, the N = 10 and N = 14 response histories are virtually indistinguishable, thus

indicating a converged solution. Compared to the benchmark histories for the 150 m depth, we see

that cavitation has more of an effect on the early-time response at θ = 0, and has a profound effect

on the late-time response at θ = 180.

-0.8

-0.4

0

0 2 4 6 8 10

-0.2

0

0.2

tc/a

wρ0c/

p0

N = 14 CASE (No Cavitation)

N = 10 CASE (1.7 × 106 Fluid DOF)

N = 14 CASE (4.6 × 106 Fluid DOF)

θ = 0

θ = 180

Figure 4.15: Nondimensional spherical-shell radial-velocity histories calculated with two super-refinedCASE-SFM fluid models coupled to the 150-element structure model. The hydrostatic pressure cor-responded to a 50 m depth, and cavitation effects were included. Also shown are the benchmarkresponses from Fig 4.5, which were calculated with cavitation effects neglected. The C-error valuesfor the N = 10 solutions relative to the N = 14 solutions are C = 0.009 and C = 0.015 for θ = 0 andθ = 180, respectively.

Figures 4.16 and 4.17 show cavitation snapshots in the axisymmetric plane at various nondi-

mensional times. As with the system at 150 m depth, we see the early-time formation of a cavitation

region at the front of the shell. For the 50 m depth, however, we see that the cavitation that occurs

at the shell’s back side is more pronounced and remains present for 2 < tc/a < 10.

56

-10 0 100

5

10

15

X (m)

Y(m

)

(a) tc/a = 0.5

-10 0 100

5

10

15

X (m)

Y(m

)

(b) tc/a = 1.0

-10 0 100

5

10

15

X (m)

Y(m

)

(c) tc/a = 1.5

-10 0 100

5

10

15

X (m)

Y(m

)

(d) tc/a = 2.0

-10 0 100

5

10

15

X (m)

Y(m

)

(e) tc/a = 2.5

-10 0 100

5

10

15

X (m)

Y(m

)

(f) tc/a = 3.0

-10 0 100

5

10

15

X (m)

Y(m

)

(g) tc/a = 3.5

-10 0 100

5

10

15

X (m)

Y(m

)

(h) tc/a = 4.0

Figure 4.16: Axisymmetric cavitation snapshots for a spherical shell at a depth of 50 m for 0.5 ≤tc/a ≤ 4.0. Results are a product of a CASE-SFM model with N = 14.

57

-10 0 100

5

10

15

X (m)

Y(m

)

(a) tc/a = 4.5

-10 0 100

5

10

15

X (m)

Y(m

)

(b) tc/a = 5.0

-10 0 100

5

10

15

X (m)

Y(m

)

(c) tc/a = 5.5

-10 0 100

5

10

15

X (m)

Y(m

)

(d) tc/a = 8.0

-10 0 100

5

10

15

X (m)

Y(m

)

(e) tc/a = 8.5

-10 0 100

5

10

15

X (m)

Y(m

)

(f) tc/a = 9.0

Figure 4.17: Axisymmetric cavitation snapshots for a spherical shell at a depth of 50 m for 4.5 ≤tc/a ≤ 9.0. Results are a product of a CASE-SFM model with N = 14.

58

With a satisfactory benchmark solution, we investigate the performance of CAFE-TFM, CASE-

TFM, and CASE-SFM for a relatively small number of fluid DOF. Figure 4.18 shows radial-velocity

response histories at the front and back of the shell that were calculated with either CAFE-TFM

or CASE-TFM and N = 1 − 3 refinement. One immediately notices that these models perform

substantially worse than for the no-cavitation results of Fig. 4.6. Although the responses appear to

be converging to the benchmark solutions, the convergence rate is slow. Further, we see that CAFE

outperforms CASE for all refinement levels shown.

Figure 4.19 shows radial-velocity response histories at the front and back of the shell that were

calculated with CASE and either TFM or SFM and N = 1 − 3 refinement. Here, we see that, except

for N = 1, the scattered-field model clearly outperforms the total-field model.

Figures 4.20-4.22 compare the performance of CAFE-TFM, CASE-TFM, and CASE-SFM in

terms of required fluid DOF, operations, and memory, respectively. In terms of required fluid DOF,

CAFE-TFM outperforms CASE-TFM, but CASE-SFM appears to be the optimal choice. In terms of

required operations, CASE-TFM performs the worst, with CASE-SFM being comparable with CAFE-

TFM for θ = 0 and performing better than CAFE for θ = 180. In terms of required memory, both

CASE methods outperform CAFE-TFM.

In the previous sections, it was found that CASE with TFM outperformed CASE-SFM for

coarse meshes and yielded similar-performing results for greater refinement. It is interesting to see

that SFM is clearly the best choice for the current system. We attempt to explain this with Fig. 4.23,

which shows the cavitation snapshots for tc/a = 1.0. The results are the product of a N = 6 CASE

model with either the TFM or SFM. For the TFM (Fig. 4.23a), we see the presence of extensive

frothing throughout the fluid domain in front of the shell, which is due to the passage of the incident

wavefront. In the fluid mesh, the discontinuous wavefront produces pressure oscillations through the

mesh, which reach the cavitation threshold. This is not seen in the SFM model. Thus, frothing

degrades incident-wave propagation and produces the poor model performance.

59

-0.8

-0.4

0

0 2 4 6 8 10

-0.2

0

0.2

0.4

wρ0c/

p0

tc/a

Benchmarkθ = 0

θ = 180

CAFE (0.090, 0.373)

CASE (0.190, 0.326)

(a) N = 1: 2,052 Fluid DOF

-0.8

-0.4

0

0 2 4 6 8 10

-0.2

0

0.2

0.4

wρ0c/

p0

tc/a

Benchmarkθ = 0

θ = 180

CAFE (0.139, 0.236)

CASE (0.154, 0.306)

(b) N = 2: 14,743 Fluid DOF

-0.8

-0.4

0

0 2 4 6 8 10

-0.2

0

0.2

0.4

wρ0c/

p0

tc/a

Benchmark

θ = 0

θ = 180

CAFE (0.092, 0.122)

CASE (0.095, 0.246)

(c) N = 3: 47,974 Fluid DOF

Figure 4.18: Nondimensional radial-velocity histories calculated with the either CAFE or CASE andthe TFM. The hydrostatic pressure corresponded to a 50 m depth, and cavitation effects were included.The benchmark solutions are the product of a CASE-SFM model with N = 14. The numbers inparentheses are the C-error values associated with the response histories at θ = 0 and θ = 180,respectively.

60

-0.8

-0.4

0

0 2 4 6 8 10

-0.2

0

0.2

0.4

wρ0c/

p0

tc/a

Benchmarkθ = 0

θ = 180

SFM (0.266, 0.127)

TFM (0.190, 0.326)

(a) N = 1: 2,052 Fluid DOF

-0.8

-0.4

0

0 2 4 6 8 10

-0.2

0

0.2

0.4

wρ0c/

p0

tc/a

Benchmarkθ = 0

θ = 180

SFM (0.123, 0.087)

TFM (0.154, 0.306)

(b) N = 2: 14,743 Fluid DOF

-0.8

-0.4

0

0 2 4 6 8 10

-0.2

0

0.2

0.4

wρ0c/

p0

tc/a

Benchmarkθ = 0

θ = 180

SFM (0.040, 0.061)

TFM (0.095, 0.246)

(c) N = 3: 47,974 Fluid DOF

Figure 4.19: Nondimensional radial-velocity histories calculated with the CASE and either the TFMor SFM. The hydrostatic pressure corresponded to a 50 m depth, and cavitation effects were included.The benchmark solutions are the product of a CASE-SFM model with N = 14. The numbers inparentheses are the C-error values associated with the response histories at θ = 0 and θ = 180,respectively.

61

1e+03 1e+04 1e+05 1e+06

0.1

C-E

rror

Fluid DOF

Unsatisfactory

Satisfactory

CAFE-TFM

CASE-TFM

CASE-SFM

(a) θ = 0

1e+03 1e+04 1e+05 1e+06

0.1C-E

rror

Fluid DOF

UnsatisfactorySatisfactory

CAFE-TFM

CASE-TFM

CASE-SFM

(b) θ = 180

Figure 4.20: C-error values as a function of total fluid DOF for radial-velocity histories produced atthe (a) front and (b) back of the spherical shell. Benchmark results were produced with a N = 14CASE-SFM model. The hydrostatic pressure corresponded to a depth of 50 m and cavitation effectswere included.

1e+07 1e+08 1e+09 1e+10 1e+11 1e+120.01

0.1

C-E

rror

Total Operations

UnsatisfactorySatisfactory

CAFE-TFM

CASE-TFM

CASE-SFM

(a) θ = 0

1e+07 1e+08 1e+09 1e+10 1e+11 1e+120.01

0.1

C-E

rror

Total Operations

UnsatisfactorySatisfactory

CAFE-TFM

CASE-TFM

CASE-SFM

(b) θ = 180

Figure 4.21: C-error values as a function of total operations required for evaluation of the matrix-vector product Hψ, for radial-velocity histories produced at the (a) front and (b) back of the sphericalshell. Benchmark results were produced with a N = 14 CASE-SFM model. The hydrostatic pressurecorresponded to a depth of 50 m and cavitation effects were included.

62

1e+00 1e+020.01

0.1

C-E

rror

Total Memory (Mb)

UnsatisfactorySatisfactory

CAFE-TFM

CASE-TFM

CASE-SFM

(a) θ = 0

1e+00 1e+01 1e+02

0.1

C-E

rror

Total Memory (Mb)

UnsatisfactorySatisfactory

CAFE-TFM

CASE-TFM

CASE-SFM

(b) θ = 180

Figure 4.22: C-error values as a function of total memory storage required for evaluation of the matrix-vector product Hψ, for radial-velocity histories produced at the (a) front and (b) back of the sphericalshell. Benchmark results were produced with a N = 14 CASE-SFM model. The hydrostatic pressurecorresponded to a depth of 50 m and cavitation effects were included.

-10 0 100

5

10

15

X (m)

Y(m

)

(a) CASE-TFM

-10 0 100

5

10

15

X (m)

Y(m

)

(b) CASE-SFM

Figure 4.23: Axisymmetric cavitation snapshots for a spherical shell at a depth of 50 m for tc/a = 1.0.Results are a product of a N = 6 CASE model with the (a) TFM and (b) SFM. The hydrostaticpressure corresponded to a depth of 50 m and cavitation effects were included.

63

4.5.1 Cavitation and Convergence

It was noted in the previous section that the performance of the coarse (N = 1 − 3) CASE

and CAFE models were significantly worse for a depth of 50 m than for a 150 m depth, or without

cavitation. Figure 4.24 shows the convergence rates of CASE-SFM as a function of fluid DOF for

the three systems discussed above. We see that, as cavitation effects become more pronounced,

the convergence rate of CASE is reduced. This is most noticeable at θ = 180. As cavitation is

a discontinuous phenomenon, this performance degradation was expected. However, for the three

systems studied, satisfactory results were produced with relatively few fluid DOF (≈ 10, 000).

1e+03 1e+04 1e+05 1e+06

0.01

0.1

C-E

rror

Fluid DOF

UnsatisfactorySatisfactory

No Cavitation

Cavitation at 150 m

Cavitation at 50 m

(a) θ = 0

1e+03 1e+04 1e+05 1e+06

0.01

0.1

C-E

rror

Fluid DOF

UnsatisfactorySatisfactory

No Cavitation

Cavitation at 150 m

Cavitation at 50 m

(b) θ = 180

Figure 4.24: C-error values as a function of fluid DOF for radial-velocity histories produced at the (a)front and (b) back of the spherical shell. Benchmark results were produced with a N = 14 CASE-SFMmodel.

4.5.2 CAFE with Reduced Integration

It has been suggested [70] that CAFE would be more competitive in terms of computational

cost if 1-point Gauss quadrature were used for evaluation of the fluid reactance matrix instead of the

full 8-point quadrature used here. Reduced integration allows efficient evaluation of matrix-vector

products at the element level. Figure 4.25 shows radial-velocity histories at the front and back of

the shell that were produced with CAFE-TFM and either 1-point or 8-point integration. For N = 3,

we see that the the histories produced with full integration for θ = 180 are much better than those

produced with reduced integration. The same is true for the histories produced with N = 6. We

note that the solution produced at θ = 180 with reduced integration and N = 6 produces a C-error

approximately equal to that for the solution produced with N = 3 and full integration. Thus, while

reduced integration allows efficient evaluation at the element level, 7.7 times as many fluid DOF were

64

required for a solution with accuracy equivalent to that produced with full integration.

-0.8

-0.4

0

0 2 4 6 8 10

-0.2

0

0.2

0.4

wρ0c/

p0

tc/a

Benchmarkθ = 0

θ = 180

1-Point Int. (0.095, 0.260)

8-Point Int. (0.092, 0.122)

(a) N = 3: 47,974 Fluid DOF

-0.8

-0.4

0

0 2 4 6 8 10

-0.2

0

0.2

0.4

wρ0c/

p0

tc/a

Benchmarkθ = 0

θ = 180

1-Point Int. (0.060, 0.121)

8-Point Int. (0.043, 0.097)

(b) N = 6: 369,907 Fluid DOF

Figure 4.25: Nondimensional radial-velocity histories calculated with CAFE-TFM and either 1- or8-point integration for evaluation of the fluid reactance matrices. The benchmark solutions are theproduct of a CASE-SFM model with N = 14. The numbers in parentheses are the C-error valuesassociated with the response histories at θ = 0 and θ = 180, respectively. The hydrostatic pressurecorresponds to a depth of 50 m and cavitation effects were included.

65

4.6 Summary and Conclusions

In this chapter we have performed a thorough analysis of the performance of the proposed CAFE

improvements when applied to the 3-D problem of a submerged spherical shell excited by a spherical,

step-exponential wave. It was found that, when cavitation is absent, the scattered-field model produces

results comparable in accuracy to those produced with the total-field model if the model is moderately

refined. This was because a too-coarse mesh was incapable of treating the discontinuous scattered field.

For the problem at a 50 m depth, in which cavitation was much more pronounced, the SFM was the

optimal choice. In this system, the TFM produced too much frothing. Overall, the SFM appears to

be the better choice due to its flexibility and superior accuracy when substantial cavitation is present.

For little or no cavitation, it was found that CASE yields significant savings in terms of required

fluid DOF and required memory relative to CAFE. When cavitation played a major role in structure

response, CAFE outperformed CASE in terms of required DOF. For most systems studied, CAFE

outperformed CASE in terms of required operations. However, CASE offered significant savings in

terms of required memory for nearly all systems. Overall, CASE appears to be a better choice than

CAFE, as the savings in memory appear more significant than the savings in required operations

for CAFE. Further, CASE is better suited to parallelization because there is no formation of global

matrices; CAFE is only competitive when a global reactance matrix is formed, which somewhat

complicates parallelization. Overall, the presence of cavitation degrades the performance of both

CAFE and CASE.

Finally, this chapter provided further validation of time-step subcycling and the non-conformal

structure-fluid coupling. With the combination of these methods, order-of-magnitude savings were

readily achieved.

Chapter 5

3-D Evaluation: Ship Structure

In the previous two chapters, we demonstrated that CASE refinement with time-step subcycling

and the scattered-field model is the best of the methods examined for the simulation of fluid-structure-

interaction problems. In this section, this method is used to calculate the response of a box-like ship

structure to an underwater explosion. We demonstrate that, with the FE/CASE/NRB method, one

can analyze realistic 3-D ship-shock problems with sufficient accuracy and manageable computational

cost. Further, we investigate the efficacy of fluid-mesh truncation in both the horizontal and vertical

directions.

5.1 Problem Description

The ship structure for this study was developed to represent a modern US frigate [24] and is

shown in Fig. 5.1. The internal point masses are such that the draft is 3.05 m; they are supported

by springs (represented by dashed lines) that yield a fixed-base natural frequency of 5 Hz for vertical

motion. The hull positions where velocities will be recorded are shown by ×’s, which are identified by

the circled numbers.

The hull bottom and sides are 12.7 mm and 9.6 mm thick, respectively. The bulkheads are

25.4 mm thick and the two decks are 12.7 mm thick. The hull is supported by an orthotropic grid

of T-type stiffeners. The longitudinal stiffeners, or stringers, are spaced every 0.610 m on the hull

sides and every 0.653 m on the hull bottom. There are eight, evenly spaced (2.743 m) large transverse

stiffeners between each bulkhead. In between each of these are three evenly spaced (0.610 m) smaller

transverse stiffeners, or intercostals. Finally, a stiffener representing the keel runs along the bottom

and ends of the hull. The web and flange dimensions for the stiffeners and keel are listed in Table

5.1. The total ship mass is 3.14 × 106 kg, of which, 16% constitutes the mass of the hull, stiffeners,

bulkheads and decks. The ship is excited by a 45.4 kg charge of TNT detonated amidships at a depth

of 15 m and offset from the keel on the starboard side by 15 m, which yields a structure standoff of

67

15.9 m.

Table 5.1: Web and flange dimensions for the ship T-type stiffeners. All dimensions are shown in mm.

Stiffeners Web Len. Web Thick. Flange Len. Flange Thick.Stringers 152 8.4 152 8.4

Large Trans. 305 10.2 102 10.2Intercostals 76 1.4 48 1.4

Keel 594 22.2 305 22.2

5.2 Discrete Model

This problem may be analyzed with the half-symmetry finite-element model shown in Fig. 5.2.

Figure 5.2(a) shows the plate elements used for the hull, decks, and bulkheads. Figure 5.2(b) shows

the beam elements that represent the hull stiffeners and the truss elements that support the internal

point masses. In Fig. 5.2(b), the two large gaps between the transverse stiffeners indicate the location

of the bulkheads, and the large gap between the stringers indicates the position of the lower deck. The

model has 5,334 4-node plate elements and 4,705 2-node beam elements, which yield a total of 31,043

DOF.

The half-symmetry base fluid model that is coupled to the structure model is shown in Fig. 5.3.

The fluid model is composed of 86,100 hexagonal elements and is bounded by a PWA NRB. The mesh

dimensions are such that V = 10 m and H = 10 m, where H and V are the horizontal and vertical

distances, respectively, of the mesh boundary from the structure. These dimensions and the charge

location are illustrated in Fig. 5.4. The structure is connected to the fluid through 2,230 wet-surface

elements, and the base fluid mesh has 9,700 PWA NRB elements.

5.3 Results

The incident-field generator for this problem was developed by Geers and Hunter [47] and Hunter

and Geers [63]. The model incorporates both the shock-wave and bubble-oscillation phases of motion.

Because the incident field propagates quasi-acoustically in the shock-wave phase, the scattered-field

model must be used. Velocity response histories at 12 points along on the ship’s hull (see Fig. 5.1) were

calculated with four CASE-refined models (N = 1 − 4) and the SFM. The structure was integrated

with ∆t = 1.99 × 10−6 s, which is approximately 75% of the experimentally determined ∆tcr, and

the fluid models were integrated with β = 0.25. Fluid-model information is shown in Table 5.3;

the subcycling ratios were chosen such that the fluid time increment was less than the critical time

68

Starboard

Port

109

21.8

12.212.212.2

4.57

1.52

9.14

B B

AA

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

Section A-A

Section B-B

Top View

Side View

Linear Spring

Internal-Structure Mass

Decks

Bulkhead

Figure 5.1: Box-like ship model. All dimensions are shown in meters. The ×’s indicate the positions where structure velocities normalto the hull are recorded; the position-identification numbers are shown in the circles next to the ×’s. The hull stiffeners are not shown.

69

(a)

(b)

Figure 5.2: Half-symmetry ship-structure finite-element model: (a) shell-element model and (b) itssupporting beam-element model.

70

Figure 5.3: Half-symmetry base fluid model coupled to the ship model shown in Fig. 5.2.

H

V15 m

15 m

Structure MeshFluid Mesh

Charge

Figure 5.4: Fluid-mesh dimensions and amidship charge location.

increment predicted with Gerschgorin’s theorem (∆tgcr). Note that α 1 for N = 1−4, which implies

substantial computational savings relative to computations performed without subcycling.

Figure 5.5 shows vertical-velocity response histories calculated at Positions 1-6 along the keel

with either N = 3 or N = 4 CASE refinement. Response histories calculated with and without

cavitation are shown for N = 3. For the histories calculated with N = 3, the C-error relative to

the N = 4 solution is shown in the legend. We see that the solutions produced with N = 3 (with

cavitation) and N = 4 CASE refinement show good agreement, thus indicating a sufficiently converged

solution. Similar agreement is seen in Fig. 5.6 for the horizontal-velocity response histories calculated

at Positions 7-12 along the hull’s starboard side. Figures 5.5 and 5.6 show that cavitation has a

substantial influence on structure response and must be considered. Figure 5.7 shows the C-error

values relative to the N = 4 response histories for histories produced at hull Positions 1-12 with

71

Table 5.2: Fluid-model properties associated with CASE refinement of the base fluid mesh shown inFig. 5.3.

N Fluid DOF ∆tgcr × 105 (s) α

1 94,320 19.7 982 721,339 6.97 343 2,397,658 3.58 174 5,639,877 2.18 10

N = 1 − 3 CASE refinement and cavitation effects included. With C = 0.15 as the limit for a

satisfactory accuracy, N = 3 refinement is required. However, the histories produced with N = 2

are quite good, with the responses at Positions 10, and 11 only slightly exceeding C = 0.15. Finally,

we note that the structure model would require 2, 230 × 42 = 35, 680 wet elements with the original

CAFE/FE approach instead of the 2,230 to accommodate N = 4 CASE refinement.

Figures 5.8 - 5.10 show cavitation snapshots on the symmetry and keel planes at several times;

the dark regions represent uncavitated fluid and the white areas below the free surface indicate the

presence of cavitation. The boundaries of these graphs correspond to the boundaries of the fluid mesh.

While it is virtually impossible to capture all cavitation in the horizontal directions (see Section 1.1),

the H = V = 10 m model captured all cavitation in the vertical direction. We see that the propagation

of cavitation is quite complicated and that significant amounts of fluid accretion occur along the

length of the ship. Because of this, Figs. 5.8 - 5.10 serve as further evidence that the wet-surface

approximations discussed in Section 1.3.2 are poorly suited to problems of this type.

Late-time cavitation occurs almost exclusively near the symmetry plane. We see that when the

cavitated volume directly beneath the keel in Figs. 5.9(b)-(d) closes, the resulting positive pressure

pulse is reflected off of the hull and causes the large cavitation zone that is seen in Figs. 5.10(a)-(b).

We note further that this pressure pulse causes the distinct jump in velocity seen at hull Position 1 at

t ≈ 34 ms (Fig. 5.5(a)). Finally, because cavitation has virtually disappeared by t ≈ 60 ms, one could

use the the results from a CASE/FE analysis for initial conditions in a FE/BE analysis of late-time

structural motion.

72

0 10 20 30 40 50 60-2

-1

0

1

2

3

N = 4Vel

ocity

(m/s

)

Time (ms)

N = 3, C = 0.043

N = 3 (No Cav.), C = 0.353

(a) Position 1

0 10 20 30 40 50 60

0

1

2

N = 4

Vel

ocity

(m/s

)

Time (ms)

N = 3, C = 0.033

N = 3 (No Cav.), C = 0.242

(b) Position 2

0 10 20 30 40 50 60-1

0

1

2

N = 4

Vel

ocity

(m/s

)

Time (ms)

N = 3, C = 0.036

N = 3 (No Cav.), C = 0.194

(c) Position 3

0 10 20 30 40 50 60

-0.4

0

0.4

0.8

N = 4

Vel

ocity

(m/s

)

Time (ms)

N = 3, C = 0.036

N = 3 (No Cav.), C = 0.243

(d) Position 4

0 10 20 30 40 50 60

-0.2

0

0.2

0.4

0.6

0.8

N = 4

Vel

ocity

(m/s

)

Time (ms)

N = 3, C = 0.027

N = 3 (No Cav.), C = 0.196

(e) Position 5

0 10 20 30 40 50 60

-0.5

0

0.5

1

N = 4

Vel

ocity

(m/s

)

Time (ms)

N = 3, C = 0.032

N = 3 (No Cav.), C = 0.191

(f) Position 6

Figure 5.5: Vertical-velocity response histories calculated with two CASE-refined fluid models. The histories pertain to Positions 1-6along the keel as illustrated in Fig. 5.1. The C-values indicate the comprehensive error of the solutions relative to the N = 4 solutions.

73

0 10 20 30 40 50 60

-2

0

2

4

6

N = 4

Vel

ocity

(m/s

)

Time (ms)

N = 3, C = 0.048

N = 3 (No Cav.), C = 0.382

(a) Position 7

0 10 20 30 40 50 60

-1

0

1

2

N = 4

Vel

ocity

(m/s

)

Time (ms)

N = 3, C = 0.053

N = 3 (No Cav.), C = 0.452

(b) Position 8

0 10 20 30 40 50 60

-1

0

1

2

3

N = 4

Vel

ocity

(m/s

)

Time (ms)

N = 3, C = 0.046

N = 3 (No Cav.), C = 0.248

(c) Position 9

0 10 20 30 40 50 60

-0.8

-0.4

0

0.4

0.8

N = 4Vel

ocity

(m/s

)

Time (ms)

N = 3, C = 0.062

N = 3 (No Cav.), C = 0.436

(d) Position 10

0 10 20 30 40 50 60

-0.5

0

0.5

1

1.5

2

N = 4

Vel

ocity

(m/s

)

Time (ms)

N = 3, C = 0.057

N = 3 (No Cav.), C = 0.288

(e) Position 11

0 10 20 30 40 50 60

-0.5

0

0.5

1

N = 4Vel

ocity

(m/s

)

Time (ms)

N = 3, C = 0.058

N = 3 (No Cav.), C = 0.327

(f) Position 12

Figure 5.6: Horizontal-velocity response histories calculated with two CASE-refined fluid models. The histories pertain to Positions7-12 along the hull’s starboard side as illustrated in Fig. 5.1. The C-values indicate the comprehensive error of the solutions relativeto the N = 4 solutions.

74

1e+05 1e+060.01

0.10

C-E

rror

Fluid DOF

N = 1

N = 2

N = 3

Unsatisfactory

Satisfactory

Pos. 1Pos. 2Pos. 3Pos. 4Pos. 5Pos. 6

(a)

1e+05 1e+060.01

0.10

C-E

rror

Fluid DOF

N = 1

N = 2

N = 3

Unsatisfactory

Satisfactory

Pos. 7Pos. 8Pos. 9Pos. 10Pos. 11Pos. 12

(b)

Figure 5.7: C-error values for velocity response histories produced at Positions (a) 1-6 and (b) 7-12on the hull with three CASE-refined fluid models (N = 1 − 3) and H = V = 10 m. The benchmarkresults were produced with N = 4 CASE refinement.

75

-60 -50 -40 -30 -20 -10 0

-10

-5

0 Hull Bulkheads

Symmetry Plane

Distance from Amidships (m)

Dep

th(m

)

-10 -5 0 5 10

-10

-5

0

-10 -5 0 5 10

-10

-5

0 Hull

Distance from Keel (m)

Dep

th(m

)

(a) t = 1.2 ms

-60 -50 -40 -30 -20 -10 0

-10

-5

0 Hull Bulkheads

Symmetry Plane

Distance from Amidships (m)

Dep

th(m

)

-10 -5 0 5 10

-10

-5

0

-10 -5 0 5 10

-10

-5

0 Hull

Distance from Keel (m)

Dep

th(m

)

(b) t = 2.1 ms

-60 -50 -40 -30 -20 -10 0

-10

-5

0 Hull Bulkheads

Symmetry Plane

Distance from Amidships (m)

Dep

th(m

)

-10 -5 0 5 10

-10

-5

0

-10 -5 0 5 10

-10

-5

0 Hull

Distance from Keel (m)

Dep

th(m

)

(c) t = 4.2 ms

-60 -50 -40 -30 -20 -10 0

-10

-5

0 Hull Bulkheads

Symmetry Plane

Distance from Amidships (m)

Dep

th(m

)

-10 -5 0 5 10

-10

-5

0

-10 -5 0 5 10

-10

-5

0 Hull

Distance from Keel (m)

Dep

th(m

)

(d) t = 8.7 ms

-60 -50 -40 -30 -20 -10 0

-10

-5

0 Hull Bulkheads

Symmetry Plane

Distance from Amidships (m)

Dep

th(m

)

-10 -5 0 5 10

-10

-5

0

-10 -5 0 5 10

-10

-5

0 Hull

Distance from Keel (m)

Dep

th(m

)

(e) t = 13.1 ms

Figure 5.8: Cavitation snapshots of the keel plane (left column) and the symmetry plane (right column)for 1.2 ≤ t ≤ 13.1 ms. Results were calculated with H = V = 10 m and N = 4.

76

-60 -50 -40 -30 -20 -10 0

-10

-5

0 Hull Bulkheads

Symmetry Plane

Distance from Amidships (m)

Dep

th(m

)

-10 -5 0 5 10

-10

-5

0

-10 -5 0 5 10

-10

-5

0 Hull

Distance from Keel (m)

Dep

th(m

)

(a) t = 17.6 ms

-60 -50 -40 -30 -20 -10 0

-10

-5

0 Hull Bulkheads

Symmetry Plane

Distance from Amidships (m)

Dep

th(m

)

-10 -5 0 5 10

-10

-5

0

-10 -5 0 5 10

-10

-5

0 Hull

Distance from Keel (m)

Dep

th(m

)(b) t = 22.1 ms

-60 -50 -40 -30 -20 -10 0

-10

-5

0 Hull Bulkheads

Symmetry Plane

Distance from Amidships (m)

Dep

th(m

)

-10 -5 0 5 10

-10

-5

0

-10 -5 0 5 10

-10

-5

0 Hull

Distance from Keel (m)

Dep

th(m

)

(c) t = 26.6 ms

-60 -50 -40 -30 -20 -10 0

-10

-5

0 Hull Bulkheads

Symmetry Plane

Distance from Amidships (m)

Dep

th(m

)

-10 -5 0 5 10

-10

-5

0

-10 -5 0 5 10

-10

-5

0 Hull

Distance from Keel (m)

Dep

th(m

)

(d) t = 31.0 ms

-60 -50 -40 -30 -20 -10 0

-10

-5

0 Hull Bulkheads

Symmetry Plane

Distance from Amidships (m)

Dep

th(m

)

-10 -5 0 5 10

-10

-5

0

-10 -5 0 5 10

-10

-5

0 Hull

Distance from Keel (m)

Dep

th(m

)

(e) t = 35.5 ms

Figure 5.9: Cavitation snapshots of the keel plane (left column) and the symmetry plane (right column)for 17.6 ≤ t ≤ 35.5 ms. Results were calculated with H = V = 10 m and N = 4.

77

-60 -50 -40 -30 -20 -10 0

-10

-5

0 Hull Bulkheads

Symmetry Plane

Distance from Amidships (m)

Dep

th(m

)

-10 -5 0 5 10

-10

-5

0

-10 -5 0 5 10

-10

-5

0 Hull

Distance from Keel (m)

Dep

th(m

)

(a) t = 40.0 ms

-60 -50 -40 -30 -20 -10 0

-10

-5

0 Hull Bulkheads

Symmetry Plane

Distance from Amidships (m)

Dep

th(m

)

-10 -5 0 5 10

-10

-5

0

-10 -5 0 5 10

-10

-5

0 Hull

Distance from Keel (m)

Dep

th(m

)

(b) t = 44.5 ms

-60 -50 -40 -30 -20 -10 0

-10

-5

0 Hull Bulkheads

Symmetry Plane

Distance from Amidships (m)

Dep

th(m

)

-10 -5 0 5 10

-10

-5

0

-10 -5 0 5 10

-10

-5

0 Hull

Distance from Keel (m)

Dep

th(m

)

(c) t = 49.0 ms

-60 -50 -40 -30 -20 -10 0

-10

-5

0 Hull Bulkheads

Symmetry Plane

Distance from Amidships (m)

Dep

th(m

)

-10 -5 0 5 10

-10

-5

0

-10 -5 0 5 10

-10

-5

0 Hull

Distance from Keel (m)

Dep

th(m

)

(d) t = 53.4 ms

-60 -50 -40 -30 -20 -10 0

-10

-5

0 Hull Bulkheads

Symmetry Plane

Distance from Amidships (m)

Dep

th(m

)

-10 -5 0 5 10

-10

-5

0

-10 -5 0 5 10

-10

-5

0 Hull

Distance from Keel (m)

Dep

th(m

)

(e) t = 57.9 ms

Figure 5.10: Cavitation-zone snapshots of the keel plane (left column) and the symmetry plane (rightcolumn) for 40.0 ≤ t ≤ 57.9 ms. Results were calculated with H = V = 10 m and N = 4.

78

5.4 Mesh Truncation

As mentioned in the Section 1.4.5, several investigators have examined the efficacy of fluid-

mesh truncation when performing near-free-surface ship-shock calculations [107, 101, 75]. However, a

satisfactory 3-D examination remains to be completed, which is the goal of this section. To this end,

we repeat the ship-response calculations of the previous section but use smaller H and V values for

the base fluid mesh and N = 3 CASE refinement with the SFM. Table 5.4 shows the properties of the

truncated fluid meshes. The elements in the fluid meshes for 2 ≤ H,V ≤ 8 are very similar to those

shown in Fig. 5.3 for the H = V = 10 m model. The elements for H,V = 0.5 m are somewhat smaller

in the direction normal to the hull.

Table 5.3: Fluid-model properties for the truncated fluid meshes and N = 3 CASE refinement.

H, V (m) Fluid DOF ∆tgcr × 105 (s) α

10 2,397,658 3.58 178 1,669,810 3.58 176 1,260,088 3.42 174 610,204 3.58 172 260,950 3.58 17

0.5 89,044 2.86 14

Figure 5.11 shows C-error values for velocity response histories produced at the 12 hull positions.

The benchmark solutions were produced with N = 3 CASE and H = V = 10 m (Fig. 5.3). With

H,V ≥ 4 m, all response histories produce C < 0.15. This indicates that the model with H = V = 10

m is sufficiently large for benchmark calculations.

The H = V = 4 m model has only 25% of the fluid DOF used in the H = V = 10 m model. For

H = V = 2 m, which has only 11% of the fluid DOF used in the benchmark model, only Position 1 has

a response histories with C > 0.15. The fluid model with H = V = 0.5 m produces response histories

with unacceptable error. Figures 5.12 and 5.13 show the velocity histories produced at Positions 1-6

and 7-12, respectively, with the H = V = 10 m and H = V = 2 m models. We see that the latter

model fails to capture the sharp velocity jump seen at Position 1 at t ≈ 34 ms. This is not surprising,

as the velocity jump is due to the closure of a cavitation region that occurs at approximately 2.5 m

below the hull.

These results support the conclusion in Sprague and Geers [103], which states that a truncated

fluid mesh will produce satisfactory structure responses if all regions of fluid accretion are enclosed in

the fluid mesh. Figures 5.8 - 5.10 indicate that accretion rarely extends below 4 m from the hull bottom.

Further, the results support the conclusion of Malone and Shin [75], which states that satisfactory

79

structure responses can be produced if the fluid mesh extends to at least half of the maximum cavitation

depth predicted with the methods in [25]. For the charge used here, the maximum cavitation depth

is approximately 9 m. Thus, half of this depth is a conservative estimate for the maximum fluid-mesh

depth for the current problem.

0 2 4 6 8 100.00

0.05

0.10

0.15

0.20

0.25

0.30C

-Err

or

H,V (m)

Unsatisfactory

Satisfactory

Pos. 1Pos. 2Pos. 3Pos. 4Pos. 5Pos. 6

(a)

0 2 4 6 8 100.00

0.05

0.10

0.15

0.20

0.25

0.30

C-E

rror

H,V (m)

Unsatisfactory

Satisfactory

Pos. 7Pos. 8Pos. 9Pos. 10Pos. 11Pos. 12

(b)

Figure 5.11: C-error for (a) vertical-velocity histories at hull Positions 1-6 and (b) horizontal-velocityhistories at hull Positions 7-12 produced with truncated CASE meshes and N = 3 refinement. Thebenchmark solutions were produced with the base fluid mesh shown in Fig. 5.3 (H = V = 10 m) withN = 4 refinement.

80

0 10 20 30 40 50 60-3

-2

-1

0

1

2

3

H = V = 10 mVel

ocity

(m/s

)

Time (ms)

H = V = 2 m

(a) Position 1 (C = 0.155)

0 10 20 30 40 50 60

-0.5

0

0.5

1

1.5

2

H = V = 10 m

Vel

ocity

(m/s

)

Time (ms)

H = V = 2 m

(b) Position 2 (C = 0.085)

0 10 20 30 40 50 60-1

-0.5

0

0.5

1

1.5

2

H = V = 10 m

Vel

ocity

(m/s

)

Time (ms)

H = V = 2 m

(c) Position 3 (C = 0.132)

0 10 20 30 40 50 60

-0.4

-0.2

0

0.2

0.4

0.6

0.8

H = V = 10 m

Vel

ocity

(m/s

)

Time (ms)

H = V = 2 m

(d) Position 4 (C = 0.057)

0 10 20 30 40 50 60-0.4

-0.2

0

0.2

0.4

0.6

0.8

H = V = 10 m

Vel

ocity

(m/s

)

Time (ms)

H = V = 2 m

(e) Position 5 (C = 0.090)

0 10 20 30 40 50 60

-0.5

0

0.5 H = V = 10 m

Vel

ocity

(m/s

)

Time (ms)

H = V = 2 m

(f) Position 6 (C = 0.147)

Figure 5.12: Vertical-velocity response histories calculated with either the benchmark model (H = V = 10 m, N = 4) or H = V = 2m with N = 3. The histories pertain to Positions 1-6 along the keel as illustrated in Fig. 5.1. The numbers in parentheses are theC-error of the solutions produced with H = V = 2 m compared to the benchmark solution.

81

0 10 20 30 40 50 60

-2

0

2

4

H = V = 10 mVel

ocity

(m/s

)

Time (ms)

H = V = 2 m

(a) Position 7 (C = 0.077)

0 10 20 30 40 50 60-1

-0.5

0

0.5

1

1.5

2

H = V = 10 m

Vel

ocity

(m/s

)

Time (ms)

H = V = 2 m

(b) Position 8 (C = 0.062)

0 10 20 30 40 50 60

0

1

2H = V = 10 m

Vel

ocity

(m/s

)

Time (ms)

H = V = 2 m

(c) Position 9 (C = 0.106)

0 10 20 30 40 50 60

-0.4

-0.2

0

0.2

0.4

0.6

H = V = 10 m

Vel

ocity

(m/s

)

Time (ms)

H = V = 2 m

(d) Position 10 (C = 0.064)

0 10 20 30 40 50 60

-0.5

0

0.5

H = V = 10 mVel

ocity

(m/s

)

Time (ms)

H = V = 2 m

(e) Position 11 (C = 0.110)

0 10 20 30 40 50 60

-0.4

-0.2

0

0.2

0.4

0.6

H = V = 10 mVel

ocity

(m/s

)

Time (ms)

H = V = 2 m

(f) Position 12 (C = 0.076)

Figure 5.13: Horizontal-velocity response histories calculated with either the benchmark model (H = V = 10 m, N = 4) or H = V =2 m with N = 3. The histories pertain to Positions 7-12 along the hull’s starboard side as illustrated in Fig. 5.1. The numbers inparentheses are the C-error of the solutions produced with H = V = 2 m compared to the benchmark solution.

Chapter 6

Conclusion

In this dissertation, we advanced the state-of-the-art for computational simulation of fluid-

structure interaction with cavitation by introducing four improvements to the original cavitating

acoustic finite element (CAFE) approach of Felippa and DeRuntz [36]. These improvements are: (i)

incorporation of field separation, (ii) replacement of CAFE with cavitating acoustic spectral elements

(CASE), (iii) incorporation of a non-conformal structure-fluid coupling method, and (iv) incorporation

of time-step subcycling. The improvements are independent of one another, thus allowing utilization

of all or some in current finite-element software.

The performance of the above methods was examined with two canonical problems: (i) a 2-

DOF floating mass-spring oscillator excited by a plane, step-exponential wave, and (ii) a submerged

spherical shell excited by a spherical, step-exponential wave. Both problems were examined with and

without cavitation. Regarding the performance of the improvements, the conclusions are as follows:

Field Separation:

Separation of the total field into equilibrium, incident, and scattered fields, i.e., the use of a scattered-

field model (SFM), is appealing because it allows accurate propagation of the incident field and admits

realistic, quasi-acoustic shock fields. When cavitation effects were not prominent, SFM performed

slightly worse than TFM in the coarsest models used. However, for modest levels of refinement, SFM

and TFM performed equally well. However, when extensive cavitation was present in the 3-D problem,

SFM performed considerably better than TFM, thereby allowing substantial reduction in the required

number of fluid DOF.

Cavitating Acoustic Spectral Elements:

For the problems studied here, CASE offered significant savings in required memory storage relative to

CAFE. This is due to the existence of a tensor-product factorization that allows evaluation of the CASE

matrices at the element level with minimal memory requirements. In terms of operations required

for a given accuracy level, CASE outperformed CAFE for the 1-D problem. For the spherical-shell

83

problem, CASE and CAFE had similar operation requirements when cavitation effects were small. For

the shallow problem, which exhibited extensive cavitation, CAFE had smaller operation requirements.

Overall, CASE appears to be the clear choice over CAFE due to similar operation requirements, much

smaller memory requirements, and ease of parallelization. The CAFE implementation used here,

which requires many fewer operations than an element-by-element formulation, is cumbersome due

to the global sparse storage scheme. CAFE can be efficiently implemented at the element level only

if reduced integration is used. However, it was demonstrated that reduced integration degrades the

method’s accuracy to an unacceptable level.

Non-conformal Fluid-Structure Coupling:

The use of non-conformal coupling produced large savings in the computational effort required for the

structure. With satisfactory refinement in the structure, a converged fluid model was found without

further structure refinement. Further, the coupling method is very simple in terms of implementation

and usage.

Time-Step Subcycling:

Time-step subcycling offered large gains in accuracy for a given number of operations. It was found

that, for a given number of fluid DOF, accuracy is improved by the incorporation of subcycling.

With the optimal approach, i.e., CASE refinement with the SFM, non-conformal coupling, and

subcycling, we examined the simulated response of a realistic ship structure excited by an underwa-

ter explosion. It was demonstrated that, with the optimal approach, one can simulate realistic 3-D

problems with satisfactory accuracy and manageable computations. For the structure, which has ap-

proximately 30,000 DOF, satisfactory responses were achieved with a fluid model of N = 3 refinement

(2.4 × 106 fluid DOF).

Finally, the efficacy of mesh truncation was investigated for the ship-shock problem. The results

support the conclusion of Sprague and Geers [103], which states that satisfactory structure-response

histories will be produced if all regions of fluid accretion are captured in the fluid mesh. The results

also support the conclusion of Malone and Shin [75], which states that satisfactory response histories

will be produced if the fluid mesh extends to at least half the maximum cavitation depth predicted

by the methods of [25]. For the ship problem studied here, a mesh with H = V = 4 m and N = 3

required only 0.6 × 106 fluid DOF, whereas the H = V = 10 m mesh, which captured all cavitation

beneath the hull, required 2.4 × 106 fluid DOF.

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Appendix A

CASE: Element-Level Matrices and Matrix-Vector Products

A.1 Explicit Matrix Formation

As discussed in Section 2.4.2, element-level capacitance and reactance matrices are required for

Gerschgorin approximation for the maximum eigenvalue of the discrete fluid system. We evaluate the

integrals in the matrix equations (2.28) with GLL quadrature and rewrite them as

Qeil

:=N∑

o,p,q=0

φi(ξo, ηp, ζq)φl(ξo, ηp, ζq)ωoωpωqJopq, (A.1)

Heil

:=N∑

o,p,q=0

~∇φi(ξo, ηp, ζq) · ~∇φl(ξo, ηp, ζq)ωoωpωqJopq, (A.2)

where

φi(ξo, ηp, ζq) := φi(ξo)φj(ηp)φk(ζq), (A.3)

a subscript with a hat denotes lexicongraphical numbering defined by

i := 1 + i+ (N + 1)j + (N + 1)2k, ∀i, j, k ∈ 0, . . . ,N,l := 1 + l + (N + 1)m+ (N + 1)2n, ∀l,m, n ∈ 0, . . . ,N,

(A.4)

Jopq is the determinant of the Jacobian matrix J(ξ, η, ζ) (generated subparametrically) evaluated at

the point (ξo, ηp, ζq), and ωi is the weight associated with the ith GLL quadrature point. After utilizing

(2.24) and performing the contractions, we find that the capacitance matrix (A.1) becomes a diagonal

matrix with entries

Qeii

:= ωiωjωkJijk, ∀i, j, k ∈ 0, . . . ,N. (A.5)

For the reactance matrix (A.2), we apply the chain rule of differentiation, which yields

Heil

=N∑

o,p,q=0

[φi,ξ

(Γ11opqφl,ξ + Γ12opqφl,η + Γ13opqφl,ζ

)

+ φi,η

(Γ21opqφl,ξ + Γ22opqφl,η + Γ23opqφl,ζ

)+ φi,ζ

(Γ31opqφl,ξ + Γ32opqφl,η + Γ33opqφl,ζ

)]ωoωpωqJopq,

(A.6)

92

where a subscript preceded by a comma denotes differentiation, e.g., φi,ξ = ∂φi∂ξ , and Γijopq is the ij

element of the 3 × 3 symmetric matrix Γ = J−1TJ−1 evaluated at the point (ξo, ηp, ζq). Because of

symmetry, Γijopq has only six unique entries.

For efficient evaluation of (A.6), we expand the basis-function terms with (A.3), utilize (2.24),

perform tensor contractions, and rearrange terms to yield

Heil

=N∑

q=0

(Ge

11qmnDiqDlqδjmδkn +Ge22lqkDjqDmqδilδkn +Ge

33lmqDkqDnqδilδjm)

+Ge12ljkDilDmjδkn +Ge

12imkDjmDliδkn +Ge23imkDjmDnkδil

+Ge23ijnDknDmjδil +Ge

13ljkDilDnkδjm +Ge13ijnDknDliδjm,

∀i, j, k, l,m, n ∈ 0, . . . ,N,

(A.7)

where Geopijk = ΓopijkωiωjωkJijk is composed of six tensors (for op = 11, 22, 33, 12, 23, 13), and

Dij =∂φi

∂ξ

∣∣∣∣ξj

. (A.8)

Geopijk contains all relevant geometric information for the element e. We note that evaluation of (A.7)

is computationally expensive as it requires O(N7) operations. However, these operations need to be

performed only once per element.

A.2 Matrix-Vector Product: Tensor-Product Factorization

As discussed in [71, 30], efficient evaluation of the matrix-vector product Hψ in (2.29) and Hψsc

in (2.31a) can be most efficiently achieved at the element level using tensor-product factorization.

Starting with (A.2), we may write the element-level matrix-vector product in its unsimplified form as

Heilψl :=

N∑l,m,n=0

N∑o,p,q=0

~∇φi(ξo, ηp, ζq) · ~∇φl(ξo, ηp, ζq)ωoωpωqJopqψl, ∀i, j, k ∈ 0, . . . ,N, (A.9)

which requires O(N9) operations to evaluate. However, the properties of the Legendre-polynomial

spectral basis functions allows substantial simplification of (A.9). To this end, we utilize (A.7), perform

93

all contractions, and rearrange terms to yield

Heilψe

l=

N∑p=0

DipG

e11pjk

N∑

q=0

Dqpψeqjk

+DjpG

e22ipk

N∑

q=0

Dqpψeiqk

+DkpGe33ijp

N∑

q=0

Dqpψeijq

+DipG

e12pjk

N∑

q=0

Dqjψepqk

+DjpGe12ipk

N∑

q=0

Dqiψeqpk

+DjpG

e23ipk

N∑

q=0

Dqkψeipq

+DkpGe23ijp

N∑

q=0

Dqjψeiqp

+DipG

e13pjk

N∑

q=0

Dqkψepjq

+DkpGe13ijp

N∑

q=0

Dqiψeqjp

.

(A.10)

For efficient evaluation, Gelmijk should be calculated and stored for all elements, which requires 6(N+1)3

positions of global storage per element. For evaluation of (A.10), one proceeds by evaluating and

storing the three tensors

γ1ijk =N∑

q=0

Dqiψeqjk, γ2ijk =

N∑q=0

Dqjψeiqk, γ3ijk =

N∑q=0

Dqkψeijq, ∀i, j, k ∈ 0, . . . ,N, (A.11)

which requires 3(N + 1)3 positions of local storage and 6(N + 1)4 operations, where an operation

constitutes an addition or multiplication. The element-level geometric information is then incorporated

into (A.11), which, after terms are grouped, yields an additional three tensors

γ1ijk = Ge11ijkγ1ijk +Ge

13ijkγ3ijk +Ge12ijkγ2ijk,

γ2ijk = Ge22ijkγ2ijk +Ge

23ijkγ3ijk +Ge12ijkγ1ijk,

γ3ijk = Ge33ijkγ3ijk +Ge

23ijkγ2ijk +Ge13ijkγ1ijk, ∀i, j, k ∈ 0, . . . ,N. (A.12)

This requires an additional 15(N + 1)3 operations and 3(N + 1)3 positions of local storage. Finally,

the element-level matrix-vector product at each node (ijk) may be calculated with

Heilψe

l=

N∑p=0

(Diq γ1ijk +Djqγ2ijk +Dkqγ3ijk) , (A.13)

which requires 6(N +1)4 operations. Thus, the element-level tensor-product matrix-vector multiplica-

tion requires 12(N + 1)4 + 15(N + 1)3 operations, 6(N + 1)3 positions of local storage, and 6(N + 1)3

positions of global storage for Gelmijk.

Appendix B

Temporal Discretization

This chapter gives a detailed presentation of the time-integration scheme used for the semi-

discrete equations presented in Chapter 2. An identical procedure is used for both CAFE and CASE.

As discussed in Section 2.4.1, we use a staggered-integration procedure with time-step subcycling.

Sections B.1 and B.2 discuss temporal discretization of the total-field and scattered-field models,

respectively. Section B.3 discusses the simultaneous pressure correction which is used to remove

unwanted oscillations that form at the CWA boundary. We note that the total-field formulation

closely follows that of Felippa and DeRuntz [36], however, we have included subcycling and use a

CWA instead of a DAA for the non-reflecting boundary.

B.1 Total-Field Model

Prior to time integration, the TFM requires initialization of the fluid response variables; the

initial conditions govern the propagation of the incident field. To this end, we

• initialize the velocity potential ψ−α/2

by integrating the incident-wave flux from t = −∞ to

t = −α∆t/2,

• initialize the displacement potential ψ0 by doubly integrating the incident-wave flux from

t = −∞ to t = 0

• initialize the absolute pressure.

We assume that we are at time t = n∆t and that all quantities are known in the fluid, NRB,

and structure equations. For each cycle, we wish to update all system values to time t+α∆t if α ≥ 1

(structure subcycling) or t+ ∆t if α < 1 (fluid subcycling), where ∆t is the time increment used for

the structure equations, and α∆t is that used for the fluid equations.

For the CWA-NRB, we use an incremental formulation with a predictor-corrector approach. The

following lists the equations used to calculate the predicted displacement (un+αpred) normal to Γnrb of all

95

nodes on the NRB:

∆∗pinc =

∗p

n+α

inc − ∗p

n

inc, ∆∗∗pinc =

α∆t2

( ∗p

n+α

inc +∗p

n

inc

),

∆∗peq = α∆tpeq,

∗p

n+α

eq =∗p

n

eq + ∆∗peq, ∆

∗∗peq = α∆t

(1 +

α∆t2t

)∗p

n+α

eq ,

∆uknown =1ρc

Θ∆∗pinc +

1ρΘR−1∆

∗∗pinc −

1ρc

(∆

∗pinc + ∆

∗peq

)− 1ρκ

(∆

∗∗pinc + ∆

∗∗peq

),

∆upred =α∆tρc

pn +α∆tρκ

∗p

n, un+α

pred = un + ∆uknown + ∆upred, (B.1)

where a superscript denotes the time position, and ∆uknown and ∆upred are the changes in normal

displacement due to known and estimated quantities, respectively, the latter of which is calculated

with the CWA.

As discussed in Section 2.4.1, the fluid-update equations require the predicted structure displace-

ment in addition to the NRB displacement predicted in (B.1). To this end, we write the boundary

interaction vector (2.43) for time t = (n+ α)∆t as

bn+α =∫

Γst

φ(xn + α∆t ˙xn)dΓ +∫

Γnrb

φφT (un+αpred − ueq)dΓ, (B.2)

where we have dropped the element-number e superscript, and we have used a forward Euler approx-

imation to predict the normal displacement of the structure wet-surface nodes.

We update the fluid equations with

s =(sn − sn−α

)/(α∆t), ψ

n+α= ψ

n+ α∆t

(pn − peq + βα∆tc2s

),

ψn+α = ψn + α∆tψn+α

, sn+α = −Q−1H(ψn+α −ψ0

)+ Q−1bn+α, (B.3)

where we have included the artificial-damping term βα∆tc2s. In the original CAFE formulation,

Felippa and DeRuntz [36] found that, with the staggered integration procedure, nonphysical pressure

oscillations develop near the NRB due to the time lag in the treatment of the PWA. They solved this

problem with a simultaneous pressure solution in which they multiplied the diagonal entries of Q for

nodes coupled to the NRB by a constant. The pressure-correction formulation modified for the CWA

may be found in Appendix B.3.

For nodes not on the free surface, the pressure is updated as

pn+αj =

c2sn+α + peqj ,

0

c2sn+αj > −peqj ,

c2sn+αj ≤ −peqj .(B.4)

For free-surface nodes, the pressure is simply atmospheric pressure.

96

We now use the known fluid pressure at t = (n+ α)∆t to calculate more accurately the normal

displacements at the NRB nodes with the following

∆∗p =

α∆t2

(pn+α + pn

),

∗p

n+α=

∗p

n+ ∆

∗p,

∆∗∗p =

α∆t2

( ∗p

n+α+

∗p

n), ∆ucor =

1ρc

∆∗p +

1ρκ∆

∗∗p,

un+α = un + ∆uknown + ∆ucor. (B.5)

If α < 1, the update of the fluid and NRB node values is repeated until their values are known for

t = (n + 1)∆t. When the fluid values are known at this time, or if α ≥ 1, the update of the fluid

equations is complete for the given cycle.

The semi-discrete structure FE equations (2.35) are also discretized in time with central differ-

ences. For each time-update cycle, the structure equations are updated to time t = (n+ 1)∆t if α ≤ 1

and t = (n+α)∆t if α > 1. To this end, the equations updating the dynamic structure displacements

are [23]

xn+µ = xn+µ−1 + ∆txn+µ−1 +∆t2

2xn+µ−1, xn+µ = −M−1Kxn+µ + M−1fn+µ,

xn+µ = xn+µ−1 +∆t2

(xn+µ−1 + xn+µ

), (B.6)

where µ = 1 if α ≤ 1, µ = (1, . . . , α) if α > 1, and fn+µ is composed of element-level translational

force vectors (see (2.42)) defined by

fn+µi = γi

∫Γwet

Npn+µdyn dΓ, i ∈ X,Y,Z. (B.7)

For α ≤ 1, the dynamic pressure is known. For α > 1, the dynamic pressure is interpolated from the

fluid pressures known at t = n∆t and t = (n+ α)∆t, i.e.,

pn+µdyn =

[(α− µ)pn

dyn + µpn+αdyn

]/α. (B.8)

If α > 1, the update of the above structure equations is repeated until time t+ α∆t. Otherwise, the

cycle is complete.

B.2 Scattered-Field Model

In this section, we discuss the time-update equations used for the scattered-field model. The

SFM time-update procedure closely follows that for the TFM. However, the SFM is simplified in that

it uses quiescent initial conditions (no initialization required), and the incident field is known at all

times and node positions.

97

As with the TFM, we assume that we are at time t = n∆t and that all quantities are known in

the fluid, NRB, and structure equations. Beginning with the NRB, a predictor-corrector incremental

formulation is used. The equations used to calculate the predicted scattered displacement (un+αsc−pred)

normal to Γnrb at all nodes on the NRB are

∆usc−pred =α∆tρc

pnsc +

α∆tρκ

∗p

n

sc, un+αsc−pred = un

sc + ∆usc−pred, (B.9)

of which, the first equation is the CWA. The boundary interaction vector (2.44) for time t = (n+α)∆t

is given by

bn+αsc =

∫Γst

φ(xnsc + α∆t ˙xn

sc)dΓ +∫

Γnrb

φφTun+αsc−preddΓ. (B.10)

The fluid equations are updated with

ssc =(snsc − sn−α

sc

)/(α∆t), ψ

n+αsc = ψ

nsc + α∆t

(pn + βα∆tc2ssc

),

ψn+αsc = ψn

sc + α∆tψn+αsc , sn+α

sc = −Q−1Hψn+αsc + Q−1bn+α

sc . (B.11)

For nodes not on the free surface, the pressure is updated as

pn+αsc j =

c2sn+α

sc j ,

−peq + pn+αinc j ,

c2sn+αsc j > −peq + pn+α

inc j ,

c2sn+αsc j ≤ −peq + pn+α

inc j .(B.12)

For free-surface nodes, the scattered pressure is −pn+αinc j .

The known fluid pressures at t = (n+α)∆t are used to correct the normal displacements at the

NRB nodes with

∆∗psc =

α∆t2

(pn+α

sc + pnsc

),

∗p

n+α

sc =∗p

n

sc + ∆∗psc,

∆∗∗psc =

α∆t2

( ∗p

n+α

sc +∗p

n

sc

), ∆usc−cor =

1ρc

∆∗psc +

1ρκ∆

∗∗psc,

un+αsc = un

sc + ∆usc−cor. (B.13)

If α < 1, the update of the fluid and NRB node values is repeated until their values are known for

t = (n + 1)∆t. When the fluid values are known at this time, or if α ≥ 1, the update of the fluid

equations is complete for the given cycle. The time-update equations for the structure model are

identical to those used for the TFM.

B.3 CWA Pressure Correction

In the implementation of Felippa and DeRuntz [36], the staggered integration procedure produces

undesirable pressure oscillations at the DAA boundary due to the time lag in the treatment of the

plane-wave portion. They solved this problem with a simultaneous pressure solution for the PWA

98

term. We follow their approach for the CWA used in the current method. The treatment is considered

at the node level, and we consider an individual node located on the NRB at ~X. We may write the

associated term of the boundary interaction vector b (with no structure forcing) as

b = −ρa (usc + uinc) , (B.14)

where a is the area associated with the NRB node. Using the CWA (2.38) for the scattered displace-

ment, we may rewrite the above as

b = −ac

∗psc − κa

∗∗p sc − ρauinc. (B.15)

We write the expression for t = (n+α)∆t, and discretize the pressure terms with the trapezoidal rule:

bn+α = − a

c

[∗p

n

sc +α∆t

2(pn

sc + pn+αsc

)]

− κa

∗∗p

n

sc +α∆t

2

[∗p

n

sc +α∆t

2(pn

sc + pn+αsc

)]− ρauinc.

(B.16)

We use pn+α = pn+αsc + peq + pn+α

inc and group terms to yield

bn+α = −(ac

+ κaα∆t) ∗p

n

sc −(aα∆t

2c+aκα2∆t2

4

)pn

sc − κa∗∗p

n

sc

−(aα∆t

2c+aκα2∆t2

4

)(pn+α − peq − pn+α

inc

) − ρauinc.

(B.17)

We write the pressure-update equation as

qpn+α =

c2 (r + bn+α)

0

c2sn+α > −peq,

c2sn+α ≤ −peq,(B.18)

where q is the diagonal value of q for the given node, r is the NRB nodal value of r = −Hψ and we

have allowed for cavitation. We first consider (B.18) without cavitation and use (B.16) to write the

pressure update equation as

qpn+α =c2 (r − ρauinc) −(ca+ c2κaα∆t

) ∗p

n

sc −(acα∆t

2+ac2κα2∆t2

4

)pn

sc

− c2κa∗∗p

n

sc −(acα∆t

2+ac2κα2∆t2

4

)(pn+α − peq − pn+α

inc

).

(B.19)

We solve for the unknown pressure term pn+α to yield

q (1 + γ) pn+α = l, (B.20)

where l represent all the known terms in (B.19), and

γ =1q

(acα∆t

2+ac2κα2∆t2

4

). (B.21)

99

The net effect of this is that the diagonal entries of Q that correspond to nodes on the NRB

must be multiplied by the correction factor (1+γ). We note that for κ = 0, γ reduces to the correction

value found in [36] for the PWA.

If we include cavitation, the pressure-update equation is simply pn+α = 0. Thus, the pressure

correction does not affect the solution when cavitation is present. A similar analysis may be carried

out for the SFM; an identical correction factor is obtained.