Modeling Submerged Structures Loaded by Underwater Explosions with ABAQUS/Explicit

15 Pages·2002·0.26 MB·English

Modeling Submerged Structures Loaded by
Underwater Explosions with ABAQUS/Explicit
David B. Woyak
ABAQUS Solutions Northeast, LLC
Abstract: Finite element analysis can be used to predict the transient response of submerged
structures that are externally loaded by an acoustic pressure shock wave resulting from an
Underwater Explosion (UNDEX). This class of problem is characterized by a strong coupling
between the structural motions and acoustic pressures at the fluid-structure wetted interface. The
structural behavior is a combination of long time (low frequency) response dominated by an
added mass effect, short time (high frequency) response dominated by radiation damping, and
intermediate time-frequency response where both added mass and radiation damping behavior
are present. For the finite element method to be useful, the analyst must develop modeling
techniques and procedures that yield accurate and computationally tractable solutions. Modeling
procedures and guidelines were developed for use with an explicit dynamics code that offers
advanced features such as: pressure formulated acoustic elements, surface based fluid-structure
coupling, surface based absorbing (radiation) boundaries, and automated incident wave loading
for the fluid-structure wetted interface. The modeling guidelines address issues such as: location
of the fluid acoustic domain outer boundary, meshing of the acoustic domain, representation of
the shock wave, and solution efficiency. These modeling procedures and guidelines are
demonstrated with an ABAQUS/Explicit analysis of an UNDEX experiment in which a submerged
test cylinder was exposed to a 60-pound HBX-1 explosive charge (Kwon & Fox, 1993).
General Background
ABAQUS/Explicit is an efficient tool for simulating the transient response of structural-acoustic
systems, of which the response of submerged structures loaded by acoustic shock waves resulting
from an Underwater Explosion (UNDEX) is an important problem class. This paper provides a
brief discussion on the general nature of the structural-acoustic interaction and describes modeling
studies that address general Finite Element Analysis (FEA) requirements for the accurate
simulation of UNDEX problems. The studies described in this report have general application to a
wide range of structural-acoustic problems, not just the analysis of submerged structures. An
example analysis of a submerged cylinder is used to illustrate an UNDEX problem.
UNDEX analyses can be generally characterized as transient simulations of acoustic scattering
behavior. However, the objective of an UNDEX analysis is to evaluate the response of the
submerged structure and not necessarily the acoustic response. The finite element model for the
external acoustic domain must be adequate to represent the influence of the water on the structural
response. The discussion herein will be restricted to those cases where the external fluid behaves
as a linear acoustic fluid with no cavitation. Therefore, the model of the external acoustic domain
need only be tailored to provide an accurate loading on the structure and does not need to
accurately represent the acoustic waves that will travel away from the structure. It should be noted
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that procedures for UNDEX analyses which include fluid cavitation will be available in
ABAQUS/Explicit with the release of Version 6.3 (Prasad & Cipolla, 2001).
The total acoustic pressure in the external fluid that results from an underwater explosion consists
of the known incident shock wave (incoming) pressure and the unknown scattered wave
(outgoing) pressure. The scattered wave pressure consists of two parts, a reflected part that is
associated with the shock wave interacting with an ideal rigid, immovable structure and a
vibratory part that results from the motions of the structure at the interface with the fluid. When
cavitation is not present, it is desirable to let the external acoustic domain represent only the
scattered portion of the total acoustic pressure. The shock wave incident pressure load is applied
directly to the structural mesh at the fluid-structure wetted interface. Acoustic loads associated
with the reflected part of the scattered pressure are applied to the fluid mesh at the wetted
interface. The full scattered pressure (reflected and vibratory) is obtained as the solution for the
acoustic element pressure degrees of freedom, and the complete scattered pressure loading on the
structure is generated through the fluid-structure coupling equations. The acoustic loads are a
characteristic of the incident shock wave, and are obtained from the fluid particle accelerations in
a direction normal to the surface that defines the fluid-structure wetted interface.
In the discussion that follows the capability of ABAQUS/Explicit to efficiently perform UNDEX
analyses is demonstrated. The ABAQUS features utilized in solving this class of problem are:
1. Computationally efficient pressure based acoustic elements.
2. Surface based automated acoustic-structure coupling.
3. Acoustic-structure coupling for mis-matched meshes.
4. Surface based impedance models for representing non-reflecting fluid boundaries.
5. Automated shock wave loading at the fluid-structure interface.
Acoustic Domain Outer Boundary
UNDEX problems are characterized by a strong coupling between the structural motions and
acoustic pressures at the wetted interface. The system response in a strongly coupled structural-
acoustic system can be described as being a combination of the following types of response:
1. Late Time - Low Frequency: Characterized by structural wavelengths that are
significantly shorter than the associated acoustic wavelengths. The effect of the external
fluid on the structure is that of adding an effective mass to the structure on the wetted
interface. The scattered energy within the acoustic domain remains near the structure
with very little energy radiating away from the structure.
2. Early Time - High Frequency: Characterized by structural wavelengths that are
significantly longer than the associated acoustic wavelengths. The effect of the external
fluid on the structure is to act as a simple radiation damper at the wetted interface. Most
of the scattered energy within the acoustic domain radiates away from the structure.
3. Intermediate Time - Frequency: Characterized by structural wavelengths that are of
similar length compared to the associated acoustic wavelengths. The effect of the
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external fluid on the structure is that of adding both effective mass and damping to the
structure. Comparable levels of scattered energy remain near the structure and are
radiated away from the structure.
Due to the high density and bulk modulus of water the finite element model for an UNDEX
analysis must be capable of accurately simulating all three ranges of structural-acoustic response.
It should be noted that for some other types of structural-acoustic analyses, where the fluid is very
light compared to the structure, not all response types may be of equal importance. For example,
the added mass effect of air acting upon heavy structures is often of no consequence.
The acoustic model for the outer boundary of the external fluid domain must provide adequate
non-reflecting behavior over all three time-frequency ranges. Non-reflecting outer boundary
models are implemented in ABAQUS/Explicit (Version 6.2) via a surface based boundary
impedance. ABAQUS has several imbedded surface impedance models, of which the circular and
sphere types were used for analyses described herein. These impedance models are based upon the
classical solutions for a 3-dimensional point source (sphere) and a 2-dimensional point source or
3-dimensioanl linear source (circular). The default impedance model corresponds to a simple
plane wave radiation condition, which is well suit to simple acoustic tube test simulations.
For the late time - low frequency response range, the extent of fluid contributing to the added mass
is largest for the longest structural response wavelength. Also, for the early time - high frequency
response range, the longest structural response wavelength has the potential to generate efficient
radiating acoustic waves with the longest wavelengths. Therefore, the location of the fluid mesh
outer boundary can be based upon the structure’s longest characteristic response wavelength. A
general guideline for locating the acoustic mesh outer boundary was developed by performing a
series of analyses representing the harmonic translational motion of a rigid infinite cylinder in an
infinite fluid domain. This type of motion is closely related to the transverse motion of a cylinder
section for beam bending response modes. A 2-dimensional rigid cylinder cross section (10”
radius) was placed within a circular fluid domain, for which the outer boundary was located at 2,
3, and 4 cylinder radii. The fluid was water with a bulk modulus of 345,600 psi and a sound speed
of 60,000 inches per second. ABAQUS/Explicit was used to drive the cylinder with a harmonic
motion until steady state conditions were achieved. Baseline analyses with very refined acoustic
meshes were used to represent the exact solutions. The baseline analyses utilized linear acoustic
triangle elements with approximately 42 element divisions per acoustic wavelength. The outer
boundary for the baseline analyses was located two full acoustic wavelengths away from the
structure, and utilized the circular type impedance boundary. The boundary evaluation models
had a minimum of 20 element divisions per acoustic wavelength for the highest driving frequency,
and the maximum acoustic element size was set at 1.5 inches for the lowest driving frequencies.
Figure 1 shows the baseline results for the complex radiation impedance of the driven cylinder
(force/velocity). The impedance values are plotted against the ratio of the structural wavelength,
which is the cylinder circumference, to the driving frequency acoustic wavelength. The radiation
reactance (imaginary) represents an added-mass effect and the radiation resistance (real)
represents the acoustic damping. Figure 1 clearly shows all three time-frequency response ranges,
with the radiation impedance transitioning from added mass at low frequencies to radiation
damping at high frequencies. Figure 2 shows a plot of the error ratio for the radiation impedance
predictions with the evaluation models. The radial thickness of the fluid mesh for the outer
boundary at 2, 3 and 4 cylinder radii, corresponds to approximately 1/6, 1/3, and 1/2 of the
structural wavelength. Setting the boundary at 2 cylinder radii (1/6 structural wavelength) works
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well at high frequencies but can introduce significant error in the added mass at low frequencies.
The error oscillations within the 5% error range at the higher driving frequencies appear to be due
to the outer boundary being placed near integer multiples of half the acoustic wavelength. Placing
the outer boundary so that the fluid domain thickness is between 1/3 and 1/2 the largest structural
wavelength provides for reasonable accuracy when using a sound source based outer boundary
surface impedance model. The performance when using a classical plane wave boundary condition
is significantly diminished, and would require at least doubling the extent of the fluid domain.
Fluid Mesh Requirements for Representing the Shock Wave
The classical representation of a spherical shock wave associated with UNDEX loading is
characterized by an instantaneous pressure rise to a peak value followed by an exponential decay.
An UNDEX analysis in which the external fluid is modeled with finite elements cannot accurately
represent a shock wave having an instantaneous pressure rise because the infinite pressure gradient
at the shock front implies infinite fluid particle acceleration, so that the acoustic loads associated
with the reflected part of the scattered pressure become indeterminate. Therefore, the shock front
must be modeled such that the pressure rise occurs over a period of time, designated the “rise
time”. A reasonable value for the rise time can usually be obtained from experimental or
anecdotal data. The pressure vs. time history of the shock wave at a known distance form the
source can be used to evaluate element size requirements for the acoustic mesh. This is
accomplished by means of a simple acoustic tube evaluation model.
A simple acoustic tube model was constructed with the linear tetrahedron acoustic elements that
will be used in the subsequent UNDEX example analyses. Acoustic loads representative of a
planar shock wave are applied at one end of the model with the ABAQUS incident wave loading
capability. The end at which the loads are applied represents a rigid immovable wall. The
resulting reflected wave travels down the acoustic tube. A simple plane wave absorbing boundary
can be applied to the opposite end of the tube, or the tube can be made of sufficient length so that
the test analysis finishes before the reflected wave reaches the opposite end.
Figure 3 shows the results for a tube analysis in which the nominal element size is equal to 1.5
times the wave propagation distance corresponding to the rise time of the shock front. The only
output quantity of concern in these acoustic tube analyses is the pressure at the rigid wall
(scattered pressure). An ideal solution for this problem would have the reflected wave being an
exact copy of the shock wave. This is clearly not the case for the Figure 3 model. Figure 4 is
another analysis with the element size set at 1/4 the rise time propagation distance. The scattered
pressure matches the shock pulse very well. However, there is still a fair degree of oscillation in
the early time solution. It should be noted that the range in the pressure oscillations can become
noticeably less when using brick type acoustic elements.
Figure 5 shows some additional solutions with the element size set at 1.5 times the rise time
distance, for which the time increment was varied via direct user control. As the time increment is
reduced the mean response approaches that of the incident shock wave. This illustrates how
simple acoustic tube models can also be used to evaluate the time increment requirements for
accurately representing the reflected wave loading. Figure 5 also suggests that a relatively coarse
fluid mesh may provide sufficient solution accuracy for the structural response as long as the
structure being analyzed responds at low frequencies relative to the reflected wave oscillations.
For these cases, the pressure impulse (time integral of pressure) associated with the reflected and
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incident shock waves should have relatively good correlation. Figure 6 shows the pressure impulse
curves generated from the Figure 5 analyses, and suggests that using a time increment that is less
than or equal to 1/20 of the rise time may provide good results with the coarse acoustic mesh for a
low frequency structural system.
Example Problem Description
The UNDEX example problem is based upon an experiment in which a submerged test cylinder
was exposed to a shock wave produced by a 60-pound HBX-1 explosive charge (Kwon & Fox,
1993). The test cylinder is made of T6061-T6 aluminum, has an overall length of 1.067 meters, an
outside diameter of 0.305 meters, a wall thickness of 6.35 millimeters and welded endcaps that are
24.5 millimeters thick. The cylinder was suspended horizontally in a 40-meter deep fresh water
test quarry (sound speed =1463 meters/second). The 60-pound HBX-1 explosive charge and the
cylinder were both placed at a depth of 3.66 meters, with the charge centered off the side of the
cylinder and located 7.62 meters from the cylinder surface. The suspension depths, charge offset
and duration of the test were selected such that cavitation of the fluid would not be significant and
no bubble pulse would occur. During the UNDEX test, two pressure transducers were positioned
7.62 meters from the charge, away from the cylinder, but at the same depth as the cylinder. These
transducers provided an experimental determination for the pressure vs. time history of the
incident spherical shock wave as it traveled by the point on the cylinder closest to the charge.
Figure 7 is a time history of the recorded shock wave pressure used as input to the
ABAQUS/Explicit analyses. Strain gages were placed at several locations on the outer surface of
the test cylinder, as shown in Figure 8. The strain gage experimental data was filtered at 2000 Hz,
with the experimental data presented herein obtained by digitizing the published Kwon & Fox
strain data.
ABAQUS/Explicit Model & Results
The test cylinder was meshed with 2400 S4R finite strain shell elements and contained 2402 nodes
(14412 dof) on 40 circumferential and 53 axial element divisions. The element connectivity is
such that each shell normal is directed into the external fluid. The shell element nodes are
positioned on the outside surface of the test cylinder. The cylinder body elements directly
adjacent to the endcaps have reduced mass & stiffness and are only used to provide a surface that
corresponds to the thickness of the endcaps. BEAM type MPCs are used to rigidly tie the
endcaps to the main cylinder body.
The external fluid is meshed with 4-noded AC3D4 acoustic tetrahedral elements. The outer
boundary of the external fluid is represented by cylindrical and spherical surfaces with the
appropriate surface impedance absorbing conditions. The characteristic radius of the fluid outer
boundary is set at 3 shell radii, thus the thickness of fluid modeled about the cylinder represents
approximately 1/3 of the cylinder’s outer circumference (rigid body translational wavelength).
Based upon the mesh boundary study this location should be sufficient to provide reasonably
accurate results. Figure 9 shows the cylinder and first acoustic mesh that was used in the analysis,
with the top half of fluid removed for clarity. The shock wave rise time is 0.0182 milliseconds,
corresponding to a wave propagation distance of 0.0266 meters. The nominal element size at the
wetted interface is also set at 0.0266 meters and increases in size to a nominal 0.080 meters at the
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outer fluid boundary. An acoustic element size of 0.080 meters corresponds to approximately 12
element divisions per acoustic wavelength for a 1500 Hz response. Figure 10 provides the results
of the acoustic tube validation for this degree of mesh refinement. Acoustic Mesh #1 contains
39186 elements and 7947 pressure degrees of freedom. Figure 11 shows the second acoustic
mesh that was used in the analysis, with the top half of fluid removed for clarity. The nominal
element size at the wetted interface is set at 0.010 meters and increases in size to a nominal 0.030
meters at the outer fluid boundary. Figure 12 provides the results of the acoustic tube validation
corresponding to acoustic Mesh #2, which contains 463114 elements and 87745 pressure degrees
of freedom.
Figure 13 shows the ABAQUS predicted axial strain response at strain gage location B1 when
using the coarse (#1) and refined (#2) acoustic meshes. The response curves are very close both in
magnitude and phasing. The close correlation between the two analyses was also apparent at the
other strain gage locations. This indicates that for the applied UNDEX loading the structural
response times are long when compared to the reflected wave oscillations obtained in the acoustic
tube validation analyses. This result was predictable when considering an eigenvalue analysis for
the cylinder with no external fluid. The modes that have the greatest potential for producing
damage have frequencies well below 1500 Hz, and will be further reduced when the cylinder is
submerged due to the added mass effect. The cylinder modes have response periods that are
significantly longer than the shock wave rise time or reflected wave pressure oscillations. Thus,
for this particular example, using an acoustic mesh and solution time increment that reasonably
captures the shock wave reflected pulse and can represent the scattered acoustic waves at the
structural response frequencies is adequate for obtaining a good solution.
The response shown in Figure 13 is dominated by the fundamental beam bending mode of the
cylinder, for which the dominant motion is transverse to the cylinder axis. At any point along the
cylinder axis the motion is dominated by a translation of the cross section through the fluid,
similar to the motion used in the infinite cylinder modeling study. The only damping mechanisms
in the analyses were due to acoustic radiation and the /Explicit default values for element bulk
viscosity. The acoustic model does not include any losses due to hydrodynamic drag (fluid
viscosity) associated with the motions of the cylinder. The effect of hydrodynamic drag on the
late time response of the cylinder is clearly shown in Figure 14, where the predicted axial strain
response is compared to the experimental data. The experimental data was digitized from a
published curve (Kwon & Fox, 1993), and was shifted by 0.2 milliseconds in order to align the
experimental and analysis time axes. The solution designated as ALPHA = 0, represents the
original analysis, whereas the analysis designated as ALPHA=750 utilized mass proportional
damping (10% critical at 600 Hz) applied to the cylinder as an approximation for the effects of
hydrodynamic drag. The application of ALPHA damping does not have an adverse effect on the
solution critical time increment. ALPHA damping does not significantly affect the early time
response (high frequency), but does significantly reduce the late time response (low frequency).
This is consistent with what is observed with the experimental data. In any event, ignoring
hydrodynamic drag in an UNDEX analysis will produce conservative (high) levels for the
structural response, which is often a desirable trait when doing a design evaluation analysis.
Figure 15 shows the levels of Accumulated Plastic Strain (PEEQ) on the outer surface of the shell
at the end of the analysis with fluid Mesh #1 and ALPHA= 750. No change occurs in the peak
plastic strain level of the cylinder wall after the first 0.34 milliseconds. Recalling that the entire
shock pulse duration is 2.0 milliseconds, this truly is an early time response. No change occurs in
the peak accumulated plastic strain of the endcaps after 2.84 milliseconds, which indicates that the
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endcap response may be influenced to a greater extent by the late time response. Unfortunately,
there were no strain gages attached to the endcaps and the cylinder wall gages located in the
regions of high plastic strain failed during the test. Table 1 compares the peak accumulated plastic
strains obtained with the two acoustic meshes, with and without ALPHA damping. The results
comparison between Mesh #1 and Mesh #2 is very good. The effect of ALPHA damping on the
cylinder wall PEEQ is very small, but is significant for the endcap response. Table 1 also provides
a comparison of the solution times for the analyses, and illustrates the solution efficiency of the
acoustic elements as compares to structural elements.
Conclusions
ABAQUS/Explicit provides an efficient means to evaluate the transient response of structural-
acoustic systems loaded by external acoustic sources. This was illustrated with the analysis of a
submerged cylinder acted upon by a shock wave generated by an underwater explosion. The
modeling studies presented in this paper indicate that sufficient accuracy for a submerged
structure’s response can be obtained when positioning the external absorbing boundary of the
acoustic domain a distance from the structure of between 1/3 to 1/2 the longest characteristic
structural wavelength. Modeling studies also indicated that the degree of refinement in the
acoustic domain mesh can be tailored to the characteristics of the shock pulse and the nature of
structural response, i.e., short vs. long response times as compared to the shock pulse transient.

Table 1. Model statistics and results comparisons.
Item Cylinder Only Acoustic Mesh #1 Acoustic Mesh #2
No. Acoustic Elements N/A 39186 463114
No. Acoustic DOF N/A 7947 87745
No. Shell Elements 2400 2400 2400
No. Shell DOF 14412 14412 14412
Solution Time Increments 4730 4730 13873
CPU Time (Seconds) 281 426 4590
CPU per Increment 0.059 0.090 0.331
PEEQ Cylinder Wall
N/A 0.00820 0.00810
(Alpha = 750)
PEEQ Cylinder Wall
N/A 0.00838 0.00828
(Alpha = 0)
PEEQ Endcap Center
N/A 0.00626 0.00611
(Alpha = 750)
PEEQ Endcap Center
N/A 0.01007 0.00985
(Alpha = 0)
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Figure 1. Baseline radiation impedance results for an infinite rigid cylinder.

Figure 2. Error ratio for the radiation impedance evaluation models.
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Figure 3. Acoustic tube with elements 1-½ the rise time propagation distance.

Figure 4. Acoustic tube with elements ¼ the rise time propagation distance.

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Figure 5. Additional results for 1-½ rise time element mesh.

Figure 6. Pressure impulse curves from Figure 5 analyses.
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