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Specify speed of sound and density for propagating wave acoustic mesh exterior surface structural mesh fluid surface solid surface reference or "standoff" node source node (where explosion charge occurs)

Figure 34.4.6–1 Incident wave loading model. In underwater explosion analyses (for example, a ship or submerged vehicle subjected to an underwater explosion loading as depicted in Figure 34.4.6–4 and Figure 34.4.6–5) the fluid is also discretized using a finite element model to capture the effects of the fluid stiffness and inertia. For these problems involving both solid and acoustic elements, two formulations of the acoustic pressure field exist. First, the acoustic elements can be used to model the total pressure in the medium, including the effects of the incident field and the overall system’s response. Alternatively, the acoustic elements can be used to model only the response of the medium to the wave loads, not the wave pulse itself. The former case will be referred to as the “total wave” formulation, the latter as the “scattered wave” formulation. Incident wave interactions are also used to model sound fields impinging on structures or acoustic domains. The acoustic field scattered by a structure or the sound transmitted through the structure may be of interest. Usually, sound scattering and transmission problems are modeled using the scattered formulation with steady-state dynamic procedures. Transient procedures can also be used, in a manner analogous to underwater explosion analysis problems. # Scattered and total wave formulations The distinction between the total wave formulation and the scattered wave formulation is relevant only when incident wave loads are applied. The total wave formulation is more closely analogous to structural loading than the scattered wave formulation: the boundary of the acoustic medium is specified as a loaded surface, and a time-varying load is applied there, which generates a response in the acoustic medium. This response is equal to the total acoustic pressure in the medium. The scattered wave formulation exploits the fact that when the acoustic medium is linear, the response in the medium can be decomposed into a sum of the incident wave and the scattered field. The total wave formulation must be used when the acoustic medium is nonlinear due to possible fluid cavitation (see “Loading due to an incident dilatational wave field,” Section 6.3.1 of the Abaqus Theory Guide). Table 34.4.6–1 describes the procedure types for which each formulation is supported. Table 34.4.6–1 Supported procedures for scattered and total wave formulations.

Procedure	Scattered	Total Wave
Steady-state dynamics	Yes	No
Transient	Yes	Yes

# Scattered wave formulation When the mechanics of a fluid can be described as linear, the observed total acoustic pressure can be decomposed into two components: the known incident wave and the “scattered” wave that is produced by the interaction of the incident wave with structures and/or fluid boundaries. When this superposition is applicable, it is common practice to seek the “scattered” wave field solution directly. When using the scattered wave formulation, the pressures at the acoustic nodes are defined to be only the scattered part of the total pressure. Both acoustic and solid surfaces at the acoustic-structural interface should be loaded in this case. When using incident wave loads in steady-state dynamic procedures, the scattered wave formulation must be used. Input File Usage: Use the following option to specify the scattered wave formulation (default): \*ACOUSTIC WAVE FORMULATION, TYPE=SCATTERED WAVE Abaqus/CAE Usage: Any module: Model→Edit Attributes→model\_name. Toggle on Specify acoustic wave formulation: select Scattered wave # Total wave formulation The total wave formulation (see “Coupled acoustic-structural medium analysis,” Section 2.9.1 of the Abaqus Theory Guide) is particularly applicable when the acoustic medium is capable of cavitation, rendering the fluid mechanical behavior nonlinear. It should also be used if the problem contains either a curved or a finite extent boundary where the pressure history is prescribed. Only the outer acoustic surfaces should be loaded with the incident wave in this case, and the incident wave source must be located exterior to the fluid model. Any impedance or nonreflecting condition that may exist on this outer acoustic boundary applies only on the part of the acoustic solution that does not include the prescribed incident wave field (that is, only the scattered field is subject to the nonreflecting condition). Thus, the applied incident wave loading will travel into the problem domain without being affected by the nonreflecting conditions on the outer acoustic surface. In the total wave formulation the acoustic pressure degree of freedom stands for the total dynamic acoustic pressure, including contributions from incident and scattered waves and, in Abaqus/Explicit, the dynamic effects of fluid cavitation. The pressure degree of freedom does not include the acoustic static pressure, which can be specified as an initial condition (see “Defining initial acoustic static pressure” in “Initial conditions in Abaqus/Standard and Abaqus/Explicit,” Section 34.2.1). This acoustic static pressure is used only in determining the cavitation status of the acoustic element nodes and does not apply any static loads to the acoustic or structural mesh at their common wetted interface. It does not apply to analyses using Abaqus/Standard. Input File Usage: Use the following option to specify the total wave formulation: \*ACOUSTIC WAVE FORMULATION, TYPE=TOTAL WAVE Abaqus/CAE Usage: Any module: Model→Edit Attributes→model\_name. Toggle on Specify acoustic wave formulation: select Total wave # Initialization of acoustic fields For transient dynamics, when the total wave formulation is used with the incident wave standoff point located inside the acoustic finite element domain, the acoustic solution is initialized to the values of the incoming incident wave. This initialization is performed automatically, for pressure-based incident wave amplitude definitions only, at the beginning of the first direct-integration dynamic step in an analysis; in restarted analyses, steps are counted from the beginning of the initial analysis. This initialization not only saves computational time but also applies the incident wave loading without significant numerical dissipation or distortion. During the initialization phase all incident wave loading definitions in the first dynamic analysis step are considered, and all acoustic element nodes are initialized to the incident wave field at time zero. Incident wave loads specified with different source locations count as separate load definitions for the purpose of initialization of the acoustic nodes. Any reflections of the incident wave loads are also taken into account during the initialization phase. # Describing incident wave loading To use incident wave loading, you must define the following: • information that establishes the direction and other properties of the incident wave, • the time history or frequency dependence of the source pulse at some reference (“standoff”) point, • the fluid and/or solid surfaces to be loaded, and • any reflection plane outside the problem domain, such as a seabed in an underwater explosion study, that would reflect the incident wave onto the problem domain. Two interfaces are available in Abaqus for applying incident wave loads: a preferred interface that is supported in Abaqus/CAE and an alternative interface that has been available in previous releases and is not supported in Abaqus/CAE. The preferred interface is conceptually the same as the alternative interface and uses essentially the same data. The preferred interface options include the term “interaction” to distinguish them from the incident wave and incident wave property options of the alternative interface. Unless otherwise specified, the discussion in this section applies to both of the interfaces. The usages for the preferred interface are included in the discussion; the usages for the alternative interface are described in “Alternative incident wave loading interface,” below. Refer to the example problems discussed at the end of this section to see how the incident wave loading is specified using the preferred interface. # Prescribing geometric properties and the speed of the incident wave You must refer to a property definition for each prescribed incident wave. Incident wave loads in Abaqus may be either planar, spherical, or diffuse. You select a planar incident wave (default), spherical incident wave, or a diffuse field in the incident wave property definition. Planar incident waves maintain constant amplitude as they travel in space; consequently, the speed and direction of travel are the critical parameters to define. The speed is defined in the incident wave interaction property definition, and the direction is determined by the locations of the source and standoff points you define as part of the incident wave interaction. For spherical incident wave definitions, the wave reduces in amplitude as a function of space. By default, the amplitude of a spherical wave is inversely proportional to the distance from the source; this behavior is called “acoustic” propagation. For the preferred interface you can modify the default propagation behavior to define spatial decay of the incident wave field. The dimensionless constants , , and are used to define the spatial decay as a function of the distance $R _ { j }$ between the source point and the loaded point and the distance $R _ { 0 }$ between the source point and the standoff point: $$ p _ {x} (R _ {0}, R _ {j}) \equiv \left(\frac {R _ {0}}{R _ {j}}\right) ^ {\left[ \frac {(A + 1) R _ {j}}{C R _ {0} + (B + 1) R _ {j}} \right]}. $$ Refer to “Loading due to an incident dilatational wave field,” Section 6.3.1 of the Abaqus Theory Guide, for details of the generalized spatial decay formulation. In Abaqus incident wave interactions can be used to simulate diffuse incident fields. Diffuse fields are characteristic of reverberant spaces or other situations in which waves from many directions strike a surface. For example, reverberant chambers are constructed intentionally in acoustic test facilities for sound transmission loss measurements. The diffuse field model used in Abaqus, as shown in Figure 34.4.6–2, allows you to specify a seed number ; deterministic incident plane waves travel along vectors distributed over a hemisphere so that the incident power per solid angle approximates a diffuse incident field. The fluid and the solid surfaces where the incident loading acts are specified in the incident wave loading definition. The incoming wave load is further described by the locations of its source point and of a reference (“standoff”) point where the wave amplitude is specified. For information on how to specify these surfaces and the standoff point, see “Identifying the fluid and the solid surfaces for incident wave loading,” and “The standoff point” below. For a planar wave the specified locations of the source and the standoff points are used to define the direction of wave propagation. 

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"Source" Unit hemisphere oriented along source-standoff vector Plane wave along one of N² directions Plane normal to source-standoff vector N seed point columns "Standoff" FE surface to be loaded N seed point rows

Figure 34.4.6–2 Diffuse loading model. The speed of the incident wave is prescribed by giving the properties for the incident wave-bearing acoustic medium. These specified properties should be consistent with the properties specified for the fluid discretized using acoustic elements. For the preferred interface you must define nodes corresponding to the source and standoff points for the incident wave; the node numbers or set names must be specified for each incident wave definition. The node set names, if used, must contain only a single node. Neither the source node nor the standoff node should be connected to any elements in the model. # Input File Usage: \*INCIDENT WAVE INTERACTION PROPERTY, NAME=wave property name, TYPE=PLANE or SPHERE speed of sound, fluid mass density, A, B, C \*INCIDENT WAVE INTERACTION, PROPERTY=wave property name fluid surface name, source node, standoff node, reference magnitude The constants A, B, and C apply only for spherical incident waves with generalized spatial decay propagation. \*INCIDENT WAVE INTERACTION PROPERTY, NAME=wave property name, TYPE=DIFFUSE speed of sound, fluid mass density \*INCIDENT WAVE INTERACTION, PROPERTY=wave property name fluid surface name, source node, standoff node, reference magnitude, N The seed number N generates planar incident waves with directions distributed on a hemisphere centered at the standoff point. # Abaqus/CAE Usage: Interaction module: Create Interaction Property: Name: wave property name and Incident wave, Speed of sound in fluid: speed of sound, Fluid density: fluid mass density Select one of the following definitions: Definition: Planar Definition: Spherical, Propagation model: Acoustic Definition: Spherical, Propagation model: Generalized decay, enter values for A, B, and C Definition: Diffuse, Seed number: N Create Interaction: Incident wave: select the source point, select the standoff point, select the region: Wave property: wave property name, Reference magnitude: reference magnitude Identifying the fluid and the solid surfaces for incident wave loading In the scattered wave formulation the incident wave loading must be specified on all fluid and solid surfaces that reflect the incident wave with two exceptions: • those fluid surfaces that have the pressure values directly prescribed using boundary conditions; and • those fluid surfaces that have symmetry conditions (the symmetry must hold for both the loading and the geometry). In problems with a fluid-solid interface both surfaces must be specified in the incident wave loading definition for the scattered formulation. See “Example: submarine close to the free surface,” shown in Figure 34.4.6–4. When the total pressure-based formulation is specified, the incident wave loading must be specified only on the fluid surfaces that border the infinite region that is excluded from the model. Typically, these surfaces have a nonreflecting radiation condition specified on them, and the implementation ensures that the radiation condition is enforced only on the scattered response of the modeled domain and not on the incident wave itself. See “Example: submarine close to the free surface,” and “Example: surface ship,” shown in Figure 34.4.6–4 and Figure 34.4.6–5, respectively. In certain problems, such as blast loads in air, you may decide that the blast wave loads on a structure need to be modeled, but the surrounding fluid medium itself does not. In these problems the incident wave loading is specified only on the solid surfaces since the fluid medium is not modeled. The distinction between the scattered wave formulation and the total wave formulation for handling the incident wave loading is not relevant in these problems since the wave propagation in the fluid medium is of no interest. # The standoff point In transient analyses the standoff point is a reference point used to specify the pulse loading time history: it is the point at which the user-defined pulse history is assumed to apply with no time delay, phase shift, or spreading loss. In steady-state analyses using discrete planar or spherical sources, the standoff point is the point at which the incident field has zero phase. In transient analyses the standoff point should be defined so that it is closer to the source than any point on the surfaces in the model that would reflect the incident wave. Doing so ensures that all the points on these surfaces will be loaded with the specified time history of the source and that the analysis begins before the wave overtakes any portion of these surfaces. To save analysis time, the standoff point is typically on or near the solid surface where the incoming incident wave would be first deflected (see “Example: submarine close to the free surface,” shown in Figure 34.4.6–4). However, the standoff point is a fixed point in the analysis: if the loaded surfaces move before the incident wave loading begins, due to previous analysis steps or geometric adjustments, the surfaces may envelop the specified standoff point. Care should be taken to define a standoff point such that it remains closer to the incident wave source point than any point on the loaded surfaces at the onset of the loading. When the total wave formulation is used and the incident wave loading is specified in the first step of the analysis in terms of pressure history, Abaqus automatically initializes the pressure and the pressure rate at the acoustic nodes to values based on the incident wave loading. This allows the acoustic analysis to start with the incident waves partially propagated into the problem domain at time zero and assumes that this propagation had taken place with negligible effect of any volumetric dissipative sources such as the fluid drag. When the incident wave loading is specified in terms of the pressure values, the recommendations given above for selecting a standoff point are valid with the total wave formulation as well. However, when the incident wave loading is specified in terms of acceleration values, the automatic initialization is not done and the standoff point should be located near the exterior fluid boundary of the model such that the standoff point is closer to the source than any point on the exterior boundary. See “Example: submarine close to the free surface,” and “Example: surface ship,” shown in Figure 34.4.6–4 and Figure 34.4.6–5, respectively. In steady-state analyses the role of the standoff point is somewhat different. When the incident wave interaction property is of planar or spherical type, you define the real and imaginary parts of the magnitude at the standoff point. Separately, the specified real and imaginary incident waves are taken to have zero phase at the standoff point (combined, these two waves could be equivalent to a single wave with nonzero phase at the standoff). Every location on the loaded surface has a phase shift in the applied pressure or acoustic traction, corresponding to the difference in propagation time between the loaded point and the standoff. This means that an incident wave defined, for example, with a pure real value at the standoff point generates both real and imaginary tractions at all the other points on the loaded surface. When the incident wave is of diffuse type, the role of the standoff and source points is primarily to orient the loaded surface with respect to the incoming reverberant field. The model used for diffuse incident wave loading applies a set of deterministically defined plane waves, whose directions are defined as vectors connecting the standoff point and an array of points on a hemisphere. This hemisphere is centered at the standoff point, and its apex is the source point. The array of points is set according to the specified seed, , and a deterministic algorithm that arranges $N ^ { 2 }$ points on the hemisphere. The algorithm concentrates the points so that the incident waves in the diffuse field model are concentrated at normal incidence, with fewer waves at oblique angles. The specified amplitude value and reference magnitude are divided equally among the $N ^ { 2 }$ incident waves. The orientation of the hemisphere containing the incident waves in the diffuse model is the same for all of the points on the loaded surface—it does not vary with the local normal vector on the surface. # Defining the amplitude of the source pulse For transient analyses the time history to be specified by the user is that observed at the standoff point: histories at a point on the loaded surface are computed from the wave type and the location of that point relative to the standoff point. The time history of the acoustic source pulse can be defined either in terms of the fluid pressure values or the fluid particle acceleration values. Pressure time histories can be used for any type of element, such as acoustic, structural, or solid elements; acceleration time histories are applicable only for acoustic elements. In either case a reference magnitude is specified for any given incident-wave-loaded surface, and a reference to a time-history data table defined by an amplitude curve is specified. The reference magnitude varies with time according to the amplitude definition. For steady-state dynamic analyses the amplitude definition specified as part of the incident wave interaction definition is interpreted as the frequency dependence of the wave at the standoff point. Currently the source pulse description in terms of fluid particle acceleration history is limited to planar incident waves acting on fluid surfaces in transient analyses. Further, if an impedance condition is specified on the same fluid surface along with incident wave loading, the source pulse is restricted to the pressure history type even for planar incident waves. The source pulse in terms of pressure history can be used without these limitations; i.e., pressure-history-based incident wave loading can be used with fluid or solid surfaces, with or without impedance, and for both planar and spherical incident waves. When the source pulse is specified using pressure values and is applied on a fluid surface, the pressure gradient is computed and applied as a pressure-conjugate load on these surfaces. Hence, it is desirable to define the pulse amplitude to begin with a zero value, particularly when the cavitation in the fluid is a concern. If the structural response is of primary concern and the scattered formulation is being used, any initial jump in the pressure amplitude can be addressed by applying additional concentrated loads on the structural nodes that are tied to the acoustic mesh, corresponding to the initial jump in the incident wave pressure amplitude. Clearly, the additional load on any given structural node should be active from the instance the incident wave first arrives at that structural node. However, the scattered wave solution in the fluid still needs careful interpretation taking the initial jump into account. # Input File Usage: Use the following option to define the time history in terms of fluid pressure values: \*INCIDENT WAVE INTERACTION, PRESSURE AMPLITUDE=amplitude data table name solid or fluid surface name, source node, standoff node, reference magnitude Use the following option to define the time history in terms of fluid particle acceleration values: \*INCIDENT WAVE INTERACTION, ACCELERATION AMPLITUDE=amplitude data table name fluid surface name, source node, standoff node, reference magnitude Use the following option to define the real part of the loading (default): \*INCIDENT WAVE INTERACTION, REAL Use the following option to define the imaginary part of the loading: \*INCIDENT WAVE INTERACTION, IMAGINARY # Abaqus/CAE Usage: Interaction module: Create Interaction: Incident wave: select the source point, select the standoff point, select the region: Reference magnitude: reference magnitude Use the following options to define the time history in terms of fluid pressure values or fluid particle acceleration values: Definition: Pressure or Acceleration, Pressure amplitude or Acceleration amplitude: amplitude data table name Use the following options to define the real or imaginary part of the loading: Toggle on Real amplitude and/or Imaginary amplitude: amplitude data table name # Defining bubble loading for spherical incident wave loading An underwater explosion forms a highly compressed gas bubble that interacts with the surrounding water, generating an outward-propagating shock wave. The gas bubble floats upward as it generates these waves changing the relative positions of the source and the loaded surfaces. The loading effects due to bubble formation can be defined for spherical incident wave loading by using a bubble definition in conjunction with the incident wave loading definition. The bubble dynamics can be described using a model internal to Abaqus or by using tabulated data. Abaqus has a built-in mechanical model of the bubble interacting with the surrounding fluid, which is simulated numerically to generate a set of data prior to running the finite element analysis. You can specify the explosive material parameters, ending time, and other parameters that affect the computation of the bubble amplitude curve used, as shown in Table 34.4.6–2. Table 34.4.6–2 Parameters that define the bubble behavior.

Name	Dimensions	Description	Default
K	$FL^{-2}(LM^{-1/3})^{1+A}$	Charge constant	None
k	$T/(M^{\frac{1-B}{3}}L^B)$	Charge constant	None
A	Dimensionless	Similitude spatial exponent	None

Name	Dimensions	Description	Default
$B$	Dimensionless	Similitude temporal exponent	None
$K_c$	$F/L^2$	Charge constant	None
$\gamma$	Dimensionless	Ratio of specific heats for explosion gas	None
$\rho_c$	$M/L^3$	Charge material density	None
$m_c$	M	Mass of charge	None
$d_I$	L	Initial charge depth	None
$\mathbf{n}_X$	Dimensionless	$X$ -direction cosine of the free surface normal	None
$\mathbf{n}_Y$	Dimensionless	$Y$ -direction cosine of the free surface normal	None
$\mathbf{n}_Z$	Dimensionless	$Z$ -direction cosine of the free surface normal	None
$g$	$L/T^2$	Acceleration due to gravity	None
$p_{atm}$	$F/L^2$	Atmospheric pressure at free surface	None
$\eta$	Dimensionless	Wave effect parameter	1.0
$C_D$	Dimensionless	Bubble drag coefficient	0.0
$E_D$	Dimensionless	Bubble drag exponent	2.0
$T_{final}$	T	Maximum allowable time in bubble simulation	None
$N_{steps}$	Dimensionless	Maximum allowable number of steps in bubble simulation	1500
$\Omega_{rel}$	Dimensionless	Relative error tolerance parameter for bubble simulation	$1 \times 10^{-11}$
$X_{abs}$	Dimensionless	Absolute error tolerance parameter for bubble simulation	$1 \times 10^{-11}$
$\beta$	Dimensionless	Error control exponent for bubble simulation	0.2
$\rho_f$	$M/L^3$	Fluid mass density	None
$c_f$	L/T	Fluid speed of sound	None