Products: ABAQUS/Standard ABAQUS/Explicit
ABAQUS provides a set of elements for modeling a fluid medium undergoing small pressure variations and interface conditions to couple these acoustic elements to a structural model. These elements are provided to model a variety of phenomena involving dynamic interactions between fluid and solid media.
Steady-state harmonic (linear) response analysis can be performed for a coupled acoustic-structural system, such as the study of the noise level in a vehicle. The steady-state procedure is based on direct solution of the coupled complex harmonic equations, as described in “Direct steady-state dynamic analysis,” Section 2.6.1; on a modal-based procedure, as described in “Steady-state linear dynamic analysis,” Section 2.5.7; or on a subspace-based procedure, as described in “Subspace-based steady-state dynamic analysis,” Section 2.6.2. Mode-based linear transient dynamic analysis is also available, as described in “Modal dynamic analysis,” Section 2.5.5.
The acoustic fluid elements can also be used with nonlinear response analysis (implicit or explicit direct integration) procedures: whether such results are useful depends on the applicability of the small pressure change assumption in the fluid. Often, in coupled fluid-solid problems the fluid forces in this linear regime are high enough that nonlinear response of the structure needs to be considered. For example, a ship subjected to underwater incident wave loads due to an explosion may experience plastic deformation, or large motions of internal machinery may occur.
The acoustic medium in ABAQUS may have velocity-dependent dissipation, caused by fluid viscosity or by flow within a resistive porous matrix material. In addition, rather general boundary conditions are provided for the acoustic medium, including impedance, or “reactive,” boundaries.
The possible conditions at the surface of the acoustic medium are:
Prescribed pressure (degree of freedom 8) at the boundary nodes. (Boundary conditions can be used to specify pressure at any node in the model.)
Prescribed inward normal derivative of pressure per unit density of the acoustic medium through the use of a concentrated load on degree of freedom 8 of a boundary node. If the applied load has zero magnitude (that is, if no concentrated load or other boundary condition is present), the inward normal derivative of pressure (and normal fluid particle acceleration) is zero, which means that the default boundary condition of the acoustic medium is a rigid, fixed wall (Neumann condition).
Acoustic-structural coupling defined either by using surface-based coupling procedures (see “Surface-based acoustic-structural medium interaction,” Section 5.2.7) or by placing ASI coupling elements on the interface between the acoustic medium and a structure.
An impedance condition, representing an absorbing boundary between the acoustic medium and a rigid wall or a vibrating structure or representing radiation to an infinite exterior.
An incident wave loading, representing the inward normal derivative of pressure per unit density of the acoustic medium resulting from the arrival of a specified wave. The formulation of this loading case is discussed in “Loading due to an incident dilatational wave field,” Section 6.3.1. It is applicable to problems involving blast loads and to acoustic scattering problems.
The equilibrium equation for small motions of a compressible, adiabatic fluid with velocity-dependent momentum losses is taken to be
where
is the excess pressure in the fluid (the pressure in excess of any static pressure), 
is the spatial position of the fluid particle, 
is the fluid particle velocity, 
is the fluid particle acceleration, 
is the density of the fluid, 
is the “volumetric drag” (force per unit volume per velocity), and 
are 
independent field variables such as temperature, humidity of air, or salinity of water on which and may depend (see “Acoustic medium,” Section 12.3.1 of the ABAQUS Analysis User's Manual). The d'Alembert term has been written without convection on the assumption that there is no steady flow of the fluid. This is usually considered sufficiently accurate for steady fluid velocities up to Mach 0.1.The constitutive behavior of the fluid is assumed to be inviscid, linear, and compressible, so
where
is the bulk modulus of the fluid.For an acoustic medium capable of undergoing cavitation, the absolute pressure (sum of the static pressure and the excess dynamic pressure) cannot drop below the specified cavitation limit. When the absolute pressure drops to this limit value, the fluid is assumed to undergo free expansion without a corresponding drop in the dynamic pressure. The pressure would rebuild in the acoustic medium once the free expansion that took place during the cavitation is reversed sufficiently to reduce the volumetric strain to the level at the cavitation limit. The constitutive behavior for an acoustic medium capable of undergoing cavitation can be stated as
, a measure of the volumetric strain, is defined as
is the fluid cavitation limit and 
is the initial acoustic static pressure. A total wave formulation is used for a nonlinear acoustic medium undergoing cavitation. This formulation is very similar to the scattered wave formulation presented below except that the pseudo-pressure, defined as the product of the bulk modulus and the compressive volumetric strain, plays the role of the material state variable instead of the acoustic excess pressure. The acoustic excess pressure is readily available from this pseudo-pressure subject to the cavitation condition.Acoustic fields are strongly dependent on the conditions at the boundary of the acoustic medium. The boundary of a region of acoustic medium that obeys Equation 2.9.1–1 and Equation 2.9.1–2 can be divided into subregions 
on which the following conditions are imposed:
, where the value of the acoustic pressure is prescribed.
, where we prescribe the normal derivative of the acoustic medium. This condition also prescribes the motion of the fluid particles and can be used to model acoustic sources, rigid walls (baffles), incident wave fields, and symmetry planes.
, the “reactive” acoustic boundary, where there is a prescribed linear relationship between the fluid acoustic pressure and its normal derivative. Quite a few physical effects can be modeled in this manner: in particular, the effect of thin layers of material, whose own motions are unimportant, placed between acoustic media and rigid baffles. An example is the carpet glued to the floor of a room or car interior that absorbs and reflects acoustic waves. This thin layer of material provides a “reactive surface,” or impedance boundary condition, to the acoustic medium. This type of boundary condition is also referred to as an imposed impedance, admittance, or a “Dirichlet to Neumann map.”
, the “radiating” acoustic boundary. Often, acoustic media extend sufficiently far from the region of interest that they can be modeled as infinite in extent. In such cases it is convenient to truncate the computational region and apply a boundary condition to simulate waves passing exclusively outward from the computational region.
, where the motion of an acoustic medium is directly coupled to the motion of a solid. On such an acoustic-structural boundary the acoustic and structural media have the same displacement normal to the boundary, but the tangential motions are uncoupled.
, an acoustic-structural boundary, where the displacements are linearly coupled but not necessarily identically equal, due to the presence of a compliant or reactive intervening layer. This layer induces an impedance condition between the relative normal velocity between acoustic fluid and solid structure and the acoustic pressure. It is analogous to a spring and dashpot interposed between the fluid and solid particles. As implemented in ABAQUS, an impedance boundary condition surface does not model any mass associated with the reactive lining; if such a mass exists, it should be incorporated into the boundary of the structure.
, a boundary between acoustic fluids of possibly differing material properties. On such an interface, displacement continuity requires that the normal forces per unit mass on the fluid particles be equal. This quantity is the natural boundary traction in ABAQUS, so this condition is enforced automatically during element assembly. This is also true in one-dimensional analysis (i.e., piping or ducts), where the relevant acoustic properties include the cross-sectional areas of the elements. Consequently, fluid-fluid boundaries do not require special treatment in ABAQUS.
In ABAQUS the finite element formulations are slightly different in direct integration transient and steady-state or modal analyses, primarily with regard to the treatment of the volumetric drag loss parameter and spatial variations of the constitutive parameters. To derive a symmetric system of ordinary differential equations for implicit integration, some approximations are made in the transient case that are not needed in steady state. For linear transient dynamic analysis, the modal procedure can be used and is much more efficient.
To derive the partial differential equation used in direct integration transient analysis, we divide Equation 2.9.1–1 by , take its gradient with respect to , neglect the gradient of 
, and combine the result with the time derivatives of Equation 2.9.1–2 to obtain the equation of motion for the fluid in terms of the fluid pressure:
is small is violated where there are discontinuities in the quantity (for example, on the boundary between two elements that have a different value).An equivalent weak form for the equation of motion, Equation 2.9.1–3, is obtained by introducing an arbitrary variational field, 
, and integrating over the fluid:
is prescribed on , the equilibrium equation, Equation 2.9.1–1, is used on the remainder of the boundary to relate the pressure gradient to the motion of the boundary: Using this equation, the term 
is eliminated from Equation 2.9.1–4 to produce where, for convenience, the boundary “traction” term has been introduced.Except for the imposed pressure on , all of the other boundary conditions described above can be formulated in terms of 
. This term has dimensions of acceleration; in the absence of volumetric drag this boundary traction is equal to the inward acceleration of the particles of the acoustic medium:
yields a lower local volumetric acceleration, due to drag losses.In direct integration transient dynamics we enforce the acoustic boundary conditions as follows:
On , is prescribed and 
.
On , where we prescribe the normal derivative of the acoustic pressure per unit density:
. An imposed 
can be used to model the oscillations of a rigid plate or body exciting a fluid, for example. A special case of this boundary condition is 
, which represents a rigid immobile boundary. As mentioned above, if the medium has nonzero volumetric drag, a unit of 
imposed at the boundary will result in a relatively lower imposed particle acceleration. Incident wave fields on a boundary of a fluid are modeled with a that varies in space and time, corresponding to the effect of the arrival of the wave on the boundary. See “Loading due to an incident dilatational wave field,” Section 6.3.1.On , the reactive boundary between the acoustic medium and a rigid baffle, we apply a condition that relates the velocity of the acoustic medium to the pressure and rate of change of pressure:
and 
are user-prescribed parameters at the boundary. This equation is in the form of an admittance relation; the impedance expression is simply the inverse. The layer of material, in admittance form, acts as a spring and dashpot in series distributed between the acoustic medium and the rigid wall. The spring and dashpot parameters are 
and 
, respectively; they are per unit area of the acoustic boundary. Using this definition for the fluid velocity, the boundary tractions in the variational statement becomeOn , the radiating boundary, we apply the radiation boundary condition by specifying the corresponding impedance:
On , the acoustic-structural interface, we apply the acoustic-structural interface condition by equating displacement of the fluid and solid, which enforces the condition
is the displacement of the structure. In the presence of volumetric drag it follows that the acoustic boundary traction coupling fluid to solid is 
. In the great majority of practical applications the acoustic tractions associated with volumetric drag are small compared to those associated with fluid inertia, 
On , the mixed impedance boundary and acoustic-structural boundary, we apply a condition that relates the relative outward velocity between the acoustic medium and the structure to the pressure and rate of change of pressure:
, we obtain for the transient case: This expression for is the sum of its definitions for and . In the steady-state case the effect of volumetric drag on the structural displacement term in the acoustic traction is included: These definitions for the boundary term, , are introduced into Equation 2.9.1–6 to give the final variational statement for the acoustic medium (this is the equivalent of the virtual work statement for the structure):
is the stress at a point in the structure, is the pressure acting on the fluid-structural interface, 
is the outward normal to the structure, 
is the density of the material, 
is the mass proportional damping factor (part of the Rayleigh damping assumption for the structure), 
is the acceleration of a point in the structure, 
is the surface traction applied to the structure, 
is a variational displacement field, and 
is the strain variation that is compatible with . For simplicity in this equation all other loading terms except the fluid pressure and surface traction have been neglected: they are imposed in the usual way.Equation 2.9.1–14 and Equation 2.9.1–15 define the variational problem for the coupled fields and . The problem is discretized by introducing interpolation functions: in the fluid 
, 
up to the number of pressure nodes and in the structure 
, 
up to the number of displacement degrees of freedom. In these and the following equations we assume summation over the superscripts that refer to the degrees of freedom of the discretized model. We also use the superscripts 
, 
to refer to pressure degrees of freedom in the fluid and 
, 
to refer to displacement degrees of freedom in the structure. We use a Galerkin method for the structural system; the variational field has the same form as the displacement: 
. For the fluid we use 
but with the subsequent Petrov-Galerkin substitution
makes the single variational equation obtained from summing Equation 2.9.1–14 and Equation 2.9.1–15 dimensionally consistent: where, for simplicity, we have introduced the following definitions: 
is the strain interpolator. This equation defines the discretized model. We see that the volumetric drag-related terms are “mass-like”; i.e., proportional to the fluid element mass matrix.The term 
is the nodal right-hand-side term for the acoustical degree of freedom 
, or the applied “force” on this degree of freedom. This term is obtained by integration of the normal derivative of pressure per unit density of the acoustic medium over the surface area tributary to a boundary node.
In the case of coupled systems where the forces on the structure due to the fluid—
are very small compared to the rest of the structural forces—the system can be solved in a “sequentially coupled” manner. The structural equations can be solved with the term omitted; i.e., in an analysis without fluid coupling. Subsequently, the fluid equations can be solved, with 
imposed as a boundary condition. This two-step analysis is less expensive and advantageous for systems such as metal structures in air.
The equations are integrated through time using the standard implicit (ABAQUS/Standard) and explicit (ABAQUS/Explicit) dynamic integration options. From the implicit integration operator we obtain relations between the variations of the solution variables (here represented by 
) and their time derivatives:
and 
are the corrections to the solution obtained from the Newton iteration, 
is the structural stiffness matrix, and 
is the structural damping matrix. These equations are symmetric if the structural stiffness is symmetric.For explicit integration the fluid mass matrix is diagonalized in a manner similar to the treatment of structural mass. The explicit central difference procedure described in “Explicit dynamic analysis,” Section 2.4.5, is applied to the coupled system of equations.
As mentioned above, derivation of symmetric ordinary differential equations in the presence of volumetric drag requires some approximations, in addition to those inherent in any finite element method. First, the spatial gradients of the ratio of volumetric drag to mass density in the fluid are neglected. This may be important in lossy, inhomogeneous acoustic media. Second, to maintain symmetry, the effect of volumetric drag on the fluid-solid boundary terms is neglected. Finally, the effect of volumetric drag on the radiation boundary conditions is approximate. If any of these effects is expected to be significant in an analysis, the user should realize that the results obtained are approximate.
From the discretized equation, Equation 2.9.1–17, neglecting any damping terms and any terms associated with a reactive surface, the eigenvalue problem can be stated as
As stated, this problem cannot be solved in ABAQUS due to the unsymmetric stiffness and mass matrices. Introducing an auxiliary variable,
, augmenting the system of equations with 
, and manipulating the equations yields This augmented system of equations is used for the eigenvalue extraction. The auxiliary variable 
is internal to ABAQUS/Standard and is not available for output. This solution method for the coupled acoustic-structural eigenvalue problem is implemented only for the Lanczos eigensolver. If 
is singular, orthogonalization against the singular acoustic modes is done in the Lanczos eigensolver.Since the original system of equations, Equation 2.9.1–18, is unsymmetric, there are left- and right-hand-side eigenvectors. It can be shown that the left and right eigenvectors are as follows:
, 
, where 
is the circular eigenfrequency. The left and right eigenvector subspaces are then used to compute modal quantities (generalized mass, participation factors, effective mass) and to project the mass, stiffness, and damping matrices in mode-based procedures (such as subspace-based steady-state dynamics or transient modal dynamic analysis) to obtain a reduced system of equations. Most of these computations are conducted in a very similar fashion to the way they are carried out in the pure structural problem and will not be discussed here. In addition, for each mode an acoustic fraction of the generalized mass is computed as the ratio between acoustic contributions to the generalized mass and the total generalized mass.The only exception worth a brief discussion is the choice for the calculation of the acoustic participation factors and effective masses, as follows. First, a “rigid body” acoustic mode, 
, analogous to the rigid body modes for the structural problem outlined in “Variables associated with the natural modes of a model,” Section 2.5.2, is chosen to be a constant pressure field of unity. A total “acoustic mass” is then defined as 
. Left and right acoustic participation factors are defined as
in the equation for 
is arbitrary. However, this scaling ensures that when all eigenmodes are extracted, the sum of all acoustic effective masses is 1.0 (minus the contributions from nodes constrained in the acoustic degree of freedom).The direct-solution steady-state dynamic analysis procedure is the preferred solution method for acoustics in ABAQUS if volumetric drag and/or acoustic radiation are significant. If these effects are not significant, the mode-based procedure is preferred because of its efficiency.
All model degrees of freedom and loads are assumed to be varying harmonically at an angular frequency 
, so we can write
is the constant complex amplitude of the variable . Thus, 
, as and, thus, write The equilibrium equation is now in a form where the density is complex and the acoustic medium velocity does not enter. We divide this equation by 
and combine it with the second time derivative of the constitutive law, Equation 2.9.1–2, to obtain We have not used the assumption that the spatial gradient of 
is small, as was done in the transient dynamics formulation.The development of the variational statement parallels that for the case of transient dynamics, as though the volumetric drag were absent and the density complex. The variational statement is
In steady state the boundary traction is defined as
On , 
is prescribed, analogous to transient analysis.
On , we prescribe
is enforced, even in the presence of volumetric drag.On , the reactive boundary between the acoustic medium and a rigid baffle, we apply
On , the radiating boundary, we apply the radiation boundary condition impedance in the same form as for the reactive boundary but with the parameters as defined in Equation 2.9.1–38 and Equation 2.9.1–39.
On , the acoustic-structural interface, we equate the displacement of the fluid and solid as in the transient case. However, the acoustic boundary traction coupling fluid to solid,
On , the mixed impedance boundary and acoustic-structural boundary, the condition
The above equation uses the complex density, 
. We manipulate it into a form that has real coefficients and an additional time derivative through the relations
Applying Galerkin's principle, the finite element equations are derived as before. We arrive again at Equation 2.9.1–17 with the same matrices except for the damping and stiffness matrices of the acoustic elements and the surfaces that have imposed impedance conditions, which now appear as
For steady-state harmonic response we assume that the structure undergoes small harmonic vibrations, identified by the prefix 
, about a deformed, stressed base state, which is identified by the subscript 
. Hence, the total stress can be written in the form
is the stress in the base state; 
is the elasticity matrix for the material; 
is the stiffness proportional damping factor chosen for the material (to give the stiffness proportional contribution to the Rayleigh damping, thus introducing the viscous part of the material behavior); and, from the discretization assumption, 
To solve the steady-state problem, we assume that the governing equations are satisfied in the base state, and we linearize these equations in terms of the harmonic oscillations. For the internal force vector this yields
We assume that the loads and (because of linearity) the response are harmonic, and, hence, we can write
, 
, 
, and 
are the real and imaginary parts of the amplitudes of the response; 
and 
are the real and imaginary parts of the amplitude of the force applied to the structure; 
and 
are the real and imaginary parts of the amplitude of the acoustic traction (dimensions of volumetric acceleration) applied to the fluid; and 
is the circular frequency. We substitute these equations into Equation 2.9.1–25 and use the time-harmonic form of Equation 2.9.1–16, 
, which yields the coupled complex linear equation system where 
is symmetric, Equation 2.9.1–26 is symmetric. The system may be quite large, because the real and imaginary parts of the structural degrees of freedom and of the pressure in the fluid all appear in the system. This set of equations is solved for each frequency requested in the direct-solution steady-state dynamics procedure. If damping is absent, the user can specify that only the real matrix equation should be factored in the analysis. Nonzero 
values for the acoustic medium and nonzero values for the impedances represent damping. As mentioned above for the transient case, the coupled system can be split into an uncoupled structural analysis and an acoustic analysis driven by the structural response, provided the fluid forces on the structure are small.The medium supporting acoustic waves may be flowing through a porous matrix, such as fiberglass used for sound deadening. In this case the parameter is the flow resistance, the pressure drop required to force a unit flow through the porous matrix. A propagating plane wave with nominal particle velocity loses energy at a rate
and bulk viscosity 
. These are proportionality constants between components of the stress and spatial derivatives of the shear strain rate and volumetric strain rate, respectively. In fluid mechanics the shear viscosity term is usually more important than the bulk term ; however, acoustics is the study of volumetrically straining flows, so both constants can be important. The linearized Navier-Stokes equations for adiabatic perturbations about a base state can be expressed in terms of the pressure field alone (Morse and Ingard, 1968):In steady state this linearized equation can be written in the form of Equation 2.9.1–21, with
If the combined viscosity effects are small,
so we can writeIn steady-state form where is the forcing frequency. This leads to the following analogy between viscous fluid losses and volumetric drag or flow resistance: with density constant with respect to frequency. The energy loss rate for a propagating plane wave in this linearized, adiabatic, small-viscosity case isSeveral secondary quantities are useful in acoustic analysis, derived from the fundamental acoustic pressure field variable. In steady-state dynamics the acoustic particle velocity at any field point is
, so this expression simplifies to
Equation 2.9.1–11 (or alternatively Equation 2.9.1–9) can be written in a complex admittance form for steady-state analysis:
where we define The term
is the complex admittance of the boundary, and 
is the corresponding complex impedance. Thus, a required complex impedance or admittance value can be entered for a given frequency by fitting to the parameters and 
using Equation 2.9.1–35.For absorption of plane waves in an infinite medium with volumetric drag, the complex impedance can be shown to be
For the impedance-based nonreflective boundary condition in ABAQUS/Standard, the equations above are used to determine the required constants 
and . They are a function of frequency if the volumetric drag is nonzero. The small-drag versions of these equations are used in the direct time integration procedures, as in Equation 2.9.1–42.
Many acoustic studies involve a vibrating structure in an infinite domain. In these cases we model a layer of the acoustic medium using finite elements, to a thickness of 
to a full wavelength, out to a “radiating” boundary surface. We then impose a condition on this surface to allow the acoustic waves to pass through and not reflect back into the computational domain. For radiation boundaries of simple shapes—such as planes, spheres, and the like—simple impedance boundary conditions can represent good approximations to the exact radiation conditions. In particular, we include local algebraic radiation conditions of the form
is the wave number and is the complex density (see Equation 2.9.1–20). 
is a geometric factor related to the metric factors of the curvilinear coordinate system used on the boundary, and 
is a spreading loss term (see Table 2.9.1–1). Comparison of Equation 2.9.1–37 and Equation 2.9.1–9 reveals that, for steady-state analysis, there exists a direct analogy to the reactive boundary equation, Equation 2.9.1–23, with and For transient procedures the treatment of volumetric drag in the acoustic equations and the radiation conditions necessitates an approximation. In the acoustics equation we use the boundary termCombining Equation 2.9.1–37 with Equation 2.9.1–40, expanding about 
, and retaining only first-order terms leads toThe Fourier inverse of the steady-state form results in the transient boundary conditionThis expression involves independent coefficients for pressure and its first derivative in time, unlike the transient reactive boundary expression (Equation 2.9.1–10), which includes independent coefficients for the first and second derivatives of pressure only. Consequently, to implement this expression, we define the admittance parameters
and so the boundary traction for the transient radiation boundary condition can be written
The values of the parameters and vary with the geometry of the boundary of the radiating surface of the acoustic medium. The geometries supported in ABAQUS are summarized in Table 2.9.1–1.
Table 2.9.1–1 Boundary condition parameters.
| Geometry | | |
|---|---|---|
| Plane |  | |
| Circle or circular cylinder | |  |
| Ellipse or elliptical cylinder |  | |
| Sphere | |  |
| Prolate spheroid |  | |
refers to the eccentricity of the ellipse or spheroid; 
refers to the radius of the circle, sphere, or the semimajor axis of the ellipse or spheroid; 
is the vector locating the integration point on the ellipse or spheroid; 
is the vector locating the center of the ellipse or spheroid; and 
is the vector that orients the major axis.These algebraic boundary conditions are approximations to the exact impedance of a boundary radiating into an infinite exterior. The plane wave condition is the exact impedance for plane waves normally incident to a planar boundary. The spherical condition exactly annihilates the first Legendre mode of a radiating spherical surface; the circular condition is asymptotically correct for the first mode (Bayliss et al., 1982). The elliptical and prolate spheroidal conditions are based on expansions of elliptical and prolate spheroidal wave functions in the low-eccentricity limit (Grote and Keller, 1995); the prolate spheroidal condition exactly annihilates the first term of its expansion, while the elliptical condition is asymptotic.
As already pointed out, the radiation boundary conditions derived in the previous section for plane waves are actually based on the presumption that the sound wave impinges on the boundary from an orthogonal direction. But this is not always the case. Figure 2.9.1–1 shows a general example for plane waves in which the sound wave direction differs from the boundary normal by an angle of 
.
is the sound speed with 
and 
is the corresponding speed normal to the boundary. This exact description of the geometry is the starting point for many developments of approximate absorbing boundary conditions (see, for example, Engquist and Majda, 1977). Thus we haveUsing the first-order expanding approximation to the second term in the square root in the above equation (similar to what we did to reach Equation 2.9.1–41), we can obtain an improved radiation boundary condition
It can be found from comparison that this equation differs from Equation 2.9.1–42 only by a factor of
for plane waves. In two dimensions the can be calculated as The normal and tangential derivatives 
and 
at the integration points can be evaluated using the corresponding element along the radiation boundary surface (see Figure 2.9.1–2); i.e., where 
are the nodal pressure values of the element.The method described in this section can be used only for direct integration transient dynamics, and it cannot be used with steady-state or modal response. In addition, it is available for planar, axisymmetric, and three-dimensional geometries.
Finally, the method makes the equilibrium equations nonlinear, as shown in Equation 2.9.1–48. Although in theory the iteration process in ABAQUS/Standard can solve the nonlinear equilibrium equations accurately, the use of a small half-step residual tolerance is strongly suggested since in many cases the pressure and its related residual along the radiation boundaries are very weak relative to the other places in the modeled domain. The computation of at the integration point is based on the nodal pressures. The nodal pressures are updated using the explicit central difference procedure described in “Explicit dynamic analysis,” Section 2.4.5.
