MultiPhysicsVault/.raw/AbaqusTheoriesManual/AbaqusTheoriesManual_019.md

Volumetric strain in the liquid and grains

The bulk behavior of the grains was discussed in ``Constitutive behavior in a porous medium, '' Section 2.8.3. From Equation 2.8.3-2,

\frac {\rho_ {g}}{\rho_ {g} ^ {0}} = \frac {1}{J _ {g}} \approx 1 + \frac {1}{K _ {g}} \left(u _ {w} + \frac {\overline {{p}}}{1 - n - n _ {t}}\right) - \varepsilon_ {g} ^ {t h}.

Combining this with Equation 2.8.1-4 and neglecting all but first-order terms in small quantities, we obtain

n \approx 1 - n _ {t} + \frac {\overline {{p}}}{K _ {g}} + \frac {1}{J} (1 - n ^ {0} - n _ {t} ^ {0}) \left(\frac {u _ {w}}{K _ {g}} - \varepsilon_ {g} ^ {t h} - 1\right).

Using Equation 2.8.3-1 and again neglecting second-order terms in small quantities, we obtain

\begin{array}{l} \frac {\rho_ {w}}{\rho_ {w} ^ {0}} n \approx 1 - n _ {t} - \frac {1}{J} (1 - n ^ {0} - n _ {t} ^ {0}) + \frac {\overline {{p}}}{K _ {g}} + u _ {w} \left(\frac {1 - n _ {t}}{K _ {w}} + \frac {(1 - n ^ {0} - n _ {t} ^ {0})}{J} \left(\frac {1}{K _ {g}} - \frac {1}{K _ {w}}\right)\right) \\ - (1 - n _ {t}) \varepsilon_ {w} ^ {t h} + \frac {1}{J} (1 - n ^ {0} - n _ {t} ^ {0}) (\varepsilon_ {w} ^ {t h} - \varepsilon_ {g} ^ {t h}). \\ \end{array}

Combining this result with Equation 2.8.3-4, and again approximating to first-order in small quantities,

Equation 2.8.4-3

\frac {\rho_ {w}}{\rho_ {w} ^ {0}} n \approx 1 - \frac {1}{J} (1 - n ^ {0} - n _ {t} ^ {0} + h _ {t}) + \frac {\overline {{p}}}{K _ {g}} + u _ {w} \left(\frac {1}{K _ {w}} + \frac {(1 - n ^ {0} - n _ {t} ^ {0})}{J} \left(\frac {1}{K _ {g}} - \frac {1}{K _ {w}}\right)\right)

- \varepsilon_ {w} ^ {t h} + \frac {1}{J} (1 - n ^ {0} - n _ {t} ^ {0}) (\varepsilon_ {w} ^ {t h} - \varepsilon_ {g} ^ {t h}).

Saturation

Because u _ { w } measures pressure in the wetting liquid and we neglect the pressure in the other fluid phase in the medium (see ``Effective stress principle for porous media,'' Section 2.8.1), the medium is fully saturated for u _ { w } > 0 . Negative values of u _ { w } represent capillary effects in the medium. For u _ { w } < 0 it is known (see, for example, Nguyen and Durso, 1983) that, at a given value of capillary pressure, - u _ { w } , the saturation lies within certain limits. Typical forms of these limits are shown in Figure 2.8.4-1.

Figure 2.8.4-1 Typical liquid absorption and exsorption behavior.

text_image

pore pressure -u_w exsorption scanning absorption 0.0 1.0 saturation

We write these limits as s ^ { a } \leq s \leq s ^ { e } , where s ^ { a } ( u _ { w } ) is the limit at which absorption will occur (so that \dot { s } > 0 ) , and s ^ { e } ( u _ { w } ) is the limit at which exsorption will occur, and thus \dot { s } < 0 . We assume that these relationships are uniquely invertible and can, thus, also be written as u _ { w } ^ { a } \left( s \right) during absorption and u _ { w } ^ { e } \left( s \right) during exsorption. We also assume that some wetting liquid will always be present in the medium: s > 0 .

Bear (1972) suggests that the transition between absorption and exsorption and vice versa takes place along "scanning" curves. We approximate these with a straight line, as shown in Figure 2.8.4-1.

Saturation is treated as a state variable that may have to change if the wetting liquid pressure is outside the range for which its value is admissible according to that actual data corresponding to Figure 2.8.4-1. The evolution of saturation as a state variable is defined as follows. Assume that the saturation at time t , s \big | _ { t } , is known. It must satisfy the constraints

s \big | _ {t} = 1. 0 \quad \text { if } \quad u _ {w} > 0. 0; \qquad s ^ {a} (u _ {w} \big | _ {t}) \leq s \big | _ {t} \leq s ^ {e} (u _ {w} \big | _ {t}) \quad \text { otherwise }.

We solve the continuity equation for \left. u _ { w } \right| _ { t + \Delta t } , initially assuming s | _ { t + \Delta t } = s | _ { t } . We then obtain s \big | _ { t + \Delta t } by the following rules:

Procedures

\left. \mathrm { i f } \quad u _ { w } \right| _ { t + \Delta t } > 0 . 0 \quad \mathrm { t h e n } \quad s \big | _ { t + \Delta t } = 1 . 0 \quad \mathrm { a n d } \quad \dot { s } = 0 . 0 ;

\mathrm { e l s e ~ i f } u _ { w } \big | _ { t + \Delta t } > u _ { w } ^ { a } ( s _ { t } ) \quad \mathrm { t h e n } \quad s \big | _ { t + \Delta t } = s ^ { a } ( u _ { w } \big | _ { t + \Delta t } ) \quad \mathrm { a n d } \quad \frac { d s } { d u _ { w } } = \frac { d s ^ { a } } { d u _ { w } } \bigg | _ { u _ { w } \big | _ { t + \Delta t } } ; duw du w ¯¯ u w ¯¯

\mathrm { e l s e ~ i f } \quad u _ { w } \big | _ { t + \Delta t } < u _ { w } ^ { e } ( s _ { t } ) \quad \mathrm { t h e n } \quad s \big | _ { t + \Delta t } = s ^ { e } ( u _ { w } \big | _ { t + \Delta t } ) \quad \mathrm { a n d } \quad \frac { d s } { d u _ { w } } = \frac { d s ^ { e } } { d u _ { w } } \bigg | _ { u _ { w } \big | _ { t + \Delta t } } ; duw duw ¯¯ uw ¯¯ t+¢t

\mathrm { \ o t h e r w i s e } \quad s \big \vert _ { t + \Delta t } = s \big \vert _ { t } + \frac { d s } { d u _ { w } } \bigg \vert _ { s } \Delta u _ { w } \quad \mathrm { a n d } \quad \frac { d s } { d u _ { w } } = \frac { d s } { d u _ { w } } \bigg \vert _ { s } , duw ¯¯ duw duw ¯¯ s

where \left( d s / d u _ { w } \right) \Big | _ { \cdot } is the slope of the scanning line. These choices are shown in Figure 2.8.4-2.

Figure 2.8.4-2 Evolution of s in unsaturated cases.

Jacobian contribution

The Jacobian contribution from the continuity equation is obtained from the variation of Equation 2.8.4-2 with respect to x and u _ { w } at time t + \Delta t .

Consider first the surface integral. The surface divides into that part across which the liquid mass flow rate, \rho _ { w } s n \mathbf { n } \cdot \mathbf { v } _ { w } , is prescribed and that part where the wetting liquid pressure, u _ { w } , is prescribed. Thus, the only contribution of this term to the Jacobian is the variation in the integral caused by change in surface area in that part where the mass flow is prescribed. We neglect this contribution.

The remaining part of the variation of Equation 2.8.4-2 is

\int_ {V} \frac {1}{J} \left[ \delta u _ {w} d \left(J \frac {\rho_ {w}}{\rho_ {w} ^ {0}} (s n + n _ {t})\right) + \frac {\Delta t}{\rho_ {w} ^ {0} g} d \left\{\frac {J k _ {s}}{(1 + \beta \sqrt {\mathbf {v} _ {w} \cdot \mathbf {v} _ {w}})} \frac {\partial \delta u _ {w}}{\partial \mathbf {x}} \cdot \mathbf {k} \cdot \left(\frac {\partial u _ {w}}{\partial \mathbf {x}} - \rho_ {w} \mathbf {g}\right) \right\} \right] d V.

Using Equation 2.8.4-3 we have

\begin{array}{l} J \frac {\rho_ {w}}{\rho_ {w} ^ {0}} s n \approx s \bigg [ J \big (1 + \frac {\overline {{p}}}{K _ {g}} + \frac {u _ {w}}{K _ {w}} - \varepsilon_ {w} ^ {t h} \big) - 1 + n ^ {0} + n _ {t} ^ {0} - h _ {t} + u _ {w} (1 - n ^ {0} - n _ {t} ^ {0}) \bigg (\frac {1}{K _ {g}} - \frac {1}{K _ {w}} \bigg) \\ \left. + (1 - n ^ {0} - n _ {t} ^ {0}) (\varepsilon_ {w} ^ {t h} - \varepsilon_ {g} ^ {t h}) \right], \\ \end{array}

and, thus, neglecting small terms compared to unity,

\begin{array}{l} d \big (J \frac {\rho_ {w}}{\rho_ {w} ^ {0}} s n \big) = s \left[ - \frac {J}{3 K _ {g}} \mathbf {I}: \overline {{\mathbf {D}}} + J \mathbf {I} \right]: d \pmb {\varepsilon} \\ + \left[ \frac {d s}{d u _ {w}} (J - 1 + n ^ {0} + n _ {t} ^ {0} - h _ {t}) - \frac {s J}{9 K _ {g}} \mathbf {I}: \overline {{\mathbf {D}}}: \mathbf {I} \left(\frac {1}{K _ {g}} + \frac {h _ {t}}{K _ {w} (1 + h _ {t} - n _ {t} ^ {0})}\right) \right. \\ \left. + \frac {s}{K _ {w}} (J - 1 + n ^ {0} + n _ {t} ^ {0}) + \frac {s}{K _ {g}} (1 - n ^ {0} - n _ {t} ^ {0}) \right] d u _ {w}. \\ \end{array}

Equation 2.8.3-5 shows that J ( \rho _ { w } / \rho _ { w } ^ { 0 } ) n _ { t } \approx h _ { t } , which is defined by the evolution equation given in ``Constitutive behavior in a porous medium, '' Section 2.8.3, and so makes no contribution to the Jacobian.

Finally, the Jacobian contribution from the permeability term is rather complex in the general case of the nonlinear Forchheimer flow law. Although we include it in the software, here we only write the linearized flow version reflecting Darcy's law (¯ = 0):

\begin{array}{l} d \left[ J k _ {s} \frac {\partial \delta u _ {w}}{\partial \mathbf {x}} \cdot \mathbf {k} \cdot \left(\frac {\partial u _ {w}}{\partial \mathbf {x}} - \rho_ {w} \mathbf {g}\right) \right] = \\ J k _ {s} \frac {\partial \delta u _ {w}}{\partial \mathbf {x}} \cdot \mathbf {k} \cdot \left(\frac {\partial u _ {w}}{\partial \mathbf {x}} - \rho_ {w} \mathbf {g}\right) \mathbf {I}: d \pmb {\varepsilon} + J \frac {d k _ {s}}{d s} \frac {d s}{d u _ {w}} \frac {\partial \delta u _ {w}}{\partial \mathbf {x}} \cdot \mathbf {k} \cdot \left(\frac {\partial u _ {w}}{\partial \mathbf {x}} - \rho_ {w} \mathbf {g}\right) d u _ {w} \\ - J k _ {s} \frac {\partial \delta u _ {w}}{\partial \mathbf {x}} \cdot \frac {\partial d \mathbf {x}}{\partial \mathbf {x}} \cdot \mathbf {k} \cdot \left(\frac {\partial u _ {w}}{\partial \mathbf {x}} - \rho_ {w} \mathbf {g}\right) - J k _ {s} \frac {\partial \delta u _ {w}}{\partial \mathbf {x}} \cdot \mathbf {k} \cdot \left(\frac {\partial u _ {w}}{\partial \mathbf {x}} \cdot \frac {\partial d \mathbf {x}}{\partial \mathbf {x}}\right) ^ {T} \\ + J k _ {s} \frac {\partial \delta u _ {w}}{\partial \mathbf {x}} \cdot \mathbf {k} \cdot \frac {\partial d u _ {w}}{\partial \mathbf {x}} \\ + J k _ {s} \frac {\partial \delta u _ {w}}{\partial \mathbf {x}} \cdot \frac {d \mathbf {k}}{d e} \cdot \left(\frac {\partial u _ {w}}{\partial \mathbf {x}} - \rho_ {w} \mathbf {g}\right) \frac {1}{(1 - n) ^ {2}} \bigg \{\frac {1}{K _ {g}} \mathbf {I}: \overline {{\mathbf {D}}}: \left[ - \frac {1}{3} d \pmb {\varepsilon} \right. \bigg \} \\ + (\frac {1}{J} (1 - n ^ {0} - n _ {t} ^ {0}) + n _ {t}) \mathbf {I}: d \pmb {\varepsilon} - \left(\frac {1}{9 K _ {g}} + \frac {h _ {t}}{9 K _ {w} (1 + h _ {t} - n _ {t} ^ {0})}\right) \mathbf {I} d u _ {w} \Biggr ] \\ \left. + \frac {(1 - n ^ {0} - n _ {t} ^ {0})}{J K _ {g}} d u _ {w} + \frac {n _ {t}}{K _ {w}} d u _ {w} \right\}. \\ \end{array}

Using these results provides the Jacobian of the continuity equation as

Procedures

\begin{array}{l} \int_ {V} \left[ \delta u _ {w} \Bigg \{s \left[ - \frac {1}{3 K _ {g}} \mathbf {I}: \overline {{\mathbf {D}}} + \mathbf {I} \right]: d \pmb {\varepsilon} \right. \\ + \left[ \frac {d s}{d u _ {w}} \frac {1}{J} (J - 1 + n ^ {0} + n _ {t} ^ {0} - h _ {t}) - \frac {s}{9 K _ {g}} \mathbf {I}: \overline {{\mathbf {D}}}: \mathbf {I} \bigg (\frac {1}{K _ {g}} + \frac {h _ {t}}{K _ {w} (1 + h _ {t} - n _ {t} ^ {0})} \bigg) \right. \\ \left. + \frac {s}{J K _ {w}} (J - 1 + n ^ {0} + n _ {t} ^ {0}) + s \frac {1 - n ^ {0} - n _ {t} ^ {0}}{J K _ {g}} \right] d u _ {w} \Biggr \} \\ + \Delta t \frac {k _ {s}}{\rho_ {w} ^ {0} g} \Bigg \{\frac {\partial \delta u _ {w}}{\partial \mathbf {x}} \cdot \Bigg [ \mathbf {k} \cdot \Bigg (\frac {\partial u _ {w}}{\partial \mathbf {x}} - \rho_ {w} \mathbf {g} \Bigg) \mathbf {I} \\ + \frac {d \mathbf {k}}{d e} \cdot \left(\frac {\partial u _ {w}}{\partial \mathbf {x}} - \rho_ {w} \mathbf {g}\right) \frac {1}{(1 - n) ^ {2}} \left(- \frac {1}{3 K _ {g}} \mathbf {I}: \overline {{\mathbf {D}}} + \left(\frac {1}{J} (1 - n ^ {0} - n _ {t} ^ {0}) + n _ {t}\right) \mathbf {I}\right)\left. \right]: d \pmb {\varepsilon} \\ + \frac {\partial \delta u _ {w}}{\partial \mathbf {x}} \cdot \mathbf {k} \cdot \frac {\partial d u _ {w}}{\partial \mathbf {x}} \\ + \left[ \frac {1}{k _ {s}} \frac {d k _ {s}}{d s} \frac {d s}{d u _ {w}} \frac {\partial \delta u _ {w}}{\partial \mathbf {x}} \cdot \mathbf {k} \cdot \left(\frac {\partial u _ {w}}{\partial \mathbf {x}} - \rho_ {w} \mathbf {g}\right) \right. \\ + \frac {1}{(1 - n) ^ {2}} \frac {\partial \delta u _ {w}}{\partial \mathbf {x}} \cdot \frac {d \mathbf {k}}{d e} \cdot \left(\frac {\partial u _ {w}}{\partial \mathbf {x}} - \rho_ {w} \mathbf {g}\right) \left(\frac {\mathbf {I} : \overline {{\mathbf {D}}} : \mathbf {I}}{9 K _ {g}} \left(- \frac {1}{K _ {g}} - \frac {h _ {t}}{K _ {w} \left(1 + h _ {t} - n _ {t} ^ {0}\right)}\right) \right. \\ \left. \left. + \frac {n _ {t}}{K _ {w}} + \frac {1 - n ^ {0} - n _ {t} ^ {0}}{J K _ {g}}\right) \right] d u _ {w} \\ \left. \left. - \left(\frac {\partial \delta u _ {w}}{\partial \mathbf {x}} \cdot \frac {\partial d \mathbf {x}}{\partial \mathbf {x}} \cdot \mathbf {k} \cdot \left(\frac {\partial u _ {w}}{\partial \mathbf {x}} - \rho_ {w} \mathbf {g}\right) + \frac {\partial \delta u _ {w}}{\partial \mathbf {x}} \cdot \mathbf {k} \cdot \left(\frac {\partial u _ {w}}{\partial \mathbf {x}} \cdot \frac {\partial d \mathbf {x}}{\partial \mathbf {x}}\right) ^ {T}\right) \right\} \right] d V. \\ \end{array}

2.8.5 Solution strategy for coupled diffusion/deformation

The governing equations of pore fluid diffusion/deformation are

equilibrium: K ^ { M N } \bar { c } _ { \delta } ^ { N } - L ^ { M P } \bar { c } _ { u } ^ { P } = P ^ { M } - { \cal I } ^ { M } ; and

pore °uid °ow: ( \widehat { B } ^ { M Q } ) ^ { T } \bar { v } ^ { M } + \widehat { H } ^ { Q P } \bar { u } ^ { P } = Q ^ { Q }

There are two common approaches to solving these coupled equations. One approach is to solve one set of equations first and then use the results obtained to solve the second set of equations. These results in turn are fed back into the first set of equations to see what changes (if any) result in the solution. This process continues until succeeding iterations produce negligible changes in the solutions obtained. This is the so-called staggered approach to the solution of coupled systems of equations. The second approach is to solve the coupled systems directly. This direct approach is used in ABAQUS/Standard because of its rapid convergence even in severely nonlinear cases.

We first introduce a time integration operator in the pore fluid flow equation. The operator chosen is the simple one-step method:

\bar {\delta} _ {t + \triangle t} ^ {N} = \bar {\delta} _ {t} ^ {N} + \triangle t [ (1 - \zeta) \bar {v} _ {t} ^ {N} + \zeta \bar {v} _ {t + \triangle t} ],

Procedures

where 0 \leq \zeta \leq 1 . In fact, to ensure numerical stability, we choose ³ = 1 (backward difference) so that

\bar {v} _ {t + \triangle t} = \frac {1}{\triangle t} (\bar {\delta} _ {t + \triangle t} ^ {N} - \bar {\delta} _ {t}).

With this operator the pore fluid flow equation at time (t + 4t) can be rewritten as

(\widehat {B} ^ {M Q}) ^ {T} \bar {\delta} _ {t + \triangle t} ^ {M} + \triangle t \widehat {H} ^ {Q P} \bar {u} _ {t + \triangle t} ^ {P} = \triangle t Q _ {t + \triangle t} ^ {Q} + (\widehat {B} ^ {M Q}) ^ {T} \bar {\delta} _ {t} ^ {M}.

Using the Newton linearization, the flow equation becomes

- (B ^ {M Q}) ^ {T} \bar {c} _ {\delta} ^ {M} - \triangle t H ^ {Q P} \bar {c} _ {u} ^ {P} = \triangle t [ - Q _ {t + \triangle t} ^ {Q} + (\widehat {B} ^ {M Q}) ^ {T} \bar {v} _ {t + \triangle t} ^ {M} + \widehat {H} ^ {Q P} \bar {u} _ {t + \triangle t} ^ {P} ].

Then the coupled system of equations to be solved is

K ^ {M N} \bar {c} _ {\delta} ^ {N} - L ^ {M P} \bar {c} _ {u} ^ {P} = P ^ {M} - I ^ {M}

and

- (B ^ {M Q}) ^ {T} \bar {c} _ {\delta} ^ {M} - \triangle t H ^ {Q P} \bar {c} _ {u} ^ {P} = R ^ {Q},

where

R ^ {Q} = \triangle t [ - Q _ {t + \triangle t} ^ {Q} + (\widehat {B} ^ {M Q}) ^ {T} \bar {v} _ {t + \triangle t} ^ {M} + \widehat {H} ^ {Q P} \bar {u} _ {t + \triangle t} ^ {P} ].

These equations form the basis of the iterative solution of a time step in a coupled flow deformation solution in ABAQUS/Standard. They are, in general, nonsymmetric. The lack of symmetry may be due to a number of effects: changes in geometry, dependence of permeability on void ratio, changes in saturation in partially saturated cases, and inclusion of fluid gravity load terms in total pore pressure analyses. The steady-state version of the coupled problem is also nonsymmetric.

ABAQUS/Standard uses the nonsymmetric equation solver by default in all steady-state or partially saturated coupled analyses; in other cases it uses the symmetric solver by default. In the latter cases, if the effects of changes in geometry or nonlinear permeability are significant, or if a total pore pressure (versus excess pore pressure) analysis is performed, the user is advised to activate the unsymmetric solver by using UNSYMM=YES on the *STEP option.

2.9 Coupled fluid-solid analysis

2.9.1 Coupled acoustic-structural medium analysis

ABAQUS/Standard provides a set of elements for modeling an acoustic medium undergoing small pressure variations and interface conditions to couple these acoustic elements to a structural model. These elements are provided primarily so that steady-state harmonic (linear) response analysis can be

performed for a coupled acoustic-structural system, such as in 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.

Since ABAQUS cannot currently extract the eigenmodes of a coupled structural-acoustic system, the modal-based and subspace-based steady-state procedures cannot be used when there is acoustic-structural coupling, although they can be used if the acoustic medium is modeled alone. The elements can also be used with nonlinear response analysis (direct integration) procedures: whether such results are useful depends on the applicability of the small pressure change assumption. The acoustic medium 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:

1. Prescribed pressure (degree of freedom 8) at the boundary nodes.
2. Prescribed inward normal derivative of pressure per unit density of the acoustic medium through the use of a *CLOAD on degree of freedom 8 of a boundary node. If the *CLOAD has zero magnitude--that is, if no *CLOAD 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).
3. Acoustic-structural coupling defined either by using surface-based contact 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.
4. 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.

The flow resistance and the properties of the absorbing boundaries may be functions of frequency in steady-state response analysis but are assumed to be constant in the direct integration procedure. This section defines the formulation used in these elements.

Acoustic equations

The equilibrium equation for small motions of a compressible, adiabatic fluid with velocity-dependent momentum losses is taken to be

Equation 2.9.1-1

\frac {\partial p}{\partial \mathbf {x}} + r (\mathbf {x}, \theta_ {i}) \dot {\mathbf {u}} ^ {f} + \rho_ {f} (\mathbf {x}, \theta_ {i}) \ddot {\mathbf {u}} ^ {f} = 0,

where p is the excess pressure in the fluid (the pressure in excess of any static pressure), x is the spatial position of the fluid particle, \dot { \mathrm { \mathbf { u } } } ^ { f } is the fluid particle velocity, \ddot { \mathrm { \mathbf { u } } } ^ { f } is the fluid particle acceleration, \rho _ { f } is the density of the fluid, r is the "volumetric drag" (force per unit volume per velocity), and \theta _ { i } are i independent field variables such as temperature, humidity (of air), or salinity (of water) on which \rho _ { f } and r may depend (see ``Acoustic medium,'' Section 12.3.1 of the ABAQUS/Standard 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

Equation 2.9.1-2

p = - \mathrm{K} _ {f} (\mathbf {x}, \boldsymbol {\theta} _ {i}) \frac {\partial}{\partial \mathbf {x}} \cdot \mathbf {u} ^ {f},

where \mathrm { K } _ { f } is the bulk modulus of the fluid.

Physical boundary conditions in acoustic analysis

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 S on which the following conditions are imposed:

S _ { \mathrm { f p } } , : where the value of the acoustic pressure p is prescribed.

S _ { \mathrm { f t } } . ,: 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), and symmetry planes.

S _ { \mathrm { f r } } , : 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, which 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."

S _ { \mathrm { { f i } } } , : 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.

S _ { \mathrm { f s } } , : 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.

S _ { \mathrm { f r s } 9 } : 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.

\begin{array} { r } { S _ { \mathrm { f f } } , \dot { \mathbf { \mu } } } \end{array} 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.

Formulation for transient dynamics

In ABAQUS the finite element formulations are slightly different in transient and steady-state analyses, primarily with regard to the treatment of the volumetric drag loss parameter and spatial variations of the constitutive paramters. To derive a symmetric system of ordinary differential equations, some approximations are made in the transient case that are not needed in steady state.

To derive the partial differential equation used in transient analysis, we divide Equation 2.9.1-1 by \rho _ { f } , take its gradient with respect to x, neglect the gradient of r / \rho _ { f } , 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:

Equation 2.9.1-3

\frac {1}{\mathrm{K} _ {f}} \ddot {p} + \frac {r}{\rho_ {f} \mathrm{K} _ {f}} \dot {p} - \frac {\partial}{\partial \mathbf {x}} \cdot \left(\frac {1}{\rho_ {f}} \frac {\partial p}{\partial \mathbf {x}}\right) = 0.

The assumption that the gradient of r / \rho _ { f } is small is violated where there are discontinuities in the quantity r / \rho _ { f } (for example, on the boundary between two elements that have a different r / \rho _ { f } value).

Variational statement

An equivalent weak form for the equation of motion, Equation 2.9.1-3, is obtained by introducing an arbitrary variational field, \delta p , , and integrating over the fluid:

\int_ {V _ {f}} \delta p \bigg (\frac {1}{\mathrm{K} _ {f}} \ddot {p} + \frac {r}{\rho_ {f} \mathrm{K} _ {f}} \dot {p} - \frac {\partial}{\partial \mathbf {x}} \cdot \bigg (\frac {1}{\rho_ {f}} \frac {\partial p}{\partial \mathbf {x}} \bigg) \bigg) d V = 0.

Green's theorem allows this to be rewritten as

Equation 2.9.1-4

\int_ {V _ {f}} \left[ \delta p \left(\frac {1}{\mathrm{K} _ {f}} \ddot {p} + \frac {r}{\rho_ {f} \mathrm{K} _ {f}} \dot {p}\right) + \frac {1}{\rho_ {f}} \frac {\partial \delta p}{\partial \mathbf {x}} \cdot \frac {\partial p}{\partial \mathbf {x}} \right] d V + \int_ {S} \delta p \left(\frac {1}{\rho_ {f}} \mathbf {n} ^ {-} \cdot \frac {\partial p}{\partial \mathbf {x}}\right) d S = 0.

Assuming that p is prescribed on S _ { \mathrm { f p } } , 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:

Equation 2.9.1-5

\mathbf {n} ^ {-} \cdot (\frac {1}{\rho_ {f}} \frac {\partial p}{\partial \mathbf {x}} + \frac {r}{\rho_ {f}} \dot {\mathbf {u}} ^ {f} + \ddot {\mathbf {u}} ^ {f}) = 0 \quad \mathrm{on} \quad S - S _ {\mathrm{fp}}.