김경종
27 KiB
Procedures
Using this equation, the term \begin{array} { r } { \mathbf { n } ^ { - } \cdot \frac { \partial p } { \partial \mathbf { x } } } \end{array} is eliminated from Equation 2.9.1-4 to produce
Equation 2.9.1-6
\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 - S _ {\mathrm{fp}}} \delta p (T (\mathbf {x})) d S = 0,
where, for convenience, the boundary "traction" term
Equation 2.9.1-7
T (\mathbf {x}) = \mathbf {n} ^ {-} \cdot (\ddot {\mathbf {u}} ^ {f} + \frac {r}{\rho_ {f}} \dot {\mathbf {u}} ^ {f}) = - \mathbf {n} ^ {-} \cdot (\frac {1}{\rho_ {f}} \frac {\partial p}{\partial \mathbf {x}}) \quad \mathrm{on} \quad S - S _ {\mathrm{fp}}
has been introduced.
Except for the imposed pressure on S _ { \mathrm { f p } } , all of the other boundary conditions described above can be formulated in terms of T ( \mathbf { x } ) . 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:
Equation 2.9.1-8
T (\mathbf {x}) = \mathbf {n} ^ {-} \cdot \ddot {\mathbf {u}} ^ {f} \quad \mathrm{on} \quad S - S _ {\mathrm{fp}}.
When volumetric drag is present, the boundary traction is the normal derivative of the pressure field, divided by the true mass density: a force per unit mass of fluid. Consequently, when volumetric drag exists in a transient acoustic model, a unit of T ( \mathbf { x } ) yields a lower local volumetric acceleration, due to drag losses.
In transient dynamics we enforce the acoustic boundary conditions as follows:
On S _ { \mathrm { f p } 9 } . p is prescribed and \delta p = 0 .
On \begin{array} { r } { S _ { \mathrm { f t } , \dot { \mathbf { \eta } } } . } \end{array} where we prescribe the outward normal derivative of the acoustic pressure per unit density:
T _ {\mathrm{ft}} (\mathbf {x}) \equiv T _ {0}.
In the absence of volumetric drag in the medium, this enforces a value of fluid particle acceleration, \mathbf { n } ^ { - } \cdot \ddot { \mathbf { u } } ^ { f } = T _ { 0 } = a _ { i n } . An imposed T _ { 0 } = a _ { i n } 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 a _ { i n } = 0 , which represents a rigid immobile boundary. As mentioned above, if the medium has nonzero volumetric drag, a unit of T _ { 0 } imposed at the boundary will result in a relatively lower imposed particle acceleration.
On S _ { \mathrm { f r } 9 } : 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:
Equation 2.9.1-9
Procedures
- \mathbf {n} ^ {-} \cdot \dot {\mathbf {u}} ^ {f} = (\frac {1}{k _ {1}} \dot {p} + \frac {1}{c _ {1}} p) \quad \mathrm{on} \quad S _ {\mathrm{fr}},
where 1 / k _ { 1 } and 1 / c _ { 1 } 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 k _ { 1 } and c _ { 1 } , respectively; they are per unit area of the acoustic boundary. Using this definition for the fluid velocity, the boundary tractions in the variational statement become
Equation 2.9.1-10
T _ {\mathrm{fr}} (\mathbf {x}) \equiv - \left(\frac {r}{\rho_ {f}} \frac {1}{c _ {1}} \right. p + \left(\frac {r}{\rho_ {f}} \frac {1}{k _ {1}} + \frac {1}{c _ {1}}\right) \left. \dot {p} + \frac {1}{k _ {1}} \ddot {p}\right).
On S _ { \mathrm { { f i } } 9 } \mathrm { { . } } the radiating boundary, we apply the radiation boundary condition by specifying the corresponding impedance:
T _ {\mathrm{fi}} (\mathbf {x}) \equiv - \left(\frac {1}{c _ {1}} \dot {p} + \frac {1}{a _ {1}} p\right),
using the admittance parameters of Equation 2.9.1-40 and Equation 2.9.1-41, defined below.
On S _ { \mathrm { f s } } \mathrm { \Omega } : --the acoustic-structural interface--we apply the acoustic-structural interface condition by equating displacement of the fluid and solid, which enforces the condition
\mathbf {n} ^ {-} \cdot \mathbf {u} ^ {f} = \mathbf {n} ^ {-} \cdot \mathbf {u} ^ {m},
where \mathbf { u } ^ { m } is the displacement of the structure. In the presence of volumetric drag it follows that the acoustic boundary traction coupling fluid to solid is
T (\mathbf {x}) = \mathbf {n} ^ {-} \cdot (\ddot {\mathbf {u}} ^ {m} + \frac {r}{\rho_ {f}} \dot {\mathbf {u}} ^ {m}).
In ABAQUS the formulation of the transient coupled problem would be made nonsymmetric by the presence of the term \begin{array} { r } { \mathbf { n } ^ { - } \cdot \left( \frac { r } { \rho _ { f } } \dot { \mathbf { u } } ^ { m } \right) } \end{array} . In the great majority of practical applications the acoustic tractions associated with volumetric drag are small compared to those associated with fluid inertia,
\ddot {\mathbf {u}} ^ {m} \gg \frac {r}{\rho_ {f}} \dot {\mathbf {u}} ^ {m} \quad \forall \mathbf {u} ^ {m} (t),
so this term is ignored in transient analysis:
T _ {\mathrm{fs}} (\mathbf {x}) \equiv \mathbf {n} ^ {-} \cdot \ddot {\mathbf {u}} ^ {m}.
On S _ { \mathrm { f r s } 9 } . the mixed impedance boundary and acoustic-structural boundary, we apply a condition
Procedures
that relates the relative outward velocity between the acoustic medium and the structure to the pressure and rate of change of pressure:
Equation 2.9.1-11
\mathbf {n} ^ {-} \cdot (\dot {\mathbf {u}} ^ {m} - \dot {\mathbf {u}} ^ {f}) = \frac {1}{k _ {1}} \dot {p} + \frac {1}{c _ {1}} p \quad \mathrm{on} \quad S _ {\mathrm{frs}}.
This relative normal velocity represents a rate of compression (or extension) of the intervening layer. Applying this equation to the definition of T ( \mathbf { x } ) , we obtain:
T _ {\mathrm{frs}} (\mathbf {x}) = \mathbf {n} ^ {-} \cdot (\ddot {\mathbf {u}} ^ {m}) - \frac {r}{\rho_ {f}} \frac {1}{c _ {1}} p - \left(\frac {r}{\rho_ {f}} \frac {1}{k _ {1}} + \frac {1}{c _ {1}}\right) \dot {p} - \frac {1}{k _ {1}} \ddot {p}.
This expression for T ( \mathbf { x } ) is the sum of its definitions for S _ { \mathrm { f s } } and S _ { \mathrm { f r } } . Again, in the transient case the effect of volumetric drag on the structural displacement term in the acoustic traction is ignored.
These definitions for the boundary term, T ( \mathbf { x } ) , 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):
\begin{array}{l} \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 _ {\mathrm{ft}}} \delta p T _ {0} d S \\ + \int_ {S _ {\mathrm{fr}}} \delta p \left(\frac {r}{\rho_ {f} c _ {1}} \right. p + \left(\frac {r}{\rho_ {f} k _ {1}} + \frac {1}{c _ {1}}\right) \left. \dot {p} + \frac {1}{k _ {1}} \ddot {p}\right) d S \\ + \int_ {S _ {\mathrm{fi}}} \delta p \left(\frac {1}{c _ {1}} \dot {p} + \frac {1}{a _ {1}} p\right) d S - \int_ {S _ {\mathrm{fs}}} \delta p \mathbf {n} ^ {-} \cdot \ddot {\mathbf {u}} ^ {m} d S \\ \end{array}
+ \int_ {S _ {\mathrm{frs}}} \delta p \left(\frac {r}{\rho_ {f} c _ {1}} p + \left(\frac {r}{\rho_ {f} k _ {1}} + \frac {1}{c _ {1}}\right) \dot {p} + \frac {1}{k _ {1}} \ddot {p} - \mathbf {n} ^ {-} \cdot \ddot {\mathbf {u}} ^ {m}\right) d S = 0.
The structural behavior is defined by the virtual work equation,
\int_ {V} \delta \varepsilon : \boldsymbol {\sigma} d V + \int_ {V} \alpha_ {c} \rho \delta \mathbf {u} ^ {m} \cdot \dot {\mathbf {u}} ^ {m} d V + \int_ {V} \rho \delta \mathbf {u} ^ {m} \cdot \ddot {\mathbf {u}} ^ {m} d V
+ \int_ {S _ {\mathrm{fs}}} p \delta \mathbf {u} ^ {m} \cdot \mathbf {n} d S - \int_ {S _ {\mathrm{t}}} \delta \mathbf {u} ^ {m} \cdot \mathbf {t} d S = 0,
Equation 2.9.1-12
Equation 2.9.1-13
where \pmb { \sigma } is the stress at a point in the structure, p is the pressure acting on the fluid-structural interface,
Procedures
n is the outward normal to the structure, \rho is the density of the material, \alpha _ { c } is the mass proportional damping factor (part of the Rayleigh damping assumption for the structure), \ddot { { \mathbf { u } } } ^ { m } is the acceleration of a point in the structure, t is the surface traction applied to the structure, \delta \mathbf { u } ^ { m } is a variational displacement field, and \delta \varepsilon is the strain variation that is compatible with \delta \mathbf { u } ^ { m } . For simplicity in this equation all other loading terms except the fluid pressure and surface traction t have been neglected: they are imposed in the usual way.
The discretized finite element equations
Equation 2.9.1-12 and Equation 2.9.1-13 define the variational problem for the coupled fields \mathbf { u } ^ { m } and p . The problem is discretized by introducing interpolation functions: in the fluid p = H ^ { P } p ^ { P } , P = 1 , 2 \dots : up to the number of pressure nodes and in the structure { \bf u } ^ { m } = { \bf N } ^ { N } u ^ { N } , n = 1 , 2 . . : 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 P , Q to refer to pressure degrees of freedom in the fluid and N , M 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: \delta \mathbf { u } ^ { m } = \mathbf { N } ^ { N } \delta u ^ { N } . For the fluid we use \delta p = H ^ { P } \delta p ^ { P } but with the subsequent Petrov-Galerkin substitution
Equation 2.9.1-14
\delta p ^ {P} = \frac {\mathrm{d} ^ {2}}{\mathrm{d} t ^ {2}} (\delta \hat {p} ^ {P}).
The new function \delta \hat { p } ^ { P } makes the single variational equation obtained from summing Equation 2.9.1-12 and Equation 2.9.1-13 dimensionally consistent:
Equation 2.9.1-15
\begin{array}{l} - \delta \hat {p} ^ {P} \Bigg \{(M _ {\mathrm{f}} ^ {P Q} + M _ {\mathrm{fr}} ^ {P Q}) \ddot {p} ^ {Q} + (C _ {\mathrm{f}} ^ {P Q} + C _ {\mathrm{fr}} ^ {P Q}) \dot {p} ^ {Q} + (K _ {\mathrm{f}} ^ {P Q} + K _ {\mathrm{fr}} ^ {P Q} + K _ {\mathrm{fi}} ^ {P Q}) p ^ {Q} - S _ {\mathrm{fs}} ^ {P M} \ddot {u} ^ {M} - P _ {f} ^ {P} \Bigg \} \\ + \delta u ^ {N} \left\{I ^ {N} + M ^ {N M} \ddot {u} ^ {M} + C _ {(m)} ^ {N M} \dot {u} ^ {M} + \left[ S _ {\mathrm{fs}} ^ {Q N} \right] ^ {T} p ^ {Q} - P ^ {N} \right\} = 0, \\ \end{array}
where, for simplicity, we have introduced the following definitions:
Procedures
M _ {\mathrm{f}} ^ {P Q} = \int_ {V _ {f}} \frac {1}{\mathrm{K} _ {f}} H ^ {P} H ^ {Q} d V,
M _ {\mathrm{fr}} ^ {P Q} = \int_ {S _ {\mathrm{fr}} \cup S _ {\mathrm{frs}}} \frac {1}{k _ {1}} H ^ {P} H ^ {Q} d S,
C _ {\mathrm{f}} ^ {P Q} = \int_ {V _ {f}} \frac {r}{\rho_ {f}} \frac {1}{\mathrm{K} _ {f}} H ^ {P} H ^ {Q} d V,
C _ {\mathrm{fr}} ^ {P Q} = \int_ {S _ {\mathrm{fr}} \cup S _ {\mathrm{frs}}} \left(\frac {r}{\rho_ {f}} \frac {1}{k _ {1}} + \frac {1}{c _ {1}}\right) H ^ {P} H ^ {Q} d S + \int_ {S _ {\mathrm{fi}}} \frac {1}{c _ {1}} H ^ {P} H ^ {Q} d S,
K _ {\mathrm{f}} ^ {P Q} = \int_ {V _ {f}} \frac {1}{\rho_ {f}} \frac {\partial H ^ {P}}{\partial \mathbf {x}} \cdot \frac {\partial H ^ {Q}}{\partial \mathbf {x}} d V,
K _ {\mathrm{fr}} ^ {P Q} = \int_ {S _ {\mathrm{fr}} \cup S _ {\mathrm{frs}}} \frac {r}{\rho_ {f}} \frac {1}{c _ {1}} H ^ {P} H ^ {Q} d S,
K _ {\mathrm{fi}} ^ {P Q} = \int_ {S _ {\mathrm{fi}}} \frac {1}{a _ {1}} H ^ {P} H ^ {Q} d S,
S _ {\mathrm{fs}} ^ {P M} = \int_ {S _ {\mathrm{fs}} \cup S _ {\mathrm{frs}}} H ^ {P} \mathbf {n} \cdot \mathbf {N} ^ {M} d S,
P _ {f} ^ {P} = \int_ {S _ {\mathrm{ft}}} H ^ {P} T _ {0} d S,
M ^ {N M} = \int_ {V} \rho \mathbf {N} ^ {N} \cdot \mathbf {N} ^ {M} d V,
C _ {(m)} ^ {N M} = \int_ {V} \alpha_ {c} \rho \mathbf {N} ^ {N} \cdot \mathbf {N} ^ {M} d V,
I ^ {N} = \int_ {V} \pmb {\beta} ^ {N}: \pmb {\sigma} d V,
P ^ {N} = \int_ {S _ {\mathrm{t}}} \mathbf {N} ^ {N} \cdot \mathbf {t} d S,
where \beta ^ { N } 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 P _ { f } ^ { P } is the nodal right-hand-side term for the acoustical degree of freedom p ^ { P } , 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-- \left[ S _ { \mathrm { f s } } ^ { Q N } \right] ^ { T } p ^ { Q } T 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 \left[ S _ { \mathrm { f s } } ^ { Q N } \right] ^ { T } p ^ { Q } term omitted; i.e., in an analysis without fluid coupling. Subsequently, the fluid equations can be solved, with \left[ S _ { \mathrm { f s } } ^ { P M } \right] \ddot { u } ^ { M } imposed as a boundary condition. This two-step analysis is less expensive and advantageous for such systems as metal structures in air.
Time integration
The equations are integrated through time using the standard implicit dynamic integration option. From the integration operator we obtain relations between the variations of the solution variables (here represented by f ) and their time derivatives:
D _ {a} \stackrel {\mathrm{def}} {=} \frac {\delta \ddot {f}}{\delta f} = \frac {\delta p ^ {P}}{\delta \hat {p} ^ {P}}; D _ {v} \stackrel {\mathrm{def}} {=} \frac {\delta \dot {f}}{\delta f}.
The equations of evolution of the degrees of freedom can be written as
\begin{array}{l} - \delta p ^ {P} \frac {1}{D _ {a}} \biggl \{(M _ {\mathrm{f}} ^ {P Q} + M _ {\mathrm{fr}} ^ {P Q}) \ddot {p} ^ {Q} + (C _ {\mathrm{f}} ^ {P Q} + C _ {\mathrm{fr}} ^ {P Q}) \dot {p} ^ {Q} + (K _ {\mathrm{f}} ^ {P Q} + K _ {\mathrm{fr}} ^ {P Q} + K _ {\mathrm{fi}} ^ {P Q}) p ^ {Q} - S _ {\mathrm{fs}} ^ {P M} \ddot {u} ^ {M} - P _ {f} ^ {P} \biggr \} \\ + \delta u ^ {N} \left\{I ^ {N} + M ^ {N M} \ddot {u} ^ {M} + C _ {(m)} ^ {N M} \dot {u} ^ {M} + \left[ S _ {\mathrm{fs}} ^ {Q N} \right] ^ {T} p ^ {Q} - P ^ {N} \right\} = 0. \\ \end{array}
The linearization of this equation is
\begin{array}{l} - \delta p ^ {P} \bigg \{(M _ {\mathrm{f}} ^ {P Q} + M _ {\mathrm{fr}} ^ {P Q}) + \frac {D _ {v}}{D _ {a}} (C _ {\mathrm{f}} ^ {P Q} + C _ {\mathrm{fr}} ^ {P Q}) + \frac {1}{D _ {a}} (K _ {\mathrm{f}} ^ {P Q} + K _ {\mathrm{fr}} ^ {P Q} + K _ {\mathrm{fi}} ^ {P Q}) \bigg \} \mathrm{d} p ^ {Q} + \delta p ^ {P} S _ {\mathrm{fs}} ^ {P M} \mathrm{d} u ^ {M} \\ + \delta u ^ {N} \left[ S _ {\mathrm{fs}} ^ {Q N} \right] ^ {T} \mathrm{d} p ^ {Q} + \delta u ^ {N} \bigg \{K ^ {N M} + D _ {a} M ^ {N M} + D _ {v} (C _ {(m)} ^ {N M} + C _ {(k)} ^ {N M}) \bigg \} \mathrm{d} u ^ {M} = 0, \\ \end{array}
where dp and du are the correction to the solution obtained from the Newton iteration, K ^ { N M } is the structural stiffness matrix, and C _ { ( k ) } ^ { N M } is the structural damping matrix. These equations are symmetric if the structural stiffness is symmetric.
Summary of additional approximations of the transient formulation
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.
Formulation for steady-state response
The *STEADY STATE DYNAMICS, DIRECT solution procedure is the preferred solution method for acoustics in ABAQUS.
All model degrees of freedom and loads are assumed to be varying harmonically at an angular frequency −, so we can write
Procedures
f = \tilde {f} \exp {i \Omega t},
where \tilde { f } is the constant complex amplitude of the variable f . Thus,
\dot {f} = i \Omega f, \quad \ddot {f} = - \Omega^ {2} f.
We begin with the equilibrium equation
\frac {\partial p}{\partial \mathbf {x}} + r \dot {\mathbf {u}} ^ {f} + \rho_ {f} \ddot {\mathbf {u}} ^ {f} = 0
and use the harmonic time-derivative relations to obtain
\frac {\partial \tilde {p}}{\partial \mathbf {x}} - \Omega^ {2} (\rho_ {f} + \frac {r}{i \Omega}) \tilde {\mathbf {u}} ^ {f} = 0.
We define the complex density, \tilde { \rho } , as
Equation 2.9.1-16
\tilde {\rho} \equiv \rho_ {f} + \frac {r}{i \Omega}
and, thus, write
Equation 2.9.1-17
\frac {\partial \tilde {p}}{\partial \mathbf {x}} - \Omega^ {2} (\tilde {\rho}) \tilde {\mathbf {u}} ^ {f} = 0.
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 \tilde { \rho } and combine it with the second time derivative of the constitutive law, Equation 2.9.1-2, to obtain
Equation 2.9.1-18
- \Omega^ {2} \frac {1}{\mathrm{K} _ {f}} \tilde {p} - \frac {\partial}{\partial \mathbf {x}} \cdot \left(\frac {1}{\tilde {\rho}} \frac {\partial \tilde {p}}{\partial \mathbf {x}}\right) = 0.
We have not used the assumption that the spatial gradient of r / \rho _ { f } is small, as was done in the transient dynamics formulation.
Variational statement
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
\int_ {V _ {f}} \delta p \left[ - \Omega^ {2} \frac {1}{\mathrm{K} _ {f}} \tilde {p} - \frac {\partial}{\partial \mathbf {x}} \cdot \left(\frac {1}{\tilde {\rho}} \frac {\partial \tilde {p}}{\partial \mathbf {x}}\right) \right] d V = 0.
Procedures
Integrating by parts, we have
- \int_ {V _ {f}} \delta p \frac {\Omega^ {2}}{\mathrm{K} _ {f}} \tilde {p} d V - \int_ {V _ {f}} \frac {1}{\tilde {\rho}} \frac {\partial \delta p}{\partial \mathbf {x}} \cdot \frac {\partial \tilde {p}}{\partial \mathbf {x}} d V + \int_ {S} \delta p \frac {1}{\tilde {\rho}} \frac {\partial \tilde {p}}{\partial \mathbf {x}} \cdot \mathbf {n} ^ {-} d S = 0.
In steady state the boundary traction is defined as
\tilde {T} (\mathbf {x}) \equiv - \frac {1}{\tilde {\rho}} \frac {\partial \tilde {p}}{\partial \mathbf {x}} \cdot \mathbf {n} ^ {-} = - \Omega^ {2} \mathbf {n} ^ {-} \cdot \tilde {\mathbf {u}} ^ {f} = \mathbf {n} ^ {-} \cdot \ddot {\mathbf {u}} ^ {f}.
This expression is not the Fourier transform of the boundary traction defined above for the transient case. The steady-state definition is based on the complex density and includes the volumetric drag effect in such a way that it is always equal to the acceleration of the fluid particles. The application of boundary conditions may be slightly different for some cases in steady state, due to this definition of the traction.
On S _ { \mathrm { f p } } , : \tilde { p } is prescribed, analogous to transient analysis.
On S _ { \mathrm { f t } } , : we prescribe
\tilde {T} _ {\mathrm{ft}} (\mathbf {x}) \equiv T _ {0}.
The condition \mathbf { n } ^ { - } \cdot \ddot { \mathbf { u } } ^ { f } = T _ { 0 } = a _ { i n } is enforced, even in the presence of volumetric drag.
On S _ { \mathrm { f r } } , : the reactive boundary between the acoustic medium and a rigid baffle, we apply
Equation 2.9.1-19
\tilde {T} _ {\mathrm{fr}} (\mathbf {x}) \equiv - \left(\frac {i \Omega}{c _ {1}} - \frac {\Omega^ {2}}{k _ {1}}\right) \tilde {p}.
On \begin{array} { r } { S _ { \hat { \mathrm { f i } } \bullet \bullet } } \end{array} 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 2.9.1-35 and Equation 2.9.1-36.
On S _ { \mathrm { f s } 9 } : 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,
\tilde {T} (\mathbf {x}) = - \Omega^ {2} \mathbf {n} ^ {-} \cdot \tilde {\mathbf {u}} ^ {m},
can be applied without affecting the symmetry of the overall formulation. Consequently, the acoustic tractions in the steady-state case make no assumptions about volumetric drag.
On S _ { \mathrm { f r s } } . ,: the mixed impedance boundary and acoustic-structural boundary, the condition
Equation -
Procedures
\mathbf {n} ^ {-} \cdot (\dot {\mathbf {u}} ^ {m} - \dot {\mathbf {u}} ^ {f}) = \frac {1}{k _ {1}} \dot {p} + \frac {1}{c _ {1}} p \quad \mathrm{on} \quad S _ {\mathrm{frs}}
results in the definition:
\tilde {T} _ {\mathrm{frs}} (\mathbf {x}) = - \Omega^ {2} \mathbf {n} ^ {-} \cdot (\tilde {\mathbf {u}} ^ {m}) - \frac {i \Omega}{c _ {1}} \tilde {p} + \frac {\Omega^ {2}}{k _ {1}} \tilde {p}.
In this case the effect of volumetric drag is included without approximation.
The final variational statement becomes
\begin{array}{l} \int_ {V _ {f}} \left[ - \Omega^ {2} \delta p \left(\frac {1}{\mathrm{K} _ {f}} \tilde {p}\right) + \frac {1}{\tilde {\rho}} \frac {\partial \delta p}{\partial \mathbf {x}} \cdot \frac {\partial \tilde {p}}{\partial \mathbf {x}} \right] d V \\ - \int_ {S _ {\mathrm{ft}}} \delta p a _ {i n} d S + \int_ {S _ {\mathrm{fr}} \cup S _ {\mathrm{fi}}} \delta p \left(\frac {i \Omega}{c _ {1}} - \frac {\Omega^ {2}}{k _ {1}}\right) \tilde {p} d S \\ + \int_ {S _ {\mathrm{fs}}} \delta p \Omega^ {2} \mathbf {n} ^ {-} \cdot \tilde {\mathbf {u}} ^ {m} d S \\ + \int_ {S _ {\mathrm{frs}}} \delta p \left(\frac {i \Omega}{c _ {1}} \tilde {p} - \frac {\Omega^ {2}}{k _ {1}} \tilde {p} + \Omega^ {2} \mathbf {n} ^ {-} \cdot \tilde {\mathbf {u}} ^ {m}\right) d S = 0. \\ \end{array}
This equation is formally identical to Equation 2.9.1-4, except for the pressure "stiffness" term, the radiation boundary conditions, and the imposed boundary traction term. Because the volumetric drag effect is contained in the complex density, the acoustic-structural boundary term in this formulation does not have the limitation that the volumetric drag must be small compared to other effects in the acoustic medium. In addition, in this formulation the applied flux on an acoustic boundary represents the inward acceleration of the acoustic medium, whether or not the volumetric drag is large. Finally, the radiation boundary conditions do not make any approximations with regard to the volumetric drag parameter.
The above equation uses the complex density, 1=½~ . We manipulate it into a form that has real coefficients and an additional time derivative through the relations
\frac {1}{\tilde {\rho}} = \frac {\rho_ {f}}{\rho_ {f} ^ {2} + r ^ {2} / \Omega^ {2}} + i \frac {r / \Omega}{\rho_ {f} ^ {2} + r ^ {2} / \Omega^ {2}}, \quad i \frac {\partial p}{\partial \mathbf {x}} = \frac {1}{\Omega} \frac {\partial \dot {p}}{\partial \mathbf {x}},
to obtain
Equation 2.9.1-21
\begin{array}{l} \int_ {V _ {f}} \left[ - \delta p \left(\frac {\Omega^ {2}}{\mathrm{K} _ {f}} \tilde {p}\right) + \frac {\rho_ {f}}{\rho_ {f} ^ {2} + r ^ {2} / \Omega^ {2}} \frac {\partial \delta p}{\partial \mathbf {x}} \cdot \frac {\partial \tilde {p}}{\partial \mathbf {x}} + (i \Omega) \frac {r / \Omega^ {2}}{\rho_ {f} ^ {2} + r ^ {2} / \Omega^ {2}} \frac {\partial \delta p}{\partial \mathbf {x}} \cdot \frac {\partial \tilde {p}}{\partial \mathbf {x}} \right] d V \\ - \int_ {S _ {\mathrm{ft}}} \delta p a _ {i n} d S + \int_ {S _ {\mathrm{fs}} \cup S _ {\mathrm{frs}}} \delta p \Omega^ {2} \mathbf {n} ^ {-} \cdot \tilde {\mathbf {u}} ^ {m} d S \\ + \int_ {S _ {\mathrm{fr}} \cup S _ {\mathrm{fi}} \cup S _ {\mathrm{frs}}} \delta p \left(\frac {i \Omega}{c _ {1}} \tilde {p} - \frac {\Omega^ {2}}{k _ {1}} \tilde {p}\right) d S = 0. \\ \end{array}
The discretized finite element equations
Applying Galerkin's principle, the finite element equations are derived as before. We arrive again at Equation 2.9.1-15 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
C _ {\mathrm{f}} ^ {P Q} = \int_ {V _ {f}} \frac {r / \Omega^ {2}}{\rho_ {f} ^ {2} + r ^ {2} / \Omega^ {2}} \frac {\partial H ^ {P}}{\partial \mathbf {x}} \cdot \frac {\partial H ^ {Q}}{\partial \mathbf {x}} d V,
K _ {\mathrm{f}} ^ {P Q} = \int_ {V _ {f}} \frac {\rho_ {f}}{\rho_ {f} ^ {2} + r ^ {2} / \Omega^ {2}} \frac {\partial H ^ {P}}{\partial \mathbf {x}} \cdot \frac {\partial H ^ {Q}}{\partial \mathbf {x}} d V,
C _ {\mathrm{fr}} ^ {P Q} = \int_ {S _ {\mathrm{fr}}} \frac {1}{c _ {1}} H ^ {P} H ^ {Q} d S,
K _ {\mathrm{fr}} ^ {P Q} = 0,
K _ {\mathrm{fi}} ^ {P Q} = 0.
The matrix modeling loss to volumetric drag is proportional to the fluid stiffness matrix in this formulation.
For steady-state harmonic response we assume that the structure undergoes small harmonic vibrations, identified by the prefix \Delta , about a deformed, stressed base state, which is identified by the subscript 0. Hence, the total stress can be written in the form
\pmb {\sigma} = \pmb {\sigma} _ {0} + \Delta \pmb {\sigma} = \pmb {\sigma} _ {0} + \mathbf {D} ^ {e l}: (\Delta \pmb {\varepsilon} + \beta_ {c} \Delta \dot {\pmb {\varepsilon}}),
where \pmb { \sigma } _ { 0 } is the stress in the base state; \mathbf { D } ^ { e l } is the elasticity matrix for the material; \beta _ { c } 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,
\Delta \varepsilon = \pmb {\beta} ^ {M} \Delta u ^ {M}.
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
\Delta I ^ {N} = K ^ {N M} \Delta u ^ {M} + C _ {(k)} ^ {N M} \Delta \dot {u} ^ {M},
and Equation 2.9.1-15 can be rewritten, using the time-harmonic relations, as
Equation 2.9.1-22