The coupling conditions on the interface between the acoustic fluid and the structure are given by:

(8–136)

(8–137)

where:

p = acoustic pressure

Equation 8–136 is a kinetic condition relating the solid stress to the pressure imposed on the interface by sound. Equation 8–137 is a kinematic condition that assumes that there is no friction between the solid and acoustic fluid on the interface.

In order to completely describe the FSI problem, the fluid pressure load acting at the interface is added to Equation 15–6. This effect is included in FLUID29, FLUID30, FLUID220, and FLUID221 only if KEYOPT(2) ≠ 1. Hence, the structural equation is rewritten as:

(8–138)

The fluid pressure load vector at the interface S is obtained by integrating the pressure over the area of the surface as follows:

(8–139)

where:

Substituting the finite element approximating function for pressure given by Equation 8–30 into Equation 8–139 leads to:

(8–140)

By comparing the integral in Equation 8–140 with the matrix definition of [R]^T in Equation 8–33, the following relation becomes clear:

(8–141)

Substituting Equation 8–141 into Equation 8–138 results in the dynamic elemental equation of the structure, expressed as:

(8–142)

Equation 8–33 and Equation 8–142 describe the complete finite element discretized equations for the FSI problem. These equations are written in assembled form as:

(8–143)

The acoustic fluid element in an FSI problem will generate all the submatrices with a superscript F in addition to the coupling submatrices , [R]^T, and [R]. Submatrices with a superscript S will be generated by the compatible structural element used in the model.

Assuming that the actual surface is at an elevation η relative to the mean surface in z-direction, the pressure for a sloshing (free) surface is given by:

(8–144)

By utilizing the definition of velocity and the momentum conservation equation in addition to Equation 8–144, pressure can be expressed as:

(8–145)

The surface integration of the "weak" form (Equation 8–33) on the sloshing surface is given by:

(8–146)

The acoustic fluid matrix equation with sloshing effect is expressed as:

(8–147)

where:

Substituting Equation 8–147 into Equation 8–143 yields:

(8–148)

If the impedance boundary is exerted on the FSI interface (input as IMPD on the SF command), the coupling condition expressed in Equation 8–137 is rewritten as:

(8–149)

Substituting Equation 8–149 into Equation 8–33 yields:

(8–150)

where:

Damping matrix [C_FSI] on the impedance FSI interface has been shown to be the same as the damping matrix in Equation 8–36. Therefore, the coupling matrix expressed in Equation 8–148 can still be the final matrix equation. In an incompressible fluid the fluid density is independent of the pressure. This implies the speed of sound equivalently tends toward infinity. The matrix in Equation 8–148 is set to zero.

The matrix in Equation 8–148 has been shown to be unsymmetric. Solving Equation 8–148 may consume more computer resources and time than solving the symmetric matrix equation. For the frequency domain, assume that:

(8–151)

Substituting Equation 8–151 into Equation 8–142 and Equation 8–33 yields:

(8–152)

(8–153)

Dividing coupled Equation 8–153 by yields the acoustic matrix equation, written as:

(8–154)

The coupled matrix equation is given by:

(8–155)

After solving Equation 8–155, the pressure is obtained using Equation 8–151.