8.2. Derivation of Acoustic Matrices

Equation 8–12 contains the fluid pressure p and the structural displacement components ux,F, uy,F, and uz,F as the dependent variables to solve. The finite element approximating shape functions for the spatial variation of the pressure and displacement components are given by:

(8–31)

(8–32)

where:

{N} = element shape function for pressure
{N'} = element shape function for displacements
{Pe} = nodal pressure vector
{ue} = {uxe},{uye},{uze} = nodal displacement component vectors

From Equation 8–31 and Equation 8–32, the second time derivative of the variables and the virtual change in the pressure can be expressed as follows:

(8–33)

(8–34)

(8–35)

After substituting Equation 8–31 and Equation 8–32 into Equation 8–12, the finite element statement of the wave Equation 8–5 is expressed as:

(8–36)

where:

{n} = outward normal vector at the fluid boundary
{q} = nodal mass source vector

Other terms are defined in Acoustic Fundamentals. Equation 8–36 can be written in matrix notation to create the following discretized wave equation:

(8–37)

where: