The previous section described a standard pressure-formulated acoustic ele-ment. These elements can be connected to structural elements so that the two become coupled—the acoustic pressure acts on a structure which causes it to vibrate, and so is the converse where a vibrating structure causes sound to be generated in an acoustic fluid.
This section describes the matrix equations for the coupled fluid–structure interaction problem. The purpose of describing these equations is to highlight:
• how the responses of the acoustic fluid and structure are connected,
14 2. Background
• the unsymmetric matrices that result from fluid–structure interaction problems, and
• how the matrices can be reformulated from unsymmetric to symmetric matrices, leading to a reduction in computation time.
The equations of motion for the structure are (for more details see the ANSYS online help manual [10]):
[Ms] { ¨U} + [Ks] {U} = {Fs} , (2.3) where [Ks] is the structural stiffness matrix, [Ms] is the structural mass ma-trix, {Fs} is a vector of applied structural loads, {U} is a vector of unknown nodal displacements and hence { ¨U} is a vector of the second derivative of dis-placements with respect to time, equivalent to the acceleration of the nodes.
The interaction of the fluid and structure occurs at the interface between the structure and the acoustic elements, where the acoustic pressure exerts a force on the structure and the motion of the structure produces a pressure. To account for the coupling between the structure and the acoustic fluid, addi-tional terms are added to the equations of motion for the structure and fluid (of density, ρ0), respectively, as
[Ms]{ ¨U} + [Ks]{U} = {Fs} + [R]{p} , (2.4) [Mf]{¨p} + [Kf]{p} = {Ff} − ρ0[R]T{ ¨U} , (2.5) where [R] is the coupling matrix that accounts for the effective surface area associated with each node on the fluid-structure interface. Equations (2.4) and (2.5) can be formed into a matrix equation including the effects of damping as
where [Cs] and [Cf] are the structural and acoustic damping matrices, respec-tively. For harmonic analyses, this equation can be reduced to an expression without differentials as The important feature to notice about Equation (2.7) is that the matrix on the left-hand side is unsymmetric and solving for the nodal pressures and dis-placements requires the inversion of this unsymmetric matrix, which requires a significant amount of computer resources. The fluid–structure interaction method described above accounts for two-way coupling between structures
2.4. Fluid–Structure Interaction 15 and fluids. This mechanism is significant if a structure is radiating into a heavier-than-air medium such as water, or if the structure is very lightweight, such as a car cabin. In some vibro-acoustic systems, an acoustic field will be dissipated by the induced vibration of a structure, which has the effect of damping the acoustic response of the system.
When using this coupling in FEA, it is necessary to carefully construct the model to accurately represent the interface between the fluid and the struc-ture. Figure 2.1 illustrates a finite element model of an acoustic duct with a structural partition. The left and right sides of the duct have acoustic elements with only pressure DOFs. The elements for the structural partition contain displacement DOFs. At the interface between the acoustic fluid and the struc-ture is a single layer of acoustic elements that have pressure and displacement DOFs. It is this thin layer of elements that enables the bi-directional coupling between the vibration of the structure and the pressure response in the fluid.
Although it is possible to use acoustic elements with both pressure and dis-placement DOFs for the entire acoustic field, this is unnecessary and would result in long solution times compared to only using this type of element at the fluid-structure interfaces and using acoustic elements with only a pressure DOF for the remainder of the acoustic field.
Structural elements
Schematic of a finite element model with fluid–structure interaction.
When using the ANSYS software it is necessary to explicitly define which surface of the structure and the fluid are in contact by using the Fluid-Structure-Interface (FSI) flag (meaning a switch to indicate the presence of FSI). Release 14.5 of the ANSYS software will try to identify and create the FSI flags at the interface of the structure (only solid elements, not beam or shell elements) and fluid if none are defined. However, it is good practice to manually define the interfaces rather than relying on the automated identi-fication. The ANSYS Mechanical APDL command that is used to define the FSI flag isSF, Nlist, FSI, 1 where Nlist is either ALL to select all the nodes currently selected orP to select the nodes individually. This FSI flag is only relevant forFLUID29, FLUID30, FLUID220, and FLUID221 acoustic elements.
16 2. Background Modeling an acoustically rigid-wall is achieved by not defining acoustic el-ements on an edge, shown in Figure 2.1 on all the exterior sides of the model.
Modeling a free surface can be achieved by setting the pressure to be zero on the nodes of pure acoustic elements (i.e., only pressure DOF). Alterna-tively, if using acoustic elements with both pressure and displacement DOFs, a free surface is modeled by not defining any loads, displacement constraints, or structure. The motion of the fluid boundary can then be obtained by ex-amining the response of the nodes on the surface.
The matrix in Equation (2.7) is unsymmetric, which means that the off-diagonal entries are not transposes of each other. The inversion of an unsym-metric matrix takes longer to compute than a symunsym-metric matrix. There is an option within ANSYS to use a symmetric formulation for the fluid–structure interaction [122, 11]. This can be accomplished by defining a transformation variable for the nodal pressures as
˙
q = jωq = p , (2.8)
and substituting this into Equation (2.7) so that the system of equations becomes
Equation (2.9) has a symmetric matrix which can be inverted and solved for the vectors of the structural nodal displacements U and the transformation variable for nodal pressures q, faster than the unsymmetric formulation in Equation (2.7). The nodal pressures p can then be calculated using Equa-tion (2.8).
An example of the use of unsymmetric and symmetric formulations is described in Section 9.4.3.
The previous discussion described how one can conduct a fluid–structure interaction analysis with bi-directional coupling. This type of analysis can re-quire large computational resources. In some situations it may be reasonable to conduct a one-way analysis where a vibrating structure induces a pres-sure response in an acoustic medium, or vice versa. For this type of analysis, one must remember that some acoustic mechanisms are being neglected such as radiation damping, mass, stiffness, and damping loading of the structure.
Hence one should be cautious if considering to conduct this type of analysis.
A procedure that can be used to conduct this type of analysis is:
• Construct the acoustic and structural models where the nodes of the struc-ture are coincident with the nodes on the exterior boundary of the acoustic domain.