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Fluid-structure interaction with acoustics

A detailed description of the acoustic FSI formulation and its verification and validation can be found in Dhulipala et al. (2022).

Fluid-structure interaction is solved by modeling the fluid as an acoustic domain and the structure as an elastic domain. The interface between the fluid and the structural domains ensures that: (1) displacements are the same; and (2) there is a continuity of stress from the structure and pressure from the fluid at the boundaries between these domains. The governing equations for the fluid domain and its weak form are discussed. Then, modeling the interface between the two domains is discussed.

Governing equation for the fluid domain

The following assumptions are made for the fluid:

  • It is inviscid

  • It is irrotational

  • It is subjected to small displacements

  • It does not lose or gain mass

Strong and weak forms

With the above assumptions, the mass conservation equation is given by (Rienstra and Hirschberg, 2004; Sandberg and Ohayon, 2009; Kohnke, 1999):

(1)

where, is the variable fluid density, is the density when the fluid is at rest, and is the fluid displacement vector. The momentum equation is given by (Rienstra and Hirschberg, 2004; Sandberg and Ohayon, 2009; Kohnke, 1999):

(2)

where, is the fluid pressure. The constitutive relation is given by:

(3)

Substituting Eq. (2) after taking the first time derivative of Eq. (1) gives:

(4)

Writing this equation in terms of pressure by using the constitutive relation:

(5)

results in the strong form.

Multiplying equation Eq. (5) with a test function and integrating over the fluid domain:

(6)

Using the Green's theorem, the weak form is given by:

(7)

Kernels and boundary conditions

The first term on the left hand side of Eq. (7) is similar to an inertia term. The AcousticInertia kernel models this term. Second term on the left hand side of Eq. (7) is a diffusion term, and the Diffusion kernel models this. The right hand side represents the boundary condition. Either a Dirichlet condition or a Neumann condition can be used for the boundary condition.

Free surface condition for the fluid domain

The fluid domain, when subjected to shaking, experiences waves on the surface due to changes in pressures. These waves are called as gravity waves. The acoustics formulation alone cannot capture these gravity waves. An additional boundary condition, called the free surface boundary condition, is needed to simulate gravity waves. The pressure at the free surface of a fluid because of waves generated due to dynamic action is given by:

(8)

where, is the height of the wave with reference to the initial free surface before applying the dynamic action. The height can be further expressed as:

(9)

where, represents the normal component of fluid displacement above/below the free surface. The pressure in equation Eq. (8) can be further expressed using equation Eq. (2) as:

(10)

The above equation is the free surface gravity condition in terms of vertical displacements. Because , it can be expressed in terms of pressures as well (Zhao et al., 2017):

(11)

The above boundary condition is like a coupled Dirchlet and Neumann conditions. Term represents a Dirichlet condition and term represents a Neumann condition. Free surface condition is implemented using the FluidFreeSurfaceBC boundary condition.

Fluid and structure interface modeling

The boundary between the fluid and structure is denoted as . At this boundary, the displacements in the normal direction for the fluid and structural domains are the same (Sandberg and Ohayon, 2009; Wang and Bathe, 1997; Bathe et al., 1995; Everstine, 1997):

(12)

where is the normal vector given by . Taking the double time derivative of Eq. (12) gives:

(13)

Using Eq. (2) in the above equation gives:

(14)

In addition, there is continuity of pressure at the boundary as expressed by the structural Cauchy stress tensor:

(15)

where is an identity matrix. The StructureAcousticInterface interface kernel enforces the normal displacements and stress-pressure continuity across the fluid and structure domains through Eq. (14) and Eq. (15), respectively.

References

  1. K. J. Bathe, C. Nitikitpaiboon, and X. Wang. A mixed displacement-based finite element formulation for acoustic fluid-structure interaction. Computers and Structures, 56(2):225–237, 1995.[BibTeX]
  2. S. L. N. Dhulipala, C. Bolisetti, L. B. Munday, W. M. Hoffman, C. Yu, F. U. H. Mir, F. Kong, D. D. Lindsay, and A. S. Whittaker. Development, verification, and validation of comprehensive acoustic fluid-structure interaction capabilities in an open-source computational platform. Earthquake Engineering and Structural Dynamics, pages 1–33, May 2022. URL: https://doi.org/10.1002/eqe.3659, doi:10.1002/eqe.3659.[BibTeX]
  3. G. C. Everstine. Finite element formulations of structural acoustics problems. Computers and Structures, 65(3):307–321, 1997.[BibTeX]
  4. P. C. Kohnke. ANSYS Theory Reference: Release 5.6. ANSYS, Incorporated., 1999.[BibTeX]
  5. S. W. Rienstra and A. Hirschberg. An introduction to acoustics. Eindhoven University of Technology, 2004.[BibTeX]
  6. G. Sandberg and R. Ohayon. Computational aspects of structural acoustics and vibration. Springer Science and Business Media, 2009.[BibTeX]
  7. X. Wang and K. J. Bathe. Displacement/pressure based mixed finite element formulations for acoustic fluid-structure interaction problems. International journal for numerical methods in engineering, 40(11):2001–2017, 1997.[BibTeX]
  8. C. Zhao, J. Chen, and N. Yu. Dynamic response of ap1000 water tank with internal ring baffles under earthquake loads. Energy Procedia, 127():407–415, 2017.[BibTeX]