The Equivalence of Standard and Mixed Finite Element Methods in Applications to Elasto-Acoustic Interaction

Abstract

Two commonly used problem formulations for the description of acoustic wave propagation are investigated; one is based on fluid displacement, and the other one is based on the velocity potential as the primary variable. Their equivalence under general Neumann boundary conditions is shown on both the continuous and the discrete level. To obtain the equivalence in the discrete setting, a nonstandard mixed finite element formulation is introduced. Thus, the transfer of an already available analysis for coupled elasto-acoustic problems in the displacement formulation to the potential formulation can be achieved. For the applications, the potential formulation is of special interest because it allows the use of standard Lagrangian elements in both the stucture and the fluid subdomain. Moreover, the approach does not require any conformity of the subdomain meshes at the interface, which considerably simplifies the physics-adapted mesh generation. Several engineering examples demonstrate the applicability and efficiency of the resulting numerical scheme.

Multiphysics simulations: challenges and opportunities.

“…Then semi-smooth Newton methods provide a superlinear solver, and an efficient algorithmic implementation can be based on the choice of biorthogonal basis functions, resulting in a diagonal mass matrix. Figure 13 illustrates the acoustic-structure interaction for a 3D loudspeaker geometry; see (Flemisch et al, 2010). The problem has been formulated in a weakly consistent mortar setting, and highly nonmatching meshes are applied for the numerical simulation.…”

Evolution of a semidiscrete system modeling the scattering of acoustic waves by a piezoelectric solid

“…This treatment of the acoustics leads to a boundary element formulation for those unknowns and we will use finite elements for the semidiscretization of the piezoelectric solid similar to [5], where the formulation considered is for a purely elastic solid. While we will not use mixed methods, we would like to remark that it has been shown by [11] that their use for the treatment of the solid results in an equivalent problem.…”

Multiphysics simulations

“…Then semi-smooth Newton methods provide a superlinear solver, and an efficient algorithmic implementation can be based on the choice of biorthogonal basis functions, resulting in a diagonal mass matrix. Figure 13 illustrates the acoustic-structure interaction for a three-dimensional loudspeaker geometry; see Flemisch et al (2010). The problem has been formulated in a weakly consistent mortar setting, and highly non-matching meshes are applied for the numerical simulation.…”

“…Figure 13 illustrates the acoustic-structure interaction for a three-dimensional loudspeaker geometry; see Flemisch et al (2010). The problem has been formulated in a weakly consistent mortar setting, and highly non-matching meshes are applied for the numerical simulation.…”