Topological design of acoustic-structure interaction structures with the mixed finite element method

We use the mixed finite element procedure with the displacement field and the pressure as primal variables to find out optimal topologies of coupled acoustic-structure systems. Changing topologies, it is challenging to alternate between the governing equations (Helmholtz equation and the linear elasticity equation) and impose the coupling boundary conditions. Thus this coupled acoustic-structure systems are usually optimized using the shape optimization scheme with the explicit boundary representation.

In this paper, in order to perform topology optimization for the coupled systems, we adopt the mixed finite element procedure called a u/p-formulation [

1

]. Compared to the standard displacement formulation, this mixed formulation is suitable to analyze the response of imcompressible media. Moreover, in the frequency domain, assigning zero shear modulus and proper values of bulk and structural density, we can derive the Helmholtz equation (Wave equation). Hence, by spatial variation of the mass density, shear and bulk moduli, we are able to simulate the coupled system in the single governing equation. For optimization, we interpolate these material properties based on the SIMP (Solid Isotropic Material with Penalization) [

2

]. Compared to shape optimization, this procedure does not require the shape parameterization. Several two-dimensional acoustic-structure problems are optimized in order to verify the proposed method. The detail numerical behavior during optimization are also studied.