Topology optimization of acoustic–structure interaction problems using a mixed finite element formulation

Abstract

SUMMARYThe paper presents a gradient based topology optimization formulation that allows to solve acousticstructure (vibro-acoustic) interaction problems without explicit boundary interface representation. In acoustic-structure interaction problems, the pressure and displacement fields are governed by Helmholtz equation and the elasticity equation, respectively. Normally, the two separate fields are coupled by surface-coupling integrals, however, such a formulation does not allow for free material redistribution in connection with topology optimization schemes since the boundaries are not explicitly given during the optimization process. In this paper we circumvent the explicit boundary representation by using a mixed finite element formulation with displacements and pressure as primary variables (a u/p-formulation). The Helmholtz equation is obtained as a special case of the mixed formulation for the elastic shear modulus equating zero. Hence, by spatial variation of the mass density, shear and bulk moduli we are able to solve the coupled problem by the mixed formulation. Using this modeling approach, the topology optimization procedure is simply implemented as a standard density approach.Several two-dimensional acoustic-structure problems are optimized in order to verify the proposed method.

Element stacking method for topology optimization with material-dependent boundary and loading conditions

“…Except for element stacking, no special numerical treatment was conducted to obtain the result. Though not explicitly shown, the result agrees well with results obtained earlier by other methods [Hammer and Olhoff 2000;Sigmund and Clausen 2007;Yoon et al 2006].…”

“…The design-dependent pressure-loaded problem is an interesting problem that has received some attention in the topology optimization community [Hammer and Olhoff 2000;Sigmund and Clausen 2007;Yoon et al 2006]. Figure 10 (top) depicts the problem under consideration.…”

“…The objective is to find an optimal structural layout in the design domain under pressure load. This can be solved as a material-selection problem [Sigmund and Clausen 2007;Yoon et al 2006], but it is considered to demonstrate that the element stacking method is capable of handling different material element types. Thus, two different elements, one a mixed finite element capable of simulating incompressible fluid and the other a structural element, will be juxtaposed on the same pixel.…”

“…Thus, two different elements, one a mixed finite element capable of simulating incompressible fluid and the other a structural element, will be juxtaposed on the same pixel. A mixed displacement-pressure element to simulate incompressible fluid 4 is the element used in Sigmund andClausen 2007 andYoon et al 2006, which uses bilinear displacements field and constant pressure field. The shape functions in the natural coordinates are illustrated in Figure 11 (left).…”

“…For instance, one can consider design-dependent pressure load problems. These problems were solved by several approaches [Hammer and Olhoff 2000;Sigmund and Clausen 2007;Yoon et al 2006], but a straightforward and easy-to-implement method may be useful.…”