For the purpose of numerically simulating continuum mechanical structures, different types of material may be represented by the extreme values $\{\epsilon,1\}$, where $0<\epsilon\ll 1$, of a varying coefficient $\alpha$ in the governing equations. The paramter $\epsilon$ is not allowed to vanish in order for the equations to be solvable, which means that the exact conditions are approximated. For example, for linear elasticity problems, presence of material is represented by the value $\alpha=1$, while $\alpha=\epsilon$ provides an approximation of void, meaning that material-free regions are approximated with a weak material. For acoustics applications, the value $\alpha=1$ corresponds to air and $\alpha=\epsilon$ to an approximation of sound-hard material using a dense fluid. Here we analyze the convergence properties of such material approximations as $\epsilon\rightarrow 0$, and we employ this type of approximations to perform design optimization.

In Paper I, we carry out boundary shape optimization of an acoustic horn. We suggest a shape parameterization based on a local, discrete curvature combined with a fixed mesh that does not conform to the generated shapes. The values of the coefficient $\alpha$, which enters in the governing equation, are obtained by projecting the generated shapes onto the underlying computational mesh. The optimized horns are smooth and exhibit good transmission properties. Due to the choice of parameterization, the smoothness of the designs is achieved without imposing severe restrictions on the design variables.

In Paper II, we analyze the convergence properties of a linear elasticity problem in which void is approximated by a weak material. We show that the error introduced by the weak material approximation, after a finite element discretization, is bounded by terms that scale as $\epsilon$ and $\epsilon^{1/2}h^s$, where $h$ is the mesh size and $s$ depends on the order of the finite element basis functions. In addition, we show that the condition number of the system matrix scales inversely proportional to $\epsilon$, and we also construct a left preconditioner that yields a system matrix with a condition number independent of $\epsilon$.

In Paper III, we observe that the standard sound-hard material approximation with $\alpha=\epsilon$ gives rise to ill-conditioned system matrices at certain wavenumbers due to resonances within the approximated sound-hard material. To cure this defect, we propose a stabilization scheme that makes the condition number of the system matrix independent of the wavenumber. In addition, we demonstrate that the stabilized formulation performs well in the context of design optimization of an acoustic waveguide transmission device.

In Paper IV, we analyze the convergence properties of a wave propagation problem in which sound-hard material is approximated by a dense fluid. To avoid the occurrence of internal resonances, we generalize the stabilization scheme presented in Paper III. We show that the error between the solution obtained using the stabilized sound-hard material approximation and the solution to the problem with exactly modeled sound-hard material is bounded proportionally to $\epsilon$.