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Consider the Helmholtz equation ∇ • α∇p + k 2 αp = 0 in a domain that contains a so-called hard scatterer. The scatterer is represented by the value α = , for 0 < 1, whereas α = 1 whenever the scatterer is absent. This scatterer model is often used for the purpose of design optimization and constitutes a fictitious domain approximation of a body characterized by homogeneous Neumann conditions on its boundary. However, such an approximation results in spurious resonances inside the scatterer at certain frequencies and causes, after discretization, ill-conditioned system matrices. Here, we present a stabilization strategy that removes these resonances. Furthermore, we prove that, in the limit → 0, the stabilized problem provides linearly convergent approximations of the solution to the problem with an exactly modeled scatterer. Numerical experiments indicate that a finite element approximation of the stabilized problem is free from internal resonances, and they also suggest that the convergence rate is indeed linear with respect to .

Abstract. Consider the solution of a boundary-value problem for steady linear elasticity in which the computational domain contains one or several holes with traction-free boundaries. The presence of holes in the material can be approximated using a weak material; that is, the relative density of material ρ is set to 0 < = ρ 1 in the hole region. The weak material approach is a standard technique in the so-called material distribution approach to topology optimization, in which the inhomogeneous relative density of material is designated as the design variable in order to optimize the spatial distribution of material. The use of a weak material ensures that the elasticity problem is uniquely solvable for each admissible value ρ ∈ [ , 1] of the design variable. A finiteelement approximation of the boundary-value problem in which the weak material approximation is used in the hole regions can be viewed as a nonconforming but convergent approximation of a version of the original problem in which the solution is continuously and elastically extended into the holes. The error in this approximation can be bounded by two terms that depend on . One term scales linearly with with a constant that is independent of the mesh size parameter h but that depends on the surface traction required to fit elastic material in the deformed holes. The other term scales like 1/2 times the finite-element approximation error inside the hole. The condition number of the weak material stiffness matrix scales like −1 , but the use of a suitable left preconditioner yields a matrix with a condition number that is bounded independently of . Moreover, the preconditioned matrix admits the limit value → 0, and the solution of corresponding system of equations yields in the limit a finite-element approximation of the continuously and elastically extended problem. 1. Introduction. Numerical design optimization has evolved into an increasingly useful tool that complements traditional methods in the engineering design of mechanical components. Such optimization can be carried out at various levels of generality. The most general case admits topological properties, such as the number of holes in the configuration, to vary during the optimization procedure; the term topology optimization is often used in order to highlight the generality. Perhaps the most common way of carrying out numerical topology optimization of linear elastic continua is through the material distribution method. In this method, the presence or absence of material is represented by an inhomogeneous relative density appearing in the coefficients of the elasticity equations, discretized on a fixed, typically uniform, mesh. This method has developed into something of a success story and is increasingly used in the design of advanced mechanical components, particularly in the automotive and aeronautical industries. For instance, three applications of topology optimization carried out on the Airbus A380 aircraft is estimated to have contributed to weight savings in the order of 1000 kg ...

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In the absence of wave propagation, transient electromagnetic fields are governed by a composite scalar/vector potential formulation for the quasistatic Darwin field model. Darwintype field models are capable of capturing inductive, resistive, and capacitive effects. To avoid possibly non-symmetric and ill-conditioned fully discrete monolithic formulations, here, a Darwin field model is presented which results in a two-step algorithm, where the discrete representations of the electric scalar potential and the magnetic vector potential are computed consecutively. Numerical simulations show the validity of the presented approach.

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