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In material distribution topology optimization, restriction methods are routinely applied to obtain well-posed optimization problems and to achieve mesh-independence of the resulting designs. One of the most popular restriction methods is to use a filtering procedure. In this paper, we present a framework where the filtering process is viewed as a quasi-arithmetic mean (or generalized f-mean) over a neighborhood with the possible addition of an extra "projection step". This framework includes the vast majority of available filters for topology optimization. The covered filtering procedures comprise three steps: (i) element-wise application of a function, (ii) computation of local averages, and (iii) element-wise application of another function. We present fast algorithms that apply this type of filters over polytope-shaped neighborhoods on regular meshes in two and three spatial dimensions. These algorithms have a computational cost that grows linearly with the number of elements and can be bounded irrespective of the filter radius.

Material distribution topology optimization problems are generally ill-posed if no restriction or regularization method is used. To deal with these issues, filtering procedures are routinely applied. In a recent paper, we presented a framework that encompasses the vast majority of currently available density filters. In this paper, we show that these nonlinear filters ensure existence of solutions to a continuous version of the minimum compliance problem. In addition, we provide a detailed description on how to efficiently compute sensitivities for the case when multiple of these nonlinear filters are applied in sequence. Finally, we present large-scale numerical experiments illustrating some characteristics of these cascaded nonlinear filters.

The archetypical topology optimization problem concerns designing the layout of material within a given region of space so that some performance measure is extremized. To improve manufacturability and reduce manufacturing costs, restrictions on the possible layouts may be imposed. Among such restrictions, constraining the minimum length scales of different regions of the design has a significant place. Within the density filter based topology optimization framework the most commonly used definition is that a region has a minimum length scale not less than D if any point within that region lies within a sphere with diameter D > 0 that is completely contained in the region. In this paper, we propose a variant of this minimum length scale definition for subsets of a convex (possibly bounded) domain We show that sets with positive minimum length scale are characterized as being morphologically open. As a corollary, we find that sets where both the interior and the exterior have positive minimum length scales are characterized as being simultaneously morphologically open and (essentially) morphologically closed. For binary designs in the discretized setting, the latter translates to that the opening of the design should equal the closing of the design. To demonstrate the capability of the developed theory, we devise a method that heuristically promotes designs that are binary and have positive minimum length scales (possibly measured in different norms) on both phases for minimum compliance problems. The obtained designs are almost binary and possess minimum length scales on both phases.

This paper presents a density-based topology optimization approach to design compact wideband coaxial-to-waveguide transitions. The underlying optimization problem shows a strong self penalization towards binary solutions, which entails mesh-dependent designs that generally exhibit poor performance. To address the self penalization issue, we develop a filtering approach that consists of two phases. The first phase aims to relax the self penalization by using a sequence of linear filters. The second phase relies on nonlinear filters and aims to obtain binary solutions and to impose minimum-size control on the final design. We present results for optimizing compact transitions between a 50-Ohm coaxial cable and a standard WR90 waveguide operating in the X-band (8-12 GHz).

All finite element methods, as well as much of the Hilbert-space theory for partial differential equations, rely on variational formulations, that is, problems of the type: find u∈V such that a(v,u)=l(v) for each v∈L, where V,L are Sobolev spaces. However, for systems of Friedrichs type, there is a sharp disparity between established well-posedness theories, which are not variational, and the very successful discontinuous Galerkin methods that have been developed for such systems, which are variational. In an attempt to override this dichotomy, we present, through three specific examples of increasing complexity, well-posed variational formulations of boundary and initial–boundary-value problems of Friedrichs type. The variational forms we introduce are generalizations of those used for discontinuous Galerkin methods, in the sense that inhomogeneous boundary and initial conditions are enforced weakly through integrals in the variational forms. In the variational forms we introduce, the solution space is defined as a subspace V of the graph space associated with the differential operator in question, whereas the test function space L is a tuple of L2 spaces that separately enforce the equation, boundary conditions of characteristic type, and initial conditions.

Galbrun's equation, which is a second order partial differential equation describing the evolution of a so-called Lagrangian displacement vector field, can be used to study acoustics in background flows as well as perturbations of astrophysical flows. Our starting point for deriving Galbrun's equation is linearized Euler's equations, which is a first order system of partial differential equations that describe the evolution of the so-called Eulerian flow perturbations. Given a solution to linearized Euler's equations, we introduce the Lagrangian displacement as the solution to a linear first order partial differential equation, where the Eulerian perturbation of the fluid velocity acts as a source term. Our Lagrangian displacement solves Galbrun's equation, provided it is regular enough and that the so-called no-resonance assumption holds. In the case that the background flow is steady and tangential to the domain boundary, we prove existence, uniqueness, and continuous dependence on data of solutions to an initial–boundary-value problem for linearized Euler's equations. For such background flows, we demonstrate that the Lagrangian displacement is well-defined, that the initial datum of the Lagrangian displacement can be chosen in order to fulfill the no-resonance assumption, and derive a classical energy estimate for (sufficiently regular solutions to) Galbrun's equation. Due to the presence of zeroth order terms of indefinite signs in the equations, the energy estimate allows solutions that grow exponentially with time.

L'équation de Galbrun, est une équation aux dérivées partielles du second ordre qui décrit l'évolution d'un champ de vecteurs déplacements, dit Lagrangien. Elle peut être utilisée pour étudier l'acoustique des écoulements à grande échelle, ainsi que les perturbations des écoulements astrophysiques. Notre point de départ, pour dériver l'équation de Galbrun, est l'équation d'Euler linéarisée, qui est un système d'équations aux dérivées partielles du premier ordre décrivant l'évolution des perturbations de l'écoulement. Une solution des équations d'Euler linéarisées étant donnée, nous introduisons le déplacement Lagrangien comme étant la solution d'une équation aux dérivées partielles linéaire du premier ordre, dont le second membre est la perturbation Eulérienne de la vitesse du fluide. Ce déplacement Lagrangien est solution de l'équation de Galbrun, à condition qu'il soit suffisamment régulier et que l'hypothèse dite de non-résonance soit satisfaite. Dans le cas où l'écoulement à grande échelle est stationnaire et tangent à la frontière du domaine, nous démontrons des résultats d'existence, d'unicité et de dépendance continue par rapport aux données, pour les solutions d'un problème aux limites avec condition initiale, pour les équations d'Euler linéarisées. Pour de tels écoulements, nous démontrons que le déplacement Lagrangien est bien défini, que la donnée initiale du déplacement Lagrangien peut être choisie afin de satisfaire à l'hypothèse de non-résonance et nous dérivons une estimation classique de l'énergie pour les solutions suffisamment régulières de l'équation de Galbrun. En raison de la présence de termes d'ordre zéro de signes indéfinis dans les équations, l'estimation de l'énergie autorise des solutions qui croissent exponentiellement avec le temps.