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Structured regularization allows machine learning models to consider spatial relationships among parameters, leading to results that generalize better and are more interpretable compared to norm penalties. In this study, we evaluated a novel structured regularization method that incorporates approximate morphology operators defined using harmonic mean-based fW-filters. We extended this method to multiclass classification and conducted experiments aimed at classifying magnetic resonance images (MRI) of subjects into four stages of Alzheimer's disease progression. The experimental results demonstrate that the novel structured regularization method not only performs better than standard sparse and structured regularization methods in terms of prediction accuracy (ACC), F1 scores, and the area under the receiver operating characteristic curve (AUC), but also produces interpretable coefficient maps.

We propose a novel way to incorporate morphology operators through structured regularization of machine learning models. Specifically, we introduce a feature map in the models that performs structured variable selection. The feature map is automatically processed by approximate morphology operators and is learned together with the model coefficients. Experiments were conducted with linear regression on both synthetic data, demonstrating that the proposed methods are effective in selecting groups of parameters with much less noise than baseline models, and on three-dimensional T1-weighted brain magnetic resonance images (MRI) for age prediction, demonstrating that the proposed methods enforce sparsity and select homogeneous regions of non-zero and relevant regression coefficients. The proposed methods improve interpretability in pattern analysis. The minimum size of features in the structured variable selection can be controlled by adjusting the structuring element in the approximate morphology operator, tailored to the specific study of interest. With these added benefits, the proposed methods still perform on par with commonly used variable selection and structured variable selection methods in terms of the coefficient of determination and the Pearson correlation coefficient.

Accurate simulations of sound propagation in narrow geometries need to account for viscous and thermal losses. In this respect, effective boundary conditions that model viscothermal losses in frequency-domain acoustics have recently gained in popularity. Here, we investigate the time-domain analogue of one such boundary condition. We find that the thermal part of the boundary condition is passive in time domain as expected, while the viscous part is not. More precisely, we demonstrate that the viscous part is responsible for exponentially growing normal modes with unbounded temporal growth rates, which indicates ill-posedness of the considered model. A finite-difference-time-domain scheme is developed for simulations of lossy sound propagation in a duct. If viscous losses are neglected the obtained transmission characteristics are found to be in excellent agreement with frequency-domain simulations. In the general case, the simulations experience an instability much in line with the theoretical findings.

This article proposes a boundary strip indicator for density-based topology optimization that can be used to estimate the design’s surface area (perimeter in 2D) or identify a coating layer. We investigate the theoretical properties of the proposed boundary strip indicator and propose a differentiable approximation that preserves key properties, such as non-negativity. Finally, we use the boundary strip indicator in a heat conduction design optimization problem for a coated structure. The resulting designs show a strong dependence on the properties of the coating.

The waveguide acoustic black hole (WAB) effect is a promising approach for controlling wave propagation in various applications, especially for attenuating sound waves. While the wave-focusing effect of structural acoustic black holes has found widespread applications, the classical ribbed design of waveguide acoustic black holes (WABs) acts more as a resonance absorber than a true wave-focusing device. In this study, we employ a computational design optimization approach to achieve a conceptual design of a WAB with enhanced wave-focusing properties. We investigate the influence of viscothermal boundary losses on the optimization process by formulating two distinct cases: one neglecting viscothermal losses and the other incorporating these losses using a recently developed material distribution topology optimization technique. We compare the performance of optimized designs in these two cases with that of the classical ribbed design. Simulations using linearized compressible Navier–Stokes equations are conducted to evaluate the wave-focusing performance of these different designs. The results reveal that considering viscothermal losses in the design optimization process leads to superior wave-focusing capabilities, highlighting the significance of incorporating these losses in the design approach. This study contributes to the advancement of WAB design and opens up new possibilities for its applications in various fields.

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.

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.

Part I. Topology optimization is the most general form of design optimization in which the optimal layout of material within a given region of space is to be determined. Filters are essential components of many successful density based topology optimization approaches. The generalized fW-mean filter framework developed in this thesis provides a unified platform for construction, analysis, and implementation of filters. Extending existing algorithms, we demonstrate that under special albeit relevant conditions, the computational complexity of evaluating generalized fW-mean filters and their derivatives is linear in the number of design degrees of freedom. We prove that generalized fW-mean filters guarantee existence of solutions to the penalized minimum compliance problem, the archetypical problem in density based topology optimization. In this problem, the layout of linearly elastic material that minimizes the compliance given static supports and loads is to be determined. We formalize the connection between mathematical morphology and the notion of minimum length scale of a layout of material and thereby provide a theoretical foundation for imposing and assessing minimum length scales in density based topology optimization. Elaborating on the fact that some sequences of generalized fW-mean filters provide differentiable approximations of morphological operators, we devise a method capable of imposing different minimum length scales on the two material phases in minimum compliance problems.

Part II. The notion of Friedrichs systems, also known as symmetric positive systems, encompasses many linear models of physical phenomena. The prototype model is Maxwell's equations, which describe the evolution of the electromagnetic field in the presence of electrical charges and currents. In this thesis, we develop well-posed variational formulations of boundary and initial–boundary value problems of Friedrichs systems on bounded domains. In particular, we consider an inhomogeneous initial–boundary value problem that models lossless propagation of acoustic disturbances in a stagnant fluid. Galbrun's equation is a linear second order vector differential equation in the so-called Lagrangian displacement, which was derived to model lossless propagation of acoustic disturbances in the presence of a background flow. Our analysis of Galbrun's equation is centered on the observation that solutions to Galbrun's equation may be formally constructed from solutions to linearized Euler's equations. More precisely, the Lagrangian displacement is constructed as the solution to a transport-type equation driven by the Eulerian velocity perturbation. We present partial results on the well-posedness of Galbrun's equation in the particular case that the background flow is everywhere tangential to the domain boundary by demonstrating mild well-posedness of an initial–boundary value problem for linearized Euler's equations and that our construction of the Lagrangian displacement is well-defined. Moreover, we demonstrate that sufficiently regular solutions to Galbrun's equation satisfy an energy estimate.

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).