Stability of mixed FEMs for non-selfadjoint indefinite second-order linear elliptic PDEs

For a well-posed non-selfadjoint indefinite second-order linear elliptic PDE with general coefficients A, b, γ in L∞ and symmetric and uniformly positive definite coefficient matrix A, this paper proves that mixed finite element problems are uniquely solvable and the discrete solutions are uniformly bounded, whenever the underlying shape-regular triangulation is sufficiently fine. This applies to the Raviart-Thomas and Brezzi-Douglas-Marini finite element families of any order and in any space dimension and leads to the best-approximation estimate in H (div) × L2 as well as in in L2 × L2 up to oscillations. This generalises earlier contributions for piecewise Lipschitz continuous coefficients to L∞ coefficients. The compactness argument of Schatz and Wang for the displacement-oriented problem does not apply immediately to the mixed formulation in H (div) × L2 . But it allows the uniform approximation of some L2 contributions and can be combined with a recent L2 best-approximation result from the medius analysis. This technique circumvents any regularity assumption and the application of a Fortin interpolation operator.Peer Reviewe

Knowing loved ones' end-of-life health care wishes: Attachment security predicts caregivers' accuracy.

Objective: At times, caregivers make life-and-death decisions for loved ones. Yet very little is known about the factors that make caregivers more or less accurate as surrogate decision makers for their loved ones. Previous research suggests that in low stress situations, individuals with high attachment-related anxiety are attentive to their relationship partners' wishes and concerns, but get overwhelmed by stressful situations. Individuals with high attachment-related avoidance are likely to avoid intimacy and stressful situations altogether. We hypothesized that both of these insecure attachment patterns limit surrogates' ability to process distressing information and should therefore be associated with lower accuracy in the stressful task of predicting their loved ones' end-of-life health care wishes. Method: Older patients visiting a medical clinic stated their preferences toward end-of-life health care in different health contexts, and surrogate decision makers independently predicted those preferences. For comparison purposes, surrogates also predicted patients' perceptions of everyday living conditions so that surrogates' accuracy of their loved ones' perceptions in nonstressful situations could be assessed. Results: Surrogates high on either type of insecure attachment dimension were less accurate in predicting their loved ones' end-of-life health care wishes. It is interesting to note that even though surrogates' attachment-related anxiety was associated with lower accuracy of end-of-life health care wishes of their loved ones, it was associated with higher accuracy in the nonstressful task of predicting their loved ones' everyday living conditions. Conclusions: Attachment orientation plays an important role in accuracy about loved ones' end-of-life health care wishes. Interventions may target emotion regulation strategies associated with insecure attachment orientations

Revisiting element removal for density-based structural topology optimization with reintroduction by Heaviside projection

We present a strategy grounded in the element removal idea of Bruns and Tortorelli [1] and aimed at reducing computational cost and circumventing potential numerical instabilities of density-based topology optimization. The design variables and the relative densities are both represented on a fixed, uniform finite element grid, and linked through filtering and Heaviside projection. The regions in the analysis domain where the relative density is below a specified threshold are removed from the forward analysis and replaced by fictitious nodal boundary conditions. This brings a progressive cut of the computational cost as the optimization proceeds and helps to mitigate numerical instabilities associated with low-density regions. Removed regions can be readily reintroduced since all the design variables remain active and are modeled in the formal sensitivity analysis. A key feature of the proposed approach is that the Heaviside functions promote material reintroduction along the structural boundaries by amplifying the magnitude of the sensitivities inside the filter reach. Several 2D and 3D structural topology optimization examples are presented, including linear and nonlinear compliance minimization, the design of a force inverter, and frequency and buckling load maximization. The approach is shown to be effective at producing optimized designs equivalent or nearly equivalent to those obtained without the element removal, while providing remarkable computational savings

Improved Two-Phase Projection Topology Optimization

Abstract Projection-based algorithms for continuum topology optimization have received considerable attention in recent years due to their ability to control minimum length scale in a computationally efficient manner. This not only provides a means for imposing manufacturing length scale constraints, but also circumvents numerical instabilities of solution mesh dependence and checkerboard patterns. Standard radial projection, however, imposes length scale on only a single material phase, potentially allowing small-scale features in the second phase to develop. This may lead to sharp corners and/or very small holes when the solid (load-carrying) phase is projected, or one-node hinge chains when only the void phase is projected. Two-phase length scale control is therefore needed to prevent these potential design issues. Ideally, the designer would be able to impose different minimum length scales on both the structural (load-carrying) and void phases as required by the manufacturing process and/or application specifications. A previously proposed algorithm towards this goal required a design variable associated with each phase to be located at every design variable location, thereby doubling the number of design variables over standard topology optimization [2]. This work proposes a two-phase projection algorithm that remedies this shortcoming. Every design variable has the capability to project either the solid or the void phase, but nonlinear, design dependent weighting functions are created to prevent both phases from being projected. The functions are constructed intentionally to resemble level set methods, where the sign of the design variable dictates the feature to be projected. Despite this resemblance to level sets, the algorithm follows the material distribution approach with sensitivities computed via the adjoint method and MMA used as the gradient-based optimizer. The algorithm is demonstrated on benchmark minimum compliance and compliant inverter problems, and is shown to satisfy length scale constraints imposed on both phases. 2. Keywords: Topology Optimization, Projection Methods, Manufacturing Constraints, Length Scale, Heaviside Projection. Introduction Topology optimization is a design tool used for determining optimal distributions of material within a domain. System connectivity and feature shapes are optimized and thus, as the initial guess need not be informed, topology optimization is capable of generating new and unanticipated designs. It is well-known, however, that this may result in impractical solutions that are difficult to fabricate or construct, such as ultra slender structural features or small scale pore spaces. A key focus of this work is to improve manufacturability of topology-optimized designs by controlling the length scale of the topological features. The length scale is generally defined as the minimum radius or diameter of the material phase of concern. It is thus a physically meaningful quantity that can be selected by the designer based on fabrication process. The fabrication process also dictates the phase (or phases) on which the restriction is applied. For example, for topologies constructed by deposition processes, it is relevant to consider constraining the minimum length scale of the solid phase. Similarly, for designs that are manufactured by removing material, for example by milling, the manufacturability constraints should include minimum length scale and maximum curvature of the voids as dictated by the machine. Moreover, it is well established that controlling the length scale has the additional advantage that it circumvents numerical instabilities, such as checkerboard patterns and mesh dependency. Several methods for controlling the length scale of a topology optimization design exist ([1], [3]). Herein, the Heaviside Projection Method (HPM) [1] is used. HPM is capable of yielding 0-1 designs in which the minimum length scale is achieved naturally, without additional constraints. In HPM, the design variables are associated with a material phase and projected onto the finite element space by a Heaviside function. This mathematical operation is independent of the problem formulation and the governing physics. The projection is typically done radially and the projection radius is chosen as the prescribed minimum length scale. In its original form [1], the method projects a single phase onto the elements and

Sexuality and Affection among Elderly German Men and Women in Long-Term Relationships: Results of a Prospective Population-Based Study

This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.The study was funded by the German Federal Ministry for Families, Senior Citizens, Women and Youth (AZ 314-1722-102/16; AZ 301-1720-295/2), the Ministry for Science, Research and Art Baden-Württemberg, and the University of Rostock (FORUN 989020; 889048)