Some nonlocal optimal design problems Fernández Bonder, Julián; Spedaletti, Juan F.
Journal of mathematical analysis and applications,
03/2018, Volume:
459, Issue:
2
Journal Article
Peer reviewed
Open access
In this paper we study two optimal design problems associated to fractional Sobolev spaces Ws,p(Ω). Then we find a relationship between these two problems and finally we investigate the convergence ...when s↑1.
The phase-field approach regularizes the variational theory of fracture by approximating cracks with a smeared damage field. In this work, the attention is focused on those formulations approximating ...mode II fractures (shear fractures). In these models, only the deviatoric part of the strain elastic energy, penalized by the phase-field, drives the crack onset and evolution, whereas the elastic hydrostatic energetic contribution has no influence on the failure process. Consequently, cracks evolves according to the von Mises–Hencky–Hüber, also known as J2, failure criterion. Unfortunately, volumetric locking problem arises in the damaged zones if classical numerical solution strategies are adopted. As a consequence, damage localization bands appear with an excessive thickness, thus overestimating the fracture energy. In addition, the crack path geometry may be erroneously described because of the loss of precision of the displacement field in damaged zones. To circumvent these drawbacks, two numerical techniques are proposed, namely selective reduced integration and mixed displacement/pressure formulation, and their effectiveness evidenced by a numerical investigation.
This paper is on Γ-convergence for degenerate integral functionals related to homogenisation problems in the Heisenberg group. Here both the rescaling and the notion of invariance or periodicity are ...chosen in a way motivated by the geometry of the Heisenberg group. Without using special geometric features, these functionals would be neither coercive nor periodic, so classic results do not apply. All the results apply to the more general case of Carnot groups.
Noting the paucity of studies of convergence in energy consumption across the US states, and the usefulness of a study that shares the spirit of the enormous research on convergence in energy-related ...variables in cross-country contexts, this paper explores convergence in per-capita energy consumption across the US states over the 44-year period 1970–2013. Several well-known parametric and non-parametric approaches are explored partly to shed light on the substantive question and partly to provide a comparative methodological perspective on these approaches. Several statements summarize the outcome of our explorations. First, the widely-used Barro-type regressions do not indicate beta-convergence during the entire period or any of several sub-periods. Second, lack of sigma-convergence is also noted in terms of standard deviation of logarithms and coefficient of variation which do not show a decline between 1970 and 2013, but show slight upward trends. Third, kernel density function plots indicate some flattening of the distribution which is consistent with the results from sigma-convergence scenario. Fourth, intra-distribution mobility (“gamma convergence”) in terms of an index of rank concordance suggests a slow decline in the index. Fifth, the general impression from several types of panel and time-series unit-root tests is that of non-stationarity of the series and thus the lack of stochastic convergence during the period. Sixth, therefore, the overall impression seems to be that of the lack of convergence across states in per-capita energy consumption. The present interstate inequality in per-capita energy consumption may, therefore, reflect variations in structural factors and might not be expected to diminish.
Convergence in electricity intensity is analyzed among a sample of IEA countries. Sigma-convergence (the narrowing of the distribution) and to a lesser degree gamma-convergence (movement within the ...distribution) are detected. However, electricity intensity convergence is less dramatic than energy intensity convergence. Convergence within the end-use sectors is more diverse: in terms of the rates, timing, extent, and ultimate modal structure of the distributions. Commercial electricity intensity has more recently converged toward a bell-shape distribution. By contrast, industry electricity intensity is largely converging toward two distinct groups of countries: one with relatively high electricity intensity and another one with relatively low electricity intensity. Different still is related residential electricity consumption per capita where a small group of countries has stopped growing; another group has slowed considerably, while a third group experienced rapid growth.
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