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   https://doi.org/10.1137/20M1322571  

   Nordström, Jan; Hagstrom, Thomas M (January 2020, SIAM Journal on Numerical Analysis)
2. Simulating hindered grain boundary diffusion using the smoothed boundary method

   https://doi.org/10.1088/1361-651X/ad4d0b  

   Hanson, Erik; Andrews, W_Beck; Powers, Max; Jenkins, Kaila_G; Thornton, Katsuyo (June 2024, Modelling and Simulation in Materials Science and Engineering)

   Abstract Grain boundaries can greatly affect the transport properties of polycrystalline materials, particularly when the grain size approaches the nanoscale. While grain boundaries often enhance diffusion by providing a fast pathway for chemical transport, some material systems, such as those of solid oxide fuel cells and battery cathode particles, exhibit the opposite behavior, where grain boundaries act to hinder diffusion. To facilitate the study of systems with hindered grain boundary diffusion, we propose a model that utilizes the smoothed boundary method to simulate the dynamic concentration evolution in polycrystalline systems. The model employs domain parameters with diffuse interfaces to describe the grains, thereby enabling solutions with explicit consideration of their complex geometries. The intrinsic error arising from the diffuse interface approach employed in our proposed model is explored by comparing the results against a sharp interface model for a variety of parameter sets. Finally, two case studies are considered to demonstrate potential applications of the model. First, a nanocrystalline yttria-stabilized zirconia solid oxide fuel cell system is investigated, and the effective diffusivities are extracted from the simulation results and are compared to the values obtained through mean-field approximations. Second, the concentration evolution during lithiation of a polycrystalline battery cathode particle is simulated to demonstrate the method’s capability. 

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3. Dataset: Simulating Hindered Grain Boundary Diffusion Using the Smoothed Boundary Method

   https://doi.org/10.13011/m3-0mxz-xd55  

   Hanson, Erik; Andrews, W Beck; Powers, Max; Jenkins, Kaila G; Thornton, Katsuyo (January 2024, Materials Commons)

   Data associated with the article "Simulating Hindered Grain Boundary Diffusion Using the Smoothed Boundary Method" [1]. 1. Hanson, E., Andrews, W. B., Powers, M., Jenkins, K. G., & Thornton, K. (2024). Simulating hindered grain boundary diffusion using the smoothed boundary method. Modelling and Simulation in Materials Science and Engineering, 32(5), 055027. 

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4. The no boundary density matrix

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   Ivo, Victor; Li, Yue-Zhou; Maldacena, Juan (February 2025, Journal of High Energy Physics)

   A<sc>bstract</sc> We discuss a no-boundary proposal for a subregion of the universe. In the classical approximation, this density matrix involves finding a specific classical solution of the equations of motion with no boundary. Beyond the usual no boundary condition at early times, we also have another no boundary condition in the region we trace out. We can find the prescription by starting from the usual Hartle-Hawking proposal for the wavefunction on a full slice and tracing out the unobserved region in the classical approximation. We discuss some specific subregions and compute the corresponding solutions. These geometries lead to phenomenologically unacceptable probabilities, as expected. We also discuss how the usual Coleman de Luccia bubble solutions can be interpreted as a possible no boundary contribution to the density matrix of the universe. These geometries lead to local (but not global) maxima of the probability that are phenomenologically acceptable. 

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5. The linearization of the boundary rigidity problem for MP-systems and generic local boundary rigidity

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   Muñoz-Thon, Sebastián (November 2024, Inverse Problems)

   Abstract We consider an $\mathrm{MP}$-system, that is, a compact Riemannian manifold with boundary, endowed with a magnetic field and a potential. On simple $\mathrm{MP}$-systems, we study the $\mathrm{MP}$-ray transform in order to obtain new boundary rigidity results for $\mathrm{MP}$-systems. We show that there is an explicit relation between the $\mathrm{MP}$-ray transform and the magnetic one, which allows us to apply results from Dairbekovet al(2007Adv. Math.216535–609) to our case. Regarding rigidity, we show that there exists a generic set $G^m$of simple $\mathrm{MP}$-systems, which is open and dense, such that any two $\mathrm{MP}$-systems close to an element in it and having the same boundary action function, must bek-gauge equivalent. 

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