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1. Validating bond-based peridynamic model using displacement potential approach

   Jared Rivera; Yuzhe Cao; Longwen Tang; Mathieu Bauchy (December 2022, Proceedings of the Institution of Mechanical Engineers Part C Journal of mechanical engineering science)

   Although peridynamics is widely used to investigate mechanical responses in materials, the ability of peridynamics to capture the main features of realistic stress states remains unknown. Here, we present a procedure that combines analytic investigation and numerical simulation to capture the elastic field in the mixed boundary condition. By using the displacement potential function, the mixed boundary condition elasticity problem is reduced to a single partial differential equation which can be analytically solved through Fourier analysis. To validate the peridynamic model, we conduct a numerical uniaxial tensile test using peridynamics, which is further compared with the analytic solution through a convergence study. We find that, when the parameters are carefully calibrated, the numerical predicted stress distribution agrees very well with the one obtained from the theoretical calculation. 

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2. Graphical solutions to one-phase free boundary problems

   https://doi.org/10.1515/crelle-2023-0067  

   Engelstein, Max; Fernández-Real, Xavier; Yu, Hui (October 2023, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We study viscosity solutions to the classical one-phase problem and its thin counterpart. In low dimensions, we show that when the free boundary is the graph of a continuous function, the solution is the half-plane solution. This answers, in the salient dimensions, a one-phase free boundary analogue of Bernstein’s problem for minimal surfaces.As an application, we also classify monotone solutions of semilinear equations with a bump-type nonlinearity. 

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3. A model of controlled growth

   https://doi.org/ISSN 2167-5163  

   Bressan, Alberto; Lewicka, Marta (January 2018, Archive for rational mechanics and analysis)

   We consider a free boundary problem for a system of PDEs, modeling the growth of a biological tissue. A morphogen, controlling volume growth, is produced by specific cells and then diffused and absorbed throughout the domain. The geometric shape of the growing tissue is determined by the instantaneous minimization of an elastic deformation energy, subject to a constraint on the volumetric growth. For an initial domain with C^{2,α} boundary, our main result establishes the local existence and uniqueness of a classical solution, up to a rigid motion. 

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4. Loss of Regularity of Solutions of the Lighthill Problem for Shock Diffraction for Potential Flow

   Chen, G-Q; Feldman, M; Hu, J; Xiang, W. (January 2020, SIAM journal on mathematical analysis)

   We are concerned with the suitability of the main models of compressible fluid dynamics for the Lighthill problem for shock diffraction by a convex corned wedge, by studying the regularity of solutions of the problem, which can be formulated as a free boundary problem. In this paper, we prove that there is no regular solution that is subsonic up to the wedge corner for potential flow. This indicates that, if the solution is subsonic at the wedge corner, at least a characteristic discontinuity (vortex sheet or entropy wave) is expected to be generated, which is consistent with the experimental and computational results. Therefore, the potential flow equation is not suitable for the Lighthill problem so that the compressible Euler system must be considered. In order to achieve the nonexistence result, a weak maximum principle for the solution is established, and several other mathematical techniques are developed. The methods and techniques developed here are also useful to the other problems with similar difficulties. 

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5. Density-constrained Chemotaxis and Hele-Shaw flow

   https://doi.org/10.1090/tran/8934  

   Kim, Inwon; Mellet, Antoine; Wu, Yijing (October 2023, Transactions of the American Mathematical Society)

   We consider a model of congestion dynamics with chemotaxis, where the density of cells follows the chemical signal it generates, while observing an incompressibility constraint (incompressible parabolic-elliptic Patlak-Keller-Segel model). We show that when the chemical diffuses slowly and attracts the cells strongly, then the dynamics of the congested cells is well approximated by a surface-tension driven free boundary problem. More precisely, we rigorously establish the convergence of the solution to the characteristic function of a set whose evolution is determined by the classical Hele-Shaw free boundary problem with surface tension. The problem is set in a bounded domain, which leads to an interesting analysis on the limiting boundary conditions. Namely, we prove that the assumption of Robin boundary conditions for the chemical potential leads to a contact angle condition for the free interface (in particular Neumann boundary conditions lead to an orthogonal contact angle condition, while Dirichlet boundary conditions lead to a tangential contact angle condition). 

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