[A4] Multiscale Techniques and Nonlocal Modeling in materials

Effective mathematical modeling and efficient model reduction techniques are needed for the success of modeling for complex physical systems. I am interested in multiscale modeling and homogenization, nonlocal-to-local coupling, and derivations of nonlocal models and nonlocal boundary conditions with applications to solid mechanics and fluids.
  1. Z. Han and X. Tian, Nonlocal half-ball vector operators on bounded domains: Poincaré inequality and its applications, accepted by Math. Models Methods Appl. Sci., 2023.

  2. Q. Du, X. Tian and Z. Zhou, Nonlocal diffusion models with consistent local and fractional limits, A³N²M: Approximation, Applications, and Analysis of Nonlocal, Nonlinear Models: Proceedings of the 50th John H. Barrett Memorial Lectures,175-213, 2023.

  3. M. D’Elia, X. Li, P. Seleson, X. Tian, and Y. Yu, A review of Local-to-Nonlocal coupling methods in nonlocal diffusion and nonlocal mechanics, J. Peridyn. Nonlocal Model., 4, 1–50, 2022.

  4. M. D’Elia, X. Tian, and Y. Yu, A physically-consistent, flexible and efficient strategy to convert local boundary conditions into nonlocal volume constraints, SIAM J. Sci. Comput., 42(4), A1935-A1949, 2020.

  5. Q. Du, B. Engquist and X. Tian, Multiscale Modeling, homogenization and nonlocal effects: mathematical and computational issues , Contemporary Mathematics, 754, 115-140, 75 Years of Mathematics of Computation, AMS, 2020.

  6. Q. Du and X. Tian, Mathematics of Smoothed Particle Hydrodynamics: A Study via Nonlocal Stokes Equations, Foundations of Computational Mathematics, 20, 801-826, 2020.

  7. Y. Tao, X. Tian and Q. Du, Nonlocal models with heterogeneous localization and their application to seamless local-nonlocal coupling, Multiscale Modeling and Simulation, 17(3), 1052-1075, 2019.

  8. Q. Du and X. Tian, Stability of nonlocal Dirichlet integrals and implications for peridynamic correspondence material modeling , SIAM J. Appl. Math., 78(3), 1536-1552 , 2018.

  9. Q. Du, X. H. Li, J. Lu, and X. Tian, A quasinonlocal coupling method for nonlocal and local diffusion models , SIAM J. Numer. Anal., 56 (3), 1386-1404, 2018.

  10. Q. Du, Y. Tao and X. Tian, A peridynamic model of fracture mechanics with bond-breaking , J. Elast.,132 (2), 197-218, 2018.

  11. Y. Tao, X. Tian and Q. Du, Nonlocal diffusion and peridynamic models with Neumann type constraints and their numerical approximations, Appl. Math. Comput., 305:282-298, 2017.