[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, preprint, 2022.

  2. Q. Du, X. Tian and Z. Zhou, Nonlocal diffusion models with consistent local and fractional limits, preprint, 2022.

  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.