### [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.
- Z. Han and X. Tian,
* Nonlocal half-ball vector operators on bounded domains: Poincaré inequality and its applications*,
preprint, 2022.

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

- 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.

- 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.

- 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.

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

- 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.

- 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.

- 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.

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

- 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.