[A3] Meshfree Methods and their Applications
Meshfree and particle methods offer many advantages over traditional mesh-based methods, particularly for problems with complex or moving geometries, large deformations of materials, or other singular behaviors of solutions.
I am interested in developing theoretical analysis for meshfree and particle methods, which could further enhance their practical performance and functionality.
- Q. Ye and X. Tian,
Monotone meshfree methods for linear elliptic equations in non-divergence form via nonlocal relaxation,
preprint, 2022. [Interactive Website Examples]
- Y. Fan, H. You, X. Tian, X. Yang, X.H. Li, N. Prakash and Y. Yu,
A meshfree peridynamic model for brittle fracture in randomly heterogeneous materials,
Comput. Methods Appl. Mech. Eng., 399, 115340, 2022.
- Y. Fan, X. Tian, X. Yang, X.H. Li, C. Webster and Y. Yu,
An asymptotically compatible probabilistic collocation method for randomly heterogeneous nonlocal problems,
J. Comput. Phys., 111376, 2022.
- Y. Leng, X. Tian, N. Trask and J. Foster,
Asymptotically compatible reproducing kernel collocation and meshfree integration for nonlocal diffusion,
SIAM J. Numer. Anal., 59(1), 88-118, 2021.
- Y. Leng, X. Tian, N. Trask and J. Foster,
Asymptotically compatible reproducing kernel collocation and meshfree integration for the peridynamic Navier equation,
Comput. Methods Appl. Mech. Eng., 370, 113264, 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. Leng, X. Tian and J. Foster,
Super-convergence of reproducing kernel approximation ,
Comput. Methods Appl. Mech. Eng., 352:488-507, 2019.