[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.
  1. Q. Ye and X. Tian, Monotone meshfree methods for linear elliptic equations in non-divergence form via nonlocal relaxation , J. Sci. Comput., 96(3), 85, 2023. [Interactive Website Examples]

  2. 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.

  3. 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.

  4. 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.

  5. 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.

  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. Leng, X. Tian and J. Foster, Super-convergence of reproducing kernel approximation , Comput. Methods Appl. Mech. Eng., 352:488-507, 2019.