209 citations · 348 across the 22 of their papers we have counts for
22 papers
Numerical analysis of a class of penalty discontinuous Galerkin methods for nonlocal diffusion problems
Qiang Du, Lili Ju, Jianfang Lu +1
In this paper, we consider a class of discontinuous Galerkin (DG) methods for one-dimensional nonlocal diffusion (ND) problems. The nonlocal models, which are integral equations, a…
Asymptotically compatible schemes for nonlinear variational models via Gamma-convergence and applications to nonlocal problems
Qiang Du, James M. Scott, Xiaochuan Tian
We present a study on asymptotically compatible Galerkin discretizations for a class of parametrized nonlinear variational problems. The abstract analytical framework is based on v…
Nonlocal half-ball vector operators on bounded domains: Poincaré inequality and its applications
Zhaolong Han, Xiaochuan Tian
This work contributes to nonlocal vector calculus as an indispensable mathematical tool for the study of nonlocal models that arises in a variety of applications. We define the non…
Monotone meshfree methods for linear elliptic equations in non-divergence form via nonlocal relaxation
Qihao Ye, Xiaochuan Tian
We design a monotone meshfree finite difference method for linear elliptic equations in the non-divergence form on point clouds via a nonlocal relaxation method. The key idea is a…
Nonlocal diffusion models with consistent local and fractional limits
Qiang Du, Xiaochuan Tian, Zhi Zhou
For some spatially nonlocal diffusion models with a finite range of nonlocal interactions measured by a positive parameter , we review their formulation defined on a bounded dom…
A Meshfree Peridynamic Model for Brittle Fracture in Randomly Heterogeneous Materials
Yiming Fan, Huaiqian You, Xiaochuan Tian +4
In this work we aim to develop a unified mathematical framework and a reliable computational approach to model the brittle fracture in heterogeneous materials with variability in m…
Efficient optimization-based quadrature for variational discretization of nonlocal problems
Marco Pasetto, Zhaoxiang Shen, Marta D'Elia +3
Casting nonlocal problems in variational form and discretizing them with the finite element (FE) method facilitates the use of nonlocal vector calculus to prove well-posedeness, co…
A Petrov-Galerkin method for nonlocal convection-dominated diffusion problems
Yu Leng, Xiaochuan Tian, Leszek Demkowicz +2
We present a Petrov-Gelerkin (PG) method for a class of nonlocal convection-dominated diffusion problems. There are two main ingredients in our approach. First, we define the norm…
An asymptotically compatible probabilistic collocation method for randomly heterogeneous nonlocal problems
Yiming Fan, Xiaochuan Tian, Xiu Yang +3
In this paper we present an asymptotically compatible meshfree method for solving nonlocal equations with random coefficients, describing diffusion in heterogeneous media. In parti…
Nonlocal Trace Spaces and Extension Results for Nonlocal Calculus
Qiang Du, Xiaochuan Tian, Cory Wright +1
For a given Lipschitz domain , it is a classical result that the trace space of is , namely any function has a well-defined $W^…
Fractional Hardy-type and trace theorems for nonlocal function spaces with heterogeneous localization
Qiang Du, Tadele Mengesha, Xiaochuan Tian
In this work, we prove some trace theorems for function spaces with a nonlocal character that contain the classical space as a subspace. The result we obtain generalizes…
Numerical methods for nonlocal and fractional models
Marta D'Elia, Qiang Du, Christian Glusa +3
Partial differential equations (PDEs) are used, with huge success, to model phenomena arising across all scientific and engineering disciplines. However, across an equally wide swa…
Asymptotically compatible reproducing kernel collocation and meshfree integration for the peridynamic Navier equation
Yu Leng, Xiaochuan Tian, Nathaniel A. Trask +1
In this work, we study the reproducing kernel (RK) collocation method for the peridynamic Navier equation. We first apply a linear RK approximation on both displacements and dilata…
A review of Local-to-Nonlocal coupling methods in nonlocal diffusion and nonlocal mechanics
Marta D'Elia, Xingjie Li, Pablo Seleson +2
Local-to-Nonlocal (LtN) coupling refers to a class of methods aimed at combining nonlocal and local modeling descriptions of a given system into a unified coupled representation. T…
Multiscale Modeling, Homogenization and Nonlocal Effects: Mathematical and Computational Issues
Qiang Du, Bjorn Engquist, Xiaochuan Tian
In this work, we review the connection between the subjects of homogenization and nonlocal modeling and discuss the relevant computational issues. By further exploring this connect…
Asymptotically compatible reproducing kernel collocation and meshfree integration for nonlocal diffusion
Yu Leng, Xiaochuan Tian, Nathaniel Trask +1
Reproducing kernel (RK) approximations are meshfree methods that construct shape functions from sets of scattered data. We present an asymptotically compatible (AC) RK collocation…
A physically-consistent, flexible and efficient strategy to convert local boundary conditions into nonlocal volume constraints
Marta D'Elia, Xiaochuan Tian, Yue Yu
Nonlocal models provide exceptional simulation fidelity for a broad spectrum of scientific and engineering applications. However, wider deployment of nonlocal models is hindered by…
Fast algorithm for computing nonlocal operators with finite interaction distance
Xiaochuan Tian, Bjorn Engquist
Developments of nonlocal operators for modeling processes that traditionally have been described by local differential operators have been increasingly active during the last few y…
Mathematics of Smoothed Particle Hydrodynamics: a Study via Nonlocal Stokes Equations
Qiang Du, Xiaochuan Tian
Smoothed Particle Hydrodynamics (SPH) is a popular numerical technique developed for simulating complex fluid flows. Among its key ingredients is the use of nonlocal integral relax…
compactness criteria with an application to variational convergence of some nonlocal energy functionals
Qiang Du, Tadele Mengesha, Xiaochuan Tian
Motivated by some variational problems from a nonlocal model of mechanics, this work presents a set of sufficient conditions that guarantee a compact inclusion in the function spac…
Stability of nonlocal Dirichlet integrals and implications for peridynamic correspondence material modeling
Qiang Du, Xiaochuan Tian
Nonlocal gradient operators are basic elements of nonlocal vector calculus that play important roles in nonlocal modeling and analysis. In this work, we extend earlier analysis on…
A quasinonlocal coupling method for nonlocal and local diffusion models
Qiang Du, Xingjie Helen Li, Jianfeng Lu +1
In this paper, we extend the idea of "geometric reconstruction" to couple a nonlocal diffusion model directly with the classical local diffusion in one dimensional space. This new…