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Geometric assumptions and notation for the corner case. Here the yellow region denotes í µí°µ(x, í µí»¿) ∩ í µí¼Ω í µí± í µí»¿ .
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In this paper we consider 2D nonlocal diffusion models with a finite nonlocal horizon parameter δ characterizing the range of nonlocal interactions, and consider the treatment of Neumann-like boundary conditions that have proven challenging for discretizations of nonlocal models. We propose a new generalization of classical local Neumann conditions...
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Context 1
... we have p 1 = (1, 0) and p 2 = (− cos í µí¼, − sin í µí¼). We illustrate geometric assumptions and notation in Figure 6. For each point x = (í µí±¥ 1 , í µí±¥ 2 ), with Taylor expansion we have the following approximation for í µí±¢(y) − í µí±¢(x) with y = (í µí±¦ 1 , í µí±¦ 2 ) ∈ í µí°µ(x, í µí»¿) ∩ í µí¼Ω í µí± í µí»¿ : ...
Context 2
... we have p 1 = (1, 0) and p 2 = (− cos í µí¼, − sin í µí¼). We illustrate geometric assumptions and notation in Figure 6. For each point x = (í µí±¥ 1 , í µí±¥ 2 ), with Taylor expansion we have the following approximation for í µí±¢(y) − í µí±¢(x) with y = (í µí±¦ 1 , í µí±¦ 2 ) ∈ í µí°µ(x, í µí»¿) ∩ í µí¼Ω í µí± í µí»¿ : ...
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Citations
... As a breakthrough, in one-dimensional [37] and two-dimensional [41] cases, the nonlocal models with O(δ 2 ) convergence rate to its local counterpart were successfully constructed under Neumann boundary condition. One year later, Lee H. and Du Q. in [25] introduced a nonlocal model under Dirichlet boundary condition by imposing a special volumetric constraint along the boundary layer, which assures O(δ 2 ) convergence rate in 1d segment and 2d plain disk. ...
In this paper, we analyze a class of nonlocal Poisson model on manifold under Dirichlet boundary condition. To get second order convergence, the geometric information of the manifold has to be incorporated in the nonlocal model. So the manifold is assumed to be given explicitly such that all geometric information can be calculated explicitly. Under this assumption, the well-posedness of nonlocal model is studied. Moreover, the second-order convergence rate of model is attained by combining the stability argument and the truncation error analysis. Such rate is currently optimal among all nonlocal models. Numerical tests are conducted to verify the convergence.
... Despite their improved accuracy, the usability of nonlocal equations is hindered by several modeling and computational challenges. Modeling challenges include the lack of a complete nonlocal theory [22][23][24], the treatment of nonlocal interfaces [25][26][27][28], and the prescription of nonlocal volume constraints (the nonlocal counterpart of boundary conditions) [29][30][31][32]. Computational challenges are due to the integral nature of nonlocal operators that yield discretization matrices that feature a much larger bandwidth compared to the sparse matrices associated with PDEs [33][34][35]. ...
We explore the approximation of feedback control of integro-differential equations containing a fractional Laplacian term. To obtain feedback control for the state variable of this nonlocal equation, we use the Hamilton–Jacobi–Bellman equation. It is well known that this approach suffers from the curse of dimensionality, and to mitigate this problem we couple semi-Lagrangian schemes for the discretization of the dynamic programming principle with the use of Shepard approximation. This coupling enables approximation of high-dimensional problems. Numerical convergence toward the solution of the continuous problem is provided together with linear and nonlinear examples. The robustness of the method with respect to disturbances of the system is illustrated by comparisons with an open-loop control approach.
... When discretizing the nonlocal models, it is desired to preserve the corresponding local limit under refinement grid size h → 0 , since analyzing consistency in the limit to the local solution provides a mathematically unambiguous means to understand accuracy and physical compatibility. Such a property is termed asymptotically compatible (AC) [37]. 1 In recent years, there has been significant work toward establishing such discretizations [37][38][39][40][41][42][43][44][45][46][47]. Broadly, strategies either involve adopting traditional finite element shape functions and carefully performing geometric calculations to integrate over relevant horizon/ element subdomains, or adopt a strong-form meshfree discretization where particles are associated with abstract measure. ...
... However, we point out that the optimizationbased quadrature rule as well as the OBMeshfree package can be readily applied to more general kernels, such as the data-driven kernel developed in [29]. 3 Neumann and Robin-type boundary conditions can be implemented with the quadrature rule provided by OBMeshfree, following [46,47,81] where L P is a nonlocal operator representing the peridynamic internal force density, is the mass density, and f is a prescribed body force density. As for the nonlocal diffusion problems, the nonlocal interactions in L P are also restricted to the nonlocal neighborhood, B (x) , characterized by the horizon size . ...
We present OBMeshfree, an Optimization-Based Meshfree solver for compactly supported nonlocal integro-differential equations (IDEs) that can describe material heterogeneity and brittle fractures. OBMeshfree is developed based on a quadrature rule calculated via an equality constrained least square problem to reproduce exact integrals for polynomials. As such, a meshfree discretization method is obtained, whose solution possesses the asymptotically compatible convergence to the corresponding local solution. Moreover, when fracture occurs, this meshfree formulation automatically provides a sharp representation of the fracture surface by breaking bonds, avoiding the loss of mass. As numerical examples, we consider the problem of modeling both homogeneous and heterogeneous materials with nonlocal diffusion and peridynamics models. Convergence to the analytical nonlocal solution and to the local theory is demonstrated. Finally, we verify the applicability of the approach to realistic problems by reproducing high-velocity impact results from the Kalthoff-Winkler experiments. Discussions on possible immediate extensions of the code to other nonlocal diffusion and peridynamics problems are provided. OBMeshfree is freely available on GitHub (Fan et al. [1]).
... With the given source item f : Ω → R and the g : Ω I → R, the problem needs to determine u : Ω → R. The second equation in (2.5) is called the nonlocal Dirichlet volume constraint. We only consider Dirichlet boundary conditions in this paper, and the implementation in this paper can be applied to problems with Neumann boundary conditions considered in [61,48] naturally. ...
Nonlocality brings many challenges to the implementation of finite element methods (FEM) for nonlocal problems, such as large number of queries and invoke operations on the meshes. Besides, the interactions are usually limited to Euclidean balls, so direct numerical integrals often introduce numerical errors. The issues of interactions between the ball and finite elements have to be carefully dealt with, such as using ball approximation strategies. In this paper, an efficient representation and construction methods for approximate balls are presented based on combinatorial map, and an efficient parallel algorithm is also designed for assembly of nonlocal linear systems. Specifically, a new ball approximation method based on Monte Carlo integrals, i.e., the fullcaps method, is also proposed to compute numerical integrals over the intersection region of an element with the ball.
... 2 Herein, we apply the assumptions in (3) so as to guarantee the compatibility of the nonlocal model and its local limit, in the limit of vanishing nonlocality, i.e., δ → 0. However, we point out that the optimization-based quadrature rule as well as the OBMeshfree package can be readily applied to more general kernels, such as the data-driven kernel developed in [76]. 3 Neumann and Robin-type boundary conditions can be implemented with the quadrature rule provided by OBMeshfree, following [17,74,75] Here, γ is the kernel function as defined in (3), and the two-point functions κ(x, y) denote the (averaged) bulk modulus property 4 . To recover parameters for linear elasticity when the nonlocal effects vanish, one should take c = 24/5 for d = 2, and c = 6 for d = 3 (see, e.g., [28,72] for further details). ...
We present OBMeshfree, an Optimization-Based Meshfree solver for compactly supported nonlocal integro-differential equations (IDEs) that can describe material heterogeneity and brittle fractures. OBMeshfree is developed based on a quadrature rule calculated via an equality constrained least square problem to reproduce exact integrals for polynomials. As such, a meshfree discretization method is obtained, whose solution possesses the asymptotically compatible convergence to the corresponding local solution. Moreover, when fracture occurs, this meshfree formulation automatically provides a sharp representation of the fracture surface by breaking bonds, avoiding the loss of mass. As numerical examples, we consider the problem of modeling both homogeneous and heterogeneous materials with nonlocal diffusion and peridynamics models. Convergences to the analytical nonlocal solution and to the local theory are demonstrated. Finally, we verify the applicability of the approach to realistic problems by reproducing high-velocity impact results from the Kalthoff-Winkler experiments. Discussions on possible immediate extensions of the code to other nonlocal diffusion and peridynamics problems are provided. OBMeshfree is freely available on GitHub.
... As a breakthrough, in one dimensional [35] and two dimensional [38] cases, the nonlocal models with O(δ 2 ) convergence rate to its local counterpart were successfully constructed under Neumann boundary condition. One year later, Lee H. and Du Q. ...
In this work, we introduced a class of nonlocal models to accurately approximate the Poisson model on manifolds that are embedded in high dimensional Euclid spaces with Dirichlet boundary. In comparison to the existing nonlocal Poisson models, instead of utilizing volumetric boundary constraint to reduce the truncation error to its local counterpart, we rely on the Poisson equation itself along the boundary to explicitly express the second order normal derivative by some geometry-based terms, so that to create a new model with $\mathcal{O}(\delta)$ truncation error along the $2\delta-$boundary layer and $\mathcal{O}(\delta^2)$ at interior, with $\delta$ be the nonlocal interaction horizon. Our concentration is on the construction and the truncation error analysis of such nonlocal model. The control on the truncation error is currently optimal among all nonlocal models, and is sufficient to attain second order localization rate that will be derived in our subsequent work.
... Although such an averaged two-point function formulation were developed for nonlocal diffusion [45,46] and peridynamics [47][48][49][50][51] models, we have for the first time provided rigorous mathematical analysis for the well-posedness of this formulation in a heterogeneous LPS model. Furthermore, an important feature of peridynamics is that when classical continuum models still apply, peridynamics revert back to classical continuum models as its horizon size δ → 0. Numerical discretizations which preserve this limit under the grid refinement h → 0 are termed asymptotically compatible (AC) [52], and there has been significant works in recent years toward establishing such discretizations [33,45,[52][53][54][55][56][57][58][59][60][61]. In this work, we have also theoretically shown that our stochastic heterogeneous LPS model guarantees consistency to the corresponding local limit, which provides a critical ingredient in achieving a convergent simulation. ...
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 material microstructures, and to provide statistic metrics for quantities of interest, such as the fracture toughness. To depict the material responses and naturally describe the nucleation and growth of fractures, we consider the peridynamics model. In particular, a stochastic state-based peridynamic model is developed, where the micromechanical parameters are modeled by a finite-dimensional random vector, or a combination of random variables truncating the Karhunen-Lo\`{e}ve decomposition or the principle component analysis (PCA). To solve this stochastic peridynamic problem, probabilistic collocation method (PCM) is employed to sample the random field representing the micromechanical parameters. For each sample, the deterministic peridynamic problem is discretized with an optimization-based meshfree quadrature rule. We present rigorous analysis for the proposed scheme and demonstrate its convergence for a number of benchmark problems, showing that it sustains the asymptotic compatibility spatially and achieves an algebraic or sub-exponential convergence rate in the random space as the number of collocation points grows. Finally, to validate the applicability of this approach on real-world fracture problems, we consider the problem of crystallization toughening in glass-ceramic materials, in which the material at the microstructural scale contains both amorphous glass and crystalline phases. The proposed stochastic peridynamic solver is employed to capture the crack initiation and growth for glass-ceramics with different crystal volume fractions, and the averaged fracture toughness are calculated. The numerical estimates of fracture toughness show good consistency with experimental measurements.
... These results would extend the knowledge on the trace space in nonlocal calculus and its connection with the trace space in classical calculus. Moreover, the trace theorem and the inverse trace theorem would also provide important mathematical tools for developing well-posed nonlocal models with volumetric boundary conditions, such as discussed in [64]. ...
For a given Lipschitz domain Ω, it is a classical result that the trace space of W 1,p (Ω) is W 1−1/p,p (∂Ω), namely any W 1,p (Ω) function has a well-defined W 1−1/p,p (∂Ω) trace on its codimension-1 boundary ∂Ω and any W 1−1/p,p (∂Ω) function on ∂Ω can be extended to a W 1,p (Ω) function. Recently, [26] characterizes the trace space for nonlocal Dirichlet problems involving integrodifferential operators with infinite interaction ranges, where the boundary datum is provided on the whole complement of the given domain R d \Ω. In this work, we study function spaces for nonlocal Dirichlet problems with a finite range of nonlocal interactions, which naturally serves a bridging role between the classical local PDE problem and the nonlocal problem with infinite interaction ranges. For these non-local Dirichlet problems, the boundary conditions are normally imposed on a region with finite thickness volume which lies outside of the domain. We introduce a function space on the volumetric boundary region that serves as a trace space for these nonlocal problems and study the related extension results. Moreover, we discuss the consistency of the new nonlocal trace space with the classical W 1−1/p,p (∂Ω) space as the size of nonlocal interaction tends to zero. In making this connection, we conduct an investigation on the relations between nonlocal interactions on a larger domain and the induced interactions on its subdomain. The various forms of trace, embedding and extension theorems may then be viewed as consequences in different scaling limits.
... These articles indicate that the nonlocal model (1.6) converges to the Poisson model in a rate of O(δ 2 ) as δ → 0 under periodic boundary condition, and has O(δ) convergence rate under Dirichlet/Neumann/Robin type boundaries due to its low accuracy near the boundary, see [4] [5] [12] [14] for the Neumann boundary case and [1] [2] [15] [26] [39] for other type of boundaries. Specially, in one dimensional [35] and two dimensional [38] cases with Neumann boundary , one can obtain O(δ 2 ) convergence by doing several modifications of (1.6) on the boundary layer. ...
Partial differential equations on manifolds have been widely studied and plays a crucial role in many subjects. In our previous work, a class of integral equations was introduced to approximate the Poisson problems on manifolds with Dirichlet and Neumann type boundary conditions. In this paper, we restrict our domain into a compact, two dimensional manifold(surface) embedded in high dimensional Euclid space with Dirichlet boundary. Under such special case, a class of more accurate nonlocal models are set up to approximate the Poisson model. One innovation of our model is that, the normal derivative on the boundary is regarded as a variable so that the second order normal derivative can be explicitly expressed by such variable and the curvature of the boundary. Our concentration is on the well-posedness analysis of the weak formulation corresponding to the integral model and the study of convergence to its PDE counterpart. The main result of our work is that, such surface nonlocal model converges to the standard Poisson problem in a rate of $\mathcal{O}(\delta^2)$ in $H^1$ norm, where $\delta$ is the parameter that denotes the range of support for the kernel of the integral operators. Such convergence rate is currently optimal among all integral models according to the literature. Two numerical experiments are included to illustrate our convergence analysis on the other side.
... Later in [28], a careful modification of the body force in a one dimensional setting is found that leads to a second order uniform convergence of solutions as the nonlocal interaction vanishes. The second order convergence result is then extended to two dimensions in [31], where the curvature of the computation domain plays an important role in the definition of the modified body force. Recently, [13] achieves the second order uniform convergence in one dimension with another approach. ...
