A Nonlocal Operator Method for Partial Differential Equations with Application to Electromagnetic Waveguide Problem

Abstract

A novel nonlocal operator theory based on the variational principle is proposed for the solution of partial differential equations. Common differential operators as well as the variational forms are defined within the context of nonlocal operators. The present nonlocal formulation allows the assembling of the tangent stiffness matrix with ease and simplicity, which is necessary for the eigenvalue analysis such as the waveguide problem. The present formulation is applied to solve the differential electromagnetic vector wave equations based on electric fields. The governing equations are converted into nonlocal integral form. An hourglass energy functional is introduced for the elimination of zeroenergy modes. Finally, the proposed method is validated by testing three classical benchmark problems.

Coupling XFEM and peridynamics for brittle fracture simulation—part I: feasibility and effectiveness

“…Various approaches have been used in the literature to obtain an equilibrium solution for the PD theory such as the Adaptive Dynamic Relaxation method (ADR) [24,55], conjugate gradient solvers [56] and implicit Newton-Raphson [57]. An implicit nonlocal operator formulation for electromagnetic problems that provides that tangent stiffness matrix is also proposed in [58].…”

A Hybrid Local/Nonlocal Continuum Mechanics Modeling and Simulation of Fracture in Brittle Materials

“…In 2000, Silling [Silling (2000)] proposed a novel model, named peridynamics, based on integral equilibrium equations. Since integral equations are mathematically compatible with discontinuities, peridynamics can handle the crack initiation and propagation without introducing a complicated failure criterion [Kilic, Agwai and Madenci (2009)], and Ha et al [Ha and Bobaru (2010); Wang, Oterkus and Oterkus (2018); Shou, Zhou and Berto (2019); Ren, Zhuang and Rabczuk (2016); Rabczuk, Ren and Zhuang (2019)] have successfully applied it to fracture simulations. However, peridynamics suffers from a high computational cost because a point in peridynamics is interacting with numerous points in a finite neighborhood, which results in the expensive computation of the resultant of forces.…”

Coupling XFEM and Peridynamics for brittle fracture simulation: part II—adaptive relocation strategy

“…For Cases 1 (the PD only model) and 2 (XFEM-PD with no relocation capabilities), the total number of dofs are 131,040 and 42,196, respectively, and remain constant throughout the simulation. Cases 3 and 4 have initially the same number of dofs (17,716) however, at the end of the simulation the dofs of Case 3 have increased to 35,000 while for Case 4 they are approximately the same (15,148). The variation in Case 4 is caused by the shape variation of PD from the initial to the final configuration (see Fig.…”

“…The dual-horizon methodology presented in [14] is such an approach that also avoids the appearance of spurious reflections. Furthermore, the nonlocal operator method, that can be considered a generalization of the dual-horizon PD model, has been proposed in [15,16]. Our study is limited to methodologies that combine the FE method with PD as it is envisaged to take advantage already established FE solvers and potentially port PD models to commercially available FE packages.…”