SUMMARYIn this paper we develop a dual-horizon peridynamics (DH-PD) formulation that naturally includes varying horizon sizes and completely solves the "ghost force" issue. Therefore, the concept of dual-horizon is introduced to consider the unbalanced interactions between the particles with different horizon sizes. The present formulation fulfills both the balances of linear momentum and angular momentum exactly. Neither the "partial stress tensor" nor the "slice" technique are needed to ameliorate the ghost force issue in [1]. We will show that the traditional peridynamics can be derived as a special case of the present DH-PD. All three peridynamic formulations, namely bond based, ordinary state based and non-ordinary state based peridynamics can be implemented within the DH-PD framework. Our DH-PD formulation allows for h-adaptivity and can be implemented in any existing peridynamics code with minimal changes. A simple adaptive refinement procedure is proposed reducing the computational cost. Both two-and threedimensional examples including the Kalthoff-Winkler experiment and plate with branching cracks are tested to demonstrate the capability of the method.

In this paper, we present a dual-horizon peridynamics formulation which allows for simulations with dual-horizon with minimal spurious wave reflection. We prove the general dual property for dual-horizon peridynamics, based on which the balance of momentum and angular momentum in PD are naturally satisfied. We also analyze the crack pattern of random point distribution and the multiple materials issue in peridynamics. For selected benchmark problems, we show that DH-PD is less sensitive to the spatial than the original PD formulation.Comment: 21 page

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.

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We propose a nonlocal operator method for solving partial differential equations (PDEs). The nonlocal operator is derived from the Taylor series expansion of the unknown field, and can be regarded as the integral form "equivalent" to the differential form in the sense of nonlocal interaction. The variation of a nonlocal operator is similar to the derivative of shape function in meshless and finite element methods, thus circumvents difficulty in the calculation of shape function and its derivatives. The nonlocal operator method is consistent with the variational principle and the weighted residual method, based on which the residual and the tangent stiffness matrix can be obtained with ease. The nonlocal operator method is equipped with an hourglass energy functional to satisfy the linear consistency of the field. Higher order nonlocal operators and higher order hourglass energy functional are generalized. The functional based on the nonlocal operator converts the construction of residual and stiffness matrix into a series of matrix multiplications on the nonlocal operators. The nonlocal strong forms of different functionals can be obtained easily via support and dual-support, two basic concepts introduced in the paper. Several numerical examples are presented to validate the method.