Periodic Boundary Conditions in Vector Nodal Meshless Method

The vector fields approximation poses a significant challenge for traditional meshless methods that use scalar-based functions. Aiming to overcome these challenges, the Vector Nodal Meshless Method (VNMM) was proposed. In the context of VNMM, a set of nodes is distributed both within the domain and along its boundaries. Each node is associated with a unit vector. The VNMM shape functions are derived from the H(curl) spaces and Nédélec’s first type elements polynomial space. The projection of VNMM shape functions verifies the delta Kronecker property, thereby facilitating the imposition of essential boundary conditions. The objective of this paper is to introduce the periodic boundary conditions in the VNMM context for application in periodic vectorial problems. The periodic boundary conditions allow to take account the periodic symmetry of the electromagnetic devices in order to reduce the study domain. However, the imposition of periodic boundary conditions in vector nodal modeling differs from that in scalar nodal analyses due to the vector nature of the problem and the method. In this work, a methodology is proposed for taking account the vector nature of the functions and imposing the periodic boundary correctly. In order to validate the strategy, the TEAM 13 problem—a three-dimensional, periodic, and nonlinear problem—is solved. In this problem, the implementation of periodic boundary conditions enables the reduction of the study domain to a quarter of its original size. Subsequently, the results are then compared with those obtained using both the entire and a half study domain. The result validates the use of periodic boundary conditions in VNMM.