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(a) Representation of Voronoï diagram (blue color) and associated Delaunay triangulation (gray color). The node, its Voronoï cell and its six natural neighbors are highlighted. (b) NEM shape function computation where S1(x) represents the subarea associated with the node .
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The purpose of this paper is to introduce periodic and anti-periodic boundary conditions in the Natural Element Method (NEM). It is shown that as the NEM shape functions verify the Kronecker delta property, the imposition of such boundary conditions can be done in an easy way as in the Finite Element Method (FEM). These boundaries conditions are im...
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... natural element method uses the concept of natural neighbors which is based on the construction of Voronoï diagram on a cloud of nodes. For instance, consider a set of nodes distributed in the whole domain in 2D space as depicted by Fig. 2(a). The Voronoï diagram is a subdivision of the domain into cells, where each cell is associated with a node such that any point in is closer to than to any other node with . The regions are the Voronoï cells of . In mathematical terms, the Voronoï cell is defined as [6,9]: with been the coordinates of the node and the distance between ...
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... any point in is closer to than to any other node with . The regions are the Voronoï cells of . In mathematical terms, the Voronoï cell is defined as [6,9]: with been the coordinates of the node and the distance between node and point . Thus, is the region of the plane that contains the points closest to node than to any other node in as shown in Fig. 2(a). The construction of each Voronoï cell can be obtained by the intersection of the segments joining the node to its neighbor nodes, and a straight line, normal to each one of these segments, traced at the central point of those segments. This diagram subdivides the studied domain into a set of polygons which defines the natural ...
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... several formulas are used to calculate this shape function [3]. Among the most used, are the Sibson functions which may be determined in analogy with classical FEM shape functions as the ratio of surfaces in the case of triangles [6]. The same principle is applied to Voronoï cells to achieve Sibson shape functions. At a point x shown by Fig. 2 (b), the Sibson shape function is given by (4) where each S i (x) represents the subarea of Voronoï cell centered on x and linked to the natural neighbor n i and S(x) is the total area of Voronoï cell linked to that point. Equation (4) verifies the same properties of FEM shape functions. The Kronecker delta, interpolation and partition of ...
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Citations
... • Periodic boundary conditions(Pbc) [38] are applied in xyz directions. 9 3 Results and Discussion ...
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... The stator slotting effect should be considered accurately to predict the magnetic field distribution in the airgap region. Indeed, the presence of stator slots has a large influence on the airgap magnetic field and therefore on the motor performances such as radial force and cogging torque which cause noise, speed ripple and vibrations (Gonçalves et al. 2017;Gonçalves 2016;Gonçalves et al. 2015;O'Connell and Krein 2010;Schmidt and Grabner 2005). Since there is no closed structure in semi-closed type slot, tangential fields interrupted at slot mouth openings. ...
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... The natural element method (NEM) is a meshless method based on the Voronoï diagram and the natural neighbor concept [1]. Due to its characteristics, the NEM is well suited to solve problems involving moving parts, like electrical machines [1][2][3]. ...
... The natural element method (NEM) is a meshless method based on the Voronoï diagram and the natural neighbor concept [1]. Due to its characteristics, the NEM is well suited to solve problems involving moving parts, like electrical machines [1][2][3]. However, in this kind of problems it is important to consider the symmetry in order to reduce the number of unknown and periodicity analysis could be very useful to explore it [1,2]. ...
... Due to its characteristics, the NEM is well suited to solve problems involving moving parts, like electrical machines [1][2][3]. However, in this kind of problems it is important to consider the symmetry in order to reduce the number of unknown and periodicity analysis could be very useful to explore it [1,2]. ...
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... However, unlike FEM the nodes present in Ω domain are influenced by nodes belonging to Ω domain [7]. To treat the materials discontinuity a constrained Voronoï diagram associated with a visibility criterion is used [9][10][11]. ...
In this paper the treatment of material discontinuities in the Natural Element Method (NEM) are discussed. A visibility criterion and the constrained Voronoï diagram are presented in order to take account this difficulty. The extension of NEM, the Constrained NEM (C-NEM), is applied to solve electromagnetic problems. An electrostatic problem is proposed and the approach is evaluated and compared with the traditional Finite Element Method (FEM).
In this paper, the interpolation functions behavior of the natural element method (NEM) and the finite element method (FEM) are compared. It is discussed how the unknown in both methods affects the stiffness matrix and contributes to NEM better accuracy. The visibility criterion and the constrained NEM are also addressed, and a pseudoalgorithm is proposed to implement the constrained Voronoï diagram, which is the base for the constrained NEM. A complex heterogeneous magnetic problem is solved by both FEM and NEM methods and their solutions are compared. It is shown that for the same discretization, the number of contributions for NEM is in general bigger than those related to the FEM and better accuracy results for NEM.
