Periodic Boundary Conditions in Natural Element ...

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TMAG.2015.2485958, IEEE Transactions on Magnetics

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Periodic Boundary Conditions in Natural Element Method Bárbara M. F. Gonçalves1, Márcio M. Afonso1, Eduardo H. R. Coppoli1, Marco Aurélio O. Schroeder1, Rafael S. Alípio1, Brahim Ramdane2 and Yves Marechal2 1

Electrical Engineering Postgraduate Program - CEFET-MG – UFSJ, Minas Gerais, Brazil 2 Electrical Engineering Laboratory of Grenoble – G2ELAB, Grenoble, France

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 important because they allow to exploit the inherent symmetry of the electromagnetic devices. So, with these techniques the NEM can now easily take advantage of electrical machines symmetry. The proposed approach is evaluated and its results are compared with those obtained by the traditional FEM. Index Terms — Natural Element Method, Periodic Boundary Conditions, Electrical Machines.

with those obtained by the traditional finite element method. I. INTRODUCTION

T

meshless methods are increasingly being used in numerical evaluation particularly in the electromagnetic field computations [1, 2]. In these methods a cloud of nodes without connectivity relations that covers the domain is used to solve the problem [2]. Due to its characteristics, these methods are well suited to solve problems involving moving parts, such as electrical machines [3, 4]. However, in this kind of problems it is important to consider the symmetry in order to reduce the number of unknown. Periodic and anti-periodic boundary conditions are useful techniques to exploit the inherent symmetry of electrical machines [5]. It is well known that meshless methods provide high accurate solutions but present some difficulties to handle interface and boundary conditions [3]. To eliminate these drawbacks, the natural element method (NEM) has been proposed [3]. The NEM approach is based on the Voronoï diagram and the natural neighbors concept. The main interest in NEM lies in its interpolation property which allows enforcing essential boundary conditions in an easy way as with the FEM [6]. Also, this method retains the natural capacity of treating heterogeneous domain and present similar numerical behavior with a better convergence compared to FEM in some cases [7]. As the natural element method shape functions verify the Kronecker delta property, the imposition of periodic and antiperiodic boundary conditions can be done with the same technique used by FEM [8]. Therefore, the aim of this paper is to introduce the periodic and anti-periodic boundary conditions in the NEM. The proposed approach is applied to periodic electromagnetic device and the results are compared HE

Manuscript received July 6, 2015; revised September 02, 2015 and September 30, 2015; accepted September 23, 2015. Date of publication July 10, 2015; date of current version September 30, 2015. (Dates will be inserted by IEEE; “published” is the date the accepted preprint is posted on IEEE Xplore®; “current version” is the date the typeset version is posted on Xplore®). Corresponding author: M. M. Afonso (e-mail: [email protected] ). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier (inserted by IEEE).

II. PROBLEM FORMULATION For the purpose of analysis, the periodic structure shown by Fig. 1 is studied. This kind of structure is common in electric machine analysis where the periodicity consideration can substantially reduce the size of the domain device [4, 8]. The periodicity in Fig. 1 is characterized by a geometric replication of the picked out domain. If there are coils and/or permanent magnets oriented in the same direction, the potentials on line c are identical to the potentials on line d [8].

Fig. 1. A simple periodic structure. Dots in the circles indicate current entrance. The Ω domain described by lines a, b, c and d define the repetitive domain. Only the Ω domain needs to be analyzed to solve the problem [8].

An anti-periodic structure is similar to the aforementioned one except that the source (current) has alternately opposing directions [8]. To formulate the problem, consider that a two dimensional magnetostatic phenomenon occurs in . So, its strong formulation can be given by the Poisson equation [5]:

on In (1) represents the z component of the scalar magnetic potential vector, is the magnetic permeability, is the current density on coils and is the magnetic potential imposed on the Dirichlet boundary (given by the lines and ). The weak formulation for (1) can be written as [4]:

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In (2) is the problem domain surrounded by the surface (given by the lines a, b, c and d), is an approximation for the scalar component of the magnetic potential vector and is the scalar weight function. Many numerical techniques can be used in the evaluation of (2). However, this paper is focused on the natural element method [9]. (a) III. THE NATURAL ELEMENT METHOD (NEM) A. Voronoï Diagram The 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 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 neighbors of the node in its center. The Delaunay triangulation, which is the dual of the Voronoï diagram, is constructed by connecting the nodes whose Voronoï cells have common boundaries [6, 9]. The interpolation of a point can be obtained by introducing a fictitious point in the Delaunay triangulation to define its natural neighbors and calculating the value of the corresponding shape function [6, 9]. B. Shape Function Based on the Voronoï diagram, the natural element shape function can be calculated. In the literature, 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 Si(x) represents the subarea of Voronoï cell centered on x and linked to the natural neighbor ni and S(x) is the total area of Voronoï cell linked to that point.

(b)

Fig. 2. (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 .

Equation (4) verifies the same properties of FEM shape functions. The Kronecker delta, interpolation and partition of unity properties are verified [3, 6]. Thus, the potential can be written as follows [6, 9]:

where N is the number of natural neighbors visible from the point x and , the value for the scalar component of the magnetic potential vector at the natural neighbors of point x. Then, the final system of equations can be obtained by substituting the discrete form of trial and test functions (5) in (2) and applying it in all domain. After numerical integration, the following linear system of equation is obtained:

where is the global simetric matrix, is an unknown vector, and is the source vector. The elements of the matrix and the vector are given by: and

C. Treatment of Material Discontinuities Due to the inherent meshless character, (5) can be computed in arbitrary clouds of nodes and these nodes can move freely on the background without any geometrical restriction [3]. To treat the materials discontinuity a constrained Delaunay triangulation associated with a visibility criterion is used [10]. On the convex boundary domain, the NEM shape functions become linear as in the FEM allowing the natural coupling of these methods. However for a non-convex boundary, this propriety is not verified. So, nodes situated on both sides of a crack would be seen as natural neighbors, and thus would have an undesirable mutual influence. To overcome this problem, the constrained Delaunay triangulation associated with a



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criterion of visibility (C-NEM) has been proposed [6, 10]. This constrained approach could also be applied on the interfaces between regions of different materials allowing all the NEM shape functions become linear and identical to FEM shape functions on these interfaces. It provides the NEM ability to naturally handle the physical discontinuities that appear on the interfaces, whereas in usual meshless methods this solution has to be enforced by other means [10].

For an anti-periodic structure the procedure is similar [5, 8]. But in this case, one should set , and . Moreover, the contributions related to the node i' must be transferred to the node i with opposite signs. It is necessary because the nodes i' and are in a sub-domain whose sources have opposite signs [5].

D. Imposing Periodic and Anti-periodic Boundary Conditions As previously mentioned, the potentials on lines and in Fig. 1 are equal and therefore only the domain needs to be analyzed [8]. A simplified representation of is given by Fig. 3 where the Voronoï cells and the nodes on the periodic boundary are highlighted. To impose periodic and antiperiodic boundary conditions the contributions related to the nodes on the boundary ( , and ) must be associated to those on the boundary ( , and ) and assembled properly in the global matrix [5].

To validate the proposed approach, the electromagnetic device shown by Fig. 5 is analyzed. The circles are aluminium conductors carrying a current density of 1MA/m2 and involved by an iron material with μr = 7000, for all results.

IV. RESULTS

h/5

FLUX

25 mm h/2

h

FLUX 75 mm h = 25 mm

Fig.5. Periodic structure and regions of potential evaluation.

In the case of a periodic structure consider the nodes and as a periodic pair ( ) [8]. The same procedure needs to be repeated for the nodes and , and as well as for all pair of nodes that are located on the lines and . Before solving (6), the contributions for the node must be assembled in the position of the contributions related to the node in order to obtain . This indicates for line the presence of an identical domain on its left [5]. Fig. 4 shows the details of this process for the periodic pair . One should make , and . It is important to note that before setting , its contribution is passed to [5]. Moreover, the values in line of the matrix must be added to the those in line . The same process needs to be done for the source terms. After that the (6) can be solved and the value of calculated considering the contributions from .

0.025

0.02

0.015

y (m)

Fig. 3. Treatment of boundary nodes by means of NEM in a periodic structure.

In Fig. 5, the periodic or anti-periodic boundary conditions are set at the left and right boundaries, while zero Dirichlet ones are defined at the top and bottom boundaries. The potential is evaluated on the dashed boundary (right) when periodic structure is analyzed and on its orthogonal line when the anti-periodic structure is considered. For a periodic analysis, the current in the conductors shown by Fig. 5 are flowing in the same direction (exiting the sheet plane). Fig. 6 shows the induction flux distribution obtained by NEM due to 1120 nodes discretization. For the same discretization, Fig. 7 shows the resulting induction flux distribution for anti-periodic boundary condition originated by a current flowing in alternating directions in each conductor.

0.01

0.005

0

0

0.01

0.02

0.03

0.04 x (m)

0.05

0.06

0.07

Fig.6. The resulting induction flux distribution for a periodic structure obtained by the NEM. Fig. 4. System assembling for the periodic pair

.

These results are validated by comparison with the traditional FEM obtained with the same number of nodes. The potential distribution on the boundary of the periodic structure



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is shown by Fig. 8. And, Fig. 9 depicts the potential distribution along the anti-periodic orthogonal line. The above results show the agreement between NEM and FEM solutions.

0.05

FEM NEM

Magnetic Potential Vector (Wb/m)

0.04

0.025

0.02

y (m)

0.015

0.02

0.01

0

0.01

-0.01 0.065 0.066 0.067 0.068 0.069 0.07 0.071 0.072 0.073 0.074 0.075 Orthogonal Line x(m)

0.005

0

0.03

Fig. 9. Potential distribution along the anti-periodic orthogonal line, for 1120 nodes discretization. 0

0.01

0.02

0.03

0.04 x (m)

0.05

0.06

0.07

Fig. 7. The induction flux distribution for an anti-periodic structure obtained by the NEM.

The relative errors obtained in each numerical method are compared. The reference solution is assumed as the FEM one with a very refined mesh (17413 nodes). The error is obtained considering the L2 norm of the solution vector. For the periodic boundary (Fig. 8), the relative error obtained by FEM is 2.28% while that obtained by NEM is 2.04%. It is observed that the error associated to the NEM is less than that one related to the FEM for the same number of unknown. This is also observed in Fig. 9 whose relative error for FEM is 2.68% against 2.27% obtained for NEM. In order to measure the number of nodes dependency, three different analysis are done. The ratios of the FEM error (eFEM) and the NEM error (eNEM) relative to the aforementioned reference solution are evaluated and presented in the Table 1.

It is possible to note from Table 1 that when the number of nodes is increased, the NEM solution becomes more accurate than FEM one. V. CONCLUSION The periodic and anti-periodic boundary conditions are adressed in this paper. These boundary contitions are introduced in the natural element method and applied to solve a periodic and anti-periodic electromagnetic problems. The analysis of the solutions have demonstrated the applicability and accuracy of NEM to treat periodicity. The results show that for the same number of nodes, the NEM is more accurate than FEM. VI. ACKNOWLEDGEMENT This work was partially supported by CEFET-MG, FAPEMIG, CAPES and CNPq. VII. REFERENCES

0.025

Magnetic Potential Vector (Wb/m)

[1] 0.02

0.015 FEM NEM 0.01

0.005

0

0

0.005

0.01 0.015 Periodic Boundary Lenght in (m)

0.02

0.025

Fig. 8. Potential distribution along the periodic boundary, for 1120 nodes discretization. TABLE 1 ERROR RATIO CONSIDERING THE NUMBER OF NODES eFEM/eNEM 291 Nodes

1120 Nodes

4395 Nodes

1.08

1.11

1.16

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