I Introduction
Electric machines are usually modelled through the magnetoquasistatic approximation of Maxwell’s equations discretised on the machine cross section, which requires the solution of a 2D Poisson problem. Its numerical solution commonly amounts to triangulating the domain and applying the *fem. The acting electromotive forces and the torque can be determined by a postprocessing procedure, e.g. invoking the Maxwell stress tensor method. A drawback of the standard approach is that a very fine mesh is needed to achieve an acceptable accuracy. Moreover, the air gap needs to be resolved properly and the solution is known to be extremely sensitive to the used discretisation
[1]. Furthermore, remeshing or reconnecting mesh parts in order to account for the machine rotation can introduce a spurious ripple on the solution for the torque [2].An exact parametrisation of the air gap is not possible within a classical *fem framework since elements of any order rely on polynomial mappings which are unable to represent conic sections such as circles and ellipses. Furthermore, the regularity of *fem solutions is typically limited by the continuity across the elements [3].
*iga [4] is able to overcome these issues. It chooses *cad basis functions such as e.g. Bsplines and *nurbs for the approximation spaces and is thus able to represent *cad geometries exactly even on the coarsest level of mesh refinement. A higher regularity at the mesh interfaces can also be obtained by applying refinement, which results in smoother basis functions compared to *fem and increases the accuracy of the solution accordingly [5]. Another benefit of IGA is that it provides an elegant way to do geometric optimisation [6] and to handle uncertainties [7].
This work aims at using *iga to model and simulate a *pmsm. The stator and rotor models parts are inevitably resolved by multiple patches, which in turn necessitates the use of a domain decomposition approach across the air gap. We employ a classical iterative procedure based on a DirichlettoNeumann map [8].
Ii Solving Maxwell’s Equations for a *pmsm
Iia Magnetostatics
In early design steps, it is sufficient to model electric machines by the 2D magnetostatic approximation of the Maxwell’s equations. Let (see Fig. 1) depict the computational domain. One has to solve the following Poisson equation
(1) 
where is the reluctivity, is the
component of the magnetic vector potential,
is the source current density in the coils and is the current density according to the magnetisation of the permanent magnets. On , a Dirichlet boundary condition is applied and on and , an antiperiodic boundary condition holds.IiB Discretisation in the Isogeometric Framework
For the discretisation of , a linear combination of basis functions is used, i.e.,
(3) 
where are the unknowns and depicts the number of unknowns. Using the RitzGalerkin approach, a system of equations is obtained, i.e.,
(4) 
where has the entries
(5a)  
and the entries for the discretised current density are  
(5b)  
(5c) 
The choice of the basis functions depends on the method one favours. The simplest case in the well established *fem is the use of linear hat functions [10]. This paper proposes the use of the IGA framework to model the electrical machine. In this method, the basis functions are defined by *nurbs [11].
Let depict the degree of the basis functions and let
(6) 
be a vector that partitions into elements, where . Then, the Coxde Boor’s formula [11] defines Bspline basis functions with . Bsplines of degree are shown in Fig. 2. *nurbs of degree are then constructed as
(7) 
with a weighting parameter associated with the th basis function. A *nurbs curve is obtained by the mapping
(8) 
with control points in and . From the curves, surfaces can be constructed by using tensor products [11], i.e.,
(9) 
The number of onedimensional basis functions along direction and their degrees are depicted by and respectively. The mapping is now redefined, namely , where is the reference square and is the physical domain. Due to the *nurbs mapping, the subdivisions constructed by the vectors are transformed to a physical mesh on .
Due to the complexity of the geometry, the cross section of the machine cannot be represented by a single map [12]. Multiple patches are constructed (Fig. 1), each one of them defined as the image of the unit square through a parametrization of the type (8) in such a way that and [12].
Using (7) and (8) enables an exact parametrization of e.g. circles and arcs. This implies that the geometry of the machine can be modelled exactly, which is particular beneficial in the air gap region. In *iga, the same set of basis functions (9) is also used to approximate the solution of (1). Hence, we can rewrite (3) as
(10) 
where we have removed the indices , for simplicity. We will always assume the same polynomial degree in both directions.
This choice of basis functions guarantees a higher regularity of the solution across the elements. The continuity at the patch boundaries reverts to the *fem case since only continuity is imposed (see [12]).
Iii Iterative StatorRotor Coupling
Since the rotor and the stator have a different topology, we parametrise them independently as multipatch *nurbs entities with nonconforming patches at the air gap interface . The idea is to follow a classical nonoverlapping domain decomposition approach based on a DirichlettoNeumann map (see e.g. [8]).
Let us consider the circular arc in the air gap and split the domain such that and (see Fig. 1). Given and an initialisation , we solve:
(11) 
(12) 
where is a unit vector perpendicular to the air gap interface. The two problems are solved iteratively with the update
(13) 
where is a relaxation parameter (in general, the method is not ensured to converge if [8]).
As a stopping criterion for the method, the error between two successive iterations is required to be below a specified tolerance both in the rotor and the stator, i.e.,
Iv Results
For testing the suitability of *iga for machine simulation, we first consider the domain and we solve a test problem with homogeneous Dirichlet boundary conditions on and antiperiodic boundary conditions connecting and . The only source present in the model is the magnetisation of the permanent magnets. The *iga solution for the magnetic vector potential is compared to the one obtained using a fine *fem discretisation with first order elements. We solve *iga basis functions of degree and and for an increasing mesh refinement, and we depict the difference to the *fem solution in in Fig. 3. For both discretisation degrees, the curves approach a fixed value, which is expected since the *iga method is solving the problem for the exact geometry in contrast to *fem which introduces an additional geometrical error related to the triangulation of the geometry.
Secondly, the full problem is considered, incorporating the DirichlettoNeumann coupling between stator and rotor introduced in the previous section. In Fig. 4, the convergence of the algorithm is shown for a simulation with degree
, with approximately 3200 total degrees of freedom. After 29 iterations, both solutions show an incremental error below the prescribed tolerance
.The simulation results are shown in Fig. 5, where the flux lines in the machine are shown. In the postprocessing the spectrum of the *emf has been calculated. In Fig. 6, the first modes of the spectrum of the *emf of the machine are shown. Both methods result in similar spectra. This is also depicted in Tab. I, where it is shown that the relative difference of the *thd is below .
*thd  
IGA (degree 2)  3224  30.6 V  6.0610 
FEM  59678  29.8 V  5.7210 
relative difference    2.5  5.7 
V Conclusions
Isogeometric Analysis (IGA) recently emerged as a promising alternative to the Finite Element Method (FEM) for highly accurate electromagnetic field simulation. IGA is capable of exactly resolving circles and, hence, avoids any geometric approximations errors in contrast to FEM, which is of particular importance in the air gap region. Our IGA model for a permanent magnet synchronous machine consists of two separate multipatch IGA discretisations for stator and rotor, glued together with a DirichlettoNeumann map in the air gap. IGA (degree 2) attains the prescribed accuracy with a number of degrees of freedom which is substantially smaller ( 20 times) than for lowest order FEM. Therefore, IGA is a valuable extension to an engineer’s simulation toolbox, especially when fast and accurate machine simulation tasks are due.
Acknowledgment
This work is supported by the German BMBF in the context of the SIMUROM project (grant nr. 05M2013), by the DFG (grant nr. SCHO1562/31) and by the ’Excellence Initiative’ of the German Federal and State Governments and the Graduate School of CE at TU Darmstadt.
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