L A Illoul

Y Le Menach

S Clenet

F Chinesta

A mixed finite element/meshless natural element method for simulating rotative electromagnetic machines

Keywords: Magnetostatics; magnetic shielding, magnetic induction, boundary-value problems, Finite-element and Galerkin methods

Efficient numerical models have been already developed in 2D to take into account the movement of electromagnetic devices with rotating parts in the framework of the Finite Element Method (FEM). When the movement becomes complex it leads to large mesh distortions. A remeshing step is then required which increases the computational complexity and can also lead, in some cases, to numerical ripples on forces and torques due to the field projections between old and new meshes. Moreover, remeshing procedures in 3D remain an open topic. Meshless methods seem an appealing choice for alleviating the mesh constraints. The Natural Element Method (NEM) which, has known a growing interest in the domain of mechanics, allows to proceed in the meshless framework, avoiding one of the main drawbacks related to the vast majority of meshless techniques, as is the imposition of essential boundary conditions. In this paper, a variant of the NEM, known as constrained natural element method (C-NEM) is applied for simulating electromagnetic machines involving rotating parts. A new mixed strategy combining the finite element and the constrained natural element methods is proposed and then tested by using an appropriate error estimator.

Introduction

Numerical modelling of electromagnetic machines

To accurately model electrical machines, one needs to solve the Maxwell equations taking into account the movement of mobile parts. Generally, an electrical rotating machine is made up of two parts. The moving part (the rotor) and the static one (the stator) are separated by a thin clearance (the air gap) to enable the rotor movement. In fact, most of the energy of the machine is concentrated in the air-gap. Thus, it is necessary to take into account very accurately the relative movement between both parts.

At present, the Finite Element Method (FEM) is widely used to solve numerically the Maxwell equations. To take the movement into account, several methods have already been proposed: the macro element [18], the moving band [5], the slip surface [16], the use of Lagrange multipliers [8], the overlapping technique [27], the nodal interpolation strategy [15], the mortar element [17], among many others. Meshless techniques could be envisaged for alleviating the computational complexity related to these approaches.

The main aim of the present work is the analysis of meshless approaches as well as the proposal and evaluation of a new mixed strategy combining the simplicity of finite element descriptions with the robustness of meshless descriptions in the regions involving large mesh deformations.

Meshless approaches

In the past years meshfree methods for the solution of partial differential equations have significantly matured and are used in various fields of applications. One of the reasons for this development is the fact that meshfree discretizations and particle models are often better suited to cope with geometrical changes of the domain of interest than mesh-based discretization techniques such as finite differences, finite elements or finite volumes. Furthermore, the computational cost associated with mesh generation is alleviated in meshfree approaches since they are based only on a cloud of nodes, without any geometrical restriction concerning their relative positioning.

Among the main meshless approaches we highlight the Smooth Particle Hydrodynamics (SPH) [10], the Diffuse Element Method (DEM) [13], the Element Free Galerkin (EFG) [1], the Reproducing Kernel Particle Method(RKPM) [9], the h-p Cloud Method [6] and the Natural Element Method (NEM) [25]. Some of these meshless techniques are still nowadays confronted to some difficulties related to: (i) the quality of the approximation as well as the conditioning of the global system is pathologically dependent on the size of the support of the shape functions; (ii) imposing of essential boundary conditions needs special treatments; (iii) integration is not accurate enough, and (iv) physical discontinuities across interfaces must be accurately enforced.

Several techniques for the analysis of problems involving discontinuities or interfaces have been proposed in the finite element [2,12] and in the meshless [14,28] frameworks. The extended finite element method (X-FEM) [12] makes use of the Partition of Unity paradigm [11,12] for approximating discontinuities. The goal of the extended finite element method is to expand the finite element approximation functions for including some features of the solution that are known.

Alternative to the aforementioned methods, the constrained natural element method (C-NEM) [29] has been proposed to handle discontinuities and moving interfaces in meshfree methods in a very simple manner [30]. The technique uses the features of the natural element method in tandem with a visibility criterion. In opposition to the vast majority of meshless strategies, the C-NEM allows to enforce exactly essential boundary conditions on the domain boundary, and then to account for transmission conditions across fixed or moving interfaces.

In this paper, the constrained natural element method will be applied to model rotating electrical machines. For this purpose, we first summarize the constrained natural element method (C-NEM). Then, the equations to be solved are given and the method to take the movement into account is described. Finally, a Variable Reluctance Machine (VRM) is modelled using the constrained natural element method and results are compared with results obtained by using the FEM. A mixed formulation combining the versatility of finite elements and the robustness of the meshless approaches to address large mesh deformations is then proposed and evaluated.

For comparison purposes we use the error estimator introduced in [3] in the natural element method framework, which is based on the solution of both complementary potential formulations. It will be applied to quantify the accuracy of the different solutions as well as to compare C-NEM and FEM solutions. A mixed approach combining the versatility of finite elements and the robustness of the meshless approaches to address large mesh deformations will be also proposed and its accuracy checked. This comparison is one of the main original contributions of this paper, an one of the first exhaustive comparative analysis of both approximations to take into account movement, at least in our knowledge.

The meshfree constrained natural element method (C-NEM) revisited

In this section, the ability of the C-NEM for describing moving interfaces and discontinuities in a fixed cloud of nodes is discussed. After a brief review of the Voronoibased interpolants, we introduce the constrained Voronoi diagram which is used to compute the shape functions in any domain. To avoid duplication with some of our former publications, different references to our former works will be addressed.

Natural neighbor interpolation

We briefly touch upon the foundation of Sibson's natural neighbor coordinates (shape functions) that are used in the natural element method. For a more in-depth discussion on the Sibson interpolant and its application for solving second-order partial differential equations, the interested reader can refer to Sambridge and Braun [19], and Sukumar et al. [25]. The NEM interpolant is constructed on the basis of the Voronoi diagram (see Fig. 1). The Delaunay tessellation is the topological dual of the Voronoi diagram.

From now on, we focus only in the 2D case, the 3D case being a direct extension. Consider a set of nodes S = {n 1 , n 2 , . . . , n N } in 2 . The Voronoi diagram is the partition of 2 into regions T i (Voronoi cells) defined ∀i ∈ S by:

T i = {x ∈ 2 : d(x, x i ) < d(x, x j ), ∀j ∈ S; j = i}, (1)
where d( ) denotes a distance. The Voronoi cells related to neighbor nodes have a common edge.

In order to define the natural neighbour coordinates it is necessary to introduce the second-order Voronoi diagram of the cloud defined for each couple of nodes (i, j) ∈ S 2 with i = j as

T ij = {x ∈ 2 : d(x, x i ) < d(x, x j ) < d(x, x k ), ∀k ∈ S; k = i, j}.
(

Sibson [START_REF] Sibson | [END_REF] defined the natural neighbor coordinates of a point x with respect to one of its neighbors n i as the ratio of the cell T i that is transferred to T x when adding x to the initial cloud of points, to the total volume of T x . In other words, if κ(x) and κ i (x) are the Lebesgue measures of T x and T xi respectively, the natural neighbor coordinates of x with respect to the node n i is defined as

φ i (x) = κ i (x) κ(x) . ( 3
)
Figure 2 illustrates the construction of φ 1 (x), that in this case is given by:

φ 1 (x) = Area(af ghe) Area(abcde) . ( 4
)
If the point x coincides with the node n i , i.e. (x = x i ), φ i (x i ) = 1, and all other shape functions are zero, i.e. φ j (x i ) = δ ij (δ ij being the Kronecker's delta). The properties of positivity, interpolation, and partition of unity are then verified [25]:

⎧ ⎨ ⎩ 0 ≤ φ i (x) ≤ 1 φ i (x j ) = δ ij N i=1 φ i (x) = 1.
(

) 5
The natural neighbor shape functions also satisfy the local coordinate property [START_REF] Sibson | [END_REF], namely:

x = N i=1 φ i (x)x i (6)
which combined with equation (5), implies that the natural neighbor interpolant spans the space of linear polynomials (linear completeness). Sibson natural neighbor shape functions are C 1 at any point except at the nodes, where they are only C 0 . The continuity can be enhanced by using special classes of natural neighbor shape functions [7].

Another important property of this interpolant is its strict linearity over the boundary of convex domains. The proof can be found in Sukumar et al. [25]. An illustration is depicted in Figure 2b: as the areas associated to points on the boundary become infinite, the contribution of internal nodes vanish in the limit when the point approaches the convex boundary, and the shape functions associated with nodes n 1 and n 2 become linear on the segment joining both nodes (S n1→n2 ). This result is essential to guarantee strict continuity of the approximation across material interfaces. The lack of this property is an important issue in most meshfree methods which require special numerical strategies to circumvent this drawback [14]. Nevertheless, the property of linearity of the interpolant is only satisfied along convex boundaries [25]. The non-convex case needs further developments that will be addressed in the next section.

Consider an interpolation scheme for a scalar function u(x) : Ω ⊂ 2 → , in the form: where u i are the nodal values of the field u at the n(x) neighbor nodes of point x, all them grouped in the set I, and φ i (x) are the shape functions at that point associated with each neighbor node. It is noted that equation ( 7) defines a local interpolation scheme. Thus, the trial and test functions used in the discretization of the variational formulation related to the electromagnetic model treated in this paper could be approximated by equation (7).

u h (x) = i∈I φ i (x) u i (7)

The constrained natural element method

Constrained Voronoi diagram

It was proved in [25,29] and [4] that spurious influences between "non-visible" nodes and loss of linearity in the interpolation along boundaries of non-convex domains appear in the framework of the NEM. In order to avoid this drawback and to recover all properties of the method for any geometry (including non-convex domains containing fixed or moving interfaces), a visibility criterion was introduced in order to restrict influent nodes among natural neighbors [29]. The computation of the shape functions is then done on the basis of the so-called constrained (or extended) Voronoi diagram (CVD), introduced by Seidel in [START_REF] Seidel | 1978-1988 Ten Years IIG[END_REF] which is the strict dual of the constrained Delaunay triangulation.

To illustrate this behavior, we consider the situation depicted in Figure 3, where the point x moves from Ω 1 to Ω 2 . If x is in Ω 1 , the interpolated field is constructed from equation ( 7) using the neighbor visible nodes from point x (Γ I is assumed opaque). If x is on Γ I the interpolated field is strictly linear because it only depends on the two neighbor nodes located on Γ I . Finally, when x is in Ω 2 , the interpolated field is defined using the neighbor nodes, visible from point x (Γ I being opaque). The continuity of the interpolated field is then guaranteed, but a discontinuity appears in the field derivatives, because of a sudden change in the neighbor nodes across the interface. Thus, we can then reproduce a continuous field with discontinuous normal derivative across the interface.

The intersection between the constrained Voronoi diagram and the domain closure is composed of cells T C i , one for each node n i , such that any point x inside T C i is closer to n i than to any other node n j visible from x. The constrained Voronoi cells are defined formally ∀i ∈ S by:

T C i = {x ∈ n : d(x, x i ) < d(x, x j ); ∀j ∈ S, j = i; S x→ni ∩ Γ = ∅; S x→nj ∩ Γ = ∅ ( 8
)
where Γ is the domain boundary and S a→b denotes the segment between the points a and b. All these constructions can be generalized to 3D [START_REF] Shewchuck | Proceedings of the fourteenth annual symposium on computational geometry[END_REF][START_REF] Shewchuck | Proceedings of the sixteenth annual symposium[END_REF].

The constrained natural element approximation

In order to solve partial differential equations defined on non-convex domains, we consider the following approximation for both the trial and the test functions:

u h (x) = i∈IV φ C i (x) u i δu h (x) = i∈IV φ C i (x) δu i (9)
where the set I V contains the V (x) natural neighbors nodes visible from point x, and φ C i represents the constrained natural neighbor shape function related to the ith node at point x. The computation of the C-n-n (constrained natural neighbor) shape functions is similar to the natural neighbor ones, when one proceeds using the constrained Voronoi diagram introduced previously. It was proved in [29] that the use of the constrained Voronoi diagram does not affect the properties of the NEM interpolation, allowing the extension of the linearity of the shape functions on the convex domain boundaries, to any geometry, convex or not.

The ability of the C-NEM to treat problems involving cracks was illustrated in [29] and for moving interfaces in thermal problems in [30]. In the present paper, the domain is partitioned in some regions with different material properties. Each subdomain is discretized using a cloud of nodes and the interfaces between the different regions are described by a polygonal curve defined by a set of nodes. Then, a constrained Voronoi diagram is defined in each subdomain with respect to the domain boundary and the interfaces. The attractive feature of the present technique is the possibility to move the interfaces without special care for the shape of the underlying Delaunay triangles because the interpolation accuracy does not depend on the geometrical quality of the Delaunay triangles, in contrast to the FEM. In this manner, the continuity of the approximation is guaranteed by the strict linearity of the interpolation across the interfaces, that are defined by a set of interface nodes.

Taking into account movement

It is well known in the context of the FEM that frequent remeshing procedures are required for accounting accurately the rotation. To model the rotation of the electrical machine different methods ara available. Remeshing can be very simple when using a regular mesh. Thus, one must shift the unknowns located on the boundary of the moving part. Nevertheless, the rotation step must be constant, that can be a drawback if one wants to model a machine working at variable speed.

To overcome this task, other methods are applied on a slip line using an interpolation technique to obtain a relation between the static and mobile unknowns. Thus, the nodal values on the slip line of the mobile part are expressed from the ones on the fixed part and then no additional unknowns are introduced. The interpolation can be linear, expressing a moving unknown from two static unknowns [15]. It can be also quadratic [START_REF] Shi | Proceeding of COMPUMAG[END_REF] where three static nodes are necessary to define the interpolation. Another more general approach is given by the "mortar" method [17] which allows expressing the moving unknowns as a function of all the static ones. The solution defined on the moving surface is projected on the fixed one using the Galerkin technique. It is obvious that the "mortar" method gives the best results but the size of the linear system bandwidth increases significantly, in contrast with the two first techniques which do not increase the linear system bandwidth but lead to less accurate solutions, in particular the linear interpolation. Still slip Lagrange [8]. test functions for minimizing the gap in the nodal values (related to moving and static nodes) computed on the slip line. In this case all the nodal unknowns must be computed, and consequently the size of the linear system increases significantly and its conditioning is degraded, like when interpolation methods are used.

Other techniques use a special mesh in the air gap such as the moving band [5]. In this method, the mesh is distorted, due to the movement, until an unacceptable degree of deformation. Then, the surface is re-meshed and the nodal connectivity updated as in the locked step approach (regular mesh in the air gap). Unfortunately some numerical ripples can be observed on the torque when remeshing is used. For the macro element technique, an analytic solution is introduced in the air-gap [18]. However, the associated linear system bandwidth increases too and the analytic solution can be difficult to obtain for some kind of machines. Finally the overlapping method uses two meshes in the air-gap [27]. One mesh is associated to the static part and the whole air-gap region. A Dirichlet condition is imposed at the nodes of the mesh that belong to the mobile part. The same operation is carried out with a second mesh (that contains the mobile part and the air-gap) prescribing the field at the nodes located in the static domain. Thus, two different meshes cover the air gap. This method leads to very accurate results but the mesh computation in the airgap requires significant efforts.

In the C-NEM context, accounting for movement is quite natural and very simple to implement. One could just move the nodes defining the interface because the geometrical quality of the resulting Delaunay triangles does not affect significantly the accuracy of the resulting approximation. In the simulations reported in this work we decided to rotate all the nodes within the inner airgap circle. Then the constrained Voronoi diagram is updated and the interpolation shape functions computed again.

Simulating rotating electrical machines

To test the capability of the C-NEM to model rotating electrical machines, we consider a 6-4 Variable Reluctance Machine (VRM). The calculation of the torque for each relative position between stator and rotor, that is parametrized using the angle θ, is a very tricky problem because for computing the torque (as we explain later) one must proceed in the air gap region where sharp angles due to the presence of rotor and stator teeth are encountered. If the mesh is not regular and fine enough, parasitic ripples can appear in the computed torque (as illustrated later) when one proceeds in the finite element framework.

In Figure 4, a 3D view of the machine is given, as well as the 2D model that is solved. The domain under study is a cross section of the machine defined as Ω = {(x, y) :

x 2 + y 2 < R 2
ext }, being R ext the stator external radius. The stator has 6 teeth and holds three phase windings. Each one is composed of two coils that are wound around one tooth. In our study only one phase is supplied by a DC current. Two regions are then defined the first is flown by a current density +J and the other by a current density -J (|J| = 1 A mm -2 ) (see Fig. 4). Elsewhere the current density is equal to zero. The eddy currents are also neglected.

The electromagnetic field distribution can be calculated by using either the vector potential or the scalar potential formulations. To introduce the notations we consider the basic electromagnetic equations:

∇B = 0 ∇ × H = J ( 10
)
where B is the magnetic flux density, H the magnetic field and J the current density. There is a constitutive equation that links B and H that writes

B = μ H (11)
where μ > 0 represents the magnetic permeability.

On the boundary ∂Ω of Ω we assume that the normal component of B vanishes, that is, Bn = 0, where n is the vector defined on ∂Ω. A field B adm it verifies the Analogously, a field H is said admissible, H adm , if it verifies the second relationship in (10). Obviously a couple (B adm , H adm ) that verifies the constitutive equation ( 11) is the exact solution of the electromagnetic problem.

In the 2D problem that we are analyzing we can assume that the magnetic flux density has a zero out-ofplane component, that is B z = 0, that implies that it exists a field A, with components A T = (0, 0, A z ) = (0, 0, A) such that B = ∇ × A. Thus the magnetic field can be defined everywhere using the constitutive equation ( 11): H = ∇ × A/μ, that introduced in the second relation of (10) leads to the 2D vector potential formulation:

-∇ 1 μ ∇A = J in Ω A = 0 on ∂Ω ( 12
)
where the vector current density J has only a zcomponent, i.e. J T = (0, 0, J z ) = (0, 0, J), with both A and J independent on the z-coordinate. Obviously, by construction, its solution allows to define an admissible magnetic flux density B adm that will be referred as B A . The associated magnetic field writes

H A = B A /μ.
In the same way, another complementary potential formulation can be introduced. Let H s be a field such that ∇ × H s = J. Since H -H s is curl free, we can introduce a scalar potential ω such that:

H = H s -∇ω. ( 13
)
Then, the equation to be solved can be written:

-∇ (μ ∇ω) = -∇ (μ H s ) in Ω μ (H s -∇ω) n = 0 on ∂Ω. ( 14
)
This formulation leads to the admissible magnetic field denoted by H ω , but the associated magnetic flux density B = μH ω is not admissible anymore because it does not verify ∇B = 0. It will be denoted in what follows by

B ω = μ H ω .
In the electromagnetic system here considered the permeability of the ferromagnetic part (stator and rotor) is assumed to be constant μ s = μ r = 4π × 10 -4 H m -1 (saturation and hysteresis effects are neglected) and the permeabilities of the air and of the coil are

μ a = μ c = 4π × 10 -7 H m -1 .
The weak formulation related to equation ( 12) results:

Find A ∈ H 1 0 (Ω) such that Ω 1 μ ∇A∇δA dΩ = Ω J δA dΩ, ∀δA ∈ H 1 0 (Ω) (15)
where H 1 0 (Ω) is the usual Sobolev functional space.

In the same way the weak formulation related to equation (14) writes: 16) In Figures 567we can observe the cloud of nodes considered for the discretization of the electromagnetic equations as well as the associated Voronoi cells. The interfaces between the different regions are described by a polygonal curve defined by a set of nodes (thick lines in Fig. 7). In order to appreciate the nodal refinement in the air gap between stator and rotor we depict in Figures 6 and7 two zooms of this area. We can observe that a constrained Voronoi diagram has been associated with each subdomain Ω s (stator), Ω c (coil), Ω a (air) and Ω r (rotor) with respect to the corresponding interfaces. Thus, the continuity of A and ω across the interfaces is guaranteed, allowing the discontinuity of the tangential component of the magnetic flux density across the interface, as discussed previously. Let D 0 be the disk centered on the shaft of the electrical machine and with a diameter equal to the diameter of the rotor as it was reported at the end of Section 2. For accounting a rotation Δθ of the rotor all nodes within D 0 rotates of that angle Δθ. At each rotation step, the constrained Voronoi diagram has to be updated, allowing the computation of the approximation shape functions and then the discretization of the variational formulations (Eqs. ( 15) and ( 16)).

Ω μ∇ω∇δω dΩ = Ω μH s ∇δω dΩ, ∀δω ∈ H 1 (Ω). (
Thus, substituting the trial and test functions (both approximated in the C-NEM framework) in the above variational formulations and using the arbitrariness of the fields δA and δω, the following linear system of equations is obtained for the vector potential formulation:

K A (θ) A(θ) = F A (θ) (17)
and for the scalar potential formulation

K ω (θ) ω(θ) = F ω (θ) (18)
where θ refers the relative position between stator and rotor, A(θ) the vector containing the nodal values of the vector potential field for that relative position θ, and ω(θ) the vector containing the nodal values of the scalar potential at position θ.

The associated torque can be calculated from:

C(θ) = 2π 0 R F t (θ, R, ϕ) dϕ ( 19
)
where R is the radius of a circle completely defined in the air gap, and F t the tangential component of the electromagnetic force density F that is computed from the Maxwell stress tensor. Thus, during a complete turn, the torque varies and due to the localization of the potential gradients significant oscillations appear when the torque is computed using a coarse mesh in the finite element framework if no remeshing is performed, as we illustrate later. Figure 8 depicts, for 4 relative positions between the rotor and the stator (θ θ 4 = 45 • ), the flux path-map given by the vector potential formulation when the phase indicated in Figure 4 (bottom) is supplied with a DC current.

1 = 0 • , θ 2 = 15 • , θ 3 = 30 • and
In order to compare the solutions related to different approximation schemes, we are using an appropriate error estimator based on the verification of the constitutive relationship. This error estimator was introduced in [26] and is often used to estimate numerical errors in static electromagnetism. This estimator was recently used to compare finite element and meshless natural element solutions in the case of magnetostatics (without taking into account movement) in [3]. In the present work we are applying this estimator in order to compare both methods when the movement is also taken into account. This estimator is defined by:

e B = B adm -μH adm = B A -B ω ( 20
)
where v = Ω (vv)/μ dΩ. It was proven in [26] that this error corresponds to the distance between the averaged magnetic flux density B = 0.5 (B adm + μH adm ) and the exact field B ex .

Obviously we can define an equivalent error norm operating on the magnetic fields:

e H = B adm μ -H adm = H A -H ω . (21)
A more appropriate expression for the error consists of normalizing the previous expressions. In that follows a relative error will be computed by using

E 2 H = 2 H A -H ω 2 H A + H ω 2 (22)
or by using the equivalent norm:

Ê2 H = 2 H A -H ω 2 max θ H A + H ω 2 . ( 23
)
As described previously the torque can be computed as soon as the fields are available. However, these fields can be computed by using either the vector potential formulation or the scalar potential one. The resulting torques will be denotes by C A and C ω respectively. Obviously all these fields, H A and H ω and the scalars C A and C ω , depend on the relative position between the rotor and stator. To define a torque relative error we must be careful because the torque vanishes for some rotor positions. To overcome this difficulty we could integrate the different norms in the angular coordinate defining the rotor position:

Ẽ2 C = 2 π 4 0 |C A (θ) -C ω (θ)| 2 dθ π 4 0 |C A (θ) + C ω (θ)| 2 dθ (24)
where the integration domain is reduced to θ ∈ [0, π 4 ] by symmetry considerations.

One could expect that the main differences between a finite element and a natural element solutions should be located in the air gap where due to the rotor movement the finite element mesh becomes very distorted. Of course in a precise implementation of the finite element method the mesh is updated locally or globally to avoid too distorted elements. However, in this work we have decided to proceed with the same cloud of nodes to perform both discretizations, the one using the FEM and the other the C-NEM. Thus, a linear finite element interpolation will be applied directly on the Delaunay's triangulation to obtain de finite element shape functions. The same triangulation was also considered to define the Voronoi diagram used to computed the C-NEM interpolation functions.

As the C-NEM becomes piecewise linear on the domain boundary or on any internal interface, as previously illustrated, one could define two interfaces, the first one located on the rotor external radius, and the other one on the internal stator radius. Thus, the region in between both radii defined the air gap, where a maximum mesh distortion is expected. It seems natural that an appealing choice for the interpolation in that region would consist of using the C-NEM approximation. On the other hand, because the CPU time related to the construction of the C-NEM shape functions is much more higher that the one required to define the finite element approximation, one could prefer to use in the other two remaining regions a linear finite element discretization constructed on the Delaunay triangulation. In this way, both interpolations connect perfectly on the interfaces between the different regions. The mixed discretization strategy uses the C-NEM in the air gap and the FEM in the remaining domain.

Figure 9 depicts the evolution of ÊH (θ) for the three functional approximations: FEM, C-NEM and the mixed FEM/C-NEM, where by symmetry considerations θ varies in the interval θ ∈ [45 • , 90 • ]. These solutions were computed using a cloud of 400 000 nodes. Now, we compute the torque for different rotor positions θ ∈ [45 • , 90 • ], different formulations (vector potential and scalar potential), different discretization techniques (C-NEM, FEM and mixed FEM-NEM) and different nodal densities.

Figure 10 compares the computed torques C(θ) using both potential formulations and the different discretization techniques (the FEM, the C-NEM and the mixed strategy), when a cloud of 6755 nodes was considered. We can notice that for coarse meshes the FEM generates the expected spurious oscillations whereas the C-NEM solution looks significantly less perturbed. All these solutions come close as the number of nodes increases, illustrating the convergence of all the used formulations. The mixed FEM/C-NEM solution shows less oscillations than the finite element solution, being its associated computing cost similar to the one related to the finite element solution.

Finally, we quantify the convergence rates for both the magnetic field and the torque. Concerning the magnetic field we consider a particular position of the rotor, in our simulation θ 1 = 0 • , evaluating the error according to equation [START_REF] Seidel | 1978-1988 Ten Years IIG[END_REF] for different nodal densities. The resulting error evolution is depicted using a log-log scale in Figure 14, where the norm [START_REF] Seidel | 1978-1988 Ten Years IIG[END_REF] was used. The slope of this representation corresponds to the convergence rate, that for both the C-NEM and the linear FE interpolation here considered is expected to be one for the energy norm here considered. We can notice that despite the same conver-gence rate that both discretizations exhibit, the C-NEM accuracy is significantly better. The convergence rate of the mixed formulation is also first order and its accuracy is in between the ones related to the FEM and the C-NEM.

Figure 15 depicts the results associated with the torque convergence analysis, where the error is computed according to equation [START_REF] Sibson | [END_REF]. The conclusions are similar to the ones just addressed in the magnetic field convergence analysis. One can notice that the mixed solution is a bit less accurate that the C-NEM solution, but it is much better that the FEM one. However, the computing time related to the mixed solution is very close to the one associated with the FEM.

Conclusions

In this paper the constrained natural element method has been applied for simulating electromagnetic rotative machines. The use of the aforementioned technique allows to alleviate the difficulties related to remeshing in the finite element context, improving significantly the solution accuracy. A method combining the versatility of the FEM and the robustness of the meshless natural element method to take into account the movement has been proposed and tested. It seems that this combination is a good compromise between computing time and accuracy.

The extension to 3D geometries and the nodal density adaptation using appropriate error estimators are in progress.

Fig. 1 .

 1 Fig. 1. Voronoi diagram.

Fig. 2 .

 2 Fig. 2. Construction of the Sibson shape functions.

Fig. 3 .

 3 Fig. 3. Reproducing discontinuous derivatives using the constrained Voronoi diagram.

Fig. 4 .

 4 Fig. 4. (Color online) Scheme of a rotative electromagnetic machine: 3D and 2D views (units in metric system). In the bottom figure we can see the regions where the current density is non zero (in red and blue). Only one phase among the three phases is supplied.

Fig. 5 .

 5 Fig. 5. (Color online) 2D scheme of a rotative electromagnetic machine partitioned with constrained Voronoi cells.

Fig. 6 .

 6 Fig. 6. (Color online) 2D scheme of a rotative electromagnetic machine: detail.

Fig. 7 .

 7 Fig. 7. (Color online) 2D scheme of a rotative electromagnetic machine: detail.

Fig. 8 .

 8 Fig. 8. (Color online) Vector potential field A(x) for four different rotor positions (θ1 = 0 • , θ2 = 15 • , θ3 = 30 • and θ4 = 45 • ).

Fig. 9 .

 9 Fig. 9. (Color online) Error in the magnetic field by using the FEM, the C-NEM and the mixed FEM-NEM discretizations calculated with a cloud of 400 000 nodes.

Figures 11 -

 11 13 compare the different formulations and discretization techniques when a cloud of N = 27 046, N = 106 804 and N = 419 116 nodes are considered respectively.

Fig. 10 .

 10 Fig. 10. (Color online) Computed torque using the vector and scalar potentials, different discretization techniques and 6755 nodes.

Fig. 11 .Fig. 12 .Fig. 13 .

 111213 Fig. 11. (Color online) Computed torque using the vector and scalar potentials, different discretization techniques and 27 046 nodes.

Fig. 14 .Fig. 15 .

 1415 Fig. 14. (Color online) Convergence analysis for the magnetic field.


Markus Endler
email: endler@inf.puc-rio.br

Jean-Pierre Briot
email: jean-pierre.briot@lip6.fr

Francisco Silva
email: fssilva@lsdi.ufma.br

E Silva

Vitor P De Almeida
email: valmeida@inf.puc-rio.br

Edward H Haeusler

An Approach for Real-time Stream Reasoning for the Internet of Things

Keywords: Internet of Things (IoT), sensors, data streams, complex event processing (CEP), semantic reasoning

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I. INTRODUCTION

Several complex IoT applications, such as of manufacturing industry, transportation systems, and healthcare, put hard real time requirements on the acquisition and processing of sensor data for identifying situations and extracting information from systems' operations and its environment. These typically require on-line processing of continuous streams of sensor data (Data Stream Processing), sensor fusion techniques, pattern recognition and timely and autonomous systems control. There are already many well-established machine learning algorithms for clustering, classification and sequence-analysis.

However, so far however in current IoT systems, sensing and actuation is mostly done at the bare bones data level, whereas many IoT applications demand higher level situation awareness of -and reasoning about -the systems' states and the physical environment where they operate. For this to be possible, it is necessary to have comprehensive semantic models for data stream analysis and actuation. Semantic models are formally defined concepts and relations on which reasoning engines can operate to derive new bits of information and knowledge about a system and its environment. The main problem is that current semantic models (mostly designed for from the Semantic Web) are not suitable for efficient continuous-mode reasoning. Current data analysis for IoT systems is either done off-line or lacks any semantic-based reasoning.

For example, consider a production plant in near future, where several -mobile or stationary -robots operate in a product assembly and interact with each other to hand over parts and tools of the assembly line. Suddenly, there is a short power outage and the assembly line stops for a few seconds, so that some robots go back to their consistent initial states, while others continue their activity (e.g. on battery power) and only stop when their sensors notice that the production line is not advancing. In this case, the robots have to "understand" what has happened, and have to "know" which of the machinery (and robots) are in which state when activity is resumed, as well as the assembly stage of items being produced. And like magic, only a few seconds after energy is back, the robots synchronize with each other, identify missed steps in the assembly process of each item, and resume cooperating again. Such knowledge and understanding is only possible because all robots have not only a semantic model of their own state, but also situational awareness, i.e. a comprehensive model of the production process as a whole and their role in the entire process. The semantic model furthermore describes possible localized and global problems of the entire production process, as well as individual and specific actuation plans for some situations. As all possible situations cannot be represented in a model, the robots have to classify features, combine situational patterns and combine parts of specific action plans. In the aforementioned IoT scenario, the robots would be capable of such fast recovery of the manufacturing process because their situational understanding (i.e. semantic-centered inference/reasoning process) is executed very fast, with almost no delay, as soon as each robot's operational capability is back.

With the goal of finding a suitable semantic model for IoT, this paper proposes a novel approach for real-time symbolic reasoning based on the concepts of Semantic Stream and Fact Stream, as natural extensions of Complex Event Processing (CEP) [START_REF] Luckham | The Power of Events: An Introduction to Complex Event Processing in Distributed Enterprise Systems[END_REF] and RDF (graph-based knowledge model) [START_REF] Candan | Resource description framework: Metadata and its applications[END_REF]. The main advantages of our approach are that: (a) it considers time marks as the link between simple events/pieces of information; (b) the processing of semantic streams can be implemented using existing CEP technology and that (c) it is generic enough to be applied to many Data Stream Management Systems (DSMS). This research is being carried out in the scope of the ESMOCYP research cooperation project between Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Universidade Federal do Maranhão (UFMA) and University of Stuttgart.

The paper is structured as follows. In Section II, we explain the basic concepts of Complex Event Processing (CEP) and list some common approaches for modeling knowledge and performing reasoning. Section III explains the two steps of semantic stream reasoning. In Section IV, we present a scenario to explain how our reasoning process would be performed using temperature and accelerometer sensors embedded into vehicles, houses and in the street. Then we present related work and in Section VI discuss the benefits of our approach and prospects. Section VII then concludes the paper.

II. FUNDAMENTALS A. Complex Event Processing

Complex Event Processing (CEP) [START_REF] Luckham | The Power of Events: An Introduction to Complex Event Processing in Distributed Enterprise Systems[END_REF] provides a rich set of concepts and operators for processing events, which include the CQL-like (Continuous Query Language) [START_REF] Arasu | The CQL continuous query language: Semantic foundations and query execution[END_REF], [START_REF] Etzion | Event Processing in Action[END_REF] queries, rules, primitive functions (aggregation, filtering, transformation, etc.) and production of derived events. A CEP workflow continuously processes incoming events, analyzes and manipulates them, and outputs derived events that are delivered to event consumers (e.g. monitoring applications). These output usually represent notifications about detected situations of interest to the the applications.

The manipulations of events are described by CEP rules, which are Event-Condition-Actions that combine continuous query primitives with context operators (e.g. temporal, logical, quantifiers) on received events, checking for correlations among these events, and generating complex (or composite) events that summarize the correlation of the input events. For example, a split rule takes an input event and creates a set of events, while a filter rule only outputs events that satisfy a given criteria. Rules can also operate on a collection of events, for example, an aggregate rule outputs a single event by executing a function on the grouped events, while a join transformation tries to correlate events from various data streams. Another important concept in CEP is that of sliding time and event windows [START_REF] Etzion | Event Processing in Action[END_REF]. A time window is a temporal context that subdivides the stream of events into intervals, where CEP rules and operators are applied only to the events within each window. CEP supports three sorts of windows, Landmark, Sliding, and Fading., the latter being a sliding window where a decay factor λ is applied to the events according to their age. Most CEP systems have the concept of Event Processing Agents (EPAs), which are software modules that implement one transformation within the event processing workflow. The type of an EPA is defined by the rules it implements, such as filtering, counting or specific event pattern detection.

B. Knowledge Representation and Reasoning Approaches

There are plenty of Semantic Models that represent knowledge about a system and its environment, but almost all of them have problems of scale (i.e. the reasoning has high computational complexity) and thus are not suitable for realtime reasoning. The main semantic approaches are:

• Frame Based Models: A frame is an artificial intelligence data structure used to divide knowledge into substructures by representing "stereotyped situations". They are used in Artificial Intelligence Frame-based representation languages. • Conceptual Graphs: Are a logical formalism that includes classes, relations, individuals and quantifiers. This formalism is based on semantic networks, but it has direct translation to the language of first order predicate logic, from which it takes its semantics. • Description Logic: Are logics serving primarily for formal description of concepts and roles (relations). These logics were created from the attempts to formalize semantic networks and frame based systems. Semantically they are based on predicate logic. • Ontologies: An ontology is a semantic/concept network that contains a body of knowledge describing some domain, typically common sense knowledge relating concepts.

• Semantic Web: RDF, RDFS and OWL: RDF is a framework for representing information about resources in a graph model, where information is represented by triples subject-predicate-object. RDFS extends RDF vocabulary to allow describing taxonomies of classes and properties. It also extends definitions for some of the elements of RDF. For example it sets the domain and range of properties and relates the RDF classes and properties into taxonomies using the RDFS vocabulary. Web Ontology Language (OWL) brings the expressive and reasoning power of Description Logic (DL) to the Semantic Web.

It is divided into two levels: OWL Lite and OWL DL, which differ in their expressive power and the deduction complexity. The problem with OWL Lite and OWL DL is that reasoning is hardly implemented in an efficient way, and it also suffers from lack of scalability.

III. GENERAL IDEA

The general idea of our semantic model and reasoning approach is to define two-level CEP transformations, each of which transforms one event flow/stream into a semantically richer one: at the first stage, we transform the stream of data events (attributes are data items) into a stream of RDF statements, and at the second stage, we transform the stream of RDF-triples into semantically richer facts, i.e combining RDF statements. The details of each step are explained in the following.

A. Mapping Data Events to Semantic Events

Our reasoning approach dictates that each simple event (i.e., an object with attributes) represents an action-based predicate (i.e. the event is the outcome of an action) and has at least one of the other two remaining RDF elements: the subject or the object. If the event has the ID of the subject and the object then we have the complete RDF triple (Subject, Predicate, Object), but otherwise, the missing third RDF element of the triple may be inferred from the shared context (i.e. the temporal and spatial correlation) of both elements, the subject and the object when these are received in separate events. For example, if we consider RDF statement (ball, kicking, in the front-yard), then the event instances represent the predicate kick. It further carries the ID of either the ball (e.g. when the ball carries an accelerometer sensor), or else the ID of the yard (e.g. the GPS-position or the street number of the yard where lawn sensors detect some kicking object). The shared context is defined by the colocation and the synchronicity of the events (detected by sensors on the yard ground and by the sensor in the ball). This contextual correlation is performed by CEP rules called "Context mappers", that analyze the streams of events and match Subjects, Objects and Predicates.

Figure 1 shows how Context mappers analyze each pair of events in the sliding time window (e.g. 60 s.) of Data Event Stream and try to identify common contexts, based on time proximity or any other data attribute.

B. Mapping Semantic Events to Knowledge Facts

The mapping from Semantic Events (i.e. RDF triples) to Facts is achieved by Semantic Event (SEv) rules. These are CEP rules that look to find causality and temporal patterns in several Semantic Event sub-Streams, where each stream comprises the Semantic events of a given context. This "context-specific splitting" is possible in most CEP engines by the concept of a stream partition (a.k.a. context). Then, depending on the SEv rule, it might consume, filter, modify or even insert new RDF triples in some SEv streams, a feature that is supported by CEP. This manipulation is achieved by querying the Knowledge base about all the concepts and relations pertaining to the sub-streams analyzed. For example, the inference might deduce that the kicking ball with a given ID has Bob as its owner, and that the yard where the ball is kicking is the one where Bob lives. By this, the new piece of knowledge may be derived such as someone is kicking Bob's ball on his house's yard. And maybe with the context information Bob has finished his homework, it is possible to deduce -with high probability -that Bob is kicking his ball in his house's yard.

The Knowledge base is organized as nested contexts [START_REF] Cafezeiro | Specifying ubiquitous systems through the algebra of contextualized ontologies[END_REF], which allows a much more efficient checking of concepts and relations when compared to single-layer (or flat) ontologies. For example, the ontology of the Knowledge Base may be organized as following nested contexts.

• Spatial nested contexts: Green Way district ⊃ house at 10 Rodeo Dr. ⊃ its yard ⊃ its lawn; • Temporal nested contexts: Bob's leisure time ⊃ Thursday ⊃ afternoon ⊃ Bob's homework finished; • Containment nested context, such as, Bob's toys ⊃ Balls ⊃ basket ball with ID, etc. Figure 2 shows how Semantic Event rules analyze all RDF triples in the sliding time window (e.g. 180 s.) of sub streams of Semantic Events, trying to find event patterns, filtering, manipulating or adding RDF triples into "their" main context sub-stream or also of sub-streams of semantically related contexts, such as, the front yard and the street in front of the yard.

C. Deriving Situations

Using the Facts of the stream and checking them against the Semantic Graph (Ontology) of the Knowledge base, complex situations may be identified such as Bob is playing basketball in the front yard, but should be notified that a strong storm is approaching his house's yard. Moreover, some of the complex facts may be used for expanding, reinforcing or removing some the knowledge about a subject, an object or a place. For example, after Bob's pen has finishes writing QED on the page with the exercises of his Math's homework notebook, the latter has been closed, and Bob house's main door has been opened and closed, sensing that someone left the house, then the Knowledge Base will be expanded with the following facts: (Bob, finished, Math homework), (Bob, left, house) and (Bob, stepped into, yard).

IV. AN EXAMPLE OF REASONING OVER DATA STREAMS

In this section we show how the aforementioned twophase reasoning could be done with off-the-shelf components and current wireless WPAN technologies, such as Bluetooth Low Energy (BLE). Consider a scenario where smart ambient sensors are everywhere: in houses, offices, public transportation, in the streets and in private cars, and that these smart devices include a temperature and an accelerometer sensor, have a unique UUID and Bluetooth Low Energy interface. Now consider a user, Silva, lives in Rio de Janeiro and carries a smartphone running our Internet of Things middleware ContextNet [START_REF] Endler | ContextNet: Context reasoning and sharing middleware for large-scale pervasive collaboration and social networking[END_REF], [START_REF] Talavera | The Mobile Hub concept: Enabling applications for the internet of mobile things[END_REF]. This middleware uses the Fig. 1. Mapping Data Events to Semantic Events Fig. 2. Mapping Semantic Events to Knowledge Facts smartphone as the bridge between Bluetooth-enabled smart devices/objects/sensors and IoT application servers executing in a cloud. The mobile middleware (Mobile Hub) periodically issues a BLE scan, discovers nearby BLE devices, connects to them, subscribes to the smart device's sensors and writes commands to the smart objects that have some actuator. Figure 3 shows the Mobile Hub with four Texas Instruments SensorTags that have 6 different sensors, including temperature and accelerometer. Assume that it is summertime and that some IoT application needs to know if Silva is in his office, if he is walking on the street or if he is in a bus or car. Whenever Silva's smartphone encounters a BLE smart sensor, it is possible to deduce if he is in an air conditioned space or not, and whether he is in movement or not (due to the smart device's accelerometer). Moreover, if the location of each deployed smart device is previously registered, it is further possible to deduce if Silva is in his office or elsewhere. And this can be deduced even without the use of GPS, either because of its signal is not available (indoors), or because Silva decides to keep it off to save the smartphone's battery.

In this case, it would be possible to deduce the RDF triple (Silva, rides, BusLine435) from the following simpler semantic events in the stream: (Silva's Portable, connectedTo, sensorX), (sensorX, in, BusKKZ8674), (sensorX, Abs(Accelerator) > 10) and (sensorX, temp=20) and from the fact that BusKKZ8674 operates the BusLine 435. Moreover, it would be possible to deduce that Silva is moving in the traffic, but that he is in an air-conditioned bus, which may be very important during Rio's summertime, when outdoor temperature can reach more than 45 Degrees Celsius.

V. RELATED WORK

In an early work, Adi et al. [START_REF] Adi | Semantic Event Model and its Implication on Situation Detection[END_REF] presents abstractions that describe semantic relationships between events, object and tasks. These are defined as generalizations and associations and through attributes that may reference events. Their abstractions are suitable for specification but cannot be computed efficiently. On the other hand, the work [START_REF] Anicic | Stream Reasoning and Complex Event Processing in ETALIS[END_REF] describes a system (ETALIS) that can perform reasoning over streaming events with respect to background knowledge, similar to our Knowledge Base. It implements two languages for specification of event patterns: the rule based ETALIS Language for Events (ELA), and Event Processing SPARQL. ETALIS can evaluate domain knowledge on-the-fly, thereby proving semantic relations among events and reasoning about them. Their semantic relations among events are time-based, but don't have the synchronicity requirement. Another difference is that they do not generate a RDF Stream which they check against a knowledge base. Thus, their inference is much simpler than the one proposed in our project.

Tachmazidis et al. [START_REF] Tachmazidis | Large-scale parallel stratified defeasible reasoning[END_REF] propose a reasoning method over RDF triples based on defeasible logic (i.e., a non-monotonic logic) which can be implemented in a massively parallel way. They used Hadoop, an open-source implementation of the MapReduce paradigm, and a stratified rule set for a more efficient processing of the knowledge base. Unlike our proposal, they do not handle Stream Processing and do not apply their method to reasoning for time-critical systems, such as CPS. Moreover, their choice for defeasible logic limits the sorts of knowledge that can be inferred by their system, as opposed to temporal logic, which shall have render highly paralleliizable implementations.

The following projects CityPulse [START_REF] Giatsoglou | CityPulse: A platform prototype for smart city social data mining[END_REF], Star-City [START_REF] Lécué | Star-City: semantic traffic analytics and reasoning for city[END_REF] and FIESTA-IoT [START_REF] Al-Hazmi | Towards semantic monitoring data collection and representation in federated infrastructures[END_REF] also present research toward the use of Semantic Stream reasoning. All of these projects use the knowledge base in order to deduce new context/facts. Also, they use a single-layer (or flat) ontology model, which differs from our ontology model that is organized as nested contexts. Moreover, none of these projects focus on the problem of delivering real-time reasoning.

The FIESTA-IoT project [START_REF] Al-Hazmi | Towards semantic monitoring data collection and representation in federated infrastructures[END_REF] integrates several other projects and one of them is the CityPulse project [START_REF] Giatsoglou | CityPulse: A platform prototype for smart city social data mining[END_REF]. The main goal of these projects is to achieve semantic interoperability at different levels (hardware, data, model, query, reasoning and application levels). The StarCity project has a similar idea, but it is aimed at using semantics to provide interoperability at the data level.

On the other hand, the work by Teymourian et al. [START_REF] Teymourian | Fusion of background knowledge and streams of events[END_REF] has the same focus as our work. They use a similar idea and combine the use of SCEP rules (Semantic Web plus CEP) with a semantic knowledge base to deliver real-time reasoning. The difference is that our work uses an ontology model organized as nested context to represent context information, rather then a flat ontology model. As a result, it could be more efficient on query processing, because when we execute a query, the query will be processed only using a sub set of the knowledge base (a partition of the knowledge base). Furthermore, other difference is that we purpose to insert new SCEP rules on-the-fly, based on new facts generated by the reasoning over the knowledge base. Consequently, it is a more efficient approach for the application to adapt to different situations. For example, in a monitoring application, we only need a CEP rule that triggers an action based on an altitude situation only if the monitored person is in a high altitude, until then this rule does not need to be there.

VI. DISCUSSION

Combining symbolic reasoning based on Ontologies with Complex Event Processing has several advantages. Firstly, it allows to leverage CEP efficient processing of dense flows of simple events, not just over raw sensor events but also over RDF triples. Secondly, CEP's ability to produce complex events is also necessary for the iterative generation of higher level information from lower level bits of information.

On the other hand, while CEP is appropriate for processing data that is carried by the incoming events, it is incapable of detecting domain-specific relationships between events that are produced by distinct entities/objects that apparently have no relation with each other, or when this relationship cannot be directly encoded by the (meta-)information carried by the events. Symbolic reasoning using ontologies, on the other hand, can very well model these "indirect" relationships among the monitored entities and/or their corresponding events. And hence, by using the results of a query over a domain-specific ontology during a CEP-based continuous processing, it becomes possible to generate new sorts of events (i.e., fact events), which are produced independently by the Semantic Event reasoners in response to the consumption of some RDF triples. These Fact events, which in some sense embody some semantic knowledge that was forked off the knowledge base, can in turn be further processed by other CEP engines and may be used to predict events that actually did not yet happen, but which are a natural consequence of initial events that have been detected by CEP.

This makes us consider the Semantic Web reasoners as a special kind of CEP engines, which have access to knowledge base, consume RDF events and eventually produce fact events that are passed onto other CEP engines in the Event Processing Network. See Figure 1 for a general architecture.

VII. CONCLUSION AND FUTURE WORK

This paper presented a real-time reasoning approach based on semantic events and fact streams for IoT systems. The reasoning approach is based on the assumptions that all objects, people, buildings, places, vehicles, environments, etc. will have many embedded tiny sensors that will emit simple events whenever some action is performed with/to it by an actor and that each event will carry the items' unique UUID and an accurate time-stamp. By enforcing the restriction that predicates in a RDF triple must be action-based, such as kick, put, grab, etc., rather than state-based, such as has, is, belongs to, etc., we are of course limiting the amount of information that the data/event streams are capable to express. However, we believe that the action-based predicates are the really important ones for reasoning in IoT applications. All the state predicates, on the other hand, should instead be represented by the nested context-based ontology in the Knowledge Base.

We are aware that this is only a first and initial step towards adding semantics to real-time reasoning over data streams and that much more theoretical and practical research is required to validate our approach, evaluate it under a broader perspective and show its feasibility for large-scale and distributed IoT applications. However, we are confident that it is a promising first step. As next steps, we will develop simple Context Mappers and Semantic Event Rules using Esper's EPL and deploy them on the mobile component Mobile Hub and on cloud-based Complex Event Processing engines in the cloud of our IoT middleware. In parallel, we will model a simple scenario and the main entities and their relationships, as described in Section IV and represent it as nested contexts.

Last, we have already started investigating various machine learning techniques (autoencoders, k-means, Markov chains, recurrent networks and inductive logic programming) to automate features extraction, clustering, temporal patterns induction and knowledge induction, as ways to automatically extract useful knowledge at various stages of our architecture. One important issue is the dynamicity of the data produced, i.e. we need to find good trade-offs depending on the usage, the nature of the data and the computing and communicating resources available, between the demand for higher level knowledge, the cost for extracting it and the risk of it being obsolete because of the constant stream of data. This preliminary study will be presented in another paper to appear.

Fig. 3 .

 3 Fig. 3. ContextNet Mobile Hub with four SensorTags

ACKNOWLEDGEMENT

Our ESMOCYP (Efficient Semantic MOdels and Faulttolerant Middleware for CYber-Physical Systems) Project is being supported by CAPES (Brazil) and DAAD (Germany) cooperation program PROBRAL (Process No 8148/2015-05) and by a CAPES PVE fellowship to J.-P. Briot. We would like to thank the researchers and students from the University of Stuttgart for fruitful discussions.