Wireless Sensor Network Analysis Using the Finite Element Boundary Integral Numerical Technique

Juliano Fujioka Mologni ESSS – Engineering Simulation & Scientific Software

São Paulo, Brazil juliano.mologni@esss.com.br

Kaku Saito Petrobras

Rio de Janeiro, Brazil kaku@petrobras.com.br

Abstract—Wireless sensor technology is becoming a practical replacement to the conventional wired sensors primarily due to the easiness of implementation and cost reduction. In spite of that, process industries are not still using such kind of solution in large scale due to lack of field-proven data related to robustness and reliability of the communication link. A wireless sensor network (WSN) consists of spatially distributed independent sensors that are connected to a wired gateway. The present study shows a full wave simulation of a WSN installed in a subsection of a refinery coupled to a circuit simulator that generates electric wave forms according to IEEE 802.11 specification. The finite element boundary integral (FEBI) approach, which is now commercially available in Ansys HFSS, was used to solve the electromagnetic model. The advantages of FEBI for solving very large field problems are presented and the simulation results were compared to the finite element method (FEM) and the method of moments (MoM). The integration of a 3D field solver and a circuit simulator enables the calculation of radiation patterns, electric field plots, bit error rate, constellation plots while incorporating the actual transmitter and receiver antennas and the electrical schematic of the sensors and gateway. The purpose of this simulation is to investigate the limitation of the technology, help engineers to create best practices for WSN installation and to give an approach for the optimization of gateways positioning.

Keywords-wireless sensor network; very large field models; finite element boundary integral; method of moments.

I. INTRODUCTION The advancement on wireless network technologies has

enabled the use of sensors on industrial environments, which in comparison to wired technology, presents many benefits including reduction of cost and installation. The first sensor networks used one simple twisted shielded–pair (TSP) as a communication medium for each sensor. Afterward, sensor networks evolved to multidrop buses (e.g., Fieldbus). Today wireless sensor network (WSN) is widely presented as a new trend and it is progressively replacing wired technology. In a WSN, the sensors can measures physical or chemical conditions such as pressure, temperature, vibration, displacement or pollutants [1-3]. An industrial WSN is comprised of "nodes" whereas each node can be connected to one or several sensors. Each node has a radio transceiver with an antenna, an electronic A/D circuit to read sensor information and an energy source. Every node is capable to communicate to

each other, and at least one of them must be connected to a gateway that needs to be wired and connected to the local area network (LAN). The layout pattern of interconnections of the various nodes and gateways is called network topology. Usual topologies for WSN are star, ring, mesh, line, tree and advanced multi-hop wireless mesh network [4]. Fig. 1 displays a fully connected network topology with one gateway and three nodes, where each node is the sensor itself. An important engineering approach during the design of a WSN is to define an optimal position and appropriate quantity of gateways inside a plant with many equipments and piping in order to reduce signal attenuation and distortion.

In order to evaluate the communication performance of a WSN, all the objects of the surrounding environment where the sensors are installed needs to be simulated with a full wave solver. The electrical size of the oil and gas refinery model is very large. Several numerical techniques are available today including the finite element method (FEM) and the method of moments (MoM). FEM requires an air box to be modeled covering all the geometries. The size of this air box depends on the frequency since the outer surfaces must be placed at least λ/4 (where λ is the wavelength) from radiating elements. FEM is capable to solve large field problems using the domain

Fig. 1. Concept of a wireless sensor network.

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decomposition method (DDM), where the whole problem is sub divided into smaller sub sections and each one of them is sent to different computers connected to a LAN and solved in a parallel way. On the other hand, the MoM is very efficient for electrically large models having only conductive materials. If any polymeric material is inserted into the model, the efficiency of the MoM solver is considerably reduced. Each technique has exclusive features and in order to take advantages of both FEM and MoM a recent hybrid technique called finite element boundary integral (FEBI) was developed. FEBI uses as boundary integral a MoM solution for Sommerfeld’s radiation condition as an interface boundary for the FEM solution. Thus, an exact mathematical and theoretical calculation of the far field radiation condition is satisfied. This technique presents a number of advantages for very large problems, such as the ability to handle complex geometries and dielectric materials using FEM solvers without having to simulate air regions, which is now calculated through MoM integral equations (IE) methods. In order to demonstrate the accuracy and efficiency of FEBI, a comparison with FEM and MoM is presented. Also, a full dynamic link between Ansys HFSS and a high level circuit simulator, Ansys Designer, enable a full WSN analysis yielding bit error rate (BER) and constellation plots using the extracted electromagnetic parameters showing the state of the art in terms of electromagnetic and circuit coupled simulation.

II. NUMERICAL TECHNIQUES In order to verify the accuracy of FEBI results, a 3D sensor model was simulated using FEM, MoM and FEBI. Reflection coefficient and radiation patterns were calculated using the three mathematical approaches and the results were compared.

A. Finite Element Method Ansys HFSS uses traditionally the FEM where a 3D model

is subdivided into a finite number of smaller subsections called elements. The elements in HFSS are tetrahedra and the whole group of tetrahedra is called a mesh. A solution is found for the fields within the finite elements, and these fields are interrelated so that Maxwell’s equations are satisfied over inter-element boundaries, yielding a field solution for the original 3D model. Once the field solution has been solved, the generalized scattering matrix (S-Matrix) solution can be calculated. HFSS solves (1), also known as wave equation, for each tetrahedra element on the model [5-8]:

(1)

where k = ω/c is the wave number of free-space; c is the speed of light; ω=2πf is the angular frequency; μr(x,y) is the complex relative permeability and εr(x,y) is the complex relative permittivity. By solving (1), the electric field mode pattern Em (x,y) and the propagation constant γm are both calculated for all the modes specified. The magnetic field is calculated according to (2):

(2)

Equation 2 implies that HFSS solve equations in terms of electric and magnetic fields and not voltages and currents. FEM requires an air box covering the geometries so (1) and (2) are used for radiation patterns calculation. The size of this air box depends on the frequency and needs to be placed at least λ/4 when using an absorbing boundary condition (ABC) and it is displayed in fig. 2 where the FEM model is applied with an air box surrounding the sensor.

B. Method of Moments The MoM (also known as IE) does not require an air region

to be modeled (see Fig. 2). The electric field is obtained from the induced surface current using (3).

(3)

where J(r´) is the unknown surface current density on the surface S of the model (r S) and E(r) is the incident electric field. Harmonic time dependence of the form j�μ is suppressed in frequency domain and k is the wave number. One can eliminate the dependence of E(r) by enforcing the boundary conditions on the tangential electric field leading to:

(4) where n(r) is the outward unit normal from the surface S. Rewriting the above equation in terms of the known incident electric field E(r) leads to:

(5)

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Equation 5 is known as the electric field integral equation (EFIE) for a perfectly conducting surface, where ∇S is the surface divergence [9]. Once solved for the unknown current J(r) the radiated field on any spatial coordinate can be calculated using Sommerfeld’s radiation condition at infinity.

C. Finite Element Boundary Integral When using FEM, an air box size needs to be modeled

surrounding the geometry. When using IE, no air box is needed, on the other hand, solver efficiency decreases when the model includes non conductive materials. To take advantage of IE and FEM, FEBI was developed. The air box in FEBI is very conformal to the geometry and its distance to the sensor model can be as close as possible as shown in Fig.2. Hence, the air region is dramatically reduced leading to a faster solution. FEBI accuracy can be observed in Fig. 3 where the S11 of the sensor shows practically no difference for the three methods. The inset in Fig. 3 shows the gain far field pattern of the antenna which is plotted in details in Fig. 4 also demonstrating a very good agreement. FEBI solver considers two different domains for a single problem. It starts by partitioning the original problem domain Ω into two non-

overlapping sub-domains Ω1 and Ω2, as shown in Fig. 5 [10].

(6)

The interface between Ω1 and Ω2 is denoted as �Ω1 in the FEM domain and �Ω2 in the IE domain. This distinction is necessary because the formulation allows non conformal coupling between two domain s. This means that the mesh and the solver can be treated independently for each domain. Based on DDM, the final system matrix can be written as [10]:

(7)

where AFE and ABI represent the system matrices of FEM and BI domains, respectively, and C is the coupling matrix between them. The coupling is calculated based only on the electric and magnetic currents at the interface; hence, it is extremely sparse. The solution of (7) is accomplished iteratively via splitting

(8)

where n is the total number of domains. By simplifying (8) we have:

(9)

The advantages of using DDM are apparent from (9). Both FEM and BI domains are decoupled so parallelization becomes trivial. The above mathematical procedure shows that BI can be used as an exact termination condition in FEM. Due to the implementation’s modularity, state-of-the-art FEM and IE solvers are easily employed in HFSS in a hybrid and robust way.

Fig. 3. Sensor reflection coefficient using FEM, MoM and FEBI. Inset shows the polar plot of the gain far field pattern.

Fig. 4. Far field pattern at =90o results when using FEM, MoM and FEBI.

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Fig. 5. Domain decomposition of the full model into FEM and IE domains.

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III. FULL WAVE ELECTROMAGNETIC MODEL In order to identify undesirable effects into the transmission

signal such as multipath, free space attenuation and phase distortion, all geometries surrounding the WSN needs to be modeled [11-14]. The complete geometry is shown in fig. 6 where is possible to observe that the gateway has a direct line of sight to sensors 1 and 2 but a metallic floor is obstructing the communication with sensor 3. The inset shows details of sensor 2 where a conformal green air box is visible and used as a boundary condition for FEBI simulation. Wireless sensors operate usually at 2.4GHz and the model needs to be solved in this frequency. If FEM is used, the volume of the air box required to cover the model is equal to (10.74m)(8.1m)(8.1m) = 704.6m3 which leads to an electrical size of 36000λ3. If the air box is replaced by a conformal air box, in the case of FEBI, the electrical size of the model is reduced down to 3680λ3. The total amount of RAM required to solve this problem was 452GB using FEM (with DDM) and 215GB using FEBI. Fig. 7 shows the electric field plot due to sensor 3 excitation and the inset details a region closer to the sensor. It is possible to observe that the fields are clearly blocked by the metallic parts of the structure.

IV. HIGH LEVEL SYSTEM DESIGN There is no official standard for WSN. IEEE focuses on the

physical and MAC layers, the Internet Engineering Task Force works on layers 3 and above. Nevertheless, there are some predominant standards used by the majority of the companies including ISA100, IEEE 1451, ZigBee / 802.15.4 and IEEE 802.11. We have based our study on the IEEE 802.11 protocol.

Ansys Designer is a circuit and system simulator that was used to create the full IEEE 802.11 system due to its complete library which already includes a complete schematic with the components already set for IEEE 802.11 (e.g. random bit generator, modulators, coders, filters and amplifiers). Some characteristics of IEEE 802.11 are fixed 22MHz channels, data rates from 6-54 Mbps depending on one of the modulations: Binary Phase Shift Keying (BPSK), Quadrature Amplitude Modulation (QAM), QAM-16 or QAM-64. Low sensitivity to time synchronization errors, 64 sub-carriers, coding rates of 1/2, 2/3 or 3/, channel correction through the use of pilot channels for magnitude and phase and cyclic prefix (CP) which

Fig. 6. Full model with one gateway and three sensors.

Fig. 7. Cross sectional electric field plot due to sensor 3 excitation.

Fig. 8. Constellations plot for the transmission between the gateway and sensor 3 for an SNR=30. Modulations: a)BPSK; b)QAM; c)QAM-16 and d)QAM-64.

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mitigates synchronization and multi-path are also features of IEEE 802.11.

The full wave electromagnetic model can be dynamically liked to the system design, meaning that the circuit simulator will take into consideration the S-matrix already solved by HFSS. The communication channel can then be fully analyzed using realistic waveforms and all the electromagnetic effects. Fig. 8 shows constellation plots for the transmission of sensor 3 and the gateway taking into consideration an ideal channel and the HFSS model. An ideal channel, where a numerical additive white Gaussian noise (AWGN) is mathematically inserted into the system is modeled for comparisons purposes. A phase distortion, frequently caused by multipath, is clearly observed in the red constellation only, where the 3D model is considered. Also, the red constellation plot shows that there is a higher probability of intersymbol interference (ISI) to occur, which accurately represents the real world environment. Bit error rate (BER) curves are calculated for BPSK, QAM, QAM-16 and QAM-64 modulations as a function of the signal to noise ratio (SNR) and are shown in fig. 9. For QAM-64 the BER is limited to 1E-3 no matter how high is the SNR. This is due to the obstruction of a metallic floor in the direct line of sight between sensor 3 and the gateway. In this case, the use of a more robust modulation (QAM-16) lowering the transmission rate is advised. This is effect is clearly visible in fig.9d where the red symbols are closer leading to a higher probability of ISI.

SUMMARY A complete simulation integrating the IEEE 802.11 standard

to a full wave 3D electromagnetic wireless sensor network is addressed in details. Multipath, wave reflections and scaterring parameters can only be obtained through the use of a full wave simulation tool, in this case Ansys HFSS, and these parameters are dynamically coupled to the system design in order to characterize the communication channel of the gateway and one of the sensors in terms of BER as a function of SNR and constellation plots. FEBI technique was employed and shows a significant reduction in computational effort when compared to FEM but at the same time keeping the accuracy of the results. The system analysis shows that robust modulations such as BPSK, QAM and QAM-16 requires lower SNR to reach

reasonable BER values (in the order of 1E-4), as expected. When using QAM-64 the performance of the communication channel between sensor 3 and the gateway is reduced and the BER is limited to 1E-3 regardless the amplification of the signal. The insight detailed in this paper is only achievable trough a dynamic link between a 3D full wave electromagnetic simulation and a system design tool. The concepts presented in this paper can also be applied to any WSN geometric model also incorporating numerous specifications such as ISA100, IEEE 1451, ZigBee / 802.15.4 among many others.

The result shows a first approach to investigate how the simulation using HFSS may be valuable for the definition of the gateway position relative to other sensors in the plant considering the geometry of surrounding equipments. The simulation also shows if the computational effort is reasonable to make possible practical works in the future.

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Fig. 9. BER curves for the transmission between the gateway and sensor 3 asa function of the SNR for BPSK, QAM, QAM-16 and QAM-64 modulations.

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