Student Member, IEEE Moussa Kafal

Senior Member, IEEE Andrea Cozza

Member, IEEE Lionel Pichon

Locating Multiple Soft Faults in Wire Networks Using An Alternative DORT Implementation

Keywords: Fault location, soft faults, multiple faults, complex wire networks, DORT imaging

Decomposition of the time reversal operator (DORT) was recently applied to the problem of detection and location of soft faults in wire networks and proved effectual when dealing with a single fault, even in the case of complex network configurations. In this paper, the case of location of multiple faults is addressed, first proving that the standard DORT formulation does not allow to take a clear decision about the individual position of each fault. An alternative version of the DORT, based on an updating procedure, is presented and demonstrated to enable accurate and selective location of multiple soft faults. The proposed procedure is also shown to allow estimating the reflection coefficient of each fault, thus giving access to their severity.

I. INTRODUCTION

Electrical cables are infrastructures widely used for distributing power and communication signals throughout nearly all modern systems [START_REF] Furse | Down to the wire [aircraft wiring[END_REF]. Their cumulative length can exceed hundreds of kilometers in many networks employed in work and transportation, e.g., 5000 km in power plants, 500 km in large aircrafts and 4 km in modern cars [START_REF] Auzanneau | Wire troubleshooting and diagnosis: Review and perspectives[END_REF].

As any system, cable networks are subject to wearing and accidents that can make them nonoperational. Open and short circuits occurring along a cable are the most well-known sources of faults; they are usually referred to as hard faults, as they stop signals from propagating further along a cable. As opposed to them, soft faults [START_REF] Griffiths | The invisible fray: A critical analysis of the use of reflectometry for fray location[END_REF] feature minor alterations, such as insulation wearing; in this case, signals can still propagate along a cable, and a network is in no danger of breaking down. Yet, a partial wearing may evolve into a stronger degradation, eventually leading to a hard fault [START_REF] Shin | Application of time-frequency domain reflectometry for detection and localization of a fault on a coaxial cable[END_REF]. Hence, developing robust procedures for the identification and location of soft faults becomes important, as an early-warning approach for ensuring the safe operation of a critical infrastructure. As a matter of fact, studies conducted on cables showed that 30% to 50% of detected wiring faults are considered to be soft.

Despite the fact that several techniques have been introduced throughout the last decades to detect and locate faults, reflectometry-based methods are still the most widely used, without having much changed in their scope and definition [START_REF] Pan | Time-domain reflectometry using arbitrary incident waveforms[END_REF]. The state-of-art of reflectometry for wiring fault detection can be categorized into two main families: Time Domain Reflectometry (TDR) [START_REF]Time-domain Reflectometry for Monitoring Cable Changes: Feasibility Study[END_REF], [START_REF] Nemarich | Time domain reflectometry liquid level sensors[END_REF], [START_REF] Paulter | An assessment on the accuracy of time-domain reflectometry for measuring the characteristic impedance of transmission lines[END_REF] and Frequency Domain Reflectometry [START_REF] Furse | Frequency-domain reflectometry for on-board testing of aging aircraft wiring[END_REF], [START_REF] Chung | Application of phase detection frequency domain reflectometry for locating faults in an f-18 flight control harness[END_REF]. Basically, both methodologies are based on an appropriate analysis of the reference and the reflected M. Kafal, A. Cozza and L. Pichon are with GeePs, UMR 8507, CEN-TRALESUPELEC -Univ Paris-Sud -CNRS, UPMC, 3 rue Joliot-Curie, 91192 Gif-sur-Yvette, France. Contact e-mail: andrea.cozza@ieee.org signal either in the time or frequency domain. Besides, they are known to perform well when dealing with hard faults, which result in large reflections, but start failing with soft ones. In fact, soft faults are characterized by a very weak reflectivity, which produce echoes that can pass unnoticed compared to those caused, e.g., by junctions within a network under test (NUT) [START_REF] Griffiths | The invisible fray: A critical analysis of the use of reflectometry for fray location[END_REF], [START_REF] Shin | Application of time-frequency domain reflectometry for detection and localization of a fault on a coaxial cable[END_REF], [START_REF] Furse | A critical comparison of reflectometry methods for location of wiring faults[END_REF], [START_REF] Zhang | Inverse scattering for soft fault diagnosis in electric transmission lines[END_REF]. A recent work [START_REF] Boudjefdjouf | Diagnosis of multiple wiring faults using time-domain reflectometry and teaching-learning-based optimization[END_REF] aiming at locating multiple faults falls in this same group, as it mainly works for relatively hard faults.

Based on a radically different concept, the DORT method, standing for decomposition of the time reversal operator, was originally developed in acoustics as a remote-sensing technique for the detection and location of scatterers within homogeneous media [START_REF] Prada | Eigenmodes of the time reversal operator: A solution to selective focusing in multiple-target media[END_REF]. Basically, it allows defining signals bound to focus over one of the scatterers, thus providing a visual method for inferring the position of a previously unknown scatterer. This idea was first transposed to guidedwave propagation, e.g., wire networks, in [START_REF] Abboud | A noniterative method for locating soft faults in complex wire networks[END_REF] and shown to precisely detect and locate single soft faults even within complex network configurations, involving a number of junctions. The proposed method presented major advantages compared to TDR-based techniques where the eventual presence of a fault is inferred from the measured scattering matrix of the NUT, rather than from echoes, as done in TDR methods. Besides, it has allowed the simultaneous analysis of all multiport measured data which are jointly processed to locate the actual position of the fault [START_REF] Abboud | A noniterative method for locating soft faults in complex wire networks[END_REF], as opposed to methods exploiting multiport data separately [START_REF] Lelong | Distributed reflectometry method for wire fault location using selective average[END_REF].

While passing from a single scatterer to several is trivial with waves propagating in homogeneous media, we show in this paper that in the case of wire networks this is no longer true. Multiple faults cannot be resolved separately, because of their strong coupling via guided propagation. An alternative formulation of the DORT based on an updating scheme is proposed and proved to allow selective focusing and location of multiple soft faults.

The paper starts by recalling the methodology of the standard DORT technique, hereafter referred to as SDORT, in locating single soft faults, in particular the main steps followed in measuring the NUT's response and the subsequent faultdetection procedure. After that, section III will be devoted to explaining the limitations SDORT faces when multiple faults occur. This is followed by presenting a solution characterized by separating the measured NUT's data allowing locating each fault selectively, thanks to a mathematical model discussed in section IV-A. The new procedure is detailed in section IV-B and will be referred to as the enhanced DORT (EDORT) method. As a by product, it allows retrieving the faults' intensities, as demonstrated in section IV-C, an important feature when needing to decide if a fault should be checked more closely by a human operator or just in order to monitor how the fault evolves in time. Experimental results dealing with coaxial cables are provided to validate the proposed method.

II. STANDARD DORT FOR FAULT LOCATION

The basic operations and properties involved in the use of the SDORT technique in fault location are here recalled; the interested reader should refer to [START_REF] Abboud | A noniterative method for locating soft faults in complex wire networks[END_REF] for more details. All subsequent quantities are functions of the frequency, except where otherwise indicated, hence the frequency variable will be dropped.

When dealing with an NUT displaying N testing ports, its scattering matrix S allows a direct computation of the output signals generated by any testing signal applied to one or more of its N testing ports. As such, SDORT applied to fault detection is based on the availability of this matrix, which can be measured, as discussed in sec. V-A, by means of a vector network analyzer (VNA). As it is often the case when dealing with soft faults, the procedure described in [START_REF] Abboud | A noniterative method for locating soft faults in complex wire networks[END_REF] requires a baselining approach, i.e., taking the difference between the response of the NUT, containing an eventual fault, and a reference response of a healthy version of it [3] [17]. This operation ideally removes the spurious echoes generated by impedance discontinuities like junctions, leaving only those echoes initially generated by the interaction between the testing signals and the faults. For the case of SDORT, the NUT response after baselining will be S = S f -S h , with S f and S h being the scattering matrices of the (eventually) faulty NUT and its healthy version, respectively.

Following the properties of the SDORT in free-space propagation, sets of input signals can be defined, such that when injected through the testing ports they will lead to waves focusing onto each fault. These signals can be computed by solving the eigenproblem SS H U n = λ n U n , where H is the Hermitian transpose. The number n < N of most significant eigenvalues hints at the number of potential faults found in the NUT; if the properties proved for SDORT [START_REF] Prada | Eigenmodes of the time reversal operator: A solution to selective focusing in multiple-target media[END_REF] hold also for NUTs, then each fault could be spotted by monitoring the propagation of input signals whose Fourier spectra would be defined by the scalar components of each eigenvector U n . This operation can be carried out by means of a numerical simulator for transmission lines, modelling the layout of the healthy NUT; the fault's position would be found by looking for maximal energy focusing. To this end, the total energy E(x) = v 2 (x, t)dt can be computed from the voltage v(x, t) observed throughout the numerical model of the NUT.

The theoretical separability of each fault response would also imply the possibility of inferring their respective severity, with clear practical implications in early-wake monitoring of a wire network. The discussions in the next section prove that this feature is not possible with SDORT; this notwithstanding, the procedure proposed in section IV will be shown to reinstate this property.

III. SDORT PERFORMANCE WITH MULTIPLE FAULTS

The limitations SDORT faced in case of multiple faults can be easily exposed by means of numerical simulations. Those shown in this paper are based on the use of an in-house code that solves networks of cables, connected by means of junctions and lumped loads; standard transmission-line theory is used to this end [START_REF] Paul | Analysis of multiconductor transmission lines[END_REF].

Numerical simulations were carried out for the NUTs presented in Fig. 1, where two different layouts were considered: a single-branch and a single-junction structure. Both involve uniform two-wire transmission lines, with the two conductors 10 mm apart, each with a 1 mm radius and with a 0.5 mm dielectric coating of relative permittivity r = 3.

In order to identify unambiguously a portion of the NUT of interest in our discussions, a number is assigned to each of the terminations serving as testing ports. We set number (1) as the origin, and distances will be measured with respect to it, and consequently all graphs will be plotted accordingly, with this distance as the independent variable. We will be interested in observing signals propagating along portions of the NUT; for the sake of simplicity these propagation paths will originate from port (1), the origin, and end at another testing port. A path can therefore be identified by referring to the far-end testing port. As an example, in Fig. 1(b), the third path is the part of the network linking the origin to the testing port numbered (3). The lengths of the wires composing the paths are chosen in accordance to the lengths of cables available for the experimental setup discussed in sec. V. Soft faults are introduced in the numerical models by removing a portion of the coating of the lines, which can model a potential pressure on the wires or a partial erosion of the coating, while in our experiments we relied on coaxial cables, where coating removal is not an option; in this case we rather crushed a portion of the cable in order to introduce a local impedance discontinuity (see sec. V).

Two such faults are introduced, where the strongest is obtained by reducing the thickness of the coating from 0.5 mm to 0.05 mm over a length of 2 cm thus corresponding to a 0.45 mm coating removal, whereas the weaker one is associated to a mere 0.05 mm coating removal. The effect of these local modifications in the coating can be quantitatively assessed by computing the relative change

R f = |Z f -Z c | / |Z c |
measuring the deviation from the characteristic impedance Z c of the line, after introducing a fault of impedance Z f .

The first issue with SDORT is the lack of selective focusing. This can be demonstrated by tracking the voltage propagation both in time and space in the NUT, hereafter referred to as ZT diagram, generated by applying to the testing ports the eigenvector corresponding to the most significant eigenvalue λ n . According to [START_REF] Prada | Eigenmodes of the time reversal operator: A solution to selective focusing in multiple-target media[END_REF], there should be a single-spot appearing at the position of the most significant fault. In fact, for the NUT in Fig. 1(a), the ZT diagram of Fig. 2 shows two focal spots where the faults were introduced.

Even though the faults are not excited individually, it could be argued that they are still visible and rightly located. That is where the second issue comes in: the ratio of the focal-spot intensities does not appear to represent the ratio of the fault severities. This means that it is not possible to use the focalspot intensity as a proxy of a fault severity. Moreover, having focal-spot intensities spanning a large dynamical range can hinder the detection of faults associated to weaker focusing, particularly in the case of noisy signals. As an example, consider the NUT in Fig. 1(a) with two faults of different intensities, placed at 1.15 m and 2 m, respectively. When choosing a weaker second fault (R f 2 goes from 4% down to 0.5%), while keeping the first one fixed (R f 1 = 6 %), the energy diagrams in Fig. 3 show a vanishingly low focusing over the weaker fault, resulting into a less likely identification. Moreover, the reduction in the energy at the second-fault position does not seem to be related to the reduction in the fault intensity.

One of the most interesting properties of SDORT was its effectiveness in locating single faults within complex NUTs, composed of several junctions and branches [START_REF] Abboud | A noniterative method for locating soft faults in complex wire networks[END_REF]. This property is no longer guaranteed in the case of multiple faults. In order to better illustrate this point, the NUTs shown in Fig. 1 were considered, each with two faults where R f 1 = 6% and R f 2 = 3%, found at the same distances with respect to the reference port (1 m for the stronger soft fault and 1.5 m for the weaker one). The objective is to assess how a changing NUT complexity impacts the focusing capabilities of SDORT. The energy diagrams in Fig. 4 show that a more complex NUT introduces an ambiguity on the weaker fault's position, due to artifacts that appear as a result of junction-related echoes, in spite of the baselining procedure. This situation is expected to worsen with NUT's composed of several junctions and branches.

IV. ALTERNATIVE ITERATIVE FORMULATION

The issues with SDORT in case of multiple faults can be ascribed to the coupling existing between the faults' responses, which in turn leads to simultaneous multiple focusing. If the signals relative to each fault could be separated, then individual focusing would be available again, making the detection and location of the faults more robust and precise. Moreover, without separation of the fault responses, it is not possible to assess their severity, a major drawback in an early-warning technique.

A. Separation of a scattering matrix

The key element in addressing separately each fault is to understand how the response an NUT featuring only one fault at the time is related to the response of the multiple-fault case. In order to get a better insight into this problem, we start by considering the simple NUT in Fig. 5(a) consisting of a single-branch with two soft faults with reflection coefficients Γ 1 and Γ 2 , respectively. Neglecting multiple reflections, its transmission coefficient can be expressed as

S f 21 = (1 + Γ 1 )(1 + Γ 2 )e -jβL , (1)
where β is the propagation constant and L the total length of the line, whereas for the case of the healthy configuration

S h 21 = e -jβL . (2)
Hence the baselined difference scattering transmission coefficient is

S 21 = S f 21 -S h 21 = (Γ 1 + Γ 2 )e -jβL + (Γ 1 Γ 2 )e -jβL . (3)
Since we are dealing with soft faults,

|Γ i | 1, i = 1, 2, so that Γ 1 Γ 2 Γ 1 + Γ 2
, allows in turn to simplify the S 21 expression to

S 21 ≈ (Γ 1 + Γ 2 )e -jβL = S 1 21 + S 2 21 . (4)
where S i 21 is the scattering transmission coefficient of the single-fault NUT with only the ith fault appearing, as in Figs. 5(b-c). The level of accuracy in neglecting the product Γ 1 Γ 2 can be better grasped by taking the special case

Γ 1 = Γ 2 = Γ, yielding |Γ 1 Γ 2 |/|Γ 1 + Γ 2 | = |Γ|/2. Typical values of |Γ|
shown in sec. V are smaller than 0.01, implying a residual error of less than 1 %.

The same result can be found for the remaining terms in the baselined scattering matrix, i.e.,

S G ≈ N f i=1 S i . ( 5
)
where N f is the number of faults in the NUT. This result is in general valid for any layout, as long as the faults can be regarded as soft, a property equivalent to the first Born approximation widely used in wave scattering [START_REF] Born | Principles of optics: electromagnetic theory of propagation, interference and diffraction of light[END_REF].

The superposition ( 5) can be exploited in order to estimate the scattering matrix of the ith fault from the measured S G matrix. To this end, let us recall that the SDORT does provide a very good estimate of the position of the faults. In particular, the position of strongest focusing typically coincides with the most severe fault. This observation can serve as a guess of the position of one of the faults. It is therefore possible to insert a dummy fault into the healthy-NUT model, previously used for baselining: the nature of this dummy fault is not important, as discussed later on. The scattering matrix of this modified NUT will be referred to as S i d , for the ith fault. This would be the response of the NUT if only one fault was present. The idea is now to remove the contribution of S i d from S G , in order to be left with a scattering matrix containing only the remaining N f -1 faults.

To this end, we can exploit a fundamental property of SDORT: in case of a single-fault, the scattering matrix has only one eigenvalue. Since each S i matrix is symmetric, it can be expressed exactly as [START_REF] Horn | Matrix analysis[END_REF].

S i = ν i w i w T i , (6)
where ν i is the only non-zero eigenvalue of S i and w i the associated eigenvector, with T is the transpose operator. As shown in [START_REF] Prada | Decomposition of the time reversal operator: Detection and selective focusing on two scatterers[END_REF], the eigenvectors only depend on the position of the fault rather than its intensity, which is coded by ν i . Therefore, assuming the guessed position of the fault is correct, the scattering matrix of the dummy-fault NUT is

S i d = µ i w i w T i (7)
i.e., correct up to a scalar factor dependent on the unknown fault reflectivity, implying that S i = α i S i d . Hence, from [START_REF] Pan | Time-domain reflectometry using arbitrary incident waveforms[END_REF],

S G = N f i=1 α i S i d = N f i=1 α i µ i w i w T i . (8)
The missing coefficients α i can be estimated by noticing that

w T i S G w i = α i µ i (w T i w i ) 2 + k =i α k µ k (w T k w i ) 2 . ( 9
)
The first term in (9) provides direct access to α i . In general, the remainder summation is different from zero when sampled at a given frequency; this is not expected to be an issue, since the scalar products lead to phase-shift angles that change with the frequency. As a result, the time-domain signals used for location involve an averaging effect that is expected to reduce the impact of the remainder, especially for complex networks.

We will therefore use

αi = w T i S G w i µ i (w T i w i ) 2 = α i + k =i α k µ i (w k T w i ) 2 µ k (w k T w k ) 2 (10)
as an estimate of α i . Notice that ( 10) is based on the measured baselined scattering matrix of the multiple fault NUT, and the eigenexpansions of the dummy single-fault NUTs. Finally, [START_REF] Chung | Application of phase detection frequency domain reflectometry for locating faults in an f-18 flight control harness[END_REF] yields an estimate of the single-fault scattering matrices, as Ŝi = αi S d i .

Access to the individual S i not only allows energy focusing on individual faults, but also estimating the fault severity, as will discussed in sec. IV-C.

B. The EDORT algorithm

The previous discussion and results can be translated into an algorithm that iteratively extracts the individual scattering matrices of single faults of an NUT. The steps required are summarized in what follows:

1) measure the global scattering matrix S 1 G of the (eventually) multiple faulty NUT.

2) apply the SDORT method on S i G , in order to estimate the position of the strongest soft fault 3) include a dummy soft fault on the guessed position and compute its scattering matrix S i d 4) compute the only significant eigenvalue of S i d and its eigenvector 5) compute αi and Ŝi , according to [START_REF] Chung | Application of phase detection frequency domain reflectometry for locating faults in an f-18 flight control harness[END_REF] has no significant eigenvalue. The stop criterion in this algorithm requires to set a threshold for negligible eigenvalues, a choice that really depends on the configuration, especially the noise level, numerical accuracy, etc. To better visualize this point, Fig. 6 shows the dominant eigenvalue of each EDORT iteration applied to the single branch NUT of Fig. 1(b). In fact, two significant eigenvalues corresponding to the first and second iteration can be recognized, whose eigenvector components located the first and second soft fault present in the NUT. On the other hand, the third iteration leads to a negligible eigenvalue compared to the first two thus indicating the absence of any new considerable discontinuity in the structure. For instance, in the case of a purely numerical computation, one could also consider the machine epsilon. Besides from a more practical point of view, as the estimated severity gets very small, the user could stop the procedure as it is no longer worth continuing, since the fault would pose no threat to the proper functioning of the NUT.

C. Estimating the fault severity

Having provided an approximate procedure for extracting the single-fault matrices S i , the severity of each fault can be assessed without ambiguity, as these matrices have rank equal to one. Since the first Born approximation underpins soft-fault

S kj = Γ i τ k τ j , (12)
where Γ i is the reflection coefficient of the ith fault. Noting as S i F the Frobenius norm of S i , (

|Γ i | = S i F N k=1 |τ k | 2 -1 . 12) yields
This last result requires a priori knowledge of the τ i functions. In fact, these are known through the healthy NUT model used for baselining. The fact that only the moduli of the τ i are required in [START_REF] Boudjefdjouf | Diagnosis of multiple wiring faults using time-domain reflectometry and teaching-learning-based optimization[END_REF] implies that errors in the positioning of the dummy fault have a minor impact on the accuracy of the estimated |Γ i |, as long as the NUT is tested at frequencies where propagation can be regarded as subject to weak dissipation.

Departure from Born first approximation are shown in sec. V-B to lead to local errors that have in fact negligible impact on the overall performance of the procedure, as these errors have zero average (i.e., turning phase-shift angles).

V. VALIDATION

The EDORT method is tested against experimental data collected according to the setup described in sec. V-A. After showing that the ability of testing each fault separately is reinstated, the accuracy of the estimation of the fault severity is tested in sec. V-B.

A. Experimental results

Experimental tests were conducted on the NUT configurations in Figs. 1(a)-(b), implemented using standard 50 Ω coaxial cables as transmission lines. Although coaxial cables are not routinely found in practical NUTs, they were chosen because of their excellent stability, in order to ensure reproducible results. The conclusions drawn in this section are not related to the type of lines used, since the proposed method is completely general.

Coaxial cables also offer another sizeable advantage, since soft faults can be introduced (and removed) by using a set of 30-cm long samples made of semi-rigid coaxial lines, as those shown in Fig. 7. Each sample was crushed by a different force over a different portion of length, thus producing a varying severity in the faults: the strongest soft fault is designated by a crushed length of 2 cm while the weaker one by 1 cm. This same approach was used in the experimental validation of the standard DORT done in [START_REF] Abboud | A noniterative method for locating soft faults in complex wire networks[END_REF], where it allowed a practical and reproducible framework. In fact, the direct generation of the faults in a transmission line is hardly reproducible, not allowing a double check of results due to the fact that it is an irreversible process.

The ends of the cables were used as testing ports, by connecting them to a Rohde & Schwarz ZVB8 VNA, capable of covering a frequency range going from 300 kHz to 8 GHz with four testing ports. The measurement of the scattering matrices was done over a total bandwidth of 2 GHz and a frequency step of 20 MHz. After calibrating the VNA using the calibration kit provided by the manufacturer, studying the network consisted of two steps:

1) measurement of S Gh of the reference healthy system (without faults), where we considered the network with unaltered 30-cm semi-rigid sections. Having measured the two scattering matrices for each NUT, the algorithm detailed in sec. IV-B was applied. Signals propagating at each step throughout the NUT were computed by means of an in-house transmission-line solver, implemented under Matlab.

The proposed procedure was first applied to the singlebranch NUT in Fig. 1(a) with two soft faults on 1.15 m and 2 m from the reference port, and implemented as shown in Fig. 8. The energy diagrams associated to the estimated singlefault scattering matrices in Fig. 9 are now characterized by a single dominant peak each, corresponding to the position of the two faults. These results show the result of selective focusing compared to the that obtained by the SDORT on the same NUT, featuring two peaks simultaneously on the faults' positions as presented also in Fig. 8.

It can be noticed that some artifacts are still present, due to the periodicity of wave propagation that is proper of singlebranch structures. Indeed, this problem disappears as soon as more complex structures are considered, where the spatial periodicity is broken. The flat background observed in the two energy distributions can be understood by recalling that, in the case of single faults, excitation signals with equivalent intensity cross the NUT and interfere constructively at the fault position [START_REF] Abboud | A noniterative method for locating soft faults in complex wire networks[END_REF]. In order to check the efficiency of the proposed method with more complex NUTs, the single-junction structure in Fig. 1(b) was considered and implemented as presented in Fig. 10 considering the first soft fault at 0.8 m from the reference port and the weaker at 2.4 m on the second path. Indeed, in this case artifacts are negligible with respect to the focusing peaks. The interest of applying the proposed procedure becomes clear when comparing the EDORT results in Fig. 11 with those from SDORT, shown in the same figure, where artifacts would have led to a wrong conclusion about the faults position. The separation of the fault responses yields similar features for the two faults, independently of their relative severity.

B. Retrieving fault reflection coefficients

In order to assess the accuracy of (13) in retrieving reflection coefficients of the faults, their values were measured by directly connecting the faulty samples to one port of the VNA, while terminating the remaining port of the samples with a Fig. 9: The total energy along the NUT in Fig. 1(a), computed after applying the EDORT separation procedure. At each iteration the dominant energy peak corresponds to a fault not identified by the previous iterations. The energy distribution obtained when using the SDORT is also shown for reference, and contains two energy peaks at the position of the two faults. 13) was subsequently applied to the previously estimated single-fault scattering matrices. Results are shown in Fig. 12, for the single-branch NUT of Fig. 8. The two sets of data closely agree.

The same operations were then carried out for the singlejunction NUT in Fig. 10. In this case, we can expect interactions between the faults and the discontinuity introduced by the junction. As a result, the simple model proposed in ( 12) is expected to hold only on average. Fig. 13 shows the comparison between the measured and ( 13)-based estimates of Since this kind of behaviour can be expected to have zeroaverage, because of phase rotation associated to frequency variations introduced by propagation delay, a simple linear regression through the estimate from ( 13) can be applied in order to reduce the impact of these oscillations. The results in Fig. 13 confirm the effectiveness this procedure. Both reflection coefficients agree with the direct measurements. Fig. 13: Same kind of results as in Fig. 12, but now dealing with the experimental setup in Fig. 10. The results estimated by means of ( 13) are here also used as the basis for a linear regression, as explained in the body of the text.

VI. CONCLUSIONS

Having pointed out the shortcomings of SDORT in the case of multiple soft faults, an alternative formulation based on an updating scheme was presented. The input signals used for imaging the faults were shown to focus selectively on single faults, allowing a clear identification. No spurious focusing was observed, thus reducing the probability of false alarms. Experimental validations also confirmed these conclusions, when considering both very weak faults and non-trivial NUT layouts. These results give credibility to the practical usefulness of the proposed method in critical real-life configurations that are hardly addressed in current literature. In addition to that, the proposed procedure allows an accurate estimate of the severity of each fault, a feature of practical interest when monitoring the state of critical cable networks.

Future work will need to deal with the robustness of EDORT with respect to errors in the baselining procedure, as due, e.g., to tolerances in the geometrical description of the reference healthy network.

Fig. 1 :Fig. 2 :

 12 Fig. 1: Layouts of the NUTs considered in the numerical and experimental studies: (a) single-branch NUT; (b) singlejunction network configuration. Both structures include the lengths of the branches and present two soft faults of different severity.

2 Fig. 3 :Fig. 4 :

 234 Fig. 3: The total energy E(x) observed along the NUT in Fig. 1(a) with two soft faults at 1.15 m and 2 m from the origin port comparing three different sets of fault severities.

Fig. 5 :

 5 Fig. 5: A single-branch NUT used for demonstrating the possibility of separating a multiple-fault scattering matrix containing two soft faults (a) into two replicas of the NUT presenting only fault 1 (b) or fault 2 (c).

7 )

 7 and (11) 6) remove the ith fault contribution by computing S i+1 G = S i G -αi S i d increase i and repeat from step 2) until the update S i G

Fig. 6 :

 6 Fig.6: The most significant eigenvalue for the TRO, calculated for the single junction NUT of Fig.1(b) after applying the EDORT steps for the fist, second and third iterations.

2 )

 2 measurement of S Gf of faulty systems considering the introduction of the two soft faults on their positions as presented in the configurations of Fig. 1(a)&(b), where the unaltered cable samples were replaced by the faulty ones.

Fig. 7 :

 7 Fig. 7: The two faulty samples used for introducing soft faults in experiments.

Fig. 8 :

 8 Fig. 8: Experimental setup for the cable NUT of Fig. 1(a), containing two faulty sections (1.15 m and 2 m from the reference port), connected to the VNA for the experimental tests.

Fig. 10 :

 10 Fig. 10: Experimental setup for the single-junction NUT in Fig. 1(b) presenting two faulty sections (0.8 m and 2.4 m from the reference port), connected to the VNA for the experimental tests.

Fig. 11 :

 11 Fig. 11: Same kind of results shown in Fig. 9, but now dealing with the single-junction NUT in Fig. 1(b).

Fig. 12 :

 12 Fig.12: Amplitude of the measured reflection coefficients of the two faulty samples in Fig.7versus frequency. Estimates of these reflection coefficients obtained by means of the EDORT and (13) applied to experimental data from VNA measurements are also shown, corresponding to the setup in Fig.8.


Kais Mekki
email: mekki.kais@gmail.com

Ahmed Zouinkhi
email: ahmed.zouinkhi@enig.rnu.tn

Mohamed Naceur Abdelkrim
email: naceur.abdelkrim@enig.rnu.tn

Wireless Sensor Network Layer Solution for Security Management

Keywords: WSN, Security, Routing protocols, Quality of Service, Simulation

Wireless sensor networks have profound effects on many application fields like security management which need an immediate, fast and energy efficient route. In this paper, we define a QoS based network layer for security management of chemical products warehouse which can be classified as real-time and mission critical application. This application generate alert packets caused by unusual events which need a short end to end delay and low packet loss rate constraints. After each node compute his hop count and build his neighbors table in the initialization phase, packets can be routed to the sink. We use Random Re-Routing protocol which dynamically transfers routine data packets to secondary paths in the network, while offering a fast track path with better QoS for the packets carrying unusual events data. We add an energy threshold for the routing protocol to control the energy consumption and we adapt the network topology changes by rerun the initialization phase when chemical units were added or removed from the warehouse. Analysis shows that the network layer is energy efficient and can meet the QoS constraints of unusual events packets.

Introduction

In the sector of the chemical industry, the priority is granted to the protection and the safety of goods and people. It is for that besides that one seeks without cease to develop increasingly reliable means ensuring safety at the level storage and handling of the dangerous chemicals, from where the integration of the Wireless Sensor Networks (WSN) in the systems design of security.

Currently, many security systems depend on safety measurements eventually exposing people lives to unpredictable environments as for examples storage and transport activities of dangerous chemical substances. This subject attracted the interest of several research projects as the research center for automatic control (CRAN) at Lorraine University in France [START_REF] Zouinkhi | Simulation and modeling of active products cooperation for active security system management[END_REF]. The project presented in [START_REF] Zouinkhi | Simulation and modeling of active products cooperation for active security system management[END_REF] [2] develops a security management application of chemical products warehouse by wireless sensor network. The purpose of this application is to monitor chemical storage because such storage management product may cause great danger if safeguards are not respected. This application generates security alerts. An alert is transferred by a high priority packet across the wireless network to achieve the sink. Hence, this real-time application requires strict constraints on both delay and packet loss rate in order to report the alert data to the sink node within certain time limits without loss. These performance metrics (delay and packet loss rate) are usually referred to as Quality of Service (QoS) requirements [START_REF] Chen | A survey on quality of service support in wireless sensor and actor networks: Requirements and challenges in the context of critical infrastructure protection[END_REF]. Therefore, enabling real-time application in wireless sensor networks requires energy and QoS awareness in network layer of the protocol stack in order to have efficient utilization of the network resources and effective access to sensors readings.

Thus, QoS routing is an important topic in sensor networks research, and it has been the focus of the research community of WSN. Many QoS based routing protocols specifically designed for WSN have been proposed, for example, SPEED [START_REF] Tian He | SPEED: A stateless protocol for realtime communication in sensor networks[END_REF], MM-SPEED [START_REF] Felemban | MMSPEED : Multipath multi-speed protocol for QoS guarantee of reliability and timeliness in wireless sensor network[END_REF], RPAR [START_REF] Chipara | Real-time power-aware routing in sensor network[END_REF], THVR [START_REF] Li | Enhancing Real-Time Delivery in Wireless Sensor Networks with Two-Hop Information[END_REF], FELGossiping [START_REF] Norouzi | A Novel Energy Efficient Routing Protocol in Wireless Sensor Networks[END_REF], M-HTR [START_REF] Nefzi | Performance analysis and improvement of zigbee routing protocol[END_REF], RRR [START_REF] Gelenbe | Adaptive QoS Routing for Significant Events in Wireless Sensor Networks[END_REF], EQSR [START_REF] Othman | Energy efficient and QoS based routing protocol for wireless sensor networks[END_REF] and MARP [START_REF] Sen | An Adaptable and QoS-Aware Routing Protocol for Wireless Sensor Networks[END_REF]. For our application, we had chosen Random Re-Routing (RRR) protocol.

In this paper, we realize energy efficient and QoS based network layer which uses RRR routing protocol for security management of chemical products warehouse, and we evaluate his adaptation to the application functionality with extensive simulations.

The rest of the paper is organized as follows. In the next section, we describe the security management application of chemical products warehouse. In section 3, we present the network layer and, in section 4, we analyze the performance of the proposed solution. Finally, we conclude the paper in section 5.

Security Management Application

Accidents in the chemical industry are becoming frequent due to the absence of adequate security measures especially in the chemical warehousing field. This topic has attracted the interest of several research projects [START_REF] Zouinkhi | Simulation and modeling of active products cooperation for active security system management[END_REF].

The application of our work has been done to meet the needs of this field, its goal is to monitor dangerous chemical products warehouse. This application was able to turn the chemical units in communicating entities by wireless sensor nodes to collect information from its environment as shown in figure 1. The sensors glued to chemical units must periodically send information (routine data) about the status of products (temperature, pressure, etc). If there is an abrupt change from an environmental or internal state of chemical products, the application at the sensors must report this alert (unusual event) to the control center via the sink node by high priority packet as shown in figure 2. The sensors must use energy efficient and QoS based network layer which gives a short delay and low packet loss rate for unusual events packets.

Wireless Sensor Nodes

Sink Node

To the control center Alert 3. Network Layer Solution

Network Initialization Phase

The network initialization phase [START_REF] Norouzi | A Novel Energy Efficient Routing Protocol in Wireless Sensor Networks[END_REF] starts after the sensor nodes are randomly distributed in the controlled area. In the beginning, the sink broadcasts a HELLO message to its neighbors. The HELLO message contains: the hop count (HC) and the sender address (SA). The hop count is used to setup the gradient to the sink which means it shows the node distance to the base station, and the sender address is used to build the neighbor table of each node. After broadcasting the HELLO message, all 1-hop neighbors will receive this message and each one will execute the following steps:

 If it doesn't have a gradient:

-Gets its hop count by saving HC in its memory.

-Saves SA and the hop count of the sender node in its neighbor table. The hop count of the sender is equal to HC-1. -Increases HC by 1. The old HC is then replaced in the HELLO message with the new one and the node will continue to broadcast this message to farther nodes. As shown in figure 3, at each stage the hop count will be incremented by 1.

 If it has a gradient:

-Gets SA and the hop count of the sender node. If SA exists in neighbor table, the corresponding hop count will be replaced with the new one. Else, information will be saved in the neighbor table. -Compares its gradient to the HC and will replace its hop count with the message's HC if the latter is smaller, and will add 1 to the HC prior to broadcasting it. However, if its gradient is smaller than or equal to the HC, it will discard the message. As a result, the gradient will keep the best route.

Finally, the process will continue until all nodes receive the HELLO message. At that time, the network initialization phase will be completed. Now each node knows its distance to the sink node and its entire neighbors and their gradient through neighbor table. Nodes can starts routing packets to the base station through a routing protocol.

Routing Protocol

We had chosen Random Re-Routing (RRR) protocol which is evaluated in [START_REF] Mekki | QoS based routing protocol for security management: Application in dangerous chemical products[END_REF] and compared with other QoS based routing protocols. Simulation results in [START_REF] Mekki | QoS based routing protocol for security management: Application in dangerous chemical products[END_REF] showed that RRR can achieve very low end to end delay and low packet loss rate, and we have to study this protocol for supporting our security management application and his network layer.

RRR [START_REF] Gelenbe | Adaptive QoS Routing for Significant Events in Wireless Sensor Networks[END_REF] is a distributed and adaptive routing algorithm which can detect the occurrence of unusual events (alert) and provide better QoS for packets that carry alert information. The WSN is composed of sensor nodes which may have both sensing and forwarding roles. RRR is implemented in each of the sensor and forwarding nodes. It distinguishes packets of routine data and unusual events, packets from unusual events are routed along preferred paths, while the routine data are randomly shunted to slower and possibly longer secondary paths.

In RRR, the sensor nodes will change their routing policy adaptively according to the current traffic level. When the overall total traffic level is low, the preferred paths will be shared by all packets. However, when the total traffic exceeds a given threshold, the preferred paths will be reserved for forwarding only the unusual events data packets and secondary paths will be used for the routine data packets. This mechanism provides significantly better QoS to unusual events packet.

In more detail, each node monitors the rate at which it receives unusual events packets, and if this rate does not exceed a threshold θu, then the node forwards all packets it receives along their preferred path towards their destinations as shown in figure 4. The preferred path will be the path with the minimum number of hops (the shortest one). In this case, RRR uses the same path for both types of packets which can consume a lot of transit nodes energy especially if it continues for long time. To

Sink

Sink Sink

1 1 1 1 1 1 1 1 1 1 Discarding X X 1 1 1 1 1 2 2 2 2 2 2 2 2 (3) (2) (1)
control this, we add an energy threshold to RRR protocol. If the energy consumed by transit node exceeds the threshold (e.g. energy consumption exceeds 85% of the initial energy), then the node is only involved in unusual events packets routing as shown in figure 5. If a node (source or transit) senses that the rate at which it forwards unusual packets exceeds the threshold θu, then it will forward all unusual events packets along the preferred path to their destination, while all routine data packets will be directed along a randomized route which leave the shortest paths to the high priority traffic (unusual events packets) as shown in figure 6.

Adding/Removing Chemical Units

If a set of chemical units are added or removed from the warehouse, a node glued to one of these chemical units broadcasts respectively a joining or leaving message to all nodes that are in its transmission range [START_REF] Zhang | Energy-Efficient Routing for Mobility Scenarios in Wireless Sensor Networks[END_REF]. A node that receives this message, routes it to the sink. Then, the sink begins running the initialization phase to rebuild the hop count and neighbor tables of all network nodes, hence adapting the network topology to change in the number of nodes.

Analysis

In this section, we evaluate the packet average delay, the packet loss rate and the energy consumption of the network layer under different sensor nodes densities. Furthermore, we use Castalia simulator [START_REF] Boulis | Castalia, a simulator for wireless sensor networks and body area networks, version 3.2[END_REF] to implement the network layer. Currently, many wireless sensor network simulators are available as NS2 and SENSE but Castalia provides realistic wireless channel and radio models, and realistic node behavior especially relating to access of the radio [16] [17].

The simulated networks consist of 100, 200, 300 and 400 nodes respectively with a single sink. The node positions are all uniformly distributed at random within a 300m x 300m square (m=meters). The communication range is 30 meters, the sink is located at the center of the square and the RRR threshold θu is 3pkt/s. The simulation parameters that we have chosen have been selected so as to be compatible with other studies of WSN [START_REF] Gelenbe | Adaptive QoS Routing for Significant Events in Wireless Sensor Networks[END_REF] We simulate the network under two state of traffic:

-Not congested traffic: the routine data packet rate is 0.2 pkt/s for each node, while the unusual event traffic rate is 1 pkt/s at 2 nodes. In this case, the unusual event traffic rate at the intermediate nodes doesn't exceed θu. -Congested traffic: the routine data packet rate is 1 pkt/s for each node, while the unusual event traffic rate is 5 pkt/s at 4 nodes. The network load becomes higher now as there are more sources of unusual event with higher rate, so the unusual event traffic rate at the intermediate nodes exceeds θu.

We simulate these two routing schemes under different node densities between 100 and 400 nodes.

Delay

Figures 7 and8 shows the packet average end to end delay for each type of packet and for four levels of network density (100, 200, 300 and 400 nodes).

Figure 7 shows that the network layer is able to meet the time constraints for alert packets (unusual event) in congested traffic, the alert packet delay is very short even with the increasing of the network density. In congested traffic, the unusual event traffic rate at the intermediate nodes exceed the threshold θu and then the unusual event packets are routed along the shortest path whereas the routine data packets are forwarded via random alternative paths. So as shown in figure 7, the unusual event data achieve lower packet delay than the routine data.

Figure 8 shows the packet average delay in not congested traffic. As expected, the routine data packet delay is very close to the unusual event packet delay because the network handles both unusual event and routine data by the shortest path. In congested traffic, the routine data packets have a medium packet loss rate as shown in figure 9 because they pass through random alternative paths that have a low quality of service and may be congested by routine data packets from other sources. The analysis of trace files showed that the majority of packets are lost due to:

Packet Loss Rate

-Overloading of nodes causing saturation of the queues.

-Interference because we have many transfers, so there is more concurrent access to the radio channel.

Reasoning according to the density of the network, figures 9 and 10 shows that the network layer was able to maintain a constant and low packet loss rate for alert packets even when the density of the network increases. For routine data packets, the number of successfully received packets increases because the node will have more choices of the next hop when the number of neighbor nodes increases.

Energy Consumption

To study the energy consumption of the network layer, we performed two different simulations. First, we compare it to another network layer which uses energy efficient and QoS based routing protocol for WSN. Second, we evaluate the energy threshold performance.

 Energy consumption: we had chosen to compare our solution with a network layer which uses FELGossiping routing protocol [START_REF] Norouzi | A Novel Energy Efficient Routing Protocol in Wireless Sensor Networks[END_REF]. We had also implemented FELGossiping with Castalia simulator. We considered nodes with an initial energy equal to 18760 Joules. We measured the average energy consumed by the network nodes in different periods of time. The simulation result is shown in figure 11. We observe that RRR achieves more energy savings than FELGossiping because RRR protocol does not use any control messages, the routing decision is done using the neighbors table built in the initialization phase. The analysis of trace files showed that FELGossiping protocol has a bigger throughput than RRR which gives a higher packet transmission number and so a higher energy consumption. Therefore, our network layer optimizes more the energy consumption and therefore ensures a long network lifetime than the FELGossiping based network layer.

 Energy threshold evaluation: now we consider nodes with limited energy and we simulate the network traffic until the first node dead first for the initial RRR protocol and then for the RRR with our energy threshold. The effect of the energy threshold is shown in table 1, the table shows death time of the first node for both protocols. Nodes tend to die faster in the initial protocol. With our threshold, the probability of choosing the same node as the next hop for long time is reduced. Thereby, the energy has been more balanced and fairly used. The threshold leads to saving energy and hence prolonging the overall network lifetime.

Conclusion

In this paper, we realized a network layer for security management application of chemical products warehouse. We used Random Re-Routing (RRR) protocol that is designed to react to network congestion so as to provide better quality of service to alert packets caused by unusual and critical events. We added an energy threshold to improve RRR protocol for saving more energy and prolonging the network lifetime. We also proposed a solution for network self-organization when chemical units are added or removed from the warehouse. Simulation results showed that the network layer can achieve short average end to end delay and low packets loss rate for alert packets, and RRR showed its improvement in the network lifetime when we used the energy threshold. As a future project, we will improve the network layer to be faulttolerant and to react when node becomes unreachable.

Fig. 1 .

 1 Fig. 1. Communicating entity for dangerous chemical products.

Fig. 2 .

 2 Fig. 2. Routing of unusual event packet.

Fig. 3 .

 3 Fig. 3. Network Initialization Phase.

Fig. 4 .

 4 Fig. 4. RRR behavior in not congested traffic for θu=3pkt/s and speed (alert packets)=1pkt/s.

Fig. 5 .Fig. 6 .

 56 Fig. 5. RRR behavior if the consumed energy of node exceeds the energy threshold.

Fig. 7 .

 7 Fig. 7. Delay performance in congested traffic.

Fig. 8 .

 8 Fig. 8. Delay performance in not congested traffic.

Figures 9

 9 Figures 9 and 10 shows that the network layer guarantees a very low packet loss rate for the alert packets in the two classes of traffic.In congested traffic, the routine data packets have a medium packet loss rate as shown in figure9because they pass through random alternative paths that have a low quality of service and may be congested by routine data packets from other sources. The analysis of trace files showed that the majority of packets are lost due to:

Fig. 9 .

 9 Fig. 9. Packet loss rate performance in congested traffic.

Fig. 10 .

 10 Fig. 10. Packet loss rate performance in not congested traffic.

Fig. 11 .

 11 Fig. 11. Energy consumption performance.

Table 1 .

 1 Death time of first node (in seconds)

		Initial RRR	RRR with energy threshold
	Death time (s)	58,309	59,15


E Guiot
email: guiot.eric_1@yahoo.fr

A classical alternative to the dark matter and dark energy hypotheses ?

Keywords: dark matter, force of gravitation, galaxies, modified gravitation

A classical alternative to the dark matter and dark energy hypotheses ?

1.Introduction

Several astronomical observations, last decades, seems in contradiction with the classical laws of gravitation. For example, the curve of rotation of galaxies can't be explain without add, inside these galaxies, a large amount of a mysterious matter, generally called «dark matter » [1]. Moreover, the expansion at an increasing rate [START_REF]Patantonopoulos The Invisible Universe[END_REF]of the universe can't be explain with these laws of gravitation, without consider another hypothetic substance, called this time « dark energy », which should be repulsive [START_REF]Patantonopoulos The Invisible Universe[END_REF][START_REF] Huterer | Prospects for probing Dark Energy via supernova distance measurements[END_REF].

However, despite all attempts of the researchers since several decades, to detect these matter and energy, it seems that there is no proof of their existence today [5]. These matter and energy could, consequently, be only a creation of the minds.

A second way, to explain these astronomical observations is to change the laws of gravitation themselves. A first problem is that these laws are well verified inside our solar system. A second problem is that the alternative theories don't do unanimity, and sometimes seems in contradiction with laws of physics [6].

Consequently, we purposed, in our turn, an alternative theory of gravitation, in two previously paper [7,8]. This theory is building with respect for classical physics. In this paper, we present a result we obtained recently. This result seems us solve the astronomical contradictions we listed before.

Our force of gravitation

We present hear the different assumptions, which lead us to our solution:

a) The classical physics can describe, with a correct approximation, the trajectories of the stars at the periphery of galaxies.

b) The trajectory of a star, which orbits around a center of mass, is a conic.

In our previously papers [7,8] we showed that the assumption b) is respected if the force is given by the relation

  e r f f r e r f f r Af F R      . 1 ' 2 1 ) ( 1 ' 2 1 2 2                   (1)
Where f is a function of r and ' f its derivate. This force is given in a polar system of coordinate, where r is the distance to the foci of the conic. The origin of the system is located at this foci and the angle  is measured from the periapsis of the orbit. Ais constant and proportional to the mass of the system.

  dt d   is given by ) ( ) 1 ( 2 r f r e Aa    
Where a is the semi-major axis of the conic and e its eccentricity. The radial speed is given by

r dt d r  
Where r is given by

 eCos e a r    1 ) 1 ( 2 For example, if we choose 2 1 ) ( r r f 
we obtain the Newton's force. If we choose

) (r f as a constant
we obtain a force of Hooke [8].

We showed that, in all the cases, the force allows well to obtain a conic trajectory. We studied also the two body-problem and the conservation of the kinetic energy.

In order to determine a correct expression of

) (r f
we list now several other assumptions: c) When the acceleration is high, this force of gravitation is the Newton's d) When the acceleration is low, and in the case of circular motion, this force leads to a "flat" curve of rotation, with respect for the Tully-Fisher law e) When the acceleration is low, and in the case of parabolic motion, this force can be attractive or repulsive We purposed several possibilities for

) (r f
, in particular, in a first time, the solution given by Because this simple solution, with a correct choice of the constant 0 r allows to respect the assumption (c) and (d). But it appears this solution doesn't respect assumption (e). At a second time, we studied the solution given by

) 1 ( 1 ) ( 2 2 kr r r f   ( 2
)
Where k is a constant of the motion. It appears today that this second solution allows to respect all our assumptions, by a correct choice of k , and we will, consequently, in this paper, study it more detailed.

By using relation (1) we obtain the force, given by

F R R     . 1 ) 1 ( 2 2              (3)
The first term of this expression is naturally the Newton's. When r is small this term is dominant, but when r increases, progressively, this term disappears. Assumption c) is well respected.

Physical interpretation of this force

This force can be decomposed in two forces, by the relations

2 1 F F F     
where the force of Newton

1 F  is given by R e r A F   2 1  
And the second part is

  e keSin e a r e r a k F R    . 1 ) 1 ( 2 2            
the magnitude of this second force is given by

ar a r e a a k F 2 ) 1 ( 2 2 2 2 2      
by noticing that the distance OM , where M is the point-particle and O the center of the conic, is given by

ar a r e a OM 2 ) 1 ( 2 2 2 2      we obtain OM a k F  2 
We can now determine the center of this force: by using the polar coordinate of the vector M O  given by We see that our solution is simply an addition on a force of Hooke and a force of Newton. The physical signification is that, when the acceleration decreases, the force of gravitation, in our model, becomes progressively a force of Hooke. The force of Newton is only valuable for small distances and high accelerations (as inside solar system or centers of galaxies). The force of Hooke can be more attractive than the Newton, but can also be repulsive, in the case of parabolic or hyperbolic motion. Note that these two well-known forces are central forces, and consequently can be derivative from a potential. Moreover, an important observation is that, when r increases, the center of the force and the center of mass aren't located at the same point.

   e aeSin e r aeCos M O R    . ) (   

And the relation of the conic

Determination of the constant k

We will now give a correct value of k , in order to respect the other assumptions. We study the two limiting cases, the circular motion and the parabolic motion.

The motion is circular

In this particular case, the eccentricity of the conic is given by 0  e and r a  . Consequently, the force given by relation

(3) becomes R R e kA e r A F       2
The curve of rotation is obtained by written the equality of the acceleration and the force

kA r A r V     2 2

Where

V is the speed of the point-particle. We obtain In order to obtain a flat curve of rotation, we see that k has to be dependant of the semi-major axis, as

kAr r A V   (4)
a r k 0 1  (5)
Where 0 r is constant. Moreover, we can determine this constant, because we have to respect the Tully Fisher law [9]. Indeed, this empirical relation suggests a correspondence between the visible mass of galaxies and the velocity as

) ( 4 M V  
Where M is the total mass of the galaxy and 

V the constant speed of the stars. We saw that A has to be proportional to the mass thus a is an acceleration. This acceleration has to be constant in the entire universe, and can be linked with the Milgrom's acceleration used in MOND. We see that now

V is constant for circular motion and is in agreement with the Tully-Fisher law. Consequently, our choice of k allows to respect assumption d).

The motion is parabolic

In this case, the relations of the conic are given by 1  e and    By using relation (3

) the force becomes R R e a r kA e r A F              1 2
And by using ( 5) we obtain

R R e a r ar A e r A F              1 0 2
Consequently, this force can be, in the case of parabolic motion, alternatively attractive or repulsive.

The sign of the acceleration is depending on the ratio a r . We can solve the equation to obtain the particular points of this evolution

0 1 0 2           a r ar A r A (7)
This equation becomes after simplification

0 2 0 2 3    a r a r r
To solve this equation we use the method of Cardan. We write thus The discriminant is given by 5 0 we have 0  D : in this case we obtain three real solutions. As previously, it appears that one of them is negative: consequently, only two solutions are possible. Thus, the force of gravitation can be successively attractive and repulsive. We present theses results in the Figure 2.

4 2 0 2 3 27 1 4 1 2 3 a r a r q p D                
We can distinguish three different cases: When A r r  the force is always attractive and can be approximately, if r is small (as for example in solar systems), considerate as a force of Newton. When

Our cosmological model

With the results we obtained, about the circular and parabolic motion, we can now present a simple and schematic cosmological model.

We consider a galaxy. At the periphery of this galaxy, the orbits of the stars are circular. The radius of their orbits are such 0 r r  . Consequently, the curve of rotation is given by Where 0 a is a constant acceleration of the universe (comparable or equal to the acceleration used in MOND), the curve of rotation is in agreement with the Tully-Fisher law.

0 r A V  Acceleration (parabolic motion) r A r B r

Force of Hooke

Force of Newton

Zone of repulsive force of Gravitation

We consider now two clusters of galaxies, which move by describing a parabolic motion. The semi major axis of this motion is a and 0 r is a constant of the system composed of the two clusters.

We have now two solutions to describe the force of gravitation between these clusters. If The force can be repulsive or attractive. If we consider now the modern results about the inflation of the Universe [2], we can suppose that it is the case today, for the majority of the interactions of the clusters between themselves. But this evolution isn't definitive: as shown on the figure 1. If our model is correct the force of gravitation could become a day, attractive.

Note that the difference of behavior of the force is only due to the difference of motion: in the first case, the motion is circular (in reality, approximately circular) and in the second case, the motion is parabolic (in reality, approximately parabolic). We show that with the solution given by (5) the same force can easily describe so different behaviors.

Discussion

A test in the solar system

As we suggested it previously [8], we can purpose a test inside the solar system, in order to valid or invalid our model. This test consists "simply" to do experiments about the free-fall motion. If our solution is correct this free fall motion should be describe, with the relation

          1 0 2 a r ar A r A r  (8)
Where r  is the radial acceleration, given by

r dt d r 2 2   
Consequently, we should obtain a modification of the Newton's acceleration, directed to the center of Mass (in the case of parabolic motion). This modification should be very small but can perhaps be detected. We can note that such modification has recently been suspected inside the solar system, in particular about the motion of several spacecrafts [10]. Therefore, in our solar system, we think that the constant 0 r can't be linked to 0 a with the simple relation 2 0 0 r A a  Because our solar system isn't an isolated system. But we can expect perhaps that 0 r is constant in all the solar system, and deduct its value from the experiments. Indeed, naturally, only experiments on the free fall motion, in order to detect a small modification of the Newton's acceleration, could give a correct result. Therefore, we don't expect modification of the trajectories of the planets, because these trajectories are always conics and consequently, not modified.

Simulations of our model

A second method to valid or invalid our model is to do simulations of it. In particular, we could test it at the periphery of spiral galaxies. We think that results could be correct. Because in fact, our solution given by It appears that this function is proximate to the "simple" interpolating function used in MOND and given by

1 0 0 ) 1 (            c c a a a a 
This function has been tested with success [11]. Consequently, in our case, we should obtain correct results. Therefore, note that in our case the function ) (r g isn't empirically.

A second important simulation should be to test our model in the case of the bullet cluster. Because in fact, the astronomical observations inside this particular cluster [12] are generally considerate as an indirect proof of the dark matter existence. Moreover, it appears that the alternative theories of gravitation, as MOND, fail to describe the dynamics of this cluster. Consequently, this test should be important in our case.

We think that this simulation could perhaps lead to correct results. Indeed in our model, the center of mass and the center of force aren't necessary located at the same point. Consequently, we won't need, perhaps, to use ad hoc hypotheses, as "dark matter" hypothesis, to explain this difference of localization.

At end, an important test will be to see if our model can be a limiting case of a relativistic theory of gravitation. This point is naturally crucial. We think it is perhaps possible. Indeed, our force of gravitation is only the sum of two central forces, which are consequently the derivative of potentials.

Is a force of Hooke possible for large distances?

In our model, we suppose that the force of Newton turns progressively into a force of Hooke, when the acceleration decreases. A problem is to know is this evolution is possible. We can give several arguments, in order to defend this idea.

Firstly, the force of Hooke exists already in the classical theory of the gravitation: it is simply the force which describes the motion inside a homogeneous sphere. Moreover, in another physical domain, this force was used, for example, to build the Thomson's model.

Secondly, we showed that the addition of these two forces leads well to conics trajectories, and allows respecting all our assumptions. In particular, we don't need to introduce a new and hypothetic force to describe the reality. All our assumptions are simply respected, and we can build a simple cosmological model, without "dark matter", and above all, without "dark energy". Our force can be repulsive, and this point is a cogent argument for it. For example, MOND, or Dark matter hypothesis, or classical theories of gravitation, can't explain the expansion of the universe at an increasing rate, without hypothetic hypotheses, as "dark energy".

Conclusion

In this paper, we present a simple classical theory of gravitation, in order to explain several anomalies which have been detected since several decades. This theory is building with respect for classical physics and is predictive. The fundamental idea is that the force of gravitation is modified when the accelerations decreases, and becomes gradually a force of Hooke. We show that this modification can easily explain the curve of rotation of the galaxies, and the inflation of the Universe.

  of Hooke and its constant factor is given bya k K we can now present our force on the figure 1.

F 2 F  1 FFigure 1 .

 211 Figure 1. Representation of our force of gravitation

  This condition, with respect for physical dimensions, leads to write the relation

  force becomes an attractive force of Hooke.

Figure 2 .

 2 Figure 2. Evolution of the force of Gravitation with r