Sébastien Serpaud
email: sebastien.serpaud@irt-saintexupery.com

Fabio Coccetti
email: fabio.coccetti@irt-saintexupery.com

Sonia Ben-Dhia
email: sonia.bendhia@insa-toulouse.fr

Alexandre Boyer
email: alexandre.boyer@laas.fr

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Keywords: EMC, near-field scanning, adaptive sampling, FPGA, fast measurement

This paper presents an optimization of the Sequential Spatial Adaptive Sampling (SSAS) algorithm to accelerate nearfield scanning of printed circuit boards or integrated circuits. The first originality of this approach is to configure this algorithm from the spatial distribution characteristic of near field on a planar surface. The second originality is to propose three selection criteria to adjust the targeted accuracy with the objective to reduce the measurement time. The low time-consuming algorithm selects only measurement points, which carry the most information, in order to reduce significantly the number of captured points without increasing the measurement error. The analysis of measurement results, based on this algorithm applied on two case studies, validates its effectiveness compared to the classical full regular grid sampling.

F

Represents a regionalized variable representing the near-field radiated by the DUT.

𝐹(𝑝 𝑖 )

Represents an achievement of F at the point 𝑝 𝑖 .

[dBA/m] 𝐹 ̂(𝑝 𝑖 ) Estimation of F value at the position 𝑝 𝑖 .

[dBA/m] 𝐹 𝑀 Final measurement dataset of F. hmeas Height of the measurement.

[mm] M Total number of points where F is measured. {𝑥, 𝑦} 𝑃 𝑐 (𝑝 𝑞𝑖 ) Local dataset of the Nc points neighboring 𝑝 𝑞𝑖 .

{𝑥, 𝑦} 𝑁𝑐 𝑃 𝑀 Final dataset of the M points where F is measured. {𝑥, 𝑦} 𝑀 𝑃 𝑀 ́

Temporary dataset of the 𝑀 ́ points where F is measured during SSAS algorithm process.

{𝑥, 𝑦} 𝑀 ́

These results have been obtained in the framework of the IRT Saint Exupery's research projects: ROBUSTESSE and FELINE. We acknowledge the financial and in-kind support (background knowledge and services) from the IRT Saint Exupery's industrial and academic members and the financial support of the French National Research Agency. October 2013.

𝑃 𝑁

Initial dataset of the N points where F is measured during the first step.

{𝑥, 𝑦} 𝑁

Q

Total number of steps.

[step] rLoop

Loop radius of the magnetic near field probe.

[mm] W Initial sampling step.

[mm] I. INTRODUCTION ITH the current industrial economic constraints, it is necessary to evaluate electromagnetic behavior of an electronic device as early as possible in its development process. If simulation tools fail to predict accurately the electromagnetic compatibility (EMC) of a complex equipment, the EMC engineer must have efficient measurement-based investigation tools. The near-field scanning (NFS) approach applied on electronic devices has been developed for several years. The experimental investigation performances allowed by NFS is triggering an increasing interest among the electronic equipment and component manufacturers for this EMC investigation measurement method. Owing to the precision and dexterity of modern automation, the NFS test bench based on a well-controlled robot moving probe, represents a powerful and cost-effective investigation tool. This contactless measurement is carried out above an electronic device under test (DUT) without the need to implement specific design constraints. Moreover, high measurement results reproducibility (<1dB) [START_REF] Serpaud | Technical improvement of the near field measurement: The vector near field scanning[END_REF] is ensured by the intrinsically weak sensitivity of the experimental read-out from the equipment set-up configuration.

Despite the aforementioned key features offered by the NFS measurements, a limiting factor, hindering a wider industrial deployment of this approach, is the effective scanning time duration. In investigation context, measuring only one field component (Hz) can be enough to identify the root cause of the EMC related issue. According to the authors' experience, gathered over more than 10 years of industrial EMC expertise, typical near-field measurement may last between 15 minutes up to 5 hours.

However, a complete characterization of the electric and magnetic fields above a complex DUT may require much longer measurement time (e.g. in device modeling context). Depending from the nature of the DUT (integrated circuit, electronic board or electronic equipment), the size of the measurement area could be rather large (1cm² up to 400cm²). As an example, for a 200x150mm aeronautic calculator scanned with 1mm sampling step, more than 30 000 points are necessary to properly execute the measurement. Let consider two magnetic field components are measured, and 3 seconds readout time for each point is within the selected frequency bandwidth (typically 30MHz-200MHz with RBW = 9KHz and 200MHz-3GHz with RBW=120KHz). The total measurement time reaches more than 25 hours. Three main factors contribute to the total duration of the near-field measurement:

 Tmeas : the time for the capture equipment (e.g. spectrum analyzer) to measure the voltage across the near-field probe in the selected frequency range.  Tmove : the time for the robot to move the probe across all points above the DUT.  Tctrl : the computer processing time to control the test bench (communication with all equipments and to process and save measurement data).

In general, with currently available robots and computers, in the worst case scenario, Tmove and Tctrl account for less than 10% of the total measurement time. Communication protocols are now optimized to ensure a rapid data transfer from the capture equipment. Thus, the main contributor left is Tmeas, that is directly linked to the acquisition speed and the number of captured points. Therefore, the most straightforward strategy to reduce the measurement time consists in decreasing the number of points.

The most effective method to reach this goal is to use a spatial adaptive sampling. If a full planar sampling is considered (for example the full sampling map in Fig. 11), most of the scanned points do not carry significant information about the radiated sources. For decades, several adaptive sampling approaches have been developed [START_REF] Rahimi | Adaptive sampling for environmental field estimation using robotic sensors[END_REF], [START_REF] Hachisuka | Multidimensional adaptive sampling and reconstruction for ray tracing[END_REF] on the wave propagation application [START_REF] Qureshi | Adaptive Sampling in Spherical and Cylindrical Near-Field Antenna Measurements[END_REF], [START_REF] Xu | Particle System Based Adaptive Sampling on Spherical Parameter Space to Improve the MDL Method for Construction of Statistical Shape Models[END_REF] and for planar surface [START_REF] Theunissen1 | Adaptive sampling in two dimensions for point-wise experimental measurement techniques[END_REF], [START_REF] Deschrijver | Automated near-field scanning algorithm for the EMC analysis of electronic devices[END_REF]. The planar sampling is suited for electronic equipment scanning. The method described in [START_REF] Deschrijver | Automated near-field scanning algorithm for the EMC analysis of electronic devices[END_REF] provides a significant reduction of the number of samples. Note that the authors present no specific indicators to quantify this observation. The choice of the full meshing resolution, taken as reference, is not justified. The main limitation of this approach is the massive used of the Kriging interpolation model. The computation of model at each iteration take a lot of time. The method described in [START_REF] Dorné | Optimized sequential sampling algorithm for EMI nearfield scanning[END_REF] partially improves this point. A second limitation is the use of a random sampling. This sampling approach significantly increases the total path between all the captured points, thereby increasing the measurement duration.

The orientation retained in this paper is to reduce the number of measurement points by capturing only the points that bring the most significant information. The advantages of the Sequential Spatial Adaptive Sampling (SSAS) proposed here are to be a very low time-consuming algorithm and configurable from the characteristics of near-field spatial distribution on a planar surface. Several indicators are also proposed to clearly quantify the performance of the SSAS and facilitate the comparison with other approaches to reduce nearfield scanning duration. The full meshing resolution is fixed by the spatial distribution characteristic of the near-field map, which defines a reference metric to analyze the reduction of measurement duration.

The first version of this algorithm, introduced in [START_REF] Serpaud | Sequential adaptive sampling algorithm to reduce the near-field measurement time[END_REF], suffers of inaccuracies to estimate the maximum emission level. In this paper, a modified version is presented to provide a faithful representation of the near-field above the DUT. Special attention is paid to improve the accuracy around the maximum and the local maxima emission levels of the near-field map.

After a description of the SSAS algorithm in section II and the interpolator used for this study in section III, the section V discuses about the benefits of using this algorithm based on both case studies described in section IV. All the symbols used to describe the algorithm are summarized in annex I.

II. DESCRIPTION OF THE SPATIAL ADAPTIVE SAMPLING ALGORITHM

Let Ω ⊂ ℝ² the space domain of the measurement defined as a rectangular planar surface above the DUT. The near-field emission above the DUT can be represented by a regionalized variable F captured by the near-field probe at several positions of Ω defined by the dataset 𝑃 𝐷 = {𝑝 𝑖 = (𝑥 𝑖 , 𝑦 𝑖 ), 𝑝 𝑖 ∈ Ω} 𝑖=1 𝐷 . 𝐹(𝑝 𝑖 ) is the measurement of F at the probe positions 𝑝 𝑖 and D is the number of samples of F in Ω. The minimum distance between each probe position along x and y axes, noted d, is the resolution of the final near-field scan. Let 𝐹 𝐷 = {𝐹(𝑃 𝐷 )} the full sampling dataset.

In this paper, only the measurement of the magnitude of the field (𝐹 ∈ ℝ) as described in the standard IEC 61967-3 is considered [START_REF]1: Integrated circuits-Measurement of electromagnetic emissions, 150KHz to 1GHz -Part 3: Measurement of radiated emissions -Surface scan method[END_REF]. The measurement is carried out at only one frequency. The purpose of the SSAS is to provide a reduced set of probe positions without introducing a significant loss of information on F. At the end, the near-field map with the final resolution d is reconstructed by interpolation.

The SSAS algorithm is based on the Multi-Level Adaptive approach [START_REF] Brandt | Multi-level adaptive solutions to boundary-value problems[END_REF] initially introduced in [START_REF] Serpaud | Sequential adaptive sampling algorithm to reduce the near-field measurement time[END_REF]. The first step consists in collecting an initial dataset of F. Let N the initial number of sampling points and 𝑃 𝑁 = {𝑝 𝑖 = (𝑥 𝑖 , 𝑦 𝑖 ), 𝑝 𝑖 ∈ Ω} 𝑖=1 𝑁 the initial dataset of probe positions. The initial dataset of near-field values captured by the probe is defined by 𝐹 𝑁 = {𝐹(𝑃 𝑁 )}. The objective is to map Ω with a limited number of points by an optimal spatial cover without any a priori knowledge about F. The initial dataset 𝐹 𝑁 has a serious impact about the optimality of the final solution. If the spatial dispersion between samples is high, it is possible to lose relevant information on F. In this condition, the adaptive algorithm may converge to a bad solution. On the contrary, a too small spatial dispersion meshing target could induce a large number of initial samples and therefore a weak reduction of the measurement time.

As seen in Fig. 1, several sampling methods [START_REF] Lavalle | On the Relationship between Classical Grid Search and Probabilistic Roadmaps[END_REF] could be used to cover Ω. Two types of sampling method exist. The pseudo-random sampling methods capture samples at random positions on Ω. The advantage is that each point of Ω has the same probability to be captured. This is the most popular approach but it can introduce a significant increase of the number of sampling points in order to minimize the spatial dispersion criterion. This parameter is linked to the standard deviation of (σg) the distribution of (W) the gap between each point and their nearest neighbors on Ω. A low spatial dispersion meshing criterion value means that Ω is uniformly meshed (σg0). The second approach is the regular sampling. The gap W is constant including a more optimal value of the spatial dispersion criterion (σg0). This approach is very easy to implement and not time-consuming. The drawback of this approach is that the sampling dataset may not be statistically representative of F. The choice of W value is critical and it must be related to the spatial characteristics of F. If the boundaries of Ω are properly defined, it is reasonable to think that sampling the boundaries of Ω does not carry a lot of information on F. The field values should be low at the boundaries. Adding to a short calculation time, the Sukharev regular sampling is used to collect the initial 𝐹 𝑁 dataset in the proposed SSAS algorithm.

SSAS algorithm is split in a sequential number of step Q as presented in Fig. 2 (Q = 5 steps in this example). At the beginning, the full regular grid is built. During the first step1, the initial dataset 𝐹 𝑁 is collected to capture some points from Ω according to the Sukharev grid with a step W. Then, step by step the spatial resolution is progressively decreased according to a selection criterion that defines whether the next point provides enough information about F to be captured.

As shown in Fig. 2, SSAS algorithm progressively meshes Ω.

Step by step, the sampling resolution starts with a value of W at the first step and it decreases linearly up to the d value. At each step q, a set of point 𝑃 𝑁𝑞 𝑚𝑎𝑥 = {𝑝 𝑞𝑖 , 𝑞 = 2. . 𝑄} 𝑖=1 𝑁 𝑞 𝑚𝑎𝑥 is defined to progressively cover Ω, where Nqmax is the maximum number of points for the step q. A selection criterion (cf §II.C) is used to define whether the point pqi must be captured during the step q. Let's define 𝑃 𝑁𝑞 = {𝑝 𝑞𝑖 } 𝑖=1 𝑁 𝑞 , as the set of 𝑁 𝑞 ≤ 𝑁 𝑞 𝑚𝑎𝑥 sampling points which are captured (according to the selection criterion) during the step q.

Step by step, the sum of the captured points 𝑀 = 𝑁 + ∑ 𝑁 𝑞 𝑄 𝑞=2

progressively increases at an exponential rate according to q as shown in Fig. 4. The dataset 𝐹 𝑀 = {𝐹(𝑃 𝑀 ), 𝑃 𝑀 = {𝑝 𝑖 } 𝑖=1 𝑀 ∈ Ω} is built from all points captured step by step, where 𝑀 defines the total number of measured points of F at all specific positions 𝑃 𝑀 . The objective is to minimize M (and so Nq) while ensuring that the dataset FM allows building an estimator that minimizes the prediction error of F at any position on Ω.

SSAS algorithm requires only three input parameters: d, W and the selection criterion. In the following, the definition and the justification of these three parameters are discussed.

A. Definition of the final sampling resolution d

The final resolution value d can be defined from the type of source. Regardless of the measurement height, the spatial distribution of the radiated emission is linked to the nature of DUT. A power electronic device or a silicon of microcontroller do not produce the same near-field spatial distribution. Consequently, the resolution of near-field probe is selected according to the type of measured sources (DUT). In this case, the final sampling resolution d is linked to the probe resolution. A commonly accepted rule-of-thumb is that a magnetic probe resolution is equal to the radius rloop of the near-field probe used to measure F. Therefore, the lower value of the final sampling resolution d will be set to the probe radius value.

d = 𝑟 𝑙𝑜𝑜𝑝 (1)

B. Definition of the initial sampling step W

The initial sampling step W is an important parameter. With

+ + + 100% 80% 60% 40% 20% 0% Step1 Step2 Step3 Step4 Step5 Step6 Step7 Step8 Step9 Step10 Step11
Max Min

Nb Points : M

𝑁 7𝑚𝑎𝑥 𝑚𝑖𝑛 Q = 3 Q = 5 Q = 9 Q = 11 𝑁 7𝑚𝑎𝑥 𝑚𝑎𝑥 N 6max N 5max N 4max N7max N 3max N 2max Q = 7
a large sampling step, relevant information about F may be missed out. A small sampling step increases the number of sampling points, which conflicts with the objective of reducing the measurement time. In this paper, the development of a relationship between the number of points N of the initial dataset and the spatial characteristics of F is proposed. In nearfield region, the distribution of the magnetic field is related to the current that flows through the DUT. That is why the magnetic near-field components are usually measured (using Hxy or/and Hz magnetic probes as presented in Fig. 5). Let's consider a current i flowing through a long thin wire above ground plane. From the spatial distribution of each magnetic near-field component, some relevant spatial characteristics about F can be extracted. For example, the maximum of Hxy magnetic field appears as a main lobe above the wire. The initial spatial sampling rate W could be defined in order to ensure that a least two measurement points are placed within this lobe. Let 𝑊 𝐻𝑦 (𝛼) and 𝑊 𝐻𝑧 (𝛼) be the width of this lobe according to the attenuation 𝛼 (dB) compared to the maximum value of magnetic field as defined in Fig. 5.

Both 𝑊 𝐻𝑦 (𝛼) and 𝑊 𝐻𝑧 (𝛼) parameters can be extracted from derivation of the radiated electric dipole equation [START_REF] Lindell | Exact image theory for the Sommerfeld halfspace problem, Part I: Vertical magnetic dipole[END_REF] summarized in Fig. 6. Let 𝜃 = 𝜋 2 ⁄ . From the radiated electric dipole equation, the expression of the three magnetic field components can be rewritten in the Cartesian coordinate system according to [START_REF] Hachisuka | Multidimensional adaptive sampling and reconstruction for ray tracing[END_REF][START_REF] Qureshi | Adaptive Sampling in Spherical and Cylindrical Near-Field Antenna Measurements[END_REF][START_REF] Xu | Particle System Based Adaptive Sampling on Spherical Parameter Space to Improve the MDL Method for Construction of Statistical Shape Models[END_REF][START_REF] Theunissen1 | Adaptive sampling in two dimensions for point-wise experimental measurement techniques[END_REF]:

𝐻 𝜑 = 𝐾 𝐻 ⋅ 𝑠𝑖𝑛(𝜃) ⋅ ( 𝑗 𝛽𝑟 + 1 𝛽 2 𝑟 2 ) (2)
𝐻 𝑥 = 0 (3) 𝐻 𝑦 = 𝐾 𝐻 ⋅ 𝛽 2 ⋅ ( 𝑗 𝛽𝑟 + 1 𝛽 2 𝑟 2 ) ⋅ 𝑠𝑖𝑛(𝜑) (4) 𝐻 𝑧 = 𝐾 𝐻 ⋅ 𝛽 2 ⋅ ( 𝑗 𝛽𝑟 + 1 𝛽 2 𝑟 2 ) ⋅ 𝑐𝑜𝑠(𝜑) (5) 𝑟 = √𝑦 2 + ℎ 𝑚𝑒𝑎𝑠 2 = 𝑦 𝑐𝑜𝑠(𝜑) = ℎ 𝑚𝑒𝑎𝑠 2 𝑠𝑖𝑛(𝜑) (6)
The modulus of magnetic field components can be written from both geometric parameters y and hmeas, as given by [START_REF] Deschrijver | Automated near-field scanning algorithm for the EMC analysis of electronic devices[END_REF][START_REF] Dorné | Optimized sequential sampling algorithm for EMI nearfield scanning[END_REF].

|𝐻 𝑦 | = 𝐾 𝐻 .ℎ 𝑚𝑒𝑎𝑠 (𝑦 2 +ℎ 𝑚𝑒𝑎𝑠 2 ) 3 2 ⁄ . √1 + 𝛽 2 . (𝑦 2 + ℎ 𝑚𝑒𝑎𝑠 2 ) (7) |𝐻 𝑧 | = 𝐾 𝐻 .𝑦 (𝑦 2 +ℎ 𝑚𝑒𝑎𝑠 2 ) 3 2 ⁄ . √1 + 𝛽 2 . (𝑦 2 + ℎ 𝑚𝑒𝑎𝑠 2 ) ( 8
)
The previous equations can be simplified if the term 𝛽 2 . (𝑦 2 + ℎ 𝑚𝑒𝑎𝑠 2 ) ≪ 1. So, in free space, if r < 3 cm and 𝑓 ≪ 15 𝐺𝐻𝑧, the parameters 𝑊 𝐻𝑦 (𝛼) and 𝑊 𝐻𝑧 (𝛼) are not significantly impacted by the frequency in the near-field region and they only depend on both geometrical parameters y and hmeas. Consequently, the maximum absolute value of magnetic field components can be extracted from the derivation of the simplified equations of Hy and Hz, as given by (9-12)

𝜕|𝐻 𝑦 | 𝑑𝑦 ≈ ( 𝑦 (𝑦 2 +ℎ 𝑚𝑒𝑎𝑠 2 ) 3 2 ⁄ ) ′ = 3𝑦 (𝑦 2 +ℎ 𝑚𝑒𝑎𝑠 2 ) 5 2 ⁄ = 0 (9)
⇨ 𝑦 𝐻 𝑦 𝑚𝑎𝑥 = 0 ⇨ |𝐻 𝑦 𝑚𝑎𝑥 | ≈ 𝐾 𝐻 . The distance y0 that separates the maximum Hy field and the maximum field attenuated by α value is given in (13) and [START_REF] Shepard | A two-dimensional interpolation function for irregularlyspaced data[END_REF]. In [START_REF] Delaunay | Sur la sphère vide, A la mémoire de Georges Voronoï[END_REF], 𝑊 𝐻𝑦 (𝛼) parameter is obtained by doubling y0, as shown in Fig. 5.

|𝐻 𝑦 𝑚𝑎𝑥 | |𝐻 𝑦 | ≈ (𝑦 2 +ℎ 𝑚𝑒𝑎𝑠 2 ) 3 2 ⁄ ℎ 𝑚𝑒𝑎𝑠 3 = 𝛼 ( 13
)
𝑦 0 𝐻𝑦 (𝛼) = ±ℎ 𝑚𝑒𝑎𝑠 √𝛼 2/3 -1 ( 14
)
𝑊 𝐻𝑦 (𝛼) = 2. |𝑦 0 𝐻𝑦 | (𝛼) (15)
The same type of development is applied to extract 𝑊 𝐻𝑧 (𝛼) from Hz expression and for the configuration where the dipole is above an ideal ground plane. In the last case, the image theory is used to take into account the ideal ground plane. With the hypothesis that the trace width (Wtrace) is lower than the height of measurement hmeas, the relationship between W(α) and hmeas can be simplified by polynomial regression as presented in Table I.

Y [mm] 4 3 2 1 0 -1 -2 -3 -4 Z [mm] Y [mm] 4 3 2 1 0 -1 -2 -3 -4 Mag [dBA/m] -20 -30 -40 -50 W Hy_noGND(10dB) W Hy_noGND(6dB) W Hy_noGND(3dB) W Hz_GND(10dB) W Hz_GND(6dB) W Hz_GND(3dB) -10dB -3dB -6dB -10dB -3dB -6dB ─ Hy_GND ─ Hz_GND ─ Hy_noGND ─ Hz_noGND

Table I

The relationship between 𝑊 𝐻 parameters and hmeas according to a specific attenuation for each magnetic field components. If α is fixed at 10 dB, the minimum sampling step 𝑊 𝐻 for all the magnetic field components, for the initial dataset, can be estimated by :

𝑊 𝐻 ≤ 2 ⋅ ℎmeas (16)
A progressive spatial sweep is done on a number of sequential step 𝑄. 𝑄 could be defined from d and 𝑊 𝐻 according to [START_REF] Serpaud | Proposal for combined conducted and radiated emission modelling for Integrated Circuit[END_REF] where I is the biggest integer that validates 𝑊 𝐻 𝑑 ⁄ ≥ 2 𝐼 or according to [START_REF][END_REF] from the setup parameters definition.

𝑄 = 2 ⋅ 𝐼 + 1 (17) 𝑄 ≤ 2 ⋅ ⌊𝑙𝑜𝑔 2 (ℎ 𝑚𝑒𝑎𝑠 𝑟 𝑙𝑜𝑜𝑝 ⁄ )⌋ + 3 (18)
According to the definition of 𝑄, the effective sampling step value W, defined by ( 19) and [START_REF]Case studies data[END_REF], determines the number of points N collected during the initial step.

𝑊 = d ⋅ 2 (𝑄-1) 2 ⁄ (19) 𝑁 = ⌈ 𝐴 𝑥 𝑊 𝑑 ⁄ -0.5⌉ ⋅ ⌈ 𝐴 𝑦 𝑊 𝑑 ⁄ -0.5⌉ (20)
If the size of Ω is close to W, the number of points collected during the first step could be low. Therefore, it is necessary to add a new constraint on W to ensure a minimum number of points N to start the algorithm. In this case, the problem may also be poorly defined and it can be necessary to increase the size of Ω to improve N.

C. Definition of selection criterion

After the collect of the initial dataset, the adaptive part of SSAS algorithm refines the sampling step by step (q>1). The definition of this selection criterion is the main point of this algorithm. The objective is to minimize M, by ensuring that the final set of M measurements of F allows building an estimator that minimizes the prediction error of F at any position on Ω. During iteration on the Nqmax points in the step q, the selection criterion validates whether the value 𝐹(𝑝 𝑞𝑖 ) must be measured in order to minimize Nq.

This criterion is based on the standard deviation approach calculated from a local estimator 𝐹 ̂(𝑝 𝑞𝑖 ) which estimates

In ( 21), the dataset 𝐹(𝑝 𝑐 ) represents all field values captured at the positions 𝑝 𝑐 of the 𝑃 𝐶 dataset. As resumed by (21), the point pqi is captured whether the field deviation between 𝐹 ̂(𝑝 𝑞𝑖 ) and 𝐹(𝑝 𝑐 ) is upper than k. k defines the maximum tolerated deviation of the field, between a point and its neighbors in M, to ensure an exact reconstruction of F. The definition of k provided in section II.C.2). Fig. 7. Definition of the datasets of points in the neighbourhood of 𝑝 𝑞𝑖

1) Definition of the local interpolator

The selection criterion is computed for each point of 𝑃 𝑁𝑞 ∈ 𝑃 𝑀 ́. It is mandatory to limit the calculation time of this criterion.

The simplest way to create an estimator is to calculate the mean value of the neighbors. To increase the effect of the closest points of pqi, the mean value of the neighboring points is weighted by the distance between pqi and each point pc. The Inverse Distance Weighting (IDW) interpolator [START_REF] Shepard | A two-dimensional interpolation function for irregularlyspaced data[END_REF] is a widely used method for the spatial estimation (22). Its very short computing time makes it a good choice to compute 𝐹 ̂(𝑝 𝑞𝑖 ).

Note that during the optimization phase, the weighting coefficient value p was fixed at 3.

𝐹 ̂(𝑝 𝑞𝑖 ) = ∑ 𝜆 𝑗 (𝑝 𝑞𝑖 ). 𝐹(𝑝 𝑗 )

𝑁 𝑐 𝑗=1

Where:

𝜆 𝑗 (𝑝 𝑞𝑖 ) = 𝑤 𝑗 (𝑝 𝑞𝑖 ) ∑ 𝑤 𝑙 (𝑝 𝑞𝑖 ) 𝑁 𝑐 𝑙=1 ; 𝑤 𝑢 (𝑝 𝑞𝑖 ) = ( 1 𝑑(𝑝 𝑢 , 𝑝 𝑞𝑖 ) ) 𝑝 (23)
2) Definition of the k parameter k defines the acceptable field deviation between two nearest points to keep enough information about F. In (21), if k value is low, the probability to capture 𝑝 𝑞𝑖 is high. Otherwise, if k value increases, the probability to capture 𝑝 𝑞𝑖 decreases. In order to adjust the algorithm performance according to the objective, three definitions of k parameter are hereafter proposed. . The section V.B presents a comparative study of the SSAS algorithm using these three k parameters.

According to the definition of W (with α=10dB), the maximum deviation from the mean was fixed at 10dB. (24) defines the first simple definition of k, used in [START_REF] Serpaud | Sequential adaptive sampling algorithm to reduce the near-field measurement time[END_REF].

𝑘 1 = 10 [𝑑𝐵] (24)
Near-field measurement is often used to localize the main root cause of EMC issues, which can be identified by a hot spot of the near-field emission map. From an industrial point of [𝑑𝐵] (25)

Where: 𝐾 𝐵 ≥ 𝐾 𝐴 ≥ 0

Note that 𝑘 1 is a special value of 𝑘 2 where 𝐾 𝐵 = 𝐾 𝐴 = 10𝑑𝐵.

As shown in Fig. 8, if 𝐹 ̂(𝑝 𝑞𝑖 ) is close to the maximum level of 𝐹 𝑀 ′ dataset, 𝑘 2 tends to 𝐾 𝐴 . Otherwise, if 𝐹 ̂(𝑝 𝑞𝑖 ) is close to the minimum level of 𝐹 𝑀 ′ dataset, 𝑘 2 tends to 𝐾 𝐵 . Both coefficients 𝐾 𝐴 and 𝐾 𝐵 must be defined in order to find a tradeoff between the number of point M and the mesh optimization around the maximum level of 𝐹 𝑀 ′ dataset. According to the tolerated deviation from the mean, 𝐾 𝐵 is set at 𝑘 1 = 10dB. 𝐾 𝐴 defines the probability that 𝑝 𝑞𝑖 is captured when 𝐹 ̂(𝑝 𝑞𝑖 ) is close to the maximum level of 𝐹 𝑀 ′ dataset. If 𝐾 𝐴 is set at zero, all the points around the maximum level will be captured, increasing the number of points M. It is accepted that the repeatability of near-field measurement is lower than 1dB [START_REF] Serpaud | Technical improvement of the near field measurement: The vector near field scanning[END_REF]. So, in order to limit the number of sampling points M, 𝐾 𝐴 is fixed to 1dB. The definition of k2 is relevant to refine the meshing around the maximum level of the near-field emission map. Note that not all the sources should produce the same maximum level. Actually, several local maxima should likely appear on the near-field emission map. With 𝑘 2 definition, measurement will be only refined around the maximum level of F and not around all these local maxima. To solve this issue, (26) proposes a last improvement of the definition of k parameter. resulting that 𝑘 3 tend to 𝐾 𝐴 . tends to max (𝐹 𝐹 (𝑝 𝑞𝑎 )), 𝑘 3 also tends to 𝐾 𝐴 . Expression of 𝑘 3 builds a tradeoff between the proximity of field value at 𝑝 𝑞𝑖 with both the maximum of 𝑃 𝑀 ′ and the local maxima. 𝑘 3 improves the meshing around all the local maximum level of 𝐹 𝑀 ′ with a meshing enhancement for the highest local maxima. Fig. 9 presents an example of the evolution of the three k parameters values according to the field distribution. For readability purpose, the field is plotted in only one spatial dimension. The difference between k2 and k3 parameters is clearer in this figure. k3 tends to zero for all local maxima (max (𝐹 𝑀 ′ ) included), leading to a large capture probability, while k2 tends to zero only for the max (𝐹 𝑀 ′ ). With k3, the meshing is more refined around all local maxima.

III. DEFINITION OF THE INTERPOLATOR

After SSAS algorithm process, M points irregularly distributed on Ω are captured. To reconstruct the near-field map with the final resolution d, it is necessary to interpolate the field from the FM dataset captured in Ω. The evaluation of the spatial sampling is linked to the performance of the interpolator. The more extensively interpolation is performed the higher is the number of constraints about spatial sampling which must be

A. Kriging interpolator

The Kriging interpolation method is used here. If we accept a slightly longer computing time, it is the most powerful spatial interpolator among all the methods cited above. The Kriging interpolator, also known as Gaussian Process regression, is an efficient spatial unbiased estimator with minimum variance. The Scikit library on Python [START_REF][END_REF] has been used to process the adaptive meshing interpolation. The interpolator is initialized with the Exponential kernel.

B. Definition of the comparison indicators

In order to compare the performance of the SSAS algorithm according to the W and Q parameters ( 19) and ( 17), and the choice of the definition of the parameter k (21), it is necessary to define some indicators related to the objective of the SSAS algorithm. The main purpose of this algorithm is to reduce the measurement time, which is directly related to the total number M of measured points. The error at interpolated points is another important indicator. It defines the loss of information resulting of the SSAS algorithm process. The logarithmic error value is computing according to (27).

𝐸𝑟𝑟𝑜𝑟 𝑑𝐵 (𝑃 𝑞𝑖 ) = 20. |𝑙𝑜𝑔 10 (𝐹 ̂(𝑝 𝑞𝑖 )) -𝑙𝑜𝑔 10 (𝐹 (𝑝 𝑞𝑖 ))| (27) In ( 27), 𝐹 ̂(𝑝 𝑞𝑖 ) represents the interpolated value of F based on Kriging interpolator and 𝐹(𝑝 𝑞𝑖 ) represents the true value of F at the position 𝑝 𝑞𝑖 . An important feature of the Kriging interpolator is that it is an unbiased estimator, so that it introduces no interpolation error on the input measurement points (𝐸𝑟𝑟𝑜𝑟 𝑑𝐵 (𝐹 𝑀 ) = 0).

The error is computed only for the highest values of F. It is not relevant to compute interpolation error for the field values close to the noise floor of the measurement receiver. So only the points having a field value greater than [max(FM)+2•min(FM)] /3 are used to evaluate the error. The number of points M, the mean error and the maximum ErrorOnLocalMax are the most important indicators to evaluate the performance of the SSAS algorithm.

Note that the maximum error value can be affected by the outlier on the upper tail of the error distribution due to the spatial location error introduced after interpolation process. Therefore, it is not a relevant indicator.

Other indicators can also be added. The 95 % quantile and the number of points affected by an error less than 3 dB (NbPtserror ≤ 3dB) are evaluated since they are not affected by the outlier. ErrorOnMax, defines the error on the maximum of F. It is relevant to validate the ability of SSAS algorithm to capture the maximum level of near-field map without significant error. This indicator must be completed by the maximum ErrorOnLocalMax to take into account the error on local maxima of F. The complete indicators list used in this study is listed in Table II. Total number of captured points.

[pts] MeanError Mean interpolation error value.

[dB] ErrorOnLocalMax Error values measured on each local maximum value of FM.

[dB]

ErrorOnMax Error value measured on the maximum value of 𝐹 𝑀 .

[dB]

Quantile@95% Maximum value of the 95% of points with the lowest error.

[dB]

NbPtserror ≤ 3dB Number of points where the interpolation error is lower than 3dB.

[%]

IV. PRESENTATION OF THE VALIDATION CASE STUDIES

A. Case study #1 from simulation data: "Dipoles"

The first proposed case study (Fig. 10) is a 75 mm x 75 mm printed circuit board (PCB) simulation model including six traces routed in air at 1mm above an ideal ground plane. The six traces have the same width equal to 0.05 mm and are routed in three different directions (0°, 45° and 90°). They are excited by a sinusoidal current source at 100 MHz. The magnitude of the three magnetic field components radiated by the wire is computed [19] from analytical equations (4)(5) at several heights hmeas (1, 5 and 18 mm). Image theory is used to take into account the ideal ground plane. The sampling resolution d is 0.5 mm.

B. Case study #2 from measurement data: "FPGA"

Fig. 14 presents the XILINX Spartan 6 FPGA device test board. The frequency range of the magnetic near-field measurements starts at 5 MHz and increases up to 1.8 GHz. Two types of magnetic field probe are used: one for the Hx and Hy components and another for the Hz component. The spatial resolution of the probes is 0.5 mm. The probes are moved at 1.1 mm above the FPGA package. The height of measurement is fixed at hmeas=1.85mm above the top of the IC substrate. More information about the configuration of the FPGA is presented in [START_REF] Serpaud | Proposal for combined conducted and radiated emission modelling for Integrated Circuit[END_REF].

For both case studies, it is possible to define the theoretical optimal configuration of W and Q parameters according to (19) and [START_REF][END_REF], as summarized in Table III. Data of both case studies, including measurements, are available [START_REF]Case studies data[END_REF] for non-commercial and academic use only.

V. VALIDATION OF THE PROPOSED SSAS ALGORITHM

A. Discussion about W and Q parameters

The parameters d, W (and Q) initialize the SSAS algorithm. W and Q are coupled and define the resolution of the first step where all points are captured. Expression (19), in section II, gives the optimal expression of W according to the measurement height hmeas. Fig. 11 and Fig. 12 present the results of the SSAS algorithm applied on case study #1 for several measurement heights and several values of the parameter W (and Q).

The configuration for the optimal value of W (and Q) is underlined in light red in Fig. 11 and Fig. 12. It can be observed that the optimal values of W and Q offer the best configuration to minimize the number of points while maintaining the lowest mean error. If W is set with a greater value than this optimal value proposed in (19), the number of measurement points is not significantly reduced but the interpolation error increases rapidly leading to an erroneous reconstruction of the near-field map. This situation is marked in Fig. 12 by 'x' including the number of points at zero and no value for the mean error. It is particularly true when the k1 criterion is used. It seems that criteria k2 and k3 are more robust than k1. These results validate the optimal choice of W and Q which are defined by the simple equations ( 19) and [START_REF][END_REF] according to the measurement configuration (hmeas and rloop).

B. Discussion about selection criteria

Three selection criteria were introduced in section II.C. The selection criterion must be chosen according to the measurement objective. The selection criterion k1 is the best choice to reduce the measurement time at the price of an increase in the interpolation error. k2 criterion reduces the interpolation error around the point of maximum level and k3 around all the local maxima. However, it leads to an increase of measurement points. This general conclusion is illustrated in Fig. 15 which shows an evaluation of the number of points M, the MeanError and the maximum ErrorOnLocalMax for both case studies using the three selection criteria. @80MHz / Q=5Steps @264.9MHz / Q=5Steps @489.9MHz / Q=5Steps @1.394GHz / Q=5Steps hmeas=5mm / Q=9Steps hmeas=18mm / Q=13Steps

Fig. 15. Comparison of the SSAS algorithm performances according to the three selection criteria for both case studies Fig. 16 completes this evaluation in a large frequency range. Using k3 criterion, the error value can be greatly reduced (the MeanError < 1 dB and the MaxErrorOnLocalMax < 2 dB) without a significant increasing of the number of points. Compared to the k1 and k2 criteria, using k3 criterion leads to an increase of 3 to 6 % in the measurement points, which never exceeds 21 % of the total number of points of the near-field map with the final resolution d. With k1, the number of measurement points never exceeds 15 % of the total number of points, the MeanError value remains less than 1.6 dB, but the MaxErrorOnLocalMax value can reach up to 8 dB. Fig. 17 presents the spatial meshing characteristics according to the three selection criteria. Using k1 reduces significantly the number of points to refine the meshing around area with high field deviation. However, due to the low field deviation around the strongest values, the actual maximum is likely going to be missed out. Using k2 improves this issue by refining the meshing around the position of the maximum of field without affecting the local maxima. With k3, the sampling around the maximum of field and all maxima is improved, leading to a significant reduction of the interpolation error as confirmed by Fig. 15.

As presented by Fig. 13 and Fig. 15, the frequency does not affect the performance of the SSAS algorithm. Only the spatial distribution of the field determines the reduction of the number of points. The larger the dispersion of the near-field values spatial distribution, the higher is the number of points increased by the SSAS algorithm in order to keep enough information on F and hence to ensure a satisfactory reconstruction of the field within Ω.

C. Discussion about the measurement time optimization

The measurement time optimization is the main objective of this study. Fig. 18 shows a comparison of the measurement time according to the selection criterion. The duration of the acquisition of 𝐹 𝑀 by measurement equipment (TMeas), the nearfield probe displacement time on 𝑃 𝑀 (TMove) and the execution time of the SSAS algorithm (TAlgo) are analyzed separately. Noteworthy is that the trends of these three durations are not impacted by the frequency. They only depend from the number of points. In order to calculate the measurement time, one second per point is taken as assumption. A constant speed of 20 cm/s is considered to compute the probe displacement time (probe acceleration and deceleration phases are not taken into account).

The SSAS algorithm reduces the number of measurement points in order to select only the points that give the most information about F. As shown in Fig. 18, the acquisition time, which is proportional to the number of measurement points, occurs for 99% of the measurement time (Tctrl is here neglected). Whatever is the configuration, the probe displacement time and especially the SSAS algorithm processing time are negligible. SSAS algorithm processing represents less than 0.1% of total measurement time. It validates our objective to develop a low time-consuming algorithm. The probe displacement time is negligible compared to the acquisition time since it represents less than 1% of the total measurement time. This is the benefit of using a regular mesh (Sukharev grid) and a progressive sequential sweep described in Fig. 2. Moreover, the distance between two consecutive points is minimized. Table IV summarizes the measurement times with the full grid measurement and using SSAS algorithm configured with the three selection criteria. On case study #1 "Dipoles", SSAS algorithm reduces measurement time to half an hour whereas the full measurement lasts more than six hours. SSAS algorithm ensures a reduction of the measurement time according to the frequency by a factor of 7 to 13 using selection criterion k1 and by a factor of 5 to 10 using k3.

VI. CONCLUSION

The main objective of this work was to reduce the measurement time of the near-field scanning of electronic equipment. To reach this goal, an existing progressive Sequential Spatial Adaptive Algorithm [START_REF] Serpaud | Sequential adaptive sampling algorithm to reduce the near-field measurement time[END_REF] has been updated and improved. The proposed SSAS algorithm presents a negligible computation time. In fact, depending from the device under test, the measurement time is reduced by a factor 5 to 13 according to the spatial distribution of F and the choice of the selected criterion. Two points ensure the accuracy of the measurement method. At first, the W parameter definition guarantees that the minimal dataset is collected during the first acquisition step. Next, this paper presents two new selection criteria to complete the first proposal method [START_REF] Serpaud | Sequential adaptive sampling algorithm to reduce the near-field measurement time[END_REF]. A special attention has been given in the selection criterion definition to limit the error around the local maxima of the near-field map. The proposed approach keeps this error lower than 1 dB, which is the estimated near-field measurement error.

The proposed selection criteria are based only on the magnitude of the field captured at one frequency. The performances of this algorithm on multi-frequency range and vector measurements have not been evaluated. The definition of a specific criterion would be certainly necessary. These questions will be addressed in future works.

N

  Number of points captured during the first step. Nq Number of points captured during the step q={2..Q} Ω Represents the space domain of measurement.𝑝 𝑞𝑖i nd position evaluated during the step q.

Fig. 1 .

 1 Fig. 1. 16-points Sampling methods (--boundary of Ω; • pi sampling points)

Fig. 2 .

 2 Fig. 2. Spatial adaptive sampling where Ω={Ax,Ay} is progressively covered in 5 steps according to the two input parameters : d and W According to the size of Ω = {Ax,Ay} and W, Nqmax is bounded by the couple {𝑁 𝑞 𝑚𝑎𝑥 𝑚𝑖𝑛, 𝑁 𝑞 𝑚𝑎𝑥 𝑚𝑎𝑥 }. Fig.3shows an

Fig. 3 .

 3 Fig. 3. {𝑁 𝑞 𝑚𝑎𝑥 𝑚𝑖𝑛 , 𝑁 𝑞 𝑚𝑎𝑥 𝑚𝑎𝑥 } definition for W=2mm and Q = 3steps.

Fig. 4 .

 4 Fig. 4. Evolution of the sampling points 𝐷 = 𝑀 𝑚𝑎𝑥 = 𝑁 + ∑ 𝑁 𝑞 𝑚𝑎𝑥 𝑄 𝑞=2

Fig. 5 .

 5 Fig.5. Spatial distribution of Hy (Hx) and Hz magnetic field above a wire (Wtrace≈0) with 1mA excitation current at 100MHz, with and without ground plane (at z=0) and hwire=1mm, hmeas=1mm

Fig. 6 .

 6 Fig. 6. Radiating electric dipole equation

  . cos(𝜃) 𝑦 = 𝑟. sin(𝜃). cos(𝜑) 𝑧 = 𝑟. sin(𝜃). cos(𝜑) Height of dipole h meas = Height of measurement

  F at the position pqi. Let 𝑃 𝑀 ́= {𝑝 𝑚 } 𝑚=1 𝑀 ́⊂ 𝑃 𝑀 the temporary dataset resuming all the measurement positions of F already captured. 𝑃 𝐶 (𝑝 𝑞𝑖 ) = {𝑝 𝑐 } 𝑐=1 𝑁 𝑐 ⊂ 𝑃 𝑀 ́ defines a local dataset including all the positions neighboring pqi as shown in Fig. 7. Selection criterion: ∃𝑝 𝑐 |𝐹(𝑝 𝑐 ) -𝐹 ̂(𝑝 𝑞𝑖 )| > 𝑘

  test during step q 𝑃 𝐶 : Closest neighbors of 𝑝 𝑞𝑖 𝑃 𝐹 ⊃ 𝑃 𝐶 : Far neighbors of 𝑝 𝑞𝑖 𝑃 𝑀 ′ ⊃ 𝑃 𝐹 : All points captured view, the maximum level of radiated emission coupled to the accuracy of measurement are the most important information. Therefore, it is mandatory to focus on both parameters during the development of the SSAS algorithm. The mean error added by the SSAS algorithm degrades the accuracy of measurement. k1 must be upgraded to improve the accuracy of measurement around the position of the maximum emission level of F. A second definition of k is proposed: k2, defined by (25). It takes into account whether the value 𝐹 ̂(𝑝 𝑞𝑖 ) is close to the maximum level of 𝐹 𝑀 ′ . 𝑘 2 = 𝐾 𝐴 + (𝐾 𝐵 -𝐾 𝐴 ) ⋅ √ max(𝐹 𝑀 ′ )-𝐹 ̂(𝑝 𝑞𝑖 ) max(𝐹 𝑀 ′ ) -min(𝐹 𝑀 ′ )

𝑘 3 = 10 )

 310 𝐾 𝐴 + (𝐾 𝐵 -𝐾 𝐴 ) max(𝐹 𝐹 (𝑝 𝑞𝑖 ))-𝐹 ̂(𝑝 𝑞𝑖 ) [𝑑𝐵] (26) With 𝐹 ̂(𝑝 𝑞𝑖 ) ≤ max (𝐹 𝐹 (𝑝 𝑞𝑖 )) ≤ max(𝐹 𝑀 ′ ) 𝑘 3 introduces a proportional coefficient relative to the gap between 𝐹 ̂(𝑝 𝑞𝑖 ) and max (𝐹 𝐹 (𝑝 𝑞𝑖 )) the local maximum value extracted from a local sub-dataset of 𝐹 𝑀 ′ centered around 𝑝 𝑞𝑖 and named 𝐹 𝐹 (𝑝 𝑞𝑖 ) = {𝐹 (𝑃 𝐹 (𝑝 𝑞𝑖 )) , 𝑃 𝐶 ⊂ 𝑃 𝐹 ⊂ 𝑃 𝑀 ′ }. Here 𝑃 𝐹 (𝑝 𝑞𝑖 ) = {𝑝 𝑓 } 𝑓=1 𝑁 𝐹 defines the set of the NF far neighboring positions from 𝑝 𝑞𝑖 as defined in Fig. 7. Fig. 8 illustrates the relationship between the different definitions of k parameter and the 𝐹 ̂(𝑝 𝑞𝑖 ) value. If 𝐹 ̂(𝑝 𝑞𝑖 ) is close to the max (𝐹 𝑀 ′ ), max (𝐹 𝐹 (𝑝 𝑞𝑖 )) tends to max(𝐹 𝑀 ′ )

Fig. 8 .Fig. 9 .

 89 Fig. 8. Relationship between the three k parameters and the 𝐹 ̂(𝑝 𝑞𝑖 ) value

  the number of samples which will be reduced. The most popular spatial interpolation methods are listed below:  Inverse Distance Weighting (IDW) interpolator [14]  Triangulated Irregular Network (TIN) interpolator also called Delaunay triangulation interpolator [15] Kriging interpolator[START_REF] Kriege | Two dimensional weighted average trend surfaces for ore evaluation[END_REF]

Fig. 10 .

 10 Fig. 10. Description of case study #1 "Dipoles"

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

 111213 Fig. 11. Evaluation of the impact of W and Q parameters : Case Study #1 "Dipoles" (Hy measurement, hmeas = 5mm, Selection criterion=k1) (Note: "KO" means that the reconstructed of the near-field map is erroneous) NbPts (M/D) "Dipoles" @100MHz MeanError NbPts (M/D) "FPGA" @ hmeas=1.85mm MeanError [%] hmeas=1mm hmeas=5mm hmeas=18mm [dB] [%] @489.9MHz @1.394GHz [dB]

Fig. 14 .

 14 Fig. 14. Description of cases studies #2 : "FPGA"

Fig. 16 .Fig. 17 .

 1617 Fig. 16. The evaluation of the errors indicators vs. frequency according to the three selection criteria on FPGA case study (Q=5-Steps)

Fig. 18 .

 18 Fig. 18. Comparison of the different contributions to the measurement duration according to the selection criterion of the SSAS algorithm

  Quantile @95% /NbPts error ≤ 3dB Error OnMax / Error OnLocalMax [dB] NbPts M (N) -(M/D) Error map |A-B| [dB]

  she was responsible for the overall leadership and management of this new engineering institute. Her research interests at LAAS -CNRS Toulouse include signal integrity in nano-scale CMOS ICs, electromagnetic compatibility and reliability of ICs. She has authored and co-authored 3 books, more than 120 publications in peer-reviewed journals & conference proceedings and supervised 14 PhD theses and 9 M.Sc. theses. Fabio Coccetti received the Laurea M.S. degree in electrical engineering from the University of Perugia, Perugia, Italy, the Ph.D. degree in high frequency engineering at the Technische Universitt Mnchen (TUM) in Munich Germany, and the Habilitation direction de recherche (HDR) from INP-Toulouse, in 1999, 2004, and 2013, respectively. He has been visiting scientist at University of Michigan USA in 2000 and from 2004 to 2008, he has been working as research scientist at the Laboratoire dAnalyse et dArchitectures des Systmes at the Centre National de la Recherche Scientifique (LAAS-CNRS) inToulouse, France. From 2008 to 2016 he has been project manager at Elemca (Toulouse) and from 2016 to 2018 general manager of RF Microtech France. Since March he joined the IRT Saint Exupery as head of the Component Modeling and Reliabilty competence center within the the More Electrical Aircraft Department. Dr. Coccetti has been involved in numerous research projects going from the theoretical and experimental investigation of power handling in RF-MEMS to design modelling and experimental validation of reliability for Microelectronics, nanosystems and more recently power electronics. Dr. Coccetti has more than 150 publications in peer reviewed conference and journal papers, and he is co-author s of two book chapters. He is has been European Microwave Lecturer (2013-2016) and member of the European Microwave Association (EUMA). He is member of the IEEE society, cofounder and chair of the RFNanotechnology Committee within the IEEE-MTT society. He was the recipient of the IHP InternationalWolfgang Mehr Fellowship Award in 2016.

Table II :

 II Indicators for SSAS Algorithm Evaluation

	Indicators	Definition	Unit
	Time	The complete time of measurement	[min:s]
		including TAlgo, TMeas and TMove.
	TAlgo	Algorithm processing time	[min:s]
	TMeas	Time to capture FM.	[min:s]
	TMove	Total time to move probe on PM.	[min:s]
	M

Table III :

 III SSAS Algorithm Parameters

	Parameters	Values for case studies Unit #1 #2
	rLoop	0.5	0.5	[mm]
	hmeas	1 / 5 / 18	1.85	[mm]
	𝑄 ≤ 3 + 2 ⋅ ⌊𝑙𝑜𝑔 2 (ℎ 𝑚𝑒𝑎𝑠 𝑟 𝑙𝑜𝑜𝑝 ⁄	)⌋ 5 / 9 / 13	5	[step]
	𝑊 = 𝑟 𝑙𝑜𝑜𝑝 ⋅ 2 (𝑄-1) 2 ⁄	2 / 8 / 32	2	[mm]

Hx

Hy Hz


Vincent Benoist
email: vincent.benoist@enit.fr

Jean- Pierre Faye
email: jean-pierre.faye@enit.fr

Improved Design methodology for Additive Manufacturing including machining load cases: Application to an aeronautical workpiece

Keywords: DFAM, Topology optimization, Support, Cutting force model, Vibrations, Stiffness

This paper focuses on the impact of post processing machining and mounting / unmounting phases on topological optimization methods. In addition to the classical load cases taken into account in optimization methods, various static and vibratory forces occurring during machining phases and manipulations, are also considered. Results of the topology optimization show that the most significant load case for the aeronautical part considered is indeed one of the post processing machining operations and it has a strong impact on the optimized shape. This result shows that such post processing phases have to be taken into account, whatever they represent supports removal or finishing surfaces. Supports are most of the time mandatory but their removal by machining operation may be require a non-negligible time, skill and cost. In this paper the machining operations are first studied through their effects on topological optimization. Then the cutting phenomenon is numerically modeled to explore various options of build supports and prevent the part from damages due to tearing. Finally, this paper demonstrates that cutting forces analysis should encourage designers (i.e. topology optimizers) and machinists to interact, in order to minimize the total cost of the additive manufactured workpiece

Introduction

For more than 20 years additive manufacturing has been used for prototyping in research and development fields. Nowadays the aeronautic industry is able to 3D print functional parts at production scale especially using SLM (Selective Laser Melting) technologies (for melting part), but mostly exclusively for high cost parts [START_REF] Barlier | Residual stress prediction in a powder bed fusion manufactured Ti6Al4V hip stem[END_REF]. Additive manufacturing allows engineers to create very innovative designs but machining post processing operations are quite always necessary, and these parts are often very difficult to machine with conventional turning and milling processes, increasing final cost. Using Design for Additive Manufacturing (DFAM) methods [START_REF] Louvigny | L'optimisation topologique et la fabrication additive amélioration de la chaine de conception[END_REF] [START_REF] Hällgren | Design for Additive Manufacturing, 26 th CIRP Design Conference[END_REF], such as topological optimization, engineers should be able to design the best possible part in order to comply technical specifications and to minimize production and postproduction complexity especially with low rigidity parts. The cost of additive manufactured parts is very strongly related to the size and the material used, but it is quite independent of the shape complexity. Machining may represent up to 50 % of part final cost, and sometimes simply reveals to be impossible without part redesign. Aeronautical parts redesigned by topology optimization still similar in size and weight compared to the subtractive manufacturing parts. This induces that the optimize part quotation target is similar to the subtractive manufacturing part, especially if there is a lack of information about the available space around the part, or the additional features that could be included in the part redesign. Furthermore, topology optimization in an aeronautical context has a strong goal of mass reduction which often leads to optimized parts with a lower stiffness and a more complex shape [START_REF] Benoist | Topological optimization design for additive manufacturing, taking into account flexion and vibrations during machining post processing operations[END_REF]. This low stiffness and organic shape have to be precisely monitored to prevent part damage and to avoid costs due to eventually additional supports. The supports, even those just needed for linking the part to the build plate, also bring another non-negligible constraint of post processing. The entry point of this study is the cutting forces, taken into account during the topology optimization. Beside the cutting forces, that may lead to an unacceptable flexure and poor accuracy, vibration must also be avoided during machining, because it would lead to poor surface quality and possible tool or workpiece break. To ensure that all the numerical models are consistent with reality, numerical results are compared to experimental tests. It is important, here to highlight that machining time, must be precisely taken into account for optimizing the total cost of the part. The machining time has been reduced here from 40 minutes, for the subtractive initial manufactured part) to only 6 minutes for the additive manufactured part). To obtain the best total cost for the additive manufactured part it is important to realize that machining time has to be maintained to the lowest possible level from this 6 minutes level. For example, increasing slightly machining allowance to increase stiffness during machining may imply to add a machining pass and thus to double the machining time. Modeling and experimentation

Cutting forces modeling

The first step, is the definition of realistic cutting forces. In order to validate a first cutting force model, finite elements calculations from the Abaqus software have been done and results compared to literature. This first 2D model is a cuboid sample representing the part, and a rigid shape representing the cutting tool (figure 3).

The literature provides the elastic material parameter for powder bed fusion manufactured Ti6Al4V. The additive manufactured titanium has the particularity to be created at the same time as the part take shape and then to be thermally tempered. So, the material parameters are slightly different from cast titanium due to the so called hyper-quenching.

The literature (Barrett, 2017) provides parameters for modelling the plastic strain during machining of 3D printed Ti6Al4V. [START_REF] Johnson | A constitutive model and data for metals subjected to large strains rates and high temperatures[END_REF] parameters are given as it is the most used material model in FE machining simulations, and defines the relationship between strain, strain rate and temperature as described in (Equ.1). The material studied from (Barrett, 2017) may be slightly different from the material we used during the printing job, because of thermal treatment.

= (! + "# $ ) %1 + & '* % ## ̇$--.1 -0 2 -2 3445 2 5678 -2 3445 9 5 : (1)
As said before, Johnson-Cook material model relates strain, strain rate and temperature to flow stress in a multiplicative form through the parameter A, B, n, C and m. These material parameters have been found in the literature (Barrett, 2017) and are summarized in (Table 1). To complete material parameters, a damage law is taken into account. The Johnson-Cook failure model defines the strain at fracture as depicted in (Equ.2) [START_REF] Johnson | Fracture characteristics of Three Metals Subjected to Various Strain[END_REF]:

# ; = (< > + < ? . exp (< B C) %1 + < D '* % ## ̇$--%1 + < E 0 2 -2 3445 2 5678 -2 3445 9- (2)
Where # ; is the equivalent fracture strain, C the stress triaxiality parameter and the parameters < > , … , < E are determined based on a series of experimental fractures tests that depends on variation of stress triaxiality, strain rate and temperatures. Damage parameters used come from [START_REF] Zhang | On the selection of Johnson-Cook constitutive model parameters for Ti-6Al-4V using three types of numerical models of orthogonal cutting[END_REF] but these parameters concern forged titanium. Nevertheless, we used these parameters because of the lack of information in the literature about 3D printed titanium damage law. The model is then created in Abaqus, with a dynamic, explicit step on a time period of 0.04 second. This time is enough for the tool to cross the sample at 0.25 m/s speed.

As shown in Figure 2 the mesh has been refined in the machined zone of the sample in order to have accurate results. In cutting modelization, chip formation induces large mesh distortions and also brings the necessity to use a separation criterion to prevent numerical problems. An Arbitrary Lagrangian Eulerian (ALE) formulation has been chosen for this simulation. It allows to integrate the advantages of Eulerian and Lagrangian representations in a single description and so reduce mesh distortions [START_REF] Hashemi | Mesa Model of Orthogonal Cutting Process[END_REF].

Finally, we used a Coulomb friction law, G = 0.2, to model the contact zone between the tool and the chip, as used in literature [START_REF] Pantalé | 2D and 3D numerical models of metal cutting with damage effects[END_REF]) [START_REF] Zhang | On the selection of Johnson-Cook constitutive model parameters for Ti-6Al-4V using three types of numerical models of orthogonal cutting[END_REF].

Support modeling

The second step of this paper is to investigate the machining of the supports and especially the plastic residual strain after support removal on the part. To answer this question a numerical model, based on the previous one, is created. The model is still a two-part assembly of a sample with supports and the tool. The only change between support removing model and machining models is the sample geometry (2D to 3D).

Three geometries have been created to test different supports orientation strategies 0°, 90° and 45° (figure 3). Each support has a width of 1 mm to represent classical support size. The supports are spaced apart of also 1 mm.

Built plate machining

The last investigation in this paper, is to look for the optimization of the built plate machining phase. In additive manufacturing, the built plate is considered as a consumable like the powder. After every printing job the supports have to be cleaned from the plate in order to be used again. The built plate has geometric constraints to respect in order to ensure a well quality printing for the next job. The three main constraints to respect are parallelism (0.05 mm), the flatness of the built face (0.05 mm) and the roughness (R ≈ 1.6 μm).

To process the machining phase, we used an J 3 = 45° milling tool with strong inserts to surface the plate.

Aside from the geometric restrictions, our observations during built plate machining show that the built plate behavior during cutting may be very different from the printed material. For example, for 316L 3D printing, the plate material is structural steel (S355) while the printed material is stainless steel (316L). This difference is strongly affecting the cutting process because the two steels have contradictory optimal cutting parameters. The optimal cutting conditions must be able to deal with both materials to obtain high quality surface and high removal rate. The cutting conditions for this machining phases are: spindle speed of 1500 rpm and a feed rate of 0.033 mm/teeth.

Cutting forces modelling results

The first 2D numerical model concerned orthogonal cutting process. The model is constituted of 16948 linear quadrilateral elements of type CPE4R and a total number of nodes of 17202. The results, (Figure 3) with Von Mises stress field, shows a classical chip formation during tool penetration with primary shears bands and is in accord with the Oxley theory [START_REF] Oxley | Mechanics of Machining: An Analytical Approach to Assessing Machinability[END_REF].

Figure 3: Creation of shear stress zones

The results show that the cutting force value is always around 2000N, with a +10% maximum at 0.002 s, i.e. when the tool is just becoming to generate the chip, as illustrated (Figure 3).

Supports removal modelling results

In this section, an extension of the 2D model presented above has been realized to perform an 3D model in order to take into account the supports geometry. The number of elements and nodes in each model, are summarized in (Table 3). For the tool, the elements and nodes description are the same for the three models. The tool is composed of 944 eight nodes thermally coupled brick explicit elements and 1269 nodes. The cutting forces results agree with 2D results for each modelling.

In figure 5, the difference between the supports are highlighted by showing the residual displacement of the parts surfaces elements and the equivalent plastic strain values (code coloured). The plastic strain and the residual displacement allows us to evaluate if the support is ripped off the sample or if it properly cut by the tool.

Figure 4: Plastic strain values for each model

Analysis of the simulations shows that only the cutting of the supports oriented at 90 ° has a weak impact on the sample.

It is important to note that the plastic strain and the residual displacement may not be the wisest criterion but further machining are still ongoing to determine the best criterion.

This first analysis allows to conclude that the support orientation is non-negligible criterion in such configurations. The designer may to take it into account in order to prevent against possible failure or poor surface quality of the part after support removal machining phase. This first result shows that the supports should be parallel to the cutting direction if possible.

More precise analysis and models are still to be done, in particular with parameters from machining tests on titanium part, like feed rate, type of cutting tool, cutting configurations etc.

Built plate machining results

A first machining test has been done on steel (S355 and 316L), to investigate the problems related to the machining of the build plate. Due to a strong material difference between the built plate and the printed part, the cutting conditions have to be optimized to ensure a good surface quality of the built plate. For our case, the plate, with a low stiffness, need to be machined at spindle speed of 1500 rpm and a feed rate of 0.033 mm/teeth. This condition allows us to have the Ra of 0.016 mm but with low removal rate.

During this machining test, we saw some changes in the metallurgical structure of the building plate due to the thermal constraints caused by the laser. These changes are located exactly where the supports and the plate are welded. The thermal effect is visible on a thickness of ≈ 0.2 mm inside the plate, and affects the surface finish during machining.

Further tests of built plate machining will allow us to study this phenomenon, i.e. the thermal effect on the plate. So, the cutting conditions will be optimized in order to make this processing as industrially as possible.

Conclusion and perspectives

The previous paper [START_REF] Benoist | Topological optimization design for additive manufacturing, taking into account flexion and vibrations during machining post processing operations[END_REF] allowed us to understand that the most significant load case, when a part is designed for additive manufacturing, may be due to machining and what we called manipulation load cases of the part (i.e. during mounting/unmounting). These results showed the need to take into account such load cases during the design process, in order to avoid unexpected breakage of the workpiece and/or complex supports during machining.

The paper presented here focus on the support removal, which is considered as a major issue in additive manufacturing. These supports are mandatories to ensure a good quality of printing but their removal for metal is not an easy post processing phase. This paper allows us to better understand how design method may include the work of the machinist and ensure better quality of the part. In this first study we have considered that during machining Ti6Al4V generated by additive manufacturing is similar to forged titanium during machining. Numerical modeling of the supports removal shows that the orientation of these supports regarding to the cutting direction may have a great impact on the surface quality and plastic strain of the surface of the part. It appears that supports built parallel to the cutting direction minimize plastic strain and surface deformation due to material pull-out.

Another part of this study is about of the build plate machining. We have observed on 316L 3D printing that a difference of material metallurgy between the plate and the part implies to adapt the cutting conditions. So, different cutting conditions have to be optimized to ensure acceptable surface quality for the part and for the build plate.

As it generates near shape parts, additive manufacturing often strongly reduces machining time. Indeed, the machinist would just need to machine functional surfaces of the part. Unfortunately, near net shape is often associated with lower rigidity parts during machining, so a classical solution is to add supports or to add thickness on the part, in order to facilitate the machining. These solutions are usually not optimized for industrial costs, because it increases the machining time. The initial subtractive manufacturing part has a machining time of 40 min and the additive manufacturing part have a machining time of 6 min. Adding for example an extra machining pass would lead to ≈ 12 min of machining, i.e. would double the machining time. So, this analysis shows the need for strong collaboration between designer and machinist.

In this paper we obtained several firsts results about machining additive manufactured context. These results will be investigated further with other new experimental tests and simulations.

Future work will concern: -a more complete comparison of different types of supports, their orientations and problems generated by their machinability -a further analysis of the problems that the machinist would encounter with additive manufacturing part, like low stiffness, vibrations, supports machinability, build plate machinability and effects of these machining phases on the tool wear. -a more precise analysis of the possible interactions between the designer and the machinist to ensure quality parts.

Figure 1 :

 1 Figure 1: Subtractive manufacturing (a) and additive manufacturing part (b)

Figure 2 :

 2 Figure 2: Geometry strategies 0 ° (a) up to 90 ° (b) with a 45 ° increment (c)

Table 1 : Johnson-Cook material model parameters

 1

	A	B	n	C	m
	970 MPa	756 MPa	0.21	0.047	1.05

Table 2 : Johnson-Cook failure material model parameters

 2

	< > -0.09	< ? 0.25	< B -0.5	< D 0.014	< E 3.87

Table 3 : Elements and nodes number in each model for the sample

 3

		Model 0°	Model 90°	Model 45°
	Number of elements 12 400	41 200	29 698
	Number of nodes	25844	67 328	46 378