2D multiple waves scattering for active detection of a dummy human body in a low frequency range and for various boundary conditions

This paper investigates the results of simulations devoted to the acoustic field variation caused by different boundary conditions applied to the backscattering of the acoustic wave in both low and moderately low frequency ranges. The goal is to analyse the object behaviour and the detection ability of the 2D acoustic field approximations when the simulation data of the scattered sound field is available. The plane waves are backscattered by a dummy human body, positioned in a cabin which has a defined size. The dummy human body is created by using clusters of finite and disjointed but tangent circular cylinders. Each cylinder is a scatterer that emulates various boundary conditions i.e., the scatterers can be sound-soft (Dirichlet boundary condition) and sound-hard scatterers (Neumann boundary condition). The cylinders are filled with biological tissue (i.e., fat). The exterior scattering problem is solved using the preconditioned Krylov subspace iterative solvers (GMRES), as a computationally cheaper alternative. We use the hypothesis that the mass density of the obstacle and wave speed in the obstacle are both constant. During the simulation experiments, we neglected the interactions among scatterers. A frequency range from 100 Hz to 1000 Hz is analysed. Both the Sound Pressure Level (SPL) and Radar Cross Section (RCS) are used to analyse the numerical simulation of the sound field.


Introduction
In recent years, inverse scattering has received substantial research attention and several models devoted to the scattering of time-harmonic acoustic waves by non-penetrable or penetrable inhomogeneous obstacles have been proposed.Multiple scattering problems were successfully applied in fields such as acoustics, medicine, electromagnetism and electrodynamics [1][2][3].
The distorted Born approximation method has been implemented to solve 2D acoustic problems [4,5].For a rigid scatterer, the boundary conditions are devoted to vanishing the total displacement field on the surface of the scatterer.Both boundary conditions together with longitudinal wave scattered by a small obstacle are analyzed under the hypothesis the size of the obstacle is smaller than the wavelength of wave [6,7].Acoustic backscattering quantifies the disturbance of an incident acoustic field due to an object's shape and surface properties [8].The 2D scattering problem contains generally-heterogeneous and spatially-bounded obstacles.
We consider acoustic scattering by a dummy human body positioned in a cabin, which has a defined size.The dummy human body is created by using clusters of finite and disjoint but tangent circular cylinders viewed as obstacles.The geometry of the human body is described with high accuracy.Each cylinder is a scatter that emulates various boundary conditions i.e., it can be either sound-soft (Dirichlet boundary condition) or sound-hard scatterer (Neumann boundary condition) [9][10][11][12][13].The boundary condition can be modelled as fixed-rigid or sound-hard and it requires the normal velocity to be zero at the boundary interface.When the boundary condition is modelled as being pressure-release or soundsoft, it requires zero acoustic pressure at the boundary interface.Both Dirichlet or Neumann boundary conditions lead to the total reflection of an impinging wave.The re-scattering from the scatterer and rereflection from the surface are not considered.Both approaches are influenced by the frequency range, the incident wave characteristics and the nature of the inhomogeneities in the propagation medium.The cylinders are filled with biological tissue (i.e., fat tissue).The exterior scattering problem is solved using preconditioned (matrix-free) Krylov subspace iterative solvers (GMRES) as a computationally cheaper alternative [13,14].The truncation procedure is used and the exterior domain is truncated to a fictitious boundary that surrounds the scatterers and bounds the computational domain with inner and outer boundaries.We use the hypothesis that the mass density of the obstacle and wave speed in the obstacle are constants.
In this paper, two analytical and numerical models were implemented to design a geometric shape similar to the human body, along with its material properties.This was done in order to better represent the human organism and to get more accurate results when the acoustic backscatter fields were investigated.The simulation experiments are performed neglecting the interactions between the scatterers.Only the absorption properties and the shape of the scatterers are considered.The frequency range is from 100 Hz to 1000 Hz.Both the total acoustic field and total scattered field are used to evaluate and compare the backscattering properties of the human body in the approaches that were used.Both the Sound Pressure Level (SPL) and Radar Cross Section (RCS) are used to analyze the numerical simulation of the sound field.
The main objectives are as follows: (1) predict the total backscattered field of the dummy human body target (absolute value of the total field and absolute value of the scattered field) as a function of frequency, (2) compare the Sound Pressure Level (SPL) and the Radar Cross Section (RCS) computed at different frequencies, using the same target and two boundary conditions, and finally to (3) summarize the target behavior in the low and moderate low frequency field.
The proposed approach could have other potential applications.It can be used during the creation of interactive games or virtual environments as a realistic generator of sound field effects.

Formulation of the problem
An incoming wave impinges upon the obstacles' surface and the scattered field have to be computed.

Target and Environment
The problem considered in this paper is depicted in Fig. 1.A dummy human body is positioned in a cabin which has a defined size.The geometry of the human body is described with high accuracy.The Matlab toolbox µ-diff is used to efficiently solve scattering by a cluster of separated circular cylinders [13].The acoustic parameters of the cylinders are as follows: the mass density ρ1 and sound speed c1 and ρ and c are the same parameters, but for the surrounding air.The mass density and speed of sound inside the cylinders were set to ρ1 = 911 kg/m 3 , and c1 = 1440 m/s, respectively.Air is the surrounding fluid with ρ = 1.20 kg/m 3 at an ambient temperature of 20 °C and the speed of sound is c = 340 m/s.The dummy body hight is 1.80 m.The cabin size is 6 m × 3 m (length × high).

Numerical results
The aim of the computations was to numerically analyse the acoustic total field (which consists of the incident field and scattering field) based on two boundary conditions assumption: sound -soft scatterers (i.e., Dirichlet boundary condition) and sound -hard scattered (i.e., Neumann boundary condition).The numerical examples are presented in a 2D space.The exterior scattering problem is solved using the preconditioned Krylov subspace iterative solvers.
The algorithm evaluates the scattering field by partitioning the homogeneous acoustic medium Ω into M patches so that, Ω = ⋃ Ω / 0 /1( .The considered integral operators involve the summation over the patches.The truncated infinite Fourier system is used to provide a finite dimensional problem.We restrict the sum over m ∈ Z to a finite number of Fourier modes that depends on  2 = )34 ! 5 , p = 1, ..., M. To truncate Fourier series related to the potential Φ 6 2 , m ∈ Z belonging to a scatterer Ω "2 , only 2 2 + 1 modes were kept.The indices m of the truncated series satisfies, ∀ p = 1, ..., M, − 2 ≤  ≤  2 [13].
The SPL (dB) and RCS (dB) curves have been plotted for a plane wave and a human body height of 1.80 m.Various frequency values (i.e., 100, 300, 500 and 1000 Hz) were chosen.The created geometry and distribution of circular scatterers are liable for generating the scattered signal.The assembling matrix of integral operators was created.It relates the scattered waves provided by scatterers to the scatterers situated at different positions.

Results and Discussion
The dummy body is immersed in a 'fluid'/acoustic field flowing parallel to the sagittal axis (i.e., xOy plane).Numerical experiments for both boundary conditions are presented, demonstrating the object behaviour in low frequency field based on the scattered sound field predicted data.
Figures 2 and 3 present the comparisons between boundary acoustical model predictions in a graphical format.In addition, a comparison between SPL and RCS values are given in Figure 4, for all considered frequency values.To the best of our knowledge, few studies have investigated predictions from numerical backscatter methods using data collected from a dummy human body.These predictions are evaluated by comparing them to empirical SPL and RCS measurements, under the assumption of some simplified hypothesis.There are various anatomical differences among different human bodies that may increase the variability within and between data sets.They may also change the results of simulations.To minimize the bias associated to these ambiguities the same anatomical features were used during the code implementation.The only variables in the proposed model are the boundary conditions.Nevertheless, some important contributions in anatomical scattering structures and model assumptions in both the numerical model and the code implementations have been achieved.All simulations reveal a backscatter shift between forward and back propagation.As we can see in Figure 4, the simulation model successfully predicts the peaks and troughs of the near-field data.The presence of RCS maxima around 180 • is associated with orientations where the dummy yields a greater insonified normal surface.For 500 Hz and 1000 Hz, small oscillations exist near the backscattering directions (around 180 • -220 • ), mostly in the Neumann approximation.These small fluctuations are generated by the complexity of decomposition of both scattered and incident signals and the complexity of overlapping at the receiver near the backscattering angles, as well.
Generally, the diagrams depict three maximum value regions of RCS and SPL.The absorption property is presented near 90 • and 270 • and a maximum value for the backscattering appears at 180 • , for all considered frequencies.Between 170 • and 210 • the maximum value is due to the contribution of the interaction of the reflected waves on the torso.The secondary peaks are attributed to the individual contributions of the arm and leg.When the frequency is gradually increased (i.e., 500 Hz and 1000 Hz) new peaks describing the absorption properties around 160 • and 220 • have appeared.When the frequency decreases, both the RCS and SPL show a broadened peak and slower oscillations.Also, the number of oscillations is reduced.

Conclusions
This paper investigated the elastic scattering problems with Dirichlet or Neumann boundary conditions for a particular physical problem, using a dummy human body and in 100 -1000 Hz frequency range.Comparisons between the scattering results for the proposed obstacle have shown almost similar patterns under the considered boundary conditions for the frequency between 100 Hz and 300 Hz and dramatic differences under the frequency of 500 Hz and 1000 Hz.This finding could be useful for further investigations devoted to the detection ability of 2D acoustic field approximations.These results show that the investigation of the human body could be relatively insensitive to the boundary conditions for the low frequency range and dependent for the moderate low frequency range.
The scattering of sound is a complex phenomenon of size, shape, and material properties of the body as well as acoustic frequency.For a sound investigation, one should consider each parameter in the scattering predictions.Also, the impedance boundary conditions (that allow the wave propagation problem separation into independent sub-problems which are solved separately) or penetrable scatterers (that ensure the continuity of the velocity field and the continuity of the pressure across the interfaces between two domains) should be considered in future studies.

1 2
,  = , where j = 0 or 1 on boundary Γ = ∂Ω " lim |x|→∞ |x| 7∇•u x |x| -iku9 =0, Sommerfeld's radiation condition here k is the real wavenumber.We restrict our study to Dirichlet (for  = − . !#$ ) and Neumann (for  = − (  !#$ ) boundary conditions, where  , is the exterior trace of order j [12].When the impedance || = ∞ ,  # u =− #  !#$ , on boundary Γ = ∂Ω " the Neumann boundary condition is in force.The acoustic wave is incident onto a rigid material and the normal component of the fluid velocity must vanish at the surface of the material.It is a no-flow condition.When the impedance Z = 0, then  = 0 which is the Dirichlet boundary condition.

Figure 1 .
Figure 1.Schematic diagram for a collection of rigid cylinders that generate a dummy human body.The source of plane wave is placed of the left side.The plane wave propagates from left to right.

Figure 2 .
Figure 2. Absolute value of the total field and absolute value of the scattered field for Dirichlet problem of scattering of a plane pressure wave by a dummy human body scatterer and frequency of 100 Hz and 300 Hz (top row), 500 Hz and 1000 Hz (bottom row).The incident angle is π.In this approximation, the amplitude of the wave is zero on the surface of the object.

Figure 3 .
Figure 3. Absolute value of the total field and absolute value of the scattered field for Neumann problem of scattering of a plane pressure wave by a dummy human body scatterer and frequency of 100 Hz and 300 Hz (top row), 500 Hz and 1000 Hz (bottom row).The incident angle is π.In this approximation, the derivative of the amplitude of the wave is zero on the surface of the object.

Figure 4 .
Figure 4.The RCS (dB) (top row) and SPL (dB) patterns (bottom row).First column depicts the Dirichlet boundary condition and the second column is for the Neumann model.