Elsevier

Wave Motion

Volume 102, April 2021, 102717
Wave Motion

A frequency domain method for scattering problems with moving boundaries

https://doi.org/10.1016/j.wavemoti.2021.102717Get rights and content

Abstract

We propose a multi-harmonic numerical method for solving wave scattering problems with moving boundaries, where the scatterer is assumed to move smoothly around an equilibrium position. We first develop an analysis to justify the method and its validity in the one-dimensional case with small-amplitude sinusoidal motions of the scatterer, before extending it to large-amplitude, arbitrary motions in one- and two-dimensional settings. We compare the numerical results of the proposed approach to standard space–time resolution schemes, which illustrates the efficiency of the new method.

Introduction

In wave scattering theory, it is well-known that the motion of the target modulates the frequency of the reflected wave, which is the so-called Doppler effect [1], [2]. For a motion with uniform velocity, the Doppler frequency shift can be computed easily [3]. The prediction of Doppler shifts for more general movements is commonly obtained through simple approximate models combined with signal processing [1], [4], [5], [6], [7], [8], [9].

The radar detection of non uniformly moving scatterers has found a tremendous number of applications in recent years, thanks to the availability of new high-frequency sensors and devices. For example, the radar frequency ranges 24–24.5 GHz and 61–61.5 GHz (the so-called ISM bands) are standard for many applications, and the new 77–81 GHz band is currently considered for automotive applications. The THz frequency range will appear in applications in the next few years, e.g. at 140 GHz. A major advantage of high-frequency radar sensing is its sensitivity to micro movements [1], [6], when the scatterer includes several moving parts. This results in the so-called micro-Doppler effect [1], [6], which is used with great success from drone detection [10] or the analysis of pedestrian movement [5], [6], [11] to the modeling of the effect of the rotation of helicopter rotor blades on the radar signature of the aircraft [12]. In the automotive industry, micro-Doppler sensing has recently been proposed [7], [8], [13], [14] for the contactless detection of vital signs like breathing of infants left alone on the back seat of overheating cars. The engineering of such very high frequency radar sensing devices entails dealing with multiple challenges, as for instance the analysis of random body movements and vehicle vibrations [7], [8], [9], [13], [15], [16], leading to radar signatures that can be classified for example by deep learning techniques [10], [17], [18]. In order to design these new sensors, an adequate full realistic simulation of the underlying high frequency scenarios is crucial.

Concretely, a suitable physical modeling of the (direct) problem leads to solve a time-dependent wave propagation problem in a complex environment (e.g. the interior of a car), which has both a complex geometrical shape and involves various materials that interact strongly with the high frequency emitted signal. In addition, the moving target is also usually of complex shape and materials (e.g. the infant on the back seat of the car), and characterized by small amplitude displacements at extremely low frequencies compared to the emitter. The natural mathematical framework to model such physical problems is the derivation of an adapted system of partial differential equations (PDEs), which are to be solved numerically when the configuration under study involves complex geometries or materials. Additional effects such as random vibrations can be handled by adding or modifying the PDE system.

The solution in a PDE setting of moving target problems has already received some attention in the mathematical and engineering communities. Analytical approaches for solving wave-like problems with simple motions have been developed e.g. for rotating obstacles [19], [20], [21], [22] or vibrating objects [9], [20], [23], [24]. In addition, numerical schemes based e.g. on FDTD [24], [25], [26] or fast integral equation solvers [27] were also investigated. More mathematical works related essentially to one-dimensional moving boundary problems have also been proposed e.g. by Fokas and his co-authors [28] to recast the problem as a Volterra integral equation in a fixed domain, or by Christov and Christov [29] for an asymptotic multiscale analysis of the Doppler effect in a half-space. To the best of the authors’ knowledge, however, the numerical solution of the micro-Doppler PDE modeling problem has not been addressed yet.

In the present paper, we propose an original frequency domain method to address this problem, which leads to the solution of coupled systems of Helmholtz-type equations. First, we transform the constant coefficients wave equation in the moving domain as a new wave-like equation in a fixed domain but with variable coefficients related to the metric change (a similar approach was used for quasi-static electromagnetic models in [30]). Since we are studying the micro-Doppler problem (small amplitude and low frequency oscillations of the scatterer) for a high frequency radar, we can then expand the solution in the Fourier domain in time as a Fourier series expansion centered around the radar frequency, modulated by the low frequency perturbation induced by the scatterer movement. For small amplitude movements, the variable coefficients wave equation can be simplified thanks to the small amplitude, and the series expansion can be truncated to keep a finite number of discrete frequency components related to the small amplitude variations. The resulting approach then yields a coupled system of Helmholtz equations for the wave numbers defined by the discrete frequency components kept by the approximation. For larger amplitude movements, a similar analysis can be developed based on adding more Fourier modes since the frequency coupling is stronger, resulting in a larger system of coupled Helmholtz equations with variable coefficients. This approach is valid for any dimension and configuration, and the resulting frequency domain coupled formulation can be solved by efficient numerical methods adapted to solving Helmholtz-type equations in the high frequency regime. In the present paper, we propose an approach based on the finite element method [31] which is known to be flexible to handle two- and three-dimensional complex engineering configurations, including complex materials and shapes. In addition, this choice allows to consider in the future an algorithmic adaptation of efficient high-order finite element solvers based on domain decomposition [32], [33], [34], [35], where only a local resolution of the problem around the moving obstacle could be resolved. Let us remark that, depending on the problem, other high frequency numerical methods may also be adapted like for example fast integral equations solvers [36] or even asymptotic approximation techniques [37]. Finally, let us notice that the finite Fourier expansion method which leads to the coupled system of PDEs has also been used in the past under the name of the harmonic-balance method or the multi-harmonic approach [30], [38], [39], [40], [41], [42], [43]. It has been proved to be particularly efficient for engineering problems, including situations related to wave-like equations [44], [45], [46].

The paper is organized as follows. In Section 2 we define a sine-motion moving boundary scattering problem and formulate it in a fixed domain. In Section 3 we develop a multi-harmonic approach to solve the scattering problem for small and large amplitudes of the boundary motion. We illustrate in Section 4 the validity of the method through numerical computations and generalize it in Section 5 to general boundary motions which are not explicitly prescribed. In Section 6, we extend the approach to higher-dimensional domains and illustrate the numerical method on a two-dimensional example. Finally, we conclude in Section 7. Appendix A details some computations.

Section snippets

The initial boundary-value problem

We assume that the bounded spatial domain is defined by: Ω(t)]0,(t)[, where tR is the time variable and x denotes the spatial variable. The modeling of the moving boundary is described by the time-dependent function (t). In the context of this work, (t) is supposed to be smooth and bounded. For R+]0,[, we introduce the unknown total wave field uu(x,t), for xΩ(t) and tR+, solution to the constant coefficients wave equation ttuc2xxu=0,where c is the wave velocity in the medium

Analysis for the case of a small amplitude boundary motion

Let us consider the case where the motion of the right endpoint (t) of the domain Ω(t) is given by a small smooth time-dependent perturbation Lϵf(ωt) of L, where f is a smooth bounded function with bounded derivatives f(p), oscillating with a frequency ν=ω(2π) small compared to the emitter frequency νf, that is 0<ννf. Therefore, we have (t)=L(1+ϵf(ωt)),with ϵ1. In the one-dimensional case, we can assume that this is given by a linear relation x=x˜(t)L. To be more explicit, we now

Numerical examples

In the following, the amplitude of emission is set to A=1, and the values of J1 and J2 (and thus the definition of I and J, see (22)) are a priori selected relatively to the reference solution û by the criterion maxjZIûh(,νf+jν),Ω˜h103×ûh(,νf),Ω˜hin such a way that I+ is minimized. This assumption ensures that ûhJ+ is restricted to the significant components of ûhνf around νf, with a normalized amplitude less than 10−3. Here, we set f,Ω˜h=maxx̃Ω˜h|f(x̃)|.

Scattering with general boundary motions

We introduce a generalization of the previous approach to a motion (t) which has no a priori explicit expression. Let us denote by u˜(x˜,t) the solution of (9). We assume that the velocity of the motion of the boundary is much smaller than the phase speed of the emitted waves. In addition, we limit the study to the case where, for fixed t, the change of variables Ω(t) to Ω˜ is linear in spatial coordinates. The motion is expected to be a C(R) periodic time-dependent function taking its

Extension to higher dimensions

We now formally extend the approach developed for (1) to the space–time dimension d+1 (i.e. d in space and 1 in time). To this end, we assume that a wave is emitted by a source Ωs, with boundary Γs=Ωs, and is scattered by an obstacle Ωobst(t) with smooth boundary Γ(t)=Ωobst(t), moving with frequency ν around an equilibrium position Γ(0). We then define the d-dimensional domain of propagation, denoted by Ωext(t), as the exterior domain with boundaries Γs and Γ(t). We schematically illustrate

Conclusion

In this paper, we presented a new numerical method for solving the scattering problem of scalar waves by a moving d-dimensional obstacle with general movement. The method is based on a change of variable which makes the moving domain fixed, and a multi-harmonic expansion of an approximate wave field. This results in the numerical solution of coupled systems of Helmholtz-type equations where optimized algorithms can be developed in the frequency domain. A preliminary numerical study is presented

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The authors thank the support of the Luxembourg National Research Fund (FNR) (2017-1 PPP 11608832). This research was funded in part through the ARC, Luxembourg grant for Concerted Research Actions (ARC WAVES 15/19-03), financed by the Wallonia-Brussels Federation of Belgium .

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