arXiv subjects

Durga Prasad Challa

Publications and source records attributed to Durga Prasad Challa.

14 recordsLinked to original sources

arXiv2024

Extraction of the mass density using only the ${\mathtt{p}}$-parts of the elastic fields generated by injected highly dense small inclusions

We propose a reconstruction method to extract the variable mass density from the elastic farfields, with a single incident direction, measured before and after injecting highly dense small scaled inclusions. We take as a model, the Lamé system where the mass density is the unknown in $Ω$ and the Lamé parameters are known constants. The injected small/dense inclusion, $D:=z +a B\, (\subset\subset Ω)$ with $z$ as its location, $a\ll 1$ as its maximum radius and $B$ of unit volume, generates a sequences of resonant frequencies. These special frequencies are related to the eigenvalues of the Lamé volume integral operator defined on the domain of the inclusion and thus are, in principle, computable. After injecting the small inclusion at a location point $z$, we send an elastic incident plane wave at an incident frequency close to one of the mentioned resonant frequencies, say $ω_{n_0}$. Contrasting the ($\mathtt{p}$-parts of the) farfields generated, at one incident direction, before and after injecting this small inclusion, we provide an explicit formula that allows us to recover the total field $V^{t,\mathtt{p}}(z,-\hat{x})$ corresponding to $\mathtt{p}$-incident waves at the location $z$. This total field is generated in the absence of the inclusion. Then we repeat the experiment by injecting more inclusions inside $Ω$. Using this reconstructed field in the Lamé PDE system, via a numerical differentiation, we recover the values of the mass density inside $Ω$. It is worth mentioning that, we use measurements of dimension 3 to recover a function of 3 dimensions freedom. This makes the inverse problem not over determined. In addition, we use only the pressure wave and the $\mathtt{p}$-part of the farfield, for the reconstruction. To our best knowledge, this is the first result using only one type of elastic waves for the parameter identification.

math.AP↗

arXiv2024

Elastic fields generated by multiple small inclusions with high mass density at nearly resonant frequencies

We derive the elastic field generated by multiple small-scaled inclusions distributed in a bounded set of $\mathbb{R}^3$. These inclusions are modeled with moderate values of the Lamé coefficients while they have a large relative mass density. These properties allow them to enjoy a sequences of resonant frequencies that can be computed via the eigenvalues of the volume integral operator having the Navier fundamental matrix as a kernel, i.e. the Navier volume operator. The dominant field, i.e. the Foldy-Lax field, models the multiple interactions between the inclusions with scattering coefficients that are inversely proportional to the difference between the used incident frequency and the already mentioned resonances. We show, in particular, that to reconstruct remotely the scattered field generated after $N$ interactions between the inclusions, one needs to use an incident frequency appropriately close to the proper resonance of the inclusions. We provide an explicit link between the order $N$ of interactions and the distance from the incident frequency to the resonance. Finally, if the cluster of the inclusions is densely distributed in a given bounded domain, then the expression of the induced dominant field suggests that the equivalent homogenized mass density can change sign depending if the used incident frequencies is smaller or larger than a certain threshold (which is explicitly given in terms of the resonant frequencies of the inclusions).

math.AP↗

arXiv2019

The equivalent media generated by bubbles of high contrasts: Volumetric metamaterials and metasurfaces

We deal with the point-interaction approximations for the acoustic wave fields generated by a cluster of highly contrasted bubbles for a wide range of densities and bulk moduli contrasts. We derive the equivalent fields when the cluster of bubbles is appropriately distributed (but not necessarily periodically) in a bounded domain $Ω$ of $\mathbb{R}^3$. We handle two situations. 1. In the first one, we distribute the bubbles to occupy a $3$ dimensional domain. For this case, we show that the equivalent speed of propagation changes sign when the medium is excited with frequencies smaller or larger than (but not necessarily close to) the Minnaert resonance. As a consequence, this medium behaves as a reflective or absorbing depending on whether the used frequency is smaller or larger than this resonance. In addition, if the used frequency is extremely close to this resonance, for a cluster of bubbles with density above a certain threshold, then the medium behaves as a 'wall', i.e. allowing no incident sound to penetrate. 2. In the second one, we distribute the bubbles to occupy a $2$ dimensional (open or closed) surface, not necessarily flat. For this case, we show that the equivalent medium is modeled by a Dirac potential supported on that surface. The sign of the surface potential changes for frequencies smaller or larger than the Minnaert resonance, i.e. it behaves as a smart metasurface reducing or amplifying the transmitted sound across it. As in the $3$D case, if the used frequency is extremely close to this resonance, for a cluster of bubbles with density above an appropriate threshold, then the surface allows no incident sound to be transmitted across the surface, i.e. it behaves as a white screen.

math.AP↗

arXiv2019

The point-interaction approximation for the fields generated by contrasted bubbles at arbitrary fixed frequencies

We deal with the linearized model of the acoustic wave propagation generated by small bubbles in the harmonic regime. We estimate the waves generated by a cluster of $M$ small bubbles, distributed in a bounded domain $Ω$, with relative densities having contrasts of the order $a^β, β>0, $ where $a$ models their relative maximum diameter, $a\ll 1$. We provide useful and natural conditions on the number $M$, the minimum distance and the contrasts parameter $β$ of the small bubbles under which the point interaction approximation (called also the Foldy-Lax approximation) is valid. With the regimes allowed by our conditions, we can deal with a general class of such materials. Applications of these expansions in material sciences and imaging are immediate. For instance, they are enough to derive and justify the effective media of the cluster of the bubbles for a class of gases with densities having contrasts of the order $a^β$, $β\in (\frac{3}{2}, 2)$ and in this case we can handle any fixed frequency. In the particular and important case $β=2$, we can handle any fixed frequency far or close (but distinct) from the corresponding Minnaert resonance. The cluster of the bubbles can be distributed to generate volumetric metamaterials but also low dimensional ones as metascreens and metawires.

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arXiv2018

Mathematical imaging using electric or magnetic nanoparticles as contrast agents

We analyse mathematically the imaging modality using electromagnetic nanoparticles as contrast agent. This method uses the electromagnetic fields, collected before and after injecting electromagnetic nanoparticles, to reconstruct the electrical permittivity. The particularity here is that these nanoparticles have high contrast electric or magnetic properties compared to the background media. First, we introduce the concept of electric (or magnetic) nanoparticles to describe the particles, of relative diameter $δ$ (relative to the size of the imaging domain), having relative electric permittivity (or relative magnetic permeability) of order $δ^{-α}$ with a certain $α>0$, as $0<δ<<1$. Examples of such material, used in the imaging community, are discussed. Second, we derive the asymptotic expansion of the electromagnetic fields due to such singular contrasts. We consider here the scalar electromagnetic model. Using these expansions, we extract the values of the total fields inside the domain of imaging from the scattered fields measured before and after injecting the nanoparticles. From these total fields, we derive the values of the electric permittivity at the expense of numerical differentiations.

math.AP↗

arXiv2017

Characterization of the equivalent acoustic scattering for a cluster of an extremely large number of small holes

We deal with the time-harmonic acoustic waves scattered by a large number of small holes, of maximal radius $a, a<<1$, arbitrary (i.e. not necessarily periodically) distributed in a bounded part of a homogeneous background. We show that as their number $M$ grows following the law $M:=M(a):=O(a^{-s}), \; a<<1$, the collection of these holes has one of the following behaviors: 1. if $s<1$, then the scattered fields tend to vanish as $a$ tends to zero, i.e. the cluster is a soft one. 2. if $s=1$, then the cluster behaves as an equivalent medium modeled by a refraction index, supported in a given bounded domain $Ω$, which is described by certain geometry properties of the holes and their local distribution. The cluster is a moderate (or intermediate) one. 3. if $s>1$, then the cluster behaves as a totally reflecting extended body, modeled by a bounded and smooth domain $Ω$, i.e. the incident waves are totally reflected by the surface of this extended body. The cluster is a rigid one. These approximations are provided with explicit error estimates in terms of $a,\; a<<1$.

math-ph↗

arXiv2016

The equivalent medium for the elastic scattering by many small rigid bodies and applications

We deal with the elastic scattering by a large number $M$ of rigid bodies, $D_m:=εB_m+z_m$, of arbitrary shapes with $ 0<\textcolor{black}ε<<1$ and with constant Lamé coefficients $λ$ and $μ$. We show that, when these rigid bodies are distributed arbitrarily (not necessarily periodically) in a bounded region $Ω$ of $\mathbb{R}^3$ where their number is $M:=M(\textcolor{black}ε):=O(\textcolor{black}ε^{-1})$ and the minimum distance between them is $d:=d(\textcolor{black}ε)\approx \textcolor{black}ε^{t}$ with $t$ in some appropriate range, as $\textcolor{black}ε \rightarrow 0$, the generated far-field patterns approximate the far-field patterns generated by an equivalent medium given by $ω^2ρI_3-(K+1)\mathbf{C}_0 $ where $ρ$ is the density of the background medium (with $I_3$ as the unit matrix) and $(K+1)\mathbf{C}_0$ is the shifting (and possibly variable) coefficient. This shifting coefficient is described by the two coefficients $K$ and $\mathbf{C}_0$ (which have supports in $\overlineΩ$) modeling the local distribution of the small bodies and their geometries, respectively. In particular, if the distributed bodies have a uniform spherical shape then the equivalent medium is isotropic while for general shapes it might be anisotropic (i.e. $\mathbf{C}_0$ might be a matrix). In addition, if the background density $ρ$ is variable in $Ω$ and $ρ=1$ in $\mathbb{R}^3\setminus{\overlineΩ}$, then if we remove from $Ω$ appropriately distributed small bodies then the equivalent medium will be equal to $ω^2 I_3$ in $\mathbb{R}^3$, i.e. the obstacle $Ω$ characterized by $ρ$ is approximately cloaked at the given and fixed frequency $ω$.

math.AP↗

arXiv2016

Extraction of the index of refraction by embedding multiple and close small inclusions

We deal with the problem of reconstructing material coefficients from the farfields they generate. By embedding small (single) inclusions to these media, located at points $z$ in the support of these materials, and measuring the farfields generated by these deformations we can extract the values of the total field generated by these media at the points $z$. The second step is to extract the values of the material coefficients from these internal values of the total field. The main difficulty in using internal fields is the treatment of their possible zeros. In this work, we propose to deform the medium using multiple (precisely double) and close inclusions instead of only single ones. By doing so, we derive from the asymptotic expansions of the farfields the internal values of the Green function, in addition to the internal values of the total fields. This is possible because of the deformation of the medium with multiple and close inclusions which generates scattered fields due to the multiple scattering between these inclusions. Then, the values of the index of refraction can be extracted from the singularities of the Green function. Hence, we overcome the difficulties arising from the zeros of the internal fields. We test these arguments for the acoustic scattering by a refractive index in presence of inclusions modeled by the impedance type small obstacles.

math.AP↗

arXiv2015

A cluster of many small holes with negative imaginary surface impedances may generate a negative refraction index

We deal with the scattering of an acoustic medium modeled by an index of refraction $n$ varying in a bounded region $Ω$ of $\mathbb{R}^3$ and equal to unity outside $Ω$. This region is perforated with an extremely large number of small holes $D_m$'s of maximum radius $a$, $a<<1$, modeled by surface impedance functions. Precisely, we are in the regime described by the number of holes of the order $M:=O(a^{β-2})$, the minimum distance between the holes is $d\sim a^t$ and the surface impedance functions of the form $λ_m \sim λ_{m,0} a^{-β}$ with $β>0$ and $λ_{m,0}$ being constants and eventually complex numbers. Under some natural conditions on the parameters $β, t$ and $λ_{m,0}$, we characterize the equivalent medium generating, approximately, the same scattered waves as the original perforated acoustic medium. We give an explicit error estimate between the scattered waves generated by the perforated medium and the equivalent one respectively, as $a \rightarrow 0$. As applications of these results, we discuss the following findings: 1. If we choose negative valued imaginary surface impedance functions, attached to each surface of the holes, then the equivalent medium behaves as a passive acoustic medium only if it is an acoustic metamaterial with index of refraction $\tilde{n}(x)=-n(x),\; x \in Ω$ and $\tilde{n}(x)=1,\; x \in \mathbb{R}^3\setminus{\overlineΩ}$. This means that, with this process, we can switch the sign of the index of the refraction from positive to negative values. 2. We can choose the surface impedance functions attached to each surface of the holes so that the equivalent index of refraction $\tilde{n}$ is $\tilde{n}(x)=1,\; x \in \mathbb{R}^3$. This means that the region $Ω$ modeled by the original index of refraction $n$ is approximately cloaked.

math.AP↗

arXiv2015

Multiscale analysis of the acoustic scattering by many scatterers of impedance type

We are concerned with the acoustic scattering problem, at a frequency $κ$, by many small obstacles of arbitrary shapes with impedance boundary condition. These scatterers are assumed to be included in a bounded domain $Ω$ in $\mathbb{R}^3$ which is embedded in an acoustic background characterized by an eventually locally varying index of refraction. The collection of the scatterers $D_m, \; m=1,...,M$ is modeled by four parameters: their number $M$, their maximum radius $a$, their minimum distance $d$ and the surface impedances $λ_m, \; m=1,...,M$. We consider the parameters $M, d$ and $λ_m$'s having the following scaling properties: $M:=M(a)=O(a^{-s})$, $d:=d(a)\approx a^t$ and $λ_m:=λ_m(a)=λ_{m,0}a^{-β}$, as $a \rightarrow 0$, with non negative constants $s, t$ and $β$ and complex numbers $λ_{m, 0}$'s with eventually negative imaginary parts. We derive the asymptotic expansion of the farfields with explicit error estimate in terms of $a$, as $a\rightarrow 0$. The dominant term is the Foldy-Lax field corresponding to the scattering by the point-like scatterers located at the centers $z_m$'s of the scatterers $D_m$'s with $λ_m \vert \partial D_m\vert$ as the related scattering coefficients.

math.AP↗

arXiv2014

Location and size estimation of small rigid bodies using elastic far-fields

We are concerned with the linearized, isotropic and homogeneous elastic scattering problem by (possibly many) small rigid obstacles of arbitrary Lipschitz regular shapes in 3D. Based on the Foldy-Lax approximation, valid under a sufficient condition on the number of the obstacles, the size and the minimum distance between them, we show that any of the two body waves, namely the pressure waves P or the shear waves S, is enough for solving the inverse problem of detecting these scatterers and estimating their sizes. Further, it is also shown that the shear-horizontal part SH or the shear vertical part SV of the shear waves S are also enough for the location detection and the size estimation. Under some extra assumption on the scatterers, as the convexity assumption, we derive finer size estimates as the radius of the largest ball contained in each scatterer and the one of the smallest ball containing it. The two estimates measure, respectively, the thickness and length of each obstacle.

math.AP↗

arXiv2014

The Foldy-Lax approximation of the scattered waves by many small bodies for the Lame system

We are concerned with the linearized, isotropic and homogeneous elastic scattering problem by many small rigid obstacles of arbitrary, Lipschitz regular, shapes in 3D case. We prove that there exists two constant $a_0$ and $c_0$, depending only on the Lipschitz character of the obstacles, such that under the conditions $a\leq a_0$ and $\sqrt{M-1}\frac{a}{d} \leq c_0$ on the number $M$ of the obstacles, their maximum diameter $a$ and the minimum distance between them $d$, the corresponding Foldy-Lax approximation of the farfields is valid. In addition, we provide the error of this approximation explicitly in terms of the three parameters $M, a$ and $d$. These approximations can be used, in particular, in the identification problems (i.e. inverse problems) and in the design problems (i.e. effective medium theory).

math.AP↗

arXiv2014

The equivalent refraction index for the acoustic scattering by many small obstacles: with error estimates

Let $M$ be the number of bounded and Lipschitz regular obstacles $D_j, j:=1, ..., M$ having a maximum radius $a$, $a<<1$, located in a bounded domain $Ω$ of $\mathbb{R}^3$. We are concerned with the acoustic scattering problem with a very large number of obstacles, as $M:=M(a):=O(a^{-1})$, $a\rightarrow 0$, when they are arbitrarily distributed in $Ω$ with a minimum distance between them of the order $d:=d(a):=O(a^t)$ with $t$ in an appropriate range. We show that the acoustic farfields corresponding to the scattered waves by this collection of obstacles, taken to be soft obstacles, converge uniformly in terms of the incident as well the propagation directions, to the one corresponding to an acoustic refraction index as $a\rightarrow 0$. This refraction index is given as a product of two coefficients $C$ and $K$, where the first one is related to the geometry of the obstacles (precisely their capacitance) and the second one is related to the local distribution of these obstacles. In addition, we provide explicit error estimates, in terms of $a$, in the case when the obstacles are locally the same (i.e. have the same capacitance, or the coefficient $C$ is piecewise constant) in $Ω$ and the coefficient $K$ is H$\ddot{\mbox{o}}$lder continuous. These approximations can be applied, in particular, to the theory of acoustic materials for the design of refraction indices by perforation using either the geometry of the holes, i.e. the coefficient $C$, or their local distribution in a given domain $Ω$, i.e. the coefficient $K$.

math.AP↗

arXiv2013

On the justification of the Foldy-Lax approximation for the acoustic scattering by small rigid bodies of arbitrary shapes

We are concerned with the acoustic scattering problem by many small rigid obstacles of arbitrary shapes. We give a sufficient condition on the number $M$ and the diameter $a$ of the obstacles as well as the minimum distance $d$ between them under which the Foldy-Lax approximation is valid. Precisely, if we use single layer potentials for the representation of the scattered fields, as it is done sometimes in the literature, then this condition is $(M-1)\frac{a}{d^2} <c$, with an appropriate constant $c$, while if we use double layer potentials then a weaker condition of the form $\sqrt{M-1}\frac{a}{d} <c$ is enough. In addition, we derive the error in this approximation explicitly in terms of the parameters $M, a$ and $d$. The analysis is based, in particular, on the precise scalings of the boundary integral operators between the corresponding Sobolev spaces. As an application, we study the inverse scattering by the small obstacles in the presence of multiple scattering.

math.AP↗