arXiv subjects

Ankit Srivastava

Publications and source records attributed to Ankit Srivastava.

At least 37 records · Page 2Linked to original sources

arXiv2021

Angle-dependent Phononic Dynamics for Deep Learning and Source Localization

In this work, a parameterized eigenvalue problem is analyzed for a phononic array in a 2D stress wave scattering setup, and a corresponding sensing application of this system is proposed to achieve source angle localization. The phononic domain consists of a periodic micro-structured medium, of which the eigen-wavevector band structure and the eigen-modes are exploited. The eigen-modes are naturally angle dependent due to changes in phases and periodic mode shapes determined by the incident angle. Intriguingly, the band exhibits angle-dependent transitions at the exceptional points (EPs) and critical angles (CAs), where the eigenvalues coincide or vanish. Coupled with these transitions, it is found that the eigen-modes switch their energy characteristics and symmetry patterns at these branch points, leading to enhanced angle dependence. Moreover, these eigen-modes also serve as the basis functions of the scattered waves. Therefore, the scattering response of the medium inherently possesses the angle-dependent properties, making this system naturally suitable for sensing applications. An artificial neural network (ANN) is trained with randomly weighted eigen-modes to achieve deep learning of the eigen features and angle dependence. The training data is derived only based on the eigen-modes of the unit cells. Nevertheless, the trained ANN can accurately identify the incident angle of an unknown scattering signal, with minimal side lobe levels and suppressed main lobe width. The ANN shows superior performance in comparison with standard delay-and-sum technique of estimating angle of arrival. The proposed application of ANN and micro-structured media highlights the physical importance of band structure topology and eigen-modes to a technological application, adds extra strength to the existing localization methods, and can be easily enhanced with the fast-growing data-driven techniques.

physics.app-ph↗

arXiv2021

The Analytical Structure of Acoustic and Elastic Material Properties

In this paper, we take an in-depth look at the analytical structure of the material transfer functions which govern acoustic and elastic response. These include wavenumber ($κ$) in such media and refractive index ($n$), density ($\boldsymbolρ$) and its inverse, stiffness ($\boldsymbol{C}$) and compliance ($\boldsymbol{D}$) tensors as well as the Bulk modulus ($B$), and finally the broader generalization of these properties which is now known as the Willis tensor ($\boldsymbol{L}$). Our goal is to clarify the appropriate dispersion relations applicable to these properties from the perspective of passivity. Under some mild assumptions, causality ensures that these properties are analytical in the upper half but deriving dispersion relations for them requires one to know how they behave in the limit $|ω|\rightarrow\infty$. Unlike electromagnetism, such a determination cannot be made on physical grounds since in that limit the continuum approximation breaks down. Instead, we can exploit the properties of the Herglotz-Nevanlinna (H-N) functions along with their tensorial counterparts which characterize the transfer functions of certain passive systems and for which the appropriate dispersion relation is known. Our aim, therefore, is to clarify the relationship that these transfer functions have with Herglotz functions, which in turn determines the appropriate dispersion relation for them. Our analysis shows that based upon passivity alone, dispersion relations of \emph{minimum} order 1 apply to the Fourier transforms of $\boldsymbol{D},\boldsymbolρ, n'$, and the inverse of $B$, order 3 apply to $\boldsymbol{C},B$, and the inverse of $\boldsymbolρ$, and order 2 applies to $κ$.

physics.class-ph↗

arXiv2020

Spectral extended finite element method for band structure calculations in phononic crystals

In this paper, we compute the band structure of one- and two-dimensional phononic composites using the extended finite element method (X-FEM) on structured higher-order (spectral) finite element meshes. On using partition-of-unity enrichment in finite element analysis, the X-FEM permits use of structured finite element meshes that do not conform to the geometry of holes and inclusions. This eliminates the need for remeshing in phononic shape optimization and topology optimization studies. In two dimensions, we adopt rational B{é}zier representation of curved (circular) geometries, and construct suitable material enrichment functions to model two-phase composites. A Bloch-formulation of the elastodynamic phononic eigenproblem is adopted. Efficient computation of weak form integrals with polynomial integrands is realized via the homogeneous numerical integration scheme -- a method that uses Euler's homogeneous function theorem and Stokes's theorem to reduce integration to the boundary of the domain. Ghost penalty stabilization is used on finite elements that are cut by a hole. Band structure calculations on perforated (circular holes, elliptical holes, and holes defined as a level set) materials as well as on two-phase phononic crystals are presented that affirm the sound accuracy and optimal convergence of the method on structured, higher-order spectral finite element meshes. Several numerical examples demonstrate the advantages of $p$-refinement made possible by the spectral extended finite element method. In these examples, fourth-order spectral extended finite elements deliver $\mathcal{O}(10^{-8})$ accuracy in frequency calculations with more than thirty-fold fewer degrees-of-freedom when compared to quadratic finite elements.

physics.comp-ph↗

arXiv2020

Causality and Passivity: from Electromagnetism and Network Theory to Metamaterials

In this review, we take an extensive look at the role that the principles of causality and passivity have played in various areas of physics and engineering, including in the modern field of metamaterials. The aim is not to provide a comprehensive list of references as that number would be in the thousands, but to review the major results and contributions which have animated these areas and to provide a unified framework from which to understand the developments in different fields. Towards these goals, we chart the early history of the field through its dual beginnings in the analysis of the Sellmeier equation and in Hilbert transforms, giving rise to the far reaching dispersion relations in the early works of Kramers, Kronig, and Titchmarsh. However, these early relations constitute a limited result as they only apply to a restricted class of transfer functions. To understand how this restriction can be lifted, we take a quick detour into the distributional analysis of Schwartz. This approach expands the reach of the dispersion analysis to distributional transfer functions and also to those functions which exhibit polynomial growth properties. To generalize the results even further to tensorial transfer functions, we consider the concept of passivity - originally studied in the theory of electrical networks. We clarify why passivity implies causality. Subsequently, as special cases, we present examples of dispersion relations from several areas of physics including electromagnetism, acoustics, seismology, reflectance measurements, and scattering theory. We discuss sum rules, derivative analyticity relations, and nearly local approximations. Finally we review the clever applications of ideas from causality and passivity to the recent field of metamaterials. These ideas have provided limits to what can be achieved in metamaterial property design and metamaterial device performance.

physics.app-ph↗

arXiv2020

Further comments on Mark Stockman's article "Criterion for Negative Refraction with Low Optical Losses from a Fundamental Principle of Causality"

We clarify the claims and errors in the paper "Criterion for Negative Refraction with Low Optical Losses from a Fundamental Principle of Causality" by Mark Stockman. Contrary to the central assertion in that paper, simple examples consistent with the basic inequality which Stockman discovered show that it is possible to have negative refraction and low loss in an arbitrarily large frequency window. Further examination of the paper reveals additional errors that invalidate his argument that active materials cannot have low loss and negative refraction in a frequency window. Also, we point out that for active materials non-analyticity of the electrical permittivity in the upper half complex frequency plane does not necessarily imply noncausality, as Stockman infers.

physics.optics↗

arXiv2020

Distributed Differentially Private Mutual Information Ranking and Its Applications

Computation of Mutual Information (MI) helps understand the amount of information shared between a pair of random variables. Automated feature selection techniques based on MI ranking are regularly used to extract information from sensitive datasets exceeding petabytes in size, over millions of features and classes. Series of one-vs-all MI computations can be cascaded to produce n-fold MI results, rapidly pinpointing informative relationships. This ability to quickly pinpoint the most informative relationships from datasets of billions of users creates privacy concerns. In this paper, we present Distributed Differentially Private Mutual Information (DDP-MI), a privacy-safe fast batch MI, across various scenarios such as feature selection, segmentation, ranking, and query expansion. This distributed implementation is protected with global model differential privacy to provide strong assurances against a wide range of privacy attacks. We also show that our DDP-MI can substantially improve the efficiency of MI calculations compared to standard implementations on a large-scale public dataset.

cs.CR↗

arXiv2020

Scattering under Linear Non Self-Adjoint Operators: Case of in-Plane Elastic Waves

In this paper, we consider the problem of the scattering of in-plane waves at an interface between a homogeneous medium and a metamaterial. The relevant eigenmodes in the two regions are calculated by solving a recently described non self-adjoint eigenvalue problem particularly suited to scattering studies. The method efficiently produces all propagating and evanescent modes consistent with the application of Snell's law and is applicable to very general scattering problems. In a model composite, we elucidate the emergence of a rich spectrum of eigenvalue degeneracies. These degeneracies appear in both the complex and real domains of the wave-vector. However, since this problem is non self-adjoint, these degeneracies generally represent a coalescing of both the eigenvalues and eigenvectors (exceptional points). Through explicit calculations of Poynting vector, we point out an intriguing phenomenon: there always appears to be an abrupt change in the sign of the refraction angle of the wave on two sides of an exceptional point. Furthermore, the presence of these degeneracies, in some cases, hints at fast changes in the scattered field as the incident angle is changed by small amounts. We calculate these scattered fields through a novel application of the Betti-Rayleigh reciprocity theorem. We present several numerical examples showing a rich scattering spectrum. In one particularly intriguing example, we point out wave behavior which may be related to the phenomenon of resonance trapping. We also show that there exists a deep connection between energy flux conservation and the biorthogonality relationship of the non self-adjoint problem. The proof applies to the general class of scattering problems involving elastic waves (under self-adjoint or non self-adjoint operators).

physics.app-ph↗

arXiv2019

Cosmological N-body simulations: a challenge for scalable generative models

Deep generative models, such as Generative Adversarial Networks (GANs) or Variational Autoencoders (VAs) have been demonstrated to produce images of high visual quality. However, the existing hardware severely limits the size of the images that can be generated. The rapid growth of high dimensional data in many fields of science therefore poses a significant challenge for generative models. In cosmology, the large-scale, three-dimensional matter distribution, modeled with N-body simulations, plays a crucial role in understanding the evolution of the universe. As these simulations are computationally very expensive, GANs have recently generated interest as a possible method to emulate these datasets, but they have been, so far, mostly limited to two dimensional data. In this work, we introduce a new benchmark for the generation of three dimensional N-body simulations, in order to stimulate new ideas in the machine learning community and move closer to the practical use of generative models in cosmology. As a first benchmark result, we propose a scalable GAN approach for training a generator of N-body three-dimensional cubes. Our technique relies on two key building blocks, (i) splitting the generation of the high-dimensional data into smaller parts, and (ii) using a multi-scale approach that efficiently captures global image features that might otherwise be lost in the splitting process. We evaluate the performance of our model for the generation of N-body samples using various statistical measures commonly used in cosmology. Our results show that the proposed model produces samples of high visual quality, although the statistical analysis reveals that capturing rare features in the data poses significant problems for the generative models. We make the data, quality evaluation routines, and the proposed GAN architecture publicly available at https://github.com/nperraud/3DcosmoGAN

physics.comp-ph↗

arXiv2019

On the Properties of Phononic Eigenvalue Problems

In this paper, we consider the operator properties of various phononic eigenvalue problems. We aim to answer some fundamental questions about the eigenvalues and eigenvectors of phononic operators. These include questions about the potential real and complex nature of the eigenvalues, whether the eigenvectors form a complete basis, what are the right orthogonality relationships, and how to create a complete basis when none may exist at the outset. In doing so we present a unified understanding of the properties of the phononic eigenvalues and eigenvectors which would emerge from any numerical method employed to compute such quantities. We show that the phononic problem can be cast into linear eigenvalue forms from which such quantities as frequencies, wavenumbers, and desired components of wavevectors can be directly ascertained without resorting to searches or quadratic eigenvalue problems and that the relevant properties of such quantities can be determined apriori through the analysis of the associated operators. We further show how the Plane Wave Expansion (PWE) method may be extended to solve each of these eigenvalue forms, thus extending the applicability of the PWE method to cases beyond those which have been considered till now. The theoretical discussions are supplemented with supporting numerical calculations. The techniques and results presented here directly apply to wave propagation in other periodic systems such as photonics.

physics.comp-ph↗

arXiv2019

On the Emergence of Negative Effective Density and Modulus in 2-phase Phononic Crystals

In this paper we report metamaterial properties including negative and singular effective properties for what would traditionally be considered non locally resonant 2-phase phononic unit cells. The negative effective material properties reported here occur well below the homogenization limit and are, therefore, acceptable descriptions of overall behavior. The material property combinations which make this possible were first revealed by a novel level set based topology optimization process which we describe. The optimization process revealed that a 2-phase unit cell in which one of the phases is simultaneously lighter and stiffer than the other results in dynamic behavior which has all the attendant characteristics of a locally resonant composite including negative effective properties far below the homogenization limit. We investigate this further using the Craig-Bampton decomposition and clarify that these properties emerge through an interplay between the fundamental internal modeshape of the unit cell and a rigid body mode. Through explicit numerical calculations on 1-D, 2-phase unit cells, we show that negative effective properties only appear for the specific material property combination mentioned above. Furthermore, we provide a proof which supports this conclusion. The concept is also shown to hold for 2-D unit cells where we show that an appropriately designed hexagonal unit cell made of 2 material phases exhibits negative effective shear modulus and density in an appropriate frequency regime in which it also exhibits negative refraction. An important conclusion of this paper is that the class of unit cells expected to result in negative properties can be expanded beyond the classic unit cell (three-phase unit cells with an explicit locally resonant phase) to include topologically simpler 2-phase unit cells as well.

physics.app-ph↗

arXiv2018

Deep Convolutional Neural Networks for Eigenvalue Problems in Mechanics

We show that deep convolutional neural networks (CNN) can massively outperform traditional densely-connected neural networks (both deep or shallow) in predicting eigenvalue problems in mechanics. In this sense, we strike out in a new direction in mechanics computations with strongly predictive NNs whose success depends not only on architectures being deep, but also being fundamentally different from the widely-used to date. We consider a model problem: predicting the eigenvalues of 1-D and 2-D phononic crystals. For the 1-D case, the optimal CNN architecture reaches $98\%$ accuracy level on unseen data when trained with just 20,000 samples, compared to $85\%$ accuracy even with $100,000$ samples for the typical network of choice in mechanics research. We show that, with relatively high data-efficiency, CNNs have the capability to generalize well and automatically learn deep symmetry operations, easily extending to higher dimensions and our 2D case. Most importantly, we show how CNNs can naturally represent mechanical material tensors, with its convolution kernels serving as local receptive fields, which is a natural representation of mechanical response. Strategies proposed are applicable to other mechanics' problems and may, in the future, be used to sidestep cumbersome algorithms with purely data-driven approaches based upon modern deep architectures.

physics.comp-ph↗

arXiv2018

Multi-Information Source Fusion and Optimization to Realize ICME: Application to Dual Phase Materials

Integrated Computational Materials Engineering (ICME) calls for the integration of computational tools into the materials and parts development cycle, while the Materials Genome Initiative (MGI) calls for the acceleration of the materials development cycle through the combination of experiments, simulation, and data. As they stand, both ICME and MGI do not prescribe how to achieve the necessary tool integration or how to efficiently exploit the computational tools, in combination with experiments, to accelerate the development of new materials and materials systems. This paper addresses the first issue by putting forward a framework for the fusion of information that exploits correlations among sources/models and between the sources and `ground truth'. The second issue is addressed through a multi-information source optimization framework that identifies, given current knowledge, the next best information source to query and where in the input space to query it via a novel value-gradient policy. The querying decision takes into account the ability to learn correlations between information sources, the resource cost of querying an information source, and what a query is expected to provide in terms of improvement over the current state. The framework is demonstrated on the optimization of a dual-phase steel to maximize its strength-normalized strain hardening rate. The ground truth is represented by a microstructure-based finite element model while three low fidelity information sources---i.e. reduced order models---based on different homogenization assumptions---isostrain, isostress and isowork---are used to efficiently and optimally query the materials design space.

physics.app-ph↗

arXiv2017

Level Repulsion and Band Sorting in Phononic Crystals

In this paper we consider the problem of avoided crossings (level repulsion) in phononic crystals and suggest a computationally efficient strategy to distinguish them from normal cross points. This process is essential for the correct sorting of the phononic bands and, subsequently, for the accurate determination of mode continuation, group velocities, and emergent properties which depend on them such as thermal conductivity. Through explicit phononic calculations using generalized Rayleigh quotient, we identify exact locations of exceptional points in the complex wavenumber domain which results in level repulsion in the real domain. We show that in the vicinity of the exceptional point the relevant phononic eigenvalue surfaces resemble the surfaces of a 2 by 2 parameter-dependent matrix. Along a closed loop encircling the exceptional point we show that the phononic eigenvalues are exchanged, just as they are for the 2 by 2 matrix case. However, the behavior of the associated eigenvectors is shown to be more complex in the phononic case. Along a closed loop around an exceptional point, we show that the eigenvectors can flip signs multiple times unlike a 2 by 2 matrix where the flip of sign occurs only once. Finally, we exploit these eigenvector sign flips around exceptional points to propose a simple and efficient method of distinguishing them from normal crosses and of correctly sorting the band-structure. Our proposed method is roughly an order-of magnitude faster than the zoom-in method and correctly identifies > 97% of the cases considered. Both its speed and accuracy can be further improved and we suggest some ways of achieving this. Our method is general and, as such, would be directly applicable to other eigenvalue problems where the eigenspectrum needs to be correctly sorted.

cond-mat.mtrl-sci↗

arXiv2017

Combining plane wave expansion and variational techniques for fast phononic computations

In this paper the salient features of the Plane Wave Expansion (PWE) method and the mixed variational technique are combined for the fast eigenvalue computations of arbitrarily complex phononic unit cells. This is done by expanding the material properties in a Fourier expansion, as is the case with PWE. The required matrix elements in the variational scheme are identified as the discrete Fourier transform coefficients of material properties, thus obviating the need for any explicit integration. The process allows us to provide succinct and closed form expressions for all the matrices involved in the mixed variational method. The scheme proposed here preserves both the simplicity of expression which is inherent in the PWE method and the superior convergence properties of the mixed variational scheme. We present numerical results and comment upon the convergence and stability of the current method. We show that the current representation renders the results of the method stable over the entire range of the expansion terms as allowed by the spatial discretization. When compared with a zero order numerical integration scheme, the present method results in greater computational accuracy of all eigenvalues. A higher order numerical integration scheme comes close to the accuracy of the present method but only with significantly more computational expense.

cond-mat.mtrl-sci↗

arXiv2017

3-D phononic crystals with ultra-wide band gaps

In this paper gradient based topology optimization (TO) is used to discover 3-D phononic structures that exhibit ultra-wide normalized all-angle all-mode band gaps. The challenging computational task of repeated 3-D phononic band-structure evaluations is accomplished by a combination of a fast mixed variational eigenvalue solver and distributed Graphic Processing Unit (GPU) parallel computations. The TO algorithm utilizes the material distribution-based approach and a gradient-based optimizer. The design sensitivity for the mixed variational eigenvalue problem is derived using the adjoint method and is implemented through highly efficient vectorization techniques. We present optimized results for two-material simple cubic (SC), body centered cubic (BCC), and face centered cubic (FCC) crystal structures and show that in each of these cases different initial designs converge to single inclusion network topologies within their corresponding primitive cells. The optimized results show that large phononic stop bands for bulk wave propagation can be achieved at lower than close packed spherical configurations leading to lighter unit cells. For tungsten carbide - epoxy crystals we identify all angle all mode normalized stop bands exceeding 100%, which is larger than what is possible with only spherical inclusions.

cond-mat.mtrl-sci↗

arXiv2016

Evanescent Wave Boundary Layers in Metamaterials and Sidestepping them through a Variational Approach

All metamaterial applications are based upon the idea that extreme material properties can be achieved through appropriate dynamic homogenization of composites. This homogenization is almost always done for infinite domains and the results are then applied to finite samples. This process ignores the evanescent waves which appear at the boundaries of such finite samples. In this paper we first clarify the emergence and purpose of these evanescent waves in a model problem consisting of an interface between a layered composite and a homogeneous medium. We show that these evanescent waves form boundary layers on either side of the interface beyond which the composite can be represented by appropriate infinite domain homogenized relations. We show that if one ignores the boundary layers then the displacement and stress fields are discontinuous across the interface. Therefore, the scattering coefficients at such an interface cannot be determined through the conventional continuity conditions involving only propagating modes. Here we propose an approximate variational approach for sidestepping these boundary layers. The aim is to determine the scattering coefficients without the knowledge of evanescent modes. Through various numerical examples we show that our technique gives very good estimates of the actual scattering coefficients beyond the long wavelength limit.

cond-mat.mtrl-sci↗

arXiv2016

Negative Refraction, Beam Steering, Mode Switching, and High-pass Filtering in a 1-D Periodic Laminate

In this paper we show that a 1-D phononic crystal (laminate) can exhibit metamaterial wave phenomenon which is traditionally associated with 2-, and 3-D crystals. Moreover, due to the absence of a length scale in 2 of its dimensions, it can outperform higher dimensional crystals on some measures. This includes allowing only negative refraction over large frequency ranges and serving as a near-omnidirectional high-pass filter up to a large frequency value. First we provide a theoretical discussion on the salient characteristics of the dispersion relation of a laminate and formulate the solution of an interface problem by the application of the normal mode decomposition technique. We present a methodology with which to induce a pure negative refraction in the laminate. As a corollary to our approach of negative refraction, we show how the laminate can be used to steer beams over large angles for small changes in the incident angles (beam steering). Furthermore, we clarify how the transmitted modes in the laminate can be switched on and off by varying the angle of the incident wave by a small amount. Finally, we show that the laminate can be used as a remarkably efficient high-pass frequency filter. An appropriately designed laminate will reflect all plane waves from quasi-static to a large frequency, incident at it from all angles except for a small set of near-normal incidences. This will be true even if the homogeneous medium is impedance matched with the laminate. Due to the similarities between SH waves and EM waves it is expected that some or all of these results may also apply to EM waves in a layered periodic dielectric.

cond-mat.mtrl-sci↗

arXiv2016

Variational Methods For Phononic Calculations

Three fundamental variational principles used for solving elastodynamic eigenvalue problems are studied within the context of elastic wave propagation in periodic composites (phononics). We study the convergence of the eigenvalue problems resulting from the displacement Rayleigh quotient, the stress Rayleigh quotient and the mixed quotient. The convergence rates of the three quotients are found to be related to the continuity and differentiability of the density and compliance variation over the unit cell. In general, the mixed quotient converges faster than both the displacement Rayleigh and the stress Rayleigh quotients, however, there exist special cases where either the displacement Rayleigh or the stress Rayleigh quotient shows the exact same convergence as the mixed-method. We show that all methods converge faster for smoother material property variations, but when density variation is rough, the difference between the mixed quotient and stress Rayleigh quotient is higher and similarly, when compliance variation is rough, the difference between the mixed quotient and displacement Rayleigh quotient is higher. Since eigenvalue problems such as those considered in this paper tend to be highly computationally intensive, it is expected that these results will lead to fast and efficient algorithms in the areas of phononics and photonics.

cond-mat.mtrl-sci↗