WAVES 2017: 13TH INTERNATIONAL CONFERENCE ON MATHEMATICAL AND NUMERICAL ASPECTS OF WAVE PROPAGATION
PROGRAM FOR THURSDAY, MAY 18TH
Days:
previous day
next day
all days

View: session overviewtalk overview

08:30-09:30 Session 18: Imaging with intensities only

Plenary Lecture

Location: Coffman Theater
08:30
Imaging with intensities only

ABSTRACT. High frequency imaging with optical, infrared or microwave systems must be done with only intensities because phases, or time resolved signals, cannot be recorded. Phase retrieval methods have been developed over a long time and are flexible and effective but depend on prior information about the image and can give uneven results. When, however, multiple illuminations of the object to be imaged are available then it may be possible to recover the missing phase information. I will present some recent results that use multiple illuminations and I will discuss associated imaging methods, their resolution and their robustness. I will also present the results of numerical simulations using these methods in optical and microwave imaging

10:00-12:00 Session 19A: Nonlinear Waves in Mechanical Metamaterials and Phononic Crystals

Minisymposium

Location: Mississippi Room
10:00
Vector solitons for elastic waves in architected soft solids

ABSTRACT. In this work, we theoretically and experimentally study nonlinear elastic wave propagation in architected soft solids. Different PDMS structures are considered based on a "rotating squares" geometry, known to exhibit an auxetic behavior upon unidirectional quasi-static loading. The nonlinear dynamics of these structures is modeled with a discrete model, accounting for both the translational and rotational degrees of freedom of the rigid square masses as well as their elastic coupling exhibiting geometrical nonlinearity. Vector soliton solutions are predicted and observed experimentally. We also demonstrate that managing the nonlinearity of these structures over a range of different nonlinearity types and amplitudes is quantitatively feasible.

10:30
Reprogrammable nonlinear phononic metamaterials
SPEAKER: Osama Bilal

ABSTRACT. A major challenge in materials design is to engineer matter that has the ability to change its mechanical properties in a predetermined manner within a practical time frame. Most of these mechanical properties are inscribed in materials' frequency dispersion spectrum, ranging form material stiffness at zero frequency to its thermal conductivity at much higher frequencies. In this work, we harness geometric and magnetic nonlinearities to tune the dispersion characteristics of matter in real-time. As a demonstration of principle, we program our nonlinear metamaterials to change the frequency range of its subwavelength band gap, where mechanical waves don’t propagate, in real-time. Using numerical modeling and experiments, we realize a meta-plate that can be re/programmed at the unit cell level (i.e., element wise) to guide elastic energy in arbitrary directions in space within fractions of a second. The realized concept can inspire the design of advanced functioning materials.

11:00
Geo-Inspired Mechanical Metamaterials
SPEAKER: Ahmed Elbanna

ABSTRACT. Friction is regarded as a source of energy dissipation. Crustal faults, however, leverage friction to grow instabilities, and localize energy. Earthquakes is the pinnacle of such phenomenon. The class of friction laws of relevance here is that in which the steady state friction force decreases with sliding velocity. Such friction may be realized on corrugated surfaces, due to flash melting of contact asperities, or during fast shearing of saturated porous media. We present a mechanical model of a chain of masses connected with linear springs and sliding on a rate weakening frictional interface. We show that the system enables the propagation of solitary waves whose characteristics are tunable by the level of the system prestress. The system is also asymmetric with respect to the direction of excitation. We discuss the implications of these observations on designing new materials that harness friction to generate unique nonlinear wave propagation features.

11:30
Domain Evolution Kinetics of Mechanical, Phase-transforming Structures

ABSTRACT. Multi-welled energy landscapes are key to micro-structural pattern formation observed in solids that undergo, e.g., phase transformations, ferroic domain switching, or diffusive phase separation. These processes necessitate the formation and movement of domain walls which separate homogeneous equilibrium states. Here, we present a purely mechanical, size-independent structure (or metamaterial) that exhibits similar domain evolution phenomena, and we demonstrate that the system obeys qualitatively and quantitatively analogous fundamental governing laws but with extreme tunability and experimental accessibility. We thus open a new chapter in mimicking atomic-scale dynamic processes at the observable metamaterial scale.

10:00-12:00 Session 19B: Contributed Talks
Location: President's Room
10:00
The Doppler Effect for SAR

ABSTRACT. We present a mathematical analysis of the start-stop approximation that is routinely used in synthetic aperture radar (SAR) imaging. The objective is to quantify the effect of those factors that the start-stop approximation neglects. They include the displacement of the antenna during the pulse round-trip time between the radar platform and the target and the Doppler frequency shift. We show that both phenomena can be accounted for by appropriately correcting the signal processing algorithm. This, in turn, requires computing the emitted and scattered field with the help of the Lorentz transform. If the correction is done, then the effect of the antenna motion on the image becomes negligibly small. Otherwise, the image gets shifted and also distorted. For some imaging settings, the distortions due to the start-stop approximation may become substantial, which is not commonly discussed in the SAR literature.

10:30
Generalized linear sampling method for active imaging of subsurface fractures

ABSTRACT. A theoretical foundation is developed for active seismic reconstruction of fractures endowed with spatially-varying interfacial condition (e.g. hydraulic fractures). The proposed indicator functional carries a superior localization property with no significant sensitivity to the fracture’s contact condition, measurement errors, and illumination frequency. This is accomplished through the paradigm of the F♯-factorization technique and the recently developed Generalized Linear Sampling Method (GLSM) applied to elastodynamics. The analysis of the well-posedness of the forward problem leads to an admissibility condition on the fracture’s (linearized) contact parameters. This in turn con- tributes toward establishing the applicability of the F♯-factorization method, and consequently aids the formulation of a convex GLSM cost functional whose minimizer can be computed without iterations. Such minimizer is then used to construct a robust fracture indicator function, whose performance is illustrated through a set of numerical experiments.

11:00
Analysis of an observers strategy for initial state reconstruction in unbounded domains

ABSTRACT. In this work, we are interested in the problem of recovering a compactly supported initial state of the wave equation in unbounded domain (such as the whole plane, a waveguide...). To this purpose, we assume that the velocity is known in a bounded observation region surrounding the support of the initial state. We consider an iterative algorithm of reconstruction based on back and forth nudging and prove the exponential convergence of this algorithm and its robustness with respect to noisy measures, at the continuous level. We also study the effect of the discretization process on the convergence of the algorithm.

11:30
Imaging polarizable dipoles

ABSTRACT. We consider the problem of imaging electric dipoles in a homogeneous medium from measurements of all three components of the electric field at an array of receivers. We show that an electromagnetic version of Kirchhoff migration can be used to recover the position and the orientation of the dipoles in the Fraunhofer asymptotic regime. We prove that the resolution estimates for the position are identical as in the acoustic case and provide error estimates for the dipole orientation. We extend these results to the case where the dipoles behave as passive sources, that is to say diffracting obstacles. In this setting, one wants to recover both the position and the polarizability tensor of each dipole in the medium.

10:00-12:00 Session 19C: Contributed Talks
Location: Room 324
10:00
A continuation method for building large invisible obstacles in waveguides
SPEAKER: Antoine Bera

ABSTRACT. In previous papers, a method has been proposed to prove the existence of invisible perturbations in waveguides. The method is constructive and has been validated numerically. But the drawback is that it is limited to small perturbations. In the present work, we show that the previous idea can be combined with a continuation method, in order to get larger invisible perturbations.

10:30
Modal expansion in elastic open waveguides with perfectly matched layers

ABSTRACT. The modal basis of elastic open waveguides contains two continua of radiation modes and a discrete set of trapped modes. A third set also exists: the leaky modes. However, they do not belong to the modal basis as they spatially grow to infinity. Herein, the excitation of elastic open waveguides is investigated. For numerical purpose, the infinite transverse direction is modelled with a truncated perfectly matched layer (PML). Indeed, the PML offers a natural way to reveal the contribution of leaky modes. The PML gives access to the improper Riemann sheets by redefining the branch cuts, yielding two rotated continua of radiation modes (PML modes). The cases of an infinite medium and an open waveguide are considered. It is shown that all sets are necessary for achieving convergence of the modal expansion.

11:00
Modeling the multimodal radiation from an open-ended waveguide
SPEAKER: Simon Félix

ABSTRACT. Using a multimodal formalism of the guided wave propagation and a complex coordinate stretching (PML), we derive algebraic solutions for the multimodal radiation impedance at the end of a waveguide open on the free space. The basic idea of the method is to turn the original unbounded problem into the problem of a cylindrical waveguide embedded in an infinite waveguide with an annular PML on the inside of its exterior wall. This method makes no assumption on the frequency range and can be applied to any cross-section geometry and wall thickness.

11:30
Identification of a time-dependent potential in a wave equation
SPEAKER: Thies Gerken

ABSTRACT. We augment the classical inhomogeneous wave equation by a zero-order term cu and consider the task of reconstructing c from the solution u. Here c is allowed to be time-dependent, which makes the problem more difficult. We present a suitable existence and uniqueness result for the wave equation and compute the Fréchet derivative of the solution operator. These results allow for the numerical reconstruction of c from artifical data, for which we apply an inexact newton method.

10:00-12:00 Session 19D: Contributed Talks
Location: Room 325
10:00
A combined Bayesian optimization-finite element approach for isoperimetric inequalities

ABSTRACT. The aim of this work is to introduce a novel numerical strategy to study problems arising in Spectral Geometry. We investigate a well-known conjecture by Polya-Szego on the Dirichlet eigenvalue problem by combining numerical strategies to approximate its eigenvalues and to find local optima of the first eigenvalue.

10:30
Fourth order explicit scheme for dissipative wave problems based on modified equation technique

ABSTRACT. In this talk with present an original numerical scheme for the explicit fourth order time discretization of linear dissipative wave equation. This scheme is based on the modified equation technique which, in general, gives implicit time discretization when dissipation is present in the propagating medium. We discuss the construction of the scheme, its stability properties and present some space/time convergence results. The scheme is shown to be more efficient for the considered cases than the fourth order explicit Runge-Kutta scheme.

11:00
High-order numerical solution of the Helmholtz equation for domains with reentrant corners
SPEAKER: Sam Magura

ABSTRACT. Standard numerical methods often fail to solve the Helmholtz equation accurately near reentrant corners, since the solution may become singular. The singularity has an inhomogeneous contribution from the boundary data near the corner and a homogeneous contribution determined by boundary conditions far from the corner. We present a regularization algorithm that uses a combination of analytical and numerical tools to distinguish between these two contributions and ultimately subtract the singularity. We then employ the method of difference potentials to numerically solve the regularized problem with high-order accuracy on a domain with a curvilinear boundary. Our numerical experiments show that the regularization successfully restores the design rate of convergence.

11:30
An efficient numerical algorithm for the 3D wave equation in domains of complex shape

ABSTRACT. We propose an efficient finite difference algorithm for the 3D wave equation in domains with curvilinear boundaries. Our approach combines the method of difference potentials for handling the complex geometries on regular grids and the Huygens’ principle for time marching.

13:30-14:30 Session 20: E.M. waves in magnetic plasmas

Plenary Lecture

Location: Coffman Theater
13:30
E.M. waves in magnetic plasmas
SPEAKER: Bruno Despres

ABSTRACT. Time harmonic waves in plasmas receive increasing interests [16] due to their scientific impor- tance for the heating of magnetic fusion plasma [Iter]. Recent progresses are reported on the de- velopment of a convenient mathematical theory for time harmonic waves in plasmas near the hybrid resonance. After presenting the cold- plasma dielectric tensor, a basic analytic solu- tion is constructed that captures the essential singularity of the problem. This information is used to construct manufactured solutions in the context of the limit absorption principle. Ma- nufactured solutions have the ability to capture the singular limit in a non singular way. Nu- merical applications are shown in the compan- ion paper [12]. In dimension two, manufactured solutions exhibit an additional highly oscillating behavior. The modeling of non linear boundary conditions is evoked.

15:00-16:30 Session 21A: Nonlinear Waves in Mechanical Metamaterials and Phononic Crystals

Minisymposium

Location: Mississippi Room
15:00
On the dynamic features of discrete lattices with nonlinear local resonators

ABSTRACT. The possibility of designing structures able to manipulate and control wave propagation --- the so-called metamaterials --- has attracted researches from different fields, from optics to mechanics, in the last decade. With respect to mechanical metamaterials, most of the works up to know have been limited to linear material behavior. In this paper, the dynamic behavior of local resonant metamaterials with nonlinear oscillators is investigated. The harmonic balance method is used to derive approximate expressions for the dispersion relations of these materials. The key idea here is to account for the effects of sub/superharmonic terms which have been neglected so far. Direct numerical simulations are also performed in order to verify the approximate solutions. From these analyses, unique features of such nonlinear metamaterials are revealed, such as: tunability, multistability and multiple band gap generation. Moreover, the possibility of designing non-reciprocal devices is shown.

15:30
Nonlinear Elastic Wave Dispersion in 1D Homogenous Media and Metamaterials

ABSTRACT. Wave motion lies at the heart of many disciplines in the physical sciences and engineering. For example, problems and applications involving light, sound, heat or fluid flow are all likely to involve wave dynamics at some level. In this extended abstract, we present our recent work on large-deformation elastic waves in solids, focusing on both homogeneous media and metamaterials.

16:00
Acoustic metamaterials in moving inhomogeneous media
SPEAKER: Wonju Jeon

ABSTRACT. During the past 10 years, most of acoustic metamaterial research has been done within a theoretical frame in which the medium is at rest. However, such acoustic metamaterials cannot preserve their unique properties or functions in the presence of flow. Therefore, in this study, we propose a theoretical framework to consider the effect of non-uniform mean flow on acoustic metamaterials for the purpose of understanding the physics of acoustic metamaterials within flow and designing a new concept of acoustic metamaterial.

15:00-16:30 Session 21B: Contributed Talks
Location: President's Room
15:00
Diffuse acoustic waves in a randomly stratified flow
SPEAKER: Etienne Gay

ABSTRACT. In this communication we develop an integral representation of the acoustic waves emitted by a source and transmitted by a randomly stratified fluid flow. The analysis is carried out in a regime of separation of scales whereby the fluctuations of the flow are much smaller than the source wavelength, which in turn is much smaller than the thickness of the flow--the so-called diffusion regime. Our aim is to subsequently develop coherent interferometric (CINT) imaging algorithms based on cross-correlation functions of the acoustic waves recorded at the bottom of the flow to possibly locate the source above it.

15:30
Wave Attenuation Along a Rough Floating Elastic Beam

ABSTRACT. Semi-analytical and numerical methods are presented to describe the attenuation of water waves in a two-dimensional fluid domain, which has its surface covered by a rough thin elastic beam and is of finite depth. Roughness of the beam is incorporated via a random process describing the variations in the properties of the beam's mass and rigidity. The semi-analytical method is based on a multiple-scale expansion of the velocity potential, from which an equation can be derived describing the attenuation of the effective wave field. The numerical results, which are obtained via a step-approximation method, validate the multiple-scale approach for small-amplitude beam roughness.

16:00
An Efficient Semi-Analytical Scheme for Determining the Reflection of Lamb Waves in a Semi-Infinite Waveguide
SPEAKER: Robert Davey

ABSTRACT. The reflection of Lamb waves from a free edge perpendicular to an elastodynamic plate is studied. It is known that extant methods for finding the reflected field have poor convergence due to irregular behaviour near corners. The form of the irregularity for an elastodynamic corner is derived asymptotically. A new method for incorporating this form of the corner behaviour is then implemented. Results are presented showing this new method improves convergence in the reflection problem.

15:00-16:30 Session 21C: Contributed Talks
Location: Room 324
15:00
Inverse scattering with iterative determination of the regularization parameter

ABSTRACT. Many applications can be described by a PDE model with a set of unknown parameters that we wish to calibrate based on measurements related to its solution. This can be seen as a constrained minimization problem where we want to minimize the mismatch between the observed data and our model prediction, including an extra regularization term, and use the PDE as a constraint. Often, a suitable regularization parameter is determined by solving the problem for a whole range of parameters -- e.g. using the L-curve -- which is computationally very expensive. In this work we present an iterative way to find a good regularization parameter based on the discrepancy principle.

15:30
Non-linear Tikhonov Regularization for Inverse Scattering from Anisotropic Media

ABSTRACT. Considering time-harmonic inverse scattering of either electromagnetic or acoustic waves from an inhomogeneous anisotropic medium, we provide Tikhonov and sparsity-promoting regularization techniques in Banach spaces. To this end we analyze the dependence of the scattered fields and their derivatives on material parameters of an admissible set equipped with L^p-topology. Therewith we first show convergence of non-linear Tikhonov regularization against a minimum-norm solution and second extend that method to a sparsity-promoting one.

16:00
Adaptive Eigenspace Method for Inverse Scattering Problems in the Frequency Domain

ABSTRACT. A nonlinear optimization method is proposed for the solution of inverse scattering problems in the frequency domain, when the scattered field is governed by the Helmholtz equation. The time-harmonic inverse medium problem is formulated as a PDE-constrained optimization problem and solved by an inexact truncated Newton-type iteration. Instead of a grid-based discrete representation, the unknown wave speed is projected to a particular finite-dimensional basis of eigenfunctions, which is iteratively adapted during the optimization. Truncating the adaptive eigenspace (AE) basis at a (small and slowly increasing) finite number of eigenfunctions effectively introduces regularization into the inversion and thus avoids the need for standard Tikhonov-type regularization. Both analytical and numerical evidence underpins the accuracy of the AE representation. Numerical experiments demonstrate the efficiency and robustness to missing or noisy data of the resulting adaptive eigenspace inversion (AEI) method.

15:00-16:30 Session 21D: Contributed Talks
Location: Room 325
15:00
Sobolev-dG a class of dG methods with tame CFL numbers

ABSTRACT. A new class of discontinuous Galerkin method with tame CFL numbers is presented.

15:30
Accuracy and Pollution Errors of HDG Methods for the Helmholtz equation involving non-smooth solutions
SPEAKER: Matthias Taus

ABSTRACT. The pollution error of Hybridizable Discontinuous Galerkin (HDG) methods is studied for problems involving non-smooth solutions. The error analysis of HDG methods is extended to obtain a wave-number explicit quasi-optimal error estimate. This estimate is used to design and analyze a HDG method involving point sources. It is proved that the proposed method converges quadratically in the $L^2$-error, and that if the order $p$ of the HDG discretization is slightly increased with the frequency $\omega$ following $p=O(\log{\omega})$, the pollution error can be eliminated, despite the non-smoothness of the problem. Results are derived for constant wave speeds but similar results can be proved for piece-wise smooth wave speeds, providing a quasi-optimal discretization, albeit with a stronger dependence of $p$ with respect to $\omega$. Numerical examples are provided to corroborate the claims.

16:00
Boundary Elements with Mesh Refinements for the Wave Equation
SPEAKER: David Stark

ABSTRACT. We discuss time domain boundary element methods for singular geometries, in particular graded meshes and adaptive mesh refinements. First, we discuss edge and corner singularities for a Dirichlet problem for the wave equation. Time independent graded meshes lead to efficient approximations, as confirmed by numerical experiments for wave scattering from screens. We briefly discuss adaptive mesh refinement procedures based on a posteriori error estimates. A modified MOT scheme provides an efficient preconditioner (or stand-alone solver) for the space-time systems obtained for the Galerkin discretisations.

15:00-16:30 Session 21E: Contributed Talks
Location: Room 326
15:00
Numerical Study of Fracture Connectivity Effect on Seismic Wave Propagation
SPEAKER: Vadim Lisitsa

ABSTRACT. According to resent theoretical study seismic wave propagating in a model with fluid-filled fractures structure may lead to the wave-induced fluid flow. Moreover, these flows are expected to depend on the connectivity of the fractures. However, this effect expected at high frequencies where scattering may dominate. In this paper we perform numerical study of this effect. We show that in case of connected fractures the wave velocity is lower than in the case of the nonintersected fractures. However, energy dissipation is mainly connected with the scattering and effect of fracture connectivity can not be estimated from full weveform simulation.

15:30
Resolution Control in Half-space Time-reversal Wave Focusing
SPEAKER: Heedong Goh

ABSTRACT. In wave focusing subsurface geophysical applications, the recordings at the mirror, situated on the surface of a half-space, may have to be time-reversed while flipping the character of the boundary conditions due to equipment/sensor limitations. For example, the recording sensors at the time-reversal mirror may record Dirichlet data, but the transmitting equipment may be able to accommodate Neumann data only. Under certain conditions, such flipping may worsen the focusing resolution. We study the relation between the wavefields generated by the recording-transmitting pairs Dr-to-Dt and Dr-to-Nt, and propose a filter to improve the resolution imposed by the aforementioned equipment constraints in the Dr-to-Nt case.

16:00
FEM-BEM coupling for transient acoustic scattering by thermoelastic obstacles

ABSTRACT. We present a combined field and boundary integral equation method for solving time-dependent scattering of acoustic waves by thermoelastic obstacles. The approach is geared towards a finite element discretization in the interior of the scatterer and a boundary element approximation of the acoust field. Using an integral representation of the solution in the infinite exterior domain occupied by the fluid, the problem is reduced to one defined only over the finite region occupied by the solid, with nonlocal boundary conditions. The resulting non local boundary problem is analized using the Laplace transform in terms of time-domain data. Existence and uniqueness results are established in the Laplace domain where Galerkin semi-discretization approximations are derived and error estimates are obtained. Full space-time discretization and time-domain error estimates based on the Convolution Quadrature method are also presented.

17:30-19:30 Session : WAVES Unplugged at the Weisman

Thursday Reception

Location: Weisman Art Museum