Abstract: It has been demonstrated that non-destructive inspection of plates can be performed by using two-dimensional maps of instantaneous out-of-plane displacements obtained with a self-developed pulsed TV-holography system. Specifically, the interaction of guided elastic waves with defects produces scattering patterns that contain information about the defects (position, dimensions, orientation, etc.). For quantitative characterization on this basis, modeling of the wave propagation and interaction with the defects is necessary. In fact, the development of models for scattering of waves in plates is yet an active research field in which the most reliable approach is usually based on the rigorous formulation of elasticity theory. By contrast, in this work the capability of a simple two-dimensional scalar model for obtaining a quantitative description of the output two-dimensional maps associated to artificial defects in plates is studied. Some experiments recording the interaction of narrowband Rayleigh waves with artificial defects in aluminum plates are presented, in which the acoustic field is obtained from the TV-holography optical phase-change maps by means of a specially developed two-step spatio-temporal Fourier transform method. For the modeling, harmonic regime and free-stress boundary conditions are assumed. Comparisons between experimental and simulated maps are included for defects with different shapes.

No.: 7389 ISSN: 0277786X

ID: CaltechAUTHORS:20170125-073103435

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Abstract: The problem of simulating the scattering of (acoustic, electromagnetic, elastic) waves has provided a particularly sustained, demanding and motivating challenge for the development of efficient and accurate numerical methods since the advent of computers. The classical issues present in most other applications, such as those related to the environmental and/or geometrical intricacies of the media in which quantities of interest are defined, are augmented in the context of wave propagation by the intrinsic complexities (i.e. oscillations) of the quantities themselves. Still, very efficient methodologies have been devised, particularly in the last twenty years, to simulate wave processes in rather complex settings. These techniques can be based, for instance, on finite elements (see e.g. [34, 35, 46] and the references therein), finite differences [40, 49] or boundary integral equations [4, 8, 11, 16, 25], and they can, today, effectively address these problems, with a high degree of accuracy, in domains that can span tens or perhaps even a few hundred wavelengths. The very nature of these classical approaches, however, limits their applicability at higher frequencies since the numerical resolution of field oscillations translates in a commensurately higher number of degrees of freedom and this, in turn, can easily lead to impractical computational times. In this chapter we review some recently proposed methodologies [5, 13–15, 29] that can overcome these limitations while retaining the mathematical rigor of classical numerical procedures.

No.: 59
ID: CaltechAUTHORS:20181105-150853442

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Abstract: A turning point method for difference equations is developed. This method is coupled with the LG-WKB method via matching to provide approximate solutions to the initial value problem. The techniques developed are used to provide strong asymptotics for Hermite polynomials.

No.: 154
ID: CaltechAUTHORS:20181105-153010735

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Abstract: We review a set of algorithms and methodologies developed recently for the numerical solution of problems of scattering by complex bodies in three-dimensional space. These methods, which are based on integral equations, high-order integration, Fast Fourier Transforms and highly accurate high-frequency integrators, can be used in the solution of problems of electromagnetic and acoustic scattering by surfaces and penetrable scatterers — even in cases in which the scatterers contain geometric singularities such as comers and edges. All of the solvers presented here exhibit high-order convergence, they run on low memories and reduced operation counts, and they result in solutions with a high degree of accuracy. In particular, our approach to direct solution of integral equations results in algorithms that can evaluate accurately in a personal computer scattering from hundred-wavelength-long objects — a goal, otherwise achievable today only by super-computing. The high-order high-frequency methods we present, in turn, are efficient where our direct methods become costly, thus leading to an overall computational methodology which is applicable and accurate throughout the electromagnetic spectrum.

No.: 31
ID: CaltechAUTHORS:20181106-073820064

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Abstract: We present a new set of algorithms and methodologies for the numerical solution of problems of scattering by complex bodies in three-dimensional space. These methods, which are based on integral equations, high-order integration, fast Fourier transforms and highly accurate high-frequency methods, can be used in the solution of problems of electromagnetic and acoustic scattering by surfaces and penetrable scatterers - even in cases in which the scatterers contain geometric singularities such as corners and edges. In all cases the solvers exhibit high-order convergence, they run on low memories and reduced operation counts, and they result in solutions with a high degree of accuracy.

ID: CaltechAUTHORS:20181102-145356744

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Abstract: We present a new algorithm for the numerical solution of problems of acoustic scattering by surfaces in three-dimensional space. This algorithm evaluates the scattered field, through fast, high-order solution of the boundary integral equation. The high-order of the solver is achieved through use of partition of unity together with analytical resolution of kernel singularities. The acceleration in turn, results from a novel approach which, based on high-order "two-face" equivalent source approximations, reduces the evaluation of far interactions to evaluation of 3-D FFTs. We demonstrate its performance with a variety of numerical results. In particular, we show that the present algorithm can evaluate accurately in a personal computer, scattering from bodies of acoustical sizes of several hundreds.

Vol.: 2
ID: CaltechAUTHORS:20181106-144801749

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Abstract: A detailed analysis of the behaviour of electromagnetic scattering from various corrugated bidimensional surfaces is presented. We show that rigorous electromagnetic computations on two dimensional surfaces can in fact yield HH/VV polarization ratios greater than one, with values consistent with those observed experimentally. We also show that HH/VV ratios greater than one are ubiquitous in the case of surfaces of the form f(x, y)=f_1/(x)+f_2(y), known as crossed gratings in optics. Theoretically and numerically, these surfaces are shown to produce backscattered returns for which the first order Rice/Valenzuela term vanishes for off axis incidence. The second order term becomes dominant and has the property that HH returns exceed VV returns for a significant range of incident angles. Our approach is based on the methods of O.Bruno and F. Reitich (see J. Opt. Soc. A., vol.10, p.2551-62, 1993) which yield accurate results for a large range of values of the surface height. In particular, these methods can be used well beyond the domain of applicability of the first order theory of S.O. Rice (1951). The error in our calculations is guaranteed to be several orders of magnitude smaller than the computed values. The high order expansions provided by these methods are essential to determining the role played by the second order terms as they show that these terms indeed dominate most of the backscattering returns for the surfaces mentioned. Classically, large HH/VV ratios were sought by means of first order approximations on one dimensional sinusoidal profiles. In that case, we show that the first order terms do not vanish and the first order theories predict the behaviour of the backscattered returns, for small values of the height to period ratio. However, in the case of a two dimensional bisinusoidal surface, strong polarization dependent anomalies appear in the scattering returns as a result of the contributions of second order terms since, in that case, the first order contributions vanish.

ID: CaltechAUTHORS:20181105-123901867

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Abstract: Perturbation theory is among the most useful and successful analytical tools in applied mathematics. Countless examples of enlightening perturbation analyses have been performed for a wide variety of models in areas ranging from fluid, solid, and quantum mechanics to chemical kinetics and physiology. The field of electromagnetic and acoustic wave propagation is certainly no exception. Many studies of these processes have been based on perturbative calculations where the role of the variation parameter has been played by the wavelength of radiation, material constants, or geometric characteristics. It is this latter instance of geometric perturbations in problems of wave propagation that we shall review in the present chapter. Use of geometric perturbation theory is advantageous in the treatment of configurations which, however complex, can be viewed as deviations from simpler ones—those for which solutions are known or can be obtained easily. Many uses of such methods exist, including, among others, applications to optics, oceanic and terrain scattering, SAR imaging and remote sensing, and diffraction from ablated, eroded, or deformed objects; see, e.g., [47, 52, 56, 59, 62]. The analysis of the scattering processes involved in such applications poses challenging computational problems that require resolution of the interplay between highly oscillatory waves and interfaces. In the case of oceanic scattering, for instance, nonlinear water wave interactions and capillarity effects give rise to highly oscillatory modulated wave trains that are responsible for the most substantial portions of the scattering returns [35].

ID: CaltechAUTHORS:20181105-144641812

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Abstract: We present a fast, high-order algorithm for the solution of problems of acoustic scattering from smooth surfaces in three dimensions. The present algorithm computes scattered fields in

ID: CaltechAUTHORS:20181102-151802755

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Abstract: Martensitic transformations are shape-deforming phase transitions which can be induced in certain alloys as a result of changes in the imposed strains, stresses or temperatures. The interest in these alloys, which undergo a shape-deforming phase transition from a high temperature phase (austenite) to a low temperature phase (martensite), stems in part from their applicability as elements in active structures. In the present text we outline three recent theories concerning the energy transfers that accompany and determine the martensitic phase change. In §1 we will thus mention the pseudoelastic hysteresis in shape-memory wires and the corresponding treatment of (Leo et al .,1993; Bruno et al .,1995). In §2 we will present some aspects of the discussion of (Bruno et al. ,1996) on equilibrium configurations in polycrystalline martensites. In §3 finally we will describe a computation (Bruno, 1997) which explains typical microstructural lengthscales observed in single-crystalline martensites as resulting from an interplay between elastic energies and dissipative mechanisms.

No.: 62
ID: CaltechAUTHORS:20190826-124739605

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Abstract: We draw attention to the problem of estimation of elastic energies in martensitic polycrystals. In particular we introduce a tensorial parameter η=η_(ijkl) which contains information about the microgeometry and disorder of the polycrystalline structure. Under the assumption of isotropic elasticity and mild hypothesis on the statistics of the polycrystal, this parameter allows for explicit calculation of rigorous and stringent upper bounds on the effective energy. For circular grains in two dimensions η gives the elastic energy resulting from transformation of a single circular inclusion in an elastic matrix and the bounds coincide with those derived recently by Bruno, Reitich and Leo. Consideration of such particular cases shows that our bounds can yield substantial improvements over those obtained under Taylor’s constant strain hypothesis. For arbitrary microgeometries the statistical parameter η can be calculated by means of two-point correlations functions.

No.: 99
ID: CaltechAUTHORS:20181106-152204596

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Abstract: Martensitic transformations are shape-deforming phase transitions which can be induced in certain alloys as a result of changes in the imposed strains, stresses or temperatures. The interest in these alloys, which undergo a shape-deforming phase transition from a high temperature phase (austenite) to a low temperature phase (martensite), stems in part from their applicability as elements in active structures. In this paper we focus on the energy transfers that accompany the martensitic phase change. We discuss, in three concrete examples, the ways in which temperature, together with the elastic and dissipated energies, determine the equilibria as well as the quasi-static dynamics in martensites. Thus, in §1 we consider the pseudoelastic hysteresis in shape-memory wires; our treatment draws from (Leo et al., 1993; Bruno et al, 1995). In §2 on the other hand, we follow (Bruno et al., 1996) and discuss equilibrium configurations in polycrystalline martensitic polycrystals. In §3 finally, we present some new theoretical computations for certain typical microstructural lengthscales, the twin widths, observed in single-crystalline martensite twinning.

No.: 60
ID: CaltechAUTHORS:20200610-144648854

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Abstract: Martensitic transformations are shape-deforming phase transitions which can be induced in certain alloys as a result of changes in the imposed strains, stresses or temperatures. The interest in these alloys, which undergo a shape-deforming phase transition form a high temperature phase to a low temperature phase, stems in part from their applicability as elements in active structures. In this paper we focus on the energy transfers that accompany the martensitic phase change. We discuss, in three concrete examples, the ways in which temperature, together with the elastic and dissipated energies, determine the equilibria as well as the quasi-static dynamics in martensites. Thus, in §1 we consider the pseudoelastic hysteresis in shape- memory wires; our treatment draws from (7, 3). In §2, on the other hand, we follow and discuss equilibrium configurations in polycrystalline martensitic polycrystals. In §3, finally, we present some new theoretical computations for certain typical microstructural lengthscales, the twin widths, observed in single- crystalline martensite twinning.

No.: 3039
ID: CaltechAUTHORS:20181105-111232917

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