Abstract: Tidal forces are generally neglected in the discussion about the mechanisms driving plate tectonics despite a worldwide geodynamic asymmetry also observed at subduction and rift zones. The tidal drag could theoretically explain the westerly shift of the lithosphere relative to the underlying mantle. Notwithstanding, viscosity in the asthenosphere is apparently too high to allow mechanical decoupling produced by tidal forces. Here, we propose a model for global scale geodynamics accompanied by numerical simulations of the tidal interaction of the Earth with the Moon and the Sun. We provide for the first time a theoretical proof that the tidal drag can produce a westerly motion of the lithosphere, also compatible with the slowing of the Earth’s rotational spin. Our results suggest a westerly rotation of the lithosphere with a lower bound of w ≈ (0.1 – 0.2)°/Myr in the presence of a basal effective shear viscosity η ≈ 10¹⁶ Pa•s, but it may rise to w > 1°/Myr with a viscosity of η ≾ 3 x 10¹⁴ Pa•s within the Low-Velocity Zone (LVZ) atop the asthenosphere. This faster velocity would be more compatible with the mainstream of plate motion and the global asymmetry at plate boundaries. Based on these computations, we suggest that the super-adiabatic asthenosphere, being vigorously convecting, may further reduce the viscous coupling within the LVZ. Therefore, the combination of solid Earth tides, ultra-low viscosity LVZ and asthenospheric polarized small-scale convection may mechanically satisfy the large-scale decoupling of the lithosphere relative to the underlying mantle. Relative plate motions are explained because of lateral viscosity heterogeneities at the base of the lithosphere, which determine variable lithosphere-asthenosphere decoupling and plate interactions, hence plate tectonics.

Publication: Geoscience Frontiers Vol.: 14 No.: 6 ISSN: 1674-9871

ID: CaltechAUTHORS:20230718-855965700.6

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Abstract: This paper presents the first parallel implementation of the novel "Interpolated Factored Green Function" (IFGF) method introduced recently for the accelerated evaluation of discrete integral operators arising in wave scattering and other areas (Bauinger and Bruno, Jour. Computat. Phys., 2021). On the basis of the hierarchical IFGF interpolation strategy, the proposed (hybrid MPI-OpenMP) parallel implementation results in efficient data communication, and it scales up to large numbers of cores—without any hard limitations on the number of cores efficiently employed. Moreover, on any given number of cores, the proposed parallel approach preserves the O(N log N) computing cost inherent in the sequential version of the IFGF algorithm. Unlike other approaches, the IFGF method does not utilize the Fast Fourier Transform (FFT), and it is thus better suited for efficient parallelization in distributed-memory computer systems. In particular, the IFGF method relies on a “peer-to-peer” strategy wherein, at every level, field propagation is directly enacted via “exchanges” between “peer” polynomials of constant degree, without data accumulation in large-scale “telephone-central” mathematical constructs such as those used in the Fast Multipole Method (FMM) and pure FFT-based approaches. A variety of numerical results presented in this paper illustrate the character of the proposed parallel algorithm, in particular demonstrating scaling from 1 to all 1,680 cores available in the High Performance Computing cluster used, and for problems of up to 4,096 wavelengths in acoustic size.

Publication: Journal of Computational Physics Vol.: 475ISSN: 0021-9991

ID: CaltechAUTHORS:20230213-460790900.1

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Abstract: The physics-informed neural operator (PINO) is a machine learning architecture that has shown promising empirical results for learning partial differential equations. PINO uses the Fourier neural operator (FNO) architecture to overcome the optimization challenges often faced by physics-informed neural networks. Since the convolution operator in PINO uses the Fourier series representation, its gradient can be computed exactly on the Fourier space. While Fourier series cannot represent nonperiodic functions, PINO and FNO still have the expressivity to learn nonperiodic problems with Fourier extension via padding. However, computing the Fourier extension in the physics-informed optimization requires solving an ill-conditioned system, resulting in inaccurate derivatives which prevent effective optimization. In this work, we present an architecture that leverages Fourier continuation (FC) to apply the exact gradient method to PINO for nonperiodic problems. This paper investigates three different ways that FC can be incorporated into PINO by testing their performance on a 1D blowup problem. Experiments show that FC-PINO outperforms padded PINO, improving equation loss by several orders of magnitude, and it can accurately capture the third order derivatives of nonsmooth solution functions.

Publication: arXiv
ID: CaltechAUTHORS:20221221-004750416

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Abstract: This paper presents an efficient parallel radiative transfer-based inverse-problem solver for time-domain optical tomography. The radiative transfer equation provides a physically accurate model for the transport of photons in biological tissue, but the high computational cost associated with its solution has hindered its use in time-domain optical-tomography and other areas. In this paper this problem is tackled by means of a number of computational and modeling innovations, including (1) A spatial parallel-decomposition strategy with perfect parallel scaling for the forward and inverse problems of optical tomography on parallel computer systems; and, (2) A Multiple Staggered Source method (MSS) that solves the inverse transport problem at a computational cost that is independent of the number of sources employed, and which significantly accelerates the reconstruction of the optical parameters: a six-fold MSS acceleration factor is demonstrated in this paper. Finally, this contribution presents (3) An intuitive derivation of the adjoint-based formulation for evaluation of functional gradients, including the highly-relevant general Fresnel boundary conditions—thus, in particular, generalizing results previously available for vacuum boundary conditions. Solutions of large and realistic 2D inverse problems are presented in this paper, which were produced on a 256-core computer system. The combined parallel/MSS acceleration approach reduced the required computing times by several orders of magnitude, from months to a few hours.

Publication: Journal of Quantitative Spectroscopy and Radiative Transfer Vol.: 290ISSN: 0022-4073

ID: CaltechAUTHORS:20220705-346498000

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Abstract: This paper presents a spectral scheme for the numerical solution of nonlinear conservation laws in non-periodic domains under arbitrary boundary conditions. The approach relies on the use of the Fourier Continuation (FC) method for spectral representation of non-periodic functions in conjunction with smooth localized artificial viscosity assignments produced by means of a Shock-Detecting Neural Network (SDNN). Like previous shock capturing schemes and artificial viscosity techniques, the combined FC-SDNN strategy effectively controls spurious oscillations in the proximity of discontinuities. Thanks to its use of a localized but smooth artificial viscosity term, whose support is restricted to a vicinity of flow-discontinuity points, the algorithm enjoys spectral accuracy and low dissipation away from flow discontinuities, and, in such regions, it produces smooth numerical solutions—as evidenced by an essential absence of spurious oscillations in level set lines. The FC-SDNN viscosity assignment, which does not require use of problem-dependent algorithmic parameters, induces a significantly lower overall dissipation than other methods, including the Fourier-spectral versions of the previous entropy viscosity method. The character of the proposed algorithm is illustrated with a variety of numerical results for the linear advection, Burgers and Euler equations in one and two-dimensional non-periodic spatial domains.

Publication: Journal of Computational Physics: X Vol.: 15ISSN: 2590-0552

ID: CaltechAUTHORS:20220705-346684000

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Abstract: A high-order method to evolve in time electromagnetic and velocity fields in conducting fluids with non-periodic boundaries is presented. The method has a small overhead compared with fast FFT-based pseudospectral methods in periodic domains. It uses the magnetic vector potential formulation for accurately enforcing the null divergence of the magnetic field, and allowing for different boundary conditions including perfectly conducting walls or vacuum surroundings, two cases relevant for many astrophysical, geophysical, and industrial flows. A spectral Fourier continuation method is used to accurately represent all fields and their spatial derivatives, allowing also for efficient solution of Poisson equations with different boundaries. A study of conducting flows at different Reynolds and Hartmann numbers, and with different boundary conditions, is presented to study convergence of the method and the accuracy of the solenoidal and boundary conditions.

Publication: Computer Physics Communications Vol.: 275ISSN: 0010-4655

ID: CaltechAUTHORS:20220209-266110000

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Abstract: This paper presents a "two-dimensional Fourier Continuation" method (2D-FC) for construction of bi-periodic extensions of smooth non-periodic functions defined over general two-dimensional smooth domains. The approach can be directly generalized to domains of any given dimensionality, and even to non-smooth domains, but such generalizations are not considered here. The 2D-FC extensions are produced in a two-step procedure. In the first step the one-dimensional Fourier Continuation method is applied along a discrete set of outward boundary-normal directions to produce, along such directions, continuations that vanish outside a narrow interval beyond the boundary. Thus, the first step of the algorithm produces "blending-to-zero along normals" for the given function values. In the second step, the extended function values are evaluated on an underlying Cartesian grid by means of an efficient, high-order boundary-normal interpolation scheme. A Fourier Continuation expansion of the given function can then be obtained by a direct application of the two-dimensional FFT algorithm. Algorithms of arbitrarily high order of accuracy can be obtained by this method. The usefulness and performance of the proposed two-dimensional Fourier Continuation method are illustrated with applications to the Poisson equation and the time-domain wave equation within a bounded domain. As part of these examples the novel "Fourier Forwarding" solver is introduced which, propagating plane waves as they would in free space and relying on certain boundary corrections, can solve the time-domain wave equation and other hyperbolic partial differential equations within general domains at computing costs that grow sublinearly with the size of the spatial discretization.

Publication: SIAM Journal on Scientific Computing Vol.: 44 No.: 2 ISSN: 1064-8275

ID: CaltechAUTHORS:20220802-839163000

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Abstract: Silicon photonics is an emerging technology which, enabling nanoscale manipulation of light on chips, impacts areas as diverse as communications, computing, and sensing. Wavelength division multiplexing is commonly used to maximize throughput over a single optical channel by modulating multiple data streams on different wavelengths concurrently. Traditionally, wavelength (de)multiplexers are implemented as monolithic devices, separate from the grating coupler, used to couple light into the chip. This paper describes the design and measurement of a grating coupler demultiplexer—a single device which combines both light coupling and demultiplexing capabilities. The device was designed by means of a custom inverse design algorithm which leverages boundary integral Maxwell solvers of extremely rapid convergence as the mesh is refined. To the best of our knowledge, the fabricated device enjoys the lowest insertion loss reported for grating demultiplexers, small size, high splitting ratio, and low coupling-efficiency imbalance between ports, while meeting the fabricability constraints of a standard UV lithography process.

Publication: Communications Physics Vol.: 5ISSN: 2399-3650

ID: CaltechAUTHORS:20220324-224059015

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Abstract: We identify a neutron-flux “skin effect” in the context of neutron transport theory. The skin effect, which emerges as a boundary layer at material interfaces, plays a critical role in a correct description of transport phenomena. A correct accounting of the boundary-layer structure helps bypass computational difficulties reported in the literature over the last several decades, and should lead to efficient numerical methods for neutron transport in two and three dimensions.

Publication: Physical Review E Vol.: 104 No.: 3 ISSN: 2470-0045

ID: CaltechAUTHORS:20210927-213255519

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Abstract: We present new methodologies for the numerical solution of problems of elastic scattering by open arcs in two dimensions. The algorithms utilize weighted versions of the classical elastic integral operators associated with Dirichlet and Neumann boundary conditions, where the integral weight accounts for (and regularizes) the singularity of the integral‐equation solutions at the open‐arc endpoints. Crucially, the method also incorporates a certain “open‐arc elastic Calderón relation” introduced in this article, whose validity is demonstrated on the basis of numerical experiments, but whose rigorous mathematical proof is left for future work. (In fact, the aforementioned open‐arc elastic Calderón relation generalizes a corresponding elastic Calderón relation for closed surfaces, which is also introduced in this article, and for which a rigorous proof is included.) Using the open‐surface Calderón relation in conjunction with spectrally accurate quadrature rules and the Krylov‐subspace linear algebra solver GMRES, the proposed overall open‐arc elastic solver produces results of high accuracy in small number of iterations, for both low and high frequencies. A variety of numerical examples in this article demonstrate the accuracy and efficiency of the proposed methodology.

Publication: International Journal for Numerical Methods in Engineering Vol.: 122 No.: 11 ISSN: 0029-5981

ID: CaltechAUTHORS:20190906-093843706

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Abstract: This paper presents a windowed Green function (WGF) method for the numerical solution of problems of elastic scattering by “locally-rough surfaces” (i.e., local perturbations of a half space), under either Dirichlet or Neumann boundary conditions, and in both two and three spatial dimensions. The proposed WGF method relies on an integral-equation formulation based on the free-space Green function, together with smooth operator windowing (based on a “slow-rise” windowing function) and efficient high-order singular-integration methods. The approach avoids the evaluation of the expensive layer Green function for elastic problems on a half-space, and it yields uniformly fast convergence for all incident angles. Numerical experiments for both two and three dimensional problems are presented, demonstrating the accuracy and super-algebraically fast convergence of the proposed method as the window-size grows.

Publication: Computer Methods in Applied Mechanics and Engineering Vol.: 376ISSN: 0045-7825

ID: CaltechAUTHORS:20200629-081223570

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Abstract: This paper presents a novel Interpolated Factored Green Function method (IFGF) for the accelerated evaluation of the integral operators in scattering theory and other areas. Like existing acceleration methods in these fields, the IFGF algorithm evaluates the action of Green function-based integral operators at a cost of O(N log N) operations for an N-point surface mesh. The IFGF strategy, which leads to an extremely simple algorithm, capitalizes on slow variations inherent in a certain Green function analytic factor, which is analytic up to and including infinity, and which therefore allows for accelerated evaluation of fields produced by groups of sources on the basis of a recursive application of classical interpolation methods. Unlike other approaches, the IFGF method does not utilize the Fast Fourier Transform (FFT), and is thus better suited than other methods for efficient parallelization in distributed-memory computer systems. Only a serial implementation of the algorithm is considered in this paper, however, whose efficiency in terms of memory and speed is illustrated by means of a variety of numerical experiments—including a 43 min., single-core operator evaluation (on 10 GB of peak memory), with a relative error of 1.5×10⁻², for a problem of acoustic size of 512 wavelengths.

Publication: Journal of Computational Physics Vol.: 430ISSN: 0021-9991

ID: CaltechAUTHORS:20210107-135537998

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Abstract: This paper introduces a high-order-accurate strategy for integration of singular kernels and edge-singular integral densities that appear in the context of boundary integral equation formulations for the problem of acoustic scattering. In particular, the proposed method is designed for use in conjunction with geometry descriptions given by a set of arbitrary non-overlapping logically-quadrilateral patches—which makes the algorithm particularly well suited for computer-aided design (CAD) geometries. Fejér's first quadrature rule is incorporated in the algorithm, to provide a spectrally accurate method for evaluation of contributions from far integration regions, while highly-accurate precomputations of singular and near-singular integrals over certain “surface patches” together with two-dimensional Chebyshev transforms and suitable surface-varying “rectangular-polar” changes of variables, are used to obtain the contributions for singular and near-singular interactions. The overall integration method is then used in conjunction with the linear-algebra solver GMRES to produce solutions for sound-soft open- and closed-surface scattering obstacles, including an application to an aircraft described by means of a CAD representation. The approach is robust, fast, and highly accurate: use of a few points per wavelength suffices for the algorithm to produce far-field accuracies of a fraction of a percent, and slight increases in the discretization densities give rise to significant accuracy improvements.

Publication: Journal of Computational Physics Vol.: 421ISSN: 0021-9991

ID: CaltechAUTHORS:20181102-090626983

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Abstract: We present a Fourier Continuation-based parallel pseudospectral method for incompressible fluids in cuboid non-periodic domains. The method produces dispersionless and dissipationless derivatives with fast spectral convergence inside the domain, and with very high order convergence at the boundaries. Incompressibility is imposed by solving a Poisson equation for the pressure. Being Fourier-based, the method allows for fast computation of spectral transforms. It is compatible with uniform grids (although refined or nested meshes can also be implemented), which in turn allows for explicit time integration at sufficiently high Reynolds numbers. Using a new parallel code named SPECTER we illustrate the method with two problems: channel flow, and plane Rayleigh-Bénard convection under the Boussinesq approximation. In both cases the method yields results compatible with previous studies using other high-order numerical methods, with mild requirements on the time step for stability.

Publication: Computer Physics Communications Vol.: 256ISSN: 0010-4655

ID: CaltechAUTHORS:20200702-121835661

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Abstract: This article presents full-spectrum, well-conditioned, Green-function methodologies for evaluation of scattering by general periodic structures, which remains applicable on a set of challenging singular configurations, usually called Rayleigh-Wood (RW) anomalies (at which the quasi-periodic Green function ceases to exist), where most existing quasi-periodic solvers break down. After reviewing a variety of existing fast-converging numerical procedures commonly used to compute the classical quasi-periodic Green-function, the present work explores the difficulties they present around RW-anomalies and introduces the concept of hybrid “spatial/spectral” representations. Such expressions allow both the modification of existing methods to obtain convergence at RW-anomalies as well as the application of a slight generalization of the Woodbury-Sherman-Morrison formulae together with a limiting procedure to bypass the singularities. (Although, for definiteness, the overall approach is applied to the scalar (acoustic) wave-scattering problem in the frequency domain, the approach can be extended in a straightforward manner to the harmonic Maxwell's and elasticity equations.) Ultimately, this thorough study of RW-anomalies yields fast and highly-accurate solvers, which are demonstrated with a variety of simulations of wave-scattering phenomena by arrays of particles, crossed impenetrable and penetrable diffraction gratings and other related structures. In particular, the methods developed in this article can be used to “upgrade” classical approaches, resulting in algorithms that are applicable throughout the spectrum, and it provides new methods for cases where previous approaches are either costly or fail altogether. In particular, it is suggested that the proposed shifted Green function approach may provide the only viable alternative for treatment of three-dimensional high-frequency configurations with either one or two directions of periodicity. A variety of computational examples are presented which demonstrate the flexibility of the overall approach.

Publication: Journal of Computational Physics Vol.: 410ISSN: 0021-9991

ID: CaltechAUTHORS:20200302-143817282

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Abstract: This paper presents novel methodologies for the numerical simulation of scattering of elastic waves by both closed and open surfaces in three-dimensional space. The proposed approach utilizes new integral formulations as well as an extension to the elastic context of the efficient high-order singular-integration methods [13] introduced recently for the acoustic case. In order to obtain formulations leading to iterative solvers (GMRES) which converge in small numbers of iterations we investigate, theoretically and computationally, the character of the spectra of various operators associated with the elastic-wave Calderón relation—including some of their possible compositions and combinations. In particular, by relying on the fact that the eigenvalues of the composite operator NS are bounded away from zero and infinity, new uniquely-solvable, low-GMRES-iteration integral formulation for the closed-surface case are presented. The introduction of corresponding low-GMRES-iteration equations for the open-surface equations additionally requires, for both spectral quality as well as accuracy and efficiency, use of weighted versions of the classical integral operators to match the singularity of the unknown density at edges. Several numerical examples demonstrate the accuracy and efficiency of the proposed methodology.

Publication: Journal of Computational Physics Vol.: 410ISSN: 0021-9991

ID: CaltechAUTHORS:20191216-160134658

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Abstract: This article concerns the asymptotic geometric character of the nodal set of the eigenfunctions of the Steklov eigenvalue problem −Δϕ_(σj) = 0, on Ω, ∂_νϕ_(σj) = σ_jϕ_(σj) on ∂Ω in two-dimensional domains Ω. In particular, this paper presents a dense family A of simply-connected two-dimensional domains with analytic boundaries such that, for each Ω∈A, the nodal set of the eigenfunction ϕ_(σj) “is not dense at scale σ_j⁻¹”. This result addresses a question put forth under “Open Problem 10” in Girouard and Polterovich (J Spectr Theory 7(2):321–359, 2017). In fact, the results in the present paper establish that, for domains Ω∈A, the nodal sets of the eigenfunctions ϕ_(σj) associated with the eigenvalue σ_j have starkly different character than anticipated: they are not dense at any shrinking scale. More precisely, for each Ω∈A there is a value r₁ > 0 such that for each j there is x_j ∈ Ω such that ϕ_(σj) does not vanish on the ball of radius r₁ around x_j.

Publication: Journal of Fourier Analysis and Applications Vol.: 26 No.: 3 ISSN: 1069-5869

ID: CaltechAUTHORS:20190906-093833395

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Abstract: This paper proposes a frequency/time hybrid integral-equation method for the time-dependent wave equation in two- and three-dimensional spatial domains. Relying on Fourier transformation in time, the method utilizes a fixed (time-independent) number of frequency-domain integral-equation solutions to evaluate, with superalgebraically small errors, time-domain solutions for arbitrarily long times. The approach relies on two main elements, namely: (1) a smooth time-windowing methodology that enables accurate band-limited representations for arbitrarily long time signals and (2) a novel Fourier transform approach which, in a time-parallel manner and without causing spurious periodicity effects, delivers numerically dispersionless spectrally accurate solutions. A similar hybrid technique can be obtained on the basis of Laplace transforms instead of Fourier transforms, but we do not consider the Laplace-based method in the present contribution. The algorithm can handle dispersive media, it can tackle complex physical structures, it enables parallelization in time in a straightforward manner, and it allows for time leaping---that is, solution sampling at any given time T at O(1)-bounded sampling cost, for arbitrarily large values of T, and without requirement of evaluation of the solution at intermediate times. The proposed frequency-time hybridization strategy, which generalizes to any linear partial differential equation in the time domain for which frequency-domain solutions can be obtained (including, e.g., the time-domain Maxwell equations) and which is applicable in a wide range of scientific and engineering contexts, provides significant advantages over other available alternatives, such as volumetric discretization, time-domain integral equations, and convolution quadrature approaches.

Publication: SIAM Journal on Scientific Computing Vol.: 42 No.: 2 ISSN: 1064-8275

ID: CaltechAUTHORS:20181102-090208948

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Abstract: In this paper, we present a new and efficient approach for optimizing the transmission signal between two points in a cavity at a given frequency, by changing boundary conditions. The proposed approach makes use of recent results on the monotonicity of the eigenvalues of the mixed boundary value problem and on the sensitivity of Green's function to small changes in the boundary conditions. The switching of the boundary condition from Dirichlet to Neumann can be performed through the use of the recently modeled concept of metasurfaces which are comprised of coupled pairs of Helmholtz resonators. A variety of numerical experiments are presented to show the applicability and the accuracy of the proposed new methodology.

Publication: SIAM Journal on Scientific Computing Vol.: 42 No.: 1 ISSN: 1064-8275

ID: CaltechAUTHORS:20200806-123557817

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Abstract: Integrated photonics is poised to become a billion-dollar industry due to its vast array of applications. However, designing and modeling photonic devices remains challenging due to the lack of analytical solutions and difficulties with numerical simulation. Recently, inverse design has emerged as a promising approach for designing photonic devices; however, the current implementations require major computational effort due to their use of inefficient electromagnetic solvers based on finite-difference methods. Here we report a new, highly efficient method for simulating devices based on boundary integral equations that is orders of magnitude faster and more accurate than existing solvers, almost achieves spectral convergence, and is free from numerical dispersion. We develop an optimization framework using our solver based on the adjoint method to design new, ready-to-fabricate devices in just minutes on a single-core laptop. As a demonstration, we optimize three different devices: a nonadiabatic waveguide taper, a 1:2 1550 nm power splitter, and a vertical-incidence grating coupler.

Publication: ACS Photonics Vol.: 6 No.: 12 ISSN: 2330-4022

ID: CaltechAUTHORS:20191120-075203260

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Abstract: We present a method for the treatment of the time dependent radiative transfer equation under the discrete ordinate approximation. The novelty of the proposed approach stems, in part, from the incorporation of a spectral method for the calculation of the spatial differential operators based on the Fourier Continuation procedure introduced recently by Bruno and co–authors. This is a spatially dispersionless and high order method, which can handle arbitrary geometries, including those encountered in the forward model of light transport in optical tomography. We validate our theoretical results by comparison with analytic and experimental outcomes of the fluence measurements on tissue-like phantoms. The method makes it possible to calculate the time of flight of photons in random media efficiently and with high accuracy.

Publication: Journal of Quantitative Spectroscopy and Radiative Transfer Vol.: 236ISSN: 0022-4073

ID: CaltechAUTHORS:20190729-151529749

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Abstract: This paper presents a new class of solvers for the subsonic compressible Navier-Stokes equations in general two- and three-dimensional multi-domains. Building up on the recent single-domain ADI-based high-order Navier-Stokes solvers (Bruno and Cubillos, Journal of Computational Physics 307 (2016) 476-495) this article presents multi-domain implicit-explicit methods of high-order of temporal accuracy. The proposed methodology incorporates: 1) A novel linear-cost implicit solver based on use of high-order backward differentiation formulae (BDF) and an alternating direction implicit approach (ADI); 2) A fast explicit solver; 3) Nearly dispersionless spectral spatial discretizations; and 4) A domain decomposition strategy that negotiates the interactions between the implicit and explicit domains. In particular, the implicit methodology is quasi-unconditionally stable (it does not suffer from CFL constraints for adequately resolved flows), and it can deliver orders of time accuracy between two and six in the presence of general boundary conditions. As demonstrated via a variety of numerical experiments in two and three dimensions, further, the proposed multi-domain parallel implicit-explicit implementations exhibit high-order convergence in space and time, robust stability properties, limited dispersion, and high parallel efficiency.

Publication: Journal of Computational Physics Vol.: 391ISSN: 0021-9991

ID: CaltechAUTHORS:20190308-103045387

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Abstract: This paper presents a fast high-order method for the solution of two-dimensional problems of scattering by penetrable inhomogeneous media, with application to high-frequency configurations containing (possibly) discontinuous refractivities. The method relies on a combination of a differential volumetric formulation and a boundary integral formulation. Thus, in the proposed approach the entire computational domain is partitioned into large numbers of volumetric spectral approximation patches which are then grouped into patch subsets for local direct solution; the interactions with the exterior domain are handled by means of a boundary integral equation. The resulting algorithm can be quite effective: after a modestly-demanding precomputation stage (whose results for a given frequency can be repeatedly used for arbitrarily chosen incidence angles), the proposed algorithm can accurately evaluate scattering by configurations including large and complex objects and/or high refractivity contrasts, including possibly refractive-index discontinuities, in fast single-core runs.

Publication: arXiv
ID: CaltechAUTHORS:20190906-093836815

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Abstract: We study the scattering of transient, high-frequency, narrow-band quasi-Rayleigh elastic waves by through-thickness holes in aluminum plates, in the framework of ultrasonic nondestructive testing (NDT) based on full-field optical detection. Sequences of the instantaneous two-dimensional (2-D) out-of-plane displacement scattering maps are measured with a self-developed PTVH system. The corresponding simulated sequences are obtained by means of an FC(Gram) elastodynamic solver introduced recently, which implements a full three-dimensional (3D) vector formulation of the direct linear-elasticity scattering problem. A detailed quantitative comparison between these experimental and numerical sequences, which is presented here for the first time, shows very good agreement both in the amplitude and the phase of the acoustic field in the forward, lateral and backscattering areas. It is thus suggested that the combination of the PTVH system and the FC(Gram) elastodynamic solver provides an effective ultrasonic inspection tool for plate-like structures, with a significant potential for ultrasonic NDT applications.

Publication: arXiv
ID: CaltechAUTHORS:20190906-093840280

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Abstract: This paper introduces a fast algorithm, applicable throughout the electromagnetic spectrum, for the numerical solution of problems of scattering by periodic surfaces in two-dimensional space. The proposed algorithm remains highly accurate and efficient for challenging configurations including randomly rough surfaces, deep corrugations, large periods, near grazing incidences, and, importantly, Wood-anomaly resonant frequencies. The proposed approach is based on use of certain “shifted equivalent sources” which enable FFT acceleration of a Wood-anomaly-capable quasi-periodic Green function introduced recently (Bruno and Delourme (2014) [4]). The Green-function strategy additionally incorporates an exponentially convergent shifted version of the classical spectral series for the Green function. While the computing-cost asymptotics depend on the asymptotic configuration assumed, the computing costs rise at most linearly with the size of the problem for a number of important rough-surface cases we consider. In practice, single-core runs in computing times ranging from a fraction of a second to a few seconds suffice for the proposed algorithm to produce highly-accurate solutions in some of the most challenging contexts arising in applications.

Publication: Journal of Computational Physics Vol.: 378ISSN: 0021-9991

ID: CaltechAUTHORS:20181101-141916128

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Abstract: This paper presents regularity results and associated high order numerical methods for one-dimensional fractional-Laplacian boundary-value problems. On the basis of a factorization of solutions as a product of a certain edge-singular weight ω times a "regular" unknown, a characterization of the regularity of solutions is obtained in terms of the smoothness of the corresponding right-hand sides. In particular, for right-hand sides which are analytic in a Bernstein ellipse, analyticity in the same Bernstein ellipse is obtained for the ``regular'' unknown. Moreover, a sharp Sobolev regularity result is presented which completely characterizes the co-domain of the fractional-Laplacian operator in terms of certain weighted Sobolev spaces introduced in (Babuška and Guo, SIAM J. Numer. Anal. 2002). The present theoretical treatment relies on a full eigendecomposition for a certain weighted integral operator in terms of the Gegenbauer polynomial basis. The proposed Gegenbauer-based Nyström numerical method for the fractional-Laplacian Dirichlet problem, further, is significantly more accurate and efficient than other algorithms considered previously. The sharp error estimates presented in this paper indicate that the proposed algorithm is spectrally accurate, with convergence rates that only depend on the smoothness of the right-hand side. In particular, convergence is exponentially fast (resp. faster than any power of the mesh-size) for analytic (resp. infinitely smooth) right-hand sides. The properties of the algorithm are illustrated with a variety of numerical results.

Publication: Mathematics of Computation Vol.: 87 No.: 312 ISSN: 0025-5718

ID: CaltechAUTHORS:20180502-133444210

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Abstract: Accuracy of Kirchhoff approximation (KA) for rough-surface electromagnetic wave scattering is studied by comparison with accurate numerical solutions in the context of three-dimensional dielectric surfaces. The Kirchhoff tangent plane approximation is examined without resorting to the principle of stationary phase. In particular, it is shown that this additional assumption leads to zero cross-polarized backscattered power, but not the tangent plane approximation itself. Extensive numerical results in the case of a bisinusoidal surface are presented for a wide range of problem parameters: height-to-period, wavelength, incidence angles, and dielectric constants. In particular, this paper shows that the range of validity inherent in the KA includes surfaces whose curvature is not only much smaller, but also comparable to the incident wavelength, with errors smaller than 5% in total reflectivity, thus presenting a detailed and reliable source for the validity of the KA in a three-dimensional fully polarimetric formulation.

Publication: Journal of the Optical Society of America A Vol.: 34 No.: 12 ISSN: 1084-7529

ID: CaltechAUTHORS:20180102-154213561

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Abstract: This paper presents a theoretical discussion as well as novel solution algorithms for problems of scattering on smooth two-dimensional domains under Zaremba boundary conditions, for which Dirichlet and Neumann conditions are specified on various portions of the domain boundary. The theoretical basis of the proposed numerical methods, which is provided for the first time in the present contribution, concerns detailed information about the singularity structure of solutions of the Helmholtz operator under boundary conditions of Zaremba type. The new numerical method is based on the use of Green functions and integral equations, and it relies on the Fourier continuation method for regularization of all smooth-domain Zaremba singularities as well as newly derived quadrature rules which give rise to high-order convergence, even around Zaremba singular points. As demonstrated in this paper, the resulting algorithms enjoy high-order convergence, and they can be used to efficiently solve challenging Helmholtz boundary value problems and Laplace eigenvalue problems with high-order accuracy.

Publication: Journal of Integral Equations and Applications Vol.: 29 No.: 4 ISSN: 0897-3962

ID: CaltechAUTHORS:20171214-155422476

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Abstract: This work, part II in a series, presents an efficient method for evaluation of wave scattering by doubly periodic diffraction gratings at or near what are commonly called ‘Wood anomaly frequencies’. At these frequencies, there is a grazing Rayleigh wave, and the quasi-periodic Green function ceases to exist. We present a modification of the Green function by adding two types of terms to its lattice sum. The first type are transversely shifted Green functions with coefficients that annihilate the growth in the original lattice sum and yield algebraic convergence. The second type are quasi-periodic plane wave solutions of the Helmholtz equation which reinstate certain necessary grazing modes without leading to blow-up at Wood anomalies. Using the new quasi-periodic Green function, we establish, for the first time, that the Dirichlet problem of scattering by a smooth doubly periodic scattering surface at a Wood frequency is uniquely solvable. We also present an efficient high-order numerical method based on this new Green function for scattering by doubly periodic surfaces at and around Wood frequencies. We believe this is the first solver able to handle Wood frequencies for doubly periodic scattering problems in three dimensions. We demonstrate the method by applying it to acoustic scattering.

Publication: Proceedings of the Royal Society A: Mathematical, physical, and engineering sciences Vol.: 473 No.: 2207 ISSN: 1364-5021

ID: CaltechAUTHORS:20171214-152953856

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Abstract: This contribution presents a novel Windowed Green Function (WGF) method for the solution of problems of wave propagation, scattering, and radiation for structures that include open (dielectric) waveguides, waveguide junctions, as well as launching and/or termination sites and other nonuniformities. Based on the use of a “slow-rise” smooth-windowing technique in conjunction with free-space Green functions and associated integral representations, the proposed approach produces numerical solutions with errors that decrease faster than any negative power of the window size. The proposed methodology bypasses some of the most significant challenges associated with waveguide simulation. In particular, the WGF approach handles spatially infinite dielectric waveguide structures without recourse to absorbing boundary conditions, it facilitates proper treatment of complex geometries, and it seamlessly incorporates the open-waveguide character and associated radiation conditions inherent in the problem under consideration. The overall WGF approach is demonstrated in this paper by means of a variety of numerical results for 2-D open-waveguide termination, launching and junction problems.

Publication: IEEE Transactions on Antennas and Propagation Vol.: 65 No.: 9 ISSN: 0018-926X

ID: CaltechAUTHORS:20170719-171226860

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Abstract: This paper presents a new methodology for the solution of problems of two- and three-dimensional acoustic scattering (and, in particular, two-dimensional electromagnetic scattering) by obstacles and defects in the presence of an arbitrary number of penetrable layers. Relying on the use of certain slow-rise windowing functions, the proposed windowed Green function approach efficiently evaluates oscillatory integrals over unbounded domains, with high accuracy, without recourse to the highly expensive Sommerfeld integrals that have typically been used to account for the effect of underlying planar multilayer structures. The proposed methodology, whose theoretical basis was presented in the recent contribution (Bruno et al. 2016 SIAM J. Appl. Math. 76, 1871–1898. (doi:10.1137/15M1033782)), is fast, accurate, flexible and easy to implement. Our numerical experiments demonstrate that the numerical errors resulting from the proposed approach decrease faster than any negative power of the window size. In a number of examples considered in this paper, the proposed method is up to thousands of times faster, for a given accuracy, than corresponding methods based on the use of Sommerfeld integrals.

Publication: Proceedings of the Royal Society A: Mathematical, physical, and engineering sciences Vol.: 473 No.: 2202 ISSN: 1364-5021

ID: CaltechAUTHORS:20170717-085623869

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Abstract: The companion paper “Higher-order in time quasi-unconditionally stable ADI solvers for the compressible Navier-Stokes equations in 2D and 3D curvilinear domains,” which is referred to as Part I in what follows, introduces ADI (alternating direction implicit) solvers of higher orders of temporal accuracy (orders s = 2 to 6) for the compressible Navier-Stokes equations in two- and three-dimensional space. The proposed methodology employs the backward differentiation formulae (BDF) together with a quasilinear-like formulation, high-order extrapolation for nonlinear components, and the Douglas-Gunn splitting. A variety of numerical results presented in Part I demonstrate in practice the theoretical convergence rates enjoyed by these algorithms, as well as their excellent accuracy and stability properties for a wide range of Reynolds numbers. In particular, the proposed schemes enjoy a certain property of “quasi-unconditional stability”: for small enough (problem-dependent) fixed values of the timestep Δt, these algorithms are stable for arbitrarily fine spatial discretizations. The present contribution presents a mathematical basis for the observed performance of these algorithms. Short of providing stability theorems for the full Navier-Stokes BDF-ADI solvers, this paper puts forth a number of stability proofs for BDF-ADI schemes as well as some related unsplit BDF schemes for a variety of related linear model problems in one, two, and three spatial dimensions. These include proofs of quasi-unconditional stability for unsplit BDF schemes of orders 2 ≤ s ≤ 6, and even a proof of a form of unconditional stability for two-dimensional BDF-ADI schemes of order 2 for both convection and diffusion problems. Additionally, a set of numerical tests presented in this paper for the compressible Navier-Stokes equation indicate that quasi-unconditional stability carries over to the fully nonlinear context.

Publication: SIAM Journal on Numerical Analysis Vol.: 55 No.: 2 ISSN: 0036-1429

ID: CaltechAUTHORS:20170605-100151973

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Abstract: This paper presents a full-spectrum Green-function methodology (which is valid, in particular, at and around Wood-anomaly frequencies) for evaluation of scattering by periodic arrays of cylinders of arbitrary cross section—with application to wire gratings, particle arrays and reflectarrays and, indeed, general arrays of conducting or dielectric bounded obstacles under both transverse electric and transverse magnetic polarized illumination. The proposed method, which, for definiteness, is demonstrated here for arrays of perfectly conducting particles under transverse electric polarization, is based on the use of the shifted Green-function method introduced in a recent contribution (Bruno & Delourme 2014 J. Computat. Phys. 262, 262–290 (doi:10.1016/j.jcp.2013.12.047)). A certain infinite term arises at Wood anomalies for the cylinder-array problems considered here that is not present in the previous rough-surface case. As shown in this paper, these infinite terms can be treated via an application of ideas related to the Woodbury–Sherman–Morrison formulae. The resulting approach, which is applicable to general arrays of obstacles even at and around Wood-anomaly frequencies, exhibits fast convergence and high accuracies. For example, a few hundreds of milliseconds suffice for the proposed approach to evaluate solutions throughout the resonance region (wavelengths comparable to the period and cylinder sizes) with full single-precision accuracy.

Publication: Proceedings of the Royal Society A: Mathematical, physical, and engineering sciences Vol.: 473 No.: 2199 ISSN: 1364-5021

ID: CaltechAUTHORS:20170424-093441215

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Abstract: This paper introduces a new windowed Green function (WGF) method for the numerical integral-equation solution of problems of electromagnetic scattering by obstacles in the presence of dielectric or conducting half-planes. The WGF method, which is based on the use of smooth windowing functions and integral kernels that can be expressed directly in terms of the free-space Green function, does not require evaluation of expensive Sommerfeld integrals. The proposed approach is fast, accurate, flexible, and easy to implement. In particular, straightforward modifications of existing (accelerated or unaccelerated) integral-equation solvers suffice to incorporate the WGF capability. The method relies on a certain integral equation posed on the union of the boundary of the obstacle and a small flat section of the interface between the penetrable media. Our analysis and numerical experiments demonstrate that both the near- and far-field errors resulting from the proposed approach decrease faster than any negative power of the window size. In the examples considered in this paper the proposed method is up to thousands of times faster, for a given accuracy, than a corresponding method based on use of Sommerfeld integrals.

Publication: SIAM Journal on Applied Mathematics Vol.: 76 No.: 5 ISSN: 0036-1399

ID: CaltechAUTHORS:20151116-094725791

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Abstract: This work, part I in a two-part series, presents: (i) a simple and highly efficient algorithm for evaluation of quasi-periodic Green functions, as well as (ii) an associated boundary-integral equation method for the numerical solution of problems of scattering of waves by doubly periodic arrays of scatterers in three-dimensional space. Except for certain ‘Wood frequencies’ at which the quasi-periodic Green function ceases to exist, the proposed approach, which is based on smooth windowing functions, gives rise to tapered lattice sums which converge superalgebraically fast to the Green function—that is, faster than any power of the number of terms used. This is in sharp contrast to the extremely slow convergence exhibited by the lattice sums in the absence of smooth windowing. (The Wood-frequency problem is treated in part II.) This paper establishes rigorously the superalgebraic convergence of the windowed lattice sums. A variety of numerical results demonstrate the practical efficiency of the proposed approach.

Publication: Proceedings of the Royal Society A: Mathematical, physical, and engineering sciences Vol.: 472 No.: 2191 ISSN: 1364-5021

ID: CaltechAUTHORS:20161014-131501720

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Abstract: This paper presents a spectral numerical algorithm for the solution of elastodynamics problems in general three-dimensional domains. Based on a recently introduced “Fourier continuation” (FC) methodology for accurate Fourier expansion of non-periodic functions, the proposed approach possesses a number of appealing properties: it yields results that are essentially free of dispersion errors, it entails mild CFL constraints, it runs at a cost that scales linearly with the discretization sizes, and it lends itself easily to efficient parallelization in distributed-memory computing clusters. The proposed algorithm is demonstrated in this paper by means of a number of applications to problems of isotropic elastodynamics that arise in the fields of materials science and seismology. These examples suggest that the new approach can yield solutions within a prescribed error tolerance by means of significantly smaller discretizations and shorter computing times than those required by other methods.

Publication: Journal of Computational Physics Vol.: 307ISSN: 0021-9991

ID: CaltechAUTHORS:20181101-152904690

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Abstract: This paper introduces alternating-direction implicit (ADI) solvers of higher order of time-accuracy (orders two to six) for the compressible Navier–Stokes equations in two- and three-dimensional curvilinear domains. The higher-order accuracy in time results from 1) An application of the backward differentiation formulae time-stepping algorithm (BDF) in conjunction with 2) A BDF-like extrapolation technique for certain components of the nonlinear terms (which makes use of nonlinear solves unnecessary), as well as 3) A novel application of the Douglas–Gunn splitting (which greatly facilitates handling of boundary conditions while preserving higher-order accuracy in time). As suggested by our theoretical analysis of the algorithms for a variety of special cases, an extensive set of numerical experiments clearly indicate that all of the BDF-based ADI algorithms proposed in this paper are “quasi-unconditionally stable” in the following sense: each algorithm is stable for all couples (h,Δt)of spatial and temporal mesh sizes in a problem-dependent rectangular neighborhood of the form (0,M_h)×(0,M_t). In other words, for each fixed value of Δt below a certain threshold, the Navier–Stokes solvers presented in this paper are stable for arbitrarily small spatial mesh-sizes. The second-order formulation has further been rigorously shown to be unconditionally stable for linear hyperbolic and parabolic equations in two-dimensional space. Although implicit ADI solvers for the Navier–Stokes equations with nominal second-order of temporal accuracy have been proposed in the past, the algorithms presented in this paper are the first ADI-based Navier–Stokes solvers for which second-order or better accuracy has been verified in practice under non-trivial (non-periodic) boundary conditions.

Publication: Journal of Computational Physics Vol.: 307ISSN: 0021-9991

ID: CaltechAUTHORS:20160218-124347606

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Abstract: We present a novel integral-equation algorithm for evaluation of Zaremba eigenvalues and eigenfunctions, that is, eigenvalues and eigenfunctions of the Laplace operator with mixed Dirichlet–Neumann boundary conditions; of course, (slight modifications of) our algorithms are also applicable to the pure Dirichlet and Neumann eigenproblems. Expressing the eigenfunctions by means of an ansatz based on the single layer boundary operator, the Zaremba eigenproblem is transformed into a nonlinear equation for the eigenvalue μ. For smooth domains the singular structure at Dirichlet–Neumann junctions is incorporated as part of our corresponding numerical algorithm—which otherwise relies on use of the cosine change of variables, trigonometric polynomials and, to avoid the Gibbs phenomenon that would arise from the solution singularities, the Fourier Continuation method (FC). The resulting numerical algorithm converges with high order accuracy without recourse to use of meshes finer than those resulting from the cosine transformation. For non-smooth (Lipschitz) domains, in turn, an alternative algorithm is presented which achieves high-order accuracy on the basis of graded meshes. In either case, smooth or Lipschitz boundary, eigenvalues are evaluated by searching for zero minimal singular values of a suitably stabilized discrete version of the single layer operator mentioned above. (The stabilization technique is used to enable robust non-local zero searches.) The resulting methods, which are fast and highly accurate for high- and low-frequencies alike, can solve extremely challenging two-dimensional Dirichlet, Neumann and Zaremba eigenproblems with high accuracies in short computing times—enabling, in particular, evaluation of thousands of eigenvalues and corresponding eigenfunctions for a given smooth or non-smooth geometry with nearly full double-precision accuracy.

Publication: Journal of Computational Physics Vol.: 298ISSN: 0021-9991

ID: CaltechAUTHORS:20150821-120443842

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Abstract: This paper presents a class of boundary integral equations for the solution of problems of electromagnetic and acoustic scattering by two-dimensional homogeneous penetrable scatterers with smooth boundaries. The new integral equations, which, as is established in this paper, are uniquely solvable Fredholm equations of the second kind, result from representations of fields as combinations of single and double layer potentials acting on appropriately chosen regularizing operators. As demonstrated in this text by means of a variety of numerical examples (that resulted from a high-order Nyström computational implementation of the new equations), these “regularized combined equations” can give rise to important reductions in computational costs, for a given accuracy, over those resulting from previous iterative boundary integral equation solvers for transmission problems.

Publication: Applied Numerical Mathematics Vol.: 95ISSN: 0168-9274

ID: CaltechAUTHORS:20150714-133922935

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Abstract: We deal with the general problem of scattering by open arcs in two-dimensional space. We show that this problem can be solved by means of certain second-kind integral equations of the form ÑS[φ] = ƒ, where Ñ and S are first-kind integral operators whose composition gives rise to a generalized Calderón formula of the form ÑS = J^τ_0 + K in a weighted, periodized Sobolev space. (Here J^τ_0 is a continuous and continuously invertible operator and K is a compact operator.) The ÑS formulation provides, for the first time, a second-kind integral equation for the open-arc scattering problem with Neumann boundary conditions. Numerical experiments show that, for both the Dirichlet and Neumann boundary conditions, our second-kind integral equations have spectra that are bounded away from zero and infinity as k → ∞; to the authors’ knowledge these are the first integral equations for these problems that possess this desirable property. This situation is in stark contrast with that arising from the related classical open-surface hypersingular and single-layer operators N and S, whose composition NS maps, for example, the function ϕ = 1 into a function that is not even square integrable. Our proofs rely on three main elements: algebraic manipulations enabled by the presence of integral weights; use of the classical result of continuity of the Cesàro operator; and explicit characterization of the point spectrum of J^τ_0, which, interestingly, can be decomposed into the union of a countable set and an open set, both of which are tightly clustered around -1/4. As shown in a separate contribution, the new approach can be used to construct simple, spectrally accurate numerical solvers and, when used in conjunction with Krylov-subspace iterative solvers such as the generalized minimal residual method, it gives rise to a dramatic reduction in the number of iterations compared with those required by other approaches.

Publication: Proceedings of the Royal Society of Edinburgh: Section A Mathematics Vol.: 145 No.: 2 ISSN: 0308-2105

ID: CaltechAUTHORS:20150508-105226858

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Abstract: We present a new computational method for the solution of elliptic eigenvalue problems with variable coefficients in general two-dimensional domains. The proposed approach is based on use of the novel Fourier continuation method (which enables fast and highly accurate Fourier approximation of nonperiodic functions in equispaced grids without the limitations arising from the Gibbs phenomenon) in conjunction with an overlapping patch domain decomposition strategy and Arnoldi iteration. A variety of examples demonstrate the versatility, accuracy, and generality of the proposed methodology.

Publication: Mathematical Problems in Engineering Vol.: 2015ISSN: 1024-123X

ID: CaltechAUTHORS:20160531-155956642

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Abstract: This paper presents a study of the absorption of electromagnetic power that results from the interaction of electromagnetic waves and cylindrical bumps or trenches on flat conducting surfaces. Configurations are characterized by means of adequately selected dimensionless variables and parameters so that applicability to mathematically equivalent (but physically diverse) systems can be achieved easily. Electromagnetic fields and absorption increments caused by such surface defects are evaluated by means of a high-order integral equation method which resolves fine details of the field near the surface, and which was validated by fully analytical approaches in a range of computationally challenging cases. The computational method is also applied to problems concerning bumps and trenches on imperfect conducting planes for which analytical solutions are not available. Typically, we find that absorption is enhanced by the presence of the defects considered, although, interestingly, absorption can also be significantly reduced in some cases—such as, e.g., in the case of a trench on a conducting plane where the incident electric field is perpendicular to the plane. Additionally, it is observed that, for some small-skin-depths large-wavelengths, the absorption increment is proportional to the increase in surface area. Significant physical insight is obtained on the heating that results from various types of electromagnetic incident fields.

Publication: Journal of Applied Physics Vol.: 116 No.: 12 ISSN: 0021-8979

ID: CaltechAUTHORS:20141106-103044717

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Abstract: This paper presents high-order integral equation methods for the evaluation of electromagnetic wave scattering by dielectric bumps and dielectric cavities on perfectly conducting or dielectric half-planes. In detail, the algorithms introduced in this paper apply to eight classical scattering problems, namely, scattering by a dielectric bump on a perfectly conducting or a dielectric half-plane, and scattering by a filled, overfilled, or void dielectric cavity on a perfectly conducting or a dielectric half-plane. In all cases field representations based on single-layer potentials for appropriately chosen Green functions are used. The numerical far fields and near fields exhibit excellent convergence as discretizations are refined—even at and around points where singular fields and infinite currents exist.

Publication: Journal of the Optical Society of America A Vol.: 31 No.: 8 ISSN: 1084-7529

ID: CaltechAUTHORS:20141009-095915180

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Abstract: We introduce a class of alternating direction implicit (ADI) methods, based on approximate factorizations of backward differentiation formulas (BDFs) of order p≥2, for the numerical solution of two-dimensional, time-dependent, nonlinear, convection-diffusion partial differential equation (PDE) systems in Cartesian domains. The proposed algorithms, which do not require the solution of nonlinear systems, additionally produce solutions of spectral accuracy in space through the use of Chebyshev approximations. In particular, these methods give rise to minimal artificial dispersion and diffusion and they therefore enable use of relatively coarse discretizations to meet a prescribed error tolerance for a given problem. A variety of numerical results presented in this text demonstrate high-order accuracy and, for the particular cases of p=2,3, unconditional stability.

Publication: Journal of Fluids Engineering Vol.: 136 No.: 6 ISSN: 0098-2202

ID: CaltechAUTHORS:20140612-100139439

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Abstract: We introduce a new methodology, based on new quasi-periodic Green functions which converge rapidly even at and around Wood-anomaly configurations, for the numerical solution of problems of scattering by periodic rough surfaces in two-dimensional space. As is well known the classical quasi-periodic Green function ceases to exist at Wood anomalies. The approach introduced in this text produces fast Green function convergence throughout the spectrum on the basis of a certain “finite-differencing” approach and smooth windowing of the classical Green function lattice sum. The resulting Green-function convergence is super-algebraically fast away from Wood anomalies, and it reduces to an arbitrarily-high (user-prescribed) algebraic order of convergence at Wood anomalies.

Publication: Journal of Computational Physics Vol.: 262ISSN: 0021-9991

ID: CaltechAUTHORS:20140313-090333919

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Abstract: We present fast, spatially dispersionless and unconditionally stable high-order solvers for partial differential equations (PDEs) with variable coefficients in general smooth domains. Our solvers, which are based on (i) A certain “Fourier continuation” (FC) method for the resolution of the Gibbs phenomenon on equi-spaced Cartesian grids, together with (ii) A new, preconditioned, FC-based solver for two-point boundary value problems for variable-coefficient Ordinary Differential Equations, and (iii) An Alternating Direction strategy, generalize significantly a class of FC-based solvers introduced recently for constant-coefficient PDEs. The present algorithms, which are applicable, with high-order accuracy, to variable-coefficient elliptic, parabolic and hyperbolic PDEs in general domains with smooth boundaries, are unconditionally stable, do not suffer from spatial numerical dispersion, and they run at Fast Fourier Transform speeds. The accuracy, efficiency and overall capabilities of our methods are demonstrated by means of applications to challenging problems of diffusion and wave propagation in heterogeneous media.

Publication: Journal of Scientific Computing Vol.: 58 No.: 2 ISSN: 0885-7474

ID: CaltechAUTHORS:20140218-114756940

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Abstract: We present a novel methodology for the numerical solution of problems of diffraction by infinitely thin screens in three-dimensional space. Our approach relies on new integral formulations as well as associated high-order quadrature rules. The new integral formulations involve weighted versions of the classical integral operators related to the thin-screen Dirichlet and Neumann problems as well as a generalization to the open-surface problem of the classical Calderón formulae. The high-order quadrature rules we introduce for these operators, in turn, resolve the multiple Green function and edge singularities (which occur at arbitrarily close distances from each other, and which include weakly singular as well as hypersingular kernels) and thus give rise to super-algebraically fast convergence as the discretization sizes are increased. When used in conjunction with Krylov-subspace linear algebra solvers such as GMRES, the resulting solvers produce results of high accuracy in small numbers of iterations for low and high frequencies alike. We demonstrate our methodology with a variety of numerical results for screen and aperture problems at high frequencies—including simulation of classical experiments such as the diffraction by a circular disc (featuring in particular the famous Poisson spot), evaluation of interference fringes resulting from diffraction across two nearby circular apertures, as well as solution of problems of scattering by more complex geometries consisting of multiple scatterers and cavities.

Publication: Journal of Computational Physics Vol.: 252ISSN: 0021-9991

ID: CaltechAUTHORS:20130829-141339834

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Abstract: In this paper we present a convergence analysis for the Nyström method proposed in [J Comput Phys 169 (1):80–110, 2001] for the solution of the combined boundary integral equation formulations of sound-soft acoustic scattering problems in three-dimensional space. This fast and efficient scheme combines FFT techniques and a polar change of variables that cancels out the kernel singularity. We establish the stability of the algorithms in the L^2 norm and we derive convergence estimates in both the L^2 and L^∞ norms. In particular, our analysis establishes theoretically the previously observed super-algebraic convergence of the method in cases in which the right-hand side is smooth.

Publication: Numerische Mathematik Vol.: 124 No.: 4 ISSN: 0029-599X

ID: CaltechAUTHORS:20130816-112500114

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Abstract: We present effcient methods for computing wave scattering by diffraction gratings that exhibit two-dimensional periodicity in three dimensional (3D) space. Applications include scattering in acoustics, electromagnetics and elasticity. Our approach uses boundary-integral equations. The quasi-periodic Green function employed is structured as a doubly infinite sum of scaled 3D free-space outgoing Helmholtz Green functions. Their source points are located at the nodes of a periodicity lattice of the grating; the scaling is effected by Bloch quasi-periodic coefficients. For efficient numerical computation of the lattice sum, we employ a smooth truncation. Super-algebraic convergence to the Green function is achieved as the truncation radius increases, except at frequency-wavenumber pairs at which a Rayleigh wave is at exactly grazing incidence to the grating. At these "Wood frequencies", the term in the Fourier series representation of the Green function that corresponds to the grazing Rayleigh wave acquires an infinite coefficient and the lattice sum blows up. A related challenge occurs at non-exact grazing incidence of a Rayleigh wave; in this case, the constants in the truncation-error bound become too large. At Wood frequencies, we modify the Green function by adding two types of terms to it. The first type adds weighted spatial shifts of the Green function to itself. The shifts are such that the spatial singularities introduced by these terms are located below the grating and therefore out of the domain of interest. With suitable choices of the weights, these terms annihilate the growing contributions in the original lattice sum and yield algebraic convergence. The degree of the algebraic convergence depends on the number of the added shifts. The second-type terms are quasi-periodic plane wave solutions of the Helmholtz equation. They reinstate (with controlled coeficients now) the grazing modes, effectively eliminated by the terms of first type. These modes are needed in the Green function for guaranteeing the well-posedness of the boundaryintegral equation that yields the scattered field. We apply this approach to acoustic scattering by a doubly periodic 2D grating near and at Wood frequencies and scattering by a doubly periodic array of scatterers away from Wood frequencies.

ID: CaltechAUTHORS:20160219-073954516

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Abstract: We present a novel approach for the numerical solution of problems of diffraction by open arcs in two dimensional space. Our methodology relies on composition of weighted versions of the classical integral operators associated with the Dirichlet and Neumann problems (TE and TM polarizations, respectively) together with a generalization to the open-arc case of the well known closed-surface Calderón formulae. When used in conjunction with spectrally accurate discretization rules and Krylov-subspace linear algebra solvers such as GMRES, the new second-kind TE and TM formulations for open arcs produce results of high accuracy in small numbers of iterations—for low and high frequencies alike.

Publication: Radio Science Vol.: 47 No.: 6 ISSN: 0048-6604

ID: CaltechAUTHORS:20130115-143636940

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Abstract: On the basis of recently developed Fourier continuation (FC) methods and associated efficient parallelization techniques, this text introduces numerical algorithms that, due to very low dispersive errors, can accurately and efficiently solve the types of nonlinear partial differential equation (PDE) models of nonlinear acoustics in hundred-wavelength domains as arise in the simulation of focused medical ultrasound. As demonstrated in the examples presented in this text, the FC approach can be used to produce solutions to nonlinear acoustics PDEs models with significantly reduced discretization requirements over those associated with finite-difference, finite-element and finite-volume methods, especially in cases involving waves that travel distances that are orders of magnitude longer than their respective wavelengths. In these examples, the FC methodology is shown to lead to improvements in computing times by factors of hundreds and even thousands over those required by the standard approaches. A variety of one-and two-dimensional examples presented in this text demonstrate the power and capabilities of the proposed methodology, including an example containing a number of scattering centers and nonlinear multiple-scattering events.

Publication: Journal of the Acoustical Society of America Vol.: 132 No.: 4 ISSN: 0001-4966

ID: CaltechAUTHORS:20130103-105523405

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Abstract: This text introduces the following: (1) new regularized combined field integral equations (CFIE-R) for frequency-domain sound-hard scattering problems; and (2) fast, high-order algorithms for the numerical solution of the CFIE-R and related integral equations. Similar to the classical combined field integral equation (CFIE), the CFIE-R are uniquely-solvable integral equations based on the use of single and double layer potentials. Unlike the CFIE, however, the CFIE-R utilize a composition of the double-layer potential with a regularizing operator that gives rise to highly favorable spectral properties—thus making it possible to produce accurate solutions by means of iterative solvers in small numbers of iterations. The CFIE-R-based fast high-order integral algorithms introduced in this text enable highly accurate solution of challenging sound-hard scattering problems, including hundred-wavelength cases, in single-processor runs on present-day desktop computers. A variety of numerical results demonstrate the qualities of the numerical solvers as well as the advantages that arise from the new integral equation formulation.

Publication: International Journal for Numerical Methods in Engineering Vol.: 91 No.: 10 ISSN: 0029-5981

ID: CaltechAUTHORS:20120927-133224529

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Abstract: We consider the problem of numerical differentiation of a function f from approximate or noisy values of f on a discrete set of points; such discrete approximate data may result from a numerical calculation (such as a finite element or finite difference solution of a partial differential equation), from experimental measurements, or, generally, from an estimate of some sort. In some such cases it is useful to guarantee that orders of accuracy are not degraded: assuming the approximating values of the function are known with an accuracy of order O(h^r), where h is the mesh size, an accuracy of O(h^r) is desired in the value of the derivatives of f. Differentiation of interpolating polynomials does not achieve this goal since, as shown in this text, n-fold differentiation of an interpolating polynomial of any degree ≥ (r − 1) obtained from function values containing errors of order O(h^r) generally gives rise to derivative errors of order O(h^(r−n)); other existing differentiation algorithms suffer from similar degradations in the order of accuracy. In this paper we present a new algorithm, the LDC method (low degree Chebyshev), which, using noisy function values of a function f on a (possibly irregular) grid, produces approximate values of derivatives f^((n)) (n = 1, 2 . . .) with limited loss in the order of accuracy. For example, for (possibly nonsmooth) O(h^r) errors in the values of an underlying infinitely differentiable function, the LDC loss in the order of accuracy is “vanishingly small”: derivatives of smooth functions are approximated by the LDC algorithm with an accuracy of order O(h^r) for all r' < r. The algorithm is very fast and simple; a variety of numerical results we present illustrate the theory and demonstrate the efficiency of the proposed methodology.

Publication: SIAM Journal on Numerical Analysis Vol.: 50 No.: 3 ISSN: 0036-1429

ID: CaltechAUTHORS:20121204-113815559

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Abstract: We introduce a multi-domain Fourier-continuation/WENO hybrid method (FC–WENO) that enables high-order and non-oscillatory solution of systems of nonlinear conservation laws, and which enjoys essentially dispersionless, spectral character away from discontinuities, as well as mild CFL constraints (comparable to those of finite difference methods). The hybrid scheme employs the expensive, shock-capturing WENO method in small regions containing discontinuities and the efficient FC method in the rest of the computational domain, yielding a highly effective overall scheme for applications with a mix of discontinuities and complex smooth structures. The smooth and discontinuous solution regions are distinguished using the multi-resolution procedure of Harten [J. Comput. Phys. 115 (1994) 319–338]. We consider WENO schemes of formal orders five and nine and a FC method of order five. The accuracy, stability and efficiency of the new hybrid method for conservation laws is investigated for problems with both smooth and non-smooth solutions. In the latter case, we solve the Euler equations for gas dynamics for the standard test case of a Mach three shock wave interacting with an entropy wave, as well as a shock wave (with Mach 1.25, three or six) interacting with a very small entropy wave and evaluate the efficiency of the hybrid FC–WENO method as compared to a purely WENO-based approach as well as alternative hybrid based techniques. We demonstrate considerable computational advantages of the new FC-based method, suggesting a potential of an order of magnitude acceleration over alternatives when extended to fully three-dimensional problems.

Publication: Journal of Computational Physics Vol.: 230 No.: 24 ISSN: 0021-9991

ID: CaltechAUTHORS:20111212-085019337

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Abstract: We present a Fourier continuation (FC) algorithm for the solution of the fully nonlinear compressible Navier–Stokes equations in general spatial domains. The new scheme is based on the recently introduced accelerated FC method, which enables use of highly accurate Fourier expansions as the main building block of general-domain PDE solvers. Previous FC-based PDE solvers are restricted to linear scalar equations with constant coefficients. The FC methodology presented in this text thus constitutes a significant generalization of the previous FC schemes, as it yields general-domain FC solvers for nonlinear systems of PDEs. While not restricted to periodic boundary conditions and therefore applicable to general boundary value problems on arbitrary domains, the proposed algorithm inherits many of the highly desirable properties arising from rapidly convergent Fourier expansions, including high-order convergence, essentially spectrally accurate dispersion relations, and much milder CFL constraints than those imposed by polynomial-based spectral methods—since, for example, the spectral radius of the FC first derivative grows linearly with the number of spatial discretization points. We demonstrate the accuracy and optimal parallel efficiency of the algorithm in a variety of scientific and engineering contexts relevant to fluid-dynamics and nonlinear acoustics.

Publication: Journal of Computational Physics Vol.: 230 No.: 16 ISSN: 0021-9991

ID: CaltechAUTHORS:20110725-065924210

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Abstract: In magnetic drug delivery, therapeutic magnetizable particles are typically injected into the blood stream and magnets are then used to concentrate them to disease locations. The behavior of such particles in-vivo is complex and is governed by blood convection, diffusion (in blood and in tissue), extravasation, and the applied magnetic fields. Using physical first-principles and a sophisticated vessel-membrane-tissue (VMT) numerical solver, we comprehensively analyze in detail the behavior of magnetic particles in blood vessels and surrounding tissue. For any blood vessel (of any size, depth, and blood velocity) and tissue properties, particle size and applied magnetic fields, we consider a Krogh tissue cylinder geometry and solve for the resulting spatial distribution of particles. We find that there are three prototypical behaviors (blood velocity dominated, magnetic force dominated, and boundary-layer formation) and that the type of behavior observed is uniquely determined by three non-dimensional numbers (the magnetic-Richardson number, mass Péclet number, and Renkin reduced diffusion coefficient). Plots and equations are provided to easily read out which behavior is found under which circumstances ([Fig. 5], [Fig. 6], [Fig. 7] and [Fig. 8]). We compare our results to previously published in-vitro and in-vivo magnetic drug delivery experiments. Not only do we find excellent agreement between our predictions and prior experimental observations, but we are also able to qualitatively and quantitatively explain behavior that was previously not understood.

Publication: Journal of Magnetism and Magnetic Materials Vol.: 323 No.: 6 ISSN: 0304-8853

ID: CaltechAUTHORS:20110207-101558443

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Abstract: We introduce a highly accurate and efficient solver for problems of scattering by vector-parametric, possibly non-smooth one-dimensional periodic surfaces. For grating-diffraction problems in the resonance regime (heights and periods of the order of the wavelength of radiation) the proposed algorithm produces solutions with full double-precision accuracy in single-processor computing times of the order of a few seconds. Our algorithm can also produce, in reasonable computing times, solutions for highly challenging TE and TM scattering problems - defined by very deep multi-valued scattering surfaces and high frequencies of radiation - including cases in which non-smooth bounding surfaces enclose open cavities.

Publication: Waves in Random and Complex Media Vol.: 20 No.: 4 ISSN: 1745-5030

ID: CaltechAUTHORS:20101214-135627503

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Abstract: Magnetic drug delivery refers to the physical confinement of therapeutic magnetic nanoparticles to regions of disease, tumors, infections and blood clots. Predicting the effectiveness of magnetic focusing in vivo is critical for the design and use of magnetic drug delivery systems. However, current simple back-of-the-envelope estimates have proven insufficient for this task. In this article, we present an analysis of nanoparticle distribution, in and around a single blood vessel (a Krogh tissue cylinder), located at any depth in the body, with any physiologically relevant blood flow velocity, diffusion and extravasation properties, and with any applied magnetic force on the particles. For any such blood vessel our analysis predicts one of three distinct types of particle behavior (velocity dominated, magnetic dominated or boundary-layer formation), which can be uniquely determined by looking up the values of three nondimensional numbers we define. We compare our predictions to previously published magnetic-focusing in vitro and in vivo studies. Not only do we find agreement between our predictions and prior observations, but we are also able to quantitatively explain behavior that was not understood previously.

Publication: Nanomedicine Vol.: 5 No.: 9 ISSN: 1743-5889

ID: CaltechAUTHORS:20110214-092710313

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Abstract: We present a novel numerical modeling of ultrasonic Lamb and Rayleigh wave propagation and scattering by through-thickness defects like holes and slots in homogeneous plates, and its experimental verification in both near and far field by a self-developed pulsed TV holography system. In contrast to rigorous vectorial formulation of elasticity theory, our model is based on the 2-D scalar wave equation over the plate surface, with specific boundary conditions in the defects and plate edges. The experimental data include complex amplitude maps of the out-of-plane displacements of the plate surface, obtained by a two-step spatiotemporal Fourier transform method. We find a fair match between the numerical and experimental results, which allows for quantitative characterization of the defects.

Publication: Optical Engineering Vol.: 49 No.: 9 ISSN: 0091-3286

ID: CaltechAUTHORS:20101025-104450688

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Abstract: A new PDE solver was introduced recently, in Part I of this two-paper sequence, on the basis of two main concepts: the well-known Alternating Direction Implicit (ADI) approach, on one hand, and a certain “Fourier Continuation” (FC) method for the resolution of the Gibbs phenomenon, on the other. Unlike previous alternating direction methods of order higher than one, which only deliver unconditional stability for rectangular domains, the new high-order FC-AD (Fourier-Continuation Alternating-Direction) algorithm yields unconditional stability for general domains—at an O(N log(N)) cost per time-step for an N point spatial discretization grid. In the present contribution we provide an overall theoretical discussion of the FC-AD approach and we extend the FC-AD methodology to linear hyperbolic PDEs. In particular, we study the convergence properties of the newly introduced FC(Gram) Fourier Continuation method for both approximation of general functions and solution of the alternating-direction ODEs. We also present (for parabolic PDEs on general domains, and, thus, for our associated elliptic solvers) a stability criterion which, when satisfied, ensures unconditional stability of the FC-AD algorithm. Use of this criterion in conjunction with numerical evaluation of a series of singular values (of the alternating-direction discrete one-dimensional operators) suggests clearly that the fifth-order accurate class of parabolic and elliptic FC-AD solvers we propose is indeed unconditionally stable for all smooth spatial domains and for arbitrarily fine discretizations. To illustrate the FC-AD methodology in the hyperbolic PDE context, finally, we present an example concerning the Wave Equation—demonstrating sixth-order spatial and fourth-order temporal accuracy, as well as a complete absence of the debilitating “dispersion error”, also known as “pollution error”, that arises as finite-difference and finite-element solvers are applied to solution of wave propagation problems.

Publication: Journal of Computational Physics Vol.: 229 No.: 9 ISSN: 0021-9991

ID: CaltechAUTHORS:20100520-133258749

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Abstract: We introduce a new methodology for the numerical solution of Partial Differential Equations in general spatial domains: our algorithms are based on the use of the well-known Alternating Direction Implicit (ADI) approach in conjunction with a certain "Fourier continuation" (FC) method for the resolution of the Gibbs phenomenon. Unlike previous alternating direction methods of order higher than one, which can only deliver unconditional stability for rectangular domains, the present high-order algorithms possess the desirable property of unconditional stability for general domains; the computational time required by our algorithms to advance a solution by one time-step, in turn, grows in an essentially linear manner with the number of spatial discretization points used. In this paper we demonstrate the FC-AD methodology through a variety of examples concerning the Heat and Laplace Equations in two and three-dimensional domains with smooth boundaries. Applications of the FC-AD methodology to Hyperbolic PDEs together with a theoretical discussion of the method will be put forth in a subsequent contribution. The numerical examples presented in this text demonstrate the unconditional stability and high-order convergence of the proposed algorithms, as well the very significant improvements they can provide (in one of our examples we demonstrate a one thousand improvement factor) over the computing times required by some of the most efficient alternative general-domain solvers.

Publication: Journal of Computational Physics Vol.: 229 No.: 6 ISSN: 0021-9991

ID: CaltechAUTHORS:20100324-103102353

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Abstract: We present a new class of integral equations for the solution of problems of scattering of electromagnetic fields by perfectly conducting bodies. Like the classical Combined Field Integral Equation (CFIE), our formulation results from a representation of the scattered field as a combination of magnetic- and electric-dipole distributions on the surface of the scatterer. In contrast with the classical equations, however, the electric-dipole operator we use contains a regularizing operator; we call the resulting equations Regularized Combined Field Integral Equations (CFIE-R). Unlike the CFIE, the CFIE-R are Fredholm equations which, we show, are uniquely solvable; our selection of coupling parameters, further, yields CFIE-R operators with excellent spectral distributions—with closely clustered eigenvalues—so that small numbers of iterations suffice to solve the corresponding equations by means of Krylov subspace iterative solvers such as GMRES. The regularizing operators are constructed on the basis of the single layer operator, and can thus be incorporated easily within any existing surface integral equation implementation for the solution of the classical CFIE. We present one such methodology: a high-order Nyström approach based on use of partitions of unity and trapezoidal-rule integration. A variety of numerical results demonstrate very significant gains in computational costs that can result from the new formulations, for a given accuracy, over those arising from previous approaches.

Publication: Journal of Computational Physics Vol.: 228 No.: 17 ISSN: 0021-9991

ID: CaltechAUTHORS:20090826-112852554

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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: We present a new algorithm, based on integral equation formulations, for the solution of constant-coefficient elliptic partial differential equations (PDE) in closed two-dimensional domains with non-smooth boundaries; we focus on cases in which the integral-equation solutions as well as physically meaningful quantities (such as, stresses, electric/magnetic fields, etc.) tend to infinity at singular boundary points (corners). While, for simplicity, we restrict our discussion to integral equations associated with the Neumann problem for the Laplace equation, the proposed methodology applies to integral equations arising from other types of PDEs, including the Helmholtz, Maxwell, and linear elasticity equations. Our numerical results demonstrate excellent convergence as discretizations are refined, even around singular points at which solutions tend to infinity. We demonstrate the efficacy of this algorithm through applications to solution of Neumann problems for the Laplace operator over a variety of domains—including domains containing extremely sharp concave and convex corners, with angles as small as π/100 and as large as 199π/100.

Publication: Computing Vol.: 84 No.: 3-4 ISSN: 1436-5057

ID: CaltechAUTHORS:20090722-113431556

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Abstract: We present a novel numerical method, based on high-frequency localization, for evaluation of electromagnetic-wave propagation through atmospheres exhibiting fully three-dimensional (height, range and cross-range) refractive index variations. This methodology, which is based on localization of Rytov-integration domains to small tubes around geometrical optics paths, can accurately solve three-dimensional propagation problems in orders-of-magnitude shorter computing times than other algorithms available presently. For example, the proposed approach can accurately produce solutions for propagation of ≈20 cm GPS signals across hundreds of kilometers of realistic, three-dimensional atmospheres in computing times on the order of 1 hour in a present-day single-processor workstation, a task for which other algorithms would require, in such single-processor computers, computing times on the order of several months.

Publication: Radio Science Vol.: 44ISSN: 0048-6604

ID: CaltechAUTHORS:20090413-104957438

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Abstract: We present a superalgebraically convergent integral equation algorithm for evaluation of TE and TM electromagnetic scattering by smooth perfectly conducting periodic surfaces z=f(x). For grating-diffraction problems in the resonance regime (heights and periods up to a few wavelengths) the proposed algorithm produces solutions with full double-precision accuracy in single-processor computing times of the order of a few seconds. The algorithm can also produce, in reasonable computing times, highly accurate solutions for very challenging problems, such as (a) a problem of diffraction by a grating for which the peak-to-trough distance equals 40 times its period that, in turn, equals 20 times the wavelength; and (b) a high-frequency problem with very small incidence, up to 0.01° from glancing. The algorithm is based on the concurrent use of Floquet and Chebyshev expansions together with certain integration weights that are computed accurately by means of an asymptotic expansion as the number of integration points tends to infinity.

Publication: Journal of the Optical Society of America A Vol.: 26 No.: 3 ISSN: 1084-7529

ID: CaltechAUTHORS:20090811-153042013

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Abstract: We investigate the statistics of the transformation strains that arise in random martensitic polycrystals as boundary conditions cause its component crystallites to undergo martensitic phase transitions. In our laminated polycrystal model the orientation of the n grains (crystallites) is given by an uncorrelated random array of the orientation angles θ_i, i = 1, . . . ,n. Under imposed boundary conditions the polycrystal grains may undergo a martensitic transformation. The associated transformation strains ε_i, i = 1, . . . ,n depend on the array of orientation angles, and they can be obtained as a solution to a nonlinear optimization problem. While the random variables θ_i, i = 1, . . . ,n are uncorrelated, the random variables ε_i, i = 1, . . . ,n may be correlated. This issue is central in our considerations. We investigate it in following three different scaling limits: (i) Infinitely long grains (laminated polycrystal of height L = ∞); (ii) Grains of finite but large height (L » 1); and (iii) Chain of short grains (L = l_0/(2n), l_0 « 1). With references to de Finetti’s theorem, Riesz’ rearrangement inequality, and near neighbor approximations, our analyses establish that under the scaling limits (i), (ii), and (iii) the arrays of transformation strains arising from given boundary conditions exhibit no correlations, long-range correlations, and exponentially decaying short-range correlations, respectively

Publication: SIAM Journal on mathematical analysis Vol.: 40 No.: 4 ISSN: 0036-1410

ID: CaltechAUTHORS:20090720-151441316

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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: It has been conjectured that, in the quasistatic regime, dielectric bodies of finite size could be perfectly cloaked by certain cylindrical arrangements of materials of positive and negative permittivities known as superlenses. We show that, although they do not cloak perfectly dielectrics objects of any size, cylindrical superlenses do cloak, to a significant extent, dielectric bodies of small size.

Publication: Journal of Applied Physics Vol.: 102 No.: 12 ISSN: 0021-8979

ID: CaltechAUTHORS:BRUjap07

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Abstract: We present a new method for construction of high-order parametrizations of surfaces: starting from point clouds, the method we propose can be used to produce full surface parametrizations (by sets of local charts, each one representing a large surface patch – which, typically, contains thousands of the points in the original point-cloud) for complex surfaces of scientific and engineering relevance. The proposed approach accurately renders both smooth and non-smooth portions of a surface: it yields super-algebraically convergent Fourier series approximations to a given surface up to and including all points of geometric singularity, such as corners, edges, conical points, etc. In view of their C^∞ smoothness (except at true geometric singularities) and their properties of high-order approximation, the surfaces produced by this method are suitable for use in conjunction with high-order numerical methods for boundary value problems in domains with complex boundaries, including PDE solvers, integral equation solvers, etc. Our approach is based on a very simple concept: use of Fourier analysis to continue smooth portions of a piecewise smooth function into new functions which, defined on larger domains, are both smooth and periodic. The “continuation functions” arising from a function f converge super-algebraically to f in its domain of definition as discretizations are refined. We demonstrate the capabilities of the proposed approach for a number of surfaces of engineering relevance.

Publication: Journal of Computational Physics Vol.: 227 No.: 2 ISSN: 0021-9991

ID: CaltechAUTHORS:20181101-152903783

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Abstract: The solid-to-solid phase transitions that result from shock loading of certain materials, such as the graphite-to-diamond transition and the alpha-epsilon transition in iron, have long been subjects of a substantial theoretical and experimental literature. Recently a model for such transitions was introduced which, based on a CS condition (CS) and without use of fitting parameters, accounts quantitatively for existing observations in a number of systems [Bruno and Vaynblat, Proc. R. Soc. London, Ser. A 457, 2871 (2001)]. While the results of the CS model match the main features of the available experimental data, disagreements in some details between the predictions of this model and experiment, attributable to an ideal character of the CS model, do exist. In this article we present a version of the CS model, the viscous CS model (vCS), as well as a numerical method for its solution. This model and the corresponding solver results in a much improved overall CS modeling capability. The innovations we introduce include: (1) Enhancement of the model by inclusion of viscous phase-transition effects; as well as a numerical solver that allows for a fully rigorous treatment of both, the (2) Rarefaction fans (which had previously been approximated by “rarefaction discontinuities”), and (3) viscous phase-transition effects, that are part of the vCS model. In particular we show that the vCS model accounts accurately for well known “gradual” rises in the alpha-epsilon transition which, in the original CS model, were somewhat crudely approximated as jump discontinuities.

Publication: Journal of Applied Physics Vol.: 102 No.: 6 ISSN: 0021-8979

ID: CaltechAUTHORS:WEAjap07

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Abstract: We present an accurate method of O(1)-complexity with respect to frequency (i.e., a method that, to achieve a prescribed error tolerance, requires a bounded computational cost for arbitrarily high frequencies) for the computation of singular oscillatory integrals arising in the boundary integral formulation of problems of acoustic scattering by surfaces in three-dimensional space. Like the two-dimensional counterpart of this algorithm, which we introduced recently and which is applicable to scattering by curves in the plane, the present method is based on a combination of two main elements: (1) a high-frequency ansatz for the unknown density in a boundary integral formulation of the problem, and (2) an extension of the ideas of the method of stationary phase to allow for O(1) (high-order-accurate) integration of oscillatory functions. The techniques we introduce to implement an efficient O(1) integrator in the present three-dimensional context differ significantly from those used in the earlier two-dimensional algorithm. In particular, in the present text, we introduce an efficient “canonical” (hybrid analytic-numerical) algorithm which, in addition to allowing for integration of oscillatory functions around both singular points and points of stationary phase, can handle the significant difficulty that arises as singular points and one or more stationary points approach each other within the two-dimensional scattering surface. We include numerical results illustrating the behavior of the integration algorithm on sound-soft spheres with diameters of up to 5000 wavelengths: in such cases, for a single integral, the algorithm yields accuracies of the order of three digits in computational times of less than two seconds. In a preliminary full scattering simulation we present, a solution with two digits of accuracy in the surface density was obtained in about three hours running time, in a single 1.5 GHz AMD Athlon processor, for a sphere of 500 wavelengths in diameter.

Publication: Journal of Computational and Applied Mathematics Vol.: 204 No.: 2 ISSN: 0377-0427

ID: CaltechAUTHORS:20100820-091539194

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Abstract: We consider the problem of evaluating the current distribution $J(z)$ that is induced on a straight wire antenna by a time-harmonic incident electromagnetic field. The scope of this paper is twofold. One of its main contributions is a regularity proof for a straight wire occupying the interval $[-1,1]$. In particular, for a smooth time-harmonic incident field this theorem implies that $J(z) = I(z)/\sqrt{1-z^2}$, where $I(z)$ is an infinitely differentiable function—the previous state of the art in this regard placed $I$ in the Sobolev space $W^{1,p}$, $p>1$. The second focus of this work is on numerics: we present three superalgebraically convergent algorithms for the solution of wire problems, two based on Hallén's integral equation and one based on the Pocklington integrodifferential equation. Both our proof and our algorithms are based on two main elements: (1) a new decomposition of the kernel of the form $G(z) = F_1(z) \ln\! |z| + F_2(z)$, where $F_1(z)$ and $F_2(z)$ are analytic functions on the real line; and (2) removal of the end-point square root singularities by means of a coordinate transformation. The Hallén- and Pocklington-based algorithms we propose converge superalgebraically: faster than $\mathcal{O}(N^{-m})$ and $\mathcal{O}(M^{-m})$ for any positive integer $m$, where $N$ and $M$ are the numbers of unknowns and the number of integration points required for construction of the discretized operator, respectively. In previous studies, at most the leading-order contribution to the logarithmic singular term was extracted from the kernel and treated analytically, the higher-order singular derivatives were left untreated, and the resulting integration methods for the kernel exhibit $\mathcal{O}(M^{-3})$ convergence at best. A rather comprehensive set of tests we consider shows that, in many cases, to achieve a given accuracy, the numbers $N$ of unknowns required by our codes are up to a factor of five times smaller than those required by the best solvers previously available; the required number $M$ of integration points, in turn, can be several orders of magnitude smaller than those required in previous methods. In particular, four-digit solutions were found in computational times of the order of four seconds and, in most cases, of the order of a fraction of a second on a contemporary personal computer; much higher accuracies result in very small additional computing times.

Publication: SIAM Journal on Scientific Computing Vol.: 29 No.: 4 ISSN: 1064-8275

ID: CaltechAUTHORS:BRUsiamjsc07

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Abstract: The numerical solution of time-dependent ordinary and partial differential equations presents a number of well known difficulties—including, possibly, severe restrictions on time-step sizes for stability in explicit procedures, as well as need for solution of challenging, generally nonlinear systems of equations in implicit schemes. In this note we introduce a novel class of explicit methods based on use of one-dimensional Padé approximation. These schemes, which are as simple and inexpensive per time-step as other explicit algorithms, possess, in many cases, properties of stability similar to those offered by implicit approaches. We demonstrate the character of our schemes through application to notoriously stiff systems of ODEs and PDEs. In a number of important cases, use of these algorithms has resulted in orders-of-magnitude reductions in computing times over those required by leading approaches.

Publication: Journal of Scientific Computing Vol.: 30 No.: 1 ISSN: 0885-7474

ID: CaltechAUTHORS:20181105-143029068

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Abstract: This paper provides a theoretical analysis of a higher-order, FFT-based integral equation method introduced recently [IEEE Trans. Antennas and Propagation, 48 (2000), pp. 1862-1864] for the evaluation of transverse electric-polarized electromagnetic scattering from a bounded, penetrable inhomogeneity in two-dimensional space. Roughly speaking, this method is based on Fourier smoothing of the integral operator and the refractive index n(x). Here we prove that the solution of the resulting integral equation approximates the solution of the exact integral equation with higher-order accuracy, even when n(x) is a discontinuous function -- as suggested by the numerical experiments contained in the paper mentioned above. In detail, we relate the convergence rates of the computed interior and exterior fields to the regularity of the scatterer, and we demonstrate, with a few numerical examples, that the predicted convergence rates are achieved in practice.

Publication: SIAM Journal on Numerical Analysis Vol.: 42 No.: 6 ISSN: 0036-1429

ID: CaltechAUTHORS:BRUsiamjna05

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Abstract: In this paper, we present a multiple-scattering solver for nonconvex geometries such as those obtained as the union of a finite number of convex surfaces. For a prescribed error tolerance, this algorithm exhibits a fixed computational cost for arbitrarily high frequencies. At the core of the method is an extension of the method of stationary phase, together with the use of an ansatz for the unknown density in a combined-field boundary integral formulation.

Publication: IEEE Transactions on Magnetics Vol.: 41 No.: 5 ISSN: 0018-9464

ID: CaltechAUTHORS:GEUieeetm05.862

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Abstract: Optical coherence tomography is a non-invasive imaging technique based on the use of light sources exhibiting a low degree of coherence. Low-coherence interferometric microscopes have been successful in producing internal images of thin pieces of biological tissue; typically samples of the order of 1 mm in depth have been imaged, with a resolution of the order of 10 µm in some portions of the sample. In this paper we deal with the imaging problem of determining the internal structure of a multi-layered sample from backscattered laser light and low-coherence interferometry. In detail, we formulate and solve an inverse problem which, using the interference fringes that result as the back scattering of low-coherence light is made to interfere with a reference beam, produces maps detailing the values of the refractive index within the imaged sample. Unlike previous approaches to the OCT imaging problem, the method we introduce does not require processing at data collection time, and it produces quantitatively accurate values of the refractive indexes within the sample from back-scattering interference fringes only.

Publication: Inverse Problems Vol.: 21 No.: 2 ISSN: 0266-5611

ID: CaltechAUTHORS:BRUip05

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Abstract: In this paper, we introduce a new fast, higher-order solver for scattering by inhomogeneous media in three dimensions. As in previously existing methods, the low complexity of our integral equation method, O(N log N) operations for an N point discretization, is obtained through extensive use of the fast Fourier transform (FFT) for the evaluation of convolutions. However, the present approach obtains significantly higher-order accuracy than these previous approaches, yielding, at worst, third-order far field accuracy (or substantially better for smooth scatterers), even for discontinuous and complex refractive index distributions (possibly containing severe geometric singularities such as corners and cusps). The increased order of convergence of our method results from (i) a partition of unity decomposition of the Green’s function into a smooth part with unbounded support and a singular part with compact support, and (ii) replacement of the (possibly discontinuous) scatterer by an appropriate “Fourier smoothed” scatterer; the resulting convolutions can then be computed with higher-order accuracy by means of O(N log N) FFTs. We present a parallel implementation of our approach, and demonstrate the method’s efficiency and accuracy through a variety of computational examples. For a very large scatterer considered earlier in the literature (with a volume of 3648λ3, where λ is the wavelength), using the same number of points per wavelength and in computing times comparable to those required by the previous approach, the present algorithm produces far-field values whose errors are two orders of magnitude smaller than those reported previously.

Publication: Journal of Computational Physics Vol.: 202 No.: 1 ISSN: 0021-9991

ID: CaltechAUTHORS:20181101-152904320

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Abstract: We consider the problem of evaluating the scattering of TE polarized electromagnetic waves by two-dimensional penetrable inhomogeneities: building upon previous work [IEEE Trans. Antennas Propag. 48 (2000) 1862] we present a practical and general fast integral equation algorithm for this problem. The contributions introduced in this text include: (1) a preconditioner that significantly reduces the number of iterations required by the algorithm in the treatment of electrically large scatterers, (2) a new radial integration scheme based on Chebyshev polynomial approximation, which gives rise to increased accuracy, efficiency and stability, and (3) an efficient and stable method for the evaluation of scaled high-order Bessel functions, which extends the capabilities of the method to arbitrarily high frequencies. These enhancements give rise to an algorithm that is much more accurate and efficient than its previous counterpart, and that allows for treatment of much larger problems than permitted by the previous approach. In one test case, for example, the present algorithm results in far-field errors of 8.9×10^(−13) in a 2.12s calculation (on a 1.7 GHz PC) whereas the original algorithm gave rise to far-field errors of 1.1×10^(−8) in 88.91s on a 400 MHz PC. In another example, the present algorithm evaluates accurately the scattering by a cylinder of acoustical size κR=256, which is of the order of 20 times larger (400 times larger in square wavelengths) than the largest scatterers that could be treated by the previous approach. Yielding, at worst, third-order far field accuracy (or substantially better, for smooth scatterers) in fast computing times (O(N log N) operations for an N point mesh) even for discontinuous and complex refractive index distributions (possibly containing severe geometric singularities such as corners and cusps), the proposed approach is the highest-order O(N log N) solver in existence for the problem under consideration.

Publication: Journal of Computational Physics Vol.: 200 No.: 2 ISSN: 0021-9991

ID: CaltechAUTHORS:20181101-152904227

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Abstract: We present a fast numerical method for the simulation of martensitic transformations in three–dimensional polycrystals. To produce the relevant overall elastic energy arising from given boundary conditions, this method proceeds by reducing the corresponding non–convex minimization problem to minimization of a certain quadratic form—over the set of arrays of transformation strains which are compatible with a given distribution of crystallite orientations. The evaluation of this quadratic form for a given array of transformation strains requires solution of certain linear elasticity problems. An acceleration strategy we use, which for a polycrystal containing N grains reduces the complexity of the algorithm from O(N^2) to O(N) operations, results from a formulation of the minimization problem which takes advantage of certain decorrelations present in the minimizing arrays of transformation strains. We illustrate our presentation with a number of examples involving cubic–to–monoclinic and cubic–to–orthorhombic polycrystalline phase transitions, such as those arising in the TiNi and CuAl shape–memory alloys. In particular, our study quantifies the effects of texture on the overall properties of such polycrystalline shape–memory alloys.

Publication: Proceedings of the Royal Society A: Mathematical, physical, and engineering sciences Vol.: 460 No.: 2046 ISSN: 1364-5021

ID: CaltechAUTHORS:20170408-135429636

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Abstract: We present a new algorithm for the numerical solution of problems of electromagnetic or acoustic scattering by large, convex obstacles. This algorithm combines the use of an ansatz for the unknown density in a boundary-integral formulation of the scattering problem with an extension of the ideas of the method of stationary phase. We include numerical results illustrating the high-order convergence of our algorithm as well as its asymptotically bounded computational cost as the frequency increases.

Publication: Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences Vol.: 362 No.: 1816 ISSN: 1364-503X

ID: CaltechAUTHORS:20181106-150424948

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Abstract: We present a new set of high-order 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 and Fast Fourier Transforms, 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. 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.

Publication: CMES: Computer Modeling in Engineering and Sciences Vol.: 5 No.: 4 ISSN: 1526-1492

ID: CaltechAUTHORS:20181102-161613781

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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 deal with the imaging problem of determining the internal structure of a body from backscattered laser light and low-coherence interferometry. Specifically, using the interference fringes that result when the backscattering of low-coherence light is made to interfere with the reference beam, we obtain maps detailing the values of the refractive index within the sample. Our approach accounts fully for the statistical nature of the coherence phenomenon; the numerical experiments that we present, which show image reconstructions of high quality, were obtained under noise floors exceeding those present for various experimental setups reported in the literature.

Publication: Optics Letters Vol.: 28 No.: 21 ISSN: 0146-9592

ID: CaltechAUTHORS:BRUol03

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Abstract: A new high-order integral algorithm for the solution of scattering problems by heterogeneous bodies is presented. Here, a scatterer is described by a (continuously or discontinuously) varying refractive index n(x) within a two-dimensional (2D) bounded region; solutions of the associated Helmholtz equation under given incident fields are then obtained by high-order inversion of the Lippmann-Schwinger integral equation. The algorithm runs in O(Nlog(N)) operations where N is the number of discretization points. A wide variety of numerical examples provided include applications to highly singular geometries, high-contrast configurations, as well as acoustically/electrically large problems for which supercomputing resources have been used recently. Our method provides highly accurate solutions for such problems on small desktop computers in CPU times of the order of seconds.

Publication: IEEE Transactions on Antennas and Propagation Vol.: 51 No.: 11 ISSN: 0018-926X

ID: CaltechAUTHORS:BRUieeetap03

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Abstract: We introduce a new fast, high-order method for scattering by inhomogeneous media in three dimensions. The O(N log N) complexity of our integral equation method is obtained through use of the fast Fourier transform in evaluating the required convolution. High-order convergence is obtained by replacing the scatterer with its truncated Fourier series and by decomposing the Green's function into a smooth part with infinite support and a singular part with compact support. Countering conventional wisdom, this Fourier smoothing of the scatterer yields high-order convergence, even in the case of discontinuous scatterers. We illustrate the performance of our parallel implementation of this method through two computational examples.

Publication: Physica B Vol.: 338 No.: 1-4 ISSN: 0921-4526

ID: CaltechAUTHORS:20181101-152903960

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Abstract: We review 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—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. A class of high-order high-frequency methods currently under development, in turn, are efficient where our direct methods become costly, and they thus lead to an overall computational methodology which is applicable and accurate throughout the electromagnetic spectrum.

Publication: Physica B Vol.: 338 No.: 1-4 ISSN: 0921-4526

ID: CaltechAUTHORS:20181101-152904409

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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: A new method is introduced for the solution of problems of scattering by rough surfaces in the high-frequency regime. It is shown that high-order summations of expansions in inverse powers of the wave number can be used within an integral equation framework to produce highly accurate results for surfaces and wavelengths of interest in applications. Our algorithm is based on systematic use and manipulation of certain Taylor-Fourier series representations and explicit asymptotic expansions of oscillatory integrals. Results with machine precision accuracy are presented which were obtained from computations involving expansions of order as high as 20.

Publication: Radio Science Vol.: 37 No.: 4 ISSN: 0048-6604

ID: CaltechAUTHORS:20141027-093609432

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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: In this paper we develop a theory for phase transitions in solids under shock loading. This theory applies, in particular, to a class of experiments which, as a result of solid–to–solid phase transitions, give rise to certain characteristic patterns consisting of two shock–like waves. We show that the single assumption that stresses in a phase cannot lie beyond its transition boundaries leads to a complete model for the physical systems at hand. This model is different from others proposed in the literature: it does not make use of kinetic relations and it accounts for the observed wave histories without parameter fitting. The first part of this paper focuses on the basic mathematical description of our model and it presents solutions to the complete set of Riemann problems which could arise as a result of dynamic interactions, including the basic two–wave structures mentioned above. In the second part of the paper we use our Riemann solver to construct general solutions for the piecewise constant initial–value problems usually arising in experiment, and we specialize our solutions to two widely studied polymorphic phase changes: the graphite–diamond transition and the α–∈ transition in iron. We show that, in the presence of well–accepted quations of state for the pure phases, our model leads to close quantitative agreement with a wide range of experimental results; possible sources of disagreement in certain fine features are also discussed. Interestingly, in some cases our theory predicts sequences of events which differ from those generally accepted. In particular, our model predicts a variety of regimes for the iron experiments which previous theoretical investigations had not surmised, and it indicates the existence of certain unexpected transformation domains in the graphite systems.

Publication: Proceedings of the Royal Society A: Mathematical, physical, and engineering sciences Vol.: 457 No.: 2016 ISSN: 1364-5021

ID: CaltechAUTHORS:20181106-145235963

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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 scattered fields through fast, high–order, accurate solution of the corresponding boundary integral equation. The high–order accuracy of our solver is achieved through use of partitions of unityI together with analytical resolution of kernel singularities. The acceleration, in turn, results from use of high–order equivalent sourceapproximations, which allow for fast evaluation of non–adjacent interactions by means of the three–dimensional fast Fourier transform (FFT). Our acceleration scheme has dramatically lower memory requirements and yields much higher accuracy than existing FFT–accelerated techniques. The present algorithm computes one matrix–vector multiply in O(N^(6/5)logN) to O(N^(4/3)logN) operations (depending on the geometric characteristics of the scattering surface), it exhibits super–algebraic convergence, and it does not suffer from accuracy breakdowns of any kind. We demonstrate the efficiency of our method through a variety of examples. In particular, we show that the present algorithm can evaluate accurately, on a personal computer, scattering from bodies of acoustical sizes (ka) of several hundreds.

Publication: Proceedings of the Royal Society A: Mathematical, physical, and engineering sciences Vol.: 457 No.: 2016 ISSN: 1364-5021

ID: CaltechAUTHORS:20181106-151356250

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Abstract: We study the macroscopic magneto-mechanical behavior of composite materials consisting of a random, statistically homogeneous distribution of ferromagnetic, rigid inclusions embedded firmly in a non-magnetic elastic matrix. Specifically, for given applied elastic and magnetic fields, we calculate the overall deformation and stress–strain relation for such a composite, correct to second order in the particle volume fraction. Our solution accounts for the fully coupled magneto-elastic interactions; the distribution of magnetization in the composite is calculated from the basic minimum energy principle of magneto-elasticity.

Publication: Journal of the Mechanics and Physics of Solids Vol.: 49 No.: 12 ISSN: 0022-5096

ID: CaltechAUTHORS:20181101-152904594

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Abstract: We consider a class of shock-loading experiments which, as a result of solid-to-solid phase transitions, give rise to certain characteristic patterns consisting of two shock-like waves. We show that the single assumption that stresses in a phase cannot lie beyond the transition boundaries leads to a complete mathematical description of the physical problem at hand. In detail, our model only requires knowledge of well-studied material observables: the equations of state (EOS) for the pure phases and the phase transition boundaries. The model presented here is different from others proposed in the literature: it does not make use of kinetic relations, and it accounts for the observed wave histories without parameter fitting. In presence of well-accepted EOS for the pure phases, our model leads to close quantitative agreement with a wide range of experimental results.

Publication: Mathematical and Computer Modelling Vol.: 34 No.: 12-13 ISSN: 0895-7177

ID: CaltechAUTHORS:20181105-130419396

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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: We present a new algorithm for the numerical solution of problems of acoustic scattering by surfaces in three-dimensional space. This algorithm evaluates scattered fields through fast, high-order solution of the corresponding boundary integral equation. The high-order accuracy of our solver is achieved through use of partitions of unity together with analytical resolution of kernel singularities. The acceleration, in turn, results from use of a novel approach which, based on high-order “two-face” equivalent source approximations, reduces the evaluation of far interactions to evaluation of 3-D fast Fourier transforms (FFTs). This approach is faster and substantially more accurate, and it runs on dramatically smaller memories than other FFT and k-space methods. The present algorithm computes one matrix-vector multiplication in O(N^(6/5)log N) to O (N^(4/3) logN) operations, where N is the number of surface discretization points. The latter estimate applies to smooth surfaces, for which our high-order algorithm provides accurate solutions with small values of N; the former, more favorable count is valid for highly complex surfaces requiring significant amounts of subwavelength sampling. Further, our approach exhibits super-algebraic convergence; it can be applied to smooth and nonsmooth scatterers, and it does not suffer from accuracy breakdowns of any kind. In this paper we introduce the main algorithmic components in our approach, and 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.

Publication: Journal of Computational Physics Vol.: 169 No.: 1 ISSN: 0021-9991

ID: CaltechAUTHORS:20181101-152903866

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Abstract: We present an algorithm which, based on certain properties of analytic dependence, constructs boundary perturbation expansions of arbitrary order for eigenfunctions of elliptic PDEs. The resulting Taylor series can be evaluated far outside their radii of convergence—by means of appropriate methods of analytic continuation in the domain of complex perturbation parameters. A difficulty associated with calculation of the Taylor coefficients becomes apparent as one considers the issues raised by multiplicity: domain perturbations may remove existing multiple eigenvalues and criteria must therefore be provided to obtain Taylor series expansions for all branches stemming from a given multiple point. The derivation of our algorithm depends on certain properties of joint analyticity (with respect to spatial variables and perturbations) which had not been established before this work. While our proofs, constructions and numerical examples are given for eigenvalue problems for the Laplacian operator in the plane, other elliptic operators can be treated similarly.

Publication: Journal of Fourier Analysis and Applications Vol.: 7 No.: 2 ISSN: 1069-5869

ID: CaltechAUTHORS:20181106-075027317

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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 new high-order integral algorithm for the solution of scattering problems by heterogeneous bodies under TE radiation. Here, a scatterer is represented by a (continuously or discontinuously) varying refractive index n(x) within a two-dimensional (2-D) bounded region. Solutions of the associated Helmholtz equation under given incident fields are then obtained by high-order inversion of the Lippmann-Schwinger integral equation. The algorithm runs in O(N log(N)) operations, where N is the number of discretization points. Our method provides highly accurate solutions in short computing times, even for problems in which the scattering bodies contain complex geometric singularities.

Publication: IEEE Transactions on Antennas and Propagation Vol.: 48 No.: 12 ISSN: 0018-926X

ID: CaltechAUTHORS:BRUieeetap00

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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: We introduce a fast numerical method for the evaluation of the effective elastic energy in martensitic polycrystals in two and three dimensions. The overall complexity of the method is O(N) operations, where N is the number of component crystallites. Upper and lower bounds on the energy are also presented which allow us to estimate the accuracy of the numerical results. Our new three-dimensional computations and bounds for random polycrystals, which are the first ones available in the literature, provide substantial insights on the behavior of polycrystalline martensites. They suggest that recoverable strains can be much larger than those attainable with zero energy.

Publication: Journal of the Mechanics and Physics of Solids Vol.: 48 No.: 6-7 ISSN: 0022-5096

ID: CaltechAUTHORS:20181102-135535088

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Abstract: We present a fast method for the solution of problems of elasticity involving microscopic misfit strains. While the main case we consider is that associated with martensitic transformations in polycrystals, our methods can be applied to a variety of systems whose constituents undergo misfit deformations, including polycrystalline magnetostriction, thermal expansion, etc., as well as mathematically analogous phenomena in ferroelectricity and ferromagnetism. The basic component of our method is an explicit solution for Eshelby–type problems on square elements. Fast computation of the polycrystal energy results through a rapidly convergent sequence of approximations which can, in fact, be interpreted as a generalization of a class of upper bounds introduced recently. The overall complexity of the method is O(N) operations, where N is the number of component crystallites. We also present a new lower bound for the energy, giving additional insights on the microscopic phenomena leading to the observed structural behaviour. The present work applies to two–dimensional polycrystals; extensions to the three–dimensional case have been implemented and will be presented elsewhere.

Publication: Proceedings of the Royal Society A: Mathematical, physical, and engineering sciences Vol.: 455 No.: 1992 ISSN: 1364-5021

ID: CaltechAUTHORS:20181106-084731697

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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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Abstract: We establish an existence result for the three‐dimensional MHD equations (∇ x B) x B = ∇p, ∇·B = 0, B·n|∂T = 0 with p ≠ const in tori T without symmetry. More precisely, our theorems insure the existence of sharp boundary solutions for tori whose departure from axisymmetry is sufficiently small; they allow for solutions to be constructed with an arbitrary number of pressure jumps.

Publication: Communications on Pure and Applied Mathematics Vol.: 49 No.: 7 ISSN: 0010-3640

ID: CaltechAUTHORS:20181106-153720174

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Abstract: We are concerned with the overall elastic energy in martensitic polycrystals. These are polycrystals whose constituent crystallites can undergo shape-deforming phase transitions as a result of changes in their stress or temperature. We approach the problem of calculation of the nonlinear overall energy via a statistical optimization method which involves solution of a sequence of linear elasticity problems. As a case study we consider simulations on a two-dimensional model in which circular randomly-oriented crystallites are arranged in a square pattern within an elastic matrix. The performance of our present code suggests that this approach can be used to compute the overall energies in realistic three-dimensional polycrystals containing grains of arbitrary shape. In addition to numerical results we present upper bounds on the overall energy. Some of these bounds apply to the square array mentioned above. Others apply to polycrystals containing circular, randomly-oriented crystallites with sizes ranging to infinitesimal, and no intergrain matrix. The square-array bounds are consistent with our numerical results. In some regimes they approximate them closely, thus providing an insight on the convergence of the numerical method. On the other hand, in the case of the random array the bounds carry substantial practical significance, since in this case the energy contains no artificial contributions from an elastic matrix. In all the cases we have considered our bounds compare favorably with those obtained under the well-known Taylor hypothesis; they show that, as far as polycrystalline martensite is concerned, calculations of the elastic energy based on the Taylor assumption may lead to substantial overestimates of this quantity.

Publication: Journal of the Mechanics and Physics of Solids Vol.: 44 No.: 7 ISSN: 0022-5096

ID: CaltechAUTHORS:20181101-152903698

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Abstract: In this paper we present an exact calculation of the transfer function associated with the nonlinear Fabry–Perot resonator. While our exact result cannot be evaluated in terms of elementary functions, it does permit us to obtain a number of simple approximate expressions of various orders of accuracy. In addition, our derivation yields criteria of validity for the approximate formulae. Our approach is to be compared with others in which approximations are introduced in the model itself, either through the equations or through the boundary conditions. Our lowest order approximate formula turns out to be identical, interestingly, with the result obtained from the slowly varying envelope approximation (SVEA). Thus, our validity criteria apply to the SVEA result, and predict well its domain of validity and its breakdown for short wavelengths and for very high intensities and nonlinearities. The simple higher order formulae we present provide improved estimations in such regimes.

Publication: Proceedings of the Royal Society A: Mathematical, physical, and engineering sciences Vol.: 447 No.: 1929 ISSN: 1364-5021

ID: CaltechAUTHORS:20181105-132848389

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Abstract: We deal with a method of enhanced convergence for the approximation of analytic functions. This method introduces conformal transformations in the approximation problems, in order to help extract the values of a given analytic function from its Taylor expansion around a point. An instance of this method, based on the Euler transform, has long been known; recently we introduced more general versions of it in connection with certain problems in wave scattering. In §2 we present a general discussion of this approach. As is known in the case of the Euler transform, conformal transformations can enlarge the region of convergence of power series and can enhance substantially the convergence rates inside the circles of convergence. We show that conformal maps can also produce a rather dramatic improvement in the conditioning of Padé approximation. This improvement, which we discuss theoretically for Stieltjes-type functions, is most notorious in cases of very poorly conditioned Padé problems. In many instances, an application of enhanced convergence in conjunction with Padé approximation leads to results which are many orders of magnitude more accurate than those obtained by either classical Padé approximants or the summation of a truncated enhanced series.

Publication: Mathematics of Computation Vol.: 63 No.: 207 ISSN: 0025-5718

ID: CaltechAUTHORS:20181106-080649656

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Abstract: We study the macroscopic mechanical behavior of materials with microscopic holes or hard inclusions. Specifically, we deal with the effective elastic moduli of composites whose microgeometry consists of either soft or hard isolated inclusions surrounded by an elastic matrix. We approach this problem by taking the stiffness of the inclusion phase to be a complex variable, which we eventually evaluate at the soft or hard limits. Our main result states that there is a certain class of non-physical, negative-definite values of the elastic moduli of the inclusion phase for which the effective tensor does not have infinities or become otherwise singular. We present applications of this result to the estimation of effective moduli and to homogenization theorems. The first application involves using complexanalytic methods to obtain rigorous and accurate bounds on the effective moduli of the high-contrast composites under consideration. We also discuss the variational estimates of Rubenfeld & Keller, which yield a complementary set of bounds on these moduli. The best bounds are given by a combination of the analytical and variational results. As a second application, we show that certain known theorems of homogenization for materials with holes are simple consequences of our main result, and in this connection we establish corresponding new theorems for materials with hard inclusions. While our rederivation of the homogenization theorems for materials with holes can be closely related to other known constructions, it appears that certain elements provided by our main result are essential in the proof of homogenization for the hard-inclusion case.

Publication: Archive for Rational Mechanics and Analysis Vol.: 121 No.: 4 ISSN: 0003-9527

ID: CaltechAUTHORS:20181106-154533730

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Abstract: Consider spherical particles of volume x having paint on a fraction y of their surface area. The particles are assumed to be homogeneously distributed at each time t, so that one can introduce the density number n (x, y, t). When collision between two particles occurs, the particles will coalesce if and only if they happen to touch each other, at impact, at points which do not belong to the painted portions of their surfaces. Introducing a dynamics for this model, we study the evolution of n (x, y, t) and, in particular, the asymptotic behavior of the mass x n (x, y, t) dx as t → ∞.

Publication: Transactions of the American Mathematical Society Vol.: 338 No.: 1 ISSN: 0002-9947

ID: CaltechAUTHORS:20120309-093931707

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Abstract: In this paper we deal with the problem of diffraction of electromagnetic waves by a periodic interface between two materials. This corresponds to a two-dimensional quasi-periodic boundary value problem for the Helmholtz equation. We prove that solutions behave analytically with respect to variations of the interface. The interest of this result is both theoretical – the legitimacy of power series expansions in the parameters of the problem has indeed been questioned – and, perhaps more importantly, practical: we have found that the solution can be computed on the basis of this observation. The simple algorithm that results from such boundary variations is described. To establish the property of analyticity of the solution for the grating f_σ(x) = δf(x) with respect to the height δ, we present a holomorphic formulation of the problem using surface potentials. We show that the densities entering into the potential theoretic formulation are analytic with respect to variations of the boundary, or, in other words, that the integral operator that results from the transmission conditions at the interface is invertible in a space of holomorphic functions of the variables (x, y, δ). This permits us to conclude, in particular, that the partial derivatives of u with respect to δ at δ = 0 satisfy certain boundary value problems for the Helmholtz equation, in regions with plane boundaries, which can be solved in a closed form.

Publication: Proceedings of the Royal Society of Edinburgh: Section A Mathematics Vol.: 122 No.: 3-4 ISSN: 0308-2105

ID: CaltechAUTHORS:20181105-103653124

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Abstract: We deal with the effective conductivity m = m(z) of two phase, ordered or disordered mixtures consisting of particles of material of conductivity z inserted in a matrix of conductivity 1. We focus on finding bounds on the set of values of z for which the function m is singular or vanishes, and we apply our results to the estimation of the effective conductivity of high contrast mixtures (z = 0 or z = ∞). We find that the zeroes and singularities of the function m lie on an interval of the negative real axis, which depends on the shape of the particles and the interparticle distances. Our results agree with previous numerical calculations for periodic arrays of spheres. In some cases we show that our estimates are optimal. We apply our results about the zeroes and singularities together with the complex variable method, and find bounds on the effective conductivity of matrix-particle random composites. These bounds give good estimations even in cases of high contrast, and, in many cases, they improve substantially over the bounds obtained by other methods, for the same types of high contrast mixtures.

Publication: Proceedings of the Royal Society A: Mathematical, physical, and engineering sciences Vol.: 433 No.: 1888 ISSN: 1364-5021

ID: CaltechAUTHORS:20181106-145800580

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Abstract: We describe a construction, based on variational inequalities, which gives a hierarchy of upper and lower bounds (of odd orders 2k+1), on the various effective moduli of random multiphase materials and polycrystals. The bounds of order 2k+1 on a given effective modulus can be explicitly evaluated if a truncated Taylor expansion of the given modulus is known to order 2k+1. Our approach is motivated by prior investigations of Beran and other authors. Our calculations do not involve Green functions or n-point correlation functions, and they are very simple. We thus rederive known and obtain new sequences of bounds on the different effective moduli. We also describe a method that, for cell materials (i.e. materials in which cells of smaller and smaller length scales cover all space, with material properties assigned at random), permits one to calculate the truncated Taylor expansions that are needed for the explicit evaluation of the bounds. In connection with this, we show that the first coefficient in the low volume fraction expansion of any effective modulus of a cell material, coincides with the corresponding low volume fraction coefficient for an array of cells randomly distributed in a matrix.

Publication: Asymptotic Analysis Vol.: 4 No.: 4 ISSN: 0921-7134

ID: CaltechAUTHORS:20181105-104502314

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Abstract: The effective conductivity σ* of an infinitely interchangeable two-component random medium is considered. This class of media includes cell materials in the continuum and the bond lattice on ℤ^d, where the cells or bonds are randomly assigned the conductivities σ_1 and σ_2 (σ_1, σ_2 ne0) with probabilities p_1 and p_2 = 1−p_1. A rigorous basis for the very old and widely used low volume fraction expansion of σ* is established, by proving that σ* is an analytic function of p_2 in a suitable domain containing [0, 1]. In the case of the bond lattice in d = 2, rigorous fourth-order upper and lower bounds on σ* valid for all p_2, σ_1, and σ_2 are derived. The four perturbation coefficients entering into the bounds are obtained from the first-order volume fraction coefficient using the method of infinite interchangeability.

Publication: Journal of Statistical Physics Vol.: 61 No.: 1-2 ISSN: 0022-4715

ID: CaltechAUTHORS:20181107-080002180

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Abstract: We are concerned with the estimation of the effective electrical conductivity of random heterogenous materials. The purpose of this paper is to discuss a property of “statistical symmetry” verified by the symmetric cell materials of Miller. This property will be referred to as infinite interchangeability. The usual way to approach cell materials is through n‐point correlation functions. The property of infinite interchangebility permits us to approach cell materials from a completely different point of view. Our main result is a simple algorithm, based on this symmetry property, for computing any coefficient of the perturbation expansion in terms of information from the dilute limit. Specifically, knowledge of the coefficients of the expansion in powers of the volume fraction up to order r allows for computation of the perturbation expansion coefficients up to order (2r + 1). This result, which was previously known for r = 2 in the isotropic case and for r = 1 in the anisotropic case, can also be obtained from the standard correlation function approach, as pointed out by Milton.

Publication: Communications on Pure and Applied Mathematics Vol.: 43 No.: 6 ISSN: 0010-3640

ID: CaltechAUTHORS:20181105-131823303

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Abstract: A third-order expansion for the effective thermal conductivity tensor κ* of anisotropic polycrystalline cell materials is derived. The coefficients of the expansion are given in terms of the average polarizability tensor, a nondimensional quantity determined from the grain shape and crystallographic orientation distributions independent of other details of the microgeometry such as two (or more) particle correlation functions. Explicit numerical results for a wide variety of microgeometries made of ellipsoidal cells are obtained. This calculation uses a new method that exploits the symmetry properties of the effective conductivity tensor of a cell material as a function of the single-crystal conductivities.

Publication: Journal of Mathematical Physics Vol.: 31 No.: 8 ISSN: 0022-2488

ID: CaltechAUTHORS:AVEjmp90

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Abstract: In this paper, we study the validity of the following two statements in the internal logic of the toposes of Synthetic Differential Geometry: 1. (1) The integral of f is non-negative if f is non-negative; 2. (2) If f=0 in the set of non-negative reals, and f=0 in the set of non-negative reals, then f=0. We find statements (1) and (2) to be true in the toposes considered. We also prove that 3. (3) For n greater than two, the arrow tn from the line to itself is not a stable effective epic. This answers a question raised by Quê-Moerdijk-Reyes.

Publication: Journal of Pure and Applied Algebra Vol.: 53 No.: 1-2 ISSN: 0022-4049

ID: CaltechAUTHORS:20181101-152904050

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Abstract: Vector fields in infinite-dimensional manifolds play an important role in differential topology-geometry. In particular the case when the manifolds are C∞-map spaces. The well developed theory modeled in Banach spaces does not apply here. Instead a theory modeled in Frechet spaces is being considered. This is a theory which seems to be a much less straightforward generalization of the finite-dimensional case. The well adapted models of S.D.G. lead naturally to treat these spaces. We investigate here the case of the 'manifold' R^R, whose space of global sections is C^∞(ℝ). We prove that to integrate a vector field in R^R is equivalent to a certain differential problem in C^∞(ℝ). To do this, we previously characterize the maps R^R → R^R, R^R X R^R in the topos by means of the functions they induce in the respective spaces of global sections.

Publication: Journal of Pure and Applied Algebra Vol.: 45 No.: 1 ISSN: 0022-4049

ID: CaltechAUTHORS:20181101-152904132

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Abstract: Let J ⊆ C^∞(R^n) be any ideal. Since a function of the variables t = (t_1,...,t_n) is a function of the variables (t,x) = (t_1,...,t_n,x_1,...x_p) which does not depend on x, we have J ⊆ C^∞(R^(n+p)). Of course, J is not an ideal of C^∞(R^(n+p), but it generates an ideal that we call J(t,x). Consider the following statement (1) on J: “Given any f ϵ C^∞ (R^(n+p), f ϵ J(t,x) if and only if for every fixed a ϵ R^p, f(t,a) ϵ J". In this paper we show that statement (1) holds for a large class of finitely generated ideals although not for all of them. We say that ideals satisfying statement (1) have line determined extensions. We characterize these ideals to be closed ideals J(t) (in the sense of Whitney) such that for all p ∈ ℕ, the ideal J(t,x) is also closed. Finally, some non-trivial examples are developed.

Publication: Bulletin of the Australian Mathematical Society Vol.: 33 No.: 2 ISSN: 0004-9727

ID: CaltechAUTHORS:20181106-160538496

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Abstract: Let X = C^∞(R^p)/J, Y = C^∞(R^n)/I be two representable objects in the Dubuc topos D (see Secsion 0) where J has line determined extensions (0.3). The main result in this paper (Theorem 1.11) says that the global section functor Γ establishes a bijection between Penon open sub-objects of Y^x and open subsets of Γ(Y^x) in the C^∞-CO topology. We show also that when I = {0}, we can assume J arbitrary (1.12). However, the restriction on J (of having line determined extensions) is seen to be unavoidable in general. We precede the article with a Section 0 where we recall all these notions and fix the notations.

Publication: Cahiers de Topologie et Géométrie Différentielle Catégoriques Vol.: 26 No.: 3 ISSN: 1245-530X

ID: CaltechAUTHORS:20181107-080545986

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