High order methods for acoustic scattering: Coupling Farfield Expansions ABC with Deferred-Correction methods

Arbitrary high order numerical methods for time-harmonic acoustic scattering problems originally defined on unbounded domains are constructed. This is done by coupling recently developed high order local absorbing boundary conditions (ABCs) with finite difference methods for the Helmholtz equation. These ABCs are based on exact representations of the outgoing waves by means of farfield expansions. The finite difference methods, which are constructed from a deferred-correction (DC) technique, approximate the Helmholtz equation and the ABCs, with the appropriate number of terms, to any desired order. As a result, high order numerical methods with an overall order of convergence equal to the order of the DC schemes are obtained. A detailed construction of these DC finite difference schemes is presented. Additionally, a rigorous proof of the consistency of the DC schemes with the Helmholtz equation and the ABCs in polar coordinates is also given. The results of several numerical experiments corroborate the high order convergence of the novel method.Comment: 36 pages, 20 figure

Non Uniform Rational B-Splines and Lagrange approximations for time-harmonic acoustic scattering: accuracy and absorbing boundary conditions

International audienceIn this paper, the performance of the finite element method based on Lagrange basis functions and the Non Uniform Rational B-Splines (NURBS) based Iso-Geometric Analysis (IGA) are systematically studied for solving time-harmonic acoustic scattering problems. To assess their performance, the numerical examples are presented with truncated absorbing boundary conditions. In the first two examples , we eliminate the domain truncation error by applying second-order Bayliss-Gunzburger-Turkel (BGT-2) Absorbing Boundary Condition (ABC) and modifying the exact solution. Hence, the calculated error is an indicator of the numerical accuracy in the bounded computational domain with no artificial domain truncation error. Next, we apply a higher order local ABC based on the Karp's and Wilcox's far-field expansions for 2D and 3D problems, respectively. The performance of both methods in solving exterior problems is compared. The introduced auxiliary surface functions are also estimated using the corresponding basis functions. The influence of various parameters, viz., order of the approximating polynomial, number of degrees of freedom, wave number and the boundary conditions (BGT-2 and number of terms in the far-field expansions) on the accuracy and convergence rate is systematically studied. It is inferred that, irrespective of the order of the polynomial, IGA yields higher accuracy per degree of freedom when compared to the conventional finite element method with Lagrange basis

Rescaling of the Roe Scheme in Low Mach-Number Flow Regions II: Artificial Speed of Sound and Low Mach Number Fix

We look at two simple modifications of the Roe scheme in the incompressible limit, based on different ideas: the Rossow's artificial speed of sound and the Rieper's low Mach number fix. Both schemes modify the eigenspaces of the dissipation matrix. The analysis emphasizes the properties of the dissipation matrix for the Von Neumann stability, the asymptotic behaviour and the solution accuracy in the incompressible limit. Numerical results in the very low-speed limit are discussed for robustness, consistency and accuracy issues of the numerical procedure. Possible occurrence of checkerboard pressure modes, when using a collocated arrangement for velocity components and pressure in the finite-volume scheme, and spurious acoustic modes, is also illustrated for both schemes

Ultrasonic transducer characterization and transducer beam modeling for applications in nondestructive evaluation

In this work, analytic models have been developed for two different nondestructive evaluation (NDE) applications: the characterization of spherically focused ultrasonic immersion transducers, and the prediction of the radiated wave fields of a variety of commonly used ultrasonic transducers;When a spherically focused probe is correctly and completely characterized, its corresponding theoretical model can accurately predict the experimentally measured structure of its incident wave field. A new and efficient method for completely characterizing a spherically focused transducer and its accompanying measurement system is described. Predicted responses that make use of this method are shown to correspond very well to measured responses for a number of different commercial transducers;The ultrasonic beam modeling work is divided into three parts. First, the problem of a planar piston ultrasonic transducer radiating at oblique incidence through a planar fluid-solid interface is studied, and two new types of beam models representing this problem are developed--a surface integral model and a boundary diffraction wave (BDW) paraxial model. The less restrictive surface integral model is used to test the validity of the BDW paraxial model, particularly in the near field and at different angles of incidence. Second, a new edge element method has been developed for numerically evaluating a variety of ultrasonic transducer beam models. This edge element model, which uses a local Fraunhofer approximation, is used to evaluate the wave fields of a focused probe in water and of a planar probe radiating at oblique incidence to a plane fluid-solid interface. Third and finally, a complete elastodynamic model of an ultrasonic angle beam shear wave transducer is presented. This model is evaluated by the edge element method, and the various transmitted mode contributions are studied and compared to more approximate models

Application of the boundary element method to elastic wave scattering problems in ultrasonic nondestructive evaluation

The boundary element method (BEM) is used to numerically simulate the interaction of ultrasonic waves with material defects such as voids, inclusions, and open cracks. The time harmonic formulation is in 3D and therefore allows flaws of arbitrary shape to be modeled. The BEM makes such problems feasible because the underlying boundary integral equation only requires a surface (2D) integration and difficulties associated with the seemingly infinite extent of the host domain are not encountered. The computer code utilized in this work is built upon recent advances in elastodynamic boundary element theory such as a scheme for self adjusting integration order and singular integration regularization. Incident fields may be taken as compressional or shear plane waves or predicted by an approximate Gauss-Hermite beam model. The code is highly optimized for voids and has been coupled with computer aided engineering packages for automated flaw shape definition and mesh generation. Subsequent graphical display of intermediate results supports model refinement and physical interpretation. Final results are typically cast in a nondestructive evaluation (NDE) context as either scattering amplitudes or flaw signals (via a measurement model based on a reciprocity integral). The near field is also predicted which allows for improved physical insight into the scattering process and the evaluation of certain modeling approximations;The accuracy of the BEM approach is first examined by comparing its predictions to those of other models for single, isolated scatterers. The comparisons are with the predictions of analytical solutions for spherical defects and with MOOT and T-matrix calculations for axisymmetric flaws. Experimental comparisons are also made for volumetric shapes with different characteristic dimensions in all three directions, since no other numerical approach has yet produced results of this type. Theoretical findings regarding the fictitious eigenfrequency difficulty are substantiated through the analytical solution of a fundamental elastodynamics problem and corresponding BEM studies;Given the confidence in the BEM technique engendered by these comparisons, it is then used to investigate the modeling of open , cracklike defects amenable to a volumetric formulation. The limits of applicability of approximate theories (e.g., quasistatic, Kirchhoff, and geometric theory of diffraction) are explored for elliptical cracks, from this basis. The problem of two interacting scatterers is then considered. Results from a fully implicit approach and from a more efficient hybrid scheme are compared with generalized Born and farfield approximate interaction theories

The design and optimisation of nanophotonic devices using the Finite Element Method

The aim of this thesis is to develop a technique which can be used in the reliable modelling, design and optimisation of practical suboptical wavelength sized photonic/plasmonic devices, which may involve arbitrary geometries on various scales. The technique involves the application of numerical electromagnetic simulation led by theoretical knowledge and physical insight to determine, design and optimise the operating mechanism of such devices. The work in this thesis contains a variety of problems/devices which involve arbitrary structures of different scales. This poses difficulties in both the fabrication and the modelling aspects of the design. The problems range in difficulty from those which can be simply and perfectly described via an analytical solution, to those which would be impractical to design using any other technique. The nature of the problems considered, i.e. the complicated geometry and the range of scales, necessitates the use of a flexible modelling technique. Finite Element Method (FEM) was found to be a valuable tool in the design and optimisation of the devices throughout this thesis, owing its success mainly to its versatility and flexible meshing abilities which allowed its operation in different length scales in an efficient manner. Three nanophotonic/plasmonic devices are considered in an effort to demonstrate the implementation and the application of the developed technique. The devices considered in this thesis demonstrate different challenges in the modelling and design while being of considerable interest in their own right as nanostructures for sensing and measurement. These devices are: A self-calibrated plasmon sensor, a plasmon resonator and an ultrahigh frequency optical acoustic surface wave detector. Whilst the first two devices are important as an application of plasmonics, the third device links the mechanical and optical processes together

Stability estimates for an inverse scattering problem at high frequencies

We consider an inverse scattering problem and its near-field approximation at high frequencies. We first prove, for both problems, Lipschitz stability results for determining the low-frequency component of the potential. Then we show that, in the case of a radial potential supported sufficiently near the boundary, infinite resolution can be achieved from measurements of the near-field operator in the monotone case

NURBS-based Isogeometric analysis of standard and phase reduction On-Surface Radiation Condition formulations for acoustic scattering

This paper is devoted to the NURBS-based Isogeometric analysis of the On-Surface Radiation Condition (OSRC) method for solving two-and three-dimensional time-harmonic acoustic scattering problems. In addition, a Phase Reduction of the OSRC formulation based on a plane wave ansatz is introduced. This leads to an efficient and accurate implementation of OSRC methods. Some numerical tests for two-and three-dimensional problems illustrate the proposed approach