On-surface Radiation Condition Research Articles

Iterative On Surface Radiation Conditions for fast and reliable single and multiple scattering analyses of arbitrarily shaped obstacles

In this paper we develop an iterative method to perform single and multiple scattering analyses of both convex and non-convex scatterers at very low computational cost and time. The computational method proposed will resolve the current inaccuracy of the On Surface Radiation Conditions (OSRC) in estimating the scattered field from non-convex obstacles and allows a fast and reliable single- and multiple-scattering analyses in reduced spatial dimensions. We demonstrate the reliability and efficiency of the proposed method in solving two- and three-dimensional time-harmonic acoustic scattering problems. Non-Uniform Rational B-Splines were used to represent the obstacle boundaries providing a seamless connection to geometrical representation. The proposed Iterative On Surface Radiation Conditions (ITOSRC) method tremendously reduces the meshing and analysis time and paves the way for shape and topology optimization of future devices relying on wave prpagation phenomena.

Method of virtual sources using on-surface radiation conditions for the Helmholtz equation

We develop a novel method of virtual sources to formulate boundary integral equations for exterior wave propagation problems. However, by contrast to classical boundary integral formulations, we displace the singularity of the Green’s function by a small distance h>0. As a result, the discretization can be performed on the actual physical boundary with continuous kernels so that any naive quadrature scheme can be used to approximate integral operators. Using on-surface radiation conditions, we combine single- and double-layer potential representations of the solution to arrive at a well-conditioned system upon discretization. The virtual displacement parameter h controls the conditioning of the discrete system. We provide mathematical guidance to choose h, in terms of the wavelength and mesh refinements, in order to strike a balance between accuracy and stability. Proof-of-concept implementations are presented, including piecewise linear and isogeometric element formulations in two- and three-dimensional settings. We observe exceptionally well-behaved spectra, and solve the corresponding systems using matrix-free GMRES iterations. The method is implemented for problems with known analytical solutions for comparison. We conclude that the proposed method leads to accurate solutions and good stability for a wide range of wavelengths and mesh refinements.

Fast and accurate boundary element methods for large-scale computational acoustics

The boundary element method (BEM) is a powerful algorithm to solve the Helmholtz equation for harmonic acoustic waves. The explicit use of Green’s functions avoids domain truncation of unbounded regions and accurately models wave propagation through homogeneous materials. Furthermore, fast multipole and hierarchical compression techniques provide efficient computations for dense matrix multiplications. However, the convergence of the iterative linear solvers deteriorates significantly when frequencies are high or materials have large contrasts in density or speed of sound. This talk presents several algorithmic improvements of the BEM. First, a preconditioner based on on-surface radiation conditions drastically reduces the iteration count of linear solvers at high frequencies. Second, anovel boundary integral formulation remains well-conditioned for high-contrast transmission problems. We used our fast and accurate BEM implementation to simulate focused ultrasound propagation in the human body, which can be translated to important biomedical applications such as the non-invasive treatment of liver cancer and neuromodulation of the brain. We validated the methodology within the benchmarking exercise of the International Transcranial Ultrasonic Stimulation Safety and Standards (ITRUSST) consortium. As a second application, we simulated the collective resonances of water-entrained arrays of air bubbles. Finally, we implemented all functionality in our open-source Python library, OptimUS.

Frequency-robust preconditioning of boundary integral equations for acoustic transmission

The scattering and transmission of harmonic acoustic waves at a penetrable material are commonly modelled by a set of Helmholtz equations. This system of partial differential equations can be rewritten into boundary integral equations defined at the surface of the objects and solved with the boundary element method (BEM). High frequencies or geometrical details require a fine surface mesh, which increases the number of degrees of freedom in the weak formulation. Then, matrix compression techniques need to be combined with iterative linear solvers to limit the computational footprint. Moreover, the convergence of the iterative linear solvers often depends on the frequency of the wave field and the objects' characteristic size. Here, the robust PMCHWT formulation is used to solve the acoustic transmission problem. An operator preconditioner based on on-surface radiation conditions (OSRC) is designed that yields frequency-robust convergence characteristics. Computational benchmarks compare the performance of this novel preconditioned formulation with other preconditioners and boundary integral formulations. The OSRC preconditioned PMCHWT formulation effectively simulates large-scale problems of engineering interest, such as focused ultrasound treatment of osteoid osteoma.

Standard and Phase Reduced Isogeometric On-Surface Radiation Conditions for acoustic scattering analyses

This paper is devoted to the Non-Uniform Rational B-Spline Isogeometric analysis of the On-Surface Radiation Condition (OSRC) method for solving two- and three-dimensional time-harmonic acoustic scattering problems. In addition, a Reduction of the OSRC formulation based on a plane wave ansatz is introduced. This development enhances the efficiency of the OSRC methods which has been demonstrated by studying a few two- and three-dimensional examples.

A Coupling between Integral Equations and On-Surface Radiation Conditions for Diffraction Problems by Non Convex Scatterers

The aim of this paper is to introduce an orignal coupling procedure between surface integral equation formulations and on-surface radiation condition (OSRC) methods for solving two-dimensional scattering problems for non convex structures. The key point is that the use of the OSRC introduces a sparse block in the surface operator representation of the wave field while the integral part leads to an improved accuracy of the OSRC method in the non convex part of the scattering structure. The procedure is given for both the Dirichlet and Neumann scattering problems. Some numerical simulations show the improvement induced by the coupling method.

Local on-surface radiation condition for multiple scattering of waves from convex obstacles

We propose a novel on-surface radiation condition to approximate the outgoing solution to the Helmholtz equation in the exterior of several impenetrable convex obstacles. Based on a local approximation of the Dirichlet-to-Neumann operator and a local formula for wave propagation, this new formulation simultaneously accounts for the outgoing behavior of the solution as well as the reflections arising from the multiple obstacles. The method involves tangential derivatives only, avoiding the use of integration over the surfaces of the obstacles. As a consequence, the method leads to sparse matrices and O(N) complexity. Numerical results are presented to illustrate the performance and limitations of the proposed formulation. Possible improvements and extensions are also discussed.

High order surface radiation conditions for time-harmonic waves in exterior domains

We formulate a new family of high order on-surface radiation conditions to approximate the outgoing solution to the Helmholtz equation in exterior domains. Motivated by the pseudo-differential expansion of the Dirichlet-to-Neumann operator developed by Antoine et al. (1999), we design a systematic procedure to apply pseudo-differential symbols of arbitrarily high order. Numerical results are presented to illustrate the performance of the proposed method for solving both the Dirichlet and the Neumann boundary value problems. Possible improvements and extensions are also discussed.

Formulation and accuracy of On-Surface Radiation Conditions for acoustic multiple scattering problems

The aim of this paper is (1) to address an extension of the On-Surface Radiation Condition (OSRC) method to multiple scattering by a cluster of convex obstacles and (2) to analyze the numerical accuracy of the formulation for some model problems. First, the acoustic scattering problem is considered and materials about integral equations and OSRCs are recalled. Then, an OSRC integral equation formulation is developed for the Dirichlet multiple scattering problems. Finally, an extensive numerical study of the method is proposed in the special case of the scattering by circular cylinders. Even if this situation is not completely general, it provides an assessment of the accuracy of the approximation, with respect to various parameters, including thanks to the choice of the OSRC operators.

A fast boundary element method for the scattering analysis of high-intensity focused ultrasound.

High-intensity focused ultrasound (HIFU) techniques are promising modalities for the non-invasive treatment of cancer. For HIFU therapies of, e.g., liver cancer, one of the main challenges is the accurate focusing of the acoustic field inside a ribcage. Computational methods can play an important role in the patient-specific planning of these transcostal HIFU treatments. This requires the accurate modeling of acoustic scattering at ribcages. The use of a boundary element method (BEM) is an effective approach for this purpose because only the boundaries of the ribs have to be discretized instead of the standard approach to model the entire volume around the ribcage. This paper combines fast algorithms that improve the efficiency of BEM specifically for the high-frequency range necessary for transcostal HIFU applications. That is, a Galerkin discretized Burton-Miller formulation is used in combination with preconditioning and matrix compression techniques. In particular, quick convergence is achieved with the operator preconditioner that has been designed with on-surface radiation conditions for the high-frequency approximation of the Neumann-to-Dirichlet map. Realistic computations of acoustic scattering at 1 MHz on a human ribcage model demonstrate the effectiveness of this dedicated BEM algorithm for HIFU scattering analysis.

Approximate local Dirichlet-to-Neumann map for three-dimensional time-harmonic elastic waves

It has been proven that the knowledge of an accurate approximation of the Dirichlet-to-Neumann (DtN) map is useful for a large range of applications in wave scattering problems. We are concerned in this paper with the construction of an approximate local DtN operator for time-harmonic elastic waves. The main contributions are the following. First, we derive exact operators using Fourier analysis in the case of an elastic half-space. These results are then extended to a general three-dimensional smooth closed surface by using a local tangent plane approximation. Next, a regularization step improves the accuracy of the approximate DtN operators and a localization process is proposed. Finally, a first application is presented in the context of the On-Surface Radiation Conditions method. The efficiency of the approach is investigated for various obstacle geometries at high frequencies.

Approximate local magnetic-to-electric surface operators for time-harmonic Maxwell's equations

The aim of this paper is to propose new local and accurate approximate magnetic-to-electric surface boundary operators for the three-dimensional time-harmonic Maxwell's equations. After their construction where their accuracy is improved through a regularization process, a localization of these operators and a full finite element approximation is introduced. Next, their numerical efficiency and accuracy is investigated in detail for different scatterers when these operators are used in the extreme situation of On-Surface Radiation Conditions methods.

On-surface radiation condition for multiple scattering of waves

The formulation of the on-surface radiation condition (OSRC) is extended to handle wave scattering problems in the presence of multiple obstacles. The new multiple-OSRC simultaneously accounts for the outgoing behavior of the wave fields, as well as, the multiple wave reflections between the obstacles. Like boundary integral equations (BIE), this method leads to a reduction in dimensionality (from volume to surface) of the discretization region. However, as opposed to BIE, the proposed technique leads to boundary integral equations with smooth kernels. Hence, these Fredholm integral equations can be handled accurately and robustly with standard numerical approaches without the need to remove singularities. Moreover, under weak scattering conditions, this approach renders a convergent iterative method which bypasses the need to solve single scattering problems at each iteration.Inherited from the original OSRC, the proposed multiple-OSRC is generally a crude approximate method. If accuracy is not satisfactory, this approach may serve as a good initial guess or as an inexpensive pre-conditioner for Krylov iterative solutions of BIE.

Numerical solution of scattering problems using pseudo‐differential impedance operators

AbstractDirect numerical solutions of scattering problems based on boundary‐integral equations are computationally costly at high frequencies. A numerical method is presented that efficiently computes accurate approximations to unknown surface quantities given known surface data (an approximate Dirichlet‐to‐Neumann map). The method is based on a pseudo‐differential impedance operator (PIO) numerically implemented using rational approximations. An example of a PIO is the so‐called on‐surface radiation condition (OSRC) method. For a convex obstacle, the method can be viewed as applying a parabolic equation directly on the surface of a scatterer. In contrast to past OSRCs, the use of rational approximations provides accuracy for wide scattering angles which is needed for grazing angles of incidence. Generalization to impedance operators for two‐dimensional acoustic scatterers with concave parts is presented. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On surface radiation conditions for high-frequency wave scattering

A new approximation of the logarithmic derivative of the Hankel function is derived and applied to high-frequency wave scattering. We re-derive the on surface radiation condition (OSRC) approximations that are well suited for a Dirichlet boundary in acoustics. These correspond to the Engquist–Majda absorbing boundary conditions. Inverse OSRC approximations are derived and they are used for Neumann boundary conditions. We obtain an implicit OSRC condition, where we need to solve a tridiagonal system. The OSRC approximations are well suited for moderate wave numbers. The approximation of the logarithmic derivative is also used for deriving a generalized physical optics approximation, both for Dirichlet and Neumann boundary conditions. We have obtained similar approximations in electromagnetics, for a perfect electric conductor. Numerical computations are done for different objects in 2D and 3D and for different wave numbers. The improvement over the standard physical optics is verified.

Generalized combined field integral equations for the iterative solution of the three-dimensional Maxwell equations

The construction of new second-kind Fredholm integral equations for the numerical solution of problems of high-frequency electromagnetic scattering by a perfect conductor is proposed. These formulations are characterized by some eigenvalue clusterings. They are especially well adapted to Krylov subspace iterative solvers. Their derivation is based on the incorporation of a sufficiently accurate approximation of the exact operator linking the Cauchy data of the scattering boundary-value problem to the classical integral relations. This operator is related to the concept of the On-Surface Radiation Condition (OSRC). These formulations can be considered as a natural generalization of the well-known Brakhage–Werner and combined field integral equations. The efficiency of the approach is established through an analytical and numerical study in the spherical case.

The performance of local absorbing boundary conditions for acoustic scattering from elliptical shapes

The computation of exterior wave problems at low wave numbers can become prohibitively expensive when higher circumferential modes are significant, yet local absorbing boundary conditions specified on simple shapes provide satisfactory performance for most scattering problems. An analysis of the effect of wave number and eccentricity on scattering problems, with local absorbing boundary conditions specified on an ellipse and a prolate spheroid as on-surface radiation conditions, shows that at moderate values of eccentricity, good performance extends to relatively low wave numbers. Excessive computational cost may be avoided for many practical applications.

An improved surface radiation condition for high-frequency acoustic scattering problems

A new artificial boundary condition for two-dimensional acoustic scattering problems is constructed in this paper. It is designed from the principal symbol of the Dirichlet-to-Neumann map so that both the propagating and evanescent fields are properly modeled. In the high-frequency regime, the new artificial boundary condition can be used as an on-surface radiation condition for efficient approximation of the wave field. The accuracy and efficiency of our method are demonstrated in numerical examples.

An improved On-Surface Radiation Condition for acoustic scattering problems in the high-frequency spectrum

This Note addresses the derivation of an improved On-Surface Radiation Condition for the numerical solution of the exterior Helmholtz equation at high-frequencies. This condition is built as an approximation of the Neumann-to-Dirichlet map by using a local regularization of its principal classical symbol in the gliding zone for modelling the creeping waves. The numerical simulation of this pseudodifferential operator is efficiently realized with a linear cost according to the dimension of the boundary element approximation space using suitable complex Padé approximants. A numerical example is provided. To cite this article: X. Antoine et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).

Alternative integral equations for the iterative solution of acoustic scattering problems

This paper addresses the first step of the derivation of well-conditioned integral formulations for the iterative solution of exterior acoustic boundary-value problems. These new formulations are designed to be implemented in a fast multipole method coupled to a Krylov subspace iterative algorithm. Their construction is based on the on-surface radiation condition formalism. Both theoretical developments and numerical aspects are examined in detail. This approach can be viewed as a generalization of the usual Burton–Miller integral equations.