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On the Numerical Solution of a Hypersingular Integral Equation for Elastic Scattering from a Planar Crack
Abstract
We describe a fully discrete quadrature method for numerical solution of a hypersingular integral equation of the first kind which describes the scattering of time-harmonic elastic waves by a cavity crack. We establish convergence of the method and prove error estimates in a Hölder space setting. Numerical examples illustrate the convergence results.
... On account of the vanishing property of the sigmoid transforms at the breakpoints T j , the weighted density φ w are 2π periodic functions whose regularity can be control by the value of p in the graded mesh used. As such, we apply the Alpert quadrature described above to produce high-order Nyström discretizations of the weighted single layer formulations (14). Finally, in order to avoid the evaluation of the density function at corner points, we employ the equispaced mesh t j = (j −1/2) 2π N , 1 ≤ j ≤ N in the Alpert Nyström method. ...
... Finally, in order to avoid the evaluation of the density function at corner points, we employ the equispaced mesh t j = (j −1/2) 2π N , 1 ≤ j ≤ N in the Alpert Nyström method. We present in Table 1 the errors in the near field achieved by the Alpert discretization of the weighted formulation (14) for real wavenumbers with parameters a = 2 and m = 3 for various Lipschitz and open arc Γ configurations. Specifically, we considered the cases when Table 1: Errors in the near field and estimated orders of convergence corresponding to the Alpert discretization of the weighted formulation (14) for real wavenumber k = 8 and plane wave normal incidence. ...
... We present in Table 1 the errors in the near field achieved by the Alpert discretization of the weighted formulation (14) for real wavenumbers with parameters a = 2 and m = 3 for various Lipschitz and open arc Γ configurations. Specifically, we considered the cases when Table 1: Errors in the near field and estimated orders of convergence corresponding to the Alpert discretization of the weighted formulation (14) for real wavenumber k = 8 and plane wave normal incidence. We used the following Alpert quadrature parameters: σ = 4 in the sigmoid transform, and respectively a = 2, m = 3. [16] (both domains (i)-(ii) have diameters equal to 2), (iii) the strip connecting (−1, 0) to (1, 0), and (iv) the V-shaped strip connecting (−1, 1) to (0, 0) and then to (1, 1). ...
We investigate high-order Convolution Quadratures methods for the solution of the wave equation in unbounded domains in two dimensions that rely on Nystr\"om discretizations for the solution of the ensemble of associated Laplace domain modified Helmholtz problems. We consider two classes of CQ discretizations, one based on linear multistep methods and the other based on Runge-Kutta methods, in conjunction with Nystr\"om discretizations based on Alpert and QBX quadratures of Boundary Integral Equation (BIE) formulations of the Laplace domain Helmholtz problems with complex wavenumbers. We present a variety of accuracy tests that showcase the high-order in time convergence (up to and including fifth order) that the Nystr\"om CQ discretizations are capable of delivering for a variety of two dimensional scatterers and types of boundary conditions.
... All the above integrals are well defined and in particular the operator S j for x ∈ is weakly singular, the operators K j , L j are singular and N j admits a hypersingular kernel. From the asymptotic behaviour of the Hankel functions we can compute also the far-field patterns of the single-(4) and double-layer potential (5) [6,20] ...
... Using this representation, applying the jump relations (6) we see that the fields given by (9) satisfy the boundary conditions (2) provided the densities ϕ, ψ satisfy the system of integral equations ...
... denotes the fundamental solution of the static (ω = 0) Navier equation [6]. Then, we obtain the forms ...
In this work we consider the inverse elastic scattering problem by an inclusion in two dimensions. The elastic inclusion is placed in an isotropic homogeneous elastic medium. The inverse problem, using the third Betti’s formula (direct method), is equivalent to a system of four integral equations that are non linear with respect to the unknown boundary. Two equations are on the boundary and two on the unit circle where the far-field patterns of the scattered waves lie. We solve iteratively the system of integral equations by linearising only the far-field equations. Numerical results are presented that illustrate the feasibility of the proposed method.
... All the above integrals are well defined and in particular the operator S j for x ∈ Γ is weakly singular, the operators K j , L j are singular and N j admits a hypersingular kernel. From the asymptotic behaviour of the Hankel functions we can compute also the far-field patterns of the single-(5) and double-layer potential (6) [7,21] ...
... For smooth kernels we use the trapezoidal rule. The exact forms of the parametrized kernels are presented in [5,7]. Thus, here we only briefly present the form of the kernel of the operator τ i N i − τ e N e appearing in (12) and in A 2 . ...
... The first term is weakly singular and the second one preserves the hypersingularity. The advantage, of this decomposition, is that the second term coming from the static case is easier to handle by a Maue-type expression [7], although it is not needed here. The integral operator with kernel the second term can be written as [5,Equation 2.6] (N ...
In this work we consider the inverse elastic scattering problem by an inclusion in two dimensions. The elastic inclusion is placed in an isotropic homogeneous elastic medium. The inverse problem, using the third Betti's formula (direct method), is equivalent to a system of four integral equations that are non linear with respect to the unknown boundary. Two equations are on the boundary and two on the unit circle where the far-field patterns of the scattered waves lie. We solve iteratively the system of integral equations by linearising only the far-field equations. Numerical results are presented that illustrate the feasibility of the proposed method.
... In this work, we extend the spectral Galerkin method [17] to the more challenging case of elastic wave scattering by multiple open-arcs. Analogous to [7], an adequate Maue's representation formula [4,30] for elastodynamic problems is used to simplify the discretization of the hyper-singular BIE. Yet, and unlike [7], the corresponding variational formulation of the hyper-singular BIE avoids the treatment of tangential derivatives of weakly-singular operators. ...
... Numerical schemes for BIEs of open arc problems have been extensively studied for Laplace/Helmholtz [2,16,17,19,25,27], elastostatic/elastodynamic [4,7,14,27] and Maxwell equations [12]. Generally, their study requires handling the following three groups of questions: ...
We study the elastic time-harmonic wave scattering problems on unbounded domains with boundaries composed of finite collections of disjoints finite open arcs (or cracks) in two dimensions. Specifically, we present a fast spectral Galerkin method for solving the associated weakly- and hyper-singular boundary integral equations (BIEs) arising from Dirichlet and Neumann boundary conditions, respectively. Discretization bases of the resulting BIEs employ weighted Chebyshev polynomials that capture the solutions' edge behavior. We show that these bases guarantee exponential convergence in the polynomial degree when assuming analyticity of sources and arcs geometries. Numerical examples demonstrate the accuracy and robustness of the proposed method with respect to number of arcs and wavenumber.
... Thus, the density function ϕ has to vanish at the end points of the arc Γ, which has to be explicitly accounted for in the numerical scheme. Following the procedure in [14], we assume a parametrization of Γ of the form γ : [−1, 1] → R 2 , followed by the cosine change of variables and an odd periodic extension of the density and the parametrized integral equations on the interval [0, 2π]. Finally, the ensuing discretized integral equations are projected back onto the grid points on the interval (0, π). ...
... The smoothness of the incident fields also manifests itself in the sparse frequency content of the boundary data in the Laplace domain [7], e.g. most boundary data defined in equations (12) and respectively (14) are small in the infinity norm. As a result, we follow the common thresholding practice in CQ and we solve only those Laplace domain Helmholtz equations whose boundary dataĝ ℓ in equations (12) The Laplace domain problems that enter the CQ methods were solved using the weighted single layer formulation whose associated BIOs were discretized using both Alpert quadratures and QBX (based on the weighted unknown defined in equation (28) and the Fejér quadratures (29)) of appropriately high order. ...
We investigate high-order Convolution Quadratures methods for the solution of the wave equation in unbounded domains in two dimensions that rely on Nyström discretizations for the solution of the ensemble of associated Laplace domain modified Helmholtz problems. We consider two classes of CQ discretizations, one based on linear multistep methods and the other based on Runge-Kutta methods, in conjunction with Nyström discretizations based on Alpert and QBX quadratures of Boundary Integral Equation (BIE) formulations of the Laplace domain Helmholtz problems with complex wavenumbers. We present a variety of accuracy tests that showcase the high-order in time convergence (up to and including fifth order) that the Nyström CQ discretizations are capable of delivering for a variety of two dimensional scatterers and types of boundary conditions.
... In this work, the classical Nyström method, which has been widely used for the acoustic and elastic problems [17,18,22], is employed for the numerical implementation of the proposed RBIE. As applying the method, we encounter the challenge of accurate evaluation of the strongly-singular and hyper-singular integrals. ...
... As a result, the Nyström method allows us to evaluate the weakly-singular integrals with spectral accuracy for analytic surfaces, and to calculate the tangential-derivative of a given function via fast Fourier transform (FFT) in GMRES iterations. Numerical tests show that the proposed scheme demonstrate a simpler and more efficient performance than some alternative numerical treatments [18,22]. ...
This paper is concerned with the boundary integral equation method for solving the exterior Neumann boundary value problem of dynamic poroelasticity in two dimensions. The main contribution of this work consists of two aspescts: the proposal of a novel regularized boundary integral equation, and the presentation of new regularized formulations of the strongly-singular and hyper-singular boundary integral operators. Firstly, turning to the spectral properties of the double-layer operator and the corresponding Calder\'{o}n relation of the poroelasticity, we propose the novel low-GMRES-iteration integral equation whose eigenvalues are bounded away from zero and infinity. Secondly, with the help of the G\"{u}nter derivatives, we reformulate the strongly-singular and hyper-singular integral operators into combinations of the weakly-singular operators and the tangential derivatives. The accuracy and efficiency of the proposed methodology are demonstrated through several numerical examples.
... The approximation of hypersingular integrals plays an important role in numerical methods for various integral equations arising in acoustics [27], electromagnetics [20,26], heat conduction [18]. Besides, equations with hypersingular integrals are also used in stress calculation [3,9], fracture mechanics [1,2,4,8] and wave scattering [2,11,12]. A special attention has been paid to quadrature formulas for hypersingular integrals, including Gaussian quadratures [10,15,17,21], composite Newton-Cotes rules [6,16,23,24] and transformation methods [5,7]. ...
... The approximation of hypersingular integrals plays an important role in numerical methods for various integral equations arising in acoustics [27], electromagnetics [20,26], heat conduction [18]. Besides, equations with hypersingular integrals are also used in stress calculation [3,9], fracture mechanics [1,2,4,8] and wave scattering [2,11,12]. A special attention has been paid to quadrature formulas for hypersingular integrals, including Gaussian quadratures [10,15,17,21], composite Newton-Cotes rules [6,16,23,24] and transformation methods [5,7]. ...
... We derive an explicit expression for it. We note first that for a function f and a matrix G there is the product rule (see [9,Sect. 2]) ...
... Following [9], we also have ...
A boundary integral based method for the stable reconstruction of missing boundary data is presented for the governing hyperbolic equation of elastodynamics in annular planar domains. Cauchy data in the form of the solution and traction is reconstructed on the inner boundary curve from the similar data given on the outer boundary. The ill-posed data reconstruction problem is reformulated as a sequence of boundary integral equations using the Laguerre transform with respect to time and employing a single-layer approach for the stationary problem. Singularities of the involved kernels in the integrals are analysed and made explicit, and standard quadrature rules are used for discretisation. Tikhonov regularization is employed for the stable solution of the obtained linear system. Numerical results are included showing that the outlined approach can be turned into a practical working method for finding the missing data.
... We derive an explicit expression for it. We note first that for a function f and a matrix G there is the product rule (see [9,Sect. 2]) ...
... Following [9], we also have ...
A boundary integral based method for the stable reconstruction of missing boundary data is presented for the governing hyperbolic equation of elastodynamics in annular planar domains. Cauchy data in the form of the solution and traction is reconstructed on the inner boundary curve from the similar data given on the outer boundary. The ill-posed data reconstruction problem is reformulated as a sequence of boundary integral equations using the Laguerre transform with respect to time and employing a single-layer approach for the stationary problem. Singularities of the involved kernels in the integrals are analysed and made explicit, and standard quadrature rules are used for discretisation. Tikhonov regularization is employed for the stable solution of the obtained linear system. Numerical results are included showing that the outlined approach can be turned into a practical working method for finding the missing data.
... Then K − K 0 and K − K 0 are compact operators. In fact, the singular items of the differences of this two operators are the product of a logarithmic function and analytic functions [8], and thus the kernels are weakly singular, which results in the compactness [23,Theorem 8.20]. ...
... Two differences need to be noted. Firstly, the boundary integral equation (11) is more complicated, which involves hypersingular integral operator [8]. Besides, the test function on the right side of the far field equation (47) should be adopted the combined far field patterns of elastic point source Γ(x, z) · p and the stress vector [T y Γ(x, y)] · p of the elastic point source, which are given by (20) ∼ (23). ...
This paper is concerned with the scattering problems of a crack with Dirichlet or mixed impedance boundary conditions in two dimensional isotropic and linearized elasticity. The well posedness of the direct scattering problems for both situations are studied by the boundary integral equation method. The inverse scattering problems we are dealing with are the shape reconstruction of the crack from the knowledge of far field patterns due to the incident plane compressional and shear waves. We aim at extending the well known factorization method to crack determination in inverse elastic scattering, although it has been proved valid in acoustic and electromagnetic scattering, electrical impedance tomography and so on. The numerical examples are presented to illustrate the feasibility of this method.
... The convergence and error analysis for this quadrature method can be established on the basis of the collective compact operator theory (see [4]) or on the basis of some estimate of the trigonometric interpolation in Hölder spaces (see, for example, [5,21]). In the latter case, this analysis is based on the estimate P n k µ − µ m,α c ln n k n ℓ−m+β−α k µ ℓ,β , k = 1, 2, (3.13) for the trigonometric interpolation which is valid for 0 m ℓ, 0 < α β < 1, and some constant c depending only on m, ℓ, α and β. ...
... We assume that the cut Γ 1 and the closed boundary part Γ 2 have the parametric representation (3.5). Using the parametrization in (3.21) and employing the cosine-substitution in combination with some further transformations (for details, see [5,21]), we obtain the following system of integral equations: ...
We consider a Cauchy problem for the Laplace equation in a bounded region containing a cut, where the region is formed by removing a sufficiently smooth arc (the cut) from a bounded simply connected domain D. The aim is to reconstruct the solution on the cut from the values of the solution and its normal derivative on the boundary of the domain D. We propose an alternating iterative method which involves solving direct mixed problems for the Laplace operator in the same region. These mixed problems have either a Dirichlet or a Neumann boundary condition imposed on the cut and are solved by a potential approach. Each of these mixed problems is reduced to a system of integral equations of the first kind with logarithmic and hypersingular kernels and at most a square root singularity in the densities at the endpoints of the cut. The full discretization of the direct problems is realized by a trigonometric quadrature method which has super-algebraic convergence. The numerical examples presented illustrate the feasibility of the proposed method.
... The exact form of the kernel of (Dϕ) can be found in [6]. For the boundary value problem (1), (2), and (4) we represent the solution in the form ...
... For the numerical calculation of the integral operators with smooth kernels we use trapezoidal rule and for those with singular and hypersingular kernels we replace the trapezoidal rule by convergent quadrature rules based on trigonometric interpolation, as suggested in [6,23] 1 2π ...
We consider the inverse scattering problem for the shape determination of a rigid scatterer or a cavity in a homogeneous and isotropic elastic medium. Both problems are formulated in R2 for time-harmonic fields and longitudinal or transversal incident plane waves. Representation of the scattered field as a single- or a double-layer potential, equivalently, leads to a system of two nonlinear integral equations for the density and the parametrization of the boundary. A detailed numerical implementation is presented for computing the corresponding solutions of both systems and numerical reconstructions are given to show the effectiveness of the method.
... Beginning with [7] in a series of papers [3,9,11] the direct and, to some extent, the inverse scattering problem for time-harmonic acoustic, electromagnetic and elastic waves from a crack in two dimensions has been considered for Dirichlet and Neumann boundary conditions by an integral equation approach in classical H older space settings. All these papers give a full treatment of the corresponding problems including the existence and uniqueness analysis of the resulting integral equations and their numerical solution via fully discrete collocation methods based on trigonometric interpolation. ...
... This ÿnally implies existence (and uniqueness) of a solution to the impedance crack problem. As opposed to the limiting case = 0 (see [3,11]), where the corresponding equation allows an odd extension to all of R, in Eq. (3.5) some terms have an odd extension whereas others have an even extension, for example if is constant. Hence, in agreement with the general results on the end point singularities (see [5,13]), we cannot improve on the regularity of the solution to (3.5). ...
For the scattering problem for time-harmonic waves from an impedance crack in two dimensions, we give a uniqueness and existence analysis via a combined single- and double-layer potential approach in a Hölder space setting leading to a system of integral equations that contains a hypersingular operator. For its numerical solution we describe a fully discrete collocation method based on trigonometric interpolation and interpolatory quadrature rules including a convergence analysis and numerical examples.
... In this work, we perform model order reduction for the elastic scattering problem by multiple shape-parametric open arcs. Firstly, as in [1,2,7,30,32,43], we cast the original boundary value problem as an equivalent system of boundary integral equations (BIEs) posed on the collection of open arcs. Then, following the approach presented in [23], we propose and thoroughly analyze a reduced basis method for this shape-parametric formulation. ...
We consider the elastic scattering problem by multiple disjoint arcs or cracks in two spatial dimensions. A key aspect of our approach lies in the parametric description of each arc's shape, which is controlled by a potentially high-dimensional, possibly countably infinite, set of parameters. We are interested in the efficient approximation of the parameter-to-solution map employing model order reduction techniques, specifically the reduced basis method. Initially, we utilize boundary potentials to transform the boundary value problem, originally posed in an unbounded domain, into a system of boundary integral equations set on the parametrically defined open arcs. Our aim is to construct a rapid surrogate for solving this problem. To achieve this, we adopt the two-phase paradigm of the reduced basis method. In the offline phase, we compute solutions for this problem under the assumption of complete decoupling among arcs for various shapes. Leveraging these high-fidelity solutions and Proper Orthogonal Decomposition (POD), we construct a reduced-order basis tailored to the single arc problem. Subsequently, in the online phase, when computing solutions for the multiple arc problem with a new parametric input, we utilize the aforementioned basis for each individual arc. To expedite the offline phase, we employ a modified version of the Empirical Interpolation Method (EIM) to compute a precise and cost-effective affine representation of the interaction terms between arcs. Finally, we present a series of numerical experiments demonstrating the advantages of our proposed method in terms of both accuracy and computational efficiency.
... Integral equations have wide-range of applications in scientific, engineering domains and various real-life problems encountered in our daily life. In particular, hypersingular integral equations (HSIEs) of the first kind arises in real-world problems like aerodynamics, electrodynamics, crack problems in fracture mechanics, potential theory, acoustics and wave scattering (see [1][2][3][4][5][6][7][8][9][10][11]). In order to properly analyze such models, it is crucial to obtain the solution of HSIEs. ...
In this article, we propose Chebyshev spectral projection methods to solve the hypersingular integral equation of first kind. The presence of strong singularity in Hadamard sense in the first part of the integral equation makes it challenging to get superconvergence results. To overcome this, we transform the first kind hypersingular integral equation into a second kind integral equation. This is achieved by defining a bounded inverse of the hypersingular integral operator in some suitable Hilbert space. Using iterated Chebyshev spectral Galerkin method on the equivalent second kind integral equation, we obtain improved convergence of O(N−2r), where N is the highest degree of Chebyshev polynomials employed in the approximation space and r is the smoothness of the solution. Further, using commutativity of projection operator and inverse of the hypersingular integral operator, we are able to obtain superconvergence of O(N−3r) and O(N−4r), by Chebyshev spectral multi-Galerkin method (CSMGM) and iterated CSMGM, respectively. Finally, numerical examples are presented to verify our theoretical results.
... We noted that these matrices N (k) j (j = 1, 2, k = 0, 1, 2) are infinitely differentiable in R 2 × R 2 ; see [14]. Once again using the the power series for the Bessel and Hankel functions to obtain the decomposition from ...
This paper is concerned with the inverse elastic scattering problem to determine the shape and location of an elastic cavity. By establishing a one-to-one correspondence between the Herglotz wave function and its kernel, we introduce the far-field operator which is crucial in the factorization method. We present a theoretical factorization of the far-field operator and rigorously prove the properties of its associated operators involved in the factorization. Unlike the Dirichlet problem where the boundary integral operator of the single-layer potential involved in the factorization of the far-field operator is weakly singular, the boundary integral operator of the conormal derivative of the double-layer potential involved in the factorization of the far-field operator with Neumann boundary conditions is hypersingular, which forces us to prove that this operator is isomorphic using Fredholm's theorem. Meanwhile, we present theoretical analyses of the factorization method for various illumination and measurement cases, including compression-wave illumination and compression-wave measurement, shear-wave illumination and shear-wave measurement, and full-wave illumination and full-wave measurement. In addition, we also consider the limited aperture problem and provide a rigorous theoretical analysis of the factorization method in this case. Numerous numerical experiments are carried out to demonstrate the effectiveness of the proposed method, and to analyze the influence of various factors, such as polarization direction, frequency, wavenumber, and multi-scale scatterers on the reconstructed results.
... To illustrate our findings, in Sect. 6 we apply these results to Helmholtz and time-harmonic elastic wave scattering by showing that the structural assumptions on the corresponding BIOs are fulfilled. Lastly, Sect. ...
We establish shape holomorphy results for general weakly- and hyper-singular boundary integral operators arising from second-order partial differential equations in unbounded two-dimensional domains with multiple finite-length open arcs. After recasting the corresponding boundary value problems as boundary integral equations, we prove that their solutions depend holomorphically upon perturbations of the arcs’ parametrizations. These results are key to prove the shape (domain) holomorphy of domain-to-solution maps associated to boundary integral equations appearing in uncertainty quantification, inverse problems and deep learning, to name a few applications.
... Given the lack of Lipchitz regularity of the domain, traditional variational formulations cannot be applied, and hence, the existence of a holomorphic extension of the domain-to-solution map does not follow from volume-based formulations, for instance, as in [10]. Consequently, we recast the volume problems as boundary integral equations (BIEs) posed on the collection of open arcs, as in [3,42,40,41,4,21,24,2,26,6], and then we extend the analysis of [18,17] to the corresponding BIEs. ...
We establish shape holomorphy results for general weakly- and hyper-singular boundary integral operators arising from second-order partial differential equations in unbounded two-dimensional domains with multiple finite-length open arcs. After recasting the corresponding boundary value problems as boundary integral equations, we prove that their solutions depend holomorphically upon perturbations of the arcs' parametrizations. These results are key to prove the shape (domain) holomorphy of the domain-to-solution maps for the associated boundary integral equations with applications in uncertainty quantification, inverse problems and deep learning.
... In the case of a discrete system like (14), the transposed matrix A acts as the adjoint operator of the matrix A. Therefore, the regularization for (14) consists in findingμ α from the system ...
The elastostatic Cauchy problem of fracture mechanics is studied in a two-dimensional bounded domain containing a crack. Given Cauchy data on the boundary of the domain, the displacement and normal stress (traction) are reconstructed on the crack. The reconstruction is done by reducing the original problem, via the elastostatic potential, to a system of integral equations to be solved for densities over the boundary of the domain and the crack. Discretization is carried out by the Nyström method using quadrature formulas adjusted for singularities manifesting at the endpoints of the crack. Tikhonov regularization is applied for the stable solution of the discretized system. The results of numerical experiments for different input data and parameters are given showing that relevant physical quantities on the crack can be stably reconstructed.
... While the regularized formula for static elasticity is easy to find in the literature [19], the elastic case is much more involved. For instance, in [7] the authors opt for a subtraction technique, where the regularized static hypersingular operator is used and then the difference between the time-harmonic and the static operators is prepared for discretization. Here we will use a formula due to Frangi and Novati [13], which we fully develop so that our results can be easily replicated. ...
In this paper we present a full discretization of the layer potentials and
boundary integral operators for the elastic wave equation on a parametrizable
smooth closed curve in the plane. The method can be understood as a
non-conforming Petrov-Galerkin discretization, with a very precise choice of
testing functions by symmetrically combining elements on two staggered grids,
and using a look-around quadrature formula. Unlike in the acoustic counterpart
of this work, the kernel of the elastic double layer operator includes a
periodic Hilbert transform that requires a particular choice of the mixing
parameters. We give mathematical justification of this fact. Finally, we test
the method on some frequency domain and time domain problems, and demonstrate
its applicability on smooth open arcs.
... In this section we will give some results of numerical experiments for identifying cracks based on the theory developed in Section 3. The far field data we use are synthetic, but corrupted by random noise added point-wise to the measurements. The forward problem is numerically solved using a quadrature method for the first kind hypersingular integral equation (2.10) as developed by Kress and co-authors in [7], [15] and [16]. The computed far field data is obtained as a trigonometric series u ∞ = N n=−N u ∞,n exp(inθ). ...
We consider the inverse scattering problem of determining the shape and the material properties of a thin dielectric infinite cylinder having an open arc as cross section from knowledge of the TM-polarized scattered electromagnetic field at a fixed frequency. We investigate two reconstruc- tion approaches, namely the linear sampling method and the reciprocity gap functional method, using far field or near field data, respectively. Numerical examples are given showing the efficaciousness of our algorithms.
Quadrature by Expansion (QBX) is a quadrature method for approximating the value of the singular integrals encountered in the evaluation of layer potentials. It exploits the smoothness of the layer potential by forming locally-valid expansion which are then evaluated to compute the near or on-surface value of the integral. Recent work towards coupling of a Fast Multipole Method (FMM) to QBX yielded a first step towards the rapid evaluation of such integrals (and the solution of related integral equations), albeit with only empirically understood error behavior. In this paper, we improve upon this approach with a modified algorithm for which we give a comprehensive analysis of error and cost in the case of the Laplace equation in two dimensions. For the same levels of (user-specified) accuracy, the new algorithm empirically has cost-per-accuracy comparable to prior approaches. We provide experimental results to demonstrate scalability and numerical accuracy.
Helmholtz decompositions of elastic fields is a common approach for the solution of Navier scattering problems. Used in the context of boundary integral equations (BIE), this approach affords solutions of Navier problems via the simpler Helmholtz boundary integral operators (BIOs). Approximations of Helmholtz Dirichlet-to-Neumann (DtN) can be employed within a regularizing combined field strategy to deliver BIE formulations of the second kind for the solution of Navier scattering problems in two dimensions with Dirichlet boundary conditions, at least in the case of smooth boundaries. Unlike the case of scattering and transmission Helmholtz problems, the approximations of the DtN maps we use in the Helmholtz decomposition BIE in the Navier case require incorporation of lower order terms in their pseudodifferential asymptotic expansions. The presence of these lower order terms in the Navier regularized BIE formulations complicates the stability analysis of their Nyström discretizations in the framework of global trigonometric interpolation and the Kussmaul–Martensen kernel singularity splitting strategy. The main difficulty stems from compositions of pseudodifferential operators of opposite orders, whose Nyström discretization must be performed with care via pseudodifferential expansions beyond the principal symbol. The error analysis is significantly simpler in the case of arclength boundary parametrizations and considerably more involved in the case of general smooth parametrizations that are typically encountered in the description of one-dimensional closed curves.
We introduce and analyse various regularized combined field integral equations (CFIER) formulations of time-harmonic Navier equations in media with piece-wise constant material properties. These formulations can be derived systematically starting from suitable coercive approximations of Dirichlet-to-Neumann operators (DtN), and we present a periodic pseudodifferential calculus framework within which the well posedness of CIER formulations can be established. We also use the DtN approximations to derive and analyse OS methods for the solution of elastodynamics transmission problems. The pseudodifferential calculus we develop in this paper relies on careful singularity splittings of the kernels of Navier boundary integral operators, which is also the basis of high-order Nyström quadratures for their discretizations. Based on these high-order discretizations we investigate the rate of convergence of iterative solvers applied to CFIER and OS formulations of scattering and transmission problems. We present a variety of numerical results that illustrate that the CFIER methodology leads to important computational savings over the classical CFIE one, whenever iterative solvers are used for the solution of the ensuing discretized boundary integral equations. Finally, we show that the OS methods are competitive in the high-frequency high-contrast regime.
This paper is concerned with the extension of the factorization method to the inverse elastic scattering problem by a mixed scatterer which is the union of a penetrable obstacle and an impenetrable object. Having established the well-posedness of the direct problem by the boundary integral equation method, we study the factorization method to reconstruct the shape of the mixed obstacle, that is, recovering the shape of the mixed scatterer from a knowledge of the far-field pattern at a fixed frequency. Some numerical examples are also presented to demonstrate the feasibility and effectiveness of the factorization method.
This paper is concerned with the inverse scattering problem of time-harmonic elastic waves by a mixed-type scatterer, which is given as the union of an impenetrable obstacle and a crack. We develop the modified factorization method to determine the shape of the mixed-type scatterer from the far field data. However, the factorization of the far field operator F is related to the boundary integral matrix operator A, which is obtained in the study of the direct scattering problem. So, in the first part, we show the well posedness of the direct scattering problem by the boundary integral equation method. Some numerical examples are presented at the end of the paper to demonstrate the feasibility and effectiveness of the inverse algorithm.
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.
This paper presents an accelerated quadrature scheme for the evaluation of layer potentials in three dimensions. Our scheme combines a generic, high order quadrature method for singular kernels called Quadrature by Expansion (QBX) with a modified version of the Fast Multipole Method (FMM). Our scheme extends a recently developed formulation of the FMM for QBX in two dimensions, which, in that setting, achieves mathematically rigorous error and running time bounds. In addition to generalization to three dimensions, we highlight some algorithmic and mathematical opportunities for improved performance and stability. Lastly, we give numerical evidence supporting the accuracy, performance, and scalability of the algorithm through a series of experiments involving the Laplace and Helmholtz equations.
Quadrature by Expansion (QBX) is a quadrature method for approximating the value of the singular integrals encountered in the evaluation of layer potentials. It exploits the smoothness of the layer potential by forming locally-valid expansion which are then evaluated to compute the near or on-surface value of the integral. Recent work towards coupling of a Fast Multipole Method (FMM) to QBX yielded a first step towards the rapid evaluation of such integrals (and the solution of related integral equations), albeit with only empirically understood error behavior. In this paper, we improve upon this approach with a modified algorithm for which we give a comprehensive analysis of error and cost in the case of the Laplace equation in two dimensions. For the same levels of (user-specified) accuracy, the new algorithm empirically has cost-per-accuracy comparable to prior approaches. We provide experimental results to demonstrate scalability and numerical accuracy.
This paper describes a collocation method for solving numerically a singular integral equation with Cauchy and Volterra operators, associated with a proper constraint condition. The numerical method is based on the transformation of the given integral problem into a hypersingular integral equation and then applying a collocation method to solve the latter equation. Convergence of the resulting method is then discussed, and optimal convergence rates for the collocation and discrete collocation methods are given in suitable weighted Sobolev spaces. Numerical examples are solved using the proposed numerical technique.
In this paper the problem of determining a screen in an isotropic and homogeneous thermoelastic medium at low frequencies is considered. We formulate the direct problem for the planar screen in the thermoelastic medium and present an equivalent model for the problem under consideration at low-frequencies based on an non - homogeneous formulation via appropriate Dirac measures. We prove that the corresponding inverse problem of reconstructing the planar screen for two important cases: the thermal stress dislocation and the thermal displacement dislocation from boundary measurements has a unique solution. Finally, we present a reconstruction method for the above cases based on a proper use of certain vector test functions and the application of the two-sided Laplace transform.
This paper presents quadrature formulae for hypersingular integrals
, where a < t < b and 0 < α ≤ 1. The asymptotic error estimates obtained by Euler–Maclaurin expansions show that, if g(x) is 2m times differentiable on [a,b], the order of convergence is O(h
2μ
) for α = 1 and O(h
2μ − α
) for 0 < α < 1, where μ is a positive integer determined by the integrand. The advantages of these formulae are as follows: (1) using the formulae to evaluate hypersingular integrals is straightforward without need of calculating any weight; (2) the quadratures only involve g(x), but not its derivatives, which implies these formulae can be easily applied for solving corresponding hypersingular boundary integral equations in that g(x) is unknown; (3) more precise quadratures can be obtained by the Richardson extrapolation. Numerical experiments in this paper verify the theoretical analysis.
Integral equation methods for the solution of partial differential equations,
when coupled with suitable fast algorithms, yield geometrically flexible,
asymptotically optimal and well-conditioned schemes in either interior or
exterior domains. The practical application of these methods, however, requires
the accurate evaluation of boundary integrals with singular, weakly singular or
nearly singular kernels. Historically, these issues have been handled either by
low-order product integration rules (computed semi-analytically), by
singularity subtraction/cancellation, by kernel regularization and asymptotic
analysis, or by the construction of special purpose "generalized Gaussian
quadrature" rules. In this paper, we present a systematic, high-order approach
that works for any singularity (including hypersingular kernels), based only on
the assumption that the field induced by the integral operator is locally
smooth when restricted to either the interior or the exterior. Discontinuities
in the field across the boundary are permitted. The scheme, denoted QBX
(quadrature by expansion), is easy to implement and compatible with fast
hierarchical algorithms such as the fast multipole method. We include accuracy
tests for a variety of integral operators in two dimensions on smooth and
corner domains.
The initial-boundary-value problem for the heat equation in the case of a toroidal surface with Dirichlet boundary conditions is considered. This problem is reduced to a sequence of elleptic boundary-value problems by a Laguerre transformation. The special integral representation leads to boundary-integral equations of the first kind and the toroidal surface gives one-dimensional integral equations with a logarithmic singularity. The numerical solution is realized by a trigonometric quadrature method in cases of open or closed smooth boundaries. The results of some numerical experiments are presented.
Three preconditioners for solving boundary hypersingular integral equations of the first kind are proposed. The first and the second preconditioners are based on circulant operators. The third is a hybrid of iterative substructuring techniques and circulant operators. Although this class of preconditioners are composition of circulant operators but we have shown that the condition number of the third preconditioner dose not exceed the condition number of two preconditioners. Numerical results are included.
In this paper, we present an algorithm for numerical solution of some type of the inclusion problem in planar linear elastostatics. This problem arises on numerical solution of an inverse problem that contains in the identification of interfaces or inclusions by elastic boundary measurements. The algorithm is based on the boundary integral equation method. By combination of the single- and double-layer potentials a boundary value problem is reduced to a system of integral equations of the first kind with logarithmic and hypersingular kernels. Full discretization is realized by trigonometric quadrature method. We establish convergence of the method and prove error estimates in a Hölder space setting. Numerical examples illustrate convergence results.
The present paper deals with the study and effective implementation for Stress Intensity Factor computation of a mixed boundary element approach based on the standard displacement integral equation and the hypersingular traction integral equation. Expressions for the evaluation of the hypersingular integrals along general curved quadratic line elements are presented. The integration is carried out by transformation of the hypersingular integrals into regular integrals, which are evaluated by standard quadratures, and simple singular integrals, which are integrated analytically. The generality of the method allows for the modelling of curved cracks and the use of straight line quarter-point elements. The Stress Intensity Factors can be computed very accurately from the Crack Opening Displacement at collocation points extremely close to the crack tip. Several examples with different crack geometries are analyzed. The computed results show that the proposed approach for Stress Intensity Factors evaluation is simple, produces very accurate solutions and has little dependence on the size of the elements near the crack tip.
A ubiquitous linear boundary-value problem in mathematical physics
involves solving a partial differential equation exterior to a thin
obstacle. One typical example is the scattering of scalar waves by a
curved crack or rigid strip (Neumann boundary condition) in two
dimensions. This problem can be reduced to an integrodifferential
equation, which is often regularized. We adopt a more direct approach,
and prove that the problem can be reduced to a hypersingular boundary
integral equation. (Similar reductions will obtain in more complicated
situations.) Computational schemes for solving this equation are
described, with special emphasis on smoothness requirements. Extensions
to three-dimensional problems involving an arbitrary smooth bounded
crack in an elastic solid are discussed.
In this paper we consider the two-dimensional problem of the diffraction of stress waves by a finite plane crack in a homogeneous isotropic elastic solid. We prove that, provided the elastic displacement is bounded, this problem has at most one solution. Also, by introducing a new class of Green functions, we are able to give a constructive proof of existence. The solution is given in terms of the solution of any one of a class of Fredholm integral equations of the second kind.
The properties of hypersingular integrals, which arise when the gradient
of conventional boundary integrals is taken, are discussed.
Interpretation in terms of Hadamard finite-part integrals, even for
integrals in three dimensions, is given, and this concept is compared
with the Cauchy Principal Value, which, by itself, is insufficient to
render meaning to the hypersingular integrals. It is shown that the
finite-part integrals may be avoided, if desired, by conversion to
regular line and surface integrals through a novel use of Stokes'
theorem. Motivation for this work is given in the context of scattering
of time-harmonic waves by cracks. Static crack analysis of linear
elastic fracture mechanics is included as an important special case in
the zero-frequency limit. A numerical example is given for the problem
of acoustic scattering by a rigid screen in three spatial dimensions.
Consider an infinite elastic solid containing a penny-shaped crack. We
suppose that time-harmonic elastic waves are incident on the crack and
are required to determine the scattered displacement field u_i. In this
paper, we describe a new method for solving the corresponding linear
boundary-value problem for u_i, which we denote by S. We begin by
defining an 'elastic double layer'; we prove that any solution of S can
be represented by an elastic double layer whose 'density' satisfies
certain conditions. We then introduce various Green functions and define
a new crack Green function, Gij, that is discontinuous across
the crack. Next, we use Gij to derive a Fredholm integral
equation of the second kind for the discontinuity in u_i across the
crack. We prove that this equation always has a unique solution. Hence,
we are able to prove that the original boundary-value problem S always
possesses a unique solution, and that this solution has an integral
representation as an elastic double layer whose density solves an
integral equation of the second kind.
The method of the regularized integral equation is applied to the second fundamental boundary-value problems of elasticity. Both static and dynamic problems are considered. It is shown that in either case the displacement vector can be expressed as a Neumann series in terms of the prescribed stress on the internal boundary.
In this paper we consider the two-dimensional scalar scattering problem
for Helmholtz's equation exterior to a smooth open arc of general shape.
The problem has a number of physical applications including the
diffraction of sound by a rigid barrier immersed in a compressible fluid
and by a crack in an elastic solid which supports a state of anti-plane
strain (SH-motion). The mathematical method used here is the crack Green
function method introduced by G. R. Wickham. This enables the scattering
problem to be reduced to the solution of a Fredholm integral equation of
the second kind with a continuous kernel. The numerical solution of this
equation is discussed and a number of examples are computed.
This paper studies the inverse scattering problem by an open sound-hard arc. We prove a uniqueness theorem for this problem. For the foundation of Newton methods we establish the Fréchet differentiability of the far-field operator with respect to the open arc and show that the Fréchet derivative can be obtained as the solution of a hypersingular integral equation. We describe regularized Newton-type methods for approximately solving the inverse scattering problem and present numerical results for a number of differently chosen ansatz functions and arcs.
A Newton method is presented for the approximate solution of the inverse problem to determine the shape of a two-dimensional crack from a knowledge of the far field pattern for the scattering of time-harmonic elastic plane waves. Fréchet differentiability with respect to the boundary is shown for the far field operator, which for a fixed incident wave maps the crack onto the far field pattern of the scattered wave. Some numerical reconstructions illustrate the feasibility of the method.
A Newton method is presented for the approximate solution of the inverse problem to determine the shape of a sound-soft or perfectly conducting arc from a knowledge of the far-field pattern for the scattering of time-harmonic plane waves. Fréchet differentiability with respect to the boundary is shown for the far-field operator, which for a fixed incident wave maps the boundary arc onto the far-field pattern of the scattered wave. For the sake of completeness, the first part of the paper gives a short outline on the corresponding direct problem via an integral equation method including the numerical solution.
We study here some integral equations linked to the Laplace or the Helmholtz equation, or to the system of elasticity equations. These equations lead to non integrable kernels only defined as finite parts, so that they are quite difficult to approximate. In each case, we introduce a variational formulation which avoids this difficulty and allow us to use stable finite element approximations for these problems
We describe a quadrature method for the numerical solution of the logarithmic integral equation of the first kind arising from the single-layer approach to the Dirichlet problem for the two-dimensional Helmholtz equation in smooth domains. We develop an error analysis in a Sobolev space setting and prove fast convergence rates for smooth boundary data.
We analyze hypersingular integral equations on a curved open smooth arc in R 2 that model either curved cracks in an elastic medium or the scattering of acoustic and elastic waves at a hard screen. By using the Mellin transformation we obtain sharp regularity results for the solution of these equations in Sobolev spaces in the form of singular expansions. In particular we show that the expansions do not contain logarithmic singularities
We consider here a Dirichlet problem for the two-dimensional linear elasticity equations in the domain exterior to an open arc in the plane. It is shown that the problem can be reduced to a system of boundary integral equations with the unknown density function being the jump of stresses across the arc. Existence, uniqueness as well as regularity results for the solution to the boundary integral equations are established in appropriate Sobolev spaces. In particular, asymptotic expansions concerning the singular behavior for the solution near the tips of the arc are obtained. By adding special singular elements to the regular splines as test and trial functions, an augmented Galerkin procedure is used for the corresponding boundary integral equations to obtain a quasi-optimal rate of convergence for the approximate solutions.
We derive a hypersingular integral equation, which is equivalent to the scattering problem. Using the cosine substitution, we obtain an integral equation which is closely related to the integral equation for the case of a closed boundary. We solve this equation approximately by a quadrature formula method and obtain pointwise error estimates in Hölder norms by using a perturbation argument.
We describe a fully discrete method for the numerical solution of the hypersingular integral equation arising from the combined double- and single-layer approach for the solution of the exterior Neumann problem for the two-dimensional Helmholtz equation in smooth domains. We develop an error analysis in a Hölder space setting with pointwise estimates and prove an exponential convergence rate for analytic boundaries and boundary data.
Collocation and quadrature methods for singular integro-differential equations of Prandtl's type are studied in weighted Sobolev spaces. A fast algorithm basing on the quadrature method is proposed. Convergence results and error estimates are given.
Numerical Analysis for Integral and Related Opera-tor Equations. Berlin: Akademie Basel: Birkhä Hypersingular quarter-point boundary elements for crack problems
- Pr Ossdorf
- S Silbermann
- B S Aez
- A Gallego
- R Dom´
PR OSSDORF, S. & SILBERMANN, B. 1991 Numerical Analysis for Integral and Related Opera-tor Equations. Berlin: Akademie Basel: Birkhä. S ´ AEZ, A., GALLEGO, R., & DOM´, J. 1995 Hypersingular quarter-point boundary elements for crack problems. Int. J. Numer. Methods Eng. 38, 1681–1701.