Boundary Element Approximation Research Articles

Acoustic scattering by impedance screens/cracks with fractal boundary: Well-posedness analysis and boundary element approximation

We study time-harmonic scattering in [Formula: see text] ([Formula: see text]) by a planar screen (a “crack” in the context of linear elasticity), assumed to be a non-empty bounded relatively open subset [Formula: see text] of the hyperplane [Formula: see text], on which impedance (Robin) boundary conditions are imposed. In contrast to previous studies, [Formula: see text] can have arbitrarily rough (possibly fractal) boundary. To obtain well-posedness for such [Formula: see text] we show how the standard impedance boundary value problem and its associated system of boundary integral equations must be supplemented with additional solution regularity conditions, which hold automatically when [Formula: see text] is smooth. We show that the associated system of boundary integral operators is compactly perturbed coercive in an appropriate function space setting, strengthening previous results. This permits the use of Mosco convergence to prove convergence of boundary element approximations on smoother “prefractal” screens to the limiting solution on a fractal screen. We present accompanying numerical results, validating our theoretical convergence results, for three-dimensional scattering by a Koch snowflake and a square snowflake.

Numerical Solution of Fractional Elliptic Problems with Inhomogeneous Boundary Conditions

The numerical solution of fractional-order elliptic problems is investigated in bounded domains. According to real-life situations, we assumed inhomogeneous boundary terms, while the underlying equations contain the full-space fractional Laplacian operator. The basis of the convergence analysis for a lower-order boundary element approximation is the theory for the corresponding continuous problem. In particular, we need continuity results for Riesz potentials and the fractional-order extension of the theory for boundary integral equations with the Laplacian operator. Accordingly, the convergence is stated in fractional-order Sobolev norms. The results were confirmed in a numerical experiment.

Galerkin approximation of holomorphic eigenvalue problems: weak T-coercivity and T-compatibility

We consider Galerkin approximations of eigenvalue problems for holomorphic Fredholm operator functions for which the operators do not have the structure “coercive+compact”. In this case the regularity (in the vocabulary of discrete approximation schemes) of Galerkin approximations is not unconditionally satisfied and the question of convergence is delicate. We report a technique to prove regularity of approximations which is applicable to a wide range of eigenvalue problems. The technique is based on the knowledge of a suitable Test function operator. In particular, we introduce the concepts of weak T-coercivity and T-compatibility and prove that for weakly T-coercive operators, T-compatibility of Galerkin approximations implies their regularity. Our framework can be successfully applied to analyze e.g. complex scaling/perfectly matched layer methods, problems involving sign-changing coefficients due to meta-materials and also (boundary element) approximations of Maxwell-type equations. We demonstrate the application of our framework to the Maxwell eigenvalue problem for a conductive material.

Two-level preconditioning for $$h$$-version boundary element approximation of hypersingular operator with GenEO

In the present contribution, we consider symmetric positive definite operators stemming from boundary integral equation (BIE), and we study a two-level preconditioner where the coarse space is built using local generalized eigenproblems in the overlap. We will refer to this coarse space as the GenEO coarse space.

A Preconditioned Krylov Subspace Iterative Methods for Inverse Source Problem by Virtue of a Regularizing LM-DRBEM

In this paper, we investigate an effectual methodology to solve the problem of identifying the source term standing on a combined discontinuous dual reciprocity boundary element method with a regularized Levenberg–Marquardt algorithm. The mathematical model is described by optimizing the variation between the measured potential in some points within the problem domain and the estimated one by the discontinuous dual reciprocity boundary element approximation. To provide a qualitative evaluation, the derived dense and non symmetric matrix system of the forward problem is solved by virtue of the LU factorization and using some preconditioned Krylov subspace iterative algorithms, expressly the biconjugate gradient stabilized, the generalized minimum residual and the squared conjugate gradient method. Due to noise interference, a regularized Levenberg–Marquardt algorithm is adopted to retrieve the unknown source term. Extensive experiments evince greater effectiveness and stability criteria of the proposed approach.

Helmholtz Scattering by Random Domains: First-Order Sparse Boundary Element Approximation

We consider the numerical solution of time-harmonic acoustic scattering by obstacles with uncertain geometries for Dirichlet, Neumann, impedance, and transmission boundary conditions. In particular, we aim to quantify diffracted fields originated by small stochastic perturbations of a given relatively smooth nominal shape. Using first-order shape Taylor expansions, we derive tensor deterministic first-kind boundary integral equations for the statistical moments of the scattering problems considered. These are then approximated by sparse tensor Galerkin discretizations via the combination technique [M. Griebel, M. Schneider, and C. Zenger, A combination technique for the solution of sparse grid problems, in Iterative Methods in Linear Algebra, P. de Groen and P. Beauwens, eds., Elsevier, Amsterdam, 1992, pp. 263--281; H. Harbrecht, M. Peters, and M. Siebenmorgen, J. Comput. Phys., 252 (2013), pp. 128--141]. We supply extensive numerical experiments confirming the predicted error convergence rates with polylogarithmic growth in the number of degrees of freedom and accuracy in approximation of the moments. Moreover, we discuss implementation details such as preconditioning to finally point out further research avenues.

A boundary element and radial basis function approximation method for a second order elliptic partial differential equation with general variable coefficients

A numerical method based on boundary integral equation and radial basis function approximation is presented for solving boundary value problems governed by a second‐order elliptic partial differential equation with variable coefficients. The equation arises in the analysis of steady‐state anisotropic heat or mass diffusion in nonhomogeneous media with properties that vary according to general smoothly varying functions of space. The method requires only the boundary of the solution domain to be discretized into elements. To check the validity and accuracy of the numerical solution, some specific problems with known solutions are solved.

Evolution of a semidiscrete system modeling the scattering of acoustic waves by a piezoelectric solid

We consider a model problem of the scattering of linear acoustic waves in free homogeneous space by an elastic solid. The stress tensor in the solid combines the effect of a linear dependence of strains with the influence of an existing electric field. The system is closed using Gauss’s law for the associated electric displacement. Well-posedness of the system is studied by its reformulation as a first order in space and time differential system with help of an elliptic lifting operator. We then proceed to studying a semidiscrete formulation, corresponding to an abstract Finite Element discretization in the electric and elastic fields, combined with an abstract Boundary Element approximation of a retarded potential representation of the acoustic field. The results obtained with this approach improve estimates obtained with Laplace domain techniques. While numerical experiments illustrating convergence of a fully discrete version of this problem had already been published, we demonstrate some properties of the full model with some simulations for the two dimensional case.

A self-adaptive projection method for contact problems with the BEM

• A new and simple computational method is developed for the contact problem. • We use the variational method to analyze the convergence of proposed method. • We give the self-adaptive rule and the boundary element approximation for the method. • The performance of the method is demonstrated by the numerical results. A self-adaptive algorithm, based on the projection and boundary integral methods, is designed and analyzed for frictionless contact problems in linear elasticity. Using the equivalence between the contact problem and a variational formulation with a projection fixed point problem of infinite dimensions, we develop an iterative algorithm that formulates the contact boundary condition into a sequence of Robin boundary conditions. In order to improve the performance of the method, we propose a self-adaptive rule which updates the penalty parameter automatically. As the iteration process is given by the displacement and the stress on the boundary of the domain, the unknowns of the problem are computed explicitly by using the boundary element method. Both theoretical results and numerical experiments show that the method presented is efficient and robust.

Projection and self-adaptive projection methods for the Signorini problem with the BEM

We propose two new projection methods for solving the Signorini problem and study the relationship between them. Since the Signorini boundary condition is equivalent to a projection fixed point problem, the methods formulate the Signorini boundary condition into a sequence of Robin boundary conditions and only require solving a variational problem with simple boundary conditions at each iterative step. The convergence speed of the first method depends on the parameter ρ heavily, and it is difficult to choose a proper ρ for individual problems. To improve the efficiency of the method, we propose an adaptive projection method which adjusts the parameter ρ automatically per iteration. We establish the convergence analysis of the methods. Because the Signorini boundary condition is given in an iterative form by the potential and its derivative on the boundary of the domain, we can easily provide the boundary element approximation of the problem. Finally, we present some numerical simulations which illustrate the performance of the methods.

Combined boundary integral equations for acoustic scattering‐resonance problems

In this paper, boundary integral formulations for a time‐harmonic acoustic scattering‐resonance problem are analyzed. The eigenvalues of eigenvalue problems resulting from boundary integral formulations for scattering‐resonance problems split in general into two parts. One part consists of scattering‐resonances, and the other one corresponds to eigenvalues of some Laplacian eigenvalue problem for the interior of the scatterer. The proposed combined boundary integral formulations enable a better separation of the unwanted spectrum from the scattering‐resonances, which allows in practical computations a reliable and simple identification of the scattering‐resonances in particular for non‐convex domains. The convergence of conforming Galerkin boundary element approximations for the combined boundary integral formulations of the resonance problem is shown in canonical trace spaces. Numerical experiments confirm the theoretical results. Copyright © 2016 John Wiley & Sons, Ltd.

A direct evaluation of stress intensity factors using the Extended Dual Boundary Element Method

We introduce an alternative method in computational fracture mechanics to evaluate Stress Intensity Factors (SIFs) directly using the Extended Dual Boundary Element Method (XBEM) for 2D problems. Like other enrichment approaches, the new approach is able to yield accurate results on coarse discretisations, and the enrichment increases the problem size by only two degrees of freedom per crack tip. The BEM equations formed by collocation at nodes are augmented by two additional equations that enforce continuity of displacement at the crack tip. The enrichment approach provides the values of SIFs KI and KII directly in the solution vector and without any need for postprocessing such as the J-integral. Numerical examples are used to compare the accuracy of these directly computed SIFs to J-integral processing of both conventional and enriched boundary element approximations.

Hybrid numerical-asymptotic approximation for high-frequency scattering by penetrable convex polygons

We consider time-harmonic scattering by penetrable convex polygons, a Helmholtz transmission problem. Standard numerical schemes based on piecewise polynomial approximation spaces become impractical at high frequencies due to the requirement that the number of degrees of freedom in any approximation must grow at least linearly with respect to frequency in order to represent the oscillatory solution. High frequency asymptotic methods on the other hand are non-convergent and may be insufficiently accurate at low to medium frequencies. Here, we design a hybrid numerical-asymptotic boundary element approximation space that combines the best features of both approaches. Specifically, we compute the classical geometrical optics solution using a beam tracing algorithm, and then we approximate the remaining diffracted field using an approximation space enriched with carefully chosen oscillatory basis functions. We demonstrate via numerical simulations that this approach permits the accurate and efficient representation of the boundary solution and the far field pattern.

Boundary element approximation for Maxwell's eigenvalue problem

We introduce a new method for computing eigenvalues of the Maxwell operator with boundary finite elements. On bounded domains with piecewise constant material coefficients, the Maxwell solution for fixed wave number can be represented by boundary integrals, which allows to reduce the eigenvalue problem to a nonlinear problem for determining the wave number along with boundary and interface traces. A Galerkin discretization yields a smooth nonlinear matrix eigenvalue problem that is solved by Newton's method or, alternatively, the contour integral method. Several numerical results including an application to the band structure computation of a photonic crystal illustrate the efficiency of this approach. Copyright © 2013 John Wiley & Sons, Ltd.

Numerical Modeling of Natural Convection-Radiation in a Vertical Vented Channel

A numerical analysis is performed for the coupling of natural convection heat transfer with surface radiation, in a vertical divided vented channel, which is heated asymmetrically (one wall is isotherm and the other is insulated). The problem has been analyzed using the finite volume method to solve differential equations governing the system. The SIMPLER algorithm was adopted for the treatment of velocity–pressure coupling, and the view factors were determined using a boundary element approximation and a Monte Carlo method. The working fluid (air) is perfectly transparent to the radiation exchange. Results have been obtained for various geometrical parameters, in particular for the size and location of the vent opening, and for different Rayleigh numbers.

Treatment of domain integrals in boundary element methods

A systematic and rigorous technique to calculate domain integrals without a volume-fitted mesh has been developed and validated in the context of a boundary element approximation. In the proposed approach, a domain integral involving a continuous or weakly-singular integrand is first converted into a surface integral by means of straight-path integrals that intersect the underlying domain. Then, the resulting surface integral is carried out either via analytic integration over boundary elements or by use of standard quadrature rules. This domain-to-boundary integral transformation is derived from an extension of the fundamental theorem of calculus to higher dimension, and the divergence theorem. In establishing the method, it is shown that the higher-dimensional version of the first fundamental theorem of calculus corresponds to the well-known Poincaré lemma.The proposed technique can be employed to evaluate integrals defined over simply- or multiply-connected domains with Lipschitz boundaries which are embedded in an Euclidean space of arbitrary but finite dimension. Combined with the singular treatment of surface integrals that is widely available in the literature, this approach can also be utilized to effectively deal with boundary-value problems involving non-homogeneous source terms by way of a collocation or a Galerkin boundary integral equation method using only the prescribed surface discretization. Sample problems associated with the three-dimensional Poisson equation and featuring the Newton potential are successfully solved by a constant element collocation method to validate this study.

Efficient coupling of finite elements and boundary elements---adaptive procedures and preconditioners

Efficient coupling of finite elements and boundary elements---adaptive procedures and preconditioners

Optimal Error Estimation for ${\bfH}({\rmcurl})$-Conformingp-Interpolation in Two Dimensions

In this paper we prove an optimal error estimate for the ${\bf H}({\rm curl})$-conforming projection-based p-interpolation operator introduced in [L. Demkowicz and I. Babuška, SIAM J. Numer. Anal., 41 (2003), pp. 1195–1208]. This result is proved on the reference element (either triangle or square) K for regular vector fields in ${\bf H}^r({\rm curl},K)$ with arbitrary $r >0$. The formulation of the result in the ${\bf H}({\rm div})$-conforming setting, which is relevant for the analysis of high-order boundary element approximations for Maxwell's equations, is provided as well.

A boundary element approximation to mixed boundary value problem of heat equation via collocation and Galerkin method

This article considers the collocation and Galerkin methods applied to boundary integral equations of transient heat problem with mixed boundary condition. The convergence rate of collocation methods is investigated and distinguished from Galerkin methods especially for negative Dirichlet data. The numerical examples of both methods are presented.

Applications of Improved 2‐D Numerical Wave Tanks in Wave Breaking

AbstractSong and Banner (2002, henceforth referred to as SB02) used a numerical wave tank (developed by Drimer and Agnon, and further refined by Segre, henceforth referred to as DAS) to study the wave breaking in deep water, and proposed a dimensionless breaking threshold based on the behaviour of the wave energy modulation and focusing during the evolution of the wave group. In this paper, two modified DAS models are used to further test the SB02's results, the first one (named to MDAS1) corrected many integral calculation errors appeared in the DAS code, and the second one (named MDAS2) replaced the linear boundary element approximation of DAS into the cubic element on the free surface. Researches show that the results of MDAS1 are the same as those of DAS for the simulations of deep water wave breaking, but the different values of the wavemaker amplitude, breaking time and the maximum local average energy growth rate ‐max for the marginal breaking cases are generated by MDAS2 in comparison with MDAS1. However, MDAS2 still satisfies the SB02's breaking threshold. Furthermore, MDAS1 is utilized to study the marginal breaking case at the intermediate water depth when wave passes over a submerged slope, where the slope is given by 1:500, 1:300, 1:150 or 1:100. It is found that the maximum local energy density μ increases significantly if the slope becomes steeper, and the ‐max decreases weakly and increases intensively for the marginal recurrence case and marginal breaking case, respectively. SB02's breaking threshold is still valid for the wave passing over a submerged slope that is gentler than 1:100 at the intermediate water depth.