Direct Boundary Integral Equation Research Articles

A coupling FEM/BEM method for acoustic-structural interaction problems in uniform relative motion between structure source and fluid

When a vibrating thin structure moves in a heavy flow field, the effect of sound waves on the structure cannot be ignored. This effect is especially pronounced when the relative velocity of the structure and the flow field increases. The hybrid numerical method of finite element method (FEM) and convective boundary element method (BEM) is proposed to solve acoustic-structural interaction problems in uniform relative motion between structure source and fluid. Besides, an acoustic-analogy Lorentz transform is proposed to handle the direct convective boundary integral equation (BIE), which makes the convective BIE has the same form as traditional BIE so that there are no more convective terms and extra singular integrals. The continuity condition at the fluid-structure interface is also proposed to generate the coupling equation. Compared with the traditional FEM, this hybrid method has the advantages of greatly reducing the number of finite elements outside the structure as well as automatically satisfying the radiation condition at far field. And compared with the direct convective BIE, this method reduces the computational complexity and time. It is potential to deliver a solver with higher accuracy and less memory consumption. Some numerical examples are designed to study the influence of the convective effect. Keywords: Acoustic-structural interaction, Finite element method, Boundary element method, FEM-BEM coupling

Force Computation for Dielectrics Using Shape Calculus

Abstract We are concerned with the numerical computation of electrostatic forces/torques in only piece-wise homogeneous materials using the boundary element method (BEM). Conventional force formulas based on the Maxwell stress tensor yield functionals that fail to be continuous on natural trace spaces. Thus their use in conjunction with BEM incurs slow convergence and low accuracy. We employ the remedy discovered in [P. Panchal and R. Hiptmair, Electrostatic force computation with boundary element methods, SMAI J. Comput. Math. 8 (2022), 49–74]. Motivated by the virtual work principle which is interpreted using techniques of shape calculus, and using the adjoint method from shape optimization, we derive stable interface-based force functionals suitable for use with BEM. This is done in the framework of single-trace direct boundary integral equations for second-order transmission problems. Numerical tests confirm the fast asymptotic convergence and superior accuracy of the new formulas for the computation of total forces and torques.

Windowed Green function method for wave scattering by periodic arrays of 2D obstacles

AbstractThis paper introduces a novel boundary integral equation (BIE) method for the numerical solution of problems of planewave scattering by periodic line arrays of two‐dimensional penetrable obstacles. Our approach is built upon a direct BIE formulation that leverages the simplicity of the free‐space Green function but in turn entails evaluation of integrals over the unit‐cell boundaries. Such integrals are here treated via the window Green function method. The windowing approximation together with a finite‐rank operator correction—used to properly impose the Rayleigh radiation condition—yield a robust second‐kind BIE that produces superalgebraically convergent solutions throughout the spectrum, including at the challenging Rayleigh–Wood anomalies. The corrected windowed BIE can be discretized by means of off‐the‐shelf Nyström and boundary element methods, and it leads to linear systems suitable for iterative linear algebra solvers as well as standard fast matrix–vector product algorithms. A variety of numerical examples demonstrate the accuracy and robustness of the proposed methodology.

Spurious Quasi-Resonances in Boundary Integral Equations for the Helmholtz Transmission Problem

We consider the Helmholtz transmission problem with piecewise-constant material coefficients, and the standard associated direct boundary integral equations. For certain coefficients and geometries, the norms of the inverses of the boundary integral operators grow rapidly through an increasing sequence of frequencies, even though this is not the case for the solution operator of the transmission problem; we call this phenomenon that of spurious quasi-resonances. We give a rigorous explanation of why and when spurious quasi-resonances occur, and propose modified boundary integral equations that are not affected by them.

Spurious Resonances in Coupled Domain-Boundary Variational Formulations of Transmission Problems in Electromagnetism and Acoustics

Abstract We develop a framework shedding light on common features of coupled variational formulations arising in electromagnetic scattering and acoustics. We show that spurious resonances haunting coupled domain-boundary formulations based on direct boundary integral equations of the first kind originate from the formal structure of their Calderón identities. Using this observation, the kernel of the coupled problem is characterized explicitly, and we show that it completely vanishes under the exterior representation formula.

Stress analysis of composites with interfacial bridged cracks and damaged zones

Stress analysis of composite structures with bridged cracks and damaged zones is performed with the direct boundary integral equations method in axisymmetric formulation. It is assumed that: a) the crack faces interact in some zones behind of the cracks edges (bridged zones); b) these zones width can be comparable to the whole crack size; c) adjacent parts of the structure can be connected by mechanical ligaments (including interfaces ahead of cracks fronts) and the relative displacements of these parts can occur under the loading (these zones consider as damaged zones). The boundary element software to solve axisymmetric problems of elasticity with interfacial bridged cracks and damaged zones was developed. The results for different properties of composite materials and bonds are presented. Applications the bridged crack approach to modeling of cracks self-healing is remarked.

Nyström method for BEM of the heat equation with moving boundaries

A direct boundary integral equation method for the heat equation based on Nystrom discretization is proposed and analyzed. For problems with moving geometries, a weakly and strongly singular Green’s integral equation is formulated. Here the hypersingular integral operator, i.e., the normal trace of the double-layer potential, must be understood as a Hadamard finite part integral. The thermal layer potentials are regarded as generalized Abel integral operators in time and discretized with a singularity-corrected trapezoidal rule. The spatial discretization is a standard quadrature rule for smooth surface integrals. The discretized systems lead to an explicit time stepping scheme and is effective for solving the Dirichlet and Neumann boundary value problems based on both the weakly and/or strongly singular integral equations.

Stress concentration analysis in technical structures with boundary integral equations

The direct boundary integral equation (DBIE) method in the multi-domain formulation is used for three-dimensional stress concentration problems solution. The computational results of several new stress concentration problems are presented and analyzed. The following problems have been studied: flat plates with oblique, conical and round-edges holes under in-plane tensile loading for different angles of holes inclination. The stress concentration problem for an aviation engine turbine disk with cooling hole has also been considered.

Analysis of Boundary-Domain Integral Equations to the mixed BVP for a compressible stokes system with variable viscosity

The mixed boundary value problem for a compressible Stokes system of partial differential equations in a bounded domain is reduced to two different systems of segregated direct Boundary Integral Equations (BDIEs) expressed in terms of surface and volume parametrix-based potential type operators. Equivalence of the BDIE systems to the mixed BVP and invertibility of the matrix operators associated with the BDIE systems are proved in appropriate Sobolev spaces.

Analysis of interaction of bridged cracks and weak interfaces

Structures with interfacial bridged cracks and weak interfaces are analyzed with the direct boundary integral equations method. Within the framework of this approach it is assumed that: a) the crack faces interact in some zones behind the crack tips (bridged zones); b) the sizes of these zones can be comparable to the whole crack length; c) different materials can be connected by mechanical ligaments and relative displacements of initially adjacent materials may occur under the loading (non-ideal or weak interfaces). The relations for the bridged crack opening and for the relative displacements at weak interface layer are defined by the transverse separation and tangential slipping to an interface surface. Several new problems for the interface bridged cracks, weak interface and interaction of bridged cracks with ideal or weak materials interface were solved and analyzed.

Boundary element modelling of dynamic behavior of piecewise homogeneous anisotropic elastic solids

A traditional direct boundary integral equations method is applied to solve three-dimensional dynamic problems of piecewise homogeneous linear elastic solids. The materials of homogeneous parts are considered to be generally anisotropic. The technique used to solve the boundary integral equations is based on the boundary element method applied together with the Radau IIA convolution quadrature method. A numerical example of suddenly loaded 3D prismatic rod consisting of two subdomains with different anisotropic elastic properties is presented to verify the accuracy of the proposed formulation.

Transient fully nonlinear ship waves using a three-dimensional NURBS numerical towing tank

A fully nonlinear Numerical Towing Tank (NTT) is developed to simulate ship waves based on non-uniform rational B-spline (NURBS) surface and potential theory in the time domain. Fully nonlinear free surface boundary conditions and the exact body boundary condition are applied to solve the boundary value problem. The free surface and the ship hull surfaces are described precisely using NURBS formulation. Mixed Eulerian–Lagrangian (MEL) method is used to update the exact free surface based on material node approach. Isoparametric direct boundary integral equation (BIE) is used to compute the flow field. The intersection of the free surface and the rigid walls are found using an adaptive methodology based on NURBS and double nodes approach at each time step. To retain the open sea condition, a sponge layer is adopted on the exact free surface adjacent the tank's walls. To examine the accuracy of NURBS surface, the NURBS description of the ship hull for different orders of the basis function and different numbers of nodal points are compared with the exact surface. To verify the present computational algorithm, the solutions are compared with the experimental records and the prior numerical study for a Wigley hull. Propagation of the ship wave within the computational domain is studied due to ship advance in different water depths.

Collocation with trigonometric polynomials for integral equations to the mixed boundary value problem

We consider the direct boundary integral equation formulation for the mixed Dirichlet–Neumann boundary value problem for the Laplace equation on a plane domain with a polygonal boundary. The resulting system of integral equations is solved by a collocation method which uses a mesh grading transformation and trigonometric polynomials. The mesh grading transformation method yields fast convergence of the collocation solution by smoothing the singularities of the exact solution. Special care is taken for handling the hypersingular operator as in Hartmann and Stephan (in: Dick, Kuo, Wozniakowski (eds) Festschrift for the 80th birthday of Ian Sloan, Springer, Berlin, 2018). With the indirect method used in Elschner et al. (Numer Math 76(3):355–381, 1997) this was avoided. Using Mellin transformation techniques a stability and solvability analysis of the transformed integral equations can be performed, in a setting in which each arc of the polygon has associated with it a periodic Sobolev space.

Structures with bridged cracks and weak interfaces

Structures with interfacial bridged cracks and weak interfaces are analyzed. In the frames of this approach is assumed that: a) the crack surfaces interact in some zones behind the crack tips (bridged zones); b) size of these zones can be comparable to the whole crack length; and c) the interface between different materials can be connected by mechanical ligaments and a relative displacement of initially adjacent materials may occur (non-ideal or weak interface). Several new problems for interface bridged cracks and interaction of bridged cracks with material interface were solved and analyzed by the direct boundary integral equations method.

A hybrid numerical–asymptotic boundary element method for high frequency scattering by penetrable convex polygons

We present a novel hybrid numerical–asymptotic boundary element method for high frequency acoustic and electromagnetic scattering by penetrable (dielectric) convex polygons. Our method is based on a standard reformulation of the associated transmission boundary value problem as a direct boundary integral equation for the unknown Cauchy data, but with a nonstandard numerical discretization which efficiently captures the high frequency oscillatory behaviour. The Cauchy data is represented as a sum of the classical geometrical optics approximation, computed by a beam tracing algorithm, plus a contribution due to diffraction, computed by a Galerkin boundary element method using oscillatory basis functions chosen according to the principles of the Geometrical Theory of Diffraction. We demonstrate with a range of numerical experiments that our boundary element method can achieve a fixed accuracy of approximation using only a relatively small, frequency-independent number of degrees of freedom. Moreover, for the scattering scenarios we consider, the inclusion of the diffraction term provides an order of magnitude improvement in accuracy over the geometrical optics approximation alone.

Degenerate scale for 2D Laplace equation with mixed boundary condition and comparison with other conditions on the boundary

It is well known that the 2D Laplace Dirichlet problem has a degenerate scale for which the direct boundary integral equation has several solutions. We study here the case of the mixed boundary condition, mainly for the exterior problem, and show that this problem has also one degenerate scale. The degenerate scale factor is a growing function of the part of the boundary submitted to Neumann condition. Different special cases are then addressed: segment, circle and symmetric problems. Some exact values of the degenerate scale factor are given for equilateral triangle and square. The numerical procedure for determining the degenerate scale factor for mixed BC is described. The comparison is made with other kinds of boundary conditions and the consequence of the choice of Green’s function when using the Boundary Element Method is studied.

Direct local boundary integral equation method for numerical solution of extended Fisher–Kolmogorov equation

In this paper, the local boundary integral equation (LBIE) method based on generalized moving least squares (GMLS) is proposed for solving extended Fisher---Kolmogorov (EFK) equation. Since the local weak forms are directly approximated from nodal data using a GMLS approximation, this method is called direct local boundary integral equation (DLBIE) method. In addition to DLBIE method, three other meshless local boundary integral equation methods such as LBIE method based on moving least squares, LBIE method based on radial point interpolation, and LBIE method based on moving Kriging are applied to find the numerical solution of EFK equation. A comparison between these methods is done from the perspective of accuracy and computational efficiency. The computational efficiency is the most significant advantage of the DLBIE method in comparison with the other local boundary integral equation methods. DLBIE shifts the numerical integrations over low-degree polynomials instead of over complicated shape functions and this reduces the computational costs, significantly. The main purpose of this paper is to show that the local boundary integral equation methods can be used for solving the fourth-order non-linear partial differential equations especially EFK equation. The numerical results confirm the good efficiency of the proposed methods for solving our model.

Modeling of Wave Propagation in the Unsaturated Soils Using Boundary Element Method

Problems of wave propagation in poroelastic bodies and media are considered. The behavior of the poroelastic medium is described by Biot theory for partially saturated material. Mathematical model is written in term of five basic functions – elastic skeleton displacements, pore water pressure and pore air pressure. Boundary element method (BEM) is used with step method of numerical inversion of Laplace transform to obtain the solution. Research is based on direct boundary integral equation of three-dimensional isotropic linear theory of poroelasticity. Green’s matrices and, based on it, boundary integral equations are written for basic differential equations in partial derivatives. Discrete analogue are obtained by applying the collocation method to a regularized boundary integral equation. To approximate the boundary consider its decomposition to a set of quadrangular and triangular 8-node biquadratic elements, where triangular elements are treated as singular quadrangular. Every element is mapped to a reference one. Interpolation nodes for boundary unknowns are a subset of geometrical boundary-element grid nodes. Local approximation follows the Goldshteyn’s generalized displacement-stress matched model: generalized boundary displacements are approximated by bilinear elements whereas generalized tractions are approximated by constant. Integrals in discretized boundary integral equations are calculated using Gaussian quadrature in combination with singularity decreasing and eliminating algorithms.

Influence of Poroelasticity on the 3D Seismic Response of Complex Geological Media

AbstractElastic wave propagation in 3D poroelastic geological media with localized heterogeneities, such as an elastic inclusion and a canyon is investigated to visualize the modification of local site responses under consideration of water saturated geomaterial. The extended computational environment herein developed is a direct Boundary Integral Equation Method (BIEM), based on the frequency-dependent fundamental solution of the governing equation in poro-visco elastodynamics. Bardet’s model is introduced in the analysis as the computationally efficient viscoelastic isomorphism to Biot’s equations of dynamic poroelasticity, thus replacing the two-phase material by a complex valued single-phase one. The potential of Bardet’s analogue is illustrated for low frequency vibrations and all simulation results demonstrate the dependency of wave field developed along the free surface on the properties of the soil material.

用于解析函数复分析的共轭边界元法

An analytic function is composed of 2 real conjugate harmonic functions, of which the complex analysis plays an important role in the fields of applied mathematics and mechanics. A set of weighted residual equations were proposed and proved to be equivalent to the approximate solution to the original problem involving 2 governing equations in the domain, the boundary condition and the CauchyRiemann equation at the boundary. 2 conventional direct boundary integral equations at the boundary collocation points were deduced from 2 of the weighted residual equations, and 1 finite difference equation was deduced from the rest one. The mathematical problem arising from the illconditioned linear equations was solved and the Cauchy integral equation was adopted for numerical calculation of the fields at the internal points inside the domain. Finally, the proposed conjugate boundary element method with constant elements was completely established. 3 examples demonstrate that, the proposed method is valid for analytic functions in terms of the power function, the exponential function and the logarithmic function in interior or exterior domains, and the error estimation of the proposed method is at the same order as that of the boundary element method for 2D potential problems.