System Of Boundary Integral Equations Research Articles

An alternative potential method for mixed steady‐state elastic oscillation problems

We consider an alternative approach to investigate three‐dimensional exterior mixed boundary value problems (BVPs) for the steady‐state oscillation equations of the elasticity theory for isotropic bodies. The unbounded domain occupied by an elastic body, , has a compact boundary surface , which is divided into two disjoint parts, the Dirichlet part and the Neumann part , where the displacement vector (the Dirichlet‐type condition) and the stress vector (the Neumann‐type condition) are prescribed, respectively. Our new approach is based on the classical potential method and has several essential advantages compared with the existing approaches. We look for a solution to the mixed BVP in the form of a linear combination of the single‐layer and double‐layer potentials with densities supported on the Dirichlet and Neumann parts of the boundary, respectively. This approach reduces the mixed BVP under consideration to a system of boundary integral equations, which contain neither extensions of the Dirichlet or Neumann data nor the Steklov–Poincaré‐type operator involving the inverse of a special boundary integral operator, which is not available explicitly for arbitrary boundary surface. Moreover, the right‐hand sides of the resulting boundary integral equations system are vector functions coinciding with the given Dirichlet and Neumann data of the problem in question. We show that the corresponding matrix integral operator is bounded and coercive in the appropriate ‐based Bessel potential spaces. Consequently, the operator is invertible, which implies unconditional unique solvability of the mixed BVP in the class of vector functions belonging to the Sobolev space and satisfying the Sommerfeld–Kupradze radiation conditions at infinity. We also show that the obtained matrix boundary integral operator is invertible in the ‐based Besov spaces and prove that under appropriate boundary data a solution to the mixed BVP possesses ‐Hölder continuity property in the closed domain with , where is an arbitrarily small number.

Numerical analysis for electromagnetic scattering with nonlinear boundary conditions

This work studies time-dependent electromagnetic scattering from obstacles whose interaction with the wave is fully determined by a nonlinear boundary condition. In particular, the boundary condition studied in this work enforces a power law type relation between the electric and magnetic fields along the boundary. Based on time-dependent jump relations of classical boundary operators, we derive a nonlinear system of time-dependent boundary integral equations that determines the tangential traces of the scattered electric and magnetic fields. These fields can subsequently be computed at arbitrary points in the exterior domain by evaluating a time-dependent representation formula. Fully discrete schemes are obtained by discretizing the nonlinear system of boundary integral equations with Runge–Kutta based convolution quadrature in time and Raviart–Thomas boundary elements in space. Error bounds with explicitly stated convergence rates are proven, under the assumption of sufficient regularity of the exact solution. The error analysis is conducted through novel techniques based on time-discrete transmission problems and the use of a new discrete partial integration inequality. Numerical experiments illustrate the use of the proposed method and provide empirical convergence rates.

Nyström discretizations of boundary integral equations for the solution of 2D elastic scattering problems

We present three high-order Nyström discretization strategies of various boundary integral equation formulations of the impenetrable time-harmonic Navier equations in two dimensions. One class of such formulations is based on the four classical Boundary Integral Operators (BIOs) associated with the Green’s function of the Navier operator. We consider two types of Nyström discretizations of these operators, one that relies on Kussmaul–Martensen logarithmic splittings (Chapko et al., 2000; Domínguez and Turc, 2000), and the other on Alpert quadratures (Alpert, 1999). In addition, we consider an alternative formulation of Navier scattering problems based on Helmholtz decompositions of the elastic fields (Dong et al., 2021), which can be solved via a system of boundary integral equations that feature integral operators associated with the Helmholtz equation. Owing to the fact that some of the BIOs that are featured in those formulations are non-standard, we use Quadrature by Expansion (QBX) methods for their high order Nyström discretization. Alternatively, we use Maue integration by parts techniques to recast those non-standard operators in terms of single and double layer Helmholtz BIOs whose Nyström discretizations is amenable to the Kussmaul–Martensen methodology. We present a variety of numerical results concerning the high order accuracy that our Nyström discretization elastic scattering solvers achieve for both smooth and Lipschitz boundaries. We also present extensive comparisons regarding the iterative behavior of solvers based on different integral equations in the high frequency regime. Finally, we illustrate how some of the Nyström discretizations we considered can be incorporated seamlessly into the Convolution Quadrature (CQ) methodology to deliver high-order solutions of the time domain elastic scattering problems.

Computation of Transmission Eigenvalues by the Regularized Schur Complement for the Boundary Integral Operators

This paper is devoted to the computation of transmission eigenvalues in the inverse acoustic scattering theory. This problem is first reformulated as a two by two boundary system of boundary integral equations. Next, utilizing the Schur complement technique, we develop a Schur complement operator with regularization to obtain a reduced system of boundary integral equations. The Nystrom discretization is then used to obtain an eigenvalue problem for a matrix. We employ the recursive integral method for the numerical computation of the matrix eigenvalue. Numerical results show that the proposed method is efficient and reduces computational costs.

Computational Modeling of Lymph Filtration and Absorption in the Lymph Node by Boundary Integral Equations

We develop a numerical method for solving three-dimensional problems of fluid filtration and absorption in a piecewise homogeneous medium by means of boundary integral equations. This method is applied to a simulation of the lymph flow in a lymph node. The lymph node is considered as a piecewise homogeneous domain containing porous media. The lymph flow is described by Darcy’s law. Taking into account the lymph absorption, we propose an integral representation for the velocity and pressure fields, where the lymph absorption imitates the lymph outflow from a lymph node through a system of capillaries. The original problem is reduced to a system of boundary integral equations, and a numerical algorithm for solving this system is provided. We simulate the lymph velocity and pressure as well as the total lymph flux. The method is verified by comparison with experimental data.

Numerical solution of the two-dimensional Steigmann–Ogden model of material surface with a boundary

In this paper, we consider an infinite isotropic elastic matrix that contains a material surface described by the Steigmann–Ogden theory, where the material surface possesses a boundary. In two-dimensional setting, the surface represents a material curve with endpoints. The presence of the surface tension introduces non-linearity in the problem since a superposition with respect to the far-field load is not allowed. We review the governing equations, the jump conditions, and the conditions at the surface tips. For solving the problem in the plane strain setting, the displacements in the matrix are sought in the complex variables form of a single layer elastic potential. The density of the potential represents the jump in complex tractions across the surface. Exact expressions for the elastic fields in the matrix are provided in terms of complex integral representations. The problem of the surface located along a straight segment is considered in detail. Its solution is reduced to that of the system of boundary integral equations using the approximations of the boundary data that involve the series of Chebyshev’s polynomials. We present numerical examples to illustrate the influence of dimensionless parameters on the solution.

Dedicated Boundary Element Modeling for Nanoparticle‐on‐Mirror Structures Incorporating Nonlocal Hydrodynamic Effects

AbstractA boundary element modeling for predicting the plasmonic response from a Nanoparticle‐on‐Mirror (NPoM) structure is proposed. The NPoM can be considered as one of the most important platforms in the research on nanophotonics. Instead of using the local response model to describe the optical response of metals, the nonlocal hydrodynamic model is used to account for the dynamics of free electrons in the metallic NP and the metallic mirror beneath. A system of boundary integral equations (BIEs) is derived to describe the interaction of light with the NPoM structure. The BIEs are solved by the boundary element method. To validate the implementation, two exemplary structures are considered: the Nanosphere‐on‐Mirror structure and the Nanocube‐on‐Mirror structure. By contrasting with the results from previous works and the results from a generalized T‐matrix method, both a qualitative and quantitative agreement in the near field and the far field regimes are observed.

A spectral boundary integral method for the elastic obstacle scattering problem in three dimensions

In this paper, we consider the scattering of a plane wave by a rigid obstacle embedded in a homogeneous and isotropic elastic medium in three dimensions. Based on the Helmholtz decomposition, the elastic scattering problem is reduced to a coupled boundary value problem for the Helmholtz and Maxwell equations. A novel system of boundary integral equations is formulated and a spectral boundary integral method is developed for the coupled boundary value problem. Numerical experiments are presented to demonstrate the superior performance of the proposed method.

Efficient simulation of rarefied gas flow past a particle: A boundary element method for the linearized G13 equations

We develop a novel boundary integral formulation for the steady linearized form of Grad's 13-moment (G13) equations applied to a uniform flow of rarefied gas past solid objects at low Mach numbers. Changing variables leads to a system of boundary integral equations that combines integral equations from Stokes flow and potential theory. The strong coupling between the stress deviator and heat flux featured by the G13 equations demands adding a boundary integral equation for the pressure. We specialize the integral equations for an axisymmetric flow with no swirl and derive the axisymmetric fundamental solutions for the pressure equation, seemingly absent in the Stokes-flow literature. Using the boundary element method to achieve a numerical solution, we apply this formulation to streaming flow of rarefied gas past prolate or oblate spheroids with their axis of symmetry parallel to the free stream, considering various aspect ratios and Knudsen numbers—the ratio of the molecules' mean free path to the macroscopic length scale. After validating the method, we obtain the surface profiles of the deviations from the unperturbed state of the traction, heat flux, pressure, temperature, and slip velocity, as well as the drag on the spheroid, observing convergence with the number of elements. Rarefaction phenomena, such as temperature jump and polarization, Knudsen effects in the drag, and velocity slippage, are predicted. This method opens a new path for investigating other gas non-equilibrium phenomena that can be modeled by the same set of equations, such as thermophoresis, and has application in nano- and microfluidics.

Inverse obstacle scattering for elastic waves in the time domain

This paper concerns an inverse elastic scattering problem which is to determine a rigid obstacle from time domain scattered field data for a single incident plane wave. By using the Helmholtz decomposition, we reduce the initial-boundary value problem for the time domain Navier equation to a coupled initial-boundary value problem for wave equations, and prove the uniqueness of the solution for the coupled problem by employing the energy method. The retarded single layer potential is introduced to establish a set of coupled boundary integral equations, and uniqueness is discussed for the solution of these boundary integral equations. Based on the convolution quadrature method for time discretization, the coupled boundary integral equations are reformulated into a system of boundary integral equations in the s-domain, and then a convolution quadrature based nonlinear integral equation method is proposed for the inverse problem. Numerical experiments are presented to show the feasibility and effectiveness of the proposed method.

On the Numerical Solution of a Hyperbolic Inverse Boundary Value Problem in Bounded Domains

We consider the inverse problem of reconstructing the boundary curve of a cavity embedded in a bounded domain. The problem is formulated in two dimensions for the wave equation. We combine the Laguerre transform with the integral equation method and we reduce the inverse problem to a system of boundary integral equations. We propose an iterative scheme that linearizes the equation using the Fréchet derivative of the forward operator. The application of special quadrature rules results to an ill-conditioned linear system which we solve using Tikhonov regularization. The numerical results show that the proposed method produces accurate and stable reconstructions.

Mathematical model of a vertical-axial wind motor in a viscous gas flow

The development of vertical axial wind turbines in Ukraine is in its infancy for many reasons: the lack of systematic theoretical and experimental studies of the aerodynamic characteristics of various schemes of wind turbines, the lack of an appropriate experimental base in technical universities, design organizations, insufficient number of available publications in foreign literature due to high competition between by monopoly firms. At present, various numerical methods are widely used to solve urgent problems of aero hydrodynamics, which are used for the approximate solution of boundary value problems in the form of differential forms of mathematical models. Their common disadvantages are the particularity and laboriousness of solutions, high requirements for computing resources, and, as a consequence, the complexity of solving optimization problems and economic feasibility. These problems can be avoided by using exact or approximate analytical dependences, which allow solving some urgent problems of studying the interaction of a viscous gas with the bearing elements of both aircraft and engineering structures. The existing methods for calculating the aerodynamic characteristics, based on the ideology of the mathematical model of the motion of an ideal medium without viscous interaction, do not correspond to the real processes and demands of practice. The article presents the ideology of determining the aerodynamic characteristics of the interacting system of solid profiles in the configuration of a vertical-axial wind turbine in a viscous gas flow. Based on generalized vector-tensor analysis, contour integral representations of solutions to the main problem of fluid and gas mechanics related to the determination of kinematic and dynamic characteristics of interaction have been constructed. In addition, the existence of a vector potential of the tensor of stresses and deformation velocities has been proved, reducing, in the simplest cases, the process of determining characteristics to integration. The limit values of these integral representations are a system of boundary integral equations, allow for elementary algorithmization, and lead to a system of linear algebraic equations having a single solution.

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.

Analysis of Eigenfrequencies of a Circular Interface Delamination in Elastic Media Based on the Boundary Integral Equation Method

The widespread of composite structures demands efficient numerical methods for the simulation dynamic behaviour of elastic laminates with interface delaminations with interacting faces. An advanced boundary integral equation method employing the Hankel transform of Green’s matrices is proposed for modelling wave scattering and analysis of the eigenfrequencies of interface circular partially closed delaminations between dissimilar media. A more general case of partially closed circular delamination is introduced using the spring boundary conditions with non-uniform spring stiffness distribution. The unknown crack opening displacement is expanded as Fourier series with respect to the angular coordinate and in terms of associated Legendre polynomials of the first kind via the radial coordinate. The problem is decomposed into a system of boundary integral equations and solved using the Bubnov-Galerkin method. The boundary integral equation method is compared with the meshless method and the published works for a homogeneous space with a circular open crack. The results of the numerical analysis showing the efficiency and the convergence of the method are demonstrated. The proposed method might be useful for damage identification employing the information on the eigenfrequencies estimated experimentally. Also, it can be extended for multi-layered composites with imperfect contact between sub-layers and multiple circular delaminations.

Mathematical Modeling of Lymph Node Drainage Function by Neural Network

The lymph node (LN) represents a key structural component of the lymphatic system network responsible for the fluid balance in tissues and the immune system functioning. Playing an important role in providing the immune defense of the host organism, LNs can also contribute to the progression of pathological processes, e.g., the spreading of cancer cells. To gain a deeper understanding of the transport function of LNs, experimental approaches are used. Mathematical modeling of the fluid transport through the LN represents a complementary tool for studying the LN functioning under broadly varying physiological conditions. We developed an artificial neural network (NN) model to describe the lymph node drainage function. The NN model predicts the flow characteristics through the LN, including the exchange with the blood vascular systems in relation to the boundary and lymphodynamic conditions, such as the afferent lymph flow, Darcy’s law constants and Starling’s equation parameters. The model is formulated as a feedforward NN with one hidden layer. The NN complements the computational physics-based model of a stationary fluid flow through the LN and the fluid transport across the blood vessel system of the LN. The physical model is specified as a system of boundary integral equations (IEs) equivalent to the original partial differential equations (PDEs; Darcy’s Law and Starling’s equation) formulations. The IE model has been used to generate the training dataset for identifying the NN model architecture and parameters. The computation of the output LN drainage function characteristics (the fluid flow parameters and the exchange with blood) with the trained NN model required about 1000-fold less central processing unit (CPU) time than computationally tracing the flow characteristics of interest with the physics-based IE model. The use of the presented computational models will allow for a more realistic description and prediction of the immune cell circulation, cytokine distribution and drug pharmacokinetics in humans under various health and disease states as well as assisting in the development of artificial LN-on-a-chip technologies.

English

Optical diffraction on a periodical interface belongs to relatively lowly exploited applications of the boundary integral equations method. This contribution presents a less frequent approach to the diffraction problem based on vector tangential fields of electromagnetic intensities. The problem is formulated as the system of boundary integral equations for tangential fields, for which existence and uniqueness of weak solution is proved. The properties of introduced boundary operators with singular kernel are discussed with regard to performed numerical implementation. Presented theoretical model is of advantage when the electromagnetic field near the material interface is studied, that is illustrated by several application outputs.

The use of the Gurtin-Murdoch theory for modeling mechanical processes in composites with two-dimensional reinforcements

This paper explores the possibility of using the Gurtin-Murdoch theory of material surface for modeling mechanical processes in nanomaterials reinforced by two-dimensional flexible and extensible nanoplatelets. In accordance with the theory, reinforcement is modeled by a vanishing thickness prestressed membrane embedded in an isotropic elastic matrix material. The governing equations for the model are reviewed with a detailed discussion of the conditions at the membrane tips. Plane strain assumption is made and with the purpose of representing the displacements in the bulk material, a single layer elastic potential is employed, with the density representing the jump in tractions across the membrane. Expressions for the remaining elastic fields are provided in terms of complex integral representations. The case of a rectilinear membrane of finite length is considered in detail. Numerical solution for this case is based on the use of approximations that automatically incorporate the membrane-tip conditions into the resulting system of boundary integral equations. Numerical examples illustrate the influence of governing dimensionless parameters and present simulations of the local elastic fields in the materials under study. Additionally, it is shown that, in absence of surface tension, the problem considered here is related to that of a membrane type elastic inhomogeneity embedded into a homogeneous elastic medium.

Inverse scattering problem for detecting a defect in a magnetoelastic layer

ABSTRACT This work is devoted to studying a direct and inverse scattering problem for a magnetoelastic layer having a defect, in the frame of the electromagnetic theory. In terms of the displacement field over the defect's contour, a coupled system of boundary integral equations is formulated, for magnetically permeable and impermeable defects. To identify the position and size of the defect, an efficient numerical algorithm is developed by using the quasi-Newton iterative method. In order to check the influence of the magnetic field upon the scattering waves from the layer, a series of numerical examples is presented with different noise levels. The results showed that the magnetic field has a sensitive effect on the identification process when the external magnetic field increases, especially for the materials having a high magnetic permeability factor . Also, a special inverse problem for predicting the external applied magnetic field, upon a copper layer having a defect with various sizes, has been performed.

Математична модель дифракції хвиль на системі стрічок з різними значеннями поверхневого імпедансу

A mathematical model of diffraction of E-polarized and H-polarized waves on a finite system of not perfectly conducting tapes is obtained. The value of the surface impedance on the two sides of the stripes is different. The initial boundary value problem for the Helmholtz equation with boundary conditions of the third kind was reduced to a system of boundary integral equations. This system of boundary integral equations consists of singular integral equations of the first kind and integral equations of the second kind with a logarithmic singularity. The method of parametric representation of integral operator was used to perform transformations. The values of the physical characteristics of the process are expressed through the solutions of the obtained systems of integral equations. Numerical solution of these equations is performed using a computational scheme based on the discrete singularities method.

On the solution of direct and inverse multiple scattering problems for mixed sound-soft, sound-hard and penetrable objects

In this work we consider a scattering problem governed by the two-dimensional Helmholtz equation, where some objects of different nature (sound-hard, sound-soft and penetrable) are present in the background medium. First we propose and analyze a system of boundary integral equations to solve the direct problem. After that, we propose a numerical method based on the computation of a multifrequency topological energy based imaging functional to find the shape of the objects (without knowing their nature) from measurements of the total field at a set of observation points. Numerical examples show that the proposed indicator function is able to detect objects of different nature and/or shape and size when processing noisy data for a rich enough range of frequencies.