Piecewise Homogeneous Domains Research Articles

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.

Benchmarking preconditioned boundary integral formulations for acoustics

AbstractThe boundary element method (BEM) is an efficient numerical method for simulating harmonic wave propagation. It uses boundary integral formulations of the Helmholtz equation at the interfaces of piecewise homogeneous domains. The discretization of its weak formulation leads to a dense system of linear equations, which is typically solved with an iterative linear method such as GMRES. The application of BEM to simulating wave propagation through large‐scale geometries is only feasible when compression and preconditioning techniques reduce the computational footprint. Furthermore, many different boundary integral equations exist that solve the same boundary value problem. The choice of preconditioner and boundary integral formulation is often optimized for a specific configuration, depending on the geometry, material characteristics, and driving frequency. On the one hand, the design flexibility for the BEM can lead to fast and accurate schemes. On the other hand, efficient and robust algorithms are difficult to achieve without expert knowledge of the BEM intricacies. This study surveys the design of boundary integral formulations for acoustics and their acceleration with operator preconditioners. Extensive benchmarks provide valuable information on the computational characteristics of several hundred different models for multiple reflection and transmission of acoustic waves.

Application of Boundary Element Method superposition technique for solving natural frequencies in piecewise homogeneous domains

In this work, the domain superposition technique of the Boundary Element Method is coupled with the direct interpolation procedure to identifying the natural frequencies spectrum in two-dimensional piecewise domains with non-regular boundaries and internal inclusions. This problem can be addressed by well-established methods as Finite Element Method; however, the proposed methodology provides better results concerning the accuracy using a simple and expedite boundary element model. Solving the eigenvalue problem, the numerical quality of the matrix related to inertia can be evaluated more precisely, since it is approximated by radial basis functions. That is very important aiming future applications approaching wave propagation problems. A scalar acoustic and isotropic model is assumed, although the methodology can extend to the anisotropic materials. For better accuracy evaluation of the proposed methodology, the solution of some test problems is compared with those obtained with the Finite Element Method using fairly refined meshes.

The degenerate scales of BIEs for conduction in piecewise homogeneous domains

The degenerate scale issue for 2D-boundary integral equations and boundary element methods has been already investigated for many cases when the properties of the medium are homogeneous. We address here the problem of several subdomains with different properties. Then, the domain decomposition into homogeneous subdomains gives rise to a system of BIEs. If there are n subdomains, (n bounded subdomains in the case of an interior problem, and n−1 bounded subdomains and one unbounded subdomain in the case of an exterior problem) there are n degenerate scales in both cases. For an interior problem, the n degenerate scales are the degenerate scales of the n subdomains. For an exterior problem, there are n−1 degenerate scales equal to the degenerate scales of the n−1 bounded subdomains and one intrinsic degenerate scale linked to the solution of a specific boundary value problem. Some properties of this intrinsic degenerate scale are investigated by analytical and numerical methods. The case of a half-plane is also studied according to the boundary condition along the line bounding the half-plane.

О решении краевых задач для уравнения Пуассона в кусочно-однородных областях, ограниченных параболами

О решении краевых задач для уравнения Пуассона в кусочно-однородных областях, ограниченных параболами

The Linear KdV Equation with an Interface

The interface problem for the linear Korteweg-de Vries (KdV) equation in one-dimensional piecewise homogeneous domains is examined by constructing an explicit solution in each domain. The location of the interface is known and a number of compatibility conditions at the boundary are imposed. We provide an explicit characterization of sufficient interface conditions for the construction of a solution using Fokas's Unified Transform Method. The problem and the method considered here extend that of earlier papers to problems with more than two spatial derivatives.

The indirect boundary element method to simulate elastic wave propagation in a 2-D piecewise homogeneous domain

SUMMARY A method is presented to simulate the propagation of elastic waves in a 2-D piecewise homogeneous domain composed of several regions of arbitrary shapes. The regions can be isolated within an elastic background or also embedded among themselves. The method is based on the indirect boundary element method (IBEM) in which the diffracted field is obtained as the contributions of distributed sources along the regions boundaries. This is somehow reminiscent of Huygens’s Principle for wave fronts. The fields emitted by the sources are computed in frequency domain using analytical expressions of Green’s functions for displacements and tractions. This approach is validated, in the antiplane case, by comparing against a semianalytical solution for a basin composed of concentric semi-circular layers and embedded in a half-space. Finally, an example of in-plane wave propagation is given for a complex configuration involving five interlocked regions.

Interface Problems for Dispersive Equations

The interface problem for the linear Schrödinger equations in one‐dimensional piecewise homogeneous domains is examined by providing an explicit solution in each domain. The location of the interfaces is known and the continuity of the wave function and a jump in their derivative at the interface are the only conditions imposed. The problem of two semi‐infinite domains and that of two finite‐sized domains are examined in detail. The problem and the method considered here extend that of an earlier paper by Deconinck et al. (2014) [1]. The dispersive nature of the problem presents additional difficulties that are addressed here.

An improved boundary distributed source method for electrical resistance tomography forward problem

This paper presents a meshless method called the improved boundary distributed source (IBDS) method to obtain the numerical solution of an electrical resistance tomography (ERT) forward problem. The ERT forward problem contains solving the Laplace equation on piece-wise homogeneous domain subjected to the mixed boundary conditions with constraints of integral form. The IBDS method is mesh-free and does not require a fictitious boundary for source points as in the case of a conventional method of fundamental solution (MFS) approach. Therefore, it can be used for a wide variety of applications involving complex shaped objects that are difficult to mesh. Also, in the IBDS method, the diagonal elements for Neumann boundary conditions are computed analytically unlike the original BDS method. Therefore, the IBDS method is computationally efficient and stable compared to the BDS method. The ERT forward problem to compute the boundary voltages is formulated using a meshless IBDS method. Several numerical examples are tested to demonstrate the feasibility and accuracy of the new formulation. The results are compared with that of standard numerical forward solvers for ERT such as the boundary element method (BEM) and the finite element method (FEM).

Ill-posed problems in geomechanics

Any inverse problem requires that its ill-posedness be overcome through regularization or derivation of precise equations. On the basis of singular integral equations, connecting boundary values of stresses and displacements, the author proposes convergence method and its numerical algorithm in terms of a piecewise-homogeneous domain (pillar) where elastic properties, boundary surfaces and the contact conditions are determined under the overdetermined boundary conditions at the available contour of the domain.

A new fast multi-domain BEM to model seismic wave propagation and amplification in 3-D geological structures

SUMMARY The analysis of seismic wave propagation and amplification in complex geological structures raises the need for efficient and accurate numerical methods. The solution of the elastodynamic equations using traditional boundary element methods (BEMs) is greatly hindered by the fully-populated nature of the matrix equations arising from the discretization. In a previous study limited to homogeneous media, the present authors have established that the fast multipole method (FMM) reduces the complexity of a 3-D elastodynamic BEM to N log N per GMRES iteration and demonstrated its effectiveness on 3-D canyon configurations. In this paper, the frequency-domain FM-BEM methodology is extented to 3-D elastic wave propagation in piecewise homogeneous domains in the form of a FM-accelerated multi-region BE–BE coupling approach. This new method considerably enhances the capability of the BEM for studying the propagation of seismic waves in 3-D alluvial basins of arbitrary geometry embedded in semi-infinite media. Several fully 3-D examples (oblique SV -waves) representative of such configurations validate and demonstrate the capabilities of the multi-domain FM approach. They include comparisons with available (low-frequency) results for various types of incident wavefields and time-domain results obtained by means of Fourier synthesis.

A linear sampling approach to inverse elastic scattering in piecewise-homogeneous domains

The focus of this study is a 3D inverse scattering problem underlying non-invasive reconstruction of piecewise-homogeneous (PH) defects in a layered semi-infinite solid from near-field, surface elastic waveforms. The solution approach revolves around the use of Green's function for the layered reference domain and a generalization of the linear sampling method to deal with the featured class of PH configurations. For a rigorous treatment of the full-waveform integral equation that is used as a basis for obstacle reconstruction, the developments include an extension of the Holmgren's uniqueness theorem to piecewise-homogeneous domains and an in-depth analysis of the situation when the sampling point is outside the support of the obstacle that employs the method of topological sensitivity. Owing to the ill-posed nature of the featured integral equation, a stable approximate solution is sought via Tikhonov regularization. A set of numerical examples is included to demonstrate the feasibility of 3D obstacle reconstruction when the defects are buried in a multi-layered elastic solid.

A Radiation Condition for Layered Elastic Media

This study aims to establish a generalized radiation condition for time-harmonic elastodynamic states in a piecewise-homogeneous, semi-infinite solid wherein the “bottom” homogeneous half-space is overlain by an arbitrary number of bonded parallel layers. To consistently deal with both body and interfacial (e.g. Rayleigh, Love and Stoneley) waves comprising the far-field patterns, the radiation condition is formulated in terms of an integral over a sufficiently large hemisphere involving elastodynamic Green's functions for the featured layered medium. On explicitly proving the reciprocity identity for the latter set of point-load solutions, it is first shown that the layered Green's functions themselves satisfy the generalized radiation condition. By virtue of this result it is further demonstrated that the entire class of layered elastodynamic solutions, admitting a representation in terms of the single-layer, double-layer, and volume potentials (distributed over finite domains), satisfy the generalized radiation condition as well. For a rigorous treatment of the problem, fundamental results such as the uniqueness theorem for radiating elastodynamic states, Graffi's reciprocity theorem for piecewise-homogeneous domains, and the integral representation theorem for semi-infinite layered media are also established.

Material constants identification in anisotropic materials using boundary element techniques

This article addresses the resolution of the inverse problem for parameter identification in anisotropic materials. It is based on Boundary Element techniques and it is approached as an optimization problem of a residual cost functional. For this purpose, a standard non-linear minimization method, the Levenberg–Marquardt algorithm, is used. This algorithm makes use of the functional derivatives with respect to the material constants, which in this article have been calculated using direct differentiation procedures. The formulation is developed for plane problems that are in general anisotropic, for homogeneous or piece-wise homogeneous domains. Several examples have been carried out in order to prove the validity of the proposed procedure. The numerical results presented demonstrate the effectiveness of the method employed.

Mapped infinite elements for three‐dimensional multi‐region boundary element analysis

AbstractA formulation for an infinite boundary element (BE) is presented, which allows the modelling of infinite surfaces. The concept is that the finite surface is mapped to an infinite surface using special mapping functions. Using such mapping functions together with linear and quadratic interpolation for the displacements and the tractions, respectively, the desired decay behaviour can be modelled. The implementation of the proposed infinite elements becomes straightforward, since the Cauchy principal value, as well as the free term, are evaluated for the finite and infinite BEs with exactly the same techniques. The element developed can be used in a multi‐region BE analysis of piecewise homogeneous domains, or for domains with joints and faults. The accuracy of the element is tested on some benchmark problems. Finally, a practical application in tunnelling is shown. Copyright © 2004 John Wiley & Sons, Ltd.

Transient heat conduction analysis in a piecewise homogeneous domain by a coupled boundary and finite element method

AbstractA coupled finite element–boundary element analysis method for the solution of transient two‐dimensional heat conduction equations involving dissimilar materials and geometric discontinuities is developed. Along the interfaces between different material regions of the domain, temperature continuity and energy balance are enforced directly. Also, a special algorithm is implemented in the boundary element method (BEM) to treat the existence of corners of arbitrary angles along the boundary of the domain.Unknown interface fluxes are expressed in terms of unknown interface temperatures by using the boundary element method for each material region of the domain. Energy balance and temperature continuity are used for the solution of unknown interface temperatures leading to a complete set of boundary conditions in each region, thus allowing the solution of the remaining unknown boundary quantities.The concepts developed for the BEM formulation of a domain with dissimilar regions is employed in the finite element–boundary element coupling procedure. Along the common boundaries of FEM–BEM regions, fluxes from specific BEM regions are expressed in terms of common boundary (interface) temperatures, then integrated and lumped at the nodal points of the common FEM–BEM boundary so that they are treated as boundary conditions in the analysis of finite element method (FEM) regions along the common FEM–BEM boundary. Copyright © 2002 John Wiley & Sons, Ltd.

Thermoelastic stress field in a piecewise homogeneous domain under non‐uniform temperature using a coupled boundary and finite element method

AbstractThis study concerns the development of a coupled finite element–boundary element analysis method for the solution of thermoelastic stresses in a domain composed of dissimilar materials with geometric discontinuities. The continuity of displacement and traction components is enforced directly along the interfaces between different material regions of the domain. The presence of material and geometric discontinuities are included in the formulation explicitly. The unknown interface traction components are expressed in terms of unknown interface displacement components by using the boundary element method for each material region of the domain. Enforcing the continuity conditions leads to a final system of equations containing unknown interface displacement components only. With the solution of interface displacement components, each region has a complete set of boundary conditions, thus leading to the solution of the remaining unknown boundary quantities. The concepts developed for the BEM formulation of a domain with dissimilar regions is employed in the finite element–boundary element coupling procedure. Along the common boundaries of FEM–BEM regions, stresses from specific BEM regions are first expressed in terms of interface displacements, then integrated and lumped at the nodal points of the common FEM–BEM boundary so that they are treated as boundary conditions in the analysis of FEM regions along the common FEM–BEM boundary. Copyright © 2002 John Wiley & Sons, Ltd.

Modelling three‐dimensional piece‐wise homogeneous domains using the α‐shape‐based natural element method

AbstractIn this paper, the application of the natural element method (NEM) to the numerical analysis of two‐ and three‐dimensional piece‐wise homogeneous domains is presented. The NEM differs from other meshless methods in its capability to accurately reproduce essential boundary conditions along convex boundaries. The α‐shape‐based extension of this method (α‐NEM) generalizes this behaviour to non‐convex domains, enables us to construct models entirely in terms of the initial cloud of points and allows us to simulate material discontinuities in a straightforward manner. In the following sections, simple and effective algorithms are presented for the construction of α‐shapes in domains composed of various materials. Examples are presented in two‐ and three‐dimensional cases in the context of linear elastostatics showing good performance even with the simple numerical quadrature used. Copyright © 2002 John Wiley & Sons, Ltd.

Electromagnetic 2.5-dimensional forward modelling with a boundary integral formulation

An effective and accurate technique for the numerical solution of 2-D electromagnetic scattering problems with 3-D sources is presented. This solution introduces a set of the usual boundary integral equations and uses a scalar Green’s function. In this scalar version, the unknowns of the problem are the boundary values of the longitudinal fields and their normal derivatives in the Fourier domain. A generalization of the usual boundary integral formulation enables us to handle a large class of models composed of piecewise homogeneous domains, including contiguous domains, multiply-connected domains and unbounded domains. This formulation involves the solution of a system of linear equations, and results in a significant saving in computation time in comparison with other rigorous methods. The requirements for the numerical implementation of this solution are described in detail. Numerical tests were carried out using the important example of electromagnetic tomography. The specific symmetry properties of the response function in this case are illustrated. Numerical accuracy is verified over a large frequency range, up to 1 MHz.

Two-Dimensional Geometrical Corner Singularities in Neutron Diffusion—Part I: Analysis

A novel analysis of the neutron multigroup diffusion equation is presented for two-dimensional piecewise homogeneous domains with interior corners that arise at the intersections between regions with distinct material properties. Using polar coordinates centered at a typical interior corner, the solution of the multigroup flux is obtained as an infinite series of products of pairs of functions such that, for every pair, one of the functions depends solely on the angular variable and a single energy group while the other function depends on the radial variable and on all energy groups. The angular functions are shown to be the eigenfunctions of a Sturm-Liouville system that admits an infinite set of discrete and, in general, noninteger eigenvalues. On the other hand, the radial functions are the solutions of an infinite system of second-order ordinary linear differential equations. Exact explicit solutions for the multigroup diffusion equation (MGDE) for two-dimensional disk-like homogeneous domains are also derived and shown to yield analytic expressions for the group fluxes. This analyticity is shown to stem from the fact that the relevant eigenvalues are positive integers, independent of material properties and/or group structure. The exact expressions for the angular eigenvalues and corresponding eigenfunctions for two-region domains are then derived and shown to depend crucially on the specific angle between the two regions. This fact is underscored by deriving the exact expressions for the complete sets of eigenvalues and eigenfunctions for two geometries of particular importance to nuclear reactors, namely the hexagonal and rectangular geometries, respectively, and by showing that they are fundamentally distinct from one another. Of course, these expressions reduce to one and the same form for both geometries when the respective two-region domains are reduced to a single-region domain. Finally, the multigroup fluxes are shown to be bounded but nonanalytic at the respective interior corners; the reason underlying this behavior is traced back to the noninteger character of the relevant eigenvalues. This nonanalyticity is shown to be the fundamental reason for the failure of conventional (e.g., finite difference, finite element) numerical methods for solving the MGDE at and around such corners.