Use Of Boundary Integral Equations Research Articles

Nonlocal impedance conditions in direct and inverse obstacle scattering

We discuss the use of nonlocal impedance conditions within the use of boundary integral equations for the solution of direct and inverse obstacle scattering problems for penetrable obstacles with constant index of refraction. In the first part, for the classical transmission problem we present an approach that leads to a two-by-two system of nonlinear integral equations in the spirit of the method initiated by Kress and Rundell (2005 Inverse Problems 21 1207–23) rather than the three-by-three system arising from the traditional boundary integral equation approach to the transmission conditions. In the second part we survey on recent work of Cakoni and Kress (2017 Appl. Anal. 96 23–38) on the use of boundary integral equations for the characterization and numerical computation of transmission eigenvalues. In particular, we modify and simplify the analysis by the use of a nonlocal rather than a local impedance condition as in the 2017 paper.

A Novel Micro-Mechanical Model for Polycrystalline Inter-Granular and Trans-Granular Fracture

In this work, a novel grain boundary formulation for inter-and trans-granular cracking of polycrystalline materials is presented. The formulation is based on the use of boundary integral equations for anisotropic solids and has the advantage of expressing the considered problem in terms of grain boundary variables only. Inter-granular cracking occurs at the grain boundaries whereas trans-granular cracking is assumed to take place along specific cleavage planes, whose orientation depends on the crystallographic orientation of the grains. The evolution of inter-and trans-granular cracks is then governed by suitably defined cohesive laws, whose parameters characterize the behavior of the two fracture mechanisms. The results show that the model is able to capture the competition between inter-and trans-granular cracking.

Non-local model for diffusion-mediated dislocation climb and cavity growth

To design efficient thermal recovery procedures for structural materials in fusion energy applications it is important to characterise quantitatively the annealing timescales of radiation-induced defect clusters. With this goal in mind, we present an extension of the Green’s function formulation of Gu et al. (2015). for the climb of curved dislocations, to include in the same framework the evaporation and growth of cavities and the effects of free surfaces. This paper focuses on the mathematical foundations of the model, which makes use of boundary integral equations (París and Cañas, 1997) to solve the steady-state vacancy diffusion problem. Numerical results are also presented in the simplified case of a dilute configuration of prismatic dislocation loops and spherical cavities in a finite-size medium, which show good agreement with experimental data on high temperature annealing in ion-irradiated tungsten (Ferroni et al., 2015).

On the coupling of DPG and BEM

In this talk we present results of our recent work [2]: We develop and analyze strategies to couple the discontinuous Petrov-Galerkin method with optimal test functions to (i) least-squares boundary elements and (ii) various variants of standard Galerkin boundary elements. The procedure (i) is somehow a natural approach, because the DPG method can also be equivalently written as a least-squares problem. However, the implementation involves, besides the computation of discrete boundary integral operators, the evaluation of non-local norms, which is not needed for the methods (ii). The derivation of the procedures (ii) relies on either one or both equations of the Calderon system and their analysis makes use of various ideas resp. results for the coupling of FEM and BEM [1, 3, 4]. The stability of these methods hinges on an additional parameter, i.e., a scaling of the trial-to-test operator or the test functions. Nevertheless, numerical experiments indicate that the stability does not depend on this parameter. An essential feature of the methods (i)–(ii) is that, despite the use of boundary integral equations, optimal test functions have to be computed only locally. We apply our findings to a standard transmission problem in full space and present numerical examples to validate our theory.

Elastic waves in continuous and discontinuous geological media by boundary integral equation methods: A review

In this review paper, we concentrate on the use of boundary integral equation (BIE) based methods for the numerical modeling of elastic wave motion in naturally occurring media. The main reason for using BIE is the presence of the free surface of the earth, whereby large categories of problems involve continua with a small surface to volume ratio. Given that under most circumstances, BIE require surface discretization only, substantial savings can be realized in terms of the size of the mesh resulting from the discretization procedure as compared to domain-type numerical methods. We note that this is not necessarily the case with man-made materials that have finite boundaries. Thus, although the emphasis here is on wave motion in geological media, this review is potentially of interest to researchers working in other scientific fields such as material science. Most of the material referenced in this reviews drawn from research work conducted in the last fifteen years, i.e., since the year 2000, but for reasons of completeness reference is made to seminal papers and books dating since the early 1970s. Furthermore, we include here methods other than the BIE-based ones, in order to better explain all the constituent parts of hybrid methods. These have become quite popular in recent years because they seem to combine the best features of surface-only discretization techniques with those of domain type approaches such as finite elements and finite differences. The result is a more rounded approach to the subject of elastic wave motion, which is the underlying foundation of all problems that have to do with time-dependent phenomena in solids.

Методика расчёта переменного во времени магнитного поля в нелинейной среде с использованием граничных интегральных уравнений

Методика расчёта переменного во времени магнитного поля в нелинейной среде с использованием граничных интегральных уравнений

Application of Fast Multipole Boundary Method in Elastostatic Problems

The boundary element method (BEM) is a numerical method for solving boundary-value or initial-value problems formulated by use of boundary integral equations. In the BEM, only the boundaries - that is, surfaces for three-dimensional problems or curves for two dimensional (2D) problems - of a problem domain need to be discretized. However the boundary element method (BEM) has been limited to solving problems with a few thousand degrees of freedom (DOFs) on a personal computer. This is because the conventional BEM, in general, produces dense and nonsymmetric matrices. The main idea of the fast multipole (FM) BEM is to employ iterative solvers to solve the BEM system of equations. Using this method we can solve models with more than one million equations on a laptop computer. In this paper, the governing equations for elasticity problems are reviewed first. Numerical examples are provided to demonstrate the accuracy and efficiencies of fast multipole method (FMM) for solving 2D elasticity problems.

Stress Analysis of Three‐Dimensional Media Containing Localized Zone by FEM‐SGBEM Coupling

This paper presents an efficient numerical technique for stress analysis of three‐dimensional infinite media containing cracks and localized complex regions. To enhance the computational efficiency of the boundary element methods generally found inefficient to treat nonlinearities and non‐homogeneous data present within a domain and the finite element method (FEM) potentially demanding substantial computational cost in the modeling of an unbounded medium containing cracks, a coupling procedure exploiting positive features of both the FEM and a symmetric Galerkin boundary element method (SGBEM) is proposed. The former is utilized to model a finite, small part of the domain containing a complex region whereas the latter is employed to treat the remaining unbounded part possibly containing cracks. Use of boundary integral equations to form the key governing equation for the unbounded region offers essential benefits including the reduction of the spatial dimension and the corresponding discretization effort without the domain truncation. In addition, all involved boundary integral equations contain only weakly singular kernels thus allowing continuous interpolation functions to be utilized in the approximation and also easing the numerical integration. Nonlinearities and other complex behaviors within the localized regions are efficiently modeled by utilizing vast features of the FEM. A selected set of results is then reported to demonstrate the accuracy and capability of the technique.

Analysis of FRP–concrete debonding via boundary integral equations

The problem of debonding of FRP plates glued over a concrete element is studied making use of boundary integral equations. Mode II cohesive crack model is adopted for the interface, whereas linear elasticity is used for the two materials outside the process zone. Symmetric Galerkin boundary element method is used, adopting the arc-length technique to follow the equilibrium path beyond its critical point. It is shown that, due to the presence of a softening branch in shear stress-slip law, the behavior of a specimen undergoing debonding may be strongly non-linear, and is associated with a very brittle failure mechanism. For bond lengths longer than minimum anchorage length, a snap-back branch typically occurs after the attainment of the maximum force. Two different test setups have been numerically simulated and results in good agreement with experimental tests are found.

Dynamics of a suspension confined in a thin cell

A suspension confined between two close parallel plates is studied in the Stokesian regime. The use of boundary integral equations and the lubrication approximation allows computation of the hydrodynamic forces acting on the particles. The forces are long ranged (decaying as R−2) and depend on the orientation of the relative position and velocity of particles. This tensorial character predicts an “antidrag” that is observed in experiments. Also, the far-field forces vanish when a particle is surrounded by an homogeneous suspension, but net forces appear in the presence of abrupt discontinuities of the suspension. The effect of the computed hydrodynamic forces is studied in the dynamics of a cluster of particles falling in a gravitational field, where the different features of the hydrodynamic forces are present. The cluster spreads and deforms from the initial circular shape due to the action of the hydrodynamic forces in the presence of the cluster boundary. The expression for the hydrodynamic forces at long distances allows application of a mean-field approximation, where the forces on a particle can be computed in terms of the particle current field. This approximation gives an excellent numerical agreement with the direct computation of all the hydrodynamic forces, being numerically much faster, yet preserving the accuracy.

Free surface instability in a confined suspension jet

A suspension confined between two close parallel plates is studied in the Stokesian regime. The use of boundary integral equations and the lubrication approximation allows to compute the hydrodynamic forces acting on the particles. The forces are long ranged and depend on the orientation of the relative position and velocity of particles. This tensorial character predicts an “antidrag” that is observed in experiments. The effect of the computed hydrodynamic forces is studied in the dynamics of a jet of particles falling by a gravitational field, which shows a surface instability similar to the Kelvin–Helmholtz one. A theoretical model, based on hydrodynamic-like equations, is able to predict the instability that is produced by the interaction of the long-range forces and the free surface.

Boundary integral-based domain decomposition technique for solution of Navier Stokes equations

This paper presents a new domain decomposition technique based on the use of Boundary Integral Equations (BIEs) for the analysis of viscous flow problems. The domain of interest is divided into a number of non-overlapping subdomains and an iterative procedure is then employed to update the boundary conditions at interfaces. The new feature in the present work is that at each iteration, the relevant two subdomains, together containing a particular interface, are assumed to satisfy the governing BI equations which they do at the end of a convergent iterative process. Hence the boundary conditions on such an interface can be updated using the interior point formulas. Updating formulas based on standard and hypersingular BIEs are derived and the final forms obtained are simple. Furthermore, the internal point formula can be used as a means to estimate the initial interface solution. The proposed method is verified in conjunction with the BEM through the simulation of Poiseuille, driven cavity and backward facing step viscous flows.

Boundary Integral Equations in Aerodynamics

A review of the use of boundary integral equations in aerodynamics is presented, with the objective of addressing what has been accomplished and, even more, what remains to be done. The paper is limited to aerodynamics of aeronautical type, with emphasis on unsteady flows (incompressible and compressible, potential and viscous). For potential flows, both incompressible and compressible flows are considered; the issue of the boundary conditions on the wake and on the trailing edge are addressed in some detail (in particular, some unresolved issues related to the impulsive start are pointed out). For incompressible viscous flows, the use of boundary integral equations in the non-primitive variable formulation are addressed: the Helmholtz decomposition and a decomposition recently introduced (and here referred to as the Poincare´ decomposition) are presented, along with their relationship. The latter is used to examine the relationship between potential and attached viscous flows (in particular, it is shown how the Poincare´ representation, for vortex layers of infinitesimal thickness, reduces to the potential-flow representation). The extension to compressible flows is also briefly outlined and the relative advantages of the two decompositions are discussed. Throughout the paper the emphasis is on the derivation and the interpretation of the boundary integral equations; issues related to the discretization (ie, panel methods, boundary element methods) are barely addressed. For numerical results, which are not included here, the reader is referred to the original references.

Boundary element method taking into account external power source

A formulation for solving two-dimensional magnetic field problems making use of boundary integral equations to extend the application of T. Nakata and N. Takashi's (1982) finite-element method including the external power source has been developed. The development of the method of analysis has made it easy to calculate practical magnetic field problems using the boundary-element method. The same procedure of the boundary-element formulation may be applied to three-dimensional magnetic field problems by using this two-dimensional boundary-element method. >

Use of boundary integral equations in deciphering interference patterns

Use of boundary integral equations in deciphering interference patterns

Progress in the use of boundary integral equations, illustrated by examples

Progress in the numerical treatment of boundary integral equations is reviewed, with particular reference to developments presented at the 1975 ASME Applied Mechanics Conference. The computational strategy of a program for three-dimensional elastostatic analysis is described. In this strategy the surface is approximated by curved quadrilateral and triangular elements. Displacement and traction are defined in terms of linear, quadratic or cubic shape functions of intrinsic coordinates. A system of simultaneous equations is obtained whose coefficients are integrals of kernel-shape function products. The elastic body is divided into subrogions so that the matrix is of banded form, and the coefficients are evaluated using Gaussian quadrature formulas. The program is designed to be capable of treating large problems without excessive core storage requirements. The structure of files created by the program and the order of calculations are such that input/output time is kept low. Two examples are presented: the first demonstrates the possibility of economically solving problems for the infinite space with cavities; the second is the solution of a problem of three-dimensional fracture mechanics and shows the performance of the program in practical use.