Boundary Integral Equation Formulation Research Articles

Numerical simulation of an extensible capsule using regularized Stokes kernels and overset finite differences

In this paper, we present a novel numerical scheme for simulating deformable and extensible capsules suspended in a Stokesian fluid. The main feature of our scheme is a partition-of-unity (POU) based representation of the surface that enables asymptotically faster computations compared to spherical-harmonics based representations. We use a boundary integral equation formulation to represent and discretize hydrodynamic interactions. The boundary integrals are weakly singular. We use the quadrature scheme based on the regularized Stokes kernels by Tlupova and Beale 2019 (given in [34]). We also use partition-of unity based finite differences that are required for the computation of interfacial forces. Given an N-point surface discretization, our numerical scheme has fourth-order accuracy and O(N) asymptotic complexity, which is an improvement over the O(N2log⁡N) complexity of a spherical harmonics based spectral scheme that uses product-rule quadratures by Veerapaneni et al. 2011 [36]. We use GPU acceleration and demonstrate the ability of our code to simulate the complex shapes with high resolution. We study capsules that resist shear and tension and their dynamics in shear and Poiseuille flows. We demonstrate the convergence of the scheme and compare with the state of the art.

Efficient convergent boundary integral methods for slender bodies

The interaction of fibers in a viscous (Stokes) fluid plays a crucial role in industrial and biological processes, such as sedimentation, rheology, transport, cell division, and locomotion. Numerical simulations generally rely on slender body theory (SBT), an asymptotic, nonconvergent approximation whose error blows up as fibers approach each other. Yet convergent boundary integral equation (BIE) methods which completely resolve the fiber surface have so far been impractical due to the prohibitive cost of layer-potential quadratures in such high aspect-ratio 3D geometries. We present a high-order Nyström quadrature scheme with aspect-ratio independent cost, making such BIEs practical. It combines centerline panels (each with a small number of poloidal Fourier modes), toroidal Green's functions, generalized Chebyshev quadratures, HPC parallel implementation, and FMM acceleration. We also present new BIE formulations for slender bodies that lead to well conditioned linear systems upon discretization. We test Laplace and Stokes Dirichlet problems, and Stokes mobility problems, for slender rigid closed fibers with (possibly varying) circular cross-section, at separations down to 1/20 of the slender radius, reporting convergence typically to at least 10 digits. We use this to quantify the breakdown of numerical SBT for close-to-touching rigid fibers. We also apply the methods to time-step the sedimentation of 512 loops with up to 1.65 million unknowns at around 7 digits of accuracy.

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.

Data-driven analysis of parametrized acoustic systems in the frequency domain

A data-driven method combined with the formulations of boundary integral equations is developed for the frequency-domain analysis of parametrized acoustic systems, arising from the spatial discretization of the linear Helmholtz equation. The method derives surrogate models for the approximation of frequency response functions at selected field points via the construction of neural networks with radial basis function interpolation. This requires the offline collection of a training database consisting of discrete frequency-parameter samples and their associated response values, whereas the information about system matrices and interference with boundary element kernels are no longer necessary, leading to a matrix-free and non-intrusive nature. To ensure accurate fitting results, a tensor decomposition technique is employed to decouple the naturally formed tensor grid between the frequency and parameter inputs in the database. After establishing two sets of radial basis networks separately, the final reduced-order solutions as a continuous function of the frequency and parameters are represented by the linear combinations of tensor products. In the online phase, the original transfer function at any testing frequency-parameter location can be quickly predicted as direct output from the compact surrogate model. With the analytical solution, the traditional boundary element method and the fast multipole boundary element method as solvers for the generation of training data, two benchmark problems and one complex system are investigated. In all cases, the simplicity, versatility and efficiency of the proposed method are demonstrated through examples of multi-frequency solutions of acoustic systems with parametrized material and geometric properties.

Boundary Integral Equation Method for Electrostatic Field Prediction in Piecewise-Homogeneous Electrolytes

This article presents a boundary integral equation formulation for the prediction of electrostatic fields, potentials, and currents in regions comprising piecewise-homogeneous electrolytes. The integral equation is formulated in terms of the boundary electric potentials and normal electric current densities and is discretized using the locally corrected Nyström method. The method is validated by comparison to analytic solution data for both linear and nonlinear canonical problems. Solution convergence is investigated with respect to mesh discretization and basis order.

Construction of the differential surface admittance operator with an extended Fokas method for electromagnetic scattering at polygonal objects with arbitrary material parameters

This article presents a novel method to accurately simulate electromagnetic scattering at homogeneous polygonal cylinders with arbitrary material properties. A single source equivalence approach is invoked, allowing to substitute the background medium for the inner material of the scatterer, provided an equivalent surface current density is introduced. We construct the pertinent differential surface admittance operator by means of the Fokas method, establishing a map between the known Dirichlet boundary values and their unknown Neumann counterparts. However, to allow for lossy materials, we extend the Fokas method to complex wavenumbers. The novel formalism, employing pulse-shaped local basis functions, natively supports combined magnetic and dielectric contrast, accurately captures the skin effect, and is conveniently integrated in traditional boundary integral equation formulations. The correctness and versatility of our technique are verified for various examples by means of analytical validation and through comparison with a Poggio-Miller-Chan-Harrington-Wu-Tsai approach, a volume integral equation method and a commercial solver.

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.

Topological Imaging Methods for the Iterative Detection of Multiple Impedance Obstacles

In this paper, we investigate shape inversion algorithms based on the computation of iterated topological derivatives for the detection of multiple particles coated by a complex surface impedance in two- and three-dimensional acoustic media. New closed-form formulae for the topological derivative of the misfit functional are derived when an approximate set of unknown particles has already been recovered. Proofs rely on the computation of shape derivatives followed by the topological asymptotic analysis of a boundary integral equation formulation of the forward and adjoint problems. The relevance of the theoretical results is illustrated by various 2D and 3D experiments using monochromatic imaging algorithms either fully or partially based on topological derivatives.

On an electrostatic micropump with a rigorous mathematical model

The novel electrostatic micropump model for applications such as in biomedical drug delivery is presented. The geometrical arrangement of the coupling rigid electrodes lets us exploit our mathematically rigorous boundary integral equation formulation and its solution. Thus, the charge densities involving the fringe effects on the plates are obtained by means of analytical regularization method (ARM) constructed for annular strips earlier. The efficiency of the constructed method is demonstrated with respect to the direct integral equation solvers implemented via the entire domain Galerkin method and point matching. The main physical characteristics of the suggested system and their deviation from that of infinitely large plane approximation are discussed.

A boundary element and radial basis function method for the Cattaneo–Vernotte equation in anisotropic media with spatially varying and temperature dependent properties

A numerical method based on boundary integral equation formulation and radial basis function approximation is presented for the numerical solution of the Cattaneo–Vernotte hyperbolic equation in anisotropic media with spatially varying and temperature dependent properties. Specific test problems are solved to verify the validity and accuracy of the proposed numerical procedure.

Density results for Sobolev, Besov and Triebel–Lizorkin spaces on rough sets

We investigate two density questions for Sobolev, Besov and Triebel–Lizorkin spaces on rough sets. Our main results, stated in the simplest Sobolev space setting, are that: (i) for an open set Ω⊂Rn, D(Ω) is dense in {u∈Hs(Rn):suppu⊂Ω‾} whenever ∂Ω has zero Lebesgue measure and Ω is “thick” (in the sense of Triebel); and (ii) for a d-set Γ⊂Rn (0<d<n), {u∈Hs1(Rn):suppu⊂Γ} is dense in {u∈Hs2(Rn):suppu⊂Γ} whenever −n−d2−m−1<s2≤s1<−n−d2−m for some m∈N0. For (ii), we provide concrete examples, for any m∈N0, where density fails when s1 and s2 are on opposite sides of −n−d2−m. The results (i) and (ii) are related in a number of ways, including via their connection to the question of whether {u∈Hs(Rn):suppu⊂Γ}={0} for a given closed set Γ⊂Rn and s∈R. They also both arise naturally in the study of boundary integral equation formulations of acoustic wave scattering by fractal screens. We additionally provide analogous results in the more general setting of Besov and Triebel–Lizorkin spaces.

“Application of boundary integral solution to thermal diffusion problem by edge based basis functions”

In this research paper, heat diffusion problem is solved using the new numerical simulation method called boundary integral equation formulation and boundary element solution method. Three dimensional irregular thermal bodies are addressed by introducing the new edge based basis and weighting functions. The three numerical formulations defined on the edges are Single Layer Formulation (EBSLF), Double Layer Formulation (EBDLF) and Combined Layer Formulation (EBCLF) are presented for higher accuracy of the results for heat dissipation. The triangular patch modeling method is used to divide the arbitrary irregular geometry of the body into triangular patches so that above edge based function are defined on those edge of the patches for numerical solution. Numerically, the solution method is developed by defining the basis functions on the Edge of the triangles, instead of on the nodes as followed in the regular BEM solutions. The temperature diffusion results from the hot surface body are plotted for analysis. Also the convergence study is done to get the result convergence towards the exact solution for the different new numerical formulations. The errors are treated in two different ways. The actual error and the absolute errors are used to determine the best model among all the combinations of numerical formulations and different mesh sizes.

Simulation of solids with multiple rectangular inhomogeneities using non-uniform eigenstrain formulation of BIEs

In the present paper, a novel computational model of non-uniform eigenstrain formulation of boundary integral equations (BIEs) with corresponding iterative solution procedures is presented for simulating solids with multiple rectangular inhomogeneities in elastic range. The computational model is developed by introducing the concepts of the equivalent inclusion of Eshelby with eigenstrains into BIEs with the removal of the constant assumption of eigenstrain in previous works that are limited to solve the elliptical or ellipsoidal inhomogeneities. The non-uniform eigenstrains are expressed by Lagrange interpolation polynomials, which are determined in an iterative way for each inclusion embedded in the matrix. Moreover, to deal with the interactions among inclusions, all of the inclusions in the matrix are divided into two groups, namely the near-field group and the far-field group, according to the distance to the current inclusion in consideration. The local Eshelby matrix is constructed over the near-field group to guarantee the convergence of iterative procedure by getting rid of the strong interactions among the inclusions in the near-field group. Due to the unknowns appear only on the boundary of the solution domain in the present model, the solution scale is effectively reduced. The results of the elastic stress distributions across the interface of inclusions are compared with the subdomain BIE method, whereas the overall effective elastic properties of the media are verified by the reference results with doubly periodic square inclusions. In addition, the overall effective elastic properties of a square representative volume element (RVE) with various inclusion distributions are also investigated in considering a variety of factors, including the properties, the aspect ratios, the orientations and the total number of inclusions. Finally, the convergence behaviors and efficiencies of the solution procedure are studied numerically, showing the validity and efficiency of the proposed computational model.

A Chebyshev-based rectangular-polar integral solver for scattering by geometries described by non-overlapping patches

This paper introduces a high-order-accurate strategy for integration of singular kernels and edge-singular integral densities that appear in the context of boundary integral equation formulations for the problem of acoustic scattering. In particular, the proposed method is designed for use in conjunction with geometry descriptions given by a set of arbitrary non-overlapping logically-quadrilateral patches—which makes the algorithm particularly well suited for computer-aided design (CAD) geometries. Fejér's first quadrature rule is incorporated in the algorithm, to provide a spectrally accurate method for evaluation of contributions from far integration regions, while highly-accurate precomputations of singular and near-singular integrals over certain “surface patches” together with two-dimensional Chebyshev transforms and suitable surface-varying “rectangular-polar” changes of variables, are used to obtain the contributions for singular and near-singular interactions. The overall integration method is then used in conjunction with the linear-algebra solver GMRES to produce solutions for sound-soft open- and closed-surface scattering obstacles, including an application to an aircraft described by means of a CAD representation. The approach is robust, fast, and highly accurate: use of a few points per wavelength suffices for the algorithm to produce far-field accuracies of a fraction of a percent, and slight increases in the discretization densities give rise to significant accuracy improvements.

Product Algebras for Galerkin Discretisations of Boundary Integral Operators and their Applications

Operator products occur naturally in a range of regularised boundary integral equation formulations. However, while a Galerkin discretisation only depends on the domain space and the test (or dual) space of the operator, products require a notion of the range. In the boundary element software package Bempp, we have implemented a complete operator algebra that depends on knowledge of the domain, range, and test space. The aim was to develop a way of working with Galerkin operators in boundary element software that is as close to working with the strong form on paper as possible, while hiding the complexities of Galerkin discretisations. In this article, we demonstrate the implementation of this operator algebra and show, using various Laplace and Helmholtz example problems, how it significantly simplifies the definition and solution of a wide range of typical boundary integral equation problems.

Sweeping Preconditioners for the Iterative Solution of Quasiperiodic Helmholtz Transmission Problems in Layered Media

We present a sweeping preconditioner for quasi-optimal domain decomposition methods (DD) applied to Helmholtz transmission problems in periodic layered media. Quasi-optimal DD (QO DD) for Helmholtz equations rely on transmission operators that are approximations of Dirichlet-to-Neumann (DtN) operators. Employing shape perturbation series, we construct approximations of DtN operators corresponding to periodic domains, which we then use as transmission operators in a non-overlapping DD framework. The Robin-to-Robin (RtR) operators that are the building blocks of DD are expressed via robust boundary integral equation formulations. We use Nystrom discretizations of quasiperiodic boundary integral operators to construct high-order approximations of RtR. Based on the premise that the quasi-optimal transmission operators should act like perfect transparent boundary conditions, we construct an approximate LU factorization of the tridiagonal QO Schwarz iteration matrix associated with periodic layered media, which is then used as a double sweep preconditioner. We present a variety of numerical results that showcase the effectiveness of the sweeping preconditioners applied to QO DD for the iterative solution of Helmholtz transmission problems in periodic layered media.

An Extended Charge-Current Formulation of the Electromagnetic Transmission Problem

A boundary integral equation formulation is presented for the electromagnetic transmission problem where an incident electromagnetic wave is scattered from a bounded dielectric object. The formulat...

A face based basis function solutions to thermal conduction problems for 3-D irregular shaped bodies using BEM

Abstract Out of all the numerical methods available to solve the engineering problems using boundary element method, the basis functions which are used to define the unknown quantity in the mathematical formulation, are based on the combination of the shape functions as in a regular finite element procedure. Also, since the BEM is based on defining the source density functions, the kernels in the integral equations become singular, strongly singular and hyper singular for which the traditional BEM does not have a common solution procedure. In this paper, it is proposed to develop boundary element solutions for the thermal conduction problems, for 3-D irregular shaped bodies, using the indirect method i.e. by defining the source density functions. Thermal conduction problem is solved using the boundary integral equation formulation and boundary element solution method. The geometry of the body is divided into triangular patches using triangular patch modeling. A numerical solution is developed effectively by defining the basis functions on the face of the triangles generated in the triangular patch modeling, in contrast to defining it on the nodes as followed in the regular BEM solutions. The temperature distribution from the surface of the hot surface of the cube and sphere is plotted. Also the convergence study is conducted to get the solution convergence towards the exact solution for the case of a sphere and comparison of these two objects for thermal distribution is done.

Isotropic Damage Analysis of Thermoelastic Problems using BIE

This work presents a quadratic boundary integral equation formulation for isotropic damage analysis of components subjected to mechanical, thermal and centrifugal loads. To evaluate domain-related integrals due to the damage effects, the radial integration method (RIM) based on the use of the approximating the normalized displacements in the domain integrals by a series of prescribed radial basis functions (RBF) is adopted. The density of micro-defects is assumed to be small in the material. The scalar damage parameter expressed in an exponential evolution equation is utilized. Numerical examples including a rectangular plate, thick-walled cylinder and rotating disk problems under the thermal loads are given.

A new boundary element method for modeling wave propagation in functionally graded materials

In this paper, a boundary element method (BEM) based on a new boundary integral equation (BIE) formulation is proposed for modeling wave propagation problems in functionally graded materials (FGM) in the frequency domain. The material properties of the considered FGMs are assumed to be graded along spatial Cartesian coordinates, and can vary continuously either in a single axis, or in multiple axes simultaneously, according to an exponential law distribution. Similar to the Somigliana's identity, a new generalized Green's identity corresponding to the elastodynamic equations for FGMs is established first, which can be used to derive the BIE for wave propagation in FGMs for either 2-D or 3-D models. For convenience, a special case with the static and isotropic fundamental solutions are adopted in applying the Green's identity of FGMs. Finally, a boundary-domain integral equation with boundary-only solution scheme is derived. The BEM is applied to solve the BIE and quadratic elements are employed in the discretization. Several test problems in 2-D domains are studied using the BEM. The effects of the material gradient, gradation direction, as well as frequencies of the incident wave on the wave propagation in the FGMs are investigated intensively. The numerical results clearly show the effectiveness and efficiency of the developed BEM in modeling the wave propagation problems in FGMs.