Boundary Integral Formulation Research Articles

Integral equation methods for vesicle electrohydrodynamics in three dimensions

In this paper, we develop a new boundary integral equation formulation that describes the coupled electro- and hydro-dynamics of a vesicle suspended in a viscous fluid and subjected to external flow and electric fields. The dynamics of the vesicle are characterized by a competition between the elastic, electric and viscous forces on its membrane. The classical Taylor–Melcher leaky-dielectric model is employed for the electric response of the vesicle and the Helfrich energy model combined with local inextensibility is employed for its elastic response. The coupled governing equations for the vesicle position and its transmembrane electric potential are solved using a numerical method that is spectrally accurate in space and first-order in time. The method uses a semi-implicit time-stepping scheme to overcome the numerical stiffness associated with the governing equations.

A Novel Numerical Formulation for Crystal Plasticity

Crystal plasticity plays a crucial role in the mechanics of polycrystalline materials and it is commonly modeled within the framework of the crystal plasticity finite element method (CPFEM). In this work, an alternative formulation for small strains crystal plasticity is presented. The method is based on a boundary integral formulation for polycrystalline problems and plasticity is addressed using an initial strains approach. Voronoi-type micro-morphologies are considered in the polycrystalline case. A general grain-boundary incremental/iterative algorithm, embedding the flow and hardening rules for crystal plasticity, is developed. The key feature of the method is the expression of the micro-mechanical problem in terms of inter-granular variables only, resulting in a reduction of the number of DoFs, which may be appealing in multi-scale applications. Some numerical results, showing the potential of the technique, are presented and discussed.

Localized Boundary-Domain Integro-partial Differential Formulations for Transient Scalar Transport Problems

Integro-partial differential equations, though very challenging can still be solved using conventional numerical techniques. Although these problems have been severally solved, it is still noticeable how little has been said about them in boundary element method (BEM) literature. A major reason for this is that for those problems where an encounter with the problem domain becomes a necessity, BEM’s inadequacies become highly apparent. Moreover for such situations, the fundamental solution is either not available in a cheaply computable form or a considerable numerical effort is required to handle this numerical challenge. Sometimes, the fundamental solutions may exist in a form that is highly non-local and lead to system of equations with a fully populated matrix that is cumbersome to handle numerically, especially for field problems. In the work reported herein, we adopt a fundamental solution of an auxiliary form of a governing partial differential equation coupled with the Green’s identity to discretize and localize an integro-partial differential transport equation by conversion into a boundary-domain form which is amenable to a hybrid boundary integral numerical formulation. It is observed that this numerical formulation is straightforward and yields accurate results when compared with those found in literature.

Fundamental solutions for general anisotropic multi-field materials based on spherical harmonics expansions

A unified method to evaluate the fundamental solutions for generally anisotropic multi-field materials is presented. Based on the relation between the Rayleigh expansion and the three-dimensional Fourier representation of a homogenous partial differential operator, the proposed technique allows to obtain the fundamental solutions and their derivatives up to the desired order as convergent series of spherical harmonics. For a given material, the coefficients of the series are computed only once, and the derivatives of the fundamental solutions are obtained without any term-by-term differentiation, making the proposed approach attractive for boundary integral formulations and efficient for numerical implementation. Useful general relationships for the computation of derivatives of various order of the fundamental solutions are presented. Furthermore, no particular treatment is needed for mathematically degenerate cases. The fundamental solutions of the Laplace equation and isotropic elastic solids are exactly retrieved as special cases. Numerical results are presented to demonstrate the accuracy of the approach for isotropic elastic, generally anisotropic elastic, transversely isotropic and generally anisotropic piezo-electric and magneto-electro-elastic materials.

Boundary integral formulation and semi-implicit scheme coupling for modeling cells under electrical stimulation

We model the electrical activity of biological cells under external stimuli via a novel boundary integral (BI) formulation together with a suitable time-space numerical discretization scheme. Ionic channels follow a non-linear dynamic behavior commonly described by systems of ordinary differential equations dependent on the electric potential jump across the membrane. Since potentials in both intra--- and extracellular domains satisfy an electrostatic approximation, we represent them using solely Dirichlet and Neumann traces over the membrane via boundary potentials. Hence, the volume problem is condensed to one posed over the cell boundary. A second-order time-stepping semi-implicit numerical Galerkin scheme is proposed and analyzed wherein BI operators are approximated by low-order basis functions, with stability independent of space discretization.

Combined boundary integral equations for acoustic scattering‐resonance problems

In this paper, boundary integral formulations for a time‐harmonic acoustic scattering‐resonance problem are analyzed. The eigenvalues of eigenvalue problems resulting from boundary integral formulations for scattering‐resonance problems split in general into two parts. One part consists of scattering‐resonances, and the other one corresponds to eigenvalues of some Laplacian eigenvalue problem for the interior of the scatterer. The proposed combined boundary integral formulations enable a better separation of the unwanted spectrum from the scattering‐resonances, which allows in practical computations a reliable and simple identification of the scattering‐resonances in particular for non‐convex domains. The convergence of conforming Galerkin boundary element approximations for the combined boundary integral formulations of the resonance problem is shown in canonical trace spaces. Numerical experiments confirm the theoretical results. Copyright © 2016 John Wiley & Sons, Ltd.

Well-posed boundary integral equation formulations and Nyström discretizations for the solution of Helmholtz transmission problems in two-dimensional Lipschitz domains

We present a comparison among the performance of solvers based on Nystr\om discretizations of several well-posed boundary integral equation formulations of Helmholtz transmission problems in two-dimensional Lipschitz domains. Specifically, we focus on the following four classes of boundary integral formulations of Helmholtz transmission problems: (1)~the classical first kind integral equations for transmission problems~\cite {costabel-stephan}, (2)~the classical second kind integral equations for transmission problems~\cite {KressRoach}, (3)~the single integral equation formulations~\cite {KleinmanMartin}, and (4)~certain direct counterparts of recently introduced generalized combined source integral equations \cite {turc2, turc3}. The former two formulations were the only formulations whose well-posedness in Lipschitz domains was rigorously established \cite {costabel-stephan, ToWe:1993}. We establish the well-posedness of the latter two formulations in appropriate functional spaces of boundary traces of solutions of transmission Helmholtz problems in Lipschitz domains. We give ample numerical evidence that Nystr\om solvers based on formulations (3) and (4) are computationally more advantageous than solvers based on the classical formulations (1) and (2), especially in the case of high-contrast transmission problems at high frequencies.

Boundary and coupled boundary–finite element methods for transient wave–structure interaction

We propose time-domain boundary integral and coupled boundary integral and variational formulations for acoustic scattering by linearly elastic obstacles. Well posedness along with stability and error bounds with explicit time dependence are established. Full discretization is achieved coupling boundary and finite elements; Convolution Quadrature is used for time evolution in the pure BIE formulation and combined with time stepping in the coupled BEM/FEM scenario. Second order convergence in time is proven for BDF2-CQ and numerical experiments are provided for both BDF2 and Trapezoidal Rule CQ showing second order behavior for the latter as well.

Concurrent finite element simulation of quadrupolar and dipolar flow noise in low Mach number aeroacoustics

The computation of flow-induced noise at low Mach numbers usually relies on a two-step hybrid methodolgy. In the first step, an incompressible fluid dynamics simulation (CFD) is performed and an acoustic source term is derived from it. The latter becomes the inhomogeneous term for an acoustic wave equation, which is solved in the second step, often resorting to boundary integral formulations. In the presence of rigid bodies, Curle’s acoustic analogy is probably the most extended approach. It has been shown that Curle’s boundary dipolar noise contribution does in fact correspond to the diffraction of the quadrupolar aerodynamic noise generated by the flow past the rigid body. In this work, advantage is taken from this fact to propose an alternative computational methodology to get the individual quadrupolar and dipolar contributions to the total acoustic pressure. For any linear acoustic wave operator, the unknown acoustic pressure can be split into its incident and diffracted components and be computed simultaneously to the incompressible flow field, in a single finite element computational run. This circumvents the problem found in Curle’s analogy of needing the total pressure at the body’s boundary, which includes the acoustic pressure fluctuations. The latter cannot be obtained from an incompressible CFD simulation. The proposed unified strategy could be beneficial for a large variety problems such as those involving noise generated from duct terminations, or those related with the simulation of fricatives in numerical voice production, among many others.

FE-BI Formulations for Characteristic Modes

Two types of finite-element boundary-integral (FE-BI) formulations are developed to determine the characteristic modes (CMs) of general composite material objects. The electromagnetic (EM) fields inside the general objects with inhomogeneous anisotropic material are flexibly formulated using the finite-element (FE) formulations, while the EM fields outside the general material objects are accurately described using the boundary-integral (BI) equations defined on the general objects’ boundary. The developed FE-BI formulations result in two generalized Hermitian eigenvalue (GHE) equations for the solution of CMs. These two novel FE-BI based formulations provide clear physical explanation for the analysis of the CMs of the general objects. It is found that their corresponding eigenvalues indicate the imaginary part of the complex power of CMs, which are zero at resonance. Simulated results for typical dielectric resonators are presented to clearly demonstrate the accuracy of the proposed FE-BI formulations.

Boundary element formulation for plane problems in size-dependent piezoelectricity

A new boundary element formulation is developed to analyze two-dimensional size-dependent piezoelectric response in isotropic dielectric materials. The model is based on the recently developed consistent couple stress theory, in which the couple-stress tensor is skew-symmetric. For isotropic materials, there is no classical piezoelectricity, and the size-dependent piezoelectricity or flexoelectricity effect is solely the result of coupling of polarization to the skew-symmetric mean curvature tensor. As a result, the size-dependent effect is specified by one characteristic length scale parameter l, and the electromechanical effect is specified by one flexoelectric coefficient f. Interestingly, in this size-dependent multi-physics model, the governing equations are decoupled. However, the problem is coupled, because of the existence of a flexoelectric effect in the boundary couple-traction and normal electric displacement. We discuss the boundary integral formulation and numerical implementation of this size-dependent piezoelectric boundary element method, which provides a boundary-only formulation involving displacements, rotations, force-tractions, couple-tractions, electric potential, and normal electric displacement as primary variables. Afterwards, we apply the resulting BEM formulation to several computational problems to confirm the validity of the numerical implementation and to explore the physics of the flexoelectric coupling. Copyright © 2016 John Wiley & Sons, Ltd.

Location and Sizing of Defects in Coated Metallic Pipes Using Limited View Scattered Data in Frequency Domain

The inverse problem of reconstructing the location and size of defects in a coated metallic pipe from single frequency limited view electromagnetic scattered field is considered. Specifically, the paper addresses the problem of assessing shape changes in the shadow region entailed by limited view data in 2D by operating in an intermediate size parameter (ka) range. The inverse scattering problem is formulated as a non-linear optimization problem solved through genetic algorithm that seeks to minimize in the least-squared sense the difference between measured data and simulated data through iterations of the solution to a forward problem. The hybrid finite element boundary integral formulations were used to solve the forward problem of coated metallic pipe with defects. The proposed inversion methodology was applied for shadow region shape change assessment to determine whether a metal pipe is coated or not, and to reconstruct the location and size of corrosion-like defect, over a range of size parameters. The study is carried out using transverse electric and transverse magnetic polarized fields. To understand the effect of coating on backscattered fields, parametric studies are conducted using numerical data. A range for size parameter \(2.0\le ka\le 4\) was found to produce the highest contrast between defect free pipe and coated metallic pipes with defects in the deep shadow region. The experiments are carried out using vector network analyzer in an anechoic chamber. The inversion results obtained using measured data were found be in good agreement with inversion results obtained using synthetic data. Estimated extent of corrosion in deep shadow region of a coated metallic pipe was found to be within 9 % of actual extent of corrosion.

Isogeometric shape design sensitivity analysis of elasticity problems using boundary integral equations

Using boundary integral equations and isogeometric approach, a shape design sensitivity analysis (DSA) method is developed for two dimensional elastic structures. In the isogeometric approach, NURBS basis functions in CAD systems are directly utilized in response analysis, which enables a seamless incorporation of exact geometry and higher continuity into computational framework. To enhance the accuracy of shape design sensitivity, the CAD-based higher-order geometric information such as curvature, normal, and tangential vector is exactly embedded in the sensitivity expressions. In boundary integral formulation, shape design velocity field is decomposed into normal and tangential components, which significantly affect the accuracy of shape design sensitivity. Also, the proposed boundary-based method does not require the tedious design parameterization of internal domain. Through the numerical examples, the developed shape DSA method turns out to be more accurate than conventional finite element based one.

Hydroelastic response analysis of pneumatically supported floating structures using a BEM-FEM coupling method

Pneumatically supported floating structures, which utilize an aircushion below the structural bottom, have been suggested to remedy the excessive hydroelastic response of the conventional pontoon-type floating structures. This paper presents an analytical study to examine the hydroelastic responses of the pneumatically supported floating structures. For the hydroelastic analysis, the fluid is modeled as a 2D-strip of finite depth seawater whereas the floating structure is modeled as a beam with zero-draft. A direct coupled model is then constructed by using the boundary integral formulation for the fluid and FEM for the structure and by incorporating a pneumatic factor at the fluid-structure interface to consider the compressibility of the aircushion. For case studies, hydroelastic responses of the pneumatically supported type are compared to those of the pontoon-type for a wide range of pneumatic factors and regular waves. It is shown that the pneumatic supports can significantly reduce the hydroelastic responses in general and that the response reduction may be greatly enhanced when the pneumatic factor is chosen as small as possible for a relatively short incident wave. This study demonstrates that the analytical technique proposed herein can be used to examine the design conditions which are essential for practical applications of the pneumatically supported floating structures.

A boundary integral method for computing forces on particles in unsteady Stokes and linear viscoelastic fluids

SummaryAccurately characterizing the forces acting on particles in fluids is of fundamental importance for understanding particle dynamics and binding kinetics. Conventional asymptotic solutions may lead to poor accuracy for neighboring particles. In this paper, we develop an accurate boundary integral method to calculate forces exerted on particles for a given velocity field. We focus our study on the fundamental two‐bead oscillating problem in an axisymmetric frame. The idea is to exploit a correspondence principle between the unsteady Stokes and linear viscoelasticity in the Fourier domain such that a unifying boundary integral formulation can be established for the resulting Brinkman equation. In addition to the dimension reduction vested in a boundary integral method, our formulation only requires the evaluation of single‐layer integrals, which can be carried out efficiently and accurately by a hybrid numerical integration scheme based on kernel decompositions. Comparison with known analytic solutions and existing asymptotic solutions confirms the uniform third‐order accuracy in space of our numerical scheme. Copyright © 2016 John Wiley & Sons, Ltd.

Grid-based volume integration for elasticity

A regular grid direct volume integral method has recently been proposed for domain integrals stemming from Poisson or nonlinear Laplace problems. The volume integral is exactly decomposed into a non-singular boundary integral, plus a remainder volume integral that can be accurately evaluated using a regular grid overlaying the problem domain. For elasticity, achieving the analogous decomposition requires the ‘Galerkin vector’ H that satisfies E(H)=U, where E is the elasticity equation and U the Kelvin fundamental solution. Herein, Fourier transforms are employed to derive formulas for H and the corresponding traction kernel TH, for both two and three dimensional isotropic elasticity. The three dimensional formulas, and their numerical implementation, are validated by solving relatively simple body force elasticity problems with known solutions. Results for a fast (P-FFT) boundary integral formulation are also presented.

Poisson–Boltzmann model for protein–surface electrostatic interactions and grid-convergence study using the PyGBe code

Interactions between surfaces and proteins occur in many vital processes and are crucial in biotechnology: the ability to control specific interactions is essential in fields like biomaterials, biomedical implants and biosensors. In the latter case, biosensor sensitivity hinges on ligand proteins adsorbing on bioactive surfaces with a favorable orientation, exposing reaction sites to target molecules. Protein adsorption, being a free-energy-driven process, is difficult to study experimentally. This paper develops and evaluates a computational model to study electrostatic interactions of proteins and charged nanosurfaces, via the Poisson–Boltzmann equation. We extended the implicit-solvent model used in the open-source code PyGBe to include surfaces of imposed charge or potential. This code solves the boundary integral formulation of the Poisson–Boltzmann equation, discretized with surface elements. PyGBe has at its core a treecode-accelerated Krylov iterative solver, resulting in O(NlogN) scaling, with further acceleration on hardware via multi-threaded execution on gpus. It computes solvation and surface free energies, providing a framework for studying the effect of electrostatics on adsorption. We derived an analytical solution for a spherical charged surface interacting with a spherical dielectric cavity, and used it in a grid-convergence study to build evidence on the correctness of our approach. The study showed the error decaying with the average area of the boundary elements, i.e., the method is O(1/N), which is consistent with our previous verification studies using PyGBe. We also studied grid-convergence using a real molecular geometry (protein GB1D4′), in this case using Richardson extrapolation (in the absence of an analytical solution) and confirmed the O(1/N) scaling. With this work, we can now access a completely new family of problems, which no other major bioelectrostatics solver, e.g. APBS, is capable of dealing with. PyGBe is open-source under an MIT license and is hosted under version control at https://github.com/barbagroup/pygbe. To supplement this paper, we prepared “reproducibility packages” consisting of running and post-processing scripts in Python for replicating the grid-convergence studies, all the way to generating the final plots, with a single command.

A Domain-Boundary Integral Treatment of Transient Scalar Transport with Memory

It is well known that the boundary element method (BEM) is capable of converting a boundary- value equation into its discrete analog by a judicious application of the Green’s identity and complementary equation. However, for many challenging problems, the fundamental solution is either not available in a cheaply computable form or does not exist at all. Even when the fundamental solution does exist, it appears in a form that is highly non-local which inadvertently leads to a sys-tem of equations with a fully populated matrix. In this paper, fundamental solution of an auxiliary form of a governing partial differential equation coupled with the Green identity is used to discretize and localize an integro-partial differential transport equation by conversion into a boundary-domain form amenable to a hybrid boundary integral numerical formulation. It is observed that the numerical technique applied herein is able to accurately represent numerical and closed form solutions available in literature.

Hypersingular boundary integral formulations for plane elasticity in terms of first-order stress functions

Hypersingular boundary integral formulations for plane elasticity in terms of first-order stress functions

A new boundary integral formulation for bending of anisotropic plates

A new boundary integral formulation for the numerical solution of bending problems of anisotropic plates is proposed in this work. The formulation is based on a Stroh-like formalism for the classical plate theory. In contrast to the conventional formulation, which utilizes Betti’s reciprocal work theorem with appropriate Green’s functions, the proposed formulation makes use of Cauchy’s integral theorem. An advantage of the new formulation is that it provides dual sets of boundary integral equations. With the dual sets, the integral equations to be solved can always be cast into the form of well-posed Fredholm integral equations of the second type for general boundary conditions. Another advantage is that all moment components on the boundary can be obtained without additional numerical differentiations. Numerical examples are given to demonstrate the effectiveness and efficiency of the proposed boundary integral formulation.