Boundary Integral Analysis Research Articles

Three-dimensional bubble jetting inside a corner formed by rigid curved plates: Boundary integral analysis

Three-dimensional dynamics of a transient bubble inside a corner formed by two rigid curved parabolic plates (walls) is studied numerically using boundary integral method (BIM) based on the potential flow theory. The bubble dynamics, including the expansion and collapse phases until the jet impact, are investigated for different corner angles associated with different focal lengths k of the parabolas. However, for all the simulations, the dimensionless initial vertical standoff distance of the bubble’s center from the corner edge (h∗) is fixed at 4. The bubble remains almost spherical during expansion, except for parts of its surface that flattens near the walls. When the bubble is initiated at the bisector plane of the two intersecting walls, it oscillates symmetrically with respect to the bisector plane and becomes oblate during the late stages of the collapse phase. A high-speed liquid jet forms towards the end of bubble collapse, pointing to the corner. If the corner angle decreases, the bubble becomes more oblate along the bisector plane making the ensuing liquid jet wider and slower. In addition, a bubble initiated closer to one of the two walls is mainly influenced by the closer wall, oscillates non-symmetrically with respect to the bisector plane and the liquid jet formed in this case is inclined towards the closer wall due to the greater Bjerknes force of that wall.

Thermotaxis of a deformable droplet in a confined Poiseuille flow

A droplet under a thermal gradient is known to migrate in a preferential direction, as governed by the variation of its interfacial tension with temperature. Contradicting the outcome of reported asymptotic analysis, here we show that the calculation of the droplet migration by considering the variation of interfacial tension with the imposed thermal field alone may be fundamentally incorrect. This error is attributed to the dynamically evolving interfacial temperature field due to a two-way coupling between the thermal field and the flow field, mediated by the droplet deformation and thermal diffusion. By directly capturing an inherent nonlinear coupling between the thermal field and the flow field using explicit interface tracking in a three-dimensional space, our boundary integral based analysis reveals that a linearly decreasing temperature profile imposed along the direction of a plane Poiseuille flow enhances the migration speed of the droplet in both the axial and cross-stream directions. This is in sharp contrast to a prediction of decelerated motion of the droplet under the same imposed thermal field, as obtained from asymptotic theory. We attribute this discrepancy to an alteration of the surface tension mediated interfacial stress due to the locally evolving temperature field, and a consequent concomitant alteration in the interfacial viscous stress to realize a tangential force balance at the interface. From scaling arguments, we show that the resulting change in the viscous drag force may occur over an order of magnitude, disrupting the outcome as compared to that obtained from asymptotic analysis. These results are likely to bear significant implications in controllable separation and sorting of deformable entities in confined fluidic media.

A standard energy eigenvalue problem for directly solving the stationary states of quantum billiards via boundary integral analysis

By using the series expansions of the Bessel functions for the real and imaginary parts of the free particle Green’s function for the two-dimensional stationary states of quantum billiards, it can be shown that some components of the Green’s function are redundant which can be eliminated to make the boundary integral equation for the wave function of free particles inside quantum billiards independent of the wave numbers. This development leads to a much faster search for the energy eigenvalues of quantum billiards by scanning their wave numbers or the formulation of the standard eigenvalue problem which can directly be solved for the energy eigenvalues. Some numerical examples were used to demonstrate that the proposed technique is accurate, computationally effective and straightforward to be applied in practice.

Efficient isogeometric boundary element method for analysis of acoustic scattering from rigid bodies.

Boundary integral analysis of scattering from rigid bodies is well known. Analysis often proceeds along the following lines: representation of the geometry using a collection of triangles, representation of physics using low order ansatz functions defined on each triangle, and then solving the resulting discrete system. This prescription for the common solution stands out in terms of the low-order approximation of both geometry and representation of physics; specifically, both are C0. Taking inspiration from computer graphics literature, a framework wherein continuity of representation (both geometry and physics) can be as high as C2 is developed. In this paper, the steps necessary to develop such a iso-geometric (i.e., using the same basis functions for representing both geometry and physics) boundary integral solver are elucidated. In doing so, an efficient method based on a wideband fast multipole method to evaluate the required inner products and matrix vector products is proposed and demonstrated. Numerous examples are presented to highlight the benefits of the proposed approach.

Boundary integral analysis for non-homogeneous, incompressible Stokes flows

A regular-grid volume-integration algorithm has been previously developed for solving non-homogeneous versions of the Laplace and the elasticity equations. This note demonstrates that the same approach can be successfully adapted to the case of non-homogeneous, incompressible Stokes flow. The key observation is that the Stokeslet (Green’s function) can be written as $\mathcal {U}=\mu \nabla ^{2}\mathcal {H}$ , where $\mathcal {H}$ has a simple analytical expression. As a consequence, the volume integral can be reformulated as an easily evaluated boundary integral, together with a remainder domain integral that can be computed using a regular cuboid grid covering the domain.

The boundary integral theory for slow and rapid curved solid/liquid interfaces propagating into binary systems.

The boundary integral method for propagating solid/liquid interfaces is detailed with allowance for the thermo-solutal Stefan-type models. Two types of mass transfer mechanisms corresponding to the local equilibrium (parabolic-type equation) and local non-equilibrium (hyperbolic-type equation) solidification conditions are considered. A unified integro-differential equation for the curved interface is derived. This equation contains the steady-state conditions of solidification as a special case. The boundary integral analysis demonstrates how to derive the quasi-stationary Ivantsov and Horvay-Cahn solutions that, respectively, define the paraboloidal and elliptical crystal shapes. In the limit of highest Péclet numbers, these quasi-stationary solutions describe the shape of the area around the dendritic tip in the form of a smooth sphere in the isotropic case and a deformed sphere along the directions of anisotropy strength in the anisotropic case. A thermo-solutal selection criterion of the quasi-stationary growth mode of dendrites which includes arbitrary Péclet numbers is obtained. To demonstrate the selection of patterns, computational modelling of the quasi-stationary growth of crystals in a binary mixture is carried out. The modelling makes it possible to obtain selected structures in the form of dendritic, fractal or planar crystals.This article is part of the theme issue 'From atomistic interfaces to dendritic patterns'.

Complexity and accuracy of the grid-based direct-volume integration BEM for quasilinear problems

In order for the boundary element method to be competitive when compared with other methods for solving nonlinear problems, the volume integral must be evaluated accurately and efficiently. The recently proposed cell-based volume integration method evaluates the volume integral on uniform Cartesian cells and therefore avoids volume discretization of the problem domain. However, this method requires the solutions of an additional integral equation; hence its efficiency must be examined. Moreover, the accuracy of the method for the boundary integral analysis of nonlinear problems needs to be studied. In this paper, we present the complexity and accuracy analysis of the BEM coupled with the cell-based volume integration method (herein termed as the grid-based direct-volume integration BEM), for solving quasilinear problems. Various numerical examples are employed to verify the theoretical findings.

Cell-based volume integration for boundary integral analysis

SUMMARY The evaluation of volume integrals that arise in boundary integral formulations for non-homogeneous problems was considered. Using the “Galerkin vector” to represent the Green's function, the volume integral was decomposed into a boundary integral, together with a volume integral wherein the source function was everywhere zero on the boundary. This new volume integral can be evaluated using a regular grid of cells covering the domain, with all cell integrals, including partial cells at the boundary, evaluated by simple linear interpolation of vertex values. For grid vertices that lie close to the boundary, the near-singular integrals were handled by partial analytic integration. The method employed a Galerkin approximation and was presented in terms of the three-dimensional Poisson problem. An axisymmetric formulation was also presented, and in this setting, the solution of a nonlinear problem was considered. Copyright © 2012 John Wiley & Sons, Ltd.

Wedge functions applied to 2D magnetostatic problems

Wedge functions applied to 2D magnetostatic problems In this paper the application of so called wedge functions is presented to solve two-dimensional simple geometries of magnetostatic and electrostatic problems, e.g. rectangles of varying aspect ratio and with different values of the magnetic permeability μ. Such problems require the use of surface charge density, or segment source, functions of the form ρs = σa-1, where the power parameters, a, have special fractional values. A methodology is presented to determine these special values of a and use them in segment sources on simple geometries, i.e. rectangles of varying aspect ratio, and with different values of the magnetic permeability μ. Wedge solutions are obtained by coupling the strength coefficients of source segments of the same power around an edge. These surface source functions have been used in the analysis of conducting and infinite permeability structures. Here we apply such functions in a boundary integral analysis method to problems having regions of finite permeability.

A hypersingular boundary integral analysis of axisymmetric steady-state heat conduction across a non-ideal interface between two dissimilar materials

Steady-state axisymmetric heat conduction across a non-ideal interface between two dissimilar materials is considered. The non-ideal interface may be either low or high conducting. The relevant interfacial conditions are formulated in terms of hypersingular boundary integral equations. A simple boundary element procedure based on the hypersingular boundary integral formulations is proposed for solving numerically the axisymmetric heat conduction problem under consideration. Numerical results for some specific problems are obtained.

Efficient preconditioner for the finite-element boundary integral analysis of electromagnetic scattering in the half space

In this study, the finite-element boundary integral (FE-BI) method is applied to analyse the electromagnetic scattering from arbitrary objects in the half-space environment. Combined with the real image method to approximate the half-space Green's function, the half-space FE-BI can be extended for electrically large problems by the fast multipole method. An efficient preconditioner is constructed with both the finite-element matrix and the near-field part of the boundary integral equation operator for the FE-BI analysis. The numerical examples demonstrate that this preconditioner is efficient for electromagnetic scattering in the half space.

Green's function expansion for exponentially graded elasticity

AbstractNew computational forms are derived for Green's function of an exponentially graded elastic material in three dimensions. By suitably expanding a term in the defining inverse Fourier integral, the displacement tensor can be written as a relatively simple analytic term, plus a single double integral that must be evaluated numerically. The integration is over a fixed finite domain, the integrand involves only elementary functions, and only low‐order Gauss quadrature is required for an accurate answer. Moreover, it is expected that this approach will allow a far simpler procedure for obtaining the first and second‐order derivatives needed in a boundary integral analysis. The new Green's function expressions have been tested by comparing with results from an earlier algorithm. Copyright © 2009 John Wiley & Sons, Ltd.

Surface Interaction Matrices for Boundary Integral Analysis of Lossy Transmission Lines

A method intended to characterize enclosed computational electromagnetic domains in terms of interaction and response of a defined set of surface excitations is described. The finite-element method is used to compute a matrix relating surface current and tangential field in terms of an appropriate basis set such that a coupled solution with a boundary integral formulation is rendered seamless. The proposed method allows for a decoupled finite-element boundary-integral system through use of a discrete-frequency surface interaction matrix, computed in an alternative way, that is still independent of the properties of the background in which the enclosed region resides. The method is applied to per-unit-length resistance and inductance extraction of a variety of multiconductor lossy transmission lines. The primary advantage the proposed method presents for this particular application is reuse of matrices given recurrence of specific conductor cross sections.

Singular boundary elements for three-dimensional elasticity problems

The focus of this paper is a set of semi-discontinuous, traction-singular surface elements introduced to help the rigorous boundary integral analysis of problems in three-dimensional solid mechanics. In contrast to the singular boundary elements developed for linear fracture mechanics where the square-root singularity is of primary interest, traction shape functions featuring the proposed four- and eight-node boundary elements can be used to represent power-type singularities of arbitrary order, such as those arising at non-smooth material boundaries and interfaces. Apart from being capable of rigorously handling traction singularities and discontinuities across the domain boundaries and interfaces, these elements also permit a smooth transition to adjacent regular elements. Complemented with a family of suitable displacement and geometry shape functions, the singular surface elements are incorporated into a regularized boundary integral equation method and shown, through a set of benchmark results, to perform well for both static and dynamic problems.

Galerkin boundary integral analysis for the axisymmetric Laplace equation

AbstractThe boundary integral equation for the axisymmetric Laplace equation is solved by employing modified Galerkin weight functions. The alternative weights smooth out the singularity of the Green's function at the symmetry axis, and restore symmetry to the formulation. As a consequence, special treatment of the axis equations is avoided, and a symmetric‐Galerkin formulation would be possible. For the singular integration, the integrals containing a logarithmic singularity are converted to a non‐singular form and evaluated partially analytically and partially numerically. The modified weight functions, together with a boundary limit definition, also result in a simple algorithm for the post‐processing of the surface gradient. Published in 2005 by John Wiley & Sons, Ltd.

On a Hermite boundary integral approximation

A cubic Hermite approximation for two-dimensional boundary integral analysis is presented. The method differs from previous Hermite interpolation algorithms in that the gradient equations are sparse, significantly reducing the computational cost, and in that information about the surface normals is incorporated to effect a cubic interpolation of the geometry. A motivation for the development of this approximation is in the solution of moving boundary problems, as high accuracy and surface gradients are required for these simulations.

Direct Evaluation of Hypersingular Galerkin Surface Integrals

A direct algorithm for evaluating hypersingular integrals arising in a three-dimensional Galerkin boundary integral analysis is presented. The singular integrals are defined as limits to the boundary, and by integrating two of the four dimensions analytically, the coincident integral is shown to be divergent. However, the divergent terms can be explicitly calculated and shown to cancel with corresponding singularities in the adjacent edge integrals. A single analytic integration is employed for the edge and vertex singular integrals. This is sufficient to display the divergent term in the edge-adjacent integral and to show that the vertex integral is finite. By explicitly identifying the divergent quantities, we can compute the hypersingular integral without recourse to Stokes's theorem or the Hadamard finite part. The algorithms are developed in the context of a linear element approximation for the Laplace equation but are expected to be generally applicable. As an example, the algorithms are applied to solve a thermal problem in an exponentially graded material.

Hybrid finite element-fast spectral domain multilayer boundary integral modeling of doubly periodic structures

Hybrid finite element (FE)-boundary integral (BI) analysis of infinite periodic arrays is extended to include planar multilayered Green's functions. In this manner, a portion of the volumetric dielectric region is modeled via the FE method whereas uniform multilayered regions are characterized using a multilayered Green's function. As such, thick uniform substrates can be modeled without loss of efficiency and accuracy. The multilayered Green's function is analytically computed in the spectral domain and the resulting BI matrix-vector products are evaluated via the fast spectral domain algorithm (FSDA) without explicit generation of the BI-matrix. Furthermore, the number of Floquet modes in the BI expansion is kept very few by appropriately placing the BI surfaces within the computational unit cell. As a result, the computational cost of the method is very little and the model can easily be adapted to various requirements. Examples of frequency selective surface (FSS) arrays are analyzed with this method to demonstrate the accuracy and capability of the approach.

Green’s Functions and Boundary Integral Analysis for Exponentially Graded Materials: Heat Conduction

Free space Green’s functions are derived for graded materials in which the thermal conductivity varies exponentially in one coordinate. Closed-form expressions are obtained for the steady-state diffusion equation, in two and three dimensions. The corresponding boundary integral equation formulations for these problems are derived, and the three-dimensional case is solved numerically using a Galerkin approximation. The results of test calculations are in excellent agreement with exact solutions and finite element simulations.

Improved quarter-point crack tip element

We present a modification to the quarter-point crack tip element and employ this element in two-dimensional boundary integral fracture analysis. The standard singular element is adjusted so that the near-tip crack opening displacement satisfies a known constraint: the coefficient of the term which is linear in the distance to the tip must vanish. Stress intensity factors calculated with the displacement correlation technique are shown to be highly accurate, and significantly more accurate than with the standard element. The improvements are especially dramatic for mixed-mode problems involving curved and interacting cracks.