Modified Boundary Integral Equation Research Articles

The Boundary Integral Equation for Kinetically Limited Dendrite Growth

The boundary integral equation defining the interface function for a curved solid/liquid phase transition boundary is analytically solved in steady-state growth conditions. This solution describes dendrite tips evolving in undercooled melts with a constant crystallization velocity, which is the sum of the steady-state and translational velocities. The dendrite tips in the form of a parabola, paraboloid, and elliptic paraboloid are considered. Taking this solution into account, we obtain the modified boundary integral equation describing the evolution of the patterns and dendrites in undercooled binary melts. Our analysis shows that dendritic tips always evolve in a steady-state manner when considering a kinetically controlled crystallization scenario. The steady-state growth velocity as a factor that is dependent on the melt undercooling, solute concentration, atomic kinetics, and other system parameters is derived. This expression can be used for determining the selection constant of the stable dendrite growth mode in the case of kinetically controlled crystallization.

Spurious Quasi-Resonances in Boundary Integral Equations for the Helmholtz Transmission Problem

We consider the Helmholtz transmission problem with piecewise-constant material coefficients, and the standard associated direct boundary integral equations. For certain coefficients and geometries, the norms of the inverses of the boundary integral operators grow rapidly through an increasing sequence of frequencies, even though this is not the case for the solution operator of the transmission problem; we call this phenomenon that of spurious quasi-resonances. We give a rigorous explanation of why and when spurious quasi-resonances occur, and propose modified boundary integral equations that are not affected by them.

Interfacial electrohydrodynamic solitary waves under horizontal electric fields

Interfacial waves between two superimposed dielectric fluid layers under a horizontal electric field are investigated from asymptotic and numerical aspects. The fluid is taken to be inviscid, incompressible and non-conducting in each layer. The competing forces resulting from gravity, surface tension and electric field are all considered. A systematic procedure is proposed to derive model equations of multiple scales in various possible limits from the electrified Euler equations in the framework of the Zakharov–Craig–Sulem formulation. Based on thorough analyses of the Dirichlet–Neumann operators in the long-wave approximation, classic weakly nonlinear models – including the Boussinesq-type system, the Korteweg–de Vries (KdV) equation and its variants, and the Benjamin-type equation – are obtained under different scaling assumptions. In addition, strongly nonlinear models (without the smallness assumption on the wave amplitude) in the Benjamin and Barannyk–Papageorgiou–Petropoulos regimes are derived. In these models, the electric effects are shown to produce dispersive regularisations of long and short waves. A modified boundary integral equation method is developed to compute solitary waves in the original electrified Euler equations. Through comparisons with solitary-wave solutions in the Euler equations, it is found that in various asymptotic regimes, weakly nonlinear models are in overall good agreement when wave amplitudes are small. In contrast, the range of validity of the strongly nonlinear model is much broader. It is shown that the horizontal electric field plays a significant role in the physical system: it expands the range of parameters for the existence of progressive waves, changes the qualitative characteristics of solitary waves and leads to a new type of solitary wave, namely the KdV–wavepacket mixed type.

Attenuated orthotropic time-domain half-space BEM for SH-wave scattering problems

SUMMARY In this paper, the complete formulation of the attenuated orthotropic time-domain half-space boundary element method was proposed to analyse the transient SH-wave scattering problems. By changing the spatial variable, the time-domain half-space Green's functions were obtained in an isotropic-like analytical process by solving the singular form of the scalar wave equation. The stress-free boundary condition of the ground surface was satisfied using the wave source image theory. To include the materials damping, the Barkan approach was used in the formulation to attenuate the half-space by applying a constant logarithmic reduction into the modified boundary integral equation. The closed-form attenuated orthotropic half-space scalar kernels were obtained in the time-domain for displacement/traction fields by analytical integration of Green's functions. The method was easily implemented in a time-domain computer code to analyse the seismic homogenous orthotropic medium. To elaborate the model of the heterogeneous problems, a substructuring approach was presented to satisfy the continuity conditions at the interface depending on the position of the node and normal. Then, several seismic problems including a surface canyon, an underground cavity, a subsurface inclusion, and an alluvial valley subjected to SH wave was solved and compared with the literature to validate the surface response in the case of isotropic convergence. To evaluate the accuracy and the time of analysis for the proposed method versus the full-space boundary element models, a comparative test was carried out for two examples. A favourite agreement was testified between the responses to verify the proposed method's capability for simple modelling of the orthotropic topographic features. Finally, the ability of the method to analyse the problems with high degrees of freedom was investigated by solving a half-space model including 25 subsurface orthotropic circular inclusions.

Rigorous accounting diffraction on non-plane gratings irradiated by non-planar waves

The modified boundary integral equation method (MIM) is considered a rigorous theoretical application for the diffraction of cylindrical waves by arbitrary profiled plane gratings, as well as for the diffraction of plane/non-planar waves by concave/convex gratings. This study investigates 2D diffraction problems of the filiform source electromagnetic field scattered by a plane lamellar grating and of plane waves scattered by a similar cylindrical-shaped grating. Unlike the problem of plane wave diffraction by a plane grating, the field of a localised source does not satisfy the quasi-periodicity requirement. Fourier transform is used to reduce the solution of the problem of localised source diffraction by the grating in the whole region to the solution of the problem of diffraction inside one Floquet channel. By considering the periodicity of the geometry structure, the problem of Floquet terms for the image can be formulated so that it enables the application of the MIM developed for plane wave diffraction problems. Accounting of the local structure of an incident field enables both the prediction of the corresponding efficiencies and the specification of the bounds within which the approximation of the incident field with plane waves is correct. For 2D diffraction problems of the high-conductive plane grating irradiated by cylindrical waves and the cylindrical high-conductive grating irradiated by plane waves, decompositions in sets of plane waves/sections are investigated. The application of such decomposition, including the dependence on the number of plane waves/sections and radii of the grating and wave front shape, was demonstrated for lamellar, sinusoidal and saw-tooth grating examples in the 0th and −1st orders as well as in the transverse electric and transverse magnetic polarisations. The primary effects of plane wave/section partitions of non-planar wave fronts and curved grating shapes on the exact solutions for 2D and 3D (conical) diffraction problems are discussed.

Investigation of impulse load attenuation in closed-cell foam in the framework of couple stress elasticity

The article analyzes the processes of attenuation of impulse loads in foam materials with closed-cell. In the framework of the couple stress elasticity, the analysis of the distribution of dynamic radial stresses in foam media with tunnel cavities of constant cross-section under the action of an impulse load of different duration applied to the cavity boundary in the radial direction is performed. An analytical-numerical approach is used for calculating of the dynamic stresses, which is based on the application of the Fourier transform over time and the modified boundary integral equation method. Numerical calculations of the dynamic stresses were performed using the method of mechanical quadrature and collocation method. The modified algorithm for discrete inverse Fourier transform was used for the calculation of the originals of the dynamic radial stresses. Based on the analysis of numerical calculations, the influence of the impulse duration on the attenuation rate of the pulse load in foam media is established.

Removing Non-Uniqueness in Symmetric Galerkin Boundary Element Method for Elastostatic Neumann Problems and its Application to Half-Space Problems

ABSTRACTWhen the Symmetric Galerkin boundary element method (SGBEM) based on full-space elastostatic fundamental solutions is used to solve Neumann problems, the displacement solution cannot be uniquely determined because of the inevitable rigid-body-motion terms involved. Several methods that have been used to remove the non-uniqueness, including additional point support, eigen decomposition, regularization of a singular system and modified boundary integral equations, were introduced to amend SGBEM, and were verified to eliminate the rigid body motions in the solutions of full-space exterior Neumann problems. Because half-space problems are common in geotechnical engineering practice and they are usually Neumann problems, typical half-space problems were also analyzed using the amended SGBEM with a truncated free surface mesh. However, various levels of errors showed for all the methods of removing non-uniqueness investigated. Among them, the modified boundary integral equations based on the Fredholm’s theory is relatively preferable for its accurate results inside and near the loaded area, especially where the deformation varies significantly.

A potential flow model with viscous dissipation based on a modified boundary element method

Conventional potential flow models are known to exaggerate predictions in water wave resonance problems, attributed to the inviscid fluid assumption in the potential theory. In this paper, we present a modified potential flow model incorporating the viscous dissipation effect. We introduce some ’dissipative surfaces’ inside the fluid domain which are defined such that the normal velocity through these surfaces remains continuous but a pressure drop occurs across them, representing physically the viscous dissipation. In formulating the boundary value problem using Green theorem by a boundary element method (BEM), modified boundary integral equations are deduced to include the integral over the dissipative surfaces. We apply this model to three cases where overestimation is reported using classical potential models: gap resonance, monocolumn moonpool resonance and tuned wave surge converter. Numerical results show that the dissipative surface is effective to dampen the responses at resonance. Validation is carried out by comparisons against either experimental data or analytical solutions. The spikes in the response amplitude operators (RAOs) produced from the non-dissipative model are removed with inclusion of dissipation. Importantly, the proposed model with dissipation effect favorably retains the same level of computational efficiency as with the classical potential flow model.

Modeling the shape of boundary using NURBS curves directly in modified boundary integral equations for Laplace\u2019s equation

The main purpose of this study is the modification of the parametric integral equations systems (PIES) method, to include the NURBS curves into its mathematical formulas. We want to create an opportunity for defining the boundary shape using NURBS curves without the classical discretization process. We present an inclusion of the NURBS curves into PIES, and its application in process of modeling and solving the boundary value problems, on Laplace’s equation example. The correctness and accuracy of such modeling is confirmed by the examples. Obtained solutions are compared with these obtained using linear segments and B-spline curves. Additionally, we verify the correctness of modeling and modifying the shape using analytical solutions. The NURBS curves significantly expand the possibilities and increase the accuracy of the boundary shape modeling which is an important aspect of the numerical solving of boundary value problems.

Cavitation Bubble in Compressible Fluid Near the Rigid Wall Subjected to the Acoustic Wave with Arbitrary Incidence Angle in Three-Dimensional

AbstractA balanced cavitation bubble is released near the rigid wall in the sound field generated by the incidence plane wave and its reflecting wave. With the modified boundary integral equation, the dynamics of bubble is solved considering the compressibility of fluid in this paper. Also the Bernoulli equation as the boundary condition for cavitation bubble in sound field is deduced using Euler equation. Since the arbitrary incidence angle of acoustic wave, the three-dimensional model is utilized. The bubble will expand or contract at first according to the initial phase of acting acoustic pressure on bubble surface. And during the contraction phase, the liquid jet with high speed will be generated pointing to rigid wall but be deflected to the incidence direction of acoustic wave. The oblique degree of jet will be affected by the incidence angle and initial distance between bubble center and rigid wall. The oscillation amplitude of bubble will be affected by the incidence amplitude and incidence frequency, but be limited by the rigid wall. Since the compressibility of fluid, the perturbation will propagate to the far-field. Thus the oscillation amplitude of bubble will be reduced.

A modified boundary integral equation for solving the exterior Robin problem for the Helmholtz equation in three dimensions

A boundary integral equation with modified fundamental solution to solve the exterior Robin problem for Helmholtz’s equation is considered. A specific choice of the coefficients of these added terms ensure the unique solvability and, in addition, they can be chosen to ensure the minimization of the least-squares difference of the modified and the exact Green’s function or the minimization of the condition number. Numerical results are reported showing the robustness and the superconvergence of the new method.

Application of the boundary integral equation method to very small wavelength-to-period diffraction problems

Diffraction from 1D multilayer gratings having arbitrary border profiles including edges is considered for small wavelength-to-period (λ/d) ratios, the most difficult case for any rigorous numerical method. The boundary integral equation theory is so flexible that we can indicate a few areas where it can be modified. In this work, special attention is paid to the main aspects of the Modified Boundary Integral Equation Method for λ/d ≪ 1 as well as to a more general treatment of the energy conservation law applicable to multilayer absorption gratings. Three types of small λ/d problems are known from optical applications: (a) shallow gratings working in the X-ray and extreme ultraviolet ranges, both at near-normal and grazing angles, (b) deep echelle gratings with a steep working facet illuminated along its normal by light of any wavelength, and (c) rough mirrors and gratings in which rough boundaries can be represented by a large-d grating, and which contain a number of random asperities illuminated at any angle and wavelength. Numerical examples of diverse in-plane diffraction problems are presented.

Modified Combined Field Integral Equations for Electromagnetic Scattering

The boundary integral formulation of exterior boundary value problems for the Maxwell system may not be equivalent to the original uniquely solvable problem if the wave number corresponds to an eigenvalue of an associated interior eigenvalue problem. To avoid these spurious modes one may use a combined boundary integral approach. To analyze the resulting boundary integral equations in the energy function spaces suitable regularizations have to be introduced. Here we formulate and analyze a modified boundary integral equation which is based on the use of standard boundary integral operators only. A first numerical example shows the applicability of the proposed approach.

An efficient BEM formulation for three-dimensional steady-state heat conduction analysis of composites

In this work, an efficient boundary element formulation has been presented for three-dimensional steady-state heat conduction analysis of fiber reinforced composites. The cylindrical shaped fibers in the three-dimensional composite matrix are represented by a system of curvilinear line elements with a prescribed diameter which facilitates efficient analysis and modeling together with the reduction in dimensionality of the problem. The variations in the temperature and flux fields in the circumferential direction of the fiber are represented in terms of a trigonometric shape function together with a linear or quadratic variation in the longitudinal direction. The resulting integrals are then treated semi-analytically which reduces the computational task significantly. The computational effort is further minimized by analytically substituting the fiber equations into the boundary integral equation of the material matrix with hole, resulting in a modified boundary integral equation of the composite matrix. An efficient assembly process of the resulting system equations is demonstrated together with several numerical examples to validate the proposed formulation. An example of application is also included.

A boundary element formulation for three-dimensional sliding wear simulation

In this paper a boundary element formulation for three-dimensional sliding wear problems is presented. In the proposed formulation, the wear problem is described by modified boundary integral equations to include sliding wear. A linear wear law is used to model wear and an incremental sliding technique is adopted to solve the non-linear wear problem. Both cases, of one and both bodies losing material are investigated. The proposed method is validated initially by analysing the classic problem of pin-on-disc wear experiment and is subsequently applied to a total hip arthroplasty wear problem. The analysis demonstrates that the proposed boundary element formulation is efficient for solving sliding wear problems and particularly for cases involving complicated geometries, such as the artificial hip joint. Due to the boundary only modelling requirement, the computational effort remains low. In case of incrementing the sliding distance in advance, some optimization of the increment size must be carried out for the total CPU time to remain low and the final geometries of the bodies to be smooth. Using an optimum increment formulation, the solution speeds up as the sliding increments increase without great loss of accuracy.

Integral Solution of a Problem with Non-Standard Boundary Conditions in an Enhanced Theory of Bending of Elastic Plates

A modified boundary integral equation method is used to solve a specific type of mixed boundary value problem in an enhanced theory of bending of elastic plates in which the effects of transverse shear deformation and transverse normal strain are taken into account. The problem considered is characterized by the fact that a combination of transverse displacement and bending and twisting moments is prescribed on the curve which bounds the middle surface of the plate. Both interior and exterior problems are formulated and the corresponding existence and uniqueness results derived.

A boundary integral equation method using auxiliary interior surface approach for acoustic radiation and scattering in two dimensions.

This paper presents an effective solution method for predicting acoustic radiation and scattering fields in two dimensions. The difficulty of the fictitious characteristic frequency is overcome by incorporating an auxiliary interior surface that satisfies certain boundary condition into the body surface. This process gives rise to a set of uniquely solvable boundary integral equations. Distributing monopoles with unknown strengths over the body and interior surfaces yields the simple source formulation. The modified boundary integral equations are further transformed to ordinary ones that contain nonsingular kernels only. This implementation allows direct application of standard quadrature formulas over the entire integration domain; that is, the collocation points are exactly the positions at which the integration points are located. Selecting the interior surface is an easy task. Moreover, only a few corresponding interior nodal points are sufficient for the computation. Numerical calculations consist of the acoustic radiation and scattering by acoustically hard elliptic and rectangular cylinders. Comparisons with analytical solutions are made. Numerical results demonstrate the efficiency and accuracy of the current solution method.

Motion of Nanobeads Proximate to Plasma Membranes During Single Particle Tracking

Drag and torque on nanobeads translating within the pericellular layer while attached to glycolipids of the plasma membrane are calculated by a novel hydrodynamic model. The model considers a bead that translates proximate to a rigid planar interface that separates two distinct Brinkman media. The hydrodynamic resistance is calculated numerically by a modified boundary integral equation formulation, where the pertinent boundary conditions result in a hybrid system of Fredholm integrals of the first and second kinds. The hydrodynamic resistance on the translating bead is calculated for different combinations of the Brinkman screening lengths in the two layers, and for different viscosity ratios. Depending on the bead–membrane separation and on the hydrodynamic properties of both the plasma membrane and the pericellular layer, the drag on the bead may be affected by the properties of the plasma membrane. The Stokes–Einstein relation is applied for calculating the diffusivity of probes (colloidal gold nanobeads attached to glycolipids) in the plasma membrane. This approach provides an alternative way for the interpretation of in vitro observations during single particle tracking procedure, and predicts new properties of the plasma membrane structure.

Modified Boundary Integral Equations for Asymmetric Composite Laminates with Singular Corner Points

AbstractThe boundary value problem for laminates under coupled bending and stretching is studied using boundary integral equations, whose kernels are modified so far, that singular terms with Kirchhoff's force become integrable.

A uniquely solvable integral equation for the neumann problem in linearelasticity

Integral equation methods are often used to treat exterior problems in acoustics, electromagnetics and linear elasticity. The simplest equations are these of the second kind which are susceptible to Fredholm type theory and fail to be uniquely solvable at the so-called irregular frequencies. Inorder to avoid the nonuniqueness problem arising in this kind of equations a modified boundary integral equation for the Neumann problem in linear elasticity in R3 is proposed. The displacement field is expressed as a linear combination of a single and a double layer potentials. The unique solvability of the boundary integral equation is demonstrated by considering commutativity properties which the involved to the problem integral operators satisfy.