Boundary Integral Technique Research Articles

An analytical study of curved frequency selective surfaces for shielding applications

AbstractThis paper demonstrates an analytical study of different nonplanar frequency selective surfaces (FSSs) for X‐band (8–12 GHz) electromagnetic (EM) shielding applications. A well‐known metallic square loop unit cell is used for the analysis, which is arranged in a periodic pattern on a three‐dimensional (3‐D) printer compatible dielectric substrate. The geometry is wrapped on various kinds of curved surfaces (cylindrical, paraboloid, and hemispherical surfaces) and investigated using the finite element boundary integral (FEBI) technique. The structure exhibits a band stop characteristic with a fractional bandwidth of 62.74% (6.68–12.66 GHz, having transmission coefficient below −10 dB), resonating around 9.53 GHz under different angles of incident EM wave. Finally, the 3‐D printing method is used to fabricate the prototypes for performing laboratory measurements. The simulated and measured results follow each other closely, thereby establishing the proposed technique in realizing nonplanar FSSs.

Separation flow through asymmetric periodic tubes

A systematic study of the impact of geometrical factors on separation flow characteristics within a longitudinally asymmetric but axially symmetric infinite periodic tube is presented throughout the main body of the paper. To disambiguate the inception of flow separation zones concerning the various geometrical aspects, we have undertaken a numerical study. The boundary integral technique is used for a periodic tube with an infinite number of wavesections to obtain the solution of the Stokes equation. For some amplitudes, we show that separation zones can be formed inside the enlarged areas of the tube. The first separation zone appears beyond a certain profile amplitude whereas the second separation zone is produced after the first separation zone at a larger amplitude. Also, the volume of the separation zone is discussed which has great physical importance, for example, it can be used in the particle separation process. The volume of the first separation zone is directly dependent on the amplitude of the tube profile. In this case, the volume increases as the tube amplitude increases.

A numerical method for the locomotion of bi-flagellated bacteria in viscous fluid

Flagellated bacteria propel themselves by rotating flexible flagella driven by independent motors. Depending on the rotation direction of the motors and the handedness of the helical filaments, the flagella either pull or push the cell body. Motivated by experimental observations of Magnetococcus marinus, we develop an elastohydrodynamic model to study the locomotion of a bi-flagellated bacterium with one puller flagellum and one pusher flagellum. In this model, the boundary integral technique and Kirchhoff rod model are employed respectively to calculate the hydrodynamic forces on the swimmer and model the elastic deformations of the flagella. Our numerical results demonstrate that the model bacterium travels along a double helical trajectory, which is consistent with the experimental observations. Varying the stiffness, orientations or positions of the flagella significantly changes the swimming characteristics. Notably, when either the applied torque is higher than a critical value or the flagellum stiffness is lower than a critical stiffness, the pusher flagellum exhibits overwhirling motion, resulting in a more complicated swimming style and a lower swimming speed. For a moderate flagellum stiffness, the swimming speed is insensitive to the rest configuration orientation over a wide range of orientation angles as the flagella deform to maintain alignment with the swimming direction.

Interfacial hydroelastic hydraulic falls and trapped waves over bottom obstacles

Steady nonlinear flexural-gravity hydraulic falls on the interface of a two-layer density stratified flow past a submerged obstruction on the bottom of a channel are considered. The fluid is assumed to be ideal, and the flow is irrotational. We extend the previous works (Dias and Vanden-Broeck, 2002; Wang et al., 2022) by including the effect of hydroelasticity. The interface is modeled as a thin elastic shell with the Cosserat theory. The boundary integral equation techniques are employed to find steady solutions by numerically solving the full Euler equations. New solutions characterized by subcritical flow upstream with different depth ratios (thick upper layer, thick bottom layer, or critical depth) are found, and the effects of the aspect ratio of obstruction are investigated. By introducing a second obstruction downstream, solutions characterized by a train of trapped waves are sought with wavelength coherent with the prediction of the linear dispersion relation. In addition, solutions with a soliton-like form and oscillatory decaying tails are found when the sheet rigidity is small and the second obstruction is placed upstream.

Analysis of Two-Dimensional Heat Transfer Problem Using the Boundary Integral Equation

In this paper, we examine the problem of two-dimensional heat equations with certain initial and boundary conditions being considered. In a two-dimensional heat transport problem, the boundary integral equation technique was applied. The problem is expressed by an integral equation using the fundamental solution in Green’s identity. In this study, we transform the boundary value problem for the steady-state heat transfer problem into a boundary integral equation and drive the solution of the two-dimensional heat transfer problem using the boundary integral equation for the mixed boundary value problem by using Green’s identity and fundamental solution.

Boundary Integral Technique of 2nd Order Partial Differential Equation by Using Radial Basis

The solution of second order partial differential equation, with continuous change in coefficients by the formation of integral equation and then using radial basis function approximation (RBSA), has been developed in this paper. Use of boundary element method (BEM), which gives the solution of heat or mass diffusion in non-homogenous medium with function varying smoothly in space is also the part of this article. Discretization of boundary of integral domain instead of entire domain of the problem concerned is also the distinction of the recent work. The numerical solution of some problems with known value of the variable has also been included at the end.

A boundary integral method to investigate pattern formation in a rotating Hele-Shaw cell with time dependent gap

In this article we present a spectrally accurate boundary integral method to investigate the evolution of a Hele Shaw blob in a rotating, radial cell with time dependent gap. The blob of the viscous fluid is affected by the shearing forces induced by a changing gap distance, however the rotation also has significant influence on the shape of the blob. Using the method of complex moments, we characterize the solution of the moving boundary problem that models the evolution of the blob and investigate the morphology using linear analysis. The results of the linear analysis indicate the de-stabilizing influence of the rotation. In the numerical methods, we develop a spectrally accurate boundary integral technique to explore the complete nonlinear dynamics of the problem. Using the boundary integral simulations we demonstrate that, for a given strength of surface tension, the rotational effect causes a more complex pattern on the fluid interface that involves the formation of narrow fingers and more splitting of the interface. The effect is more pronounced if the angular speed increases. We also prescribe a method for controlling the viscous fingering using insights from linear analysis. Our numerical simulations validate the proposed controlling mechanism.

Viscoelastic dynamic response of asphalt pavement with imperfect interface bonding base on transfer matrix

AbstractThe interface bonding condition between asphalt pavement layers is always imperfect because of the differing material properties of different layers and the limitation of construction techniques. Therefore, designing asphalt pavement by considering the interface to be completely continuous is not precise. The viscoelastic character of the asphalt layer contributes significantly to the performance of asphalt pavement. To explore the influence of the imperfect interface on the viscoelastic dynamic response of asphalt pavement, the Hankel–Laplace transformation was applied to convert the partial differential equations to ordinary differential equations and a transfer matrix was introduced to describe the behavior of the imperfect interface. Based on the boundary condition and integral inverse transformation technique, an analytical solution for the displacement, stress, and strain of the pavement can be obtained. By considering the variation in the interface condition and the viscoelastic characteristics of the asphalt layer, the dynamic response of the pavement was calculated based on a practical example. The results show that the interface condition between the asphalt layer and the base course has a marked influence on the viscoelastic dynamic response of the pavement and that enhancing the strength of this interface can reduce the deflection of the pavement. The series elastic component in the modified Burgers model is another significant factor that affects the calculation results of the pavement dynamic response.

Interactive Fluid Coupling Effects of Non-Neighbouring Members.

Broadband, multi-functional and parallel-processing devices are often built on coupled oscillators or arrays of resonators. Different length scales and applications determine the dominating coupling mechanism of the device. In this paper we investigate the effects of interactive fluid coupling between members of a one-dimensional array wherein only one member is actuated. We are specifically interested in studying the influence of non-neighbouring members in small-size arrays comprising of three and five members for different Reynolds numbers and gap widths between members. Our model and analysis is based on the Navier–Stokes equation for incompressible flow which is solved using a boundary integral technique resulting in the hydrodynamic coupling matrix through which added mass and damping effects are inferred. Results clearly suggest that non-neighbouring members play a significant role for most typical array configurations and therefore cannot be ignored. In particular, arrays with more than three members must account for the behaviour of such a device with all member interactions. Thus, predicting the performance of most new and emerging technologies such as sensors and biomedical devices is determined by array effects rather than local, nearest neighbour influences.

Numerical solution of a cavity problem under surface tension effect

Free‐surface flow past two inclined plate is considered. The “Riabouchinsky model” has been chosen to close the cavitiy. The fluid is assumed to be inviscid and incompressible and the flow to be two dimensional, steady, and irrotational. Surface tension is included in the dynamic boundary conditions, but the effects of gravity are neglected. The problem is solved numerically using boundary integral equation techniques. More specifically, the numerical method used is based on an integro‐differential equation reformulation. Numerical solutions are found for different values of the angle of inclination γ and for various values of the inverse Weber number δ.

Minuet motion of a pair of capsules interacting in simple shear flow

We study the three-dimensional hydrodynamic interaction of a pair of identical, initially spherical capsules freely suspended in a simple shear flow under Stokes flow conditions. The capsules are filled with a Newtonian liquid (same density and viscosity as the suspending fluid). Their membranes satisfy the neo-Hookean constitutive law. We consider the rarely studied case where the capsule centres are initially located in (or near) the plane defined by the flow direction and the vorticity vector, i.e. in two different shear planes. The motion and deformation of the capsules are modelled by means of a boundary integral technique to compute the flows, coupled to a finite element method to calculate the force exerted by the membranes on the fluids. We follow the motion and deformation of the capsules as they are convected towards each other after a sudden start of the flow. Our main finding is that, depending on their initial position and deformability, the two capsules may oscillate slowly about the flow gradient axis, get nearer to each other at each oscillation to finally interact strongly and separate. This minuet motion had not been identified previously. We identify the regions of space where either simple crossing or minuet occurs. This phenomenon has a marked influence on the irreversible trajectory drift of two capsules after crossing: the minuet process leads to a significant trajectory displacement along the flow gradient when none was expected, based on the previous studies where the two capsules had a significant relative velocity.

Numerical Discrete-Domain Integral Formulations for Generalized Burger-Fisher Equation

In this study we use a boundary integral element-based numerical technique to solve the generalized Burger-Fisher equation. The essential feature of this method is the fundamental integral representation of the solution inside the problem domain by means of both the boundary and domain values. The occurrences of domain integrals within the problem arising from nonlinearity as well as the temporal derivative are not avoided or transferred to the boundary. However, unlike the classical boundary element approach, they are resolved within a finite-element-type discrete domain. The utility and correctness of this formulation are proved by comparing the results obtained herein with closed form solutions.

Free surface flows over a successive obstacles with surface tension and gravity effects

The problem of steady two-dimensional flow of a fluid of finite depth over a successive obstacles is considered. Both gravity and surface tension are taken into account in the dynamic boundary conditions. The fluid is assumed to be inviscid, incompressible and the flow to be irrotational. The flow is characterized by the two parameters, the Froude number $Fr$ and the inverse Weber number $\delta$. The fully non-linear problem is solved numerically by using the boundary integral equation technique. The numerical solutions for sub-critical ($Fr 1)$ are presented for various values of $Fr$ and $\delta$. The effects of surface tension and gravity on the shape of the free surface are discussed, and solution diagrams for all flow regimes are presented.

Simplified Integral Calculations for Radial Fin with Temperature-Dependent Thermal Conductivity

Numerical solution of a radiative radial fin with temperature-dependent thermal conductivity is presented. Calculations are implemented along the lines of a boundary integral technique coupled with domain discretization. Localized solutions of the nonlinear governing differential equation are sought on each element of the problem domain after enforcing inter-nodal connectivity as well as the boundary conditions for the dependent variables. A finite element-type assembly of the element equations and matrix solution yield the scalar profile. Comparison of the numerical results with those found in literature validates the formulation. The effects of such problem parameters as radiation-sink temperature, thermal conductivity, radiation-conduction fin parameter, volumetric heat generation, on the scalar profile were found to be in conformity with the physics of the problem. We also observed from this study that the volumetric heat generation plays a significant role in the overall heat transfer activity for a fin. For relatively high values of internal heat generation, a situation arises where a greater percentage of this energy can not escape to the environment and the fin ends up gaining energy instead of losing it. And the overall fin performance deteriorates. The same can also be said for the radiation-conduction parameter , whose increases can only give physically realistic results below a certain threshold value.

Analyzing bowtie structures with sharp tips by a vertical mode expansion method.

Bowtie structures of metallic nanoparticles are very effective in producing strong local fields needed in many applications. Existing numerical studies on bowtie structures are limited to those with rounded tips. Due to the field singularities at sharp edges and corners, accurate numerical solutions for bowtie structures with mathematically sharp tips are difficult to obtain. Based on an improved vertical mode expansion method (VMEM) that incorporates boundary integral equation techniques for domains with corners, we analyze bowtie structures with truly sharp tips. Numerical results are presented to reveal the effects of a few key factors including the distance between the tips, the apex angle and the substrate.

Solitary flexural-gravity waves in three dimensions.

The focus of this work is on three-dimensional nonlinear flexural–gravity waves, propagating at the interface between a fluid and an ice sheet. The ice sheet is modelled using the special Cosserat theory of hyperelastic shells satisfying Kirchhoff's hypothesis, presented in (Plotnikov & Toland. 2011 Phil. Trans. R. Soc. A 369, 2942–2956 (doi:10.1098/rsta.2011.0104)). The fluid is assumed inviscid and incompressible, and the flow irrotational. A numerical method based on boundary integral equation techniques is used to compute solitary waves and forced waves to Euler's equations.This article is part of the theme issue ‘Modelling of sea-ice phenomena’.

An integral equation method for the numerical solution of a Dirichlet problem for second-order elliptic equations with variable coefficients

We develop a numerical approximation involving boundary integral techniques for the solution of the Dirichlet problem for second-order elliptic equations with variable coefficients. Using the concept of a parametrix, the problem is reduced to a boundary-domain integral equation to be solved for two unknown densities. Via a change of variables based on shrinkage of the boundary curve of the solution domain a parameterised system of boundary-domain integrals is obtained. It is shown how to write the singularities in this system in an explicit form such that boundary integral techniques can be applied for analysis and discretisation. An effective discretisation involving the Nystrom method is given, together with numerical experiments showing that the proposed approach can be turned into a practical working method.

Nonlinear studies of tumor morphological stability using a two-fluid flow model.

We consider the nonlinear dynamics of an avascular tumor at the tissue scale using a two-fluid flow Stokes model, where the viscosity of the tumor and host microenvironment may be different. The viscosities reflect the combined properties of cell and extracellular matrix mixtures. We perform a linear morphological stability analysis of the tumors, and we investigate the role of nonlinearity using boundary-integral simulations in two dimensions. The tumor is non-necrotic, although cell death may occur through apoptosis. We demonstrate that tumor evolution is regulated by a reduced set of nondimensional parameters that characterize apoptosis, cell-cell/cell-extracellular matrix adhesion, vascularization and the ratio of tumor and host viscosities. A novel reformulation of the equations enables the use of standard boundary integral techniques to solve the equations numerically. Nonlinear simulation results are consistent with linear predictions for nearly circular tumors. As perturbations develop and grow, the linear and nonlinear results deviate and linear theory tends to underpredict the growth of perturbations. Simulations reveal two basic types of tumor shapes, depending on the viscosities of the tumor and microenvironment. When the tumor is more viscous than its environment, the tumors tend to develop invasive fingers and a branched-like structure. As the relative ratio of the tumor and host viscosities decreases, the tumors tend to grow with a more compact shape and develop complex invaginations of healthy regions that may become encapsulated in the tumor interior. Although our model utilizes a simplified description of the tumor and host biomechanics, our results are consistent with experiments in a variety of tumor types that suggest that there is a positive correlation between tumor stiffness and tumor aggressiveness.

Overresolving in the Laplace Domain for Convolution Quadrature Methods

Convolution quadrature (CQ) methods have enjoyed tremendous interest in recent years as an efficient tool for solving time-domain wave problems in unbounded domains via boundary integral equation techniques. In this paper we consider CQ type formulations for the parallel space-time evaluation of multistep or stiffly accurate Runge--Kutta rules for the wave equation. In particular, we decouple the number of Laplace domain solves from the number of time steps. This allows us to overresolve in the Laplace domain by computing more Laplace domain solutions than there are time steps. We use techniques from complex approximation theory to analyze the error of the CQ approximation of the underlying time-stepping rule when overresolving in the Laplace domain and show that the performance is intimately linked to the location of the poles of the solution operator. Several examples using boundary integral equation formulations in the Laplace domain are presented to illustrate the main results.

Investigation of breaking and non-breaking solitary waves and measurements of swash zone dynamics on a 5° beach

This study presents an experimental investigation of plunging breakers on a sloping beach with an inclination of 5.1°. The incident waves are solitary waves with various amplitudes from non-breaking waves to plunging breakers, and the area investigated is the swash zone. PIV (Particle Image Velocimetry) is performed on images captured at four different field of views (FOV). Shoreline position and maximum runup are measured, and are repeatable in both time and height, although cross-sectional variations of the shoreline shape are observed at maximum runup. For non-breaking waves the runup and fluid flow is computed by a boundary integral technique combined with a boundary layer model. Then, there is excellent agreement between the experimental and the computed velocity profiles at the lower region of the beach, while the boundary integral technique overpredicts the maximum runup height severely. For breaking waves the experiments indicate that the motion becomes more irregular as we move further up the beach. In addition, there are more irregularities present for waves with larger amplitude. Length and velocity of air bubbles entrapped by the plunging breakers are extracted from an image series captured with a large FOV. The images showed that a large air bubble remains intact for a time period during runup for the breaking waves.