arXiv subjects

Evert Klaseboer

Publications and source records attributed to Evert Klaseboer.

16 recordsLinked to original sources

arXiv2024

Efficient Field-Only Surface Integral Equations for Electromagnetics

In a recent paper, Klaseboer et al. (IEEE Trans. Antennas Propag., vol. 65, no. 2, pp. 972-977, Feb. 2017) developed a surface integral formulation of electromagnetics that does not require working with integral equations that have singular kernels. Instead of solving for the induced surface currents, the method involves surface integral solutions for 4 coupled Helmholtz equations: 3 for each Cartesian component of the electric E field plus 1 for the scalar function r*E on the surface of scatterers. Here we improve on this approach by advancing a formulation due to Yuffa et al. (IEEE Trans.Antennas Propag., vol. 66, no. 10, pp. 5274-5281, Oct. 2018) that solves for E and its normal derivative. Apart from a 25% reduction in problem size, the normal derivative of the field is often of interest in micro-photonic applications.

physics.comp-ph↗

arXiv2022

Helmholtz equation and non-singular boundary elements applied to multi-disciplinary physical problems

The famous scientist Hermann von Helmholtz was born 200 years ago. Many complex physical wave phenomena in engineering can effectively be described using one or a set of equations named after him: the Helmholtz equation. Although this has been known for a long time from a theoretical point of view, the actual numerical implementation has often been hindered by divergence free and/or curl free constraints. There is further a need for a numerical method that is accurate, reliable and takes into account radiation conditions at infinity. The classical boundary element method (BEM) satisfies the last condition, yet one has to deal with singularities in the implementation. We review here how a recently developed singularity-free three-dimensional (3D) boundary element framework with superior accuracy can be used to tackle such problems only using one or a few Helmholtz equations with higher order (quadratic) elements which can tackle complex curved shapes. Examples are given for acoustics (a Helmholtz resonator among others) and electromagnetic scattering.

math.NA↗

arXiv2022

A universal model for drag on a spherical bubble

A theoretical expression for the drag on a spherical bubble is derived for the entire range from very viscous to inertial flow conditions. It is based on a solution for only that part of the velocity profile that determines the drag. It is assumed the surface of the bubble has a zero tangential stress condition. Excellent agreement with a previously proposed empirical model by Mei et al. is obtained. This shows that a theoretical framework with relatively simple physics can still predict the terminal velocity of a spherical bubble accurately. To the best of our knowledge, this is one of the few models in fluid dynamics to predict drag on an object for a range of Reynolds numbers that spans many orders of magnitude.

physics.flu-dyn↗

arXiv2022

Analytical solution for a vibrating rigid sphere with an elastic shell in an infinite linear elastic medium

The analytical solution is given for a vibrating rigid core sphere, oscillating up and down without volume change, situated at the center of an elastic material spherical shell, which in turn is situated inside an infinite (possible different) elastic medium. The solution is based on symmetry considerations and the continuity of the displacement both at the core and the shell - outer medium boundaries as well as the continuity of the stress at the outer edge of the shell. Furthermore, a separation into longitudinal and transverse waves is used. Analysis of the solution shows that a surprisingly complex range of physical phenomena can be observed when the frequency is changed while keeping the material parameters the same, especially when compared to the case of a core without any shell. With a careful choice of materials, shell thickness and vibration frequency, it is possible to filter out most of the longitudinal waves and generate pure tangential waves in the infinite domain (and vice-versa, we can filter out the tangential waves and generate longitudinal waves). When the solution is applied to different frequencies and with the help of a fast Fourier transform (FFT), a pulsed vibration is shown to exhibit the separation of the longitudinal (L) and transverse (T) waves (often called P- and S-waves in earthquake terminology).

physics.geo-ph↗

arXiv2022

A non-singular, field-only surface integral method for interactions between electric and magnetic dipoles and nano-structures

With the development of condensed-matter physics and nanotechnology, attention has turned to the fields near and on surfaces that result from interactions between electric dipole radiation and mesoscale structures. It is hoped that studying these fields will further our understanding of optical phenomena in nano-optics, quantum mechanics, electromagnetics and sensing using solid-state photon emitters. Here, we describe a method for implementing dynamic electric and magnetic dipoles in the frequency domain into a non-singular field-only surface integral method. We show that the effect of dipoles can conveniently be described as a relatively simple term in the integral equations, which fully represents how they drive the fields and interactions. Also, due to the non-singularity, our method can calculate the electric and magnetic fields on the surfaces of objects in both near and far fields with the same accuracy, which makes it an ideal tool to investigate nano-optical phenomena. The derivation of the framework is given and tested against a Mie theory alike formula. Some interesting examples are shown involving the interaction of dipoles with different types of mesoscale structures including parabolic nano-antennas and gold probes.

physics.optics↗

arXiv2020

Analytical solution for an acoustic boundary layer around an oscillating rigid sphere

Analytical solutions in fluid dynamics can be used to elucidate the physics of complex flows and to serve as test cases for numerical models. In this work, we present the analytical solution for the acoustic boundary layer that develops around a rigid sphere executing small amplitude harmonic rectilinear motion in a compressible fluid. The mathematical framework that describes the primary flow is identical to that of wave propagation in linearly elastic solids, the difference being the appearance of complex instead of real valued wave numbers. The solution reverts to well-known classical solutions in special limits: the potential flow solution in the thin boundary layer limit, the oscillatory flat plate solution in the limit of large sphere radius and the Stokes flow solutions in the incompressible limit of infinite sound speed. As a companion analytical result, the steady second order acoustic streaming flow is obtained. This streaming flow is driven by the Reynolds stress tensor that arises from the axisymmetric first order primary flow around such a rigid sphere. These results are obtained with a linearization of the non-linear Navier-Stokes equations valid for small amplitude oscillations of the sphere. The streaming flow obeys a time-averaged Stokes equation with a body force given by the Nyborg model in which the above mentioned primary flow in a compressible Newtonian fluid is used to estimate the time-averaged body force. Numerical results are presented to explore different regimes of the complex transverse and longitudinal wave numbers that characterize the primary flow.

physics.flu-dyn↗

arXiv2019

Eliminating the fictitious frequency problem in BEM solutions of the external Helmholtz equation

The problem of the fictitious frequency spectrum resulting from numerical implementations of the boundary element method for the exterior Helmholtz problem is revisited. When the ordinary 3D free space Green's function is replaced by a modified Green's function, it is shown that these fictitious frequencies do not necessarily have to correspond to the internal resonance frequency of the object. Together with a recently developed fully desingularized boundary element method that confers superior numerical accuracy, a simple and practical way is proposed for detecting and avoiding these fictitious solutions. The concepts are illustrated with examples of a scattering wave on a rigid sphere.

physics.comp-ph↗

arXiv2019

Robust Field-Only Surface Integral Equations: Scattering from a Dielectric Body

A robust and efficient field-only nonsingular surface integral method to solve Maxwell's equations for the components of the electric field on the surface of a dielectric scatterer is introduced. In this method, both the vector Helmholtz equation and the divergence-free constraint are satisfied inside and outside the scatterer. The divergence-free condition is replaced by an equivalent boundary condition that relates the normal derivatives of the electric field across the surface of the scatterer. Also, the continuity and jump conditions on the electric and magnetic fields are expressed in terms of the electric field across the surface of the scatterer. Together with these boundary conditions, the scalar Helmholtz equation for the components of the electric field inside and outside the scatterer is solved by a fully desingularized surface integral method. Comparing with the most popular surface integral methods based on the Stratton-Chu formulation or the PMCHWT formulation, our method is conceptually simpler and numerically straightforward because there is no need to introduce intermediate quantities such as surface currents and the use of complicated vector basis functions can be avoided altogether. Also, our method is not affected by numerical issues such as the zero frequency catastrophe and does not contain integrals with (strong) singularities. To illustrate the robustness and versatility of our method, we show examples in the Rayleigh, Mie, and geometrical optics scattering regimes. Given the symmetry between the electric field and the magnetic field, our theoretical framework can also be used to solve for the magnetic field.

physics.optics↗

arXiv2019

Robust Field-Only Surface Integral Equations: Scattering from a Perfect Electric Conductor

A robust field-only boundary integral formulation of electromagnetics is derived without the use of surface currents that appear in the Stratton-Chu formulation. For scattering by a perfect electrical conductor (PEC), the components of the electric field are obtained directly from surface integral equation solutions of three scalar Helmholtz equations for the field components. The divergence-free condition is enforced via a boundary condition on the normal component of the field and its normal derivative. Field values and their normal derivatives at the surface of the PEC are obtained directly from surface integral equations that do not contain divergent kernels. Consequently, high-order elements with fewer degrees of freedom can be used to represent surface features to a higher precision than the traditional planar elements. This theoretical framework is illustrated with numerical examples that provide further physical insight into the role of the surface curvature in scattering problems.

physics.comp-ph↗

arXiv2018

Helmholtz Decomposition and Boundary Element Method applied to Dynamic Linear Elastic Problems

The displacement field for three dimensional dynamic elasticity problems in the frequency domain can be decomposed into a sum of a longitudinal and a transversal part known as a Helmholtz decomposition. The Cartesian components of both the longitudinal and transverse fields satisfy scalar Helmholtz equations that can be solved using a desingularized boundary element method (BEM) framework. The curl free longitudinal and divergence free transversal conditions can also be cast as additional scalar Helmholtz equations. When compared to other BEM implementations, the current framework leads to smaller matrix dimensions and a simpler conceptual approach. The numerical implementation of this approach is benchmarked against the 3D elastic wave field generated by a rigid vibrating sphere embedded in an infinite linear elastic medium for which the analytical solution has been derived. Examples of focussed 3D elastic waves generated by a vibrating bowl-shaped rigid object with convex and concave surfaces are also considered. In the static zero frequency limit, the Helmholtz decomposition becomes non-unique, and both the longitudinal and transverse components contain divergent terms that are proportional to the inverse square of the frequency. However, these divergences are equal and opposite so that their sum, that is the displacement field that reflects the physics of the problem, remains finite in the zero frequency limit.

physics.comp-ph↗

arXiv2018

A Robust Multi-Scale Field-Only Formulation of Electromagnetic Scattering

We present a boundary integral formulation of electromagnetic scattering by homogeneous bodies that are characterized by linear constitutive equations in the frequency domain. By working with the Cartesian components of the electric, E and magnetic, H fields and with the scalar functions (r*E) and (r*H), the problem is cast as solving a set of scalar Helmholtz equations for the field components that are coupled by the usual electromagnetic boundary conditions at material boundaries. This facilitates a direct solution for E and H rather than working with surface currents as intermediate quantities in existing methods. Consequently, our formulation is free of the well-known numerical instability that occurs in the zero frequency or long wavelength limit in traditional surface integral solutions of Maxwell's equations and our numerical results converge uniformly to the static results in the long wavelength limit. Furthermore, we use a formulation of the scalar Helmholtz equation that is expressed as classically convergent integrals and does not require the evaluation of principal value integrals or any knowledge of the solid angle. Therefore, standard quadrature and higher order surface elements can readily be used to improve numerical precision. In addition, near and far field values can be calculated with equal precision and multiscale problems in which the scatterers possess characteristic length scales that are both large and small relative to the wavelength can be easily accommodated. From this we obtain results for the scattering and transmission of electromagnetic waves at dielectric boundaries that are valid for any ratio of the local surface curvature to the wave number. This is a generalization of the familiar Fresnel formula and Snell's law, valid at planar dielectric boundaries, for the scattering and transmission of electromagnetic waves at surfaces of arbitrary curvature.

physics.comp-ph↗

arXiv2018

Non-singular field-only surface integral equations for electromagnetic scattering

A boundary integral formulation of electromagnetics that involves only the components of $\boldsymbol{E}$ and $\boldsymbol{H}$ is derived without the use of surface currents that appear in the classical PMCHWT formulation. The kernels of the boundary integral equations for $\boldsymbol{E}$ and $\boldsymbol{H}$ are non-singular so that all field quantities at the surface can be determined to high precision and also geometries with closely spaced surfaces present no numerical difficulties. Quadratic elements can readily be used to represent the surfaces so that the surface integrals can be calculated to higher numerical precision than using planar elements for the same numbers of degrees of freedom.

physics.comp-ph↗

arXiv2018

Field-only integral equation method for time domain scattering of electromagnetic pulses

The scattering of electromagnetic pulses is described using a non-singular boundary integral method to solve directly for the field components in the frequency domain, and Fourier transform is then used to obtain the complete space-time behavior. This approach is stable for wavelengths both small and large relative to characteristic length scales. Amplitudes and phases of field values can be obtained accurately on or near material boundaries. Local field enhancement effects due to multiple scattering of interest to applications in microphotonics are demonstrated.

physics.optics↗

arXiv2017

Space-time domain solutions of the wave equation by a non-singular boundary integral method and Fourier transform

The general space-time evolution of the scattering of an incident acoustic plane wave pulse by an arbitrary configuration of targets is treated by employing a recently developed non-singular boundary integral method to solve the Helmholtz equation in the frequency domain from which the fast Fourier transform is used to obtain the full space-time solution of the wave equation. The non-singular boundary integral solution can enforce the radiation boundary condition at infinity exactly and can account for multiple scattering effects at all spacings between scatterers without adverse effects on the numerical precision. More generally, the absence of singular kernels in the non-singular integral equation confers high numerical stability and precision for smaller numbers of degrees of freedom. The use of fast Fourier transform to obtain the time dependence is not constrained to discrete time steps and is particularly efficient for studying the response to different incident pulses by the same configuration of scatterers. The precision that can be attained using a smaller number of Fourier components is also quantified.

physics.comp-ph↗

arXiv2009

Dynamic interactions between deformable drops in the Hele-Shaw geometry

A model has been developed to describe the collision and possible coalescence of two driven deformable drops in the Hele-Shaw cell. The interdependence between hydrodynamic effects and interfacial deformations is characterised by a film capillary number: Ca_f = (μv/σ)(R_o/h_o)^(3/2) as revealed by an analytic perturbation solution of the governing equations for a system with continuous phase viscosity μ, interfacial tension σ, drop radius R_o, characteristic relative velocity v and separation h_o between the drops. Numerical solutions of the model demonstrate the importance of the full dynamic history of the interacting drops in determining stability or coalescence. The geometry of the Hele-Shaw cell allows for the possibility of using the model to infer the time dependent force between colliding drops by measuring their separation.

cond-mat.soft↗

arXiv2009

Dynamic Deformations and Forces in Soft Matter

The ability of soft matter such as drops and bubbles to change shape dynamically during interaction can give rise to counter-intuitive behaviour that may be expected of rigid materials. Here we show that dimple formation on approach and the possibility of coalescence on separation of proximal drops in relative motion are examples of this general dynamic behaviour of soft matter that arise from the coupling between hydrodynamic forces and geometric deformations. The key parameter is a film capillary number Caf = (muVo/sigma)(R/Ho)2 that depends on viscosity mu, interfacial tension sigma, the Laplace radius R, characteristic film thickness Ho and velocity Vo.

cond-mat.soft↗