Solving the problem of diffraction of an optical-band electromagnetic wave by metal nanostructured aperture arrays with the use of the method of impedance boundary conditions

The problem of diffraction of an optical wave by a 2D periodic metal aperture array with square, circular, and ring apertures is solved with allowance for the finite permittivity of a metal in the optical band. The correctness of the obtained results is verified through comparison with experimental data. It is shown that the transmission coefficient can be substantially greater than the corresponding value reached in the case of diffraction by a grating in a perfectly conducting screen.

A high frequency boundary element method for scattering by convex polygons with impedance boundary conditions

We consider scattering of a time harmonic incident plane wave by a convex polygon with piecewise constant impedance boundary conditions. Standard finite or boundary element methods require the number of degrees of freedom to grow at least linearly with respect to the frequency of the incident wave in order to maintain accuracy. Extending earlier work by Chandler-Wilde and Langdon for the sound soft problem, we propose a novel Galerkin boundary element method, with the approximation space consisting of the products of plane waves with piecewise polynomials supported on a graded mesh with smaller elements closer to the corners of the polygon. Theoretical analysis and numerical results suggest that the number of degrees of freedom required to achieve a prescribed level of accuracy grows only logarithmically with respect to the frequency of the incident wave.

Stability results for the time-harmonic Maxwell equations with impedance boundary conditions

We consider the time-harmonic Maxwell equations with constant coefficients in a bounded, uniformly star-shaped polyhedron. We prove wavenumber-explicit norm bounds for weak solutions. This result is pivotal for convergence proofs in numerical analysis and may be a tool in the analysis of electromagnetic boundary integral operators.

Fluid-solid interaction in arteries incorporating the autoregulation concept in boundary conditions

In pre-surgery decisions in hospital emergency cases, fast and reliable results of the solid and fluid mechanics problems are of great interest to clinicians. In the current investigation, an iterative process based on a pressure-type boundary condition is proposed in order to reduce the computational costs of blood flow simulations in arteries, without losing control of the important clinical parameters. The incorporation of cardiovascular autoregulation, together with the well-known impedance boundary condition, forms the basis of the proposed methodology. With autoregulation, the instabilities associated with conventional pressure-type or impedance boundary conditions are avoided without an excessive increase in computational costs. The general behaviour of pulsatile blood flow in arteries, which is important from the clinical point of view, is well reproduced through this new methodology. In addition, the interaction between the blood and the arterial walls occurs via a modified weak coupling, which makes the simulation more stable and computationally efficient. Based on in vitro experiments, the hyperelastic behaviour of the wall is characterised and modelled. The applications and benefits of the proposed pressure-type boundary condition are shown in a model of an idealised aortic arch with and without an ascending aorta dissection, which is a common cardiovascular disorder.

Influence of Outlet Boundary Conditions on Cerebrovascular Aneurysm Hemodynamics

Computational fluid dynamic (CFD) studies of blood flow in cerebrovascular aneurysms have potential to improve patient treatment planning by enabling clinicians and engineers to model patient-specific geometries and compute predictors and risks prior to neurovascular intervention. However, the use of patient-specific computational models in clinical settings is unfeasible due to their complexity, computationally intensive and time-consuming nature. An important factor contributing to this challenge is the choice of outlet boundary conditions, which often involves a trade-off between physiological accuracy, patient-specificity, simplicity and speed. In this study, we analyze how resistance and impedance outlet boundary conditions affect blood flow velocities, wall shear stresses and pressure distributions in a patient-specific model of a cerebrovascular aneurysm. We also use geometrical manipulation techniques to obtain a model of the patient’s vasculature prior to aneurysm development, and study how forces and stresses may have been involved in the initiation of aneurysm growth. Our CFD results show that the nature of the prescribed outlet boundary conditions is not as important as the relative distributions of blood flow through each outlet branch. As long as the appropriate parameters are chosen to keep these flow distributions consistent with physiology, resistance boundary conditions, which are simpler, easier to use and more practical than their impedance counterparts, are sufficient to study aneurysm pathophysiology, since they predict very similar wall shear stresses, time-averaged wall shear stresses, time-averaged pressures, and blood flow patterns and velocities. The only situations where the use of impedance boundary conditions should be prioritized is if pressure waveforms are being analyzed, or if local pressure distributions are being evaluated at specific time points, especially at peak systole, where the use of resistance boundary conditions leads to unnaturally large pressure pulses. In addition, we show that in this specific patient, the region of the blood vessel where the neck of the aneurysm developed was subject to abnormally high wall shear stresses, and that regions surrounding blebs on the aneurysmal surface were subject to low, oscillatory wall shear stresses. Computational models using resistance outlet boundary conditions may be suitable to study patient-specific aneurysm progression in a clinical setting, although several other challenges must be addressed before these tools can be applied clinically.

Numerical approximation of a fractional-in-space diffusion equation (II) - with nonhomogeneous boundary conditions

In this paper, a space fractional di®usion equation (SFDE) with non- homogeneous boundary conditions on a bounded domain is considered. A new matrix transfer technique (MTT) for solving the SFDE is proposed. The method is based on a matrix representation of the fractional-in-space operator and the novelty of this approach is that a standard discretisation of the operator leads to a system of linear ODEs with the matrix raised to the same fractional power. Analytic solutions of the SFDE are derived. Finally, some numerical results are given to demonstrate that the MTT is a computationally e±cient and accurate method for solving SFDE.

Natural convection in attic-shaped spaces subject to sudden and ramp heating boundary conditions

In this study, a discussion of the fluid dynamics in the attic space is reported, focusing on its transient response to sudden and linear changes of temperature along the two inclined walls. The transient behaviour of an attic space is relevant to our daily life. The instantaneous and non-instantaneous (ramp) heating boundary condition is applied on the sloping walls of the attic space. A theoretical understanding of the transient behaviour of the flow in the enclosure is performed through scaling analysis. A proper identification of the timescales, the velocity and the thickness relevant to the flow that develops inside the cavity makes it possible to predict theoretically the basic flow features that will survive once the thermal flow in the enclosure reaches a steady state. A time scale for the heating-up of the whole cavity together with the heat transfer scales through the inclined walls has also been obtained through scaling analysis. All scales are verified by the numerical simulations.

Natural convection in attics subject to instantaneous and ramp cooling boundary conditions

A fundamental study of the fluid dynamics inside an attic shaped triangular enclosure with cold upper walls and adiabatic horizontal bottom wall is reported in this study. The transient behaviour of the attic fluid which is relevant to our daily life is examined based on a scaling analysis. The transient phenomenon begins with the instantaneous cooling and the cooling with linear decreases of temperature up to some specific time (ramp time) and then maintain constant of the upper sloped walls. It is shown that both inclined walls develop a thermal boundary layer whose thicknesses increase towards steady-state or quasi-steady values. A proper identification of the timescales, the velocity and the thickness relevant to the flow that develops inside the cavity makes it possible to predict theoretically the basic flow features that will survive once the thermal flow in the enclosure reaches a steady state. A time scale for the cooling-down of the whole cavity together with the heat transfer scales through the inclined walls has also been obtained through scaling analysis. All scales are verified by the numerical simulations.

An analytical framework for quantifying aquifer response time scales associated with transient boundary conditions

A major challenge in studying coupled groundwater and surface-water interactions arises from the considerable difference in the response time scales of groundwater and surface-water systems affected by external forcings. Although coupled models representing the interaction of groundwater and surface-water systems have been studied for over a century, most have focused on groundwater quantity or quality issues rather than response time. In this study, we present an analytical framework, based on the concept of mean action time (MAT), to estimate the time scale required for groundwater systems to respond to changes in surface-water conditions. MAT can be used to estimate the transient response time scale by analyzing the governing mathematical model. This framework does not require any form of transient solution (either numerical or analytical) to the governing equation, yet it provides a closed form mathematical relationship for the response time as a function of the aquifer geometry, boundary conditions, and flow parameters. Our analysis indicates that aquifer systems have three fundamental time scales: (i) a time scale that depends on the intrinsic properties of the aquifer; (ii) a time scale that depends on the intrinsic properties of the boundary condition, and; (iii) a time scale that depends on the properties of the entire system. We discuss two practical scenarios where MAT estimates provide useful insights and we test the MAT predictions using new laboratory-scale experimental data sets.

Fluid-structure interaction study on human arterial plaque with patient specific geometry and boundary conditions

Atherosclerotic plaque rupture has been extensively considered as the leading cause of death in western countries. It is believed that high stresses within plaque can be an important factor on triggering the rupture of the plaque. Stress analysis in the coronary and carotid arteries with plaque have been developed by many researchers from 2D to 3-D models, from structure analysis only to the Fluid-Structure Interaction (FSI) models[1].

The effect of closed boundary conditions on a stationary dynamo

One of two boundary conditions generally assumed in solutions of the dynamo equation is related to the disappearance of the azimuthal field at the boundary. Parker (1984) points out that for the realization of this condition the field must escape freely through the surface. Escape requires that the field be detached from the gas in which it is embedded. In the case of the sun, this can be accomplished only through reconnection in the tenuous gas above the visible surface. Parker concludes that the observed magnetic activity on the solar surface permits at most three percent of the emerging flux to escape. He arrives at the conclusion that, instead of B(phi) = 0, the partial derivative of B(phi) to r is equal to zero. The present investigation is concerned with the effect of changing the boundary condition according to Parker's conclusion. Implications for the solar convection zone are discussed.

A Direct Approach to the Problem of Diffraction by a Half-Plane under Mixed Boundary Conditions

A general direct technique of solving a mixed boundary value problem in the theory of diffraction by a semi-infinite plane is presented. Taking account of the correct edge-conditions, the unique solution of the problem is derived, by means of Jones' method in the theory of Wiener-Hopf technique, in the case of incident plane wave. The solution of the half-plane problem is found out in exact form. (The far-field is derived by the method of steepest descent.) It is observed that it is not the Wiener-Hopf technique which really needs any modification but a new technique is certainly required to handle the peculiar type of coupled integral equations which the Wiener-Hopf technique leads to. Eine allgemeine direkte Technik zur Lösung eines gemischten Randwertproblems in der Theorie der Beugung an einer halbunendlichen Ebene wird vorgestellt. Unter Berücksichtigung der korrekten Eckbedingungen wird mit der Methode von Jones aus der Theorie der Wiener-Hopf-Technik die eindeutige Lösung für den Fall der einfallenden ebenen Welle hergeleitet. Die Lösung des Halbebenenproblems wird in exakter Form angegeben. (Das Fernfeld wurde mit der Methode des steilsten Abstiegs bestimmt.) Es wurde bemerkt, daß es nicht die Wiener-Hopf-Technik ist, die wirklich irgend welcher Modifikationen bedurfte. Gewiß aber wird eine neue Technik zur Behandlung des besonderen Typs gekoppelter Integralgleichungen benötigt, auf die die Wiener-Hopf-Technik führt.