Three-dimensional matched interface and boundary (MIB) method for treating geometric singularities

This paper reports the three-dimensional (3D) generalization of our previous 2D higher-order matched interface and boundary (MIB) method for solving elliptic equations with discontinuous coefficients and non-smooth interfaces. New MIB algorithms that make use of two sets of interface jump conditions are proposed to remove the critical acute angle constraint of our earlier MIB scheme for treating interfaces with sharp geometric singularities, such as sharp edges, sharp wedges and sharp tips. The resulting 3D MIB schemes are of second-order accuracy for arbitrarily complex interfaces with sharp geometric singularities, of fourth-order accuracy for complex interfaces with moderate geometric singularities, and of sixth-order accuracy for curved smooth interfaces. A systematical procedure is introduced to make the MIB matrix optimally symmetric and banded by appropriately choosing auxiliary grid points. Consequently, the new MIB linear algebraic equations can be solved with fewer number of iterations. The proposed MIB method makes use of Cartesian grids, standard finite difference schemes, lowest order interface jump conditions and fictitious values. The interface jump conditions are enforced at each intersecting point of the interface and mesh lines to overcome the staircase phenomena in finite difference approximation. While a pair of fictitious values are determined along a mesh at a time, an iterative procedure is proposed to determine all the required fictitious values for higher-order schemes by repeatedly using the lowest order jump conditions. A variety of MIB techniques are developed to overcome geometric constraints. The essential strategy of the MIB method is to locally reduce a 2D or a 3D interface problem into 1D-like ones. The proposed MIB method is extensively validated in terms of the order of accuracy, the speed of convergence, the number of iterations and CPU time. Numerical experiments are carried out to complex interfaces, including the molecular surfaces of a protein, a missile interface, and van der Waals surfaces of intersecting spheres.     

Time-Domain Numerical Solutions of Maxwell Interface Problems with Discontinuous Electromagnetic Waves

This paper is devoted to time domain numerical solutions of two-dimensional (2D) material interface problems governed by the transverse magnetic (TM) and transverse electric (TE) Maxwell's equations with discontinuous electromagnetic solutions. Due to the discontinuity in wave solutions across the interface, the usual numerical methods will converge slowly or even fail to converge. This calls for the development of advanced interface treatments for popular Maxwell solvers. We will investigate such interface treatments by considering two typical Maxwell solvers – one based on collocation formulation and the other based on Galerkin formulation. To restore the accuracy reduction of the collocation finite-difference time-domain (FDTD) algorithm near an interface, the physical jump conditions relating discontinuous wave solutions on both sides of the interface must be rigorously enforced. For this purpose, a novel matched interface and boundary (MIB) scheme is proposed in this work, in which new jump conditions are derived so that the discontinuous and staggered features of electric and magnetic field components can be accommodated. The resulting MIB time-domain (MIBTD) scheme satisfies the jump conditions locally and suppresses the staircase approximation errors completely over the Yee lattices. In the discontinuous Galerkin time-domain (DGTD) algorithm – a popular Galerkin Maxwell solver, a proper numerical flux can be designed to accurately capture the jumps in the electromagnetic waves across the interface and automatically preserves the discontinuity in the explicit time integration. The DGTD solution to Maxwell interface problems is explored in this work, by considering a nodal based high order discontinuous Galerkin method. In benchmark TM and TE tests with analytical solutions, both MIBTD and DGTD schemes achieve the second order of accuracy in solving circular interfaces. In comparison, the numerical convergence of the MIBTD method is slightly more uniform, while the DGTD method is more flexible and robust.     

Adaptively deformed mesh based interface method for elliptic equations with discontinuous coefficients

Mesh deformation methods are a versatile strategy for solving partial differential equations (PDEs) with a vast variety of practical applications. However, these methods break down for elliptic PDEs with discontinuous coefficients, namely, elliptic interface problems. For this class of problems, the additional interface jump conditions are required to maintain the well-posedness of the governing equation. Consequently, in order to achieve high accuracy and high order convergence, additional numerical algorithms are required to enforce the interface jump conditions in solving elliptic interface problems. The present work introduces an interface technique based adaptively deformed mesh strategy for resolving elliptic interface problems. We take the advantages of the high accuracy, flexibility and robustness of the matched interface and boundary (MIB) method to construct an adaptively deformed mesh based interface method for elliptic equations with discontinuous coefficients. The proposed method generates deformed meshes in the physical domain and solves the transformed governed equations in the computational domain, which maintains regular Cartesian meshes. The mesh deformation is realized by a mesh transformation PDE, which controls the mesh redistribution by a source term. The source term consists of a monitor function, which builds in mesh contraction rules. Both interface geometry based deformed meshes and solution gradient based deformed meshes are constructed to reduce the L(∞) and L(2) errors in solving elliptic interface problems. The proposed adaptively deformed mesh based interface method is extensively validated by many numerical experiments. Numerical results indicate that the adaptively deformed mesh based interface method outperforms the original MIB method for dealing with elliptic interface problems.     

A matched interface and boundary method for solving multi-flow Navier–Stokes equations with applications to geodynamics

We have developed a second-order numerical method, based on the matched interface and boundary (MIB) approach, to solve the Navier–Stokes equations with discontinuous viscosity and density on non-staggered Cartesian grids. We have derived for the first time the interface conditions for the intermediate velocity field and the pressure potential function that are introduced in the projection method. Differentiation of the velocity components on stencils across the interface is aided by the coupled fictitious velocity values, whose representations are solved by using the coupled velocity interface conditions. These fictitious values and the non-staggered grid allow a convenient and accurate approximation of the pressure and potential jump conditions. A compact finite difference method was adopted to explicitly compute the pressure derivatives at regular nodes to avoid the pressure–velocity decoupling. Numerical experiments verified the desired accuracy of the numerical method. Applications to geophysical problems demonstrated that the sharp pressure jumps on the clast-Newtonian matrix are accurately captured for various shear conditions, moderate viscosity contrasts and a wide range of density contrasts. We showed that large transfer errors will be introduced to the jumps of the pressure and the potential function in case of a large absolute difference of the viscosity across the interface; these errors will cause simulations to become unstable.     

An immersed interface method for Stokes flows with fixed/moving interfaces and rigid boundaries

We present an immersed interface method for solving the incompressible steady Stokes equations involving fixed/moving interfaces and rigid boundaries (irregular domains). The fixed/moving interfaces and rigid boundaries are represented by a number of Lagrangian control points. In order to enforce the prescribed velocity at the rigid boundaries, singular forces are applied on the fluid at these boundaries. The strength of singular forces at the rigid boundary is determined by solving a small system of equations. For the deformable interfaces, the forces that the interface exerts on the fluid are calculated from the configuration (position) of the deformed interface. The jumps in the pressure and the jumps in the derivatives of both pressure and velocity are related to the forces at the fixed/moving interfaces and rigid boundaries. These forces are interpolated using cubic splines and applied to the fluid through the jump conditions. The positions of the deformable interfaces are updated implicitly using a quasi-Newton method (BFGS) within each time step. In the proposed method, the Stokes equations are discretized via the finite difference method on a staggered Cartesian grid with the incorporation of jump contributions and solved by the conjugate gradient Uzawa-type method. Numerical results demonstrate the accuracy and ability of the proposed method to simulate incompressible Stokes flows with fixed/moving interfaces on irregular domains.     

MIB method for elliptic equations with multi-material interfaces

Elliptic partial differential equations (PDEs) are widely used to model real-world problems. Due to the heterogeneous characteristics of many naturally occurring materials and man-made structures, devices, and equipments, one frequently needs to solve elliptic PDEs with discontinuous coefficients and singular sources. The development of high-order elliptic interface schemes has been an active research field for decades. However, challenges remain in the construction of high-order schemes and particularly, for nonsmooth interfaces, i.e., interfaces with geometric singularities. The challenge of geometric singularities is amplified when they are originated from two or more material interfaces joining together or crossing each other. High-order methods for elliptic equations with multi-material interfaces have not been reported in the literature to our knowledge. The present work develops matched interface and boundary (MIB) method based schemes for solving two-dimensional (2D) elliptic PDEs with geometric singularities of multi-material interfaces. A number of new MIB schemes are constructed to account for all possible topological variations due to two-material interfaces. The geometric singularities of three-material interfaces are significantly more difficult to handle. Three new MIB schemes are designed to handle a variety of geometric situations and topological variations, although not all of them. The performance of the proposed new MIB schemes is validated by numerical experiments with a wide range of coefficient contrasts, geometric singularities, and solution types. Extensive numerical studies confirm the designed second order accuracy of the MIB method for multi-material interfaces, including a case where the derivative of the solution diverges.     

Augmented coupling interface method for solving eigenvalue problems with sign-changed coefficients

In this paper, we propose an augmented coupling interface method on a Cartesian grid for solving eigenvalue problems with sign-changed coefficients. The underlying idea of the method is the correct local construction near the interface which incorporates the jump conditions. The method, which is very easy to implement, is based on finite difference discretization. The main ingredients of the proposed method comprise (i) an adaptive-order strategy of using interpolating polynomials of different orders on different sides of interfaces, which avoids the singularity of the local linear system and enables us to handle complex interfaces; (ii) when the interface condition involves the eigenvalue, the original problem is reduced to a quadratic eigenvalue problem by introducing an auxiliary variable and an interfacial operator on the interface; (iii) the auxiliary variable is discretized uniformly on the interface, the rest of variables are discretized on an underlying rectangular grid, and a proper interpolation between these two grids are designed to reduce the number of stencil points. Several examples are tested to show the robustness and accuracy of the schemes.     

Full-vectorial mode solver for anisotropic optical waveguides using multidomain spectral collocation method

A full-vectorial mode solver in terms of the transverse magnetic field components for optical waveguides with transverse anisotropy is described by using the multidomain spectral collocation method based on Chebyshev polynomials. The waveguide cross section surrounded by the perfectly matched layers is divided into suitable number of homogeneous rectangles, and then connected with by imposing the continuities of the longitudinal field components at the dielectric interfaces shared by the adjacent rectangles, resulting in a generalized matrix eigenvalue problem. To validate the established method, results of an anisotropic square waveguide and a magnetooptic rib waveguide are presented and compared with those from the full-vectorial finite difference method, full-vectorial beam propagation method, and the experimental data.     

Optical vortices in low-mode fibers: III. Dislocation reactions, phase transitions, and topological birefringence

Topological birefringence of waves in optical fibers resulting from the spin-orbit interaction in the field of optical vortices is manifested, as a rule, in the form of Rytov-Magnus unified optical effect. At the same time, the field transformations caused by this effect are not explicitly related to the evolution of phase dislocations of longitudinal and transverse components of the electric and the magnetic fields. This relation can be provided by the dislocation reactions proposed by Berry. As opposed to the Berry’s approach, where dislocation reactions at the wavefront surface are considered, it is suggested in this work that topological reactions at the specific characteristic surface of the wave field formed by the coordinate representation of the transverse components of the Poynting vector be considered. Using the action of topological birefringence in a low-mode optical fiber as an example, it is shown that the course of a topological reaction in a vector optical field is accompanied by rigorous conservation of the total topological index of the characteristic surface and does not depend on the presence of an interface (where topological charges can originate and annihilate). The total topological index of a dislocation reaction is found to be equal to the absolute value of the sum of the topological charge and the spirality of the vector wave field.     

Method of integral equations in the theory of microstructured optical fibers

A method for designing microstructured optical fibers that is based on exact integral equations for the transverse components of the magnetic field of the mode is proposed. A solution to the vector waveguide problem for fibers with a finite number of circular capillaries in the round cavity of the cladding can be refined successively. Quartz fibers with hexagonal capillary rings are also studied.     

A second order virtual node method for elliptic problems with interfaces and irregular domains

We present a second order accurate, geometrically flexible and easy to implement method for solving the variable coefficient Poisson equation with interfacial discontinuities or on irregular domains, handling both cases with the same approach. We discretize the equations using an embedded approach on a uniform Cartesian grid employing virtual nodes at interfaces and boundaries. A variational method is used to define numerical stencils near these special virtual nodes and a Lagrange multiplier approach is used to enforce jump conditions and Dirichlet boundary conditions. Our combination of these two aspects yields a symmetric positive definite discretization. In the general case, we obtain the standard 5-point stencil away from the interface. For the specific case of interface problems with continuous coefficients, we present a discontinuity removal technique that admits use of the standard 5-point finite difference stencil everywhere in the domain. Numerical experiments indicate second order accuracy in L^∞.     

Electron–Nuclear Spin Dynamics in Semiconductor QDs

This work presents an overview of investigations of the nuclear spin dynamics in nanostructures with negatively charged InGaAs/GaAs quantum dots characterized by strong quadrupole splitting of nuclear spin sublevels. The main method of the investigations is the experimental measurements and the theoretical analysis of the photoluminescence polarization as a function of the transverse magnetic field (effect Hanle). The dependence of the Hanle curve profile on the temporal protocol of optical excitation is examined. Experimental data are analyzed using an original approach based on separate consideration of behavior of the longitudinal and transverse components of the nuclear polarization. The rise and decay times of each component of the nuclear polarization and their dependence on transverse magnetic field strength are determined. To study the role of the Knight field in the dynamic of nuclear polarization, a weak additional magnetic field parallel to the optical axis is used. We have found that, only taking into account the nuclear spin fluctuations, we can accurately describe the measured Hanle curves and evaluate the parameters of the electron–nuclear spin system in the studied quantum dots. A new effect of the resonant optical pumping of nuclear spin polarization in an ensemble of the singly charged (In,Ga)As/GaAs quantum dots subjected to a transverse magnetic field is discussed. Nuclear spin resonances for all isotopes in the quantum dots are detected in that way. In particular, transitions between the states split off from the ±1/2 doublets by the nuclear quadrupole interaction are identified.     

Revisit to the THINC scheme: A simple algebraic VOF algorithm

This short note presents an improved multi-dimensional algebraic VOF method to capture moving interfaces. The interface jump in the THINC (tangent of hyperbola for INterface capturing) scheme is adaptively scaled to a proper thickness according to the interface orientation. The numerical accuracy in computing multi-dimensional moving interfaces is significantly improved. Without any geometrical reconstruction, the proposed method is extremely simple and easy to use, and its numerical accuracy is superior to other existing methods of its kind and comparable to the conventional PLIC (piecewise linear interface calculation) type VOF schemes.     

Numerical approach and experimental verification for interface shape determination between layered fluids subject to a magnetic field

Various applications of magnetic fluids involve interface phenomena. The analysis of the hydrostatic interface shape between two immiscible liquid layers, especially under magnetic field influence, is the first step to understand the accompanying complex dynamic phenomena as well as to providing reliable numerical capabilities for their accurate prediction. This study presents a relatively simple numerical approach, and the accompanying theory, to reliably define the meniscus shape in a two-layered fluid system in presence of a horizontal magnetic field with a vertical gradient. In the course of the study, two dimensionless parameters have been derived to describe the magnetic pressure jump at the interface and the magnetic body force throughout the volume. These parameters are used to interpret the results of the analysis and to show that a horizontal magnetic field tends to flatten the meniscus shape at the interface despite of the direction of its vertical gradient.     

Magnetic Resonance Imaging Measurement of Volume Magnetic Susceptibility Using a Boundary Condition

A magnetic resonance imaging method is described for measuring the magnetic susceptibility difference between two homogeneous macroscopic compartments in contact with each other. A boundary condition is derived for the interface of the two compartments. This boundary condition predicts that across the interface there is a resonant frequency jump, which is a function of interfacial orientation relative to B0 field and the difference in susceptibility of the two sides. Based on this relationship, the magnetic susceptibility difference between two materials can be obtained from MR gradient echo imaging using signals from both sides in the vicinity of the boundary. This method is demonstrated by solution phantom experiments.     

铁磁性加载复合管壳电子注聚焦性能的研究

大功率行波管通常利用复合管壳提升高频系统的集成度和散热特性。宽带行波管采用复合管壳高频制造工艺时,由于加载翼片含有铁磁性材料(纯铁)使得聚焦系统的横向磁场分量变大,径向和角向磁场分量呈非均匀性,电子注聚焦困难。本文研究了周期永磁聚焦系统横向磁场产生的原因并建立理论模型,并对磁场分量和其对电子注形态的影响进行了仿真,仿真结果与理论计算结果一致。根据横向磁场分布模型对加载翼片的形状和数量进行优化仿真,结果表明9片齿形加载翼片方案可在保持慢波电路参数的同时,降低聚焦系统的横向磁场分量,改善电子注聚焦效果。     

Maximizing band gaps in two-dimensional photonic crystals by using level set methods

The optimal design of photonic band gaps for two-dimensional square lattices is considered. We use the level set method to represent the interface between two materials with two different dielectric constants. The interface is moved by a generalized gradient ascent method. The biggest gap of GaAs in air that we found is 0.4418 for TM (transverse magnetic field) and 0.2104 for TE (transverse electric field).     

Jump relations for shocks in an anisotropic magnetized plasma

The jump relations for shocks moving into a collision-free anisotropic magnetized plasma are investigated under the assumption of isotropy of the plasma behind the shock front. The plasma ahead of the shock is assumed to be stable against the fire-hose instability and the mirror instability. In order to facilitate comparison with the work of Bazer and Ericson on isotropic shocks our nomenclature has been adapted to theirs. It turns out that as in the case of isotropic shocks the density ratio can be at most four corresponding to γ=5/3, that the change in magnetic field is bounded and that except for the case of Alfvén shocks the transverse parts of the magnetic field are collinear. It is further shown that the influence of the anisotropy is greatest for nearly equal thermal and magnetic energy densities. In the case of negative anisotropy no compressive shocks are possible with a major decrease in magnetic field if the thermal energy density much exceeds the magnetic energy density. A new kind of shock is shown to result from the analysis, the major effect of which is to destroy the anisotropy with only small changes in density, magnetic field and velocity vector. Its propagation speed is unbounded. Furthermore it has turned out that compressive, magnetic field increasing shocks have lower bounds in the density jump and magnetic field change for negative and positive anisotropy, respectively. In the collision-free case no unique entropy condition depending only on the total pressure components and densities can be given before the solution of the problem of shock structure. Therefore even expansive shocks may be admissible. The applicability of the isotropy assumption and ad-hoc-assumptions of other authors are briefly discussed.     

A low numerical dissipation immersed interface method for the compressible Navier–Stokes equations

A numerical method to solve the compressible Navier–Stokes equations around objects of arbitrary shape using Cartesian grids is described. The approach considered here uses an embedded geometry representation of the objects and approximate the governing equations with a low numerical dissipation centered finite-difference discretization. The method is suitable for compressible flows without shocks and can be classified as an immersed interface method. The objects are sharply captured by the Cartesian mesh by appropriately adapting the discretization stencils around the irregular grid nodes, located around the boundary. In contrast with available methods, no jump conditions are used or explicitly derived from the boundary conditions, although a number of elements are adopted from previous immersed interface approaches. A new element in the present approach is the use of the summation-by-parts formalism to develop stable non-stiff first-order derivative approximations at the irregular grid points. Second-order derivative approximations, as those appearing in the transport terms, can be stiff when irregular grid points are located too close to the boundary. This is addressed using a semi-implicit time integration method. Moreover, it is shown that the resulting implicit equations can be solved explicitly in the case of constant transport properties. Convergence studies are performed for a rotating cylinder and vortex shedding behind objects of varying shapes at different Mach and Reynolds numbers.