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In previous works, it was shown that the propagation of magnetic spin waves in thin films can be approximated by a nonlinear Schrodinger-type equation. The formulation begins with the magnetostatic equations (Gauss and Ampere’s laws of magnetism) and the Landau–Lifshitz equation. The solution of this system is a potential function whose dimensionless amplitude is the solution of a nonlinear Schrodinger. In the current work, we are demonstrating an efficient infinite series solution using the Christov functions. This is the first time the functions are used in problems involving complex arithmetics. The solutions of the time-independent and time-dependent problems are given in complex series form.

Liquid crystal compounds are increasingly used as tunable materials for a plethora of microwave and millimeter-wave devices such as phase shifters and printed antennas. Modeling of liquid crystals mandate the solution of the directors’ field under an externally biased electric field which is governed by the Oseen-Frank free-energy functional. Minimization of this functional results in a nonlinear partial differential equation which is often simplified by applying the one-constant approximation, where the splay and bend elastic constants are set equal to each other. The effects of this approximation on the radiation characteristics of a microstrip patch antenna built on top of a liquid-crystal substrate are not well-studied. In this work, we adopt this approximation, along with neglecting the off-diagonal entries of the corresponding dielectric tensor, and compare the results with the original model. The reduced model results in a more computationally efficient algorithm for the characterization of liquid crystal materials; however, there are substantial discrepancies in the simulated antenna figures of merit for intermediate bias voltages.

Tunable devices and materials have always been extremely important for a variety of applications in the optical as well as the µ-wave and mm-wave bands. With the arrival of 5G communications networks and the ubiquitous use of wireless gadgets and applications ranging from smartphones to the implementation of the Internet of Things (IoT), the need for compact, versatile, and lightweight devices is imperative. Liquid Crystals (LCs) are anisotropic and tunable, and these properties make them an attractive proposition for use in portable devices and communication hubs. In this paper, three such LC-based devices operating in the 5GHz I 30GHz (5G) bands are presented: A frequency-agile patch antenna, a variable phase shifter, and a beam­ steerable leaky wave antenna. In all cases, tunability is achieved via the application of a low-strength external bias electric field. This affects the dielectric properties of the crystal by re-orienting its molecules, the macroscopic orientation of which is denoted by a unit vector called the director. The dielectric properties of the LC-cell are characterized by its relative permittivity tensor, which is a function of the directors' orientation. The latter is determined at every point of the cell by solving a coupled system of partial differential equations (PDEs) numerically. The obtained relative permittivity tensor is input into a high-frequency full­ wave electromagnetic simulator based on the finite-element method (FEM). Finally, the simulation results are analyzed and the performance and capabilities of the applications are discussed.

In this Letter, an investigation is performed on the utilization of nematic liquid crystal (NLC) cells in the design of leaky-wave antennas (LWAs) for millimeter-wave (mm-wave) radiation in order to dynamically control its beam scanning capability at a single frequency. A NLC compound is sandwiched between two single-sided copper-plated substrates allowing a traveling wave to be guided through a substrate-integrated waveguide. The tuning capabilities of the structure, based on the use of K15 or GT7-29001 as the middle layer, were evaluated for different biasing conditions demonstrating the associated dynamic scanning of the main beam. A quasi-periodic LWA was designed to operate in the 5G mm-wave band, thus supporting a fast-wave propagation with tunable phase constant and dynamic beam steering at a single frequency. The simulated results clearly illustrate a dynamic beam scanning range of 45° through the use of an external bias voltage ranging between 0 and 40 V. These results are quite promising creating a fertile ground for the utilization of NLCs in the design and fabrication of LWAs for 5G wireless communication networks.

A robust two-dimensional (2D) formulation for the electrical characterization of nematic liquid crystals (N-LCs) under low-frequency (LF) AC biasing conditions is proposed. The finite-difference (FD) method is first implemented to solve Poisson's equation in the domain of interest in order to obtain the governing LF electric field, which affects the local dielectric properties of the anisotropic material. Then, the nonlinear Euler–Lagrange partial differential equation (PDE), governing the orientation of the directors, is solved using one of three FD schemes with relaxation proposed in this paper. Once the N-LC layer is characterized, the average refractive index as a function of the x -coordinate is calculated assuming a normally incident monochromatic laser beam. The results are compared with published data in the literature obtained using a finite element method (FEM). Solution of the PDE governing the orientation of the directors in a non-uniform 2D electric field is obtained using either strong anchoring or soft anchoring. An investigation of the effects of boundary conditions on the average refractive index is presented.

We apply the Christov-Galerkin spectral method for the numerical investigation of the interaction of solitons in the Cubic Nonlinear Schrodinger Equation. The issues of convergence are addressed and an algorithm is devised for the application of the method. Results are obtained for the interaction of solitons with different phase velocities and different carrier frequencies. The interactions are shown to be elastic, save for the phase shifts. The latter are extracted from the numerical solution and discussed.

A mathematical model is proposed to investigate the effect of microgravity on biofilm growth. We examine the case of biofilm suspended in a quiescent aqueous nutrient solution contained in a rectangular tank. The bacterial colony is assumed to follow logistic growth whereas nutrient absorption is assumed to follow Monod kinetics. The problem is modeled by a coupled system of nonlinear partial differential equations in two spatial dimensions solved using the Discontinuous Galerkin Finite Element method. Nutrient and biofilm concentrations are computed in microgravity and normal gravity conditions. A preliminary quantitative relationship between the biofilm concentration and the gravity field intensity is derived.

A nematic liquid crystal (N-LC) cell is used as a substrate to a microstrip patch antenna in order to control its resonant frequency through the use of a DC or low-frequency AC bias voltage. The dielectric tensor properties of the liquid crystal (LC) are dependent on the orientation of the LC molecules, called the directors. The directors' tilt angle is controlled by the amplitude of an externally applied electric field. A finite-difference (FD) scheme with relaxation is formulated in order to solve the highly nonlinear partial differential equation (PDE) that models the directors' tilt angle in the LC. The problem of solving for the directors' field is coupled to an electrostatic boundary value problem (BVP) in a nonhomogeneous anisotropic medium. The orientation of the directors determines the constitutive parameters of the material. The HFSS is then utilized to obtain the radiation characteristics of the antenna under various bias conditions. Comparisons with measurements are provided in order to validate the proposed numerical approach and illustrate the potential use of LC's as tunable materials in microwave and millimeter-wave frequencies.

We develop a robust Christov-Galerkin spectral technique for computing interacting localized wave solutions of and fourth and sixth-order generalized wave equations. To this end, a special complete orthonormal system of functions in L^2(-~,~) is used whose rate of convergence is shown to be exponential for the cases under consideration. For the time-stepping, an implicit algorithm is chosen which makes use of the banded structure of the matrices representing the different spatial derivatives. As featuring examples, the head-on collision of solitary waves is investigated for a sixth-order generalized Boussinesq equation and a fourth-order Boussinesq type equation with a linear term. Its solutions comprise monotone shapes (sech-es) and damped oscillatory shapes (Kawahara solitons). The numerical results are validated against published data in the literature using the method of variational imbedding.

A frequency-agile microstrip patch antenna printed on a biased nematic liquid crystal (N-LC) substrate is accurately modelled. The dielectric properties of the N-LC are controlled through the strength of the bias voltage. The model is based on the solution of Poisson's equation coupled to a nonlinear partial differential equation (PDE) describing the orientation of the directors in a non-homogeneous electric field. The coupled problem was solved using an iterative finite difference (FD) scheme. The obtained numerical results are compared and verified against measurements already published in the literature.

This paper presents a robust numerical method for the analysis of wave propagation in nematic liquid crystals. The structure is excited by a plane wave incident at an oblique angle with respect to the normal to the liquid-crystal cell. The underlined formulation is based on an eigenvalue problem which is solved analytically in order to obtain the governing field expressions inside a homogeneous, thin crystal layer. The liquid-crystal cell is comprised of N such layers. Enforcing the continuity of the tangential electric and magnetic fields at the interfaces formed by the various layers, a matrix system is generated. Solution of the linear system of equations results in the light intensity inside the liquid crystal, which is coupled to a non-linear differential equation for the director tilt angle. This equation is solved using either an explicit or implicit finite-difference scheme. An iteration process continues until convergence is reached for the coupled problem. The proposed numerical method was validated against published results that were generated by approximate analytical methods. Further simulations and studies were conducted emphasizing on the physics of the problem and related interesting phenomena.

A Nematic Liquid Crystal (N-LC) compound may be injected into a cavity beneath a printed patch antenna to act as a tunable material whose dielectric properties are controlled by an externally applied electric field. The strength and direction of the applied low-frequency field affects the orientation of the LC molecules known as directors. The orientation of the directors determines the dielectric tensor entries of the LC compound which, in general, is anisotropic and lossy. The directors' tilt angle is governed by a Partial Differential Equation (PDE) which is obtained through minimization of the Oseen-Frank free-energy functional and solved using a Finite-Difference (FD) scheme. The profile of the directors' tilt angle underneath the patch follows a flattened sinusoidal shape along the normal-to-the-patch direction. In previous work by the authors, this non-uniform dielectric profile was averaged out, thus treating the LC substrate as a homogeneous but lossy material. In this paper, we are investigating the accuracy of this model as compared to a more realistic representation of the sinusoidal dielectric profile using multiple homogeneous layers. In addition, we are proposing ways to improve the existing design of the tunable LC patch antenna in order to obtain enhanced radiation characteristics such as gain and radiation efficiency.

In this paper, we present a computationally efficient and highly accurate numerical method for the analysis of electromagnetic wave propagation in nematic liquid crystal (N-LC) cells. An iterative procedure is employed where the mode-matching technique (MMT) is used to solve the time-harmonic Maxwell equations inside the N-LC cell, whereas a finite-difference method (FDM) with relaxation is utilized to treat the nonlinear stationary Ginzburg-Landau equation for the director field. The angular distortion of the directors in the N-LC cell depends on the applied electric field which, in turn, affects the anisotropic dielectric properties of the medium. Numerical results are obtained for various values of the governing parameters. These simulations provide further insight into the Freedericksz transition with special emphasis on resonances, bi-stability, hysteresis, phase shift between ordinary and extraordinary waves (birefringence), and soft anchoring effects. Obtained results are compared and validated against measurements and data published in the literature.

We study the effects of the Maxwell–Cattaneo (MC) law of heat conduction on the flow of a Newtonian fluid in a vertical slot subject to both vertical and horizontal temperature gradients. Working in one spatial dimension (1D), we employ a spectral expansion involving Rayleigh’s beam functions as the basis set, which are especially well-suited to the fourth order boundary value problem (b.v.p.) considered here, and the stability of the resulting dynamical system for the Galerkin coefficients is investigated. It is shown that the absolute value of the (negative) real parts of the eigenvalues are reduced, while the absolute values of the imaginary parts are somewhat increased, under the MC law. This means that the presence of the time derivative of the heat flux increases the order of the system, thus leading to more oscillatory regimes in comparison with the usual Fourier case. Moreover, no eigenvalues with positive real parts were found, which means that in this particular situation, the inclusion of thermal relaxation does not lead to destabilization of the motion.

TM, electromagnetic scattering from a perfectly conducting wedge with a cylindrical tip is formulated using a mode-matching technique. The eigenmode expansion is then written in a convenient form, where the total field is expressed as a superposition of a wedge-diffraction term based on the uniform theory of diffraction, a geometrical optics term, and a Correction Field term due to the presence of the cylindrical tip. The obtained diffraction coefficient can be easily incorporated into existing ray-tracing, high-frequency codes for the prediction of scattered fields from electrically large structures. The underlined formulation and obtained expressions are verified by comparing numerical results with the finite element method.

A Fourier–Galerkin spectral technique for solving coupled higher-order initial-boundary value problems is developed. Conjugated systems arising in thermoconvection that involve both equations of fourth and second spatial orders are considered. The set of so-called beam functions is used as basis together with the harmonic functions. The necessary formulas for expressing each basis system into series with respect to the other are derived. The convergence rate of the spectral solution series is thoroughly investigated and shown to be fifth-order algebraic for both linear and nonlinear problems. Though algebraic, the fifth-order rate of convergence is fully adequate for the generic problems under consideration, which makes the new technique a useful tool in numerical approaches to convective problems. An algorithm is created for the implementation of the method and the results are thoroughly tested and verified on different model examples. The spatial and temporal approximation of the scheme is tested. To further validate the scheme, a singular asymptotic expansion is derived for small values of the modulation frequency and amplitude and the numerical and analytic results are found to be in good agreement. The new technique is applied to the G-jitter flow, and the Floquet stability diagrams are produced. We obtain the expected alternating isochronous and subharmonic branches and find that stable motions are always isochronous while unstable motions can be either isochronous or subharmonic. The numerical investigation also leads to novel conclusions regarding the dependence of the amplitude of the solutions on some of the governing parameters. Copyright © 2008 John Wiley & Sons, Ltd.

In this work, we develop a High-Order Symmetric Interior Penalty (SIP) Discontinuous Galerkin (DG) Finite Element Method (FEM) to investigate convective flows in a rectangular cavity subject to both vertical and horizontal temperature gradients. The whole cavity is subject to gravity modulation (g-jitter), simulating a microgravity environment. The sensitivity of the bifurcation problem makes the use of a high-order accurate and efficient technique essential. Our method is validated by solving the plane-parallel flow problem and the results were found to be in good agreement with published results. The numerical method was designed to be easily extendable to even more complex flows.

A design for a tunable, frequency-agile antenna is proposed. The structure consists of a microstrip patch and a nematic liquid crystal (N-LC) cell which serves as substrate. The N-LC cell is subject to an applied DC or low-frequency AC electric field. The strength of the bias field alters the tilt-angle of the directors, and thus, affects the relative dielectric tensor of the anisotropic material. The non-linear partial differential equation (PDE) governing the behavior of the directors' field is coupled to Poisson's equation for the electric potential in a non-homogeneous anisotropic medium. The coupled problem is solved iteratively using a finite difference (FD) numerical scheme. It is shown that the resonant frequency of the antenna can be altered using relatively low applied voltages. The obtained results are compared with measurements illustrating the potential use of LCs as tunable materials in microwave and millimeter-wave frequencies.

In the current work we examine the application of Nematic Liquid Crystals (N-LCs) to frequency-agile antennas. A patch antenna design with a liquid crystal base is proposed. N-LCs are anisotropic and their electrical properties are determined by the macroscopic orientation of their molecules (director tilt-angle). However, these depend on the applied electric field, which means that the electric properties of the N-LC base can be effectively controlled. The above described problem is governed by a coupled system of PDEs. It is solved iteratively using a finite-difference scheme with relaxation. Once the director field is obtained, the dielectric properties of the material are determined for each value of the bias voltage. The proposed antenna is then simulated using HFSS. The return loss and resonant frequency are computed for each of value of the applied voltage. It is shown that the antennas under consideration can be tuned using relatively low applied voltages. This demonstrates the potential of liquid crystal based antennas in frequency-agile antenna design.

In the current work we propose an efficient and accurate numerical approach to the problem of electrical characterization of Nematic Liquid Crystal (N-LC) microstructures under the influence of low-frequency AC electric fields. N-LCs are anisotropic and their electrical properties are determined by the directors' tilt angles which in turn depend on the applied electric field. Therefore, the problem is governed by a coupled system of two-dimensional PDEs: A Poisson equation with variable coefficients for the electric potential and a highly nonlinear second-order equation for the tilt angle of the directors. Both equations are solved using finite-difference schemes with relaxation and the results are found to be in good agreement with the literature. Various 2-D geometries are considered and it is shown that a low DC voltage is sufficient to tune the average refractive index of the N-LC structures under consideration.

Cholesteric Liquid Crystals (Ch-LC) are anisotropic and inhomogeneous materials with intriguing and useful properties in the visible region of the electromagnetic spectrum. In this paper, we use eigenmode analysis to obtain the supported field expressions in a thin, homogeneous sub-layer of the Ch-LC cell; the liquid crystal is made of multiple sub-layers. The cell is sandwiched between layers of dielectric and is excited by an elliptically polarized plane wave at an oblique incidence. The governing field expressions in the dielectric layers are also obtained using eigenmode analysis. A mode-matching technique (MMT) is then employed to enforce the continuity of the tangential electric and magnetic fields at the interface of two neighboring layers, thus, resulting in a matrix system representative of the problem at hand. Solution of the linear system of equations yields the reflection and transmission coefficients on the two principal planes as well as the expansion coefficients for the modal fields inside the Ch-LC and the dielectric layers. The underlined formulation is verified by comparing results to the open literature.

This work reports on the implementation of an an iterative procedure to solve the non-linear problem of wave propagation in homeotropic Nematic Liquid Crystals (N-LC). The nematic structure of the crystal molecules is strongly dependent on the applied external electromagnetic field. In our case, a monochromatic plane wave is normally incident on the liquid crystal, which is sandwiched between two glass layers. The orientation of these molecules, called the directors, determine the dielectric tensor properties of the medium. A Mode-Matching Technique (MMT) was used to accurately solve for the governing fields in each of the subdivided layers composing the crystal, whereas an explicit finite-difference scheme with relaxation was implemented to solve for the directors' orientation. The non-linear problem was also solved using a more efficient implicit finite difference scheme characterized by a faster convergence rate. Obtained computational results were compared to published data indicating good agreement.

In the current work, the nonlinear problem of electromagnetic wave propagation in a Nematic Liquid Crystal (NLC) cell is solved numerically. The LC is sandwiched between two glass layers of finite thickness and a linearly polarized beam is obliquely incident to the cell. The dielectric properties of N-LCs depend on the tilt angle of the directors. When the excitation beam enters the cell, and providing the incident intensity is above the Freedericksz threshold, the directors reorient themselves changing the LC's relative permittivity tensor. In turn, this affects beam propagation throughout the crystal. The electromagnetic field is modeled by the time-harmonic Maxwell equations whereas the director field is governed by a nonlinear ordinary differential equation (ODE). Our solution method is iterative, consistently taking into account this interaction between the excitation beam and the director field. The Maxwell equations are solved employing the Mode-Matching Technique (MMT). The solution of the nonlinear differential equation for the director field is obtained with the aid of a finite difference (FD) scheme.

The reflection and transmission spectra from right-handed cholesteric liquid crystals are computed in the visible region for a linearly, circularly, or elliptically polarized incident plane wave at oblique incidence. The liquid crystal cell is sandwiched between dielectric layers of certain thickness and refractive index. The underlined formulation is based on a modal analysis of the governing field expressions in the dielectric and liquid-crystal regions. A representative matrix system is obtained after enforcing the continuity of the tangential electric and magnetic fields at the material interfaces. Solution of the governing matrix system results in the reflectance and transmittance for a given wavelength. Numerical results for both normal and oblique incidence were obtained and compared with data published in the literature. The underlined formulation is effective, accurate, robust, versatile, and computationally efficient.

In the current work, we develop a numerical method suitable for treating the problem of nonlinear two‐dimensional flows in rectangular domains. For the spatial approximation we employ the Fourier‐Galerkin approach. More specifically, our basis functions are products of trigonometric and Beam functions. This choice means that the solutions automatically satisfy the boundary and periodic conditions in the x and y directions respectively.The accuracy of the method is assessed by applying it to model problems which admit exact analytical solutions. The numerical and analytic solutions are found to be in good agreement. The convergence rate of the spectral coefficients is found to be fifth‐order algebraic in the x‐direction and y‐direction, confirming the efficiency and speed of our technique.

In this work, we investigate the 2D flow in a rectangular cavity subject to both vertical and horizontal temperature gradients. The linearized model is studied and the effect of thermal relaxation, as described by the Maxwell‐Cattaneo law of heat conduction is examined. To this end, a spectral numerical model is created based on a Galerkin expansion. The basis is the Cartesian product of systems of beam functions and trigonometric functions.The natural modes of the system are derived for both the Fourier and non‐Fourier models. The results are compared to earlier works for the plain Fourier law. Our computations show that for the same set of parameters, the Maxwell‐Cattaneo law yields modes which are quantitatively different from the Fourier. It is found that the real parts of the eigenvalues increase with the Straughan number Sg, which quantifies the non‐Fourier effects. This confirms the destabilizing effect of the MC‐law on the convective flow.

We study the effect of the Maxwell‐Cattaneo law of heat conduction (MCHC) on the 1D flow in a vertical slot subject to both vertical and horizontal temperature gradients. The gravitational acceleration is allowed to oscillate, which provides an opportunity to investigate the quantitative contribution of thermal inertia as epitomized by MCHC. The addition of the time derivative in MCHC increases the order of the system. We use a spectral expansion with Rayleigh’s beam functions as the basis set, which is especially suited to fourth order boundary value problems (BVP). We show that the time derivative (relaxation of the thermal flux) has a dissipative nature and leads to the appearance of purely real negative eigenvalues. Yet it also increases the absolute value of the imaginary part and decreases the absolute value of the real part of the complex eigenvalues. Thus, the system has a somewhat more oscillatory behavior than the one based on Fourier’s heat conduction law (FHC).

In this paper we develop further the Galerkin technique based on the so-called beam functions with application to nonUnear problems. We make use of the formidas expressing a product of two beam functions into a series with respect to the system. First we prove that the overall convergence rate for a fourth-order linear b.v.p is algebraic fifth order, provided that the derivatives of the sought function up to fifth order exist. It is then shown that the inclusion of a quadratic nonlinear term in the equation does not degrade the fifth-order convergence. We validate our findings on a model problem which possesses analytical solution in the linear case. The agreement between the beam-Galerkin solution and the analytical solution for the linear problem is better than 10^'^ for 200 terms. We also show that the error introduced by the expansion of the nonlinear term is lesser than 10^'. The foeam-Galerkin method outperforms finite differences due to its superior accuracy whilst its advantage over the Chebyshev-tau method is attributed to the smaller condition number of the matrices involved in the former