Open Boundary Problems Research Articles

Mathematical Assessment of Pollution Factors from Eddy Currents in the Durrës Seaport

Eddy currents, a significant phenomenon in electromagnetic fields, were first distinguished from general electric currents by Foucault. These currents, influenced by the magnetic field vector (H), magnetic flux density (B), and electric field (E), are present in various environments, including homogeneous and non-homogeneous areas, linear and non-linear environments, and open boundary problems. In marine environments, these currents, referred to as planetary currents, contribute to electrical pollution, impacting maritime infrastructure, machinery, and ecosystems. This study aims to assess the impact of eddy currents in the Durrës Seaport, Albania, focusing on how meteorological factors such as temperature, atmospheric pressure, and humidity influence pollution levels. Using the TES-92 Electro-smog meter and the PCE-THB 40 Thermo-hygrometer and barometer, we measured electromagnetic field characteristics and meteorological parameters. Our results highlight the significant impact of stray currents on marine environments, emphasizing the need for effective measures to minimize their effects and improve the quality and selection of materials and equipment in port facilities. This study provides valuable insights into the environmental and infrastructural challenges posed by electromagnetic pollution and proposes recommendations for mitigating these impacts.

Eshelby's method for unidirectional periodic composites

Open boundary conditions are always used to investigate the effective properties of composites. In this paper, Eshelby's transformation field method is developed to deal with the effective responses of the unidirectional periodic composites having an open boundary. In this method, Hermite polynomials are chosen to cope with the open boundary conditions of the perturbation fields induced by inclusions. The transformation fields are introduced in the composite system to meet the interface conditions between the complex structure inclusions and the matrix. As an example, the formulas of Eshelby's method are derived for calculating the effective responses of the two-dimensional unidirectional periodic dielectric composites having an open boundary. The validity is verified by comparing the effective responses calculated by the method with the exact solutions of dilute limit. It is shown that the method is valid to solve the open boundary problem of the unidirectional periodic composites having complex geometric inclusions.

An efficient numerical scheme for the FE-approximation of magnetic stray fields in infinite domains

In this contribution we propose an efficient and simple finite-element procedure for the approximation of open boundary problems for applications in magnetostatics. In these problems, the interaction of the solid with external space plays a crucial role because of the magnetic stray fields that arise. For this purpose, the infinite region under consideration is approximated by a sufficiently large domain. This region is then divided into a so-called interior domain and an exterior domain. As an essential prerequisite, we assume linear behavior of the (large) exterior domain. The latter is then reduced to the degrees of freedom of the connecting line (2D)/connecting surface (3D) of both domains via static condensation. The proposed finite element scheme can be seen as an alternative to established methods for infinite domains. These methods often require semi-analytical solutions to describe the behavior in the exterior domain, which can be difficult to obtain if heterogeneous structures are present. The proposed finite element procedure is not subject to any restrictions with regard to the topology of the exterior space. After a general introduction of the numerical scheme, we apply the method to problems of magnetostatics with nonlinear behavior in the interior domain.

The Hybrid FEM-DBCI for the Solution of Open-Boundary Low-Frequency Problems

This paper describes a particular use of the hybrid FEM-DBCI, for the computation of low-frequency electromagnetic fields in open-boundary domains. Once the unbounded free space enclosing the system has been truncated, the FEM is applied to the bounded domain thus obtained, assuming an unknown Dirichlet condition on the truncation boundary. An integral equation is used to express this boundary condition in which the integration surface is selected in the middle of the most external layer of finite elements, very close to the truncation boundary, so that the integral equation becomes quasi-singular. The method is described for the computation of electrostatic fields in 3D and of eddy currents in 2D, but it is also applicable to the solution of other kinds of electromagnetic problems. Comparisons are made with other methods, concluding that FEM-DBCI is competitive with the well-known FEM-BEM and coordinate transformations for what concerns accuracy and computing time.

Isogeometric symmetric FE-BE coupling method for acoustic-structural interaction

The problems of Acoustic-Structural Interaction (ASI) are commonly encountered in the simulation of the thin-walled structures immersed in the fluid. This interaction can affect the acoustic properties of the fluid and the dynamic characteristics of the structure. In this paper, a Non-Uniform Rational B-Splines (NURBS)-based Isogeometric Finite Element (FE) - Boundary Element (BE) symmetric coupling method is developed to study the ASI problems and the free vibration of the elastic structures submerged in the fluid. The geometry of the thin-walled structures is exactly modelled and discretized by utilizing the Isogeometric Reissner-Mindlin shell elements. This shell theory simplifies the continuity enforcement between NURBS patch boundaries. The Isogeometric Boundary Element Method (IGABEM) is applied to describe the acoustic field. In order to impose coupling constraints, a transformation equation of unknown variables on the interface between sound field and structure field is established, that is, sound pressure versus external force and particle velocity versus displacement. Different from the non-symmetric matrix obtained by the traditional direct BEM, a new variational formulation is introduced to obtain the symmetric coupling coefficient matrix, which maks this coupling approach suitable for a broad class of solvers and improves the robustness of the computation. The reliability and stability of the symmetric coupling method are verified by numerical examples considering acoustic and structural loadings. It is concluded that the symmetric coupling method possesses high accuracy, and can circumvent the limitation of traditional coupling method in solving open boundary problems.

Variational assimilation of temperature for the model of hydrodynamics of the Baltic Sea: the solution of the open boundary problem

Variational assimilation of temperature for the model of hydrodynamics of the Baltic Sea: the solution of the open boundary problem

Applicability and Optimization of the Alternating-Direction-Implicit Iterative Method for the 2-D Finite-Difference Frequency-Domain Solution of Scattering Problems

The applicability of the alternating-direction-implicit (ADI) iterative method for the 2-D finite-difference frequency-domain (FDFD) solution of electromagnetic problems is studied, evaluated, and optimized. The ADI is used as an iterative solver of the system of linear equations resulting from the FDFD formulation. The physical limitations of applying the ADI method with a single acceleration parameter are examined in terms of its convergence behavior. In order to overcome these limitations, multiple acceleration parameters are necessary. An optimization procedure based on the simulated annealing algorithm is presented in order to obtain the optimum or near-optimum acceleration parameters. The approach is applied on the scattering problem by a dielectric cylinder as a test. The optimized acceleration parameters are compared with the ones suggested by Hadley. The acceleration parameters depend on the type of transparency conditions used to treat open-boundary problems. A simple absorbing layer (ABL), the stretched-coordinated perfectly matched layer (SC-PML), and the uniaxial PML are tested for both TE and TM polarizations. The SC-PML absorbing condition seems to be the best solution for the ADI iterative implementation of FDFD. Finally, empirical acceleration parameters are proposed for arbitrary scatterers for ABL and SC-PML absorbing conditions.

Self-Energy Concept for the Numerical Solution of the Liouville-von Neumann Equation

A new numerical approach is presented for the determination of the statistical density matrix as a solution of the Liouville-von Neumann equation in center-mass coordinates. The numerical discretization is performed by utilizing a finite volume method, which leads to a discretized drift and diffusion operator. The solution is based on the eigenvector basis of the discretized diffusion operator with its corresponding eigenvalues and on the introduction of the self-energy concept. More specifically, the self-energy concept is essential to describe open-boundary problems adequately. Furthermore, this approach allows the definition of inflow and outflow conditions. The method presented is investigated with regard to the conventional Wigner transport equation and the quantum transmitting boundary method, when investigating coherent effects.

Solution of Open-Boundary Problems by Means of the Hybrid FEM-GDBCI Method

In this paper, the hybrid finite-element method-Galerkin Dirichlet boundary condition iteration method is described for the finite-element solution of a variety of low-frequency electromagnetic field problems in open boundary domains. The method assumes a Dirichlet boundary condition on the truncation boundary and its imposition is made by means of a Galerkin integral equation over another surface strictly enclosed by the truncation boundary.

An investigation of the unstable phenomena of natural convection in parallel square plates by multi-GPU implementation

ABSTRACTUnstable phenomena induced by natural convection of parallel square plates are investigated numerically. The geometry belongs to the open boundary problem, and the compressibility of fluids is taken into consideration. The absorbing boundary condition and modified local one-dimensional inviscid relation method are adopted for open boundaries. Numerical methods of the Roe scheme, preconditioning, and dual time stepping matching the data-parallel lower-upper relaxation method are coordinated with multi-GPU implementation. Unstable phenomena of natural convection caused by the buoyancy force can be observed. Present results have good agreement with the experimental results in the overlap range of the existing study.

Simulated Experiments for Teaching Mutually-Coupled Circuits CAD Techniques Using Analytic and Finite Element Solutions

The paper describes an approach to teaching mutually-coupled circuits CAD techniques to undergraduate students pursuing a degree course in electrical engineering or physics, and explains how a series of simulated experiments may be incorporated into the existing subjects. The simulated experiments make use of a two-dimensional open-access software based on the finite-element method. At the laboratory meetings, the students learn how to set up field problems for solution, and how to examine the results. Simulation tasks based on three axisymmetric open-boundary problems are used to introduce different numeric techniques to compute inductance and magnetic forces. The paper takes the reader to a step-by-step simulation journey, and provides all the basic elements required for further exploration of axially-symmetric systems.

Implementation of Open Boundary Problems in Photo-Conductive Antennas by Using Convolutional Perfectly Matched Layers

A method to simulate an open boundary problem within the finite difference time domain approach for the emission of photo-conductive antennas is presented here. For this purpose, we use convolutional perfectly matched layers (CPMLs). In these devices, the semiconductor region, where transient currents are present in simulation time, is considered to be an “active” medium. This medium is extended virtually beyond its boundaries or the computational domain limits. We explain in this communication how to simulate the transient state of a semiconductor in a CPML region as well as the potential of the method developed to solve conventional practical applications.

An isogeometric indirect boundary element method for solving acoustic problems in open-boundary domains

This work introduces an indirect Boundary Element Method (BEM) within a NURBS-based isogeometric framework for solving three-dimensional acoustic problems in the frequency domain. Developments of isogeometric boundary elements so far have focused on the direct BEM. Yet, a multitude of common problems in acoustics involves open-boundary surfaces, which require a more involved, indirect, boundary element formulation. The current work presents an indirect variational BEM which makes use of NURBS shape functions. Additionally, a novel technique for coupling (strongly) non-conforming patches is introduced to allow the analysis of more complex geometries. The proposed isogeometric indirect boundary element method is verified against analytical solutions and benchmarked against the conventional polynomial-based indirect BEM. Also two open-boundary problems are studied, including analyses over wider frequency ranges, and one industrial-type, complex geometry containing multiple non-conforming patches. The proposed method is found to be not only significantly more efficient than its polynomial-based counterpart, but also very robust against strong non-conformities in the NURBS descriptions.

FEM-DBCI Solution of Open-Boundary Electrostatic Problems in the Presence of Floating Potential Conductors

This paper extends the hybrid finite-element method–Dirichlet boundary condition iteration method for the solution of open-boundary electrostatic problems to the case, in which some floating potential conductors are present in the system. The iterative solution scheme of the basic method is modified in order to deal with the unknown values of the potential of these conductors.

Efficient Analysis of Grounding Systems by Means of the Hybrid FEM–DBCI Method

In the design of a grounding system, technical and normative constraints have to be met. Therefore, it is necessary to use a tool that is able to accurately estimate the values of the significant parameters of the grounding system, such as earth resistance and touch-and-step voltages. The computation of such quantities calls for the solution of a 3-D open-boundary static electromagnetic field problem, which should take into account the nonhomogeneous structure of soil. The hybrid finite-element method–Dirichlet boundary condition iteration method is applied to the solutions of such problems. A great advantage of this method is that it directly provides the electric potential distribution on the earth surface as part of the solution. Some grounding systems are analyzed, and the comparisons of the results with those obtained by other methods are made.

Development of Moving Particle Explicit (MPE) method for incompressible flows

A new particle method is proposed for computing incompressible flows. Using Moving Particle Semi-implicit (MPS) and Smoothed Particle Hydrodynamics (SPH) approaches, a new Moving Particle Explicit (MPE) method is developed. The concept, like the MPS method, is based on the weighted averaging scheme. Introducing a directional particle number density, the derivation of spatial discretization is performed without imposing any simplification. As a consequence, the formulation becomes more generic and covers the MPS scheme. The solution algorithm is similar to the weakly compressible SPH method. Considering slight compressibility for an incompressible flow, the continuity equation for compressible flow together with an equation of state is used to decouple the velocity and pressure of the Navier–Stokes equations. Consequently, the method becomes fully explicit, and the problems related to solving a set of equations in an implicit procedure are removed. Easy application of the solution algorithm, in comparison with the MPS method, as well as the generic discretization scheme, makes the MPE method more efficient for simulating incompressible flows. The proposed method is validated using a Poiseuille flow and two dam break problems; furthermore, a submerged hydraulic jump is investigated to evaluate the method’s capability in simulating open boundary problems. In all cases, good agreement between numerical results and analytical/experimental data is observed.

Effect and Optimization Study of Composite Insulator Sheds Radius on Electric Filed Distribution

A method to optimize the structure parameters of sheds has been put forward for transmission line composite insulator. Fist of all, the unbounded domain is simulated by far field element to handle the open boundary problem of the insulator electric field model. Then, the finite element method (FEM) is employed to calculate the electrical field. The influence of sheds radius on composite insulator electric filed distribution is analyzed. The sheds radius is selected as optimization variable, and the optimization objective is to make the electric field more uniform. Moreover, the parameter analysis method is employed, and the optimal solution of the sheds radius is obtained. The optimized sheds radius make the electric field range and gradient along the insulator surface both tend to the minimum.

Improvised Asymptotic Boundary Conditions for Electrostatic Finite Elements

Improvised asymptotic boundary conditions are a sparse but approximate method of solving open boundary problems. Since this method approximates unbounded space as series of isotropic shells, it can be readily implemented in virtually any finite element solver without the need for additional code. A previous paper addressed application of this technique to magnetic problems described with a vector potential formulation. This paper extends the method to electrostatics and related scalar potential formulations.

Non-uniform FFT for the finite element computation of the micromagnetic scalar potential

We present a quasi-linearly scaling, first order polynomial finite element method for the solution of the magnetostatic open boundary problem by splitting the magnetic scalar potential. The potential is determined by solving a Dirichlet problem and evaluation of the single layer potential by a fast approximation technique based on Fourier approximation of the kernel function. The latter approximation leads to a generalization of the well-known convolution theorem used in finite difference methods. We address it by a non-uniform FFT approach. Overall, our method scales O(M+N+NlogN) for N nodes and M surface triangles. We confirm our approach by several numerical tests.

Initial Value Problem Formulation of 3-D Time Domain Boundary Element Method

A time domain boundary element method (TDBEM) gives us another possibility of time domain microwave simulations in addition to a finite difference time domain (FDTD) method. In particular, the TDBEM has good advantages in analysis of coupling problems with charged particle motion such as in a particle accelerator. However, it is known that time domain microwave simulations in the particle accelerator by the conventional TDBEM often encounter numerical instability and inaccuracy because of its bad matrix property. To avoid the numerical instability and inaccuracy caused by the conventional open boundary problem formulation of the TDBEM, an initial value problem formulation of 3-D TDBEM is presented in this paper.