Open Boundary Problems Research Articles

A comparison between hybrid methods: FEM-BEM versus FEM-DBCI

Purpose – The purpose of this paper is to compare the hybrid FEM-BEM and FEM-DBCI methods for the solution of open-boundary static and quasi-static electromagnetic field problems. Design/methodology/approach – After a brief review of the two methods (both coupling a differential equation for the interior problem with an integral equation for the exterior one), they are compared in terms of accuracy, memory and computing time requirements by means of a set of simple examples. Findings – The comparison suggests that FEM-BEM is more accurate than FEM-DBCI but requires more computing time. Practical implications – Then FEM-DBCI appears more appropriate for applications which require a shorter computing time, for example in the stochastic optimization of electromagnetic devices. Conversely, FEM-BEM is more appropriate in cases in which a high level of precision is required in a single computation. Originality/value – Note that the FEM-BEM considered in this paper is a non standard one in which the nodes of the normal derivative on the truncation boundary are placed in positions different from those of the potential.

Coupling of Boundary Element and Wave Based Methods for the efficient solution of complex multiple scattering problems

A novel hybrid method for the efficient solution of complex acoustic multiple scattering problems is proposed in this paper. The Wave Based Method and the Boundary Element Method are coupled to benefit from the strengths of both. The former is an indirect Trefftz approach, which has a faster convergence rate and lower computational load compared to element based methods when applied on geometries of moderate complexity. The latter is the state-of-the-art technique for unbounded acoustic problems and can handle very complex geometries. The idea behind the hybrid method is to take advantage of the fast WBM solution for scatterers of moderate complexity and take advantage of the BEMʼs capability for handling scatterers of high complexity. For the BEM part, the indirect variational formulation is used which allows modeling of open boundary problems (zero thickness walls). In addition, the WBM makes it possible to easily add heterogeneities (domain inclusions) to the problem. Therefore, the hybrid method does not only aim for better efficiency but also for extending the variety of configurations that can be tackled by both methods with ease. The accuracy and the performance of the method are demonstrated with three examples, both in 2D and 3D. It is shown that when complex and simple scatterers coexist, the hybrid method is more efficient than the WBM, the BEM and the Fast-Multipole BEM while it is able to provide accuracy of similar level.

Performance of 3-D Infinite Elements for High-Frequency Electromagnetic Fields

The infinite elements for edge based finite-element methods (FEMs) have been shown effective for open boundary problems. In the infinite elements, electromagnetic fields are expressed in terms of radially decaying basis functions. On the other hand, the perfect matched layer has widely been used for FEMs for high-frequency problems. In this paper, numerical performance of both methods is comparably discussed. The numerical experiments show that the former has higher computational efficiency.

A scaled boundary finite element method applied to electrostatic problems

Abstract The scaled boundary finite element method (SBFEM) is a novel semi-analytical technique, combining the advantages of the finite element and the boundary element methods with unique properties of its own. In this paper, the SBFEM is firstly extended to solve electrostatic problems. Two new SBFE coordination systems are introduced. Based on Laplace equation of electrostatic field, the derivations (based on a new variational principle formulation) and solutions of SBFEM equations for both bounded domain and unbounded domain problems are expressed in details, the solution for the inclusion of prescribed potential along the side-faces of bounded domain is also presented in details, then the total charges on the side-faces can be semi-analytically solved, and a particular solution for the potential field in unbounded domain satisfying the constant external field is solved. The accuracy and efficiency of the method are illustrated by numerical examples with complicated field domains, potential singularities, inhomogeneous media and open boundaries. In comparison with analytic solution method and other numerical methods, the results show that the present method has strong ability to resolve singularity problems analytically by choosing the scaling centre at the singular point, has the inherent advantage of solving the open boundary problems without truncation boundary condition, has efficient application to the problems with inhomogeneous media by placing the scaling centre in the bi-material interfaces, and produces more accurate solution than conventional numerical methods with far less number of degrees of freedom. The method in electromagnetic field calculation can have broad application prospects.

Singular-Boundary Reductions of Type-Q ABS Equations

We study the fully discrete elliptic integrable model Q4 and its immediate trigonometric and rational counterparts (Q3, Q2, and Q1). Singular boundary problems for these equations are systematized in the framework of global singularity analysis. We introduce a technique to obtain solutions of such problems, in particular, constructing the exact solution on a regular singularity-bounded strip. The solution technique is based on the multidimensional consistency and uses new insights into these equations related to the singularity structure in multidimensions and the identification of an associated tau-function. The obtained special solutions can be identified with open boundary problems of the associated Toda-type systems, and have interesting application to the construction of periodic solutions.

Orthogonalized Infinite Edge Element Method—Convergence Improvement by Orthogonalization of Hilbert Matrix in Infinite Edge Element Method

This paper proposes an orthogonalization of the Hilbert matrix in element matrices of the infinite edge elements. The validity of the infinite edge element is demonstrated in previous researches, but the Hilbert matrix results in extremely slow convergence in the ICCG method, especially when using higher order expansions. The proposed orthogonalization technique improves the convergence drastically and it makes the infinite elements practical in the electromagnetic FEM analysis of the open boundary problems in quasi-static magnetic fields.

Experimental analysis of a Fast Adaptive Composite‐based grid expansion scheme for open boundary problems

SUMMARYPoisson's equation on unbounded domains was solved with a hierarchical grid coarsening approach on the basis of Stephen F. McCormick's Fast Adaptive Composite grid method. We were especially interested in the efficient interface treatment that is enabled by regular stencils. The error behavior of a second‐order scheme was empirically analyzed for an exemplary setup from molecular dynamics. Comparisons with previous methods were also performed and showed that our approach is competitive. Copyright © 2012 John Wiley & Sons, Ltd.

Asymptotic boundary conditions with immersed finite elements for interface magnetostatic/electrostatic field problems with open boundary

Many of the magnetostatic/electrostatic field problems encountered in aerospace engineering, such as plasma sheath simulation and ion neutralization process in space, are not confined to finite domain and non-interface problems, but characterized as open boundary and interface problems. Asymptotic boundary conditions (ABC) and immersed finite elements (IFE) are relatively new tools to handle open boundaries and interface problems respectively. Compared with the traditional truncation approach, asymptotic boundary conditions need a much smaller domain to achieve the same accuracy. When regular finite element methods are applied to an interface problem, it is necessary to use a body-fitting mesh in order to obtain the optimal convergence rate. However, immersed finite elements possess the same optimal convergence rate on a Cartesian mesh, which is critical to many applications. This paper applies immersed finite element methods and asymptotic boundary conditions to solve an interface problem arising from electric field simulation in composite materials with open boundary. Numerical examples are provided to demonstrate the high global accuracy of the IFE method with ABC based on Cartesian meshes, especially around both interface and boundary. This algorithm uses a much smaller domain than the truncation approach in order to achieve the same accuracy.

Finite-Element Analysis of Magnetic Field Problem With Open Boundary Using Infinite Edge Element

This paper proposes a novel effective method for open boundary problems by using the finite-element method (FEM) with a new formulation of the infinite edge element. We developed the formulation of the infinite element by adopting the characteristic of conventional edge elements, resulting in a sparse and symmetric matrix. This proposed technique with the infinite edge element has the advantage of drastically reducing the computational burden with keeping the analyses' accuracy. Some numerical examples which demonstrate the validity of the proposed method are presented.

The scaled boundary finite element method applied to electromagnetic field problems

Computation electromagnetic is an important research field of electromagnetic fields and microwave technology subjects. In this paper, the scaled boundary finite element method (SBFEM) is extended to solve one type of electromagnetic field problems-electrostatic field problems. Based on Laplace equation of electrostatic field, the derivations and solutions of SBFEM equations for both bounded and unbounded domain problems are expressed in details, and the solution for the inclusion of prescribed potential along the side-faces of bounded domain is also presented in details, then the total charges on the side-faces can be semi-analytically solved. The accuracy and efficiency of the method are illustrated by numerical examples of electromagnetic field problems with complicated field domains, potential singularities, inhomogeneous media and open boundaries. In comparison with analytic solution method and other numerical methods, the results show that the present method has strong ability to resolve potential field singularities analytically by choosing the scaling centre at the singular point, has the inherent advantage of solving the open boundary problems without truncation boundary condition, has efficient application to the problems with inhomogeneous media by placing the scaling centre in the bi-material interfaces, and produces more accurate solution than conventional numerical methods with far less number of degrees of freedom. The method in electromagnetic field calculation can have broad application prospects.

Transparent boundary conditions for quantum-waveguide simulations

We consider the two-dimensional, time-dependent Schrödinger equation discretized with the Crank–Nicolson finite difference scheme. For this difference equation we derive discrete transparent boundary conditions (DTBCs) in order to get highly accurate solutions for open boundary problems. We apply inhomogeneous DTBCs to the transient simulation of quantum waveguides with a prescribed electron inflow.

Eigenstates and scattering solutions for billiard problems: A boundary wall approach

It was proposed about a decade ago [M.G.E. da Luz, A.S. Lupu-Sax, E.J. Heller, Phys. Rev. E 56 (1997) 2496] a simple approach for obtaining scattering states for arbitrary disconnected open or closed boundaries C , with different boundary conditions. Since then, the so called boundary wall method has been successfully used to solve different open boundary problems. However, its applicability to closed shapes has not been fully explored. In this contribution we present a complete account of how to use the boundary wall to the case of billiard systems. We review the general ideas and particularize them to single connected closed shapes, assuming Dirichlet boundary conditions for the C ’s. We discuss the mathematical aspects that lead to both the inside and outside solutions. We also present a different way to calculate the exterior scattering S matrix. From it, we revisit the important inside–outside duality for billiards. Finally, we give some numerical examples, illustrating the efficiency and flexibility of the method to treat this type of problem.

Discrete transparent boundary conditions for the Schrodinger equation -- a compact higher order scheme

We consider the two-dimensional time-dependent Schrödinger equation with the new compact nine-point scheme in space and the Crank-Nicolson difference scheme in time. For the resulting difference equation we derive discrete transparent boundary conditions in order to get highly accurate solutions for open boundary problems. Numerical experiments illustrate the perfect absorption of outgoing wave independently of their impact angle at the boundary. Finally, we apply inhomogeneous discrete transparent boundary conditions to the transient simulation of quantum waveguides.

An Iterative Solution to FEM-BEM Algebraic Systems for Open-Boundary Electrostatic Problems

This paper describes an iterative method to solve efficiently the global algebraic system of equations obtained by applying the hybrid finite element method-boundary element method (FEM-BEM) to the solution of open-boundary electrostatic problems. In the proposed approach, a conjugate gradient (CG) solver is used to solve the FEM equations, whereas the BEM equations are solved by making use of the direct LU solver. To do so, it is convenient to implement the BEM equations in a nonconventional way, by choosing the nodes of the potential as not coinciding with those of its normal derivative

A multi-purpose Schrödinger-Poisson Solver for TCAD applications

We present the Vienna Schrodinger-Poisson Solver (VSP), a multi-purpose quantum mechanical solver for investigations on nano-scaled device structures. VSP includes a quantum mechanical solver for closed as well as open boundary problems on fairly arbitrary one-dimensional cross sections within the effective mass framework. For investigations on novel gate dielectrics VSP holds models for bulk and interface trap charges, and direct and trap assisted tunneling. Hetero-structured semiconductor devices, like resonant tunneling diodes (RTD), can be treated within the closed boundary model for quick estimation of resonant energy levels. The open boundary model allows evaluation of current voltage characteristics.

Confinement loss in adiabatic photonic crystal fiber tapers

We numerically study confinement loss in photonic crystal fiber (PCF) tapers and compare our results with previously published experimental data. Agreement between theory and experiment requires taking into account hole shrinkage during the tapering process, which we measure by using a noninvasive technique. We show that losses are fully explained within the adiabatic approximation and that they are closely linked to the existence of a fundamental core-mode cutoff. This cutoff is equivalent to the core-mode cutoff in depressed-cladding fibers, so that losses in PCF tapers can be obtained semiquantitatively from an equivalent depressed-cladding fiber model. Finally, we discuss the definition of adiabaticity in this open boundary problem.

COUPLED SIMULATION OF 3D ELECTRO-MAGNETO-FLOW FIELD IN HALL-HEROULT CELLS USING FINITE ELEMENT METHOD

Two full 3D steady mathematical models are developed by finite element method (FEM) to calculate coupled physics fields: the electro-magnetic model is built and solved first and so is the fluid motion model with the acquired electromagnetic force as source body forces in Navier-Stokes equations. Effects caused by the ferromagnetic shell, busbar system around, and open boundary problem as well as inside induced current were considered in terms of the magnetic field. Furthermore, a new modeling method is found to set up solid models and then mesh them entirely with so-called structuralized grids, namely hex-mesh. Examples of 75kA prebaked cell with two kinds of busbar arrangements are presented. Results agree with those disclosed in the literature and confirm that the coupled simulation is valid. It is also concluded that the usage of these models facilitates the consistent analysis of the electric field to magnetic field and then flow motion to the greater extent, local distributions of current density and magnetic flux density are very much dependent on the cell structure, the steel shell is a shield to reduce the magnetic field and flow pattern is two dimensional in the main body of the metal pad.

Local Approximation Matching for Open Boundary Problems

A local approximation matching method is used to derive a computational scheme for static field problems with open boundaries. This scheme results in a conventional finite difference method applied in the interior and a new difference scheme applied to nodes on the exterior boundary. Numerical examples are provided indicating that errors due to mesh truncation are low.

Improvement of Open Boundary Magnetic Field Analysis of Air-Core Magnet Using High Order Boundary Sensitivity

This paper presents a new approach to the finite element open boundary problems using high order boundary sensitivity. The finite element method (FEM) gives the solution with boundary truncation error in the open boundary problems like air-core magnet. And it doesn't inform us of the precision of solution. In order to overcome the weakness of FEM, the high-order energy sensitivity method is proposed. It is based on the sensitivity analysis technique. It can be used for the check on the precision of solution and suggest the proper boundary. It was applied to the finite element analysis of air-core magnet to validate the proposed method.

Computation of axisymmetric open-boundary eddy-current field problems

Based on the idea of region division a technique known as the hybrid finite-element analytical approach is developed for treating axisymmetric open-boundary eddy-current field problems. The general principle of the technique first introduces a spherical fictitious surface boundary called a coupling boundary to enclose the complex-shaped devices in the presence of an inhomogeneous, anisotropic or even nonlinear medium. In the interior of the spherical fictitious surface boundary the field is formulated by the finite-element method (FEM). Meanwhile, in the exterior region the field is presented by an eigenfunction expression (in Legendre functions). Taking the exterior region as a macroelement, it requires consideration of only the potential continuity at the fictitious boundary nodes with interior finite elements, so the FEM is extended to solve open-boundary problems. The results were compared with both the analytical solution and experimental cases, showing good agreement. The new hybrid scheme combining an analytical solution with the finite-element method offers several advantages: symmetrical matrices; much fewer elements and nodes for the same level at accuracy; and an easy means for calculating the external region field.