Order Transmission Conditions Research Articles

Twofold Domain Decomposition Method for the Analysis of Multiscale Composite Structures

A nonconformal twofold domain decomposition method (TDDM) based on the hybrid finite element method-boundary element method (FEM-BEM) is proposed for analyzing 3-D multiscale composite structures. The proposed TDDM starts by partitioning the composite object into a closed exterior boundary domain and an interior volume domain. The interior and exterior boundary value problems are coupled to each other through the Robin transmission conditions (RTCs). Both domains are then independently decomposed into subregions to facilitate computation. Specifically, FEM-DDM with the second order transmission conditions (SOTCs) is employed for the interior domain, and BEM-discontinuous Galerkin (BEM-DG) based on the combined field integral equation (CFIE) is applied for the exterior boundary domain. The proposed TDDM allows for nonconformal discretization between any touching subdomains. Without the introduction of a stabilization term that relies on a line integral over intersection of nonmatching meshes and relevant terms involving surface-line integrals, the proposed TDDM provides an effective domain decomposition (DD) preconditioner for the global system. Numerical examples are presented, and the comparisons of the simulation results with FEM-BEM confirm the validity and accuracy of TDDM. Moreover, its ability to model practical large-scale and multiscale targets is also demonstrated.

A boundary integral equation domain decomposition method for electromagnetic scattering from large and deep cavities

Electromagnetic scattering analysis of large and deep cavities embedded in an arbitrarily shaped host body is of high interest to the engineering community. The objective of this work is to investigate an effective boundary integral equation domain decomposition method for solving the cavity scattering problems. The key features of the proposed work include: (i) the introduction of individual electric and magnetic traces as unknowns for each sub-region, (ii) the development of a multi-trace combined field integral equation formulation for decomposed boundary value problem, and (iii) the derivation of optimized multiplicative Schwarz preconditioning using complete second order transmission condition. The proposed method can be viewed as an effective preconditioning scheme for the integral equation based solution of the cavity scattering problems. The strength and flexibility of the proposed method will be illustrated by means of several representative numerical examples.

A Scalable Nonoverlapping and Nonconformal Domain Decomposition Method for Solving Time-Harmonic Maxwell Equations in $\mathbb{R}^3$

We present a novel nonoverlapping and nonconformal domain decomposition method (DDM) for solving the time-harmonic Maxwell equations in $\mathbb{R}^3$. There are three major technical ingredients in the proposed nonconformal DDM: (a) a true second order transmission condition (SOTC) to enforce fields continuities across domain interfaces; (b) a corner edge penalty term to account for corner edges between neighboring subdomains; and (c) a global plane wave deflation technique to further improve the convergence of DDM for electrically large problems. It has been shown previously that a SOTC, which involves two second order transverse derivatives, facilitates convergence in the conformal domain decomposition method for both propagating and evanescent electromagnetic waves across domain interfaces. However, the discontinuous nature of the cement variables across the corner edges between neighboring subdomains remains troublesome. To mitigate the technical difficulty encountered and to enforce the needed divergence-free condition, we introduced a corner edge penalty term into the interior penalty formulation for the nonconformal DDM. The introduction of the corner edge penalty term successfully restored the superior performance of the SOTC. Finally, through an analysis of the DDM with the SOTC, we show that there still exists a weakly convergent region where the convergence in the DDM can still be unbearably slow for electrically large problems. Furthermore, it is found that the weakly convergent region is centered at the cutoff modes, or electromagnetic waves propagate in parallel to the domain interfaces. Subsequently, a global plane wave deflation technique is utilized to derive an effective global-coarse-grid preconditioner to promote fast convergence of the cutoff or near cutoff modes in the vicinity of domain interfaces. Finally, the strength of the proposed method is illustrated by means of three numerical examples.

An Optimal Block Iterative Method and Preconditioner for Banded Matrices with Applications to PDEs on Irregular Domains

Classical Schwarz methods and preconditioners subdivide the domain of a PDE into subdomains and use Dirichlet transmission conditions at the artificial interfaces. Optimized Schwarz methods use Robin (or higher order) transmission conditions instead, and the Robin parameter can be optimized so that the resulting iterative method has an optimized convergence factor. The usual technique used to find the optimal parameter is Fourier analysis; but this is applicable only to certain regular domains, for example, a rectangle, and with constant coefficients. In this paper, we present a completely algebraic version of the optimized Schwarz method, including an algebraic approach to finding the optimal operator or a sparse approximation thereof. This approach allows us to apply this method to any banded or block banded linear system of equations, and in particular to discretizations of PDEs in two and three dimensions on irregular domains. With the computable optimal operator, we prove that the optimized Schwarz method converges in no more than two iterations, even for the case of many subdomains (which means that this optimal operator communicates globally). Similarly, we prove that when we use an optimized Schwarz preconditioner with this optimal operator, the underlying minimal residual Krylov subspace method (e.g., GMRES) converges in no more than two iterations. Very fast convergence is attained even when the optimal transmission operator is approximated by a sparse matrix. Numerical examples illustrating these results are presented.

High order transmission conditions for thin conductive sheets in magneto-quasistatics

We propose transmission conditions of order 1, 2 and 3 approximating the shielding behaviour of thin conducting curved sheets for the magneto-quasistatic eddy current model in 2D. This model reduction applies to sheets whose thicknesses $\eps$ are essentially smaller or at the order of the skin depth. The sheet has itself not to be resolved, only its midline is represented by an interface. The computation is directly in one step with almost no additional cost. We prove the well-posedness w.r.t.~to the small parameter $\eps$ and obtain optimal bound for the modelling error outside the sheet of order $\eps^{N+1}$ for the condition of order N. Numerical experiments with high order finite elements for sheets with varying curvature verify the theoretical findings.

Non-Conformal Domain Decomposition Method With Mixed True Second Order Transmission Condition for Solving Large Finite Antenna Arrays

A non-overlapping and non-conformal domain decomposition method (DDM) is presented for modeling large finite antenna arrays. Two major benefits of the proposed DDM: (a) A mixed true second order transmission condition (SOTC) with corner edge penalty terms is developed to facilitate fast convergence in the DDM iterations. Numerical experiments indicate that the convergence of DDM with the aforementioned SOTC is insensitive to the sizes of arrays; and, (b) The non-conformal property of the proposed DDM permits the use of completely independent discretization for each of the sub-domains. We demonstrate the performance of the proposed approach through several large-scale problems.

Non-conformal domain decomposition method with second-order transmission conditions for time-harmonic electromagnetics

Non-overlapping domain decomposition (DD) methods provide efficient algorithms for solving time-harmonic Maxwell equations. It has been shown that the convergence of DD algorithms can be improved significantly by using high order transmission conditions. In this paper, we extend a newly developed second-order transmission condition (SOTC), which involves two second-order transverse derivatives, to facilitate fast convergence in the non-conformal DD algorithms. However, the non-conformal nature of the DD methods introduces an additional technical difficulty, which results in poor convergence in many real-life applications. To mitigate the difficulty, a corner-edge penalty method is proposed and implemented in conjunction with the SOTC to obtain truly robust solver performance. Numerical results verify the analysis and demonstrate the effectiveness of the proposed methods on a few model problems. Finally, drastically improved convergence, compared to the conventional Robin transmission condition, was observed for an electrically large problem of practical interest.

Nonoverlapping Domain Decomposition with Second Order Transmission Condition for the Time-Harmonic Maxwell's Equations

Nonoverlapping domain decomposition methods have been shown to provide efficient iterative algorithms for the solution of the time-harmonic Maxwell's equations. Convergence of the algorithms depends strongly upon the nature of the transmission conditions that communicate information between adjacent subdomains. In this work, we introduce a new second order transmission condition that improves convergence. In contrast to previous high order interface conditions, the new condition uses two second order transverse derivatives to improve the convergence of evanescent modes without deteriorating the convergence of propagating modes. An analysis using a splitting of the field into traverse electric and magnetic components demonstrates the improved convergence provided by the transmission condition. Numerical experiments demonstrate the effectiveness of the algorithm.

One way domain decomposition method with second order transmission conditions for solving electromagnetic wave problems

Non-overlapping domain decomposition (DD) methods with complex first order Robin-type transmission conditions (TCs) provide an efficient iterative solution for Maxwell’s equation. Unfortunately, the first order TCs do not effectively account for some eigenmodes of the system matrix, which limits the scalability of the methods. In this work, we examine two TCs with a second order transverse derivative to improve the method’s performance. A detailed convergence analysis of the two TCs is presented. We then investigate the use of the two second order TCs in non-conformal and non-overlapping one way DD methods. Numerical results illustrate the effectiveness of the proposed methods on some model problems and on several problems of practical interest.

A homographic best approximation problem with application to optimized Schwarz waveform relaxation

We present and study a homographic best approximation problem, which arises in the analysis of waveform relaxation algorithms with optimized transmission conditions. Its solution characterizes in each class of transmission conditions the one with the best performance of the associated waveform relaxation algorithm. We present the particular class of first order transmission conditions in detail and show that the new waveform relaxation algorithms are well posed and converge much faster than the classical one: the number of iterations to reach a certain accuracy can be orders of magnitudes smaller. We illustrate our analysis with numerical experiments.

A Domain Decomposition Method With Nonconformal Meshes for Finite Periodic and Semi-Periodic Structures

We present a domain decomposition method as a preconditioner for Krylov-type solvers to model complex electromagnetic problems containing periodicities. The method reduces memory requirements by decomposing the original problem into several nonoverlapping sub-domains. The 1st order transmission condition is employed on interfaces between adjacent sub-domains to enforce continuity of electromagnetic fields and to ensure the sub-domain problems are well-posed. By following the spirit of duality paring a symmetric system is obtained. To reduce the computational burdens of the present method, the finite element tearing and interconnecting like algorithm is adopted. This algorithm results in the computation of the so-called Green's function, which can be compressed efficiently via a rank-revealing matrix factorization algorithm. The final system matrix is solved by Krylov solvers instead of classical stationary solvers. To improve the convergence of iterative solvers, several robust implementation details are discussed and the choice of some popular Krylov-subspace solvers is studied through numerical examples.

Optimized Schwarz methods with an overset grid for the shallow-water equations: preliminary results

The overset grid nicknamed “Yin–Yang” grid is singularity free and has quasi-uniform grid spacing. It is composed of two identical latitude/longitude orthogonal grid panels that are combined to cover the sphere with partial overlap on their boundaries. The system of shallow-water equations (SWEs) is a hyperbolic system at the core of many models of the atmosphere. In this paper, the SWEs are solved on the Yin–Yang grid by using an implicit and semi-Lagrangian discretization on a staggered mesh. The resulting scalar elliptic equation is solved using a Schwarz-type domain decomposition method, known as the optimized Schwarz method, which gives better performance than the classical Schwarz method by using specific Robin or higher order transmission conditions.

2402 Three dimensional optimized Schwarz methods without overlap for the numerical simulation of urban acoustic pollution

In this paper we investigate three dimensional optimized Schwarz methods for the numerical solution of urban acoustic pollution within a city. Optimized Schwarz methods are using new transmission conditions which are then optimized for the efficiency of the Schwarz algorithm without overlap. We perform several calculations in three dimension on parallel computer and compare the speed and accuracy of the optimized Schwarz methods without overlap equipped with three dimensional zeroth order and second order transmission conditions. We show that the optimized Schwarz method applied to large scale exterior acoustics problems is strongly robust and scalable.

A non-overlapping domain decomposition method with non-matching grids for modeling large finite antenna arrays

A non-overlapping domain decomposition method (DDM) is proposed herein to solve Maxwell equations in R 3 . In this work, the Maxwell equations are discretized using a vector finite element method with hierarchical H(curl) vector basis functions. There are two major ingredients in the proposed non-overlapping DDM: (a) A proper 1st order transmission condition to enforce field continuity across domain boundaries and (b) A cement technique to allow non-matching grids for neighboring domains. Moreover, a detail Fourier analysis of the transmission condition for a canonical half-space example is presented. The analysis provides significant insights into the convergence behavior of the proposed non-overlapping DDM for solving electromagnetic radiation problems, such as the large finite antenna arrays. Particularly for the antenna arrays, the proposed non-overlapping DDM is extremely efficient since the formulation can easily incorporate geometrical repetitions. Exponentially tapered notch (Vivaldi) antenna arrays with size up to 100 × 100 elements are solved on a common PC to validate the proposed non-overlapping DDM.

A Transmission Problem with a Fractal Interface

In this paper we study a transmission problem with a fractal interface K , where a second order transmission condition is imposed. We consider the case in which the interface K is the Koch curve and we prove existence and uniqueness of the weak solution of the problem in V (\Omega, K) , a suitable ”energy space”. The link between the variational formulation and the problem is possible once we recover a version of the Gauss-Green formula for fractal boundaries, hence a definition of ”normal derivative”.

Macroscopic models for semiconductor heterostructures

The semiconductor Boltzmann equation is considered within a material with a space-dependent band characteristic. Its fluid limit under strong elastic collisions leads to the so-called spherical harmonics expansion (SHE) model. For this model, the influence of smoothly space-dependent or abruptly discontinuous band characteristics is analyzed. For the latter case which occurs at heterojunction interfaces, kinetic boundary layers are analyzed and higher order transmission conditions are derived. In a further step, transmission conditions through heterojunctions for the energy-transport and drift-diffusion models (which are obtained as moments of the SHE model) are constructed.