Non-conformal Domain Decomposition Method Research Articles

A non-conformal domain decomposition method utilizing rotating subdomains and non-matching grids for periodic metamaterial simulation

Electromagnetic metamaterials possess characteristics such as having large-scale, periodic, multi-media, and complex geometric structures, making them highly suitable for simulation using the finite element domain decomposition method (DDM). This work proposes a non-conformal DDM for simulating finite-period metamaterials, achieving simulation with a minimal number of domains while significantly reducing computational resource consumption. First, the computational domain of the periodic metamaterial is partitioned into several non-conformal hexahedral subdomains. Subsequently, a limited number of non-repetitive subdomain models with non-matching grids are constructed, leveraging the three-dimensional rotational and translational properties of these subdomains to facilitate simulation. By eliminating the necessity to construct models for every subdomain, a substantial reduction in memory consumption is achieved. Second, an iteration method for solving the matrix equations of subdomains is enhanced by introducing a multifrontal block incomplete Cholesky decomposition preconditioner, thereby enhancing the computational efficiency of matrix equations with a large number of unknowns. Meanwhile, parallel computing techniques are employed to accelerate the proposed method. Finally, we integrate the aforementioned method into a solver and leverage it to develop an electromagnetic simulation tool tailored for metamaterials. The tool is employed to simulate metamaterial structures of varying scales, resulting in notable reductions in both memory and time consumption while maintaining accuracy comparable to commercial software.

Simulation of Electromagnetic Waves in Plasma by Subdomain-Level Nonconformal DGTD Method

The subdomain-level discontinuous Galerkin time-domain (SL-DGTD) method in a 3-D case is proposed in this article to simulate electromagnetic waves in plasma. An auxiliary differential equation (ADE) technique is employed to manipulate the dispersive characteristic of plasma. A hybrid nonconformal domain decomposition method is used to improve computational efficiency and modeling accuracy. The fourth-order explicit Runge–Kutta integration scheme is implemented to update the fields in each time step. Finally, the proposed algorithm is validated by studying the cutoff frequency of plasma, the transmission coefficients of a plasma slab, and the scattering characteristics of a plasma-coated sphere.

Resistive Sheet Boundary Condition-Based Nonconformal Domain Decomposition FE-BI-MLFMA for Electromagnetic Scattering From Inhomogeneous Objects With Honeycomb Structures

A flexible and efficient resistive sheet boundary condition (RSBC)-based hybrid finite element-boundary integral-multilevel fast multipole algorithm (FE-BI-MLFMA) is presented for computing electromagnetic scattering from inhomogeneous objects with microwave-absorbing honeycomb structures. In the proposed algorithm, each nonmagnetic and high lossy material coated unit cell wall of the honeycomb is first approximated by the multilayered RSBC as a zero-thickness resistive sheet to eliminate the computational burden due to the extremely thin and multilayered characteristics of the coated unit cell wall. Then, the RSBC is incorporated into the FE-part of the FE-BI-MLFMA formulation. To further reduce the burden of meshing complicated objects involving cellular structures after RSBC approximation, the hybrid conformal and nonconformal domain decomposition method (DDM) of the FE-BI-MLFMA, which integrates the nonconformal Schwarz DDM-FE and the simplified discontinuous Galerkin (S-DG), is employed to bring significant flexibility and versatility in geometry modeling and mesh generating. An effective block low-rank multifrontal solver-based domain decomposition finite-element method (FEM)-absorbing boundary condition (ABC) preconditioner is constructed to speed up the solution of the FE-BI equations using locally approximated integral operators for the BI part. Numerical examples are given to demonstrate the accuracy, capability, and performance of the proposed algorithm, including a high-definition complicated fighter model with antenna array, multilayer dielectric radome, and microwave-absorbing honeycomb structures.

A Flexible FEM-BEM-DDM for EM Scattering by Multiscale Anisotropic Objects

In this article, a nonconformal domain decomposition method (DDM) based on the hybrid finite element method (FEM) and boundary element method (BEM) is proposed to solve arbitrary 3-D anisotropic (including bianisotropic and single anisotropic) multiscale composite objects. The object is divided into interior and exterior regions. The interior region is solved by the bianisotropic finite element domain decomposition method (FEM-DDM), and the exterior region is solved by the discontinuous Galerkin (DG) method based on the combined field integral equation (CFIE). In order to decouple the electric and magnetic fields of the bianisotropic object, a new finite element–boundary element–domain decomposition method (FEM-BEM-DDM) formulation is derived. To ensure the continuity of electric current and electric field between anisotropic FEM subdomains, a new auxiliary current is introduced into the interface of different subdomains of FEM and the interface of FEM subdomains with BEM subdomains. The former uses second-order transmission conditions (SOTCs) to ensure continuity, whereas the latter uses Robin transmission conditions (RTCs) to ensure continuity. The BEM dense matrix–vector multiplication is accelerated by a multilevel fast multipole algorithm (MLFMA). The proposed FEM-BEM-DDM allows for nonconformal discretization between any touching subdomains and provides an effective preconditioner. In numerical examples, the accuracy and effectiveness of the method are verified.

Study of accuracy of a non-conformal finite element domain decomposition method

Domain Decomposition Methods (DDM) have been widely used in the Computational Electromagnetics (CEM) community in the last years to tackle large-scale problems with Finite Element Methods (FEM). Non-conformal DDM is more flexible (e.g., independently created meshes for different parts of the problem under analysis are supported) but may introduce an approximation error. In this communication, a thorough study of the accuracy of the solutions when using non-conformal DDM is presented. Three experiments are analyzed showing the verification of the implementation and that the accuracy is acceptable with different numbers of discontinuities in the propagation direction and various aspect ratios of the mesh on the interface. The numerical results use three different shapes (tetrahedra, prisms, and hexahedra) and up to order three in the basis functions that approximate the field. These studies are relevant for the introduction of non-conformal DDM with real problems or scalable (in the parallel sense) implementation of adaptive mesh techniques.

Solving EM Scattering From Complex Thin Dielectric/PEC Composite Targets by a VSIE-Based Method

A nonconformal and nonoverlapping domain decomposition method (DDM) based on hybrid volume and surface integral equation (VSIE) is presented for modeling complex thin dielectric/perfect electric conductor (PEC) composite targets. To reduce the unknown amount of the conventional volume integral equation (VIE) for thin dielectric bulks, the improved simplified prism vector (ISPV) basis function is developed. By using the basis function, the volume integrals can be simplified into surface integrals or line integrals, which significantly reduce the simulation time. Prism grids can also avoid the overmeshing problem for thin dielectrics. Moreover, they can be perfectly matched to the triangular meshes widely used in surface integral equation (SIE) hence assures a simple and robust VSIE implementation. Next, in order to increase the flexibility of geometrical modeling and accelerate the convergence of the presented VSIE system for electrically large and multiscale composite targets, a DDM is further developed. In this DDM scheme, discontinuous Galerkin approaches are, respectively, employed for VIE and SIE, accounting for conformal/nonconformal volume grids and surface grids between adjacent subdomains. In addition, an interior penalty term is introduced to stabilize the DDM solution. Finally, some numerical results are considered and analyzed to show the validity, efficiency, and applicability of the presented scheme.

Parallel Integral Equation-Based Nonoverlapping DDM for Solving Challenging Electromagnetic Scattering Problems of Two Thousand Wavelengths

In this paper, a parallel nonoverlapping and nonconformal domain decomposition method (DDM) is proposed for fast and accurate analysis of electrically large objects in the condition of limited resources. The formulation of nonoverlapping DDM for PEC bodies is derived from combined-field integral equation (CFIE), and an explicit boundary condition is applied to ensure the continuity of electric currents across the boundary. A parallel multilevel fast multipole algorithm (MLFMA) is extended to accelerate matrix-vector multiplications of subdomains as well as the coupling between them, and the coupling between different subdomains is computed in the manner of near field to avoid the storage of the mutual impedance. An improved adaptive direction partitioning scheme is applied to the oct-tree of MLFMA to achieve high parallel efficiency. Numerical examples demonstrate that the proposed method is able to simulate realistic problems with a maximum dimension greater than 2000 wavelengths.

Parallel Integral Equation Based Nonoverlapping DDM for Solving Electrically-Large Scattering Problems in Half Space

In this paper, a parallel nonoverlapping non-conformal domain decomposition method using combined-field integral equation is proposed for fast and accurate analysis of electrically large objects situated in half space. This method decomposes the whole model into several easily solvable closed subdomains in a flexible way, and an explicit boundary condition is applied to ensure the continuity of electric currents across the boundary. The adaptive direction partitioning parallelization scheme for the oct-tree of a multilevel fast multipole algorithm is adopted to accelerate matrix-vector multiplications of subdomains as well as the coupling between them. The introducing of real image source to account for far interaction matrix greatly accelerates the process and reduces memory requirement with precision guaranteed. Numerical examples demonstrate that the proposed method is able to simulate realistic problems with a maximum dimension greater than 1000 wavelengths.

Nonconformal FETI-DP Domain Decomposition Methods for FE-BI-MLFMA

The nonconformal domain decomposition methods (DDMs) of the hybrid finite element-boundary integral-multilevel fast multipole algorithm (FE-BI-MLFMA) are presented for computing large electromagnetic scattering/radiation problems. The numerical performance of different transmission conditions between finite element method (FEM) subdomains, and FEM and BI domains is studied in this paper. The Dirichlet transmission condition (DTC) employed at the interface between FEM and BI domains has faster convergence than the Robin-type transmission condition, and this finding is different from the conclusions for the FEM DDMs published in the literature. Furthermore, the cement-element (CE)-based dual-primal finite element tearing and interconnecting (FETI-DP) DDM has faster convergence than the Lagrange-multiplier-based FETI-DP method under the FE-BI system, and is also different from that under the system of the FEM with the absorbing boundary condition (FEM-ABC). It shows that the DDMs of FE-BI-MLFMA have some essential differences from those of FEM-ABC. The analysis of these numerical phenomena is presented in this paper. Further comparison between the DDMs of FE-BI-MLFMA and FEM-ABC demonstrates that the DDMs of FE-BI-MLMFA have essential advantages over those of FEM-ABC. Various numerical experiments confirm the accuracy, efficiency, and flexibility of the nonconformal DDM of FE-BI-MLFMA, which is the CE-based FETI-DP DDM with DTC connecting FEM and BI.

Domain Decomposition Preconditioning for Surface Integral Equations in Solving Challenging Electromagnetic Scattering Problems

We propose and study a nonoverlapping and nonconforming domain decomposition method for the integral-equation-based solution of large, complex electromagnetic (EM) scattering problems. The continuity of the electric surface current across the boundary between adjacent subdomains is enforced by a skew-symmetric interior penalty formulation. A nonoverlapping additive Schwarz preconditioner is designed and analyzed for the solution of the linear system of equations resulting from Galerkin boundary-element discretization. We show that the preconditioned system exhibits a uniformly confined eigenspectrum with respect to changing problem and discretization parameters. Numerical examples are presented to demonstrate the fast convergence of iterative solvers and the superior accuracy of the solutions obtained by our method. The proposed work can be viewed as an effective preconditioning scheme that reduces the condition number of very large systems of equations in challenging EM scattering problems. The strength and capability of the proposed method will be illustrated by means of several examples of practical interest.

FDFD Nonconformal Domain Decomposition Method for the Electromagnetic Modeling of Interconnections in Silicon Interposer

This paper proposes an efficient method to model interconnections with through-silicon vias (TSVs) in silicon interposer for 3-D systems. The proposed method uses 3-D finite-difference frequency-domain nonconformal domain decomposition method to model the redistribution layer transmission lines on lossy silicon interposer. Using the nonconformal domain decomposition approach, the interposer can be divided into separate subdomains. Individual subdomains can be discretized independently based on its feature size. Field continuity at interfaces between domains is enforced by introducing Lagrange multiplier and vector basis functions at the interfaces. TSVs are modeled using an integral-equation-based solver, which uses cylindrical modal basis functions. The formulation on incorporating multiport network into nonconformal domain decomposition is presented to include the parasitic effects of TSV arrays into the system matrix. By comparing with 3-D full-wave simulations and commercial solvers, this paper validates the accuracy and efficiency of the proposed modeling approach.

Efficient 3-D Domain Decomposition With Dual Basis Functions

Novel basis functions are proposed for enforcing continuity constraints in 3-D elliptic problems discretized by non-conforming domain decomposition methods. The major advantage over standard coupling methods is that the projection matrix, mapping degrees of freedom from master to slave surface, can be constructed with minimum computing effort since the slave matrix is diagonal. The accuracy of matching conditions and convergence properties of the method are tested on a benchmark problem.

Fast Integral Equation Solution of Scattering of Multiscale Objects by Domain Decomposition Method with Mixed Basis Functions

Nonconformal nonoverlapping domain decomposition method (DDM) with mixed basis functions is presented to realize fast integral equation solution of electromagnetic scattering of multiscale objects. The original multiscale objects are decomposed into several closed subdomains. The higher order hierarchical vector basis functions are used in the electrically large smooth subdomains to significantly reduce the number of unknowns, while traditional Rao-Wilton-Glisson basis functions are used for subdomains with tiny structures. A well-posed matrix is successfully derived by the present DDM. Besides, the nonconformal property of DDM allows flexible mesh generation for complicated objects. Numerical results are presented to validate the proposed method and illustrate its advantages.

Efficient Modeling of Large-Scale Electromagnetic Well-Logging Problems Using an Improved Nonconformal FEM-DDM

The numerical modeling of the advanced electromagnetic well-logging problems is very challenging because it requires tremendous workload of grid meshing and computation. For example, global meshing and simulation have to be done repeatedly for every logging position, because the model includes complex sensor arrays moving continuously along the borehole. In this paper, an efficient nonconformal finite element domain decomposition method is developed to solve these problems efficiently. First, the well-logging model is divided into two nonconformal subdomains so that the sensing arrays inside the borehole are separated from the beds outside. Since the mesh size between different subdomains can be inconsistent, well-logging modeling can significantly reduce the unknowns and computational cost. Second, a second-order transmission condition is implemented successfully in a quasi-symmetrical form via a novel treatment of Gaussian integration on the nonconformal interface. Finally, a hierarchical hexahedral basis function is introduced to further reduce the unknowns and extend the applicability of the method to multiscale problems. Numerical examples show that this method greatly reduces the memory cost and speeds up the computation compared with the traditional finite element method, especially when the response needs to be simulated repeatedly while the logging tools keep moving along the borehole.

A posteriori error analysis for a boundary element method with nonconforming domain decomposition

AbstractWe present and analyze an a posteriori error estimator based on mesh refinement for the solution of the hypersingular boundary integral equation governing the Laplacian in three dimensions. The discretization under consideration is a nonconforming domain decomposition method based on the Nitsche technique. Assuming a saturation property, we establish quasireliability and efficiency of the error estimator in comparison with the error in a natural (nonconforming) norm. Numerical experiments with uniform and adaptively refined meshes confirm our theoretical results. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 947–963, 2014

A NEW INTERFACE CEMENT EQUILIBRATED MORTAR (NICEM) METHOD WITH ROBIN INTERFACE CONDITIONS: THE P1 FINITE ELEMENT CASE

We design and analyze a new non-conforming domain decomposition method, named the NICEM method, based on Schwarz-type approaches that allows for the use of Robin interface conditions on non-conforming grids. The method is proven to be well posed. The error analysis is performed in 2D and in 3D for P1 elements. Numerical results in 2D illustrate the new method.

Thermal Analysis of High-Power Integrated Circuits and Packages Using Nonconformal Domain Decomposition Method

A nonconformal domain decomposition method (DDM) is proposed to solve moderately stiff parabolic partial differential equations in inhomogeneous domains. The proposed nonconformal DDM decomposes the entire problem domain into many nonoverlapping subdomains. Consequently, it is effective in addressing complex thermal problems of electronic systems with multiscaled features. Moreover, the time discretization employed is based on an unconditionally stable and implicit Euler scheme. The unconditionally stable time-marching algorithm is beneficial since the time-step size is no longer governed by the spatial discretization of the mesh, but rather by the desired accuracy. Additionally, this paper includes numerical investigations of the convergence properties of the proposed nonconformal DDM. Finally, numerical results are shown for a chip-package-printed circuit board example with thermal cooling by both natural convection and forced convection of heat sinks.

Non-conformal domain decomposition methods for time-harmonic Maxwell equations

We review non-conformal domain decomposition methods (DDMs) and their applications in solving electrically large and multi-scale electromagnetic (EM) radiation and scattering problems. In particular, a finite-element DDM, together with a finite-element tearing and interconnecting (FETI)-like algorithm, incorporating Robin transmission conditions and an edge corner penalty term, are discussed in detail. We address in full the formulations, and subsequently, their applications to problems with significant amounts of repetitions. The non-conformal DDM approach has also been extended into surface integral equation methods. We elucidate a non-conformal integral equation domain decomposition method and a generalized combined field integral equation method for modelling EM wave scattering from non-penetrable and penetrable targets, respectively. Moreover, a plane wave scattering from a composite mockup fighter jet has been simulated using the newly developed multi-solver domain decomposition method.

Thermal-aware DC IR-drop co-analysis using non-conformal domain decomposition methods

Almost all practical engineering applications are multi-physics in nature, and various physical phenomena usually interact and couple with each other. For instance, the resistivity of most conducting metals increases linearly with increases in the surrounding temperature resulting from Joule heating by electrical currents flowing through conductors. Therefore, in order to accurately characterize the performance of high-power integrated circuits (ICs), packages and printed circuit boards (PCBs), it is essential to account for both electrical and thermal effects and the intimate couplings between them. In this paper, we present non-conformal, non-overlapping domain decomposition methods (DDMs) for thermal-aware direct current (DC) IR drop co-analysis of high-power chip-package-PCBs. Here, IR stands for the finite resistivity (R) of metals and current (I) drawn off from the power/ground planes. The proposed DDM starts by partitioning the composite device into inhomogeneous sub-regions with temperature-dependent material properties. Subsequently, each sub-domain is meshed independently according to its own characteristic features. As a consequence, the troublesome mesh-generation task for complex ICs can be greatly subdued. The proposed thermal-aware DC IR drop co-analysis applies the non-conformal DDM for both conduction and steady-state heat-transfer analyses with a two-way coupling between them. Numerical examples, including an IC package and a chip-package-PCB, demonstrate the flexibility and potential of the proposed thermal-aware DC IR-drop co-analysis using non-conformal DDMs.

Signal Integrity Analysis of High-Speed Interconnects by Using Nonconformal Domain Decomposition Method

3-D interconnect techniques in multilayer very large scale integration design, such as stacked layers and various chip-stacking systems, have paved the way for the improvement of integrated circuit density and operation speed. However, these techniques often accompany with impedance discontinuities that induce signal integrity (SI)/power integrity and electromagnetic interference effects. Consequently, SI analysis serves as one of the key guidelines of chip-package-board design in deep submicrometer technology. However, full-wave numerical analysis of 3-D high-speed and high-density interconnects is also considered to be a great challenge. In this paper, a nonconformal domain decomposition method with second-order transmission condition for SI analysis of high-speed interconnects on a multiscale multilayer printed circuit board is presented. The accuracy and robustness of the proposed method are first witnessed by modeling a six-layer two-trace differential pair. Then we study a 14-layer 16-trace interconnect model to demonstrate the efficiency of the method. Eye diagrams of the two models are also presented in the time-domain SI analysis.