Finite Element Domain Decomposition Method Research Articles

Analysis of conductive filament heat transfer in TiO2-based RRAM device

Abstract Resistive random access memory (RRAM) has developed into a new type of non-volatile memory that has attracted much attention for its high density and low power; TiO2 has been studied to make RRAM because of its high conductivity. Based on the COMSOL Multiphysics finite element method and domain decomposition method, an electrically-thermally coupled model of TiO2-based RRAM with oxygen vacancy (Vo) conduction mechanism is constructed, the formation and cutting process of RESET and SET conductive filament (CF) under different voltages is simulated. Moreover, the characteristics of the internal temperature distribution of the CF are explored. The results show that the RESET process is more sensitive to heat changes, the transverse direction thermal value of the CF is more obvious than the longitudinal direction, and the rate of change is faster. Additionally, the electric field induces the migration of the Vo in filament, which affects the enthalpy change of the device’s heat transfer and resistive properties. This is an important reference for a deeper understanding of the switching behavior of RRAM devices and the control mechanisms for thermal studies.

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.

3D MCSEM modeling based on domain-decomposition finite-element method

Domain decomposition approach is effective in converting large modeling problems into many small ones, so that the parallel computation can be easily facilitated and the memory consumption can be largely reduced. In this paper, we developed a domain-decomposition method for 3D time-domain marine controlled-source (MCSEM) modeling based on Dual-Primal Finite-Element Tearing and Interconnecting technique (FETI-DP). We use tetrahedral grids to do the spatial discretization first and then partition the mesh into subdomains. In each subdomain, the finite-element equations are independently constructed using the first-order vector shape functions and boundary conditions. To build the interconnections among these subdomains, we establish the interface equation for the Lagrange multipliers that are used as the boundary conditions for the subdomains. After solving the equations, the electric field for all subdomains can be obtained from the Lagrange multipliers. For the time discretization, we use unconditionally stable backward Euler scheme. We validate our algorithm by comparing with a semi-analytical solutions for a uniform half-space model under the ocean. Numerical experiments show that compared to the conventional finite-element method, our domain-decomposed method can save large amount of memory, render it possible to run 3D MCSEM modeling in small computational equipment.

An Explicit Time-Domain Finite-Element Boundary Integral Method for Analysis of Electromagnetic Scattering

A numerical scheme, which hybridizes the element-level dual-field time-domain finite-element domain decomposition method (ELDDM) and the time-domain boundary integral (TDBI) method to accurately and efficiently analyze open-region transient electromagnetic scattering problems, is proposed. Element-level decomposition decouples Maxwell equations on a discretization element from those on its neighboring elements using equivalent currents defined on their faces. For any element inside the computation domain, the equivalent currents are obtained from fields in the neighboring elements. For any element on the boundary of the computation domain, the equivalent currents are obtained using the fields generated by TDBI. To generate these fields, TDBI “radiates” equivalent currents on a Huygens surface enclosing the scatterer. This approach when combined with a leapfrog-type time updates results in a fully explicit numerical scheme that allows ELDDM and TDBI to use different time steps. Numerical results that demonstrate the applicability of the proposed method to concave and disconnected scatterers are presented.

Physics-based modeling of sea clutter phenomenon by a full-wave numerical solver

The sea clutter phenomenon is investigated from a different perspective by using the finite element domain decomposition (FEDD) method, which is a full-wave numerical method based on the decomposition of the problem into sub-problems with the help of the locally-conformal perfectly matched layer (LC-PML) approach. The numerical model developed in this work provides the means to investigate the sea clutter phenomenon by a full-wave Maxwell solver, although the electrical size of computational domain is formidably large. Monte Carlo simulations are performed to obtain some statistical properties of reflectivity. The results are compared with those obtained from existing empirical models and measured data for different polarizations, frequencies, grazing angles and wind speeds (or sea states). It is numerically shown that the probability distribution of the clutter signal power appears to follow the gamma distribution. Furthermore, the concept of ergodicity, which means that the statistical results obtained by ensemble averaging in a single cell are equivalent to those obtained by spatial averaging over a set of consecutive range cells in a radar system, is examined. It is observed from the numerical results that the behavior of the sea clutter random field is consistent with the ergodicity hypothesis. Some conclusions are drawn about the statistical behavior of the sea surface reflectivity.

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.

Parallel hierarchical decomposition of finite element method with block diagonal symmetric Gauss‐Seidel preconditioner for solving 3D problems with over ten billion unknowns

AbstractWe present in this paper an efficient parallelization approach of the finite element domain decomposition method (FEM‐DDM) for solving large‐scale 3D scattering problems with over ten billion unknowns. In this parallel FEM‐DDM, the whole solution domain is decomposed in a two‐level hierarchical way to minimize the inter process communication. A communication‐avoided block diagonal symmetric Gauss‐Seidel (BD‐SGS) preconditioner is specially designed to accelerate the iterative solution of the interface problem from FEM‐DDM and reduce the CPU time, as well as the peak memory. Besides, a communication strategy is carefully designed to make computation and communication overlapped via nonblocking sending and receiving to further improve the parallel scalability. Numerical examples are presented to demonstrate the accuracy, scalability, and capability of the proposed parallel FEM‐DDM algorithm. The solutions of scattering by a practical aircraft model with a three‐layer antenna radome on its nose section is presented, which are discretized with more than ten billion unknowns. To the best of our knowledge, this is the largest number of unknowns solved by parallel FEM‐DDM in computational electromagnetics to date, for this type of simulation.

Simulation and computational heat transfer in the human eye with Dirichlet–Neumann domain decomposition approximation

In this paper, a bioheat model of temperature distribution in the human eye is studied, the mathematical formulation of this model is described using adequate mathematical tools. The existence and the uniqueness of the solution of this problem is proven and four algorithms based on finite element method approximation and domain decomposition methods are presented in details. The validation of all algorithm is done using a numerical application for an example where the analytical solution is known. The properties and parameters reported in the open literature for the human eye are used to approximate numerically the temperature for bioheat model by finite element approximation and nonoverlapping domain decomposition method. The obtained results that are verified using the experimental results recorded in the literature revealed a better accuracy by the use of algorithm proposed.

An Iteration-Free Domain Decomposition Method for the Fast Finite Element Analysis of Electromagnetic Problems

An iteration-free finite element domain decomposition method (DDM) is proposed for the simulation of 3-D electromagnetic problems. The finite element tearing and interconnecting (FETI) with Robin-type transmission condition is introduced to construct a domain decomposition framework. Based on hierarchical ( $\mathcal {H}$ -) matrix algorithm, the numerical Green’s function of each subdomain can be represented by a data-sparse matrix form. Subsequently, the original 3-D problem is reduced to a 2-D problem on the skeleton. Derived from the finite element discretization of pseudo-differential operator, the skeleton matrix can also be efficiently approximated with an $\mathcal {H}$ -matrix by assembling all the numerical Green’s functions of subdomains. Based on $\mathcal {H}$ -LU factorization algorithm, the computational complexity and memory requirement for the direct solution of the skeleton equation can be reduced to be logarithmic linear. The resulting direct FETI (D-FETI) method completely depends on direct method without any iterative process. Hence, it does not suffer from the issue of convergence rate, and avoids redundant computations for multiple right-hand side problems. Numerical examples demonstrate the effectiveness of the proposed method.

A Domain Decomposition Finite-Element Method for Modeling Electromagnetic Scattering From Rough Sea Surfaces With Emphasis on Near-Forward Scattering

A high-fidelity full-wave simulator is presented to perform numerical experiments for rough sea scattering problem by considering different polarizations, frequencies, grazing angles, wind speeds, and sea surface spectra. The simulator is based on a novel finite-element domain decomposition (FEDD) method for solving the problem of 2-D electromagnetic scattering over 1-D sea surface. This noniterative method partitions the computational domain into a number of overlapping subdomains and solves each domain individually by employing the locally conformal perfectly matched layer (LC-PML) at the truncation boundaries. LC-PML has a unique feature such that it can be applied to irregular domains on the contrary to standard PML methods, and hence inspired the birth of FEDD. The FEDD method is used at each Monte Carlo realization corresponding to a sample from random rough surfaces and decreases the computational load, especially for electrically large problems. The accuracy and computational efficiency of the method are investigated through several simulations. Using the FEDD method, the statistical behavior of the bistatic radar cross section (RCS) is obtained for both the horizontal and vertical polarizations. A special emphasis is given to forward-scattered RCS and the mean reflection coefficient for sea surface, especially at low grazing angles, and it is shown that the simulator produces results in agreement with the Ament and Miller–Brown approximations, and experimental data, proving the reliability of the simulation approach. The results are also compared with the standard finite-element method and method of moments. Rough sea surfaces are created by using both the Pierson–Moskowitz and Elfouhaily spectra.

Fast Modeling and Practical Inversion of Laterolog-Type Downhole Resistivity Measurements

In this paper, we present a fast modeling and practical inversion scheme for the laterolog-type measurements that includes dual-laterolog and array laterolog methods. A modified domain decomposition finite-element method (DDFEM) is proposed to implement the fast modeling of the laterolog-type measurements in 2-D cylindrical coordinates. This method enables the simulation of the tool’s responses at multiple measuring points along the well trajectory all at once, and thus accelerates the simulation process significantly. A Levenberg–Marquardt method is employed to derive the radial resistivity profile from the laterolog-type measurements. The kernel of the inversion is that the full 2-D response is separated into a fast-preliminary inversion process followed by a rigorous 2-D inversion. A 1-D radial response is used in the fast-preliminary inversion and the difference between the 1-D radial response and the strict 2-D response is attributed to the shoulder bed effects. The full 2-D inversion is only necessary at the end of the preliminary inversion process. Synthetic and field data are processed to illustrate the detailed implementations as well as the performance of the proposed method. Full 3-D modeling and inversion examples are also presented to show that the DDFEM method can be readily extended to 3-D. The resistivity anisotropy effects and invasion effects are also discussed.

Multilevel compressed block decomposition-based finite-element domain decomposition method for the fast analysis of finite periodic structures

An efficient finite-element domain decomposition method based on the multilevel compressed block decomposition (MLCBD) algorithm is presented. The dual-primal finite element tearing and interconnecting (FETI-DP) method is first introduced to generate a nonoverlapping domain decomposition method with good convergence properties. A second-order transmission condition is adopted to further accelerate the convergence rate of the FETI-DP. With the MLCBD algorithm, the computational complexity for the direct solution of FETI-DP subdomain equations can be reduced to be almost logarithmic linear. The accuracy of the MLCBD algorithm is highly adjustable according to practical requirements. Exploiting the repetitiveness of finite periodic structures, the proposed method can significantly reduce the computational costs. Numerical examples demonstrate the effectiveness of the proposed method for the electromagnetic simulation of finite periodic structures.

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.

국부 및 혼합 Lagrange 승수법을 이용한 영역분할 기반 유한요소 구조해석 기법 개발

In this paper, a finite element domain decomposition method using local and mixed Lagrange multipliers for a large scal structural analysis is presented. The proposed algorithms use local and mixed Lagrange multipliers to improve computational efficiency. In the original FETI method, classical Lagrange multiplier technique was used. In the dual-primal FETI method, the interface nodes are used at the corner nodes of each sub-domain. On the other hand, the proposed FETI-local analysis adopts localized Lagrange multipliers and the proposed FETI-mixed analysis uses both global and local Lagrange multipliers. The numerical analysis results by the proposed algorithms are compared with those obtained by dual-primal FETI method.

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.

The modified method of characteristics with mixed finite element domain decomposition procedures for the transient behavior of a semiconductor device

AbstractFor the transient behavior of a semiconductor device, the modified method of characteristics with mixed finite element domain decomposition procedures applicable to parallel arithmetic is put forward. The electric potential equation is described by the mixed finite element method, and the electric, hole concentration and heat conduction equations are treated by the modified method of characteristics finite element domain decomposition methods. Some techniques, such as calculus of variations, domain decomposition, characteristic method, energy method, negative norm estimate and prior estimates and techniques are employed. Optimal order estimates in L2 norm are derived for the error in the approximation solution. Thus the well‐known theoretical problem has been thoroughly and completely solved.© 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 353–368 2012

Full-Wave Real-Life 3-D Package Signal Integrity Analysis Using Nonconformal Domain Decomposition Method

Advances in interconnect technologies, such as the increase of the number of metal layers and 3-D stacking technique, have paved the way for higher functionality and superior performance while reducing size, power, and cost in today's integrated circuit and package products. However, whether or not the package preserves signal integrity (SI) has become a crucial concern for system designers. In this study, a systematic full-wave numerical approach, based on a nonconformal finite-element domain decomposition method (DDM), is proposed for 3-D real-life circuit/package simulations. First, an automatic domain partitioning strategy is utilized to divide the entire model into a number of sub-domains. Each sub-domain is then meshed independently and an <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h</i> -version of adaptive mesh refinement is employed. Next, a nonoverlapping DDM is adopted to efficiently solve the finite-element matrix equation. Afterwards, a model-order reduction technique is exploited to compute the multiport spectral responses. SI effects such as signal delay, coupling, and reflection are simulated on a product-level package benchmark. Finally, numerical results verify the analysis and demonstrate the effectiveness of the proposed method.

An overlapping domain decomposition method for a polymer exchange membrane fuel cell model

Abstract The structure of channels, cell material and operation parameters are usually implemented in different ways from anode to cathode in the actual work to seek better performance for polymer exchange membrane fuel cell (PEMFC). However, the asymmetrical structure of such specific design often leads to non-matching grids in the numerical simulation of PEMFC, especially when the structured grids are used. Besides, different mesh sizes are often used in the simulation for anode and cathode in order to obtain both high accuracy and relatively low computational cost. In this paper, an overlapping domain decomposition finite element method is studied to deal with the non-matching grids for a simplified two-dimensional single-phase PEMFC model. Numerical experiments demonstrate that our methods are able to deal with PEMFC simulation on the non-matching grids with fast convergence and low mass balance error.

A dual-field domain-decomposition method for the time-domain finite-element analysis of large finite arrays

A novel dual-field time-domain finite-element domain-decomposition method is presented for an efficient and broadband numerical simulation of electromagnetic properties of large finite arrays. Instead of treating the entire array as a single computation domain, the method considers each array element as a smaller subdomain and computes both the electric and magnetic fields inside each subdomain. Adjacent subdomains are related to each other by the equivalent surface currents on the subdomain interfaces in an explicit manner. Furthermore, the method exploits the identical geometry of the array elements and further reduces the memory requirement and CPU time. The proposed method is highly efficient for the simulation of large finite arrays. Numerical stability and computational performance of the method are discussed. Several radiation examples are presented to demonstrate the accuracy and efficiency of the method.